Function to determine number of trees surviving forest fire
Function representing a forest fire after one time step
What is Stack Overflow's business model? I once , but that was . Now, , and they have , but where's the business model? The  its own Q&amp;A about its site (since it wasn't included in the ), and just about the only dig in there that halfway has merit is the business model dig:  We switched to a premium model to keep out of the Venture Capital business (been there, done that, got the t-shirt). Companies like StackExchange couldn't do it without the VC cookie jar. Where's the model? Huh, Spolsky? (I’m sorry. Did that slip out?)  It hurt, but I can't help but wonder if they're on to something.  How would Stack Overflow Internet Services pay back the VCs? Ad revenue alone?  There is only one 'paid' product, Stack Overflow careers, but it doesn't seem like that could sustain a business as large as Stack Overflow (Joel once mentioned on his blog that his jobs board brought in an extra million a year in revenue). So what does Stack Overflow do when the VC cash runs out? Is this a get-big-and-take-the-company-public maneuver? Is that the only viable business model for Stack Exchange? Edit: With the size of Stack Overflow's team, it isn't a stretch to say that payroll is at least $2 million  a year, probably as much as 4 when you think about benefits (average 120K for a developer in New York City; x2 for benefits/taxes/all-that-other-good-stuff). Office space can't be cheap, so how long does $6 million last before you need another VC infusion? Find the angle ADE of the given triangle. Given :  Angle ABE = 50, Angle DAC = 20, Angle BAD = 60, Angle ACB = 20 and Side CD = Side DA.Unit of angle values given is degree.  Find : Angle ADE.    I'm stuck and I need help here. Consistent, complete, and generalized description of the quantum harmonic oscillator If I want to add more RAM to my computer, how can I buy something compatible? What kinds are there? How do I determine which kind my computer will accept and how much I can add? Should VMware Tools always be installed in headless Linux guests? We have a deployment with what I would call an Enterprise Cloud Provider. It's VMWare Infrastructure 3.5.x and includes VMotion / VMWare HA instances, etc.  At Provider 1 VMWare Tools was installed:   vmware-guestd is running and; at least two kernel modules are loaded, vmmemctl &amp; vmhgfs   Provider 1 does nightly VM Snapshots.  We are beginning a new deployment at Provider 2 (also an Enterprise Cloud Provider). They did not install VMWare Tools. Provider 2 does not do nightly VM Snapshots but rather file-system backups (using an agent).  From my reading it seems like at the very least VMWare Tools should be installed for the Memory Balloon Driver (vmmemctl) and from reading sources like  it suggests that certain VirtualCenter management operations also depend upon VMWare Tools being installed in the Linux guest.  Note that all of our instances are headless (default runlevel is 3) and RHEL.  It is hard to find authoritative information on this question and the aforementioned source is somewhat dated.  Can someone shed some light on this? What are the factors/criteria that would lead to an answer in the affirmative or negative?  TIA  Cheers Effect of elementary row operations on determinant? If Cooper wanted to stay, why did he send the NASA Coordinates? In Interstellar, after Cooper arrives in the Tesseract, Cooper tries to send the message STAY to his daughter in the past so she won't let him go on the mission. If that was his intention then why did he send the coordinates of NASA afterwards? If he didn't send the coordinates, he wouldn't have found NASA or go into space, he would have stayed. I remember reading a series when I was around 7th or 8th grade about aliens who had taken over Earth by putting metal caps on people when they were "entering adulthood" to control them (not direct control, but it would keep them from rebelling). There was a faction of people who able to escape and had fake caps and would save teens who were about to be capped.  The main protagonist was a teen boy who was a boxer (or training to be a boxer or something). The aliens would hold tournaments for the humans and whoever won would be "worthy" to serve as personal slaves to the aliens.     The protagonist won and was able to get inside to try and learn more about the aliens from the inside. Types of circular acceleration? To my knowledge there are three types of acceleration when a body (e.g. a rod) is moving in a circle about an axis. These are:   Angular acceleration : this is the rate of change of angular velocity. Tangential acceleration : this is the linear acceleration of the system in a tangential direction to the circle and equals the radius times the angular acceleration. Radial/centripetal acceleration : this is the linear acceleration of the system that is directed inwards towards the center of the circle.   I also think there are two types of velocity:   Linear velocity: this is its velocity in the tangential direction and is constantly changing Angular velocity or angular frequency: this is the rate of change of angle.   Is the above correct? And have I missed anything? What are the advantages and disadvantages of using ARC? Replace one Ubuntu with another from USB pen drive Should we restrict signup for a service to Facebook accounts only? An ideal whose radical is maximal is primary I've got to prove that an ideal $Q$ whose radical is a maximal ideal is a primary ideal. That is, I want to prove that if $xy\in Q$, then $x\in Q$ or $y^n\in Q$ for some $n&gt;0$.    I've been trying for a while and I'm not sure where to begin. All I've got is that if $\text{Rad}(Q)$ is maximal and by definition it's the intersection of all prime ideals $P_i$ containing $Q$, then it must be equal to each of these $P_i$. So there is only 1 prime ideal containing $Q$, namely $\text{Rad}(Q)$.  Could anyone point me in the right direction? Thanks for any replies. Defining font color inside highlight fails Since I wanted a better readability of highlights of a would be black &amp; white-printed document, I needed to change both the highlighting color (gray) AND the font color (white).  Depending which command you implement inside the other will compile or not: \hl{\textcolor{white}{Fail}} does not work, compiling returns Package xcolor Error: Undefined color '{white}'.  Here's the mwe:  \documentclass{article} \usepackage{xcolor} \usepackage{color} \usepackage{soul} % Enables highlighting \sethlcolor{gray}  \begin{document}      \textcolor{white}{\hl{Success}}     %% Error:     %% Package xcolor Error: Undefined color `{white}'.     \hl{\textcolor{white}{Fail}}  \end{document}   So, in the end, the problem is sorta solved but I thought you might have an answer as to what happens. Prove this inequality $\sum \cos{A}\ge\frac{1}{4}(3+\sum\cos{(A-B)})$ An annulus in $\mathbb R^2$ is path connected Let $A = \{(x,y) \in\mathbb{R}^2: a \leq (x-c)^2+(y-d)^2 \leq b\}$ for given $a,b,c, d$ real numbers. I want to show that $A$ is path-connected.  How can I do that?  I know that every open subset of $\mathbb R^2$ that is connected is path connected. But this is obviously not open so I cannot use that. Then I thought of multiple cases. If we take arbitrary $x$ and $y$ and draw the line between them and they do not intersect with the circle centred at $(c,d)$ then we can obviously draw a line between the points which is still in the set, so we can then define the function. I am stuck on the other case. With services like Google Finance or Yahoo Finance it is very convenient to find the historical stock price metric of a listed company, but if the company is not listed, I can't find the data.  For example, I want to see the price metrics for Macromedia, which had been listed in 1993 on NASDAQ (MACR), then acquired by Adobe in 2005. How could I find its historical price metrics? How to zero out a drive? Question  The following is from Bartle:  Give an example of a function $f$ on $X$ to $\mathbb{R}$  which is not $X-$measurable but is such that $|f|,$ and $f^2$ are measurable.  Relavant Information  A family $\textbf{X}$ of subsets of $X$ is a $\sigma$ algebra in case:   $\phi, \mathbb{R} \in \textbf{X}$ $X \setminus A \in \textbf{X}$ if $A \in \textbf{X}$ If $(A_n) \in \textbf{X}$, then the union $\bigcup_{n=1}^{\infty} A_n \in \textbf{X}$  Any set in $\textbf{X}$ is called $X-$ measurable.   A function is $X-$ measurable if for every real $\alpha$, the set $\{x \in X : f(x)&gt;\alpha\}$ belongs to $\textbf{X}$  Attempt  Consider the following function: $$f=\begin{cases} 1 &amp;\mbox{ if} \ x \not\in\mathbb{Q} \\                   -1 &amp;\mbox{ if} \ x \in \mathbb{Q}  \end{cases}$$ defined from $\mathbb{R}$ to $\mathbb{R}$ with $\textbf{X}$ being the Borel algebra (the $\sigma$ algebra generated by all open intervals $(a,b)$) Then if $\alpha \in [-1,1)$, the set $\{x: f(x)&gt;\alpha\}$ consists of the irrational numbers. If I can show that the irrationals are not in the Borel Algebra, then I would be done since $|f|=f^2$ is a constant and easily shown to be measurable. Unfortunately I do not know how to show that the irrationals are not in the Borel Algebra.  Edit it has been brought to my attention that the irrationals are in the Borel algebra. I think this is because the rationals, being countable, can be shown to be in the Borel algebra by constructing appropriate small neighborhoods around each rational and then taking the intersections. I now seek to construct a set that is not in the Borel algebra  Alternatively I have tried to define a set other than the irrationals that is not in the Borel Algebra and define a similar function but to no avail. I am currently self studying this book so it would be of great help to me if the answer could contain heuristics on how to tell when a set if borel measurable (although I do not require this in the answer). What follows if we fail to reject the null hypothesis? What conclusions can we draw if $p&gt;\alpha$? Does not rejecting the $H_0$ mean anything?