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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*}

$$

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.)

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Blog:Fabrice

$$

 \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)

Unesco

Spanish cities

#	Municipality	Autonomous community	Pop. (2009)
1	Madrid		3,255,944
2	Barcelona		1,621,537
3	Valencia		852,208
4	Seville		703,206
5	Zaragoza		674,317
6	Málaga		568,305
7	Murcia		436,870
8	Palma		401,270
9	Las Palmas		381,847
10	Bilbao		354,860
11	Alicante		334,757
12	Córdoba		328,428
13	Valladolid		317,864
14	Vigo		297,332
15	Gijón		277,554
16	L'Hospitalet de Llobregat		257,038
17	A Coruña		246,056
18	Vitoria-Gasteiz		235,661
19	Granada		234,325
20	Elche		230,112
21	Oviedo		224,005
22	Santa Cruz de Tenerife		222,417
23	Badalona		219,547
24	Cartagena		211,996
25	Terrassa		210,941
26	Jerez de la Frontera		207,532
27	Sabadell		206,493
28	Móstoles		206,478
29	Alcalá de Henares		204,574
30	Pamplona		198,491
31	Fuenlabrada		197,836
32	Almería		188,810
33	Leganés		186,066
34	Donostia-San Sebastián		186,066
35	Santander		182,700
36	Castellón de la Plana		180,005
37	Burgos		178,966
38	Albacete		169,716
39	Alcorcón		167,967
40	Getafe		167,164
41	Salamanca		155,619
42	Logroño		152,107
43	San Cristóbal de La Laguna		150,661
44	Huelva		148,806
45	Badajoz		148,334
46	Tarragona		140,323
47	Lleida		138.416
48	Marbella		134,623
49	León		134,305
50	Cádiz		126,766

Shorter

Spanish cities

#	Municipality	Pop. (2009)
1	Madrid	3,255,944
2	Barcelona	1,621,537
3	Valencia	852,208
4	Seville	703,206
5	Zaragoza	674,317
6	Málaga	568,305
7	Murcia	436,870
8	Palma	401,270
9	Las Palmas	381,847
10	Bilbao	354,860
11	Alicante	334,757
12	Córdoba	328,428
13	Valladolid	317,864
14	Vigo	297,332
15	Gijón	277,554
16	L'Hospitalet de Llobregat	257,038
17	A Coruña	246,056
18	Vitoria-Gasteiz	235,661
19	Granada	234,325
20	Elche	230,112
21	Oviedo	224,005
22	Santa Cruz de Tenerife	222,417
23	Badalona	219,547
24	Cartagena	211,996
25	Terrassa	210,941
26	Jerez de la Frontera	207,532
27	Sabadell	206,493
28	Móstoles	206,478
29	Alcalá de Henares	204,574
30	Pamplona	198,491
31	Fuenlabrada	197,836
32	Almería	188,810
33	Leganés	186,066
34	Donostia-San Sebastián	186,066
35	Santander	182,700
36	Castellón de la Plana	180,005
37	Burgos	178,966
38	Albacete	169,716
39	Alcorcón	167,967
40	Getafe	167,164
41	Salamanca	155,619
42	Logroño	152,107
43	San Cristóbal de La Laguna	150,661
44	Huelva	148,806
45	Badajoz	148,334
46	Tarragona	140,323
47	Lleida	138.416
48	Marbella	134,623
49	León	134,305
50	Cádiz	126,766