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Revision as of 18:00, 15 August 2012
¿Hey, whatcha doin on this page?
It's just where I put stuff that I'm experimenting on for possible f¯uture use.
Contents |
¿Hey? 
$$ \begin{align*} a\ket{n}&=\sqrt{n}\ket{n-1}\,,&\bra{n}\,&a=\bra{n+1}\sqrt{n+1}\,,\\ \ud{a}\ket{n}&=\sqrt{n+1}\ket{n+1}\,,&\bra{n}\,&\ud{a}=\bra{n-1}\sqrt{n}\,, \end{align*} $$
$$ \begin{align*} \kern-1cm{(\mathrm{for}~i\le n+j)}\kern1cm a^i{\ud{a}}^j\ket{n}&={(n+j)!\over\sqrt{n!}\sqrt{(n+j-i)!}}\ket{n+j-i}\,,\\ \kern-1cm{(\mathrm{for}~i\le n)}\kern1cm a^{\dagger j}a^i\ket{n}&={\sqrt{n!}\sqrt{(n+j-i)!}\over(n-i)!}\ket{n+j-i}\,. \end{align*} $$
playing with faces
arial font algerian font bookman font braggadocio font courier font desdemona font garamond font modern font symbol font (These are pretty silly.) wingdings font (As are these.)
Blog
Ramblings
$$ \newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)} $$
We consider, for various values of $s$, the $n$-dimensional integral \begin{align} \tag{1} W_n (s) &:= \int_{[0, 1]^n} \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} \end{align} % which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral (1) expresses the $s$-th moment of the distance to the origin after $n$ steps.
By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer \begin{align} \tag{2} W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. \end{align} Appropriately defined, (2) also holds for negative odd integers. The reason for (2) was long a mystery, but it will be explained at the end of the paper.
\[ \begin{aligned}
\label{def:1}
\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}
\]
\begin{aligned} \tag{3} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}
<google1 style="2"></google1>
Do you know this formula of mine <m>\frac{2\pi^2}{q}\int_0^\infty f(r)J_1(qr)rdr</m>?
21, May (2010) 17, August (2010)
<plot> set pm3d at s solid set palette rgb -6,-15,-7 unset colorbox set ticslevel 0 unset ztics unset surface set samples 70 set isosamples 70,70 complex(x,y)=x*{1,0}+y*{0,1} mandel(x,y,z,n) = (abs(z)>2.0 || n>=1000)? log(n): mandel(x,y,z*z+complex(x,y),n+1) a=-0.38 b=-0.612 set multiplot set origin 0,0 set size 0.55,0.77 splot [-0.5:0.5][-0.5:0.5] mandel(a,b,complex(x,y),0) set origin 0.35,-0.15 set size 0.7,0.96 set view 0,0,,, splot [-0.5:0.5][-0.5:0.5] mandel(a,b,complex(x,y),0) </plot>
<music>
\relative c' {
e16-.->a(b gis)a-.->c(d b)c-.->e(f dis)e-.->a(b a)
gis(b e)e,(gis b)b,(e gis)gis,(b e)e,(gis? b e)
}
</music>
<music> \new Pianostaff << \new Staff { \time 2/2 \clef violin \key cis \minor \relative c \context Staff << \new Voice { \voiceOne
r4 cis8 dis e4 fis gis8 fis gis a gis fis e gis fis e fis gis fis e dis fis e dis e fis e d cis e d cis d e d cis b d cis b cis d cis b a cis b a b cis b a gis b a2 r cis2.
} \new Voice { \voiceTwo
e,8 gis a b cis dis bis cis dis4 r r2 r1 r1 r4 fis, b b b a8 gis a2 gis1~ gis8 gis fis eis fis2 gis2.
} \new Voice { \voiceThree \stemDown
s1 s s s s2. fis4 eis2 fis
} >> } \new Staff { \clef bass \time 2/2 \key cis \minor \relative c' \context Staff << \new Voice { \voiceOne
s1 r4 gis cis cis cis bis8 ais bis2 cis1 b2. s4 s1 b2 cis~ cis~ cis8 cis b a gis2.
} \new Voice { \voiceTwo
\stemUp cis,1 bis2 e dis1 \stemDown cis4 e a a a gis8 fis gis2~ \stemUp gis fis gis1 a2 fis~ fis8 fis e dis e4
} \new Voice { \voiceThree
\stemDown cis4 b a2 gis4 r4 g2\rest e1\rest e1\rest e1\rest r4 cis' fis fis fis eis8 dis eis2 fis r r
} >> } >> </music>
Trips
Excursions
- fuente de la Campanilla, the last walk with Nanou.