I have been playing around with the Ackermann function a bit and realized that it gets very big very fast. (Im going to use $A$ for $Ackermann$ throughout this question)  Already $A(5,1)$ is (according to ) an integer too large to represent. It also presents me with a representation that looks like this:   $$ A(5,1) = 2 \uparrow \uparrow \uparrow 4 - 3 = 2 \uparrow^{3} 4 - 3 $$  After playing around a little I found out that it always represents it like this:  $$ A(n,m) = 2 \uparrow^{n-2} (m+3)-3 $$  After seeing this I started to wonder what would happen if you use Graham's number as the arguments. Since the value of it would uncomprehensible I tried to find something to compare it against. Remembering how the number is defined I asked myself if $G_{G_{64}}$ is bigger than $A(G_{64},G_{64})$?  Or in other words:  $$ A(G_{64},G_{64}) = 2 \uparrow^{G_{64} - 2} (G_{64} + 3) - 3 $$ vs $$ G_{G_{64}} = 3 \uparrow^{G_{G_{64}} - 1} 3 $$  I personally supspect it is $G_{G_{64}}$ since it  has a lot more arrows but I'm not entirely sure.
In  it states that      [xkcd] means calling the Ackermann function with Graham's number as the arguments just to horrify mathematicians.   $A(g_{64},g_{64})$  In  it states that      Even simply making $g_{65}$ dwarfs the number $A( A(g_{64}, g_{64}), A(g_{64}, g_{64}))$ into insignificance.   So:  Is $g_{65}$ greater than $A(g_{64}, g_{64})$?  And is it greater than $A( A(g_{64}, g_{64}), A(g_{64}, g_{64}))$?
Orthographic scale of camera in Blender could somebody explain how orthographic scale is calculated in Blender? I am trying to match an orthographic view that I get from another software with that of Blender, but I am not able. Camera gets in the correct position but the zoom level is wrong.  Thanks, byte Add space between headings and body in the home of the iPad app It's a pain in the ass for me to quit the app, draw a red circle in another app and return to the app, so...    If you notice the three questions in the bottom, you will notice that the emboldened headings (New Answer, Comment) are stick to the next character. Imperfect.    App Version: 1.4.0.249 Device: iPad 2 (GSM) OS Version: Version 9.1 (Build 13B143) USB is a popular standard and I was wondering whether there is some way to connect two or more computers using USB ports rather than the Ethernet ports. I've got a project where I'm looking to tweak the standard lead conversion process.   Triggers and lead conversion has always been funky in past projects so I was hoping to start the design off by reviewing the order of execution, but after a long google search I'm coming up empty handed.   Anyone out there know the specific trigger order for lead conversion?    Edit: I'm not asking about trigger execution order for any object, but all the objects created in the lead conversion process.  I'd think it would be something like this:        Lead Before Update Triggers (isConverted = false, convertedAccountId = null, convertContactId = null, convertedOpportunityId = null)         Account Before Insert Triggers (not fired if merged with existing account)   Account After Insert Triggers (not fired if merged with existing account)   Contact Before Insert Triggers (not fired if merged with existing contact)   Contact After Insert Triggers (not fired if merged with existing contact)   Opportunity Before Insert Triggers (not fired if no opportunity) (OpportunityContactRoles null)   Opportunity After Insert Triggers (not fired if no opportunity) (OpportunityContactRoles = convertedContactId)   Lead After Update Triggers (isConverted = true, convertedAccountId = XXX, convertedContactId = XXX, convertedOpportunityId = XXX) Electric circuits in TeX, LaTeX, and Friends What are some ways to draw electric circuits in TeX systems?  I'm making this community wiki since I haven't found a question related to this.  I was thinking along the lines of . What do you think of that? Other examples are more than welcome, of course. Why do Lagrange Multipliers work? I know that the Lagrange multiplier method helps us evaluate critical points of $f$ on the closed boundary of the restriction.   In other words we solve:$$\nabla f=\lambda \nabla g$$  But why does this method actually work? Can someone please give me an explanation. Thank you! how do you make clothes move along with your model? Do you rig the model first and then parent it to the body of the character then click cloth simulation? I am a little bit confused with the process. Thanks for the help. efficient and accurate approximation of error function I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z  e^{-t^2} \,\mathrm dt$$ How to show that all points are inside of unit circle? There are $n$ points on the plane. Any $3$ of them are inside of a unit circle. How to show that all points are inside of unit circle?  It is needed to prove that if there is a unit circle for each $3$  points which contains them, there is also a unit circle which contains all $n$ points. Is the Nintendo Switch region locked? Many Nintendo devices often come with , which means if you purchase a device in one country, you won't be able to play games from another (generally).  Is the Nintendo Switch also region locked like many of its predecessors such as the Wii U and Wii? Consider a system of the two identical positive point charges situated in free space (isolated from the influence of any other external fields) as shown in the attached diagram. Particle $1$ is at $(a,a,0)$ and particle $2$ is at $(0,a,0)$. Their velocities, at the considered instant of time, are as shown in the diagram ($\mathbf{v}_1$ along the $+x$ axis, $\mathbf{v}_2$ along the $+z$ axis).    Now, by applying the Biot-Savart law, we find that the magnetic field due to particle $2$ at the position of particle $1$ is along $+y$ axis, which means that the force acting on particle $1$ is along $+z$ axis according to Fleming’s left-hand rule. A similar analysis shows that there is no magnetic force on particle $2$ as the magnetic field of particle $1$ should vanish at positions (relative to particle $1$) located along its velocity vector.  Now, if we observe the net torque on the system of two particles about the $y$ axis then it is non-zero and is directed along the $-y$ axis. Here, there are no external forces or torques acting on the isolated two-particle system, and yet the net torque, as well as the net force on the particles, are nonzero. Why? Also, Newton's third law of motion seems to be broken in this scenario. Why?    Edit $1$  I have come to know from the responses that the electromagnetic field itself takes away some momentum and angular momentum about the considered axis. However, I think that if I consider only two charged particles as my system then the Abraham-Lorentz force can be assumed to be acting upon the system, and that is sufficient to make sure that we have considered the momentum being carried away by the electromagnetic field itself.   Even after considering the action of the Abraham-Lorentz force for the two-particle system, the scenario breaks both Newton's third law and the conservation of linear and angular momentum. This is because the Abraham Lorentz forces do not exactly counterbalance the force and the torques, on the two-particle system under consideration, due to the magnetic field.    Edit $2$  The previous edit was a result of confusion and misunderstanding on my part. The force associated with the momentum carried away by the electromagnetic field as a result of the electromagnetic interaction described in the question is the Lorentz force itself which simply doesn't obey the third law of motion. The Abraham-Lorentz force is a different story. It is associated with the momentum carried away by the radiation emitted by the accelerated charged particles. This is an additional force apart from the Lorentz force and corresponds to an additional carriage of momentum by the electromagnetic field. The momentum carried away by the electromagnetic field in correspondence with the Lorentz force doesn't correspond to radiation. Commutator of two normal subgroups with trivial intersections So I tried looking everywhere but I'm really lost with this proof. I need to to show that:  Let $H$, $K$ be normal subgroups of a group $G$, satisfying $H\cap K = \{e\}$. Let $h\in H$, $k \in K$. (a) Prove that the commutator $w = hkh^{-1}k^{−1}$ satisfies $w \in H$ and $w \in K$. Then conclude that $w = e$ (b) Prove that $hk = kh$, for all $h \in H$, $k \in K$. Allow horizontal scrolling of long math expressions Stack Exchange formats math, but often the formatted math is wider than the screen. It would be nice if formatted math were scrollable the same way that formatted code wider than the screen is scrollable. How to safely remove proprietary Intel driver I recently installed the proprietary Intel drivers for my Toshiba Satellite A105 s4284 running Ubuntu 14.10. Nouveau was working just fine, I was just attempting to make it so that I was able to run CS:S (no need for explanation on that.)  Long story short, I went and downloaded the Intel Graphic Drivers for Linux* from . After downloading and installing the .deb, I ran intel-linux-graphics-installer which opened up a window that downloaded &amp; installed the open Intel drivers. I then rebooted, and it's slow. Very slow, especially on 3D effects and when I hit Alt+Tab.  Now, I don't know if the previously installed Nouveau drivers are conflicting with these new Intel drivers (if that's the issue, what should I do about fixing it?) What I do know is that my system isn't behaving right and I want it to (obviously.) If removing the Nouveau drivers isn't the solution, how can I safely remove the just-installed Intel drivers?  Does anyone have any suggestions? Partial Fraction Decomposition? How do I deal with 'sudoers.so must be only be writable by owner'? Whenever I try to install or remove something using terminal it gives the following error message:  bluebird@bluebird:~$ sudo apt-get update sudo: /usr/lib/sudo/sudoers.so must be only be writable by owner sudo: fatal error, unable to load plugins bluebird@bluebird:~$ sudo apt-get remove vlc sudo: /usr/lib/sudo/sudoers.so must be only be writable by owner sudo: fatal error, unable to load plugins bluebird@bluebird:~$ sudo apt-get install apt-file sudo: /usr/lib/sudo/sudoers.so must be only be writable by owner sudo: fatal error, unable to load plugins How can I render bokeh circles?  Nothing I do seems to work.   I tried using an empty as the focus point for a cube, with some point lamps in the background with about 2000 strength, but it doesn't work. It just makes everything blurry.  I also raised the blades to 4/5/6/12, and it doesn't make a difference.  I'm using Blender 2.74. How to download Ubuntu Server backups automatically? I'm thinking of an investment in a new root server running ubuntu. Due to the fact, that the provider won't have extra backup-space, I need an other solution.  Is it possible to download the server backups automatically when my desktop machine is connected to the internet and in idle state? Why do networks need a common ground cable? How to determine if .NET code is running in an ASP.NET process? I have an instance of a general purpose class that will be executed both under ASP.NET and a stand alone program. This code is sensative to the process where it is being run - that is, there are certin methods that should not called if running under ASP.NET. How do you determine if the code is executing in an ASP.NET process?  The solution I am currently using is answered below.    I wish someone would add a comment as to why this question has gotten downvoted and/or propose a better way to ask it! I can only assume at least some folks have looked at the question and said "what an idiot, ASP.NET code is .NET code".