How are quarks elementary when they can become leptons? From a recently reignited [casual] curiosity into particle physics thanks to the Fermilab YouTube channel, I read about the , followed by muons, naturally. Muons, it turns out have short lives, and they decay into an electron (and antineutrino), for example. Reading on how muons are created, I learned about the role of the cosmic rays, and intuitively I understand how 3 quarks can end up as 3 quarks and a quark–antiquark pion. [Charged] pions decay to muons, so my curiosity was satisfied, but only momentarily . . .  Until I realized this muon creation and the subsequent electron originated from quarks.  Q: How are quarks elementary when they can become leptons?  I'm either misunderstanding the meaning of a particle being elementary, or I'm missing something. (I can't get my head around that point; I also casually understand that the weak interaction is involved.)  Does it work the opposite way as well (lepton → quark)? Or, given enough time, will all elementary particles eventually decay to leptons? (Thinking out loud; not necessarily extra questions.)    I could not find the answer to my question on Wikipedia or via Google. I checked the related topics, for example,  But it's a different question about what makes up quarks/leptons (I'm content with the fact they're as small as they get according to the Standard Model).  Thanks to @AccidentalFourierTransform for the following links (and their links):       As far as I can see those posts do not address the quark → lepton decay. All the examples given are quark → another quark (or lepton → another lepton), which I have no issue with (the muon → electron example I gave). The same for the decays of virtual/mediating particles, e.g., photons and Higgs bosons. My issue is (was) the class-changing decay of the non-virtual particles.
Is the Higgs boson an elementary particle? If so, why does it decay? The  is an excitation of the Higgs field and is very massive and short lived. It also interacts with the Higgs field and thus is able to experience mass.   Why does it decay if it is supposed to be an elementary particle according to the standard model?
$f(z)=\frac1{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}d\zeta$ if $\lim_{z\to\infty}f(z)=0$ and $f$ is analytic in $|z|\ge1$ Let $f$ be analytic in $|z|\ge1$ and  $\lim_{z\to\infty}f(z)=0$. Then is $f(z)=\frac1{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}d\zeta$,$\forall |z|\ge2$, where $\Gamma$ is the circle of radius $2$ with centre at the origin.   I think yes, by the Cauchy integral formula. But, how do we ensure that the non analyticity of $f$ in $|z|&lt;1$ dosent affect the formula? Thanks beforehand. Fields finitely generated as $\mathbb Z$-algebras are finite? Suppose $k$ is a field that is finitely generated as a ${\mathbb Z}$-algebra.  (That is, $k$ is a quotient of ${\mathbb Z}[X_1,\dots,X_n]$ for some $n$). Does it follow that $k$ is finite? Creating a node fitting the horizontal width of two other nodes I have the two PGF nodes foo and bar positioned in a row.  \documentclass{article} \usepackage{tikz} \usetikzlibrary{positioning} \begin{document}   \begin{tikzpicture}[node distance=0.2cm,mynode/.style={rectangle,draw}]     \node[mynode] (foo)                {foo};     \node[mynode] (bar) [right=of foo] {bar};   \end{tikzpicture} \end{document}   How can I create a third node below those two nodes with the width of foo, bar and the node distance between them? Is there a way to do this with the fit library or is this the wrong approach? Do electrical devices "take what they need" One concept of electronics I have a hard time understanding is if things like motors, actuators, solenoids etc. use as much power as they need or what you give them.   If a motor needs 12 volts and 500ma and I supply it with 12 volts and 3000ma will it only consume 500ma? Also, if I supply it with 15volts and 500ma what will happen?   It seems logical that an LED and a DC motor are way different when it comes to requiring/using electric where as an led has to be completely regulated and (I assume) a DC motor doesn't.   Is my understanding wrong? What is the most convenient way to change options for Graph[] objects? Determine the number of functions $f: \{1,2,3....,1999\}\to \{2000,2001,2002,2003\}$satisfying the condition that $f(1)+f(2)+...f(1999)$ is odd. Problems:Determine the number of functions $$f: \{1,2,3....,1999\}\to \{2000,2001,2002,2003\}$$ satisfying the condition that $f(1)+f(2)+...f(1999)$ is odd.   My Attempt: Each integer in domain has $4$ choices and therefore the total number of functions is $f$ is $4^{1999}.$ Since there are an equal number of functions that yield odd or even result, we can directly write that the number of functions satisfying the above condition to be $2*4^{1998}.$ I am unsure about this last claim and would like to prove it (if it is true.)Also is the answer to this problem correct?  Edit $1$: I tried to proceed further in the following manner:   For a given mapping $f$ let the number of integers assigned to $2000,2001,2002$ amd $2003$ be $p,q,r$ and $s$ respectively, where $p,q,r,s\in \mathbb{Z}^+$. Therefore the sum in question can be written as $f(1)+f(2)+...+f(1999)=2000p+2001q+2002r+2003s.$ Clearly the sum will be odd iff $q+s$  is odd. Also note that $p+q+r+s=1999\Rightarrow p+r$ is even. On the other hand if we let $q+s$ as even then $p+r$ must be even. Now the number of values of $p,q,r$ and $s$ for which the above two lemmas hold is the same and therefore if we let $f$ to be the function which maps elements form $\{1,2,3,...,1999\}$ to $\{2000,2001,2002,2003\}$ such that $q+s$ is even and $g$ to be the function  hich maps elements form $\{1,2,3,...,1999\}$ to $\{2000,2001,2002,2003\}$ such that $p+q$ is even we will get a bijection.(Maybe!) How to sort a character vector where elements contain letters and numbers in R? Learning special relativity and working with Minkowski diagrams for a while now i am still trying to get my head around some oddity.   From my understanding time can often be seen as the dependent variable in those diagrams, so why put it to the vertical axis? What confuses me even more is the apparent convention to denote a point in space time $(t,x)$ even though $x$ is on the horizontal and $t$ is on the vertical axis, so the usual notation would be to refer to that vector as $(x, t)$.  Are there mostly historical reasons or why is it beneficial to use this notation and to violate usual conventions? Approximating commuting matrices by commuting diagonalizable matrices Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A_n$ and $B_n$ of matrices such that   $A_n \rightarrow A$, $B_n \rightarrow B$.  Each $A_n$ is diagonalizable  and the same for each $B_n$.  For every $n$, $A_n$ commutes with $B_n$.    Moreover, it would be nice if the following property was additionally satisfied: if $A,B$ are real, then $A_n,B_n$ can be chosen to be real as well. What is the origin of "that's using your noggin"? Validation Rule on picklist under opportunities There are 2 requirements:    standard users can’t update the status of a record beyond “open won” (so that only SF Admins or auto processes update “registered” and the stages beyond registration. Validation rule for this:    ~   AND(       $Profile.Name &lt;&gt; "System Administrator",      ISCHANGED(StageName),      OR(TEXT(StageName) = "Registered",     TEXT(StageName) = "Pre-Certified",     TEXT(StageName) = "Documentation, Submission",     TEXT(StageName) = "Performance verified",     TEXT(StageName) = "Appeals",     TEXT(StageName) = "Certification/ Closed Won")      )   Requirement 2: Standard users cannot update the status of a record that is already Registered (or post-registered phase). For us, some Opp was already “Registered” but some standard user  was able to change the record to “Open Won.”    I am looking for the validation rule of requirement 2 and which should align with first requirement also.  Thanks, Please guide. Calling Python in PHP I have a Python script I recently wrote that I call using the command line with some options. I now want a very thin web interface to call this script locally on my Mac.  I don't want to go through the minor trouble of installing mod_python or mod_wsgi on my Mac, so I was just going to do a system() or popen() from PHP to call the Python script.  Any better ideas? Thanks in advance! Why is $M\frac{dv}{dt} = v_{rel} .\frac{dm}{dt}$ correct and $(M - dm)\frac{dv}{dt} = v_{rel} .\frac{dm}{dt}$ wrong? Newton's 2nd law of motion can't be applied for mass-varying systems. Another force, known as Thrust must come to play. It can be measured using law of conservation of linear momentum. $$\text{Thrust} M.\dfrac{dv}{dt} = v_{rel} .\dfrac{dm}{dt}.$$   where $M$ is the mass of the system;   $\frac{dm}{dt}$ is the rate at which the mass changes;    &amp;    $v_{rel}$ is the relative velocity of the exhausted mass w.r.t to the main system.    This is what my book writes. Now, the left hand side of the eqn. represents force on mass $M$ whereas the right hand side represents the force due to variable mass. How can they be equal? During the application of thrust, $M$ does change and the equation had to be $$ (M - dm). \dfrac{dv}{dt} = v_{rel} .\dfrac{dm}{dt}$$ But that is not the book's equation. Why is mine wrong? What is the intuition? What is the correct way to use infinitive after the verb "help": with or without "to"? I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and Minkowski tensors, instead of the more traditional "beginners" approach. Would anyone have a recommendation for such a textbook ?  I already have decent knowledge of the physics and maths of both SR and GR ( including vector and tensor calculus ), but would like to take a step back and expand and broaden my intuition of the geometry underlying SR, as described by 4-vectors and tensors. What I do not need is another "and here is the formula for time dilation..." type of text, of which there are thousands out there, but something much more geometric and in-depth.  Thanks in advance. Can I use the "Scan" button with SANE? I have an HP Deskjet with integrated scanner. It has a "Scan" button on its top, but this button does not interact with the xsane program on my Debian machine: To get a page scanned, I have to initiate the scanning process from xsane.  Is there a setting (or a scan program) that allows me to press the scan button and cause the program to scan a page? I'm trying to determine the best way to truncate or drop extra decimal places in SQL without rounding.  For example:  declare @value decimal(18,2)  set @value = 123.456   This will automatically round @value to be 123.46, which is good in most cases.  However, for this project, I don't need that.  Is there a simple way to truncate the decimals I don't need?  I know I can use the left() function and convert back to a decimal.  Are there any other ways? Linear regression not fitting well I do a linear regression using R lm function:  x = log(errors) plot(x,y) lm.result = lm(formula = y ~ x) abline(lm.result, col="blue") # showing the "fit" in blue     but it does not fit well. Unfortunately I can't make sense of the manual.  Can someone point me in the right direction to fit this better?  By fitting I mean I want to minimize the Root Mean Squared Error (RMSE).    Edit: I have posted a related question (it's the same problem) here:   and the raw data here:    except that on that  x is what is called errors on the present page here, and there are less samples (1000 vs 3000 in the present page plot). I wanted to make things simpler in the other question. Also how do LEFT JOIN, RIGHT JOIN and FULL JOIN fit in? When is JavaScript's eval() not evil? I'm writing some JavaScript code to parse user-entered functions (for spreadsheet-like functionality). Having parsed the formula I could convert it into JavaScript and run eval() on it to yield the result.  However, I've always shied away from using eval() if I can avoid it because it's evil (and, rightly or wrongly, I've always thought it is even more evil in JavaScript, because the code to be evaluated might be changed by the user).  So, when it is OK to use it?