|
|
Line 4: |
Line 4: |
| | | |
| == ¿Hey? [[Image:Thorne-spider.jpeg|50px|link=http://www.27bslash6.com/overdue.html]] == | | == ¿Hey? [[Image:Thorne-spider.jpeg|50px|link=http://www.27bslash6.com/overdue.html]] == |
− |
| |
− | === Answers ===
| |
− |
| |
− | ==== Question 1 ====
| |
− |
| |
− | Calculate:
| |
− |
| |
− | $$\int_{-\infty}^\infty\frac{dx}{1+x^2+x^4}\,.$$
| |
− |
| |
− | The poles are easily find by noticing that $(1+x^2+x^4)(1-x^2)=1-x^6$, i.e., they are the sixth root of unity which are not solutions of $x^2=1$, which leaves:
| |
− |
| |
− | \begin{equation}
| |
− | e^{ik\pi/3}\quad\mathrm{for\ }k\in\{1,2,4,5\}
| |
− | \end{equation}
| |
− |
| |
− | We can now calculate the integral by computing the residues and summing those on the upper half-plane (times $2i\pi$):
| |
− |
| |
− | \begin{multline}
| |
− | \mathrm{Res}_{k=1}\frac{1}{(z-e^{i\pi/3})(z-e^{i2\pi/3})(z-e^{-i\pi/3})(z-e^{-i2\pi/3})}
| |
− | =\\
| |
− | \frac{1}{(e^{i\pi/3}-e^{i2\pi/3})(e^{i\pi/3}-e^{-i\pi/3})(e^{i\pi/3}-e^{-i2\pi/3})}\,.
| |
− | \end{multline}
| |
− |
| |
− | The denominator is calculated easily by dealing with the geometry of
| |
− | the unit circle (one is a sine, the other is after computing a
| |
− | product, the translated sum of a number with its opposite). One thus
| |
− | finds the residue as:
| |
− |
| |
− | $$1/(2i\sqrt{3}e^{i\pi/3})$$
| |
− |
| |
− | The other residue is found similarly as
| |
− |
| |
− | $$1/(2i\sqrt{3}e^{-i\pi/3})$$
| |
− |
| |
− | so that the integral is finally, since $1/e^{-i\pi/3}+1/e^{i\pi/3}=e^{-i\pi/3}+e^{i\pi/3}=1$, simply:
| |
− |
| |
− | $$\frac{\pi}{\sqrt3}$$
| |
− |
| |
− | ==== Question 2 ====
| |
− |
| |
− | We apply the Cauchy-Riemann criterion:
| |
− |
| |
− | \begin{align}
| |
− | \partial_x\cos x&=-\sin x&\, \partial_x(-\sinh y)&=0\\
| |
− | \partial_y\cos x&=0&\, \partial_y(-\sinh y)&=-\cosh y
| |
− | \end{align}
| |
− |
| |
− | While one condition ($\partial_x v=-\partial_y u$) is always
| |
− | satisfied, the other ($\partial_x u=\partial_y v$) is only for points
| |
− | where both the cosine and the hyperbolic sine are unity, since one is
| |
− | less than one and the other larger than one otherwise. The function is
| |
− | therefore derivable at points $\pi/2+2k\pi$ for $k\in\mathbf{Z}$ on
| |
− | the real line.
| |
− |
| |
− | ==== Question 3 ====
| |
− |
| |
− | A function $f$ is harmonic of $\nabla^2 f=0$, i.e.,
| |
− | $(\partial_x^2+\partial_y^2)f=0$, which in our case translates as:
| |
− |
| |
− | \begin{equation}
| |
− | n(n-1)(x^{n-2}+y^{n-2})=0
| |
− | \end{equation}
| |
− |
| |
− | for all $x$ and $y\in\mathbf{R}$. This is achieved when $n=0$, and 1
| |
− | by direct cancellation of the prefix, but also for $n=2$ by
| |
− | cancellation of the function itself $x^0=y^0=1$. For other integer
| |
− | values of $n$, the Laplacian is a nonzero function of $x$ and $y$ and
| |
− | the function therefore not harmonic.
| |
− |
| |
− | ==== Question 4 ====
| |
− |
| |
− | We write $\sin(z)=\sin(x+iy)$ with $x$ and $y$ real, and expand
| |
− | trigonometrically:
| |
− |
| |
− | \begin{equation}
| |
− | \sin(x+iy)=\sin x\cosh y+i\cos x\sinh y
| |
− | \end{equation}
| |
− |
| |
− | so that
| |
− |
| |
− | \begin{align}
| |
− | |\sin(x+iy)|^2&=\sin^2x\cosh^2y+\cos^2x\sinh^2y\\
| |
− | &=\sin^2x\cosh^2y+\sin^2x\sinh^2y-\sin^2x\sinh^2y+\cos^2x\sinh^2y\\
| |
− | &=\sin^2x+\sinh^2y
| |
− | \end{align}
| |
− |
| |
− | here we have added and removed $\sin^2x\sinh^2y$ and used
| |
− | $\sin^2+\cos^2=1$ and $\cosh^2-\sinh^2=1$.
| |
− |
| |
− | Now $\sinh(y)$ is zero only for $y=0$ (this is the only solution to
| |
− | $e^y=e^{-y}$), therefore all the zeros of $\sin(z)$ are real, and are
| |
− | given by the zeros of $\sin(x)$, i.e., $z=\pi\mathbf{Z}$.
| |
| | | |
| == ¿Hey? [[Image:Thorne-spider.jpeg|50px|link=|]][[Image:Thorne-spider.jpeg|40px|link=|]][[Image:Thorne-spider.jpeg|30px|link=|]][[Image:Thorne-spider.jpeg|20px|link=|]][[Image:Thorne-spider.jpeg|10px|link=|]] == | | == ¿Hey? [[Image:Thorne-spider.jpeg|50px|link=|]][[Image:Thorne-spider.jpeg|40px|link=|]][[Image:Thorne-spider.jpeg|30px|link=|]][[Image:Thorne-spider.jpeg|20px|link=|]][[Image:Thorne-spider.jpeg|10px|link=|]] == |
Revision as of 07:52, 4 July 2014
¿Hey, whatcha doin on this page?
It's just where I put stuff that I'm experimenting on for possible f¯uture use.
¿Hey?
¿Hey?
Script error
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$$
\begin{align*}
\tag{1}
a\ket{n}&=\sqrt{n}\ket{n-1}\,,&\bra{n}\,&a=\bra{n+1}\sqrt{n+1}\,,\\
\ud{a}\ket{n}&=\sqrt{n+1}\ket{n+1}\,,&\bra{n}\,&\ud{a}=\bra{n-1}\sqrt{n}\,,
\end{align*}
$$
$$
\begin{align*}
\kern-1cm{(\mathrm{for}~i\le n+j)}\kern1cm a^i{\ud{a}}^j\ket{n}&={(n+j)!\over\sqrt{n!}\sqrt{(n+j-i)!}}\ket{n+j-i}\,,\\
\kern-1cm{(\mathrm{for}~i\le n)}\kern1cm a^{\dagger j}a^i\ket{n}&={\sqrt{n!}\sqrt{(n+j-i)!}\over(n-i)!}\ket{n+j-i}\,.
\end{align*}
$$
fuck this shit
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.)
Blog
Blog:Sandbox
File:laussy-jornada-divulgacion.ppt
$$f(z) = \left( \prod_{j=1}^n \frac{z - z_j}{1 - \overline{z_j}z} \right) \left( \prod_{j=1}^m \frac{z - w_j}{1 - \overline{w_j}z} \right)^{-1} g(z)$$
Ramblings
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!
<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
Cool videos
http://www.youtube.com/watch?v=MZAKjKC7Gho
http://www.youtube.com/watch?v=2_HXUhShhmY
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
Template:MunicipalitiesinMurcia
GPS
File:AbedularCanencia-27Oct2013.gpx
File:AbedularCanencia-27Oct2013.gpx
Download the gpx file for this track.
Download the gpx file for this track.
Download the gpx file for this track.
Colors in MathJaX
$$ax^2+bx+c$$
$$\color{red}{ax^2}+bx+c$$