m (→Madrid) |
m (→Unesco) |
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* <sup>3</sup> Shared with [[Template:World Heritage Sites in Slovenia|Slovenia]] | * <sup>3</sup> Shared with [[Template:World Heritage Sites in Slovenia|Slovenia]] | ||
}} | }} | ||
+ | |||
+ | {| class="wikitable sortable" | ||
+ | |- | ||
+ | !# | ||
+ | !Municipality | ||
+ | !Autonomous community | ||
+ | !Pop. (2009)<ref name="INE"/> | ||
+ | |- | ||
+ | |1 | ||
+ | |[[Madrid]] | ||
+ | |{{flag|Madrid|name=Community of Madrid}} | ||
+ | |3,255,944 | ||
+ | |- | ||
+ | |2 | ||
+ | |[[Barcelona]] | ||
+ | |{{flag|Catalonia}} | ||
+ | |1,621,537 | ||
+ | |- | ||
+ | |3 | ||
+ | |[[Valencia, Spain|Valencia]] | ||
+ | |{{flag|Valencian Community}} | ||
+ | |852,208 | ||
+ | |- | ||
+ | |4 | ||
+ | |[[Seville]] | ||
+ | |{{flag|Andalusia}} | ||
+ | |703,206 | ||
+ | |- | ||
+ | |5 | ||
+ | |[[Zaragoza]] | ||
+ | |{{flag|Aragon}} | ||
+ | |674,317 | ||
+ | |- | ||
+ | |6 | ||
+ | |[[Málaga]] | ||
+ | |{{flag|Andalusia}} | ||
+ | |568,305 | ||
+ | |- | ||
+ | |7 | ||
+ | |[[Murcia]] | ||
+ | |{{flag|Murcia|name=Region of Murcia}} | ||
+ | |436,870 | ||
+ | |- | ||
+ | |8 | ||
+ | |[[Palma, Majorca|Palma]] | ||
+ | |{{flag|Balearic Islands}} | ||
+ | |401,270 | ||
+ | |- | ||
+ | |9 | ||
+ | |[[Las Palmas]] | ||
+ | |{{flag|Canary Islands}} | ||
+ | |381,847 | ||
+ | |- | ||
+ | |10 | ||
+ | |[[Bilbao]] | ||
+ | |{{flag|Basque Country}} | ||
+ | |354,860 | ||
+ | |- | ||
+ | |11 | ||
+ | |[[Alicante]] | ||
+ | |{{flag|Valencian Community}} | ||
+ | |334,757 | ||
+ | |- | ||
+ | |12 | ||
+ | |[[Córdoba, Spain|Córdoba]] | ||
+ | |{{flag|Andalusia}} | ||
+ | |328,428 | ||
+ | |- | ||
+ | |13 | ||
+ | |[[Valladolid]] | ||
+ | |{{flag|Castile and León}} | ||
+ | |317,864 | ||
+ | |- | ||
+ | |14 | ||
+ | |[[Vigo]] | ||
+ | |{{flag|Galicia}} | ||
+ | |297,332 | ||
+ | |- | ||
+ | |15 | ||
+ | |[[Gijón]] | ||
+ | |{{flag|Asturias}} | ||
+ | |277,554 | ||
+ | |- | ||
+ | |16 | ||
+ | |[[L'Hospitalet de Llobregat]] | ||
+ | |{{flag|Catalonia}} | ||
+ | |257,038 | ||
+ | |- | ||
+ | |17 | ||
+ | |[[A Coruña]] | ||
+ | |{{flag|Galicia}} | ||
+ | |246,056 | ||
+ | |- | ||
+ | |18 | ||
+ | |[[Vitoria-Gasteiz]] | ||
+ | |{{flag|Basque Country}} | ||
+ | |235,661 | ||
+ | |- | ||
+ | |19 | ||
+ | |[[Granada]] | ||
+ | |{{flag|Andalusia}} | ||
+ | |234,325 | ||
+ | |- | ||
+ | |20 | ||
+ | |[[Elche]] | ||
+ | |{{flag|Valencian Community}} | ||
+ | |230,112 | ||
+ | |- | ||
+ | |21 | ||
+ | |[[Oviedo]] | ||
+ | |{{flag|Asturias}} | ||
+ | |224,005 | ||
+ | |- | ||
+ | |22 | ||
+ | |[[Santa Cruz de Tenerife]] | ||
+ | |{{flag|Canary Islands}} | ||
+ | |222,417 | ||
+ | |- | ||
+ | |23 | ||
+ | |[[Badalona]] | ||
+ | |{{flag|Catalonia}} | ||
+ | |219,547 | ||
+ | |- | ||
+ | |24 | ||
+ | |[[Cartagena, Spain|Cartagena]] | ||
+ | |{{flag|Region of Murcia}} | ||
+ | |211,996 | ||
+ | |- | ||
+ | |25 | ||
+ | |[[Terrassa]] | ||
+ | |{{flag|Catalonia}} | ||
+ | |210,941 | ||
+ | |- | ||
+ | |26 | ||
+ | |[[Jerez de la Frontera]] | ||
+ | |{{flag|Andalusia}} | ||
+ | |207,532 | ||
+ | |- | ||
+ | |27 | ||
+ | |[[Sabadell]] | ||
+ | |{{flag|Catalonia}} | ||
+ | |206,493 | ||
+ | |- | ||
+ | |28 | ||
+ | |[[Móstoles]] | ||
+ | |{{flag|Madrid|name=Community of Madrid}} | ||
+ | |206,478 | ||
+ | |- | ||
+ | |29 | ||
+ | |[[Alcalá de Henares]] | ||
+ | |{{flag|Madrid|name=Community of Madrid}} | ||
+ | |204,574 | ||
+ | |- | ||
+ | |30 | ||
+ | |[[Pamplona]] | ||
+ | |{{flag|Navarre}} | ||
+ | |198,491 | ||
+ | |- | ||
+ | |31 | ||
+ | |[[Fuenlabrada]] | ||
+ | |{{flag|Madrid|name=Community of Madrid}} | ||
+ | |197,836 | ||
+ | |- | ||
+ | |32 | ||
+ | |[[Almería]] | ||
+ | |{{flag|Andalusia}} | ||
+ | |188,810 | ||
+ | |- | ||
+ | |33 | ||
+ | |[[Leganés]] | ||
+ | |{{flag|Madrid|name=Community of Madrid}} | ||
+ | |186,066 | ||
+ | |- | ||
+ | |34 | ||
+ | |[[Donostia-San Sebastián]] | ||
+ | |{{flag|Basque Country}} | ||
+ | |186,066 | ||
+ | |- | ||
+ | |35 | ||
+ | |[[Santander, Cantabria|Santander]] | ||
+ | |{{flag|Cantabria}} | ||
+ | |182,700 | ||
+ | |- | ||
+ | |36 | ||
+ | |[[Castellón de la Plana]] | ||
+ | |{{flag|Valencian Community}} | ||
+ | |180,005 | ||
+ | |- | ||
+ | |37 | ||
+ | |[[Burgos]] | ||
+ | |{{flag|Castile and León}} | ||
+ | |178,966 | ||
+ | |- | ||
+ | |38 | ||
+ | |[[Albacete]] | ||
+ | |{{flag|Castile–La Mancha}} | ||
+ | |169,716 | ||
+ | |- | ||
+ | |39 | ||
+ | |[[Alcorcón]] | ||
+ | |{{flag|Madrid|name=Community of Madrid}} | ||
+ | |167,967 | ||
+ | |- | ||
+ | |40 | ||
+ | |[[Getafe]] | ||
+ | |{{flag|Madrid|name=Community of Madrid}} | ||
+ | |167,164 | ||
+ | |- | ||
+ | |41 | ||
+ | |[[Salamanca]] | ||
+ | |{{flag|Castile and León}} | ||
+ | |155,619 | ||
+ | |- | ||
+ | |42 | ||
+ | |[[Logroño]] | ||
+ | |{{flagcountry|La Rioja (Spain)}} | ||
+ | |152,107 | ||
+ | |- | ||
+ | |43 | ||
+ | |[[San Cristóbal de La Laguna]] | ||
+ | |{{flag|Canary Islands}} | ||
+ | |150,661 | ||
+ | |- | ||
+ | |44 | ||
+ | |[[Huelva]] | ||
+ | |{{flag|Andalusia}} | ||
+ | |148,806 | ||
+ | |- | ||
+ | |45 | ||
+ | |[[Badajoz]] | ||
+ | |{{flag|Extremadura}} | ||
+ | |148,334 | ||
+ | |- | ||
+ | |46 | ||
+ | |[[Tarragona]] | ||
+ | |{{flag|Catalonia}} | ||
+ | |140,323 | ||
+ | |- | ||
+ | |47 | ||
+ | |[[Lleida]] | ||
+ | |{{flag|Catalonia}} | ||
+ | |138.416 | ||
+ | |- | ||
+ | |48 | ||
+ | |[[Marbella]] | ||
+ | |{{flag|Andalusia}} | ||
+ | |134,623 | ||
+ | |- | ||
+ | |49 | ||
+ | |[[León, Spain|León]] | ||
+ | |{{flag|Castile and León}} | ||
+ | |134,305 | ||
+ | |- | ||
+ | |50 | ||
+ | |[[Cádiz]] | ||
+ | |{{flag|Andalusia}} | ||
+ | |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] |
---|---|---|---|
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 | Template:Flagcountry | 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 |
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