m (→Spanish cities) |
m (→Spanish cities) |
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Line 639: | Line 639: | ||
|- | |- | ||
|11 | |11 | ||
− | |[[Alicante]] | + | |[[Alicante]] {{flag|Valencian Community|size=30px}} |
− | + | ||
|334,757 | |334,757 | ||
|- | |- | ||
|12 | |12 | ||
− | |[[Córdoba, Spain|Córdoba]] | + | |[[Córdoba, Spain|Córdoba]] {{flag|Andalusia|size=30px}} |
− | + | ||
|328,428 | |328,428 | ||
|- | |- | ||
|13 | |13 | ||
− | |[[Valladolid]] | + | |[[Valladolid]] {{flag|Castile and León|size=30px}} |
− | + | ||
|317,864 | |317,864 | ||
|- | |- | ||
|14 | |14 | ||
− | |[[Vigo]] | + | |[[Vigo]] {{flag|Galicia|size=30px}} |
− | + | ||
|297,332 | |297,332 | ||
|- | |- | ||
|15 | |15 | ||
− | |[[Gijón]] | + | |[[Gijón]] {{flag|Asturias|size=30px}} |
− | + | ||
|277,554 | |277,554 | ||
|- | |- | ||
|16 | |16 | ||
− | |[[L'Hospitalet de Llobregat]] | + | |[[L'Hospitalet de Llobregat]] {{flag|Catalonia|size=30px}} |
− | + | ||
|257,038 | |257,038 | ||
|- | |- | ||
|17 | |17 | ||
− | |[[A Coruña]] | + | |[[A Coruña]] {{flag|Galicia|size=30px}} |
− | + | ||
|246,056 | |246,056 | ||
|- | |- | ||
|18 | |18 | ||
− | |[[Vitoria-Gasteiz]] | + | |[[Vitoria-Gasteiz]] {{flag|Basque Country|size=30px}} |
− | + | ||
|235,661 | |235,661 | ||
|- | |- | ||
|19 | |19 | ||
− | |[[Granada]] | + | |[[Granada]] {{flag|Andalusia|size=30px}} |
− | + | ||
|234,325 | |234,325 | ||
|- | |- | ||
|20 | |20 | ||
− | |[[Elche]] | + | |[[Elche]] {{flag|Valencian Community|size=30px}} |
− | + | ||
|230,112 | |230,112 | ||
|- | |- | ||
|21 | |21 | ||
− | |[[Oviedo]] | + | |[[Oviedo]] {{flag|Asturias|size=30px}} |
− | + | ||
|224,005 | |224,005 | ||
|- | |- | ||
|22 | |22 | ||
− | |[[Santa Cruz de Tenerife]] | + | |[[Santa Cruz de Tenerife]] {{flag|Canary Islands|size=30px}} |
− | + | ||
|222,417 | |222,417 | ||
|- | |- | ||
|23 | |23 | ||
− | |[[Badalona]] | + | |[[Badalona]] {{flag|Catalonia|size=30px}} |
− | + | ||
|219,547 | |219,547 | ||
|- | |- | ||
|24 | |24 | ||
− | |[[Cartagena, Spain|Cartagena]] | + | |[[Cartagena, Spain|Cartagena]] {{flag|Region of Murcia|size=30px}} |
− | + | ||
|211,996 | |211,996 | ||
|- | |- | ||
|25 | |25 | ||
− | |[[Terrassa]] | + | |[[Terrassa]] {{flag|Catalonia|size=30px}} |
− | + | ||
|210,941 | |210,941 | ||
|- | |- | ||
|26 | |26 | ||
− | |[[Jerez de la Frontera]] | + | |[[Jerez de la Frontera]] {{flag|Andalusia|size=30px}} |
− | + | ||
|207,532 | |207,532 | ||
|- | |- | ||
|27 | |27 | ||
− | |[[Sabadell]] | + | |[[Sabadell]] {{flag|Catalonia|size=30px}} |
− | + | ||
|206,493 | |206,493 | ||
|- | |- | ||
|28 | |28 | ||
− | |[[Móstoles]] | + | |[[Móstoles]] {{flag|Madrid|name=Community of Madrid|size=30px}} |
− | + | ||
|206,478 | |206,478 | ||
|- | |- | ||
|29 | |29 | ||
− | |[[Alcalá de Henares]] | + | |[[Alcalá de Henares]] {{flag|Madrid|name=Community of Madrid|size=30px}} |
− | + | ||
|204,574 | |204,574 | ||
|- | |- | ||
|30 | |30 | ||
− | |[[Pamplona]] | + | |[[Pamplona]] {{flag|Navarre|size=30px}} |
− | + | ||
|198,491 | |198,491 | ||
|- | |- | ||
|31 | |31 | ||
− | |[[Fuenlabrada]] | + | |[[Fuenlabrada]] {{flag|Madrid|name=Community of Madrid|size=30px}} |
− | + | ||
|197,836 | |197,836 | ||
|- | |- | ||
|32 | |32 | ||
− | |[[Almería]] | + | |[[Almería]] {{flag|Andalusia|size=30px}} |
− | + | ||
|188,810 | |188,810 | ||
|- | |- | ||
|33 | |33 | ||
− | |[[Leganés]] | + | |[[Leganés]] {{flag|Madrid|name=Community of Madrid|size=30px}} |
− | + | ||
|186,066 | |186,066 | ||
|- | |- | ||
|34 | |34 | ||
− | |[[Donostia-San Sebastián]] | + | |[[Donostia-San Sebastián]] {{flag|Basque Country|size=30px}} |
− | + | ||
|186,066 | |186,066 | ||
|- | |- | ||
|35 | |35 | ||
− | |[[Santander, Cantabria|Santander]] | + | |[[Santander, Cantabria|Santander]] {{flag|Cantabria|size=30px}} |
− | + | ||
|182,700 | |182,700 | ||
|- | |- | ||
|36 | |36 | ||
− | |[[Castellón de la Plana]] | + | |[[Castellón de la Plana]] {{flag|Valencian Community|size=30px}} |
− | + | ||
|180,005 | |180,005 | ||
|- | |- | ||
|37 | |37 | ||
− | |[[Burgos]] | + | |[[Burgos]] {{flag|Castile and León|size=30px}} |
− | + | ||
|178,966 | |178,966 | ||
|- | |- | ||
|38 | |38 | ||
− | |[[Albacete]] | + | |[[Albacete]] {{flag|Castile–La Mancha|size=30px}} |
− | + | ||
|169,716 | |169,716 | ||
|- | |- | ||
|39 | |39 | ||
− | |[[Alcorcón]] | + | |[[Alcorcón]] {{flag|Madrid|name=Community of Madrid|size=30px}} |
− | + | ||
|167,967 | |167,967 | ||
|- | |- | ||
|40 | |40 | ||
− | |[[Getafe]] | + | |[[Getafe]] {{flag|Madrid|name=Community of Madrid|size=30px}} |
− | + | ||
|167,164 | |167,164 | ||
|- | |- | ||
|41 | |41 | ||
− | |[[Salamanca]] | + | |[[Salamanca]] {{flag|Castile and León|size=30px}} |
− | + | ||
|155,619 | |155,619 | ||
|- | |- | ||
|42 | |42 | ||
− | |[[Logroño]] | + | |[[Logroño]] {{flag|La Rioja|size=30px}} |
− | + | ||
|152,107 | |152,107 | ||
|- | |- | ||
|43 | |43 | ||
− | |[[San Cristóbal de La Laguna]] | + | |[[San Cristóbal de La Laguna]] {{flag|Canary Islands|size=30px}} |
− | + | ||
|150,661 | |150,661 | ||
|- | |- | ||
|44 | |44 | ||
− | |[[Huelva]] | + | |[[Huelva]] {{flag|Andalusia|size=30px}} |
− | + | ||
|148,806 | |148,806 | ||
|- | |- | ||
|45 | |45 | ||
− | |[[Badajoz]] | + | |[[Badajoz]] {{flag|Extremadura|size=30px}} |
− | + | ||
|148,334 | |148,334 | ||
|- | |- | ||
|46 | |46 | ||
− | |[[Tarragona]] | + | |[[Tarragona]] {{flag|Catalonia|size=30px}} |
− | + | ||
|140,323 | |140,323 | ||
|- | |- | ||
|47 | |47 | ||
− | |[[Lleida]] | + | |[[Lleida]] {{flag|Catalonia|size=30px}} |
− | + | ||
|138.416 | |138.416 | ||
|- | |- | ||
|48 | |48 | ||
− | |[[Marbella]] | + | |[[Marbella]] {{flag|Andalusia|size=30px}} |
− | + | ||
|134,623 | |134,623 | ||
|- | |- | ||
|49 | |49 | ||
− | |[[León, Spain|León]] | + | |[[León, Spain|León]] {{flag|Castile and León|size=30px}} |
− | + | ||
|134,305 | |134,305 | ||
|- | |- | ||
|50 | |50 | ||
− | |[[Cádiz]] | + | |[[Cádiz]] {{flag|Andalusia|size=30px}} |
− | + | ||
|126,766 | |126,766 | ||
|} | |} |
¿Hey, whatcha doin on this page?
It's just where I put stuff that I'm experimenting on for possible f¯uture use.
Contents |
$$ \begin{align*} \tag{1} a\ket{n}&=\sqrt{n}\ket{n-1}\,,&\bra{n}\,&a=\bra{n+1}\sqrt{n+1}\,,\\ \ud{a}\ket{n}&=\sqrt{n+1}\ket{n+1}\,,&\bra{n}\,&\ud{a}=\bra{n-1}\sqrt{n}\,, \end{align*} $$
$$ \begin{align*} \kern-1cm{(\mathrm{for}~i\le n+j)}\kern1cm a^i{\ud{a}}^j\ket{n}&={(n+j)!\over\sqrt{n!}\sqrt{(n+j-i)!}}\ket{n+j-i}\,,\\ \kern-1cm{(\mathrm{for}~i\le n)}\kern1cm a^{\dagger j}a^i\ket{n}&={\sqrt{n!}\sqrt{(n+j-i)!}\over(n-i)!}\ket{n+j-i}\,. \end{align*} $$
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.)
$$ \newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)} $$
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}
<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>
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# | 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 |
# | 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 |