¿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. 45.588245,2.992312, Saint-Nectaire, November 2006. 51.451727,-2.601619, Bristol, 28 November (2006). 51.124659,1.334354, Dover, 16 December (2006). 38.888972,-3.715144, Almagro, 23 Enero (2007). 41.305647,13.025673, Sabaudia, 31 March (2007). 23.140888,-82.351673, la Habana, 10 April (2007). Last time on 15 January (2009). 23.143624,-82.380670, Hotel Nacional, 10 April (2007). 22.619636,-83.742771, Mural de la prehistoria, 14 April (2007). 22.676217,-83.705464, Cueva del Indio, 14 April (2007). 21.805123,-79.984585, Trinidad, 17 April (2007). 21.821832,-79.998572, parque el cubano, 18 April (2007). 21.737847,-80.011847, playa Ancon, 19 April (2007). 21.797615,-80.035639, la Boca, , 19 April (2007). 21.835437,-80.009945, Topes de Collantes, 20 April (2007). 21.932245,-79.444588, Sancti Spíritus, 20 April (2007). 21.944896,-79.168042, Jatibonico, 21 April (2007). 22.108604,-78.627472, Morón, 21 April (2007). 22.347296,-78.484554, Road to Cayo Coco, 21 April (2007). 22.546057,-78.427110, playa Flamenco, 21 April (2007). 22.553109,-78.409919, playa Prohibida, 22 April (2007). 22.195102,-78.914919, Chambas, 22 April (2007). 22.494822,-79.54509, Remedios, 22 April (2007). 22.405856,-79.962941, Santa Clara, 22 April (2007). 22.241409,-80.392951, Palmira, 23 April (2007). 22.121346,-80.451497, Palacio del Valle in Cienfuegos, 23 April (2007). 22.525134,-79.472615, Caibarién, 23 April (2007). 22.124467,-80.325241, Jardín Botánico de Cienfuego, 23 April (2007). 22.402613,-79.979391, Memorial Emesto Che Guevara, 24 April (2007). 23.046288,-81.576909, Matanzas, 24 April (2007). 23.169569,-81.222029, Varadero, 24 April (2007). 45.453991,3.171775, Vallée des Saints, Boudes, 11 June (2007). -15.755209,-47.876182, Brasilia, Colinas, 1 July (2007). -15.798382,-47.875546, Cathedral Nossa Senhora Aparecida, July (2007). -15.767686,-47.868889, CIFMC, July—August (2007). -15.790526,-47.893032, Torre de TV de Brasília, July (2007). -15.817302,-47.930916, Brasilia cemetery, July (2007). -15.800666,-47.894781, Santuário Dom Bosco, August (2007). -15.823635,-47.829571, Juscelino Kubitschek bridge, August (2007). -15.368528,-47.454767, Indaiá, 8 July (2007). -14.120141,-47.786624, Chapada dos Veadeiros, 13 July (2007). -14.176814,-47.814946, Vila de São Jorge, 14 July (2007). -23.563146,-46.758680, São Paulo, 27 July (2007). -25.535461,-54.590685, Foz do Iguaçu, 30 July (2007). -25.695587,-54.436832, Iguazu falls, 31 July (2007). -25.407417,-54.589296, Itaipu, 1 August (2007). -25.509489,-54.607664, Ciudad del Este, 1 August (2007). -22.983136,-43.201380, Rio de Janeiro, 1 August (2007). -22.949617,-43.156485, Pão de Açúcar, 3 August (2007). -22.951910,-43.210385, Corcovado, 2 August (2007). -15.841095,-48.767585, Salto de Corumbá, 29 August (2007). -15.926341,-48.809903, Corumbá de Goiás, 29 August (2007). -15.849643,-48.959022, Pirenópolis, 30 August (2007). -15.839146,-48.885098, Cachoeira do Abade, 30 August (2007). 40.420322,-3.747043, Casa del Campo, 30 September (2007). 40.657963,-3.766269, Colmenar Viejo, 19 October (2007). 44.974461,3.177332, Viaduc de Garabit, 31 October (2007). 44.085239,3.021769, Viaduc de Millau, 31 October (2007). 43.975457,2.990938, Roquefort-sur-Soulzon, 31 October (2007). 42.699582,2.898359, Perpignan, 31 October (2007). 41.656327,-0.878905, Zaragoza, 1 November (2007). 41.352365,-1.642744, Calatayud, 1 November (2007). 40.888415,-3.887633, Santa Maria de el Paular, 11 November (2007). 39.856962,-4.024620, Toledo, 19 January (2008). 41.403397,2.174068, Sagrada Familia, 27 January (2008). 41.275636,1.989522, ICFO, 28 January (2008). 41.500462,2.108038, Universitat Autònoma de Barcelona, 29 January (2008). 40.419181,-3.684308, El Retiro, 23 February (2008). 40.618458,-3.694303, La Hoya del Tocon, 2 March (2008). 40.482097,-3.367056, Alcalá de Henares, 7 March (2008). 40.658481,-4.700189, Ávila, 15 March (2008). 40.660765,-4.710109, Los Cuatro Postes, 15 March (2008). 40.965079,-5.663571, Salamanca, 15 March (2008). 40.960564,-5.666570, Catedral Nueva de Salamanca, 16 March (2008). 41.330047,-5.080367, Nava del Rey, 17 March (2008). 41.309109,-4.908476, Castillo de la Mota, 17 March (2008). 41.308081,-4.915533, Medina del Campo, 17 March (2008). 41.500914,-5.000444, Tordesillas, 17 March (2008). 41.651951,-4.728709, Valladolid, 17 March (2008). 42.011050,-4.537398, Catedral San Antolín de Palencia, 17 March (2008). 42.007481,-4.534275, Parroquia de San Miguel, 18 March (2008). 42.268108,-4.405064, Fromista, 18 March (2008).‎ 42.291681,-4.136172, Castillo de Castrojeriz, 18 March (2008). 42.288309,-4.141803, Castrojeriz, 18 March (2008). 42.573710,-4.385177, Calahorra de Boedo, 19 March (2008). 42.603098,-4.359975, Zorita del Páramo, 19 March (2008). 42.726628,-4.273834, Santa María del Mave, 19 March (2008). 42.710098,-4.303928, Becerril del Carpio, 19 March (2008). 42.763856,-4.297212, Lomilla, 19 March (2008). 42.745033,-4.285831, Iglesia rupestre de Olleros de Pisuerga, 19 March (2008). 42.767922,-4.275631, Valoria de Aguilar, 19 March (2008). 42.795669,-4.262392, Aguilar del Campoo, 19 March (2008). 42.794272,-4.290567, Embalse de Aguilar, 19 March (2008). 42.773780,-4.367486, Vallespinoso de Aguilar, 19 March (2008). 42.340741,-3.704026, Burgos, 20 March (2008). 40.994589,-3.633969, Buitrago de Lozoya, 20 March (2008).

41.889879,12.491368, Roma, 5 April (2008)

last time on 6 April (2012). 35.665391,139.716298, Tokyo, 6 April (2008). 35.659505,139.700534, Shibuya, 7 April (2008). 35.685780,139.690660, Park Hyatt Tokyo, 8 April (2008). 35.331889,139.555108, Kenchō-ji, 11 April (2008). 35.337705,139.547498, Engaku-ji, 11 April (2008). 35.334921,139.547406, Hachinoki restaurant, 11 April (2008). 35.316854,139.535734, Koutokuin, 11 April (2008). 35.325366,139.555973, Tsurugaoka Hachiman-gū, 11 April (2008). 34.297298,132.319051, Miyajima (宮島町), 12 April (2008). 34.279631,132.319497, Mount Misen, 13 April (2008). 34.391801,132.452102, Hiroshima, 14 April (2008). 35.014198,135.747509, Nijo castle, 15 April (2008). (F) 35.029439,135.771596, Kamo River, 15 April (2008). 35.022878,135.795550, Philosopher's path, 15 April (2008). 35.003491,135.778538, Gion, 15 April (2008). 34.685163,135.837380, Isui-in garden, 16 April (2008). 34.689071,135.839904, Todai-ji temple, 16 April (2008). 34.681332,135.848307, Kasuga shrine, 16 April (2008). 34.994877,135.785021, Kiyomizu, 17 April (2008). 35.727550,139.801004, Aizuya Inn, 17 April (2008). 35.661498,139.769568, Tsukiji fish market, 18 April (2008). 35.659803,139.763232, Hama garden, 18 April (2008). 35.658596,139.745428, Tokyo tower, 18 April (2008). 35.718949,139.776557, National museum of Japan, 19 April (2008). 35.714784,139.796666, Senso-ji temple, 19 April (2008). 43.315502,-1.991898, Donostia, 8 May (2008). 43.324827,-1.989073, Castillo de la Mota, 9 May (2008). 43.321737,-2.005587, Peines al viento, 9 May (2008). 42.685099,-2.948746, Miranda de Ebro, 9 May (2008). 40.423907,-3.717378, Templo de Debod, 21 June (2008). 40.879564,-3.608288, Pico de la Miel, 6 July (2008). 40.864394,-3.787032, Chorrera de Mojonavalle, 6 July (2008). 42.140353,-0.408609, Coffee in Huesca, 29 July (2008). 43.294750,-0.374864, Pau, 30 July (2008). 43.608092,1.441239, Toulouse, 30 July (2008). 43.928446,2.142457, Sainte Cécile d'Albi, 30 July (2008). 44.351037,2.573348, Rodez, 31 July (2008). 44.522462,2.763694, Espalion, 31 July (2008). 44.684255,2.846982, Laguiole, 31 July (2008). 44.853778,3.002727, Chaudes-Aigues, 31 July (2008). 45.475315,3.445141, Plan d'eau du Vernet, 5 August (2008). (E) 45.772264,2.965464, Puy-de-Dôme, 8 August (2008). (E) 45.621827,2.820653, Les Roches Tuilière et Sanadoire, 8 August (2008). (E) 45.569337,2.929159, Lac Chambon, 8 August (2008). (E) 45.588269,2.99218, Saint-Nectaire, 8 August (2008). 45.512393,3.451117, Église de Chaméane, 13 August (2008). 45.403739,3.126151, Ardes-sur-Couze, 17 August (2008). 45.394702,3.116040, Parc animalier du Cézallier, 17 August (2008). 45.473458,3.27819, Nonette, 17 August (2008). 28.655762,77.241364, Red Fort, 31 December (2008). 28.645685,77.216747, New Delhi, 31 December (2008). 28.524450,77.185357, Qutb complex, 1 January (2009). 28.593140,77.250723, Humayun's tomb, 1 January (2009). 28.650692,77.233484, Jama Masjid, 1 January (2009). 25.307612,83.011179, Varanasi, 2 January (2009). 25.449429,81.83248, Allahabad, 5 January (2009). 25.460429,81.851462, Allahabad University, 6 January (2009). 20.026060,-75.811091, Santiago de Cuba, 9 January (2009). 20.050816,-75.948937, Santuario de La Virgen de La Caridad del Cobre, , 10 January (2009). 19.946045,-75.6545, Valle de la prehistoria, Baconao, 10 January (2009). (F) 19.959227,-75.704116, A concrete platform, 10 January (2009). 19.960155,-75.703982, Beach near Siboney, 10 January (2009). 19.968502,-75.870228, Castillo del morro, 15 January (2009). 19.969873,-75.867776, Beach near Castillo del morro, 15 January (2009). 23.150398,-82.357314, Castillo del Morro, 17 January (2009). 50.908749,-1.410237, Mayflower theatre, 27 January (2009). 42.833729,-3.793137, Orbaneja del castillo, 19 March (2009). 42.423662,-3.682888, rio Ubierna, 19 March (2009). 43.389006,-4.108359, Santillana del Mar, 19 March (2009). (E) 41.390586,2.183675, Barcelona, 26 March (2009). (E) 51.541311,-0.14514, Camden town, 3 April (2009). 40.637669,17.945029, Brindisi, 13 April (2009). (E) 40.732087,17.578106, Ostuni, 13 April (2009). (E) 39.796499,18.368072, Santa Maria di Leuca, 14 April (2009). (E) 40.053035,17.973876, Gallipoli, 14 April (2009). 40.559813,17.807413, Mesagne, 15 April (2009). 40.709704,17.798719, Beach near the Torre Guaceto, 15 April (2009). 40.352554,18.171555, Lecce, 15 April (2009). 40.352293,18.174879, Castello di Carlo V, 16 April (2009). 40.875428,17.148386, Grotte di Castellana, 19 April (2009). 40.782077,17.236934, Alberobello, 19 April (2009). 64.139436,-21.952966, Reykjavik, 21 April (2009). 64.14172,-21.926687, Hallgrímskirkja, April 2009. 63.879722,-22.448402, Blue Lagoon, 25 April (2009). 63.836649,-22.437803, Grindavik, 25 April (2009). 63.801806,-22.699359, Reykjanesviti, 25 April (2009). 63.934621,-22.687469, Hafnir, 25 April (2009). 64.065463,-21.955111, Hafnarfjörður, 25 April (2009). 64.253794,-21.037119, Þingvellir, 26 April (2009). 64.279508,-20.361379, Geysir, 26 April (2009). 64.327662,-20.123029, Gullfoss, 26 April (2009). 64.125736,-20.527105, Skálholt, 26 April (2009). </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