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typo-fixed: removed to many "must" employed

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Giù Marcer 2020-05-19 17:07:23 +02:00 committed by rnhmjoj
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4
notes/sections/3.md
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@ -3,7 +3,7 @@
## Meson decay events generation ## Meson decay events generation
A number of $N = 50000$ points on the unit sphere, each representing a A number of $N = 50000$ points on the unit sphere, each representing a
particle detection event, must be generated according to then angular particle detection event, is to be generated according to then angular
probability distribution function $F$: probability distribution function $F$:
\begin{align*} \begin{align*}
F (\theta, \phi) = &\frac{3}{4 \pi} \Bigg[ F (\theta, \phi) = &\frac{3}{4 \pi} \Bigg[
@ -105,7 +105,7 @@ a single point, the effect of this omission is negligible.
## Parameters estimation ## Parameters estimation
The sample must now be used to estimate the parameters $\alpha$, $\beta$ and The sample is now used to estimate the parameters $\alpha$, $\beta$ and
$\gamma$ of the angular distribution $F$. The correct set will be referred to $\gamma$ of the angular distribution $F$. The correct set will be referred to
as {$\alpha_0$, $\beta_0$, $\gamma_0$}. as {$\alpha_0$, $\beta_0$, $\gamma_0$}.

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notes/sections/5.md
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@ -1,6 +1,6 @@
# Exercise 5 # Exercise 5
The following integral must be evaluated comparing different Monte Carlo The following integral is to be evaluated comparing different Monte Carlo
techniques. techniques.
\begin{figure} \begin{figure}
@ -39,7 +39,7 @@ being implemented in the GSL libraries `gsl_monte_plain`, `gsl_monte_miser` and
## Plain Monte Carlo ## Plain Monte Carlo
When an integral $I$ over a $n-$dimensional space $\Omega$ of volume $V$ of a When an integral $I$ over a $n-$dimensional space $\Omega$ of volume $V$ of a
function $f$ must be evaluated, that is: function $f$ has to be evaluated, that is:
$$ $$
I = \int\limits_{\Omega} dx \, f(x) I = \int\limits_{\Omega} dx \, f(x)
\with V = \int\limits_{\Omega} dx \with V = \int\limits_{\Omega} dx
@ -111,7 +111,7 @@ $$
$$ $$
if an error of $\sim 1^{-n}$ is required, a number $\propto 10^{2n}$ of if an error of $\sim 1^{-n}$ is required, a number $\propto 10^{2n}$ of
function calls must be executed, meaning that for $\sigma \sim 1^{-10} function calls should be executed, meaning that for $\sigma \sim 1^{-10}
\rightarrow C = \SI{1e20}{}$, which would be impractical. \rightarrow C = \SI{1e20}{}$, which would be impractical.
@ -251,7 +251,7 @@ probability distribution $f$ itself, so that the points cluster in the regions
that make the largest contribution to the integral. that make the largest contribution to the integral.
Remind that $I = V \cdot \langle f \rangle$ and therefore only $\langle f Remind that $I = V \cdot \langle f \rangle$ and therefore only $\langle f
\rangle$ must be estimated. Consider a sample of $n$ points {$x_i$} generated \rangle$ is to be estimated. Consider a sample of $n$ points {$x_i$} generated
according to a probability distribution function $P$ which gives thereby the according to a probability distribution function $P$ which gives thereby the
following expected value: following expected value:
$$ $$

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notes/sections/6.md
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@ -2,7 +2,7 @@
## Generating points according to Fraunhöfer diffraction ## Generating points according to Fraunhöfer diffraction
The diffraction of a plane wave thorough a round slit must be simulated by The diffraction of a plane wave through a round slit must be simulated by
generating $N =$ 50'000 points according to the intensity distribution generating $N =$ 50'000 points according to the intensity distribution
$I(\theta)$ on a screen at a great distance $L$ from the slit itself: $I(\theta)$ on a screen at a great distance $L$ from the slit itself:
@ -263,7 +263,7 @@ stored. This works for all lengths: when the length is even, the middle value
is real. Thus, only $n$ real numbers are required to store the half-complex is real. Thus, only $n$ real numbers are required to store the half-complex
sequence (half for the real part and half for the imaginary). sequence (half for the real part and half for the imaginary).
If the bin width is $\Delta \theta$, then the DFT domain ranges from $-1 / (2 If the bin width is $\Delta \theta$, then the DFT domain ranges from $-1 / (2
\Delta \theta)$ to $+1 / (2 \Delta \theta$). The GSL functions aforementioned \Delta \theta)$ to $+1 / (2 \Delta \theta$). The aforementioned GSL functions
store the positive values from the beginning of the array up to the middle and store the positive values from the beginning of the array up to the middle and
the negative backwards from the end of the array (see @fig:reorder). the negative backwards from the end of the array (see @fig:reorder).
@ -321,7 +321,7 @@ shown in [@fig:results1; @fig:results2; @fig:results3].
## Unfolding with Richardson-Lucy ## Unfolding with Richardson-Lucy
The Richardson–Lucy (RL) deconvolution is an iterative procedure usually used The Richardson–Lucy (RL) deconvolution is an iterative procedure tipically used
for recovering an image that has been blurred by a known point spread function. for recovering an image that has been blurred by a known point spread function.
It is based on the fact that an ideal point source does not appear as a point It is based on the fact that an ideal point source does not appear as a point
@ -391,7 +391,7 @@ width of the original histogram, which is the one previously introduced in
histogram deconvolved with the FFT method is in the middle and the one histogram deconvolved with the FFT method is in the middle and the one
deconvolved with RL is located below. deconvolved with RL is located below.
As can be seen, increasig the value of $\sigma$ implies a stronger smoothing of As can be seen, increasing the value of $\sigma$ implies a stronger smoothing of
the curve. The FFT deconvolution process seems not to be affected by $\sigma$ the curve. The FFT deconvolution process seems not to be affected by $\sigma$
amplitude changes: it always gives the same outcome, which is exactly the amplitude changes: it always gives the same outcome, which is exactly the
original signal. In fact, the FFT is the analitical result of the deconvolution. original signal. In fact, the FFT is the analitical result of the deconvolution.
@ -406,7 +406,7 @@ convolved is less smooth, it is less smooth too.
The original signal is shown below for convenience. The original signal is shown below for convenience.
![Example of an intensity histogram.](images/fraun-original.pdf){#fig:original} ![Example of an intensity histogram.](images/fraun-original.pdf)
<div id="fig:results1"> <div id="fig:results1">
![Convolved signal.](images/fraun-conv-0.05.pdf){width=12cm} ![Convolved signal.](images/fraun-conv-0.05.pdf){width=12cm}

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notes/sections/8.md
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@ -1 +1,4 @@
# Bibliography # Bibliography
The usage and a brief description of the theory underneath all the GLS functions
employed in this report were found in [@GSL].