Dec 09 2012

HOCE LI ISTA O MENI ZNATI


Hoće li išta
о meni znati onaj
što noću će, uz lampu i mrak uokolo,
ove redove čitati?


Hoće li čuti drhat, i strepnju
srca koje misli, dok biskao bude
slovo svako i redak, ко brižna majka
kosu dječiju, češljem najgušćim?


Hoće li išta, išta, uistinu znati?


Il jednakće pustoš
u domu njegovom da zjapi i sja,
nad stolom i knjigom, dok sabran,
i težak od tuge, svoje,
ovu moju bude razgrtao,
vlat po vlat,
mrak po mrak?

A.Sidran


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

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