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¿Hey, whatcha doin on this page?

It's just where I put stuff that I'm experimenting on for possible f¯uture use.

¿Hey?

$$ \begin{align*} \label{eq:MonAug20130247CEST2012}

 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*}

$$

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$$

 \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}

 \label{def:Wns}
 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 \eqref{def:Wns} 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}

 \label{eq:W3k}
 W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.

\end{align} Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \eqref{eq:W3k} 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} \label{eq: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}

That's \eqref{eq:1} or \eqref{eq:W3k} above!

<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

• Visit of Lionel, Georges and Antoine (April 2012)
• Visit of Elena's parents (August 2012)