And when they are paralleled, orbital angular momentum is 0.
Why is orbital angular momentum not 0 when spin and linear momentum are not collinear?   Why can it be 0 when spin and linear momentum are parallel?  Like in the example of a scalar field at rest decaying into an electron and a positron via interaction $\frac{1}{2}g\phi \bar{\Psi}\Psi $   If spin is along x axis and three-momenta is along z axis, decay rate will be 0 if two spins are opposite because orbital momentum is not 0 after decaying.
the and thee (I prefer to pronounce it as thuh all the time) My question is can I always pronounce THE with thuh instead of thee? Because unlike "a" "an" rule, pronouncing "thee" seems cumbersome for some people (including me) Note that I know the "emphasis" and "vowel, consonant" rule about THE but I still prefer to pronounce it as thuh everytime. So can I?? How can I add left aligned text to an equation? I have a document I am trying to copy to learn TeX. Here is what I have encountered: How can I have the "or" in the equation. This is what I have right now:  \[ f = ma; \]  But $a$ is the change in velocity,  \[ f = m \frac{dv}{dt};\] \[ f = m  \frac{d^2y}{dt^2};\]   Edit: Trying out Stefan's answer independent variables in Random forest model How to reset Canon 70D WiFi? When I go to the WiFi Settings in Camera (Canon 70D) all I see is:  Connection:   Exit Confirm Set Error Details (greyed out)   I can't recall the password or find any way to reset it. Some I found () say the Camera should say Infrastructure mode. Inside the Confirm Settings I can see on page 3 it shows:     Connect'n method: Camera access point mode   But again I see no way to change this.  On my Smartphone EOS Remote app there's a Connection Guide with completely different menus from what I'm seeing on the Camera.  Any idea how I can reset the WiFi Name and Password on the Canon 70D? Is it possible to search for down voted questions? I hope this qualifies as a programming question, as in any programming tutorial, you eventually come across 'foo' in the code examples. (yeah, right?)  what does 'foo' really mean?  If it is meant to mean nothing, when did it begin to be used so? How can a company use money from stock investors when they are constantly being bought and sold? Capture console exit C# How to set page geometry for a single page only? Generating list of manually installed packages and querying individual packages Absolutely continuity and Lipschitz-like continuity I'm trying to figure out if the following is true or false:  Let $\mu$ and $\nu$ be finite measures on Borel subsets of $[0,1]$,and $\nu \ll \mu$ ($\nu$ is absolutely cts with respect to $\mu$). Does there exist always a $C \in \mathbb{R}$ such that $\forall A \subseteq [0,1]$ that is Borel, we have $\nu(A)$ $\lt$ $C \mu (A)$.  The Radon-Nykodym theorem seems to almost suggest that is true, if I knew it was bounded, then I could say $\nu (A) \le \int_{A} g \ d\mu \le \int_{A} C \ d\mu \le C\mu (A)$, where $C$ is the bound. However, I don't know if it is bounded and in fact, I could take an unbounded function that is still integrable on $[0,1]$, such as $\frac{1}{\sqrt x}$.  I can't even be sure if it is true or false.   Thank you! prove every connected manifold is path connected manifold .       my thought:      connected space : Let $ X$  be a topological space. A separation of $ X $ is a pair $U, V$ of disjoint    nonempty open subsets of $ X $ whose union is $X$. The space $ X $ is said to be connected    if there does not exist a separation of  $X$ .      Components: Given  $X$ , define an equivalence relation on $X$ by setting x~y if there    is a connected subspace of $X$  containing both $ x$  and $ y$ . The equivalence classes are    called the components (or the "connected components") of $ X$.       Path Component: I define another equivalence relation on the space $ X$  by defining x ~ у    if there is a path in $ X$  from $ x$  to $ y$ . The equivalence classes are called the path components of $ X$ .       Theorem  : The path components of $ X$  are path connected disjoint subspaces    of $X$  whose union is $X$ , such that each nonempty path connected subspace of $X$  intersects only one of them.       thank you so much Set limit to one author when using "et al." in biblatex In the header of my document I set the maximal author names to 2:  \usepackage[style=authoryear-icomp, maxbibnames=9, maxcitenames=2, backend=biber]{biblatex}   Now, when I cite texts with two authors, LaTeX gives me the following:     (Author_A/ Author_B 2012: 232)   When I have three authors or more, LaTeX makes it:     (Author_A/ Author_B et al. 2012: 232)   But I want LaTeX to just name the first author if there are more than two authors ... like this:     (Author_A et al. 2012: 232)   Any tips on how to achieve this ? We all know some curves can be described by $y=f(x)$ and some surfaces can be described by $z=f(x,y)$ However, there exists curves and surfaces which cannot be described by those, such as a circle and a sphere. Therefore, we introduce parameterized vector equations, which can describe them. For example, circle: $\vec r(t)=r\cos(t)\hat i+r\sin(t)\hat j$ sphere: $\vec r(u,v)=\rho\cos(u)\sin(v)\hat i+\rho\sin(u)\sin(v)\hat j+\rho\cos(v)\hat k$ However, all curves described by $y=f(x)$ and  $z=f(x,y)$ can be parameterized. For curves, $$\vec r(t)=t\hat i+f(t)\hat j$$ For surfaces, $$\vec r(u,v)=u\hat i+v\hat j+f(u,v)\hat k$$ Therefore, I think this suggests the set of all parameterized surfaces (or curves) is the super-set of the set of all surfaces (or curves) described by $z=f(x,y)$ (or $y=f(x)$). Is that correct? Now, here comes the the real challenge. A curve can also be described by an implicit function $f(x,y)=0$ and a surface can also be described by an implicit function $f(x,y,z)=0$ I have 3 questions regarding this.   Can all surfaces (curves) described by an implicit function be parameterized? (If yes, then what is the general way?) Can all surfaces (curves) described by parametric vector equations be represented using implicit function? (If yes, then what is the general way?) Compare the set of all parameterized surfaces (curves) and the set of all surfaces (curves) represented by implicit function. (which is which super-set?)   Sorry for the use of nontechnical terms. I use them because I don't know the technical ones. I have only started learning vector calculus last year in university.  EDIT: I think my question is not too clear, so I will give an example of writing the surface $f(x,y,z)=0$ into $\vec r(u,v)$ We want to parameterize a sphere. $$x^2+y^2+z^2-\rho^2=0$$ Let $x=\rho\cos(u)\sin(v)$, $y=\rho\sin(u)\sin(v)$, $z=\rho\cos(v)$, $$\rho^2\cos^2(u)\sin^2(v)+\rho^2\sin^2(u)\sin^2(v)+\rho^2\cos^2(v)-\rho^2$$ $$=\rho^2\sin^2(v)(\cos^2(u)+\sin^2(u))+\rho^2\cos^2(v)-\rho^2=\rho^2-\rho^2=0$$ I want to know if there is a general way of finding $x=x(u,v)$, $y=y(u,v)$ and $z=z(u,v)$ for any given $f(x,y,z)=0$ Hibernate is disabled, hiberfil.sys gets recreated at each boot I have Windows 7 running on a small (40GB) partition, with 4GB ram. This means that the hiberfil.sys file created by Hibernate takes up a significant portion of the available diskspace. I would like to remove it.  I am aware that I can disable Hibernate and remove hiberfil.sys by entering powercfg -h off in an elevated command prompt. This works -- the file is immediately removed, and after doing so, the HKEY_LOCAL_MACHINE\SYSTEM\ControlSet001\Control\Power\HibernateEnabled key is (correctly) set to 0.  However, the next time I reboot the PC, hiberfil.sys returns from the dead, Hibernate is reenabled, and that registry key has returned to 1.  I'm pretty much at my wits' end with this. Almost everything I can find online related to removing the hiberfil.sys file simply suggests using powercfg to turn off hibernation, and that appears to work for just about everyone. But it just keeps coming back for me! (Like a vampire, sucking up my disk space.) I did find  from someone who seems to have had the same issue, but none of the suggestions there worked for the original poster (or for me). Still, I have tried everything listed there, including:   Disabling hybrid sleep Disabling Hibernate through the command prompt, through the Power Options GUI, and through both (in both orders) Manually changing the HKEY_LOCAL_MACHINE\SYSTEM\ControlSet001\Control\Power\HibernateEnabled key Pretty much everything else I can think of!   I do want to reiterate that I have no problem removing the file -- that works great. It just comes back after every reboot. I'm about ready to throw in the towel and just run a script on login to disable Hibernate each time, even though that seems like a crazily hacky "solution" . . . but I was hoping someone here could suggest something else, first. How can I decode HTML characters in C#? I have email addresses encoded with HTML character entities. Is there anything in .NET that can convert them to plain strings? How to turn hundreds of text URLs in Excel into clickable hyperlinks? Some of the git commands have many options, and it would often be useful to search through them for the one I need - I was just looking for the option which controls the TAB width in git-gui, but there are about 200 completions for git config. An obvious workaround is to copy all the completions into an editor and search through them, but I'd rather do  [something] | grep tab   There are no man or info pages for compgen, help compgen doesn't even explain its own options, and there's no auto-complete for compgen (how's that for irony?).  PS:  doesn't work.  PPS: This is not a question about git-gui - The  was elsewhere.  PPPS: This is not about auto-completing commands, only command parameters. $\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$ Consider a finite-dimensional vector space $V$ over the field $\mathbb{F}$ with a linear operator $T.$ Prove that we have $\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2 \operatorname{rank}(T^2).$   First, let us assume that $\dim_{\mathbb{F}}(V) = n.$ Our immediate observations include that $T^3(V) \subseteq T^2(V) \subseteq T(V)$ so that $\operatorname{rank}(T^3) \leq \operatorname{rank}(T^2) \leq \operatorname{rank}(T)$ and $\operatorname{rank}(T^3) \geq \operatorname{rank}(T) + \operatorname{rank}(T^2) - n$ by Sylvester's Rank Inequality. Can we do some manipulations with what we have to get the desired inequality, or do we require something else? We have a new application that requires glibc 2.4 (from gcc 4.1). The machine we have runs on has gcc 3.4.6.   We can not upgrade, and the application must be run on this machine.    We installed gcc 4.1, however, when it comes to compile time it is using all the includes, etc, from 3.4.6.    How do we get around this?  Any suggestions on using 4.1 for just this application?