Dec 09 2012

Znalo je biti vremena

Znalo je biti vremena kad duša ciči.
Oči, srce i ruke, prepuni budu straha.
 Na mrtve svoje u trenu počnem da ličim,
tijelo mi lomača biva, gnijezdo sveopšteg praha.


Znalo je biti vremena kad duša pjeva.
Vjetru i kiši, trnu i cesti, nasmijan pruža naklon,
kao da nikad nesretan ne bjeh ni gnjevan,
niti me ikad slamao očaj i stvari crni zakon.


Znalo je biti vremena kad duše nema.
U domu njenom stoglavo ništa zjapi
i tamne posvuda pipke širi nezbivanja neman,
a žestoke kiše udaraca niotkud ni kapi.


Tek tad je urliko u oku ustrijeljen vepar!
Svemu je duša sestrom znala biti, i bila,
al nje kad nema, dok svijetom vlada kosmički nepar 
- blago je i toplo strašno lice ništavila.

A.Sidran


Nov 10 2012

Photo

Testing new server. :)))

Testing new server. :)))


Nov 08 2012

Bosnian language

Here is what wikipedia says about Bosnian language:  http://en.wikipedia.org/wiki/Bosnian_language :)


Nov 06 2012

Lectures digital image processing

https://rapidshare.com/files/2968184648/lec01.mp4;

https://rapidshare.com/files/1095912649/lec02.mp4;

https://rapidshare.com/files/2361663880/lec03.mp4;

https://rapidshare.com/files/3795140206/lec04.mp4;

https://rapidshare.com/files/1772240076/lec05.mp4;

https://rapidshare.com/files/929959307/lec06.mp4;

https://rapidshare.com/files/3855895699/lec07.mp4;

https://rapidshare.com/files/3220084299/lec08.mp4;

https://rapidshare.com/files/1537614407/lec09.mp4;

https://rapidshare.com/files/3873529871/lec10.mp4;

https://rapidshare.com/files/887631679/lec11.mp4;

https://rapidshare.com/files/154278693/lec12.mp4;

https://rapidshare.com/files/954016232/lec13.mp4;

https://rapidshare.com/files/715910894/lec14.mp4;

https://rapidshare.com/files/2019258102/lec15.mp4;

https://rapidshare.com/files/2154454705/lec16.mp4;

https://rapidshare.com/files/4240804980/lec17.mp4;

https://rapidshare.com/files/3274204480/lec18.mp4;

https://rapidshare.com/files/3253339427/lec19.mp4;

https://rapidshare.com/files/382858833/lec20.mp4;

https://rapidshare.com/files/925947051/lec21.mp4;

https://rapidshare.com/files/2010012729/lec22.mp4;

https://rapidshare.com/files/148899075/lec23.mp4;

https://rapidshare.com/files/3461564047/lec24.mp4;

https://rapidshare.com/files/3515715742/lec25.mp4;

https://rapidshare.com/files/268087098/lec26.mp4;

https://rapidshare.com/files/2225718863/lec27.mp4;

https://rapidshare.com/files/3785410777/lec28.mp4;

https://rapidshare.com/files/3512592569/lec29.mp4;

https://rapidshare.com/files/4239789250/lec30.mp4;

https://rapidshare.com/files/1101258555/lec31.mp4;

https://rapidshare.com/files/2071664244/lec32.mp4;

https://rapidshare.com/files/1199200952/lec33.mp4;

https://rapidshare.com/files/2431935799/lec34.mp4;

https://rapidshare.com/files/3202796057/lec35.mp4;

https://rapidshare.com/files/2373613967/lec36.mp4;

https://rapidshare.com/files/3233880022/lec37.mp4;

https://rapidshare.com/files/2003063087/lec38.mp4;

https://rapidshare.com/files/261013949/lec39.mp4;

https://rapidshare.com/files/1068646618/lec40.mp4;

These are lectures from nptel on digital image processing.

Nov 01 2012

Video

Free open source RTS game


Nov 01 2012
3 notes

Fourier series

Fourier Series: Basic Results

Recall that the mathematical expression 

\begin{displaymath}A_0 + \sum_{n = 1}^{\infty} (A_n\cos(nx) + B_n\sin(nx)).\end{displaymath}


is called a Fourier series
Since this expression deals with convergence, we start by defining a similar expression when the sum is finite. 

Definition. A Fourier polynomial is an expression of the form 

\begin{displaymath}F_n(x) = a_0 + \Big(a_1\cos(x) +b_1\sin(x)\Big)+ \cdots + \Big(a_n\cos(nx) + b_n\sin(nx)\Big)\end{displaymath}


which may rewritten as 

\begin{displaymath}F_n(x) = a_0 + \sum_{k= 1}^{k=n} \Big(a_k\cos(kx) + b_k\sin(kx)\Big).\end{displaymath}


The constants a0ai and bi$i=1,\cdots,n$, are called the coefficients of Fn(x). 

The Fourier polynomials are $2\pi$-periodic functions. Using the trigonometric identities 

\begin{displaymath}\begin{array}{lcr}
\sin(mx)\cos(nx) &=&\displaystyle \frac{1}...
...e \frac{1}{2}\Big[\cos((m-n)x) - \cos((m+n)x) \Big]
\end{array}\end{displaymath}


we can easily prove the integral formulas(1)for $n \geq 0$, we have 

\begin{displaymath}\int_{-\pi}^{\pi} \cos(nx)dx = 0,\;\;\;\mbox{and}\;\; \int_{-\pi}^{\pi}\sin(nx)
dx = 0,\end{displaymath}


(2)for m et n, we have 

\begin{displaymath}\int_{-\pi}^{\pi} \sin(mx) \cos(nx)dx = 0,\end{displaymath}


(3)for $n \neq m$, we have 

\begin{displaymath}\int_{-\pi}^{\pi} \cos(mx)\cos(nx)dx = 0,\;\;\mbox{and}\;\;
\int_{-\pi}^{\pi}\sin(mx) \sin(nx)dx=0,\end{displaymath}


(4)for $ n \geq 1$, we have 

\begin{displaymath}\int_{-\pi}^{\pi}\cos^2(nx)dx = \pi,\;\;\mbox{and}\;\;
\int_{-\pi}^{\pi} \sin^2(nx)dx = \pi.\end{displaymath}

Using the above formulas, we can easily deduce the following result: 

Theorem. Let 

\begin{displaymath}F_n(x) = a_0 + \sum_{k= 1}^{k=n} \Big(a_k\cos(kx) + b_k\sin(kx)\Big).\end{displaymath}


We have 

\begin{displaymath}\left\{\begin{array}{lclr}
a_0 &=&\displaystyle \frac{1}{2\pi...
...\pi} F_n(x) \sin(kx)dx,& 1 \leq k \leq n.\\
\end{array}\right.\end{displaymath}

This theorem helps associate a Fourier series to any $2\pi$-periodic function. 

Definition. Let f(x) be a $2\pi$-periodic function which is integrable on $[-\pi, \pi]$. Set 

\begin{displaymath}\left\{\begin{array}{lclr}
a_0 &=& \displaystyle \frac{1}{2\p...
..._{-\pi}^{\pi} f(x) \sin(nx)dx,& 1 \leq n.\\
\end{array}\right.\end{displaymath}


The trigonometric series 

\begin{displaymath}a_0 + \sum \Big(a_n\cos(nx) + b_n\sin(nx)\Big)\end{displaymath}


is called the Fourier series associated to the function f(x). We will use the notation 

\begin{displaymath}f(x) \sim a_0 + \sum_{n=1}^{\infty} \Big(a_n\cos(nx) + b_n\sin(nx)\Big).\end{displaymath}

Example. Find the Fourier series of the function 

\begin{displaymath}f(x) = x, \;\;\; -\pi \leq x \leq \pi.\end{displaymath}


Answer. Since f(x) is odd, then an = 0, for $n \geq 0$. We turn our attention to the coefficients bn. For any $ n \geq 1$, we have 

\begin{displaymath}b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x \sin(nx)dx =
\frac{1...
...\frac{x\cos(nx)}{n} + \frac{\sin(nx)}{n^2}\right]^{\pi}_{-\pi}.\end{displaymath}


We deduce 

\begin{displaymath}b_n = -\frac{2}{n}\cos(n\pi) = \frac{2}{n}(-1)^{n+1}.\end{displaymath}


Hence 

\begin{displaymath}f(x) \sim 2\left(\sin(x) - \frac{\sin(2x)}{2} + \frac{\sin(3x)}{3} ....\right).\end{displaymath}

Example. Find the Fourier series of the function 

\begin{displaymath}f(x) = \left\{ \begin{array}{lll}
0,& -\pi \leq x < 0\\
\pi, & 0 \leq x \leq \pi
\end{array} \right.\end{displaymath}


Answer. We have 

\begin{displaymath}a_0 = \frac{1}{2\pi}\left(\int_{-\pi}^{0} 0dx + \int_{0}^{\pi...
...\;\;\; a_n = \int_{0}^{\pi} \pi\cos(nx)dx = 0, \;\;\; n \geq 1,\end{displaymath}


and 

\begin{displaymath}b_n = \int_{0}^{\pi} \pi\sin(nx)dx = \frac{1}{n}(1-\cos(n\pi)) =
\frac{1}{n}(1-(-1)^n).\end{displaymath}


We obtain b2n = 0 and 

\begin{displaymath}b_{2n+1} =\frac{2}{2n+1}.\end{displaymath}


Therefore, the Fourier series of f(x) is 

\begin{displaymath}f(x) \sim \frac{\pi}{2} + 2 \left(\sin(x) + \frac{\sin(3x)}{3} +
\frac{\sin(5x)}{5}+\ldots\right).\end{displaymath}

Example. Find the Fourier series of the function function 

\begin{displaymath}f(x) = \left\{ \begin{array}{rrr}
-{\displaystyle \frac{\pi}...
...ystyle \frac{\pi}{2}}, & 0 \leq x \leq \pi
\end{array} \right.\end{displaymath}


Answer. Since this function is the function of the example above minus the constant $\displaystyle \frac{\pi}{2}$. So Therefore, the Fourier series of f(x) is 

\begin{displaymath}f(x) \sim 2 \left(\sin(x) + \frac{\sin(3x)}{3} + \frac{\sin(5x)}{5}+\ldots
\right).\end{displaymath}

Remark. We defined the Fourier series for functions which are $2\pi$-periodic, one would wonder how to define a similar notion for functions which are L-periodic. 

Assume that f(x) is defined and integrable on the interval [-L,L]. Set 

\begin{displaymath}F(x) = f\left(\frac{Lx}{\pi}\right).\end{displaymath}


The function F(x) is defined and integrable on $[-\pi, \pi]$. Consider the Fourier series ofF(x

\begin{displaymath}F(x) = f\left(\frac{Lx}{\pi}\right) \sim \frac{a_0}{2} +
\sum_{n=1}^{\infty} \Big(a_n\cos(nx) + b_n\sin(nx)\Big).\end{displaymath}


Using the substitution $t =\displaystyle \frac{Lx}{\pi}$, we obtain the following definition: 

Definition. Let f(x) be a function defined and integrable on [-L,L]. The Fourier series of f(x) is 

\begin{displaymath}f(t) \sim a_0 + \sum_{n=1}^{\infty} \left(a_n\cos\left(n\frac{\pi t}
{L}\right) + b_n\sin\left(n\frac{\pi t}{L}\right)\right)\end{displaymath}


where 

\begin{displaymath}\left\{\begin{array}{lclr}
a_0 &=& \displaystyle \displaystyl...
...f(x) \sin\left(n\frac{\pi x}{L}\right)dx,\\
\end{array}\right.\end{displaymath}


for $1\leq n$

Example. Find the Fourier series of 

\begin{displaymath}f(x) = \left\{ \begin{array}{lll}
0,& -2 \leq x < 0\\
x, & 0 \leq x \leq 2
\end{array} \right.\end{displaymath}


Answer. Since L = 2, we obtain 

\begin{displaymath}\begin{array}{lll}
a_0 &=& \displaystyle \frac{1}{4} \int_{0...
...{n\pi} = \displaystyle \frac{2}{n\pi}(-1)^{n+1}\\
\end{array}\end{displaymath}


for $ n \geq 1$. Therefore, we have 

\begin{displaymath}f(x) \sim \frac{1}{2} +
\sum_{n=1}^{\infty} \left[\frac{2}{n...
...rac{2}{n\pi}(-1)^{n+1}\sin\left(n\frac{\pi x}{2}\right)\right].\end{displaymath}


Nov 01 2012

Nov 01 2012

Abdulah Sidran Hrast i knjiga

HRAST I KNJIGA

«Ja sa ljudima nemam više o čemu razgovarati»
Andrej Rubljov

Došao sam miru da me učiš. Soba je
prepuna tako dragih stvari: hrast i knjiga 
(ko duša i um!), sat i svijećnjak, spužva
kocka i sehara – sve mirom miruje,
osim srca, što bije i bije, u velikom
drugom velikom srcu – srcu vaseljene.

Sve je zapravo muzika sama. Izlišna 
postaje moć govora; sve počinje, i svršava,
šutnjom, iz koje će, kroz trpnju,
liti sjaj čiste blagosti. Ničeg, inače,
dobrog međ mirnim stvarima. Jer, napolju, 
gladan je svijet, i ljudske su oči 
bunari prastari – prazno, žeđ i tama.

Zar je to zbilja moguće!? Putevima
različitim, istom se cilju bližimo!

Preduboka je zamisao Tvorca. Da bih
shvatio 
taj svijet, iznova bih morao, stalno
i stalno, živjeti ljubav. S manje bih 
zlojeda, straha i žestine, potom, 
pod zvijezdom kročio, trpnju
ko mlijeko majčinu kušajuć.

Komad se neba kroz prozor vidi, 
i neke žene dolje, sasvim nadrealne,
eno, rublje razastiru. U sobi, odasvud,
maslačak cvate! Djevojčice naše obrašca
nadimaju, u krunice se dah dječiji sasipa
i cvjetno perje zrakom lebdi! Prah i 
pelud lice nam mije, i prepuna je
soba snijega i topline!

O hoću li išta
naučiti iz svega!? Sada je, evo, mirno
moje srce. Ali, ko će, umjesto mene,
iz ove sobe izići? I čije će oči
k nebu se podić? U čijoj će to duši 
iz starih knjiga glas da zvoni: «O Bože,
pun neka je svjetla Ibrahimov grob!»?

Nov 01 2012

Nov 01 2012

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