¿Hey?

$$ \begin{align*} \tag{1} a\ket{n}&=\sqrt{n}\ket{n-1}\,,&\bra{n}\,&a=\bra{n+1}\sqrt{n+1}\,,\\ \ud{a}\ket{n}&=\sqrt{n+1}\ket{n+1}\,,&\bra{n}\,&\ud{a}=\bra{n-1}\sqrt{n}\,, \end{align*} $$

$$ \begin{align*} \kern-1cm{(\mathrm{for}~i\le n+j)}\kern1cm a^i{\ud{a}}^j\ket{n}&={(n+j)!\over\sqrt{n!}\sqrt{(n+j-i)!}}\ket{n+j-i}\,,\\ \kern-1cm{(\mathrm{for}~i\le n)}\kern1cm a^{\dagger j}a^i\ket{n}&={\sqrt{n!}\sqrt{(n+j-i)!}\over(n-i)!}\ket{n+j-i}\,. \end{align*} $$

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} \tag{2} 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 (2) 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{3} W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. \end{align} Appropriately defined, (3) also holds for negative odd integers. The reason for (3) 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{4} \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 (4) or (3) above!

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

NavBox

<googlemap lat="32.546813" lon="11.953125" type="terrain" zoom="2" controls="small" width="730" height="400"> (F) 16.846091,-99.915515, La Quebrada, 17 March (2005). (F) 16.856761,-99.867810, Acapulco, 16 March (2005). 40.031002, 15.675370, Maratea, 16 September (2005) (F) on 16 September (2009). 51.504639,-0.171509, Hyde park, 5 November (2005). 51.503475,-0.119663, Eye of London, 3 November (2005). 51.515356,-0.173908, Cardiff hotel, 2 November (2005). 48.858235,2.294569, Tour Eiffel, 30 December (2005). 40.638484,-4.049767, Collado Villalba, 19 February (2006). 40.415408,-3.707155, Plaza Mayor, February 2006. 39.4739,-0.375728, Valencia, 15 March (2006). 45.527860,3.339433, Usson, 7 April (2006), last on 23 April (2011). 45.621629,3.205498, Montpeyroux, 14 April (2006). 45.517621,3.437181, Pierre du Moine Blanc, 18 April (2006). 40.414004,-3.681951, Parque del Retiro, 27 April (2006). 40.947857,-4.117859, Segovia, 30 April (2006); (E) on 9 September (2009). 40.418084,-3.714258, Palacio Real, 3 May (2006). 40.641771,-4.155211, Valle de los Caídos, 8 July (2006). 40.589118,-4.147448, El Escorial, 8 July (2006). 48.208547,16.373010, Vienna, 23 July (2006). 48.055350,16.130969, Heiligenkreuz Abbey, 27 July (2006). 48.04534,16.096109, Mayerling, 27 July (2006). 48.842444,2.321878, Tour Montparnasse, 2 August (2006). 48.881082,2.383109, les Buttes Chaumont, 2 August (2006). 48.861816,2.398036, Père Lachaise, 2 August (2006). 48.886218,2.343130, Sacré Cœur, 2 August (2006). 45.047265,3.885179, Puy-en-Velay, August (2006). 44.853547,0.483402, Bergerac, 18 August (2006). 43.708769,-1.052911, Dax, 19 August (2006). 43.776447,-1.418469, Vieux Bouceau, 19 August (2006). 43.653637,-0.684931, Saint-Cricq-Chalosse, 20 August (2006). 44.830789,-0.572673, Bordeaux, 21 August (2006). 45.321166,3.696032, La Chaise-Dieu, 27 August (2006). 51.894643,-0.418401, Luton's Wardown Park Museum, 1 October (2006). (F) 48.857039, 2.362410, Hotel Carnavalet, 20 October (2006). 51.525972,-0.124799, London, 4 November (2006). 50.920355,-1.293576, Botleigh Grange Hotel, 5 November (2006). 53.380466,-1.470289, Sheffield, 7 November (2006). 41.414044,2.152387, Parque Güell, 17 November (2006). 45.587339,3.128500, Champeix, November 2006. </googlemap>

week-end paris

Days	Pet (salida-llegada ida) (salida-llegada vuelta)	Beb (salida-llegada ida) (salida-llegada vuelta)	Prices	Total (€)
15-18	20:30-22:30 10:15-12:20	06:35-08:10 21:30-23:00	103+109	212
15-18	20:30-22:30 10:15-12:20	12:55-14:35 10:25-11:55	103+136	239
15-18	18:30-20:35 10:00-12:10	12:55-14:35 10:25-11:55	126+136	262
22-25	7:05-9:10 12:30-14:35	06:35-08:10 21:30-23:00	113+109	222
22-25	7:05-9:10 12:30-14:35	10:35-12:15 16:05-17:35	113+136	249

Excursions

fuente de la Campanilla, the last walk with Nanou.

Literature

Ora 25 by Virgil Gheorghiu.

Contents