text:
\section{Introduction}

Pulsar timing from long observations is a current topic of astrophysics.
This is especially due to the discovery of an unprecedented population of gamma-ray pulsars by the Large Area Telescope (LAT) on board the {\it Fermi} satellite \citep{psrCatalog}.
The low rate of gamma-rays implies the need of very long observations so as to reach a significant detection of the pulsations. The survey mode of {\it Fermi}-LAT perfectly matches this need, and a timing consortium of radio and X-ray observers has been formed in order to build long ephemeris  to adequately extract pulsations \citep{TimeCons}.
More recently, adapting the radio techniques, \cite{PRay} presented a method to build long ephemeris using gamma-ray data. Since {\it Fermi}-LAT continuously monitors all the sky,  these ephemeris have an accuracy that is even better than the radio ones for  bright gamma-ray pulsars.
Efforts were also spent in order to find new techniques to  make the blind search for gamma-ray pulsations more efficient and computationally feasible. Indeed, due to the long integration times, a full Fourier analysis is computationally too demanding
for the search of pulsations in gamma-rays, particularly when lacking knowledge from other wavelengths (a blind search).
A sub-population of radio quiet gamma ray pulsars (\cite{CTA1}, \cite{blindpsr}) has been discovered with LAT data thanks to the `time differencing technique'  \cite{Atwood2006}.
Recently, a new technique dedicated especially to the search of millisecond pulsars has also been developed \cite{Pletsh}.

The cited works on pulsar timing have greatly improved the analysis of isolated pulsars, and their blind search on gamma-rays. On the other hand, the search of gamma-ray pulsars in binary systems is performed only when their radio ephemeris are known. Indeed, the first pulsar catalog of {\it Fermi}-LAT \citep{psrCatalog} includes only five radio loud millisecond pulsars in binary systems. The reason is mainly that the Doppler shift due to the orbital motion of the pulsar wash out the pulsations at the observer's frame of reference. The correction for the orbital motion (as well as for other effects due to the companion star) need a very precise knowledge of the orbit. Since there are several parameters that define the orbit, uncertainties in one or more of them can vanish the effect of the correction.
We have recently presented a systematic study on how the uncertainties of the orbital parameters affect the pulsed signal \cite{Cali12}. In that paper we already made use of a new method to compute the expectation value of the power spectrum at the signal and nearby frequencies, based on a statistical approach, which we will detail and prove here.
The rest of the paper is organized as follows: In Section 2 we recall the main approaches used in the literature to compute the expectation value of a time series up to now.
In Section 3 we demonstrate our method, which we validate it in Section 4 by means of numerical  simulations and by reproducing known features of the power spectrum of a periodic times series.
In sections 5 and 6, respectively, we apply the method to analytically solve the power spectrum in the case the first derivative of the frequency is uncertain and not negligible, and in the case of a blind search where both the frequency and its derivative are unknown.
Finally, in Section 7 we conclude underlining some interesting aspects of this approach.

\section{Context}

In our work we calculate the expectation value of the power spectrum following a statistical approach.
Specifically, we treat the phase associated to each time stamp as a random variable, and the power spectrum as a function of it.
Therefore, starting from the probability density function of the phases, we make use of the known rules of statistics in order to derive the properties of the power spectrum.
We will show that with this approach we reach the same results obtained by other authors in different ways. But the novelty of our approach is that we introduce a different concept of the event phases and a different way  to treat them. We found that the approach we introduce can be particularly useful to solve specific problems that can arise in a timing analysis.
In order to highlight the difference of our method with respect to what can be found in literature, we first summarize other approaches in what follows.

\subsection{Lomb's approach}
In 1975, Lomb \cite{Lomb} undertook the study of the frequency analysis by means of the least-square method, which consists in fitting  sine waves
to the data ($y_i$, taken at times $t_i, i=1,2,...,n$) in the time domain,
and plotting versus the frequency ($\nu$) the reduction ($\Delta R$) in the sum of the residual squared respect to fitting a constant value (see also \cite{Barning})
\begin{equation}
\Delta R = \sum_i (y_i - c)^2 - \sum_i \left[ y_i - (a\,{\rm cos} (2\pi \nu t_i) + b\,{\rm sin} (2\pi \nu t_i) +c)\right] ^2.
\label{ls}
\end{equation}
Lomb \cite{Lomb} (see his Section 2) found the formula of the least-squared spectrum (a posteriori called Lomb-Scargle periodogram) and demonstrates that the formula of the classical periodogram is an approximation for it. From this base the statistical properties of the power spectrum of random noise were examined, as well as the effect of noise on the spectrum of a sine wave.

\subsection{van der Klis' approach}

van der Klis \cite{Klis} (see his Section 2.5) explained the relation between the continuous Fourier transform and the Discrete Fourier Transform (DFT) introducing the concepts of windowing and sampling.
The Fourier transform of an infinitely extended continuous function $x(t)$ ($-\infty <t< \infty$) is defined as
\begin{equation}
a(\nu) = \int_{-\infty}^{\infty} x(t) e^{2\pi i \nu t} dt.
\label{Fourier}
\end{equation}
When a real instrument is used to detect the phenomena described by the function $x(t)$, we need to take into account two effects. The observation will be finite in time, and the instrument hardware will detect discrete samples of the function $x(t)$. In other words, what we detect is the function $x(t)$ multiplied by both the window function, which describe the structure and the duration of the observations, and by the characteristic sampling function of the instrument.
The simplest window function is
\begin{eqnarray}
\omega(t)=\begin{cases} 1, & \mbox{if }\, 0\leq t<T \\ 0, & \mbox{otherwise }
\end{cases},
\end{eqnarray}
where $T$ is the duration of the observation.
The sampling function is
\begin{equation}
s(t) = \sum_{k=-\infty}^{\infty} \delta\left( t-\frac{kt}{N}\right),
\end{equation}
where $N$ is the number of bins in the observation time $T$.
The observed function is then
\begin{equation}
\tilde{x}(t) = x(t) \cdot \omega(t) \cdot s(t).
\label{xtilde}
\end{equation}
Substituting (\ref{xtilde}) into (\ref{Fourier}) we end up with the definition of the DFT.
A powerful theorem of the Fourier analysis states that the Fourier transform of the product of two functions is the convolution of the Fourier transforms of these two functions. So, if $a(\nu)$ is the Fourier transform of $x(t)$, and $b(\nu)$ that of $y(t)$, then the Fourier transform of $x(t)y(t)$ is $a(\nu)*b(\nu) \equiv \int_{-\infty}^{\infty} a(\nu')b(\nu - \nu')d\nu'$.
Making use of this theorem applied to Eq. \ref{xtilde}, van der Klis \cite{Klis} derived the main properties of the power spectrum calculated for an event data time series.

\subsection{Ransom et al.'s approach}

In an alternative way, Ransom et al. \cite{Ransom2002} (see their Section 2.1) derived the properties of the Fourier power of a time series from a direct study of the Discrete Fourier Transform. The $k^{th}$ element of a DFT of a uniformly-spaced time series $n_j$ ($j=0,1,....,N-1$) is defined as
\begin{equation}
A_k = \sum_{j=0}^{N-1} n_j e^{-2\pi ijk/N},
\end{equation}
where for a time spacing $dt$, and a total observation length $T$, the frequency of the $k^{th}$ element is $f_k = k/T$, and the index $j$ indicate the time stamp $t=j dt$.
If the DFT summation is drawn on a complex plane, it appears as a simple vector addition with each element rotated by $-2\pi k/N$ from the previous element.
Starting from this representation of the DFT summation, \cite{Ransom2002} derived the Fourier response and its power to full noise time series as well as to periodic signals.
\\

To describe the statistical properties of the power spectrum of a time series composed by noise and a periodic signal, both \cite{Klis}, and \cite{Ransom2002} follow the description given by Groth \cite{Groth}. He considers the Fourier power of a single frequency bin as a random variable. Invoking the central limit theorem, Groth \cite{Groth} demonstrates in a few steps that the Fourier power of a noisy time series has a $\chi^2$ distribution with 2 degrees of freedom. As one can notice, none of the authors cited above directly define the phase of the time stamps ($\theta = 2 \pi \nu t$), nor use it as a random variable,
in contrast with what follows.

\section{Power spectrum expectation value}

\subsection{First approach}

Following the definition given in \cite{Scargle1982}, the power spectrum of a
sample data set $\{X(t_i), i=1,2,....,N_0\}$ calculated at an arbitrary test frequency $\omega$ is given by
\begin{equation}
\! \!
P(\omega) \!=\! \frac{1}{N_0} \!
\left[ \left( \sum_{i=1}^{N_0} X_i {\rm cos}(\omega t_i)\right) ^2
\! \!  + \!
\left( \sum_{i=1}^{N_0} X_i {\rm sin}(\omega t_i)\right) ^2\right] . \!
\label{2.1}
\end{equation}
Here, we consider the series of the arrival times of single events on a detector, i.e. the sample data set $X$ is such that $X_i=1$ for each $i$.
Attributing a phase value to each event $\theta_i = \omega t_i$ we have for the power spectrum
\begin{equation}
P(\omega) = \frac{1}{N_0}\left[ \left( \sum_{i=1}^{N_0} {\rm cos}(\theta_i)\right) ^2+\left( \sum_{i=1}^{N_0} {\rm sin}(\theta_i)\right) ^2\right].
\label{pow}
\end{equation}
This definition corresponds to the Rayleigh power, and it differs by just a factor 2 with respect to
the $Z_1^2$ test introduced by \cite{Buccheri1983}.
The variable $Z_1^2$ has a probability density function
(pdf) for the null hypothesis equal to that of a $\chi^2$ with 2 degrees of freedom, where null hypothesis is used to signify that there is no periodic signal at frequency $\omega$.
In what follows we focus on how to calculate the expectation
value of the power spectrum in Eq. (\ref{pow}) using the statistical
properties of the phase distribution $P_{\theta}(\theta)$.

Here and throughout the paper $\theta$ does not refer to phase of the
pulsar, but to a phase relative to an arbitrary test frequency, as defined above.
$P_{\theta}(\theta)$ refers to the theoretical continuous distribution of the phases.

For a large number of events ($N_0 \gtrsim 100$) in Eq. (\ref{pow}),
the sums of the trigonometric functions of the phases,
cos$(\theta_i)$ and sin$(\theta_i)$, are well approximated by
the expectation value of their continuous distributions times $N_0$. Thus,
\begin{equation}
\sum_{i=1}^{N_0} {\rm cos}(\theta_i) \longrightarrow N_0  \left\langle {\rm cos}(\theta_i) \right\rangle
:= N_0\int_{-1}^1 {\rm cos}(\theta) \cdot P_{\rm cos}(\theta) d{\rm cos}(\theta),
\label{2.3}
\end{equation}
\begin{equation}
\sum_{i=1}^{N_0} {\rm sin} (\theta_i) \longrightarrow N_0  \left\langle {\rm sin}(\theta_i) \right\rangle
:= N_0\int_{-1}^1 {\rm sin} (\theta) \cdot P_{\rm sin} (\theta) d{\rm sin}(\theta)  ,
\label{2.4}
\end{equation}
where $P_{\rm cos}(\theta)$ and $P_{\rm sin}(\theta)$ are the continuous distributions of ${\rm cos}(\theta)$ and ${\rm sin}(\theta)$ expressed as function of $\theta$, respectively.
Substituting these in Eq. (\ref{pow}), the power spectrum is given by
\begin{equation}
P(\omega) = N_0\left[ \left\langle {\rm cos}(\theta_i) \right\rangle^2 + \left\langle {\rm sin}(\theta_i) \right\rangle^2\right] .
\label{pow2}
\end{equation}
Since we used the mean values of cosine and sine, Eq. (\ref{pow2}) is the expectation value of the power spectrum at the frequency $\omega$.

\subsection{A more rigorous approach}

This can be demonstrated more rigorously in the following way. Take a set of $N$ independent random variables $z_i$, all with the same variance $\sigma$ and the same average $\left\langle z \right\rangle$. The expectation value of their squared sum is
\begin{eqnarray}
&& \left\langle \left( \sum z_i \right)^2    \right\rangle
=\left\langle \sum z_i^2 + 2\sum_{i\neq j}z_i z_j \right\rangle =
\nonumber \\ &&
\sum \left\langle z_i^2 \right\rangle + 2\sum_{i\neq j} \left\langle z_i \right\rangle \left\langle z_j \right\rangle=
\nonumber \\ &&
 \sum \left( \left\langle z_i^2 \right\rangle - \left\langle z_i \right\rangle^2 \right) + 2\sum_{i\neq j} \left\langle z_i \right\rangle \left\langle z_j \right\rangle + \sum  \left\langle z_i \right\rangle^2=
\nonumber \\&&
 N\sigma^2 + 2\left( N^2 - \sum_{i=1}^N i \right) \left\langle z \right\rangle^2 + N \left\langle z \right\rangle^2 =
\nonumber \\ &&
 N\sigma^2 + N^2 \left\langle z \right\rangle^2 \xrightarrow{N\rightarrow\infty, \left\langle z \right\rangle\neq 0} N^2 \left\langle z \right\rangle ^2
\label{proof}
\end{eqnarray}
where we used the commutative properties of the average operator ($\left\langle \right\rangle $), and we expressed the number of the double products from the square of a sum as $\sum_{i\neq j} 1 = \left( N^2 - \sum_{i=1}^N i \right)$.
Averaging Eq. (\ref{pow}), and doing  algebra as above on $\left( \sum {\rm cos}(\theta_i)\right) ^2$ and $\left( \sum {\rm sin}(\theta_i)\right) ^2$, it can be proven that the expectation value of the power spectrum is given by Eq. (\ref{pow2}).

In the last step of Eq. (\ref{proof}) it is shown that $\left\langle \left( \sum z_i \right)^2  \right\rangle = N^2 \left\langle z \right\rangle^2$ only if $\left\langle z \right\rangle\neq 0$, whereas in the opposite case, the term with the variance ($N\sigma^2$) is dominant.
This represents a limitation of our method, which is correct only for frequencies $\omega$ which are close to the signal ($\omega_0$). Indeed, for frequencies far from the signal, the distribution of the phases is flat and the power spectrum is dominated by noise.
Several works dedicated to the study of the noise in the power spectrum can be found in literature (e.g. \cite{Scargle1981}). Here, we focus on the power spectrum close to the signal peak, with the aim of using
the results for a future development of optimized software dedicated to the search of periodic signals.

\subsection{Distributions}

In the next paragraphs we will find a formula for $P_{\rm cos} (\theta)$ and $P_{\rm sin} (\theta)$, in order to solve the integrals in Eqs. (\ref{2.3}) and (\ref{2.4}), and calculate Eq. (\ref{pow2}).
We shall use that the probability density function (pdf) of a variable $z=f(x)$ that is function of a random variable $x$ whose pdf $P_x(x)$ is known, can be calculated as
\begin{equation}
P_{z}(z) = \sum_i \frac{P_x(x_{2\_i})}{f^{\prime}(x_{2\_i})} - \frac{P_x(x_{1\_i})}{f^{\prime}(x_{1\_i})}
\label{rotondi}
\end{equation}
(Eq. \ref{rotondi} is taken from \cite{Rotondi}. It can also be found in e.g. \cite{Miller}).
Here, the prime (as in $f^{\prime}$) represents a derivative with respect to $x$, and the intervals $\left[ x_{1\_i}, x_{2\_i}\right] $ are those for which for a given $z=z_0$, $f(x)<z_0$. At the first or the last interval (with respect to the validity range of $x$),  it could happen that the edges, $x_{1\_i}$ and $x_{2\_i}$, respectively, can not be properly defined.
The corresponding term in Eq. (\ref{rotondi}) is null in these cases.

In our case, $P_z$ in Eq. (\ref{rotondi}) corresponds to $P_{\rm cos}$ or $P_{\rm sin}$, while $P_x$ is the phase distribution $P_{\theta}$.
The phases $\theta=\omega t$ are defined in the full range $(0, \omega T_{\rm obs})$, where $T_{\rm obs}$ is the total length of the observation. We will find useful to express $\omega T_{\rm obs}$ as
\begin{equation}
\omega T_{\rm obs} = 2\pi N  + R
\label{11}
\end{equation}
where $N$ is the number of entire cycles of $2\pi/\omega$ contained in the observation time $T_{\rm obs}$ and $R$ is the fractional part of the last cycle ($R = {\rm fmod}(\omega T_{\rm obs}, 2\pi)$) expressed in phase.

\subsection{$P_{\rm cos}$}

We shall start with the calculation of $P_{\rm cos}$. Figure \ref{cos} shows the cosine function. We marked a generic value $z_0^+>0$ and  found the edges of the intervals for which cos$(\theta) < z_0^+$. They are marked in the figure as $\theta^{+}_{1\_i}$, and $\theta^{+}_{2\_i}$, where the subindex $_i$ indicates the cycle number of the cosine function. In the same way we marked a generic value lower than 0 ($z_0^-<0$), and the respective intervals $\left[ \theta^{-}_{1\_i}, \theta^{-}_{2\_i} \right] $ for which cos$(\theta) < z_0^-$.
 Since the codomain of the inverse function ${\rm acos}(z)$ is $[0, \pi]$, it is convenient to express the edges of the intervals marked in figure \ref{cos} in the following way
\begin{align}
  &\theta^{+/-}_{1\_0} = {\rm acos}(z_0), \nonumber \\
  &\theta^{+/-}_{2\_0} = 2\pi - \theta^{+/-}_{1\_0} , \\
  &\theta^{+/-}_{1\_i} = 2\pi i + \theta^{+/-}_{1\_0}, \nonumber \\
  &\theta^{+/-}_{2\_i} = 2\pi (i+1) - \theta^{+/-}_{1\_0} .\nonumber
\end{align}
Substituting these values in Eq. (\ref{rotondi}), the most general expression for the pdf of ${\rm cos}(\theta)$ is
\begin{eqnarray}
P_{\rm cos}(\theta) = \frac{1}{{\rm sin}(\theta)}\left\lbrace \sum_{i=0}^{N-1}\left[ P_{\theta}(2\pi i + \theta) + \right. \right.
 \left. \left.
 P_{\theta}(2\pi(i+1) - \theta) \right]_{0,\pi}  \right\rbrace + B(\theta; R).
 \label{Pcos}
\end{eqnarray}
Here we have substituted $\theta^{+/-}_{1\_0}$ with $\theta$, which for this expression is constrained in the range $[0, \pi]$, as is specified in the formula at the feet of the closing of the square bracket. In Eq. (\ref{Pcos}), sin$(\theta)$ is the derivative of the cosine calculated at $(2\pi i + \theta)$ and $(2\pi (i+1) - \theta)$.
The sum is over the number $N$ of entire cycles contained within the observation (see Eq. \ref{11}), while the term $B(\theta; R)$ takes into account the fractional part of the last cycle.
This term is in general negligible, because commonly one would have $N \gg 1$.
It strongly depends on how long is the fraction of the last cycle, thus, on $R$, defined by Eq. (\ref{11}).
For example, if $R<\pi$ the interval for which cos$(\theta) < z_0$ is $\left[ \theta_{1\_N}, R \right] $. Since the second edge of the interval is not properly defined, but it is set to the end of the observation, the term $P_x(x_{2\_N})/f^{\prime}(x_{2\_N})$ in Eq. (\ref{rotondi}) has to be set to zero. Finally, $B(\theta; R)$ is equal in this case to $[P_{\theta}(2\pi N + \theta)/{\rm sin}(\theta)]_{0,R}$, where the subindices indicate the range of validity of $\theta$ only in this last cycle.

\subsection{$P_{\rm sin}$}

The solution for $P_{\rm sin} (\theta)$ is similarly found.
Figure \ref{sin} plots the sine function. We marked there a generic value $z_0^+>0$, and the edges of the intervals for which sin$(\theta) < z_0^+$ ($\theta^{+}_{1\_i}$ and $\theta^{+}_{2\_i}$, where the subindex $_i$ indicates the cycle number of the sine function). In the same way we marked a generic value lower than 0 ($z_0^-<0$), and the respective intervals $\left[ \theta^{-}_{1\_i}, \theta^{-}_{2\_i} \right] $ for which sin$(\theta) < z_0^-$. Since in this work $\theta=\omega t$ is positively defined, the phase $\theta_0$ also marked in the figure is not a solution of interest for us, but we need it because the inverse function ${\rm asin}(z)$ has the codomain $[-\pi/2, \pi/2]$.
The intervals defined for $z_0^+>0$ are symmetric respect to $\pi/2 + 2\pi i$, while the intervals defined for $z_0^-<0$ are symmetric respect to $\frac{3}{2}\pi + 2\pi i$. For this reason it is not immediate to join the two cases. Indeed, we will split them defining $P^+_{\rm sin}$ and $P^-_{\rm sin}$.
For positive values of $z_0$, the edges of the intervals can be expressed as
\begin{align}
  &\theta^{+}_{1\_0} = {\rm as in}(z_0^+)        ,     \nonumber\\
  &\theta^{+}_{2\_0} = \pi - \theta^{+}_{1\_0}   ,    \\
  &\theta^{+}_{1\_i} = 2\pi i + \theta^{+}_{1\_0} ,    \nonumber\\
  &\theta^{+}_{2\_i} = 2\pi(i+1/2) - \theta^{+}_{1\_0} .  \nonumber
\end{align}
Substituting in Eq. (\ref{rotondi}), we get
\begin{eqnarray}
P^+_{\rm sin}(\theta) =  \frac{1}{{\rm cos}(\theta)}\left\lbrace \sum_{i=0}^{N-1}\left[ P_{\theta}(2\pi i + \theta) +
\right. \right.
\left. \left.
 P_{\theta}(2\pi (i + 1/2) - \theta) \right]_{0,\frac{\pi}{2}}  \right\rbrace ,
\label{Psin+}
\end{eqnarray}
where here $\theta=\theta^{+}_{1\_0}$ is defined in the range $\left[ 0,\pi/2 \right] $, as indicated by the subindices of the closing square bracket.
For negative $z_0$, the edges of the intervals are
\begin{align}
& \theta_0 = {\rm as in}(z_0^-) ,  \nonumber\\
& \theta^{-}_{1\_0} = \pi - \theta_0  ,   \nonumber\\
& \theta^{-}_{2\_0} = 2\pi + \theta_0 ,  \\
& \theta^{-}_{1\_i} = \pi(i+1/2) - \theta_0  ,  \nonumber\\
& \theta^{-}_{2\_i} = 2\pi(i+1) + \theta_0  .  \nonumber
\end{align}
Substituting in Eq. (\ref{rotondi}), we obtain
\begin{eqnarray}
P^-_{\rm sin}(\theta) =  \frac{1}{{\rm cos}(\theta)}\left\lbrace \sum_{i=0}^{N-1}\left[ P_{\theta}(2\pi(i+1) + \theta) + \right. \right.
\left. \left. P_{\theta}(2\pi (i + 1/2) - \theta) \right]_{-\frac{\pi}{2},0}  \right\rbrace ,
\label{Psin-}
\end{eqnarray}
where here $\theta=\theta_0$ is defined in the range $\left[-\pi/2, 0 \right] $, as indicated by the subindices of the closing square bracket.
In Eqs. (\ref{Psin+}) and  (\ref{Psin-}) cos$(\theta)$ at the denominator is the derivative of the sine calculated at
$(2\pi i+\theta)$, $(2\pi(i+1/2)-\theta)$, and $(2\pi(i+1)+\theta)$.

Finally, we can sum the Eqs. (\ref{Psin+}) and (\ref{Psin-}) to get the complete solution for $P_{\rm sin}$,
\begin{eqnarray}
P_{\rm sin}(\theta) = && \frac{1}{{\rm cos}(\theta)}\sum_{i=0}^{N-1} \left\lbrace
\left[ P_{\theta}(2\pi(i+1/2) - \theta) \right]_{-\frac{\pi}{2},\frac{\pi}{2}} + \right.  \left.
\left[ P_{\theta}(2\pi i + \theta) \right]_{0,\frac{\pi}{2}}  + \right.
\nonumber \\ && \left.
\left[ P_{\theta}(2\pi(i+1) + \theta) \right]_{-\frac{\pi}{2},0}  \right\rbrace
+ B(\theta; R)
\label{Psin}
\end{eqnarray}
Also here the term $B(\theta; R)$ takes into account the fraction of the last cycle of the sine function, and it is in general negligible.

We have now all the ingredients to solve the integrals in Eqs. (\ref{2.3}) and (\ref{2.4}), and
calculate the expectation value of the power spectrum in Eq. (\ref{pow2}).
In both, Eqs. (\ref{Pcos}) and (\ref{Psin}), a key role is played by the sum of the terms $\sum_{i=0}^{N-1} P_{\theta}(2\pi(i+k) \pm \theta)$, where $k = 0, 1/2, 1$, as well as the sign of $\theta$ depend on the cases. This corresponds to nothing more than the distribution of the phases folded by $2 \pi$. This will be commented in the last Section. The solutions we found here are free from any assumption on $P_{\theta}$, and in this sense they are universal.

\begin{figure}[t]
\center
\includegraphics[width=0.8\columnwidth]{Figures/cos.eps}
\caption{The function $z={\rm cos}(\theta)$.
The edges of the intervals for which cos$(\theta) < z_0^+$ are marked with $\theta^{+}_{1\_i}$, and $\theta^{+}_{2\_i}$, where the subindex $_i$ indicates the cycle number of the sine function.
Similarly, the edges of the intervals for which cos$(\theta) < z_0^-$ are marked with $\theta^{-}_{1\_i}$, and $\theta^{-}_{2\_i}$.\label{cos}}
\end{figure}

\begin{figure}
\center
\includegraphics[width=0.8\columnwidth]{Figures/sin.eps}
\caption{The function $z={\rm sin}(\theta)$.
The edges of the intervals for which sin$(\theta) < z_0^+$ are marked with $\theta^{+}_{1\_i}$, and $\theta^{+}_{2\_i}$, where the subindex $_i$ indicates the cycle number of the sine function.
Similarly, the edges of the intervals for which sin$(\theta) < z_0^-$ are marked with $\theta^{-}_{1\_i}$, and $\theta^{-}_{2\_i}$. The phase $\theta_{0}$ is useful in the computations, as described in the text.\label{sin}}
\end{figure}

\section{ Numerical validation}

In this section we are going to validate in two different ways the method formulated to evaluate the expectation value of the power spectrum. First, the results found for the pdf of the sine and cosine (Eqs. \ref{Psin}, \ref{Pcos}) will be checked with simulations. Secondly, we shall show that using our approach we can derive all the already known features of the power spectrum.

\subsection{Simulations}

In order to check the formulae we have found in the previous Section, we wrote a code to simulate the arrival time series from a sinusoidal signal of frequency $\omega_0$ (Eq. \ref{8}), rate $r$, observed for a time $T_{\rm obs}$ and having a distribution given by
\begin{equation}
 P_t(t) = 1+{\rm sin}(\omega_0 t).
 \label{8}
\end{equation}
The arrival time series is simulated in two steps. First, all the time stamps are simulated as random numbers in the range $0,2\pi/\omega_0$, following the distribution of Eq. (\ref{8}). Then, in order to cover the full duration of the observation, each time stamp has been randomly delayed adding a value $({2\pi}/{\omega_0}) n$, where $n$ is a random integer number uniformly distributed in $0, \omega_0 T_{\rm obs}/2\pi$.
Figure \ref{simulation} shows the distributions of the time stamps in the first step for $\omega_0=1.0$ s$^{-1}$ (left panel), and the distribution of $n$ for the second step, for $T_{\rm obs}=1.0$E+4 s (right panel). This procedure requires that the observation time $T_{\rm obs}$ is exactly an integer multiple of the signal period. This implies that for $\omega=\omega_0$ the term $R$ in equation \ref{11} is null, while it is in general not true for $\omega \neq \omega_0$.
Once the arrival time series has been fully simulated, we calculate the phases $\theta_i = \omega t_i$, their  cosine, and sine value, and finally the power at the frequency $\omega$ from Eq. (\ref{pow}).

Figure \ref{sc@peak} shows the distributions of $\cos(\theta)$ and $\sin(\theta)$ (left and right panel, respectively) obtained for $\omega=\omega_0$. In this case $P_{\theta}(\theta) = 1 + {\rm sin}(\theta)$, and it allows for a simplification of  the terms $2\pi(i + k)$  in Eqs. (\ref{Pcos}, \ref{Psin}). The pdf $P_{\rm cos}$ and $P_{\rm sin}$, depicted in red on the plots of Figure \ref{sc@peak} are equal to
\begin{eqnarray}
P_{\rm cos}(\theta) &=& \frac{1}{{\rm sin}(\theta)} \Rightarrow \left\langle {\rm cos}(\theta) \right\rangle = 0  , \\
P_{\rm sin}(\theta) &=& \frac{1+{\rm sin}(\theta)}{{\rm cos}(\theta)} \Rightarrow \left\langle {\rm sin}(\theta) \right\rangle = 0.5.
\end{eqnarray}
More generally, for $\omega \neq \omega_0$, the phase distribution is equal to
\begin{equation}
 P_{\theta}(\theta) = 1+{\rm sin}(\frac{\omega_0}{\omega} \theta),
 \label{9}
\end{equation}
and Eqs. (\ref{Pcos}, \ref{Psin}) can not be simplified. Specifically, when the terms $B(\theta; R)$ are not negligible it is particularly hard to analytically reproduce the pdf $P_{\rm cos}$ and $P_{\rm sin}$, because they have different points of discontinuity. In Figure \ref{SimulationPanels}, we choose these cases that most severely test our formulae.  The phase distribution folded in $2\pi$ is plotted in the left panels of the figure.
Middle and right panels plot the pdf of cos$(\theta)$ and sin$(\theta)$, respectively. Each row in the figure corresponds to a different simulation with observation time $T_{\rm obs}$ chosen such that $N$ from Eq. (\ref{11}) is equal to zero or one, and the phases are calculated for different $\omega \neq \omega_0$, as specified in the caption. The red line in each plot correspond to the analytical solutions. The perfect agreement with the simulations also in reproducing the discontinuities validates our computations in the previous Section.
In the search of pulsations, the condition $N\gg 1$ is always satisfied, even for only few hours of observations. In this most common case, then, the distributions we plotted appear smoother (as shown in Figure \ref{SimulationPanels2}), and the term $B(\theta; R)$ is completely negligible, because its weight is $1/N \sim 0$.

\begin{figure*}
 \centering
\begin{minipage}{1.0\textwidth}
\centering
 \includegraphics[width=0.4\textwidth]{Figures/Ptime.eps}
 \hspace{0.5 cm}
 \includegraphics[width=0.4\textwidth]{Figures/n-distrib.eps}
\end{minipage}
\caption{Left panel: distribution of the time stamps after the first step in the simulation. Right panel: distribution of the integers n among $0$ and $\omega_0 T_{\rm obs}/2\pi$.
\label{simulation}}
\end{figure*}

\begin{figure*}
 \centering

\includegraphics[width=0.49\textwidth]{Figures/Pcos-lstat.eps}
\includegraphics[width=0.49\textwidth]{Figures/Psin-lstat.eps}
 \caption{Distribution of sin$(\theta)$ (left), and cos$(\theta)$ (right) for the case $\omega=\omega_0$.
The red lines represent the analytical formulae for the pdf. \label{sc@peak}}
 \end{figure*}

\begin{figure*}
 \centering
 \includegraphics[width=0.95\textwidth]{Figures/SimulationPanels.eps}
 \caption{Distributions of $\theta$ (left), cos$(\theta)$ (middle), and sin$(\theta)$ (right) from the simulation. The analytical pdf is over imposed (red lines). First row: $\omega=0.3\omega_0$, $N=0$, $R=5.6$. 2nd row: $\omega=0.5\omega_0$, $N=0$, $R=3.14$. 3rd row: $\omega=0.7\omega_0$, $N=0$, $R=4.39$. 4th row: $\omega=0.4\omega_0$, $N=1$, $R=1.25$. \label{SimulationPanels}}
\end{figure*}

\begin{figure*}
 \centering
 \includegraphics[width=0.32\textwidth]{Figures/Ptheta035.eps}
 \includegraphics[width=0.32\textwidth]{Figures/cos035.eps}
 \includegraphics[width=0.32\textwidth]{Figures/sin035.eps}
 \caption{From the simulation of an observation lasting $T_{\rm obs}$ = 3 hours, are showed the distributions of $\theta$ (left), cos$(\theta)$ (middle), and sin$(\theta)$ (right) for $\omega = \omega_0 + 0.35 \cdot 2\pi/T_{\rm obs}$.
\label{SimulationPanels2}}
\end{figure*}

\subsection{Power spectrum features}

Now we are going to demonstrate that the power spectrum of a sinusoidal signal observed for a finite time has a sinc squared shape ($[{\rm sin}(x)/x]^2$) centered on its proper frequency.
With this aim, the power spectrum calculated by Eq. (\ref{pow2}) needs the evaluation of the average values of sin$(\theta)$ and cos$(\theta)$,
\begin{eqnarray}
\left\langle {\rm sin}(\theta) \right\rangle &=& \int{\left[ {\rm sin}(\theta)P_{\rm sin}(\theta) \right] {\rm cos}(\theta)d\theta }  , \label{20}
\\
\left\langle {\rm cos}(\theta) \right\rangle &=& \int{ - \left[ {\rm cos}(\theta)P_{\rm cos}(\theta) \right] {\rm sin}(\theta)d\theta } .  \label{21}
\end{eqnarray}
The pdf $P_{\rm cos}(\theta)$ and $P_{\rm sin}(\theta)$ are given by Eqs. (\ref{Pcos}) and (\ref{Psin}), respectively. As we already underlined, they are composed by sums of the terms $P_{\theta}(2\pi(i+k) \pm \theta)$. For a sinusoidal signal, which general distribution of the phases is given by Eq. (\ref{9}), these terms are equal to
\begin{equation}
P_{\theta}(2\pi(i+k) \pm \theta) = 1+{\rm sin}\left( 2\pi i \frac{\omega_0}{\omega} + 2\pi k \frac{\omega_0}{\omega} \pm \frac{\omega_0}{\omega} \theta \right) .
\label{22}
\end{equation}
The power will be not null at frequencies close enough to the signal, such that $\omega_0/\omega \sim 1$.
This approximation can be adopted in Eq.~(\ref{22}), but just for the terms that do not contain the integer $i$, which is the index of the sums in Eqs. (\ref{Pcos}) and (\ref{Psin}). Indeed, for large values of $i$, the term $2\pi i \omega_0/\omega$ can significantly differ by an integer multiple of $2\pi$, and so can not be simplified. Eq. (\ref{22}) is then approximately equal to
\begin{eqnarray}
P_{\theta}(2\pi(i+k) \pm \theta) \sim 1+{\rm sin}\left( 2\pi i \frac{\omega_0}{\omega} + 2\pi k \pm \theta \right) =
\nonumber \\
 1 \pm {\rm sin}\left( 2\pi i \frac{\omega_0}{\omega}\right)  {\rm cos}(\theta)  \pm {\rm cos}\left( 2\pi i \frac{\omega_0}{\omega}\right)  {\rm sin}(\theta).
\label{23}
\end{eqnarray}

To calculate $\left\langle {\rm sin}(\theta) \right\rangle$, only the term containing sin$(\theta)$ in Eq. (\ref{23})
will lead to a not null quantity when substituted in Eq. (\ref{Psin}),
and this one into Eq. (\ref{20}). The average value of sin$(\theta)$ is then equal to
\begin{eqnarray}
\left\langle {\rm sin}(\theta) \right\rangle = \frac{1}{2} \left[ \frac{1}{N} \sum_{i=0}^{N-1} {\rm cos}(\frac{\omega_0}{\omega}2\pi i) \right] ,
\label{24}
\end{eqnarray}
where $1/N$ come from the normalization of $P_{\rm sin}(\theta)$.
For a random value of the ratio $\omega_0/\omega$ the former sum is negligible,
but when the ratio is close to 1 we can write it as
\begin{eqnarray}
\frac{\omega_0}{\omega} = 1 - \frac{\Delta \omega}{\omega},
\label{24b}
\end{eqnarray}
and all the values in the sum will be positive until $2\pi N |\Delta \omega|/\omega \leq \pi/2$, which neglecting $R$ in Eq. (\ref{11}) becomes
\begin{eqnarray}
|\Delta \omega| \leq \frac{1}{4} \omega_T ,
\label{25}
\end{eqnarray}
where $\omega_T = 2\pi/T_{\rm obs}$.
With good approximation, the term in square brackets in Eq. (\ref{24})
is equal to the following integral expression
\begin{eqnarray}
 \frac{1}{N} \sum_{i=0}^{N-1} {\rm cos}(\frac{\omega_0}{\omega}2\pi i) \rightarrow \frac{1}{2\pi N \varepsilon}\int_{0}^{2\pi N \varepsilon} {{\rm cos}(x) dx} \nonumber \\ \hspace{2cm}  = \frac{{\rm sin}(2\pi N \varepsilon)}{2\pi N \varepsilon},
 \label{26}
\end{eqnarray}
with $\varepsilon = \Delta \omega/\omega$.

In the same way, the average value of cos$(\theta)$ is evaluated substituting Eq. (\ref{23}) in
Eq. (\ref{Pcos}), and this one into Eq. (\ref{21}), leading to
\begin{eqnarray}
\left\langle {\rm cos}(\theta) \right\rangle =  - \frac{1}{2} \left[ \frac{1}{N} \sum_{i=0}^{N-1} {\rm sin}(\frac{\omega_0}{\omega}2\pi i) \right]  .
\label{27}
\end{eqnarray}
In this case, all the terms in the sum have the same sign when $2\pi N |\Delta \omega|/\omega \leq \pi$. This condition is less constraining with respect to Eq. (\ref{25}), and can be adopted to define the half peak width ($HPW$) in the power spectrum around $\omega_0$
\begin{eqnarray}
HPW = \frac{1}{2} \omega_T .
\end{eqnarray}
The integral expression for the term in square brackets in Eq. (\ref{27}) is
\begin{eqnarray}
\frac{1}{N} \sum_{i=0}^{N-1} {\rm sin}(\frac{\omega_0}{\omega}2\pi i) \rightarrow \frac{1}{2\pi N \varepsilon}\int_{0}^{2\pi N \varepsilon} {{\rm sin}(x) dx} \nonumber \\ \hspace{2cm} = \frac{1-{\rm cos}(2\pi N \varepsilon)}{2\pi N \varepsilon} .
\label{29}
\end{eqnarray}
From Eq. (\ref{pow2}), the power spectrum is calculated adding the squares of the sine and cosine averages.
Correspondingly, from Eqs. (\ref{26}) and (\ref{29}) we have
\begin{eqnarray}
\left[ \frac{{\rm sin}(2\pi N \varepsilon)}{2\pi N \varepsilon} \right]^2  + \left[ \frac{1-{\rm cos}(2\pi N \varepsilon)}{2\pi N \varepsilon} \right]^2 = \left[ \frac{{\rm sin}(\pi N \varepsilon)}{\pi N \varepsilon} \right]^2
\label{30}
\end{eqnarray}
where $\pi N \varepsilon = \pi \Delta \omega / \omega_T$.
Figure \ref{SingleP} shows the power spectrum (black curve) calculated from Eq. (\ref{pow})
for a sinusoidal signal, centered at its proper frequency ($\omega_0$), and in units of $\omega_T$.
The contribution of the sine sum is shown in blue $\left( \sum {\rm sin}(\theta_i)\right) ^2$ and that of the cosine sum is shown in green $\left( \sum {\rm cos}(\theta_i)\right) ^2$, which are equal to the first and second term on the left hand of Eq. (\ref{30}), respectively. The right hand term of Eq. (\ref{30}) is a squared sinc function centered on the signal frequency, and with width inversely proportional to the observation time.

In Figure \ref{SingleP} the power spectrum is normalized so that the peak is equal to 1.
We have considered in this demonstration a  100\% pulsed sinusoidal signal (see Eq. (\ref{8})).
In contrast, a signal partially pulsed can be represented by the following distribution of the arrival times
\begin{equation}
 P_t(t) = 1 + a \, {\rm sin}(\omega_0 t),
 \label{fracP}
\end{equation}
where $0 \leq a \leq 1$ determines the fraction of the signal that is pulsed.
Then, in Eqs. (\ref{24}) and (\ref{27}) the term multiplying the square brackets is $a/2$,
which substituting in Eq. (\ref{pow2}) results in the power spectrum being proportional to $a^2$.
The peak power in Figure \ref{SingleP} would be equal to $a^2$.
On the other hand, the mean power at frequencies far away from $\omega_0$ remains unchanged. Then, the signal to noise ratio in the power spectrum is proportional to $a^2$. Specifically, it is
$P(\omega_0)/\left\langle P(\omega\neq\omega_0)\right\rangle = N_0 a^2/4$.

\begin{figure}
\center
\includegraphics[width=0.75\textwidth]{Figures/SinglePeak-v2.eps}
\caption{Black: power spectrum of a sinusoidal signal. Blue: the contribution of the sine sum $\left( \sum {\rm sin}(\theta_i)\right) ^2$ to the power spectrum. Green: the contribution of the cosine sum $\left( \sum {\rm cos}(\theta_i)\right) ^2$. The half width of the peak (HPW) is also indicated by the double arrow. \label{SingleP}}
\end{figure}

\section{Pulsar frequency derivative}

We are going to apply the method developed in this paper to the practical case of observations so long that the first derivative of the pulsar frequency can not be neglected.
The time series of the emitted photons ($t_e$) by an isolated pulsar can be corrected for the first frequency derivative as:
\begin{equation}
\omega_0 t_c = \omega_0 t_e + \frac{1}{2} \dot{\omega}_0 t_e^2 ,
\label{Fdot-1}
\end{equation}
where $t_c$ is the corrected time series, and $\dot{\omega}_0$ is the frequency derivative.
If the frequency derivative is un-known, one should try different values of $\dot{\omega}$, which will affect the correction of time series
\begin{equation}
\omega_0 t_w = \omega_0 t_e + \frac{1}{2} \dot{\omega} t_e^2 .
\label{Fdot-2}
\end{equation}
Here $t_w$ stays for generally corrected time series, while $t_c$ is the properly corrected time series. Anyway, once corrected the phase assigned to each photon is
\begin{equation}
\theta = \omega_0 t_w .
\label{Fdot-6}
\end{equation}
In order to apply our method, we need to evaluate the distribution of the phases $P_{\theta}$.
With this aim we have first to find the relationship between $t_c$ and $t_w$.
From Eqs. \ref{Fdot-1} and \ref{Fdot-2}
\begin{equation}
\omega_0 t_w = \omega_0 t_c + \frac{1}{2} \delta\dot{\omega} t_e^2 ,
\label{Fdot-3}
\end{equation}
where $\delta\dot{\omega} = \dot{\omega}-\dot{\omega}_0$.
Solving Eq. \ref{Fdot-2} for $t_e$ we have
\begin{equation}
t_e = - \frac{\omega_0}{\dot{\omega}} \left[ 1 - \sqrt{1+2\frac{\dot{\omega}}{\omega_0}t_w} \right] ,
\label{Fdot-4}
\end{equation}
where we choose the solution with the negative sign of the square root because this satisfies the condition that $t_e = 0$ when  $t_w=0$.
Squaring Eq. \ref{Fdot-4}, expanding the root square in the Taylor series until the third term ($\dot{\omega}/\omega_0 t_w \ll 1$ for all the pulsars),
and substituting in Eq. \ref{Fdot-3} we have
\begin{equation}
t_c \sim t_w -\frac{1}{2} \frac{\delta\dot{\omega}}{\omega_0} t_w^2 .
\label{Fdot-5}
\end{equation}
and its inverse
\begin{equation}
t_w = \frac{1- \sqrt{1-2\frac{\delta\dot{\omega}}{\omega_0} t_c}}{\frac{\delta\dot{\omega}}{\omega_0}} .
\label{Fdot-7}
\end{equation}

The distribution of the corrected time series $P_{t_w}$ can be calculated applying the formula in Eq. \ref{rotondi}. Since $t_w$ is a monotonic function of $t_c$, the evaluation of $P_{t_w}$ is simplified as
\begin{equation}
P_{t_w}(t_c) = U \frac{P_{t_c}(t_c)}{dt_w/dt_c} ,
\label{Fdot-8}
\end{equation}
where here --and hereafter-- $U$ indicates a normalisation factor.
In the same way, the distribution of the phase assigned to each photon can be caculated considering Eq. \ref{Fdot-6}.
Since in Eq. \ref{Fdot-6} $\omega_0$ acts like a constant, $P_{\theta}$ has the same form as $P_{t_w}$
\begin{equation}
P_{\theta}(t_c) = U \frac{P_{t_c}(t_c)}{dt_w/dt_c} .
\label{Fdot-9}
\end{equation}
We assume that the properly corrected times have a sinusoidal distribution
\begin{equation}
P_{t_c}(t_c) = 1 + {\rm sin}(\omega_0 t_c) .
\label{Fdot-10}
\end{equation}
Substituting in Eq. \ref{Fdot-9}
\begin{equation}
P_{\theta}(t_c) = U \frac{1 + {\rm sin}(\omega_0 t_c)}{1/\sqrt{1-2\frac{\delta\dot{\omega}}{\omega_0} t_c}}
\label{Fdot-11}
\end{equation}
The square root at the denominator can be approximated to one, since $2\frac{\delta\dot{\omega}}{\omega_0} t_c \ll 1$ for all the pulsars even for observations as long as some years. Then, substituting $t_c$ with Eq. \ref{Fdot-5} in the argument of the sine we have
\begin{equation}
P_{\theta}(t_w) \sim U \left[ 1 + {\rm sin}(\omega_0 t_w - \frac{1}{2} \delta\dot{\omega} t_w^2) \right] .
\label{Fdot-12}
\end{equation}
Finally, the distribution $P_{\theta}$ as function of $\theta$ is obtained substituting $t_w = \theta / \omega_0$
\begin{equation}
P_{\theta}(\theta) \sim U \left[ 1 + {\rm sin}(\theta - \frac{1}{2} \frac{\delta\dot{\omega}}{\omega_0^2} \theta^2) \right] .
\label{Fdot-13}
\end{equation}

We should substitute Eq. \ref{Fdot-13} in Eqs. \ref{Pcos} and \ref{Psin} to calculate $P_{\rm cos}(\theta)$ and $P_{\rm sin}(\theta)$, which are composed by sums of the terms $P_{\theta}(2\pi (i+k) \pm \theta)$. In this case these terms are equal to
\begin{equation}
P_{\theta}(2\pi (i+k) \pm \theta) = 1 + {\rm sin}(2\pi i + 2\pi k \pm \theta - \frac{1}{2} \frac{\delta\dot{\omega}}{\omega_0^2} (2\pi i + 2\pi k \pm \theta)^2 ) .
\label{Fdot-14}
\end{equation}
Since here $0 \leqslant \theta < 2\pi$ while $2\pi i$ can be as large as $\omega_0 T_{obs}$ (see Eq. 3.8), then in the squared term $\pm \theta$ can be neglected. Thus
\begin{eqnarray}
P_{\theta}(2\pi (i+k) \pm \theta) \sim 1 + {\rm sin}(2\pi k \pm \theta - 2\pi^2 \frac{\delta\dot{\omega}}{\omega_0^2} (i + k)^2 ) = \nonumber \\
1 \pm {\rm sin}(2\pi^2 \frac{\delta\dot{\omega}}{\omega_0^2} (i + k)^2){\rm cos}(\theta) \pm {\rm cos}(2\pi^2 \frac{\delta\dot{\omega}}{\omega_0^2} (i + k)^2){\rm sin}(\theta) .
\label{Fdot-15}
\end{eqnarray}
To calculate $\left\langle {\rm sin} \theta \right\rangle$ , only the term containing sin$(\theta)$ in Eq. (\ref{Fdot-15}) will lead to a not null quantity
when substituted in Eq. (\ref{Psin}), and this one into Eq. (\ref{20}). The average value of sin$(\theta)$ is then equal
to
\begin{equation}
\left\langle {\rm sin} \theta \right\rangle = \frac{1}{2}\left[ \frac{1}{N} \sum_{i=0}^{N-1} {\rm cos}(2\pi^2 \frac{\delta\dot{\omega}}{\omega_0^2} (i + k)^2)\right]
\label{Fdot-16}
\end{equation}
$k = 0, 1/2, 1$ can be neglected.
With good approximation, the term in square brackets in Eq. (\ref{Fdot-16}) is equal to
the following integral expression
\begin{equation}
\frac{1}{N} \sum_{i=0}^{N-1} {\rm cos}\left( 2\pi^2 \frac{\delta\dot{\omega}}{\omega_0^2} i^2\right)  \rightarrow
\frac{1}{\sqrt{\pi y}} \int_{0}^{\sqrt{\pi y}} {{\rm cos}(x^2) dx} ,
\label{Fdot-17}
\end{equation}
where
\begin{equation}
y = 2 \pi N^2 \frac{\delta\dot{\omega}}{\omega_0^2} = 2 \pi \frac{\delta\dot{\omega}}{\omega_T^2} .
\label{Fdot-18}
\end{equation}
Then
\begin{equation}
\left\langle {\rm sin} \theta \right\rangle = \frac{1}{2} C(\sqrt{\pi y})
\label{Fdot-19}
\end{equation}
where $C(x) = \int_{0}^{x} {{\rm cos}(t^2) dt}$ is the cosine Fresnel integral.

In the same way, the average value of cos$(\theta)$ is evaluated substituting Eq. (\ref{Fdot-15}) in
Eq. (\ref{Pcos}), and this one into Eq. (\ref{21}), leading to
\begin{eqnarray}
\left\langle {\rm cos}(\theta) \right\rangle =  - \frac{1}{2} \left[ \frac{1}{N} \sum_{i=0}^{N-1} {\rm sin}(2\pi^2 \frac{\delta\dot{\omega}}{\omega_0^2} i^2) \right]  .
\label{Fdot-20}
\end{eqnarray}
All the terms in the sum are positive until $ 2\pi^2 \frac{\delta\dot{\omega}}{\omega_0^2} N^2 \leq \pi$. This condition can be adopted to define the width of the peak in the power spectrum at variance of $\delta\dot{\omega}$
\begin{eqnarray}
HPW_{\delta\dot{\omega}} = \frac{\omega_T^2}{2\pi}  .
\label{Fdot-21}
\end{eqnarray}
The integral expression for the term in square brackets in Eq. (\ref{Fdot-20}) is
\begin{equation}
\frac{1}{N} \sum_{i=0}^{N-1} {\rm sin}\left( 2\pi^2 \frac{\delta\dot{\omega}}{\omega_0^2} i^2\right)  \rightarrow
\frac{1}{\sqrt{\pi y}} \int_{0}^{\sqrt{\pi y}} {{\rm sin}(x^2) dx} .
\label{Fdot-22}
\end{equation}
Then
\begin{equation}
\left\langle {\rm cos} \theta \right\rangle = \frac{1}{2} S(\sqrt{\pi y})
\label{Fdot-23}
\end{equation}
where $S(x) = \int_{0}^{x} {{\rm sin}(t^2) dt}$ is the sine Fresnel integral.

From Eq. (\ref{pow2}), the power spectrum is calculated adding the squares of the sine and cosine averages.
Correspondingly, from Eqs. (\ref{Fdot-19}) and (\ref{Fdot-23}) we have
\begin{eqnarray}
P(\omega) = U \left[ \left\langle {\rm sin} \theta \right\rangle^2 + \left\langle {\rm cos} \theta \right\rangle^2 \right] =
U\frac{S(\sqrt{\pi y})^2 + C(\sqrt{\pi y})^2} {\pi y}.
\label{Fdot-24}
\end{eqnarray}
where $U$ is a normalization factor, which in Figure \ref{FdotShape} is choosen so that the power peak is equal to 1.
Figure \ref{FdotShape} shows the shape of the power spectrum at variance of $\delta\dot{\omega}$ as function of the variable $y = 2\pi \delta\dot{\omega} / \omega_T^2 $. In these units the width of the peak is equal to $y=1$, and the first minimum is at $y \sim 1.8$. Figure \ref{FdotShape} and Eq. (\ref{Fdot-21}) show that in a pulsation search the first frequency derivative can not be neglected when $\dot{\omega}_0$ is of the order of magnitude of $1/T_{obs}^2$, or greater.

\begin{figure}
\center
\includegraphics[width=0.75\textwidth]{Figures/Pdot.eps}
\caption{Power spectrum of a sinusoidal signal at variance of $\delta\dot{\omega}$ in units of $2\pi \delta\dot{\omega} / \omega_T^2$.
\label{FdotShape}}
\end{figure}

\section{Blind search}

A more general case happens when both the pulsar frequency and its first derivative are unknown.
Of course, this happen every time one search for new pulsars,
but there are at least two situations where the first frequency derivative can not be neglected in the search. On the one hand,
the search for radio quiet $\gamma$-ray pulsars with $\gamma$-ray data needs integration times of few weeks or more, so that
$\dot{\omega}_0$ is not negligible.
On the other hand, in radio searches for pulsars with fast spin down $\dot{\omega}_0$, it is important even for observations of few hours.

The general form of Eqs. (\ref{Fdot-1}), and (\ref{Fdot-2}) is:
\begin{eqnarray}
t_c = t_e + \frac{1}{2} \frac{\dot{\omega}_0}{\omega_0} t_e^2
\label{B1} \\
t_w = t_e + \frac{1}{2} \frac{\dot{\omega}}{\omega} t_e^2.
\label{B2}
\end{eqnarray}
Following the same steps and approximations from Eq. (\ref{Fdot-3}) to Eq. (\ref{Fdot-6}), and from Eq. (\ref{Fdot-8}) to Eq. (\ref{Fdot-13}) we get for the general case
\begin{eqnarray}
&& t_c = t_w - \frac{\delta R}{2} t_w^2
\label{B3} \\ &&
P_{\theta}(\theta) = U \left[ 1 + {\rm sin}\left( \frac{\omega_0}{\omega} \theta - \frac{\delta R}{2} \frac{\omega_0}{\omega^2}  \theta^2\right)  \right] .
\label{B4}
\end{eqnarray}
where
\begin{eqnarray}
\delta R = \frac{\dot{\omega}}{\omega} - \frac{\dot{\omega}_0}{\omega_0} \sim
\frac{1}{\omega_0^2}(\omega_0 \delta \dot{\omega} - \dot{\omega}_0 \Delta \omega) .
\label{B5}
\end{eqnarray}
To calculate $P_{\rm cos}(\theta)$, and $P_{\rm sin}(\theta)$ we substitute in Eq. (\ref{B4}) $\theta$ with $2 \pi (i+k) \pm \theta$, and we apply the same approximations as in Eq. (\ref{Fdot-15}) (neglecting $k$ and $\theta$ when possible):
\begin{eqnarray}
P_{\theta}(2\pi (i+k) \pm \theta) \sim 1 + {\rm sin}(\frac{\omega_0}{\omega} (2\pi i \pm \theta) - \delta R \frac{\omega_0}{\omega^2} 2\pi^2 i^2 ).
\label{B6}
\end{eqnarray}
The first term within the sine is equal to
\begin{eqnarray}
\frac{\omega_0}{\omega} (2\pi i \pm \theta) = \left( 1-\frac{\Delta \omega}{\omega}\right) (2\pi i \pm \theta) \sim
2\pi i \pm \theta - \frac{\Delta \omega}{\omega} 2\pi i.
\label{B7}
\end{eqnarray}
Substituting in Eq. (\ref{B5}) we have
\begin{eqnarray}
&& P_{\theta}(2\pi (i+k) \pm \theta) \sim
\nonumber \\ &&
1 \pm {\rm sin}(\frac{\Delta \omega}{\omega} 2\pi i + \delta R \frac{\omega_0}{\omega^2} 2\pi^2 i^2){\rm cos}(\theta) \pm {\rm cos}(\frac{\Delta \omega}{\omega} 2\pi i + \delta R \frac{\omega_0}{\omega^2} 2\pi^2 i^2){\rm sin}(\theta) .
\label{B8}
\end{eqnarray}
Only the term multiplying sin$(\theta)$ in Eq. (\ref{B8}) gives a not null contribution to $\left\langle {\rm sin}(\theta) \right\rangle$
\begin{eqnarray}
\left\langle {\rm sin}(\theta) \right\rangle = \frac{1}{2}\left[ \frac{1}{N}\sum_{i=0}^{N-1} {\rm cos}\left( \frac{2\pi i}{\omega}\left[ \Delta \omega + \frac{\delta R}{2} \omega_0 \frac{2\pi i}{\omega}\right] \right) \right] .
\label{B9}
\end{eqnarray}
Setting $z = 2\pi i/\omega$, the term within the square brackets can be approximated with
\begin{eqnarray}
&&  \frac{1}{N}\sum_{i=0}^{N-1} {\rm cos}\left( \frac{2\pi i}{\omega}\left[ \Delta \omega + \frac{\delta R}{2} \omega_0 \frac{2\pi i}{\omega}\right] \right) \rightarrow
\frac{\omega}{2 \pi N} \int_{0}^{2\pi N/\omega} {{\rm cos}\left(\Delta \omega z + \frac{\delta R}{2} \omega_0 z^2 \right) dz} =
\nonumber \\ &&
\frac{\omega}{2N \sqrt{\pi \delta R \omega_0} } \left\lbrace  {\rm cos} \left( \frac{\Delta \omega^2}{2 \delta R \omega_0} \right)
\left[ -C\left( \frac{\Delta \omega}{\sqrt{\pi \delta R \omega_0}} \right) + C\left( \frac{\Delta \omega + 2N\pi \delta R
\omega_0/\omega}{\sqrt{\pi \delta R \omega_0}}\right) \right] +\right.
\nonumber \\ && \left.
{\rm sin} \left( \frac{\Delta \omega^2}{2 \delta R \omega_0} \right)
\left[ -S\left( \frac{\Delta \omega}{\sqrt{\pi \delta R \omega_0}} \right) + S\left( \frac{\Delta \omega + 2N\pi \delta R \omega_0/\omega}{\sqrt{\pi \delta R \omega_0}}\right)   \right]  \right\rbrace
\label{B10}
\end{eqnarray}
where $S(x)$ and $C(x)$ are the sine and cosine Fresnel integrals, respectively.
Similarly, the average value of cos$(\theta)$ is
\begin{eqnarray}
\left\langle {\rm cos}(\theta) \right\rangle = \frac{1}{2}\left[ \frac{1}{N}\sum_{i=0}^{N-1} {\rm sin}\left( \frac{2\pi i}{\omega}\left[ \Delta \omega + \frac{\delta R}{2} \omega_0 \frac{2\pi i}{\omega}\right] \right) \right] ,
\label{B11}
\end{eqnarray}
and the term within the square brackets can be approximated with
\begin{eqnarray}
&&  \frac{1}{N}\sum_{i=0}^{N-1} {\rm sin}\left( \frac{2\pi i}{\omega}\left[ \Delta \omega + \frac{\delta R}{2} \omega_0 \frac{2\pi i}{\omega}\right] \right) \rightarrow
\frac{\omega}{2 \pi N} \int_{0}^{2\pi N/\omega} {{\rm sin}\left(\Delta \omega z + \frac{\delta R}{2} \omega_0 z^2 \right) dz} =
\nonumber \\ &&
\frac{\omega}{2N \sqrt{\pi \delta R \omega_0} } \left\lbrace  {\rm cos} \left( \frac{\Delta \omega^2}{2 \delta R \omega_0} \right)
\left[ -S\left( \frac{\Delta \omega}{\sqrt{\pi \delta R \omega_0}} \right) + S\left( \frac{\Delta \omega + 2N\pi \delta R
\omega_0/\omega}{\sqrt{\pi \delta R \omega_0}}\right) \right] -\right.
\nonumber \\ && \left.
{\rm sin} \left( \frac{\Delta \omega^2}{2 \delta R \omega_0} \right)
\left[ -C\left( \frac{\Delta \omega}{\sqrt{\pi \delta R \omega_0}} \right) + C\left( \frac{\Delta \omega + 2N\pi \delta R \omega_0/\omega}{\sqrt{\pi \delta R \omega_0}}\right)   \right]  \right\rbrace .
\label{B12}
\end{eqnarray}
Before writing the formula of the expectation value of the power spectrum is useful to make the following simplifications.
\begin{eqnarray}
&& \left[ \frac{2N \sqrt{\pi \delta R \omega_0}}{\omega} \right]^2 =
\frac{4\pi}{\omega_T^2 \omega_0}(\omega_0 \delta \dot{\omega} - \dot{\omega_0} \Delta \omega) =
\nonumber \\
&& \frac{4\pi \delta \dot{\omega}}{\omega_T^2} - \frac{4\pi}{\omega_T} \frac{\dot{\omega_0}}{\omega_0} \frac{\Delta \omega}{\omega_T} = 2y - Kx ,
\label{B13}
\end{eqnarray}
where from Eq. (\ref{11}) $N=\omega/\omega_T$, $\delta R$ is given by Eq. (\ref{B5}), and we set
\begin{eqnarray}
&& y = \frac{2\pi \delta \dot{\omega}}{\omega_T^2}
\label{B14} \\
&& x = \frac{\Delta \omega}{\omega_T}
\label{B15} \\
&& K = \frac{4\pi}{\omega_T} \frac{\dot{\omega_0}}{\omega_0} .
\label{B16}
\end{eqnarray}
With this notation the arguments of the Fresnel integrals in Eqs. (\ref{B10}), and (\ref{B12}) are
\begin{eqnarray}
&& \frac{\Delta \omega}{\sqrt{\pi \delta R \omega_0}} = \frac{2N\Delta \omega /\omega}{2N \sqrt{\pi \delta R \omega_0}/\omega}
= \frac{2x}{\sqrt{2y-Kx}}
\label{B17} \\
&& \frac{\Delta \omega + 2N\pi \delta R \omega_0/\omega}{\sqrt{\pi \delta R \omega_0}} =
\frac{\Delta \omega}{\sqrt{\pi \delta R \omega_0}} + \frac{2N \sqrt{\pi \delta R \omega_0}}{\omega} =
\frac{2x}{\sqrt{2y-Kx}} + \sqrt{2y-Kx} .
\label{B18}
\end{eqnarray}
Finally, the power spectrum given by the sum of the squares of Eqs.(\ref{B10}), and (\ref{B12}) is
\begin{eqnarray}
P(x, y) = \frac{1}{2y-Kx} && \left\lbrace \left[ C\left( \frac{2x}{\sqrt{2y-Kx}} \right) - C\left( \frac{2x}{\sqrt{2y-Kx}} + \sqrt{2y-Kx}\right) \right]^2 + \right.
\nonumber \\ && \left.
 \left[ S\left( \frac{2x}{\sqrt{2y-Kx}} \right) - S\left( \frac{2x}{\sqrt{2y-Kx}} + \sqrt{2y-Kx}\right) \right]^2 \right\rbrace .
\label{B19}
\end{eqnarray}

Figure \ref{FBlind} shows the power spectrum at variance of both $\Delta \omega$ and $\delta \dot{\omega}$.
The shape of the power spectrum follow an oblique structure, which is the typical one observed in plots produced
for example by the program {\it PRESTO} when a blind search is performed. The diagonal axis of the structure has a weak dependence by the parameter $K$ of Eq. (\ref{B18}) when it is lower than 1. For $K\rightarrow 0$ the diagonal axis has the equation $y=-2x$, that means
\begin{eqnarray}
\delta \dot{\omega} = - \Delta \omega \frac{\omega_T}{\pi} .
\label{B20}
\end{eqnarray}

\begin{figure}
\center
\includegraphics[width=0.75\textwidth]{Figures/FigureBlind-200dpi.eps}
\caption{Contour plot of the power spectrum of a sinusoidal signal at variance of $\Delta \omega$ and $\delta\dot{\omega}$ in units of $\Delta \omega/\omega_T$, $2\pi \delta\dot{\omega} / \omega_T^2$, respectively.
The maximum power is equal to 1 at the origin of the axes. The contours range from 0.1 to 0.9 in steps of 0.1.
In this plot the parameter $K$ of Eq. (\ref{B18}) is set $K=0.1$.
\label{FBlind}}
\end{figure}

\section{Conclusions}

In this paper we describe and  validate a method to calculate the expectation value of the power spectrum.
Adopting the definition given by \cite{Scargle1982} (see Eq. \ref{pow}) we calculate the expectation value making use of the statistical properties of the arrival time series, and consequently of the phases attributed to each event.
Our results are summarized by Eqs. (\ref{pow2}), (\ref{Pcos}), and (\ref{Psin}).

We validate the method focusing on the simple case of a sinusoidal signal assumed to come from an isolated pulsar. But since the solutions in Eq. (\ref{Pcos}) and (\ref{Psin}) are free from any assumption on the event phase distribution, the method can be generalized to any situation.

As noticed at the end of Section 2, a key ingredient of our method is the sum of the terms $\sum_{i=0}^{N-1} P_{\theta}(2\pi(i+k) \pm \theta)$, which corresponds to the distribution of the event phases folded by $2\pi$. At the proper frequency $\omega_0$, the folded distribution is equivalent to the pulse profile, but this is not true anymore at a different frequency $\omega \neq \omega_0$, as shown for example by Eq. (\ref{9}) in the case of a sinusoidal signal. There are several factors that can modify the folded distribution of the phases.
In this paper we applied our method to the case in which the folded distribution is perturbed by the first derivative of the pulsar frequency. Also, we considered the power spectrum expected in a blind search, in which both the frequency and its first derivative are uncertain. The analytical descriptions of the power spectra in these cases are given by Eq. (\ref{Fdot-24}) and Eq. (\ref{B19}), respectively. These are novel results in the field of timing.
In a separate paper we make direct use of the method developed here to evaluate the effects of the uncertainties of orbital parameters in the timing of pulsars in binary systems \cite{Cali12}.

\section*{Acknowledgments}

This work was supported by the  grants AYA2012-39303, SGR2009-811, and iLINK2011-0303. DFT was additionally supported by a Friedrich Wilhelm Bessel Award of the Alexander von Humboldt Foundation.

redpajama_set_name: RedPajamaArXiv


text:
title: Gospel Grammar
testament: New
pastor: Leon Coates
book: 1 John 2:12-17
audio: http://www.mecgoodnews.org/audio/161220181.mp3
service: Morning
---

redpajama_set_name: RedPajamaGithub


The 16th biannual International Conference on Perception and Action (ICPA) will provide an opportunity for researchers from multidisciplinary backgrounds to discuss issues broadly related to perception and action.
- intentionality and issues in philosophy of mind.
As with previous ICPAs, long abstracts of posters accepted for presentation will be made available in book form in Studies in Perception and Action XI (published by Taylor & Francis). Abstracts of accepted oral presentations for Invited Symposia and Open Forums will be made available in the program book of ICPA16.


text:
Caratteristiche tecniche
È una esterno sinistro.

Carriera
Cresciuto nelle giovanili del , ha esordito in prima squadra il 30 novembre 2016 in occasione del match di Coppa del Re vinto 5-1 contro il .

Nel gennaio 2018 è stato ceduto in prestito all'.

Collegamenti esterni

redpajama_set_name: RedPajamaWikipedia


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The first night of Wellfleet's recent Town Meeting, FinCom, trying hard to curb spending, suggested cutting the proposed $101,000 budget for "legal services" in half. In the debate, one selectman pointed out that we spent about that amount just in the lalte 2016 court case trying (unsuccessfully) to prevent Cumberland Farms' makeover of its existing store and we might be in for more expense if the company appeals the selectmen's recent vote not to permit a Cumby's gas station.
If you're foolishly idealist enough to take on a large, deep-pocketed corporation intent on having its way, he seemed to be saying, you shouldn't be talking about reducing the legal services budget.
The selectman was one of the minority of two who voted in favor of the gas station proposal.
There's something wrong about this either/or. Whether you vote for the full $101,000 or vote to halve it, a fixed amount seems insufficiently flexible. And unrealistic.
Just about all those who have expressed an opinion at hearings about Cumby's p lans have been against both the bigger store and the gas station as being bad for our town. In the first court case, after our formula business bylaw was found by a judge to be invalid as a basis for denying the glorified store, the selectmen voted to cave because we had already spent $97,000. Why throw good money after bad?
But what if most of of us wanted to dig deeper and put up more of a fight against this company's hostile makeover? $97,000 is about $20 of a median home's tax bill, or a loaded pizza or a couple of mixed drinks or five fancy coffees. What if we had couple of drinks' worth more to spend in defense of what we know to be right for our town?
Since Town Meeting, Cumby's has, as expected, decided to appeal the town's rejection of its unwanted gas station. This should be a whole different matter from the earlier court case, nothing to do with a legally questionable formula business bylaw. It's hard to see the basis for the company's appeal, given that it is the selectmen's job to decide about matters of public health and safety. If, nevertheless, Cumby's should once again succeed in court, do we just accept the unwanted gas station, widely deemed both redundant and dangerous in that congested stretch of Route 6, with its history of bad accidents? I would personally have a few more bucks for my share of keeping that gas station out of town.
Fighting such fights seems more important than many other budget items we rubber-stamp at TM.
There ought to be some more expressive and flexible way, when we feel strongly enough about something, to put our money where our hearts are. To tell the selectmen when necessary: Yes, of course we want in general to keep expenses down, but this is one we need you to fight. And we have the price of more than one pizza per tax bill to put into such a fight.
One way or another, if we're serious about resisting the effect we see large corporations having on most other towns, we need to vote ourselves a bigger gun. How about a reserve Goliath Budget funded by the price of several pizzas per tax bill to fight for the soul and the future of our town?


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Currently, when issuing a tender invitation, Authorities must disclose the criteria, weights and articulate how the best value supplier will be determined. Further disclosure requirements could include why certain criteria were selected, why weighting is being applied to certain factors or why a particular approach to value for money is being used. All too often the mechanics/mathematics of the assessment scheme are poorly understood or inadequately tested. Maintaining a record of why criteria were selected, the analysis that has been undertaken and how best value will be determined could become the norm. As thought leaders in this space, Commerce Decisions is seeing better outcomes from engaging industry more closely in this way. Auditors are also positive about the rigour of the approaches.
In the debrief process, Authorities are expected to provide the scores of each bidder, the reasons/rationale for their scores and the relative advantages of the winning bidder. The scores provided are very often the Authority view – i.e. a harmonised score following consensus or averaging. Authorities have resisted giving the independent scores of the individual evaluators, in case these conflict or show uncertainties in the decision making. In our experience, these independent scores reflect the different viewpoints of the team and following discussion lead to a better educated Authority view point. Use of averaging would be very difficult to sustain going forward and may introduce significant risk of protest/challenge.
If there were a requirement for further disclosure, it could legitimately be expected that a bidder could request more detail in a debrief regarding their competitiveness. Where they were strong or weak in relation to all competitors – not just the winner.
These pressures aren't limited to the UK; it's a conundrum for government authorities internationally. The rise of access to information legislation in Europe, Australia, Canada, USA and elsewhere may develop into full disclosure – i.e. bidders are given all the information they need to make informed decisions… An interesting thought!
This guidance highlights the need for a "proactive approach to information management", something to which Commerce Decisions, through its methods and software, is very well aligned. For more information on our solutions, contact us or take a look at our the Solutions and Services areas of our website.


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text:
Council of Europe Releases Report on Azerbaijan Corruption
Written by Katarina Sabados
An independent investigation found that some current and former members of the Parliamentary Assembly (PACE) may have been bribed to soften the body's criticism of the serious human rights violations in Azerbaijan, a report published on Sunday said.
The Azerbaijani Laundromat is a complex money-laundering operation and slush fund that handled $2.9 billion over a two-year period through four shell companies registered in the UK."As to the corruptive activities in favour of Azerbaijan, the Investigation Body established that there was a strong suspicion that certain current and former members of PACE had engaged in activity of a corruptive nature," the report says.
PACE launched the investigation in April 2017 after several NGO reports claimed Azerbaijan tried to silence criticism by PACE in exchange for gifts and money, which became known as "caviar diplomacy."
The Berlin-based think tank, the European Stability Initiative, ESI, said in 2016 that politicians were given "luxury electronic goods, watches and jewellery, paid holidays, prostitutes and also large sums of money given in cash."
The independent report, however, concluded that there was no substantial evidence to suggest that these gifts constituted bribes and that only two Azerbaijani officials and former Italian MP, Luca Volonte, had breached the body's Code of Conduct and were directly involved in facilitating corruption within PACE.
Among others, the Council of Europe cited the Organized Crime and Corruption Reporting Project (OCCRP) investigation, the Azerbaijani Laundromat.
In September 2017, the OCCRP and its partners revealed a multi-billion dollar money laundering and corruption scheme and detailed how the Azerbaijani government used a slush fund to bribe European politicians, including members of the European Parliament and the Parliamentary Assembly (PACE), in addition to laundering money through major European financial institutions and U.K.-based shell companies.
The OCCRP report found European MPs, PACE members, UNESCO officials' family members, Slovenian politicians, and media figures all received money via the Azerbaijani laundromat.
The EC investigation confirmed that illicit lobbying by the country and "ongoing conflicts of interest" regarding European officials involved in a number of election observation missions in Azerbaijan had occurred.
Council of Europe	Azerbaijani laundromat	Parliamentary Assembly

redpajama_set_name: RedPajamaCommonCrawl


text:
Recep Tayyip Erdogan: Turkey's Saviour or Sultan?
Fanack Published on March 14, 2020
Turkish President Recep Tayyip Erdogan (C) visits the Anitkabir, the mausoleum of the founder of the Turkish Republic Mustafa Kemal Ataturk, during a ceremony marking the 94th anniversary of Republic Day on 29 October 2017, in Ankara. Photo AFP
From humble beginnings, Recep Tayyip Erdogan has grown into a political giant, reshaping Turkey more than any leader since Mustafa Kemal Ataturk, the revered father of the modern republic.
Now he is the proud creator of a muscular presidency, wielding unrivalled power and heading a political system focused not on a party or bloc but on one man, Erdogan himself. To his supporters, Erdogan has brought Turkey years of sustained economic growth, increased Ankara's international clout and championed a strident Sunni Islam, Ottoman-nostalgic nationalism. To his critics, he is an autocrat, intolerant of dissent from his vision of Turkey's future and his methods of achieving it.
In partnership with Abdullah Gül who preceded Erdogan as president, he founded the AKP in 2001, a centre-right conservative party that sought to bring Islamist politics into the Turkish mainstream. He has achieved this goal with stunning success, winning general elections in 2002, 2007, 2011 and two in 2015.
Not since Ataturk has a figure dominated Turkish politics for so long. Yet his meteoric rise has been marred by a string of allegations of misconduct, human rights abuses, repression of free speech and nation-wide purges.
Born in 1954 in Istanbul and raised on Turkey's Black Sea coast, Erdogan, the son of a coastguard, is a graduate of an Islamic high school and the management programme at Marmara University.
While at university, Erdogan met Necebettin Erbakan, who would become the country's first Islamist prime minister and a mentor to the young would-be politician. Following the military coup of 1980, Erdogan joined Erbakan's Welfare Party, rising through its ranks until his election to parliament in 1991 (he was prevented from taking his seat on a technicality). A more resounding success came in 1994, when he was elected Istanbul's mayor, sparking fears among the city's elites that he would try to impose restrictions on alcohol and 'non-Islamic' ways of life. Although he proved to be a capable and pragmatic manager of the city, his time as an elected official was short-lived. In 1997, he was stripped of his position and sentenced to ten months in prison for reading an Islamic poem in a public address.
The poem included the line, 'The mosques are our barracks, the domes our helmets, the minarets our bayonets and the faithful our soldiers…' and was considered a violation of the country's Kemalist principles by the secular judiciary. He served only four months behind bars, but this was to be a seminal moment for the future president, who used his trial and the attention it garnered as a lightning rod for Islamist-minded Turks angered by the oppressive secularism of the country's constitution.
Forced to give up his position as Istanbul's mayor, he was also banned from parliamentary elections for five years, a ban that the AKP government promptly overturned in 2003, after it became Turkey's largest party in the 2002 elections, winning over a third of the votes cast. In March 2003, Erdogan contested, and won, a by-election in Siirt, his wife's home city, completing a long-awaited turnaround in Islamist politics in Turkey.
Throughout his career, Erdogan has been careful to maintain perceptions of his closeness to the people, relying on a style of populist politics heavy on rallies and public speeches to connect with voters. His speeches are typically laden with daily Islamic terms, a conversational tone and a healthy disrespect for institutional authority at home and abroad. His oratory, Islamic and Ottoman-friendly politics, and even his short career as a professional footballer have all endeared him to his largely working-class, conservative voter base. That said, as Turkey's economy has blossomed, so have the fortunes of Turks. A self-made success story himself, Erdogan has become a symbol of good fortune for many from Turkey's growing conservative middle class.
The core of Erdogan's AKP party was the breakaway Virtue Party, which the courts shuttered in 2001 for its 'anti-secular' activities. Ironically, it was by distancing himself from Turkey's previous Islamist movements and embracing a more centrist line that Erdogan came to the power with the AKP.
Nevertheless, he has carried his grievances against the old-guard of Kemalist Turkey with him. Still smarting from the humiliation of a prison sentence and a political ban for his religious beliefs, on coming to power he made curbing the clout of the once-powerful military something of a personal crusade. Starting with the Ergenekon trials in 2003, which implicated dozens of senior military officers in a coup plot, and continuing through to the purge of the past year, Erdogan has firmly recentred the country's power in the hands of the parliament, and now in his hands, through the executive branch. The military has been forced to toe the line of parliamentary rule, with their top ranks seeded with loyalist figures who have a similar background to Erdogan. These are the kind of men who, like the president, would never have risen so high under the old system.
The centrepiece of Erdogan's politics has been his vision for the 'New Turkey'. Although a vague concept, Erdogan has claimed it refers to a democratic Turkey free from military coups. In hindsight, a more accurate description might be the upturning of the past centres of power in Turkey.
Few could argue that when the AKP swept to power, Erdogan presided over a government that implemented much-needed liberalizing reforms far more quickly and effectively than almost any of its predecessors. His parliament opened up everything from schools, the army and even parliament to women wearing headscarves, heralding a new acceptance of religion in official Turkish life.
A non-smoking, teetotal father of four, Erdogan has been criticized for imposing on Turkey the Islamic values that underpin his own life. His government brought in tough regulations on alcohol sales in 2013, and three years later he claimed that no Muslim family should consider birth control. Critiques of gender equality and feminism have also marked his rhetoric. He went so far as to say that men and women cannot be treated equally.
That said, from his earliest days in politics he encouraged female participation in the political sphere. At that time, the inclusion of veiled women in Turkish public affairs was a key goal of Islamist politicians, as they represented a powerful symbol of the Muslim day-to-day life of most Turks within an ardently secular system of governance.
Under Erdogan's reign, Turkey has developed into a manufacturing and export powerhouse, allaying fears of financial crisis and putting the country's soaring inflation rates of the 1990s firmly in the past.
Yet this 'New Turkey' holds some disturbing hallmarks of the old, many of which have resurfaced as Erdogan's hold on power has tightened. His ongoing conflict with exiled cleric and former ally Fethullah Gülen was supposedly behind the 2014 allegations of massive corruption at the highest levels of government, which implicated Erdogan and his sons.
However, far more insidious has been the crackdown on human rights, freedom of the press, freedom of speech and even democracy. Scores of opposition politicians, journalists from across the media spectrum and tens of thousands of public servants, military personnel and citizens have been jailed on charges of involvement in terrorism. Last year's attempted coup, allegedly by Gülenist groups, can certainly be seen as a response to Erdogan's concentration of power in his own hands, which reached its zenith with his victory in the constitutional referendum last April 2017.
Erdogan has cemented his place in the history books as modern Turkey's second-most notable ruler, but he seems determined to better that. Even after last year's referendum, he has orchestrated a purge of elected AKP officials in cities where results from the vote were lower than expected. Loyalty to Erdogan seems to be the sole determinant of survival in this latest reshuffle of Turkish politics. If Turkey's future seems unsure, one thing is certain: Recep Tayyip Erdogan will be at its centre.
ADEM ALTAN ©AFP
Turkey Faces


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More and more people are choosing e-bikes for the shorter journeys. Though some cyclists may be put off by the weight and expense, there are many reasons why you should choose an e-bike for your commute.... So if you love cycling, the fresh air and free transport an e-bike could be for you. You will most likely find that your journey door-to-door is shortened in length because of the extra help from the motor.
An electric bike can be a faster alternative in for busy urban areas and with the average speed of cars often falling below 10mph. in cities. Being able maintain a higher average speed than a bicycle, yet take advantage of the full network of cycle facilities giving access to routes that cars cannot reach. The result is often a faster door-to-door journey time on shorter journeys than any other transport mode. And by nipping along the relatively uncongested cycle network, electric bikes tend to be the most consistent mode of travel.
Most people who rule out commuting to work by bike do so because of the mess factor. Getting to work all sweaty and sweating through your clothes is not a good look for the office. But the combination of cruising and cycling that e-bikes allow, will save you some effort (and some sweat), so you can burn some calories but prevent your ride from becoming a workout you're not dressed for.
That may sound obvious, but it's the primary advantage of a good electric bike is that it effectively 'flattens' hills, helping to increase your average speed as you go up. Provided you supply a reasonable amount of effort, you can expect to climb hills. In hilly areas the effect is nothing short of miraculous.
The extra speed and acceleration means that you can easily stop and start in traffic and at junctions, which helps to keep you out of danger. The faster your acceleration at junctions and roundabouts the safer you'll be able to navigate them.
An Electric Bike will cost a little more than a conventional bike and the mechanical wear and tear would be about the same as any other bike. The cost to charge an electric bike is so cheap as to be largely irrelevant, there is an extra expense in terms of battery depreciation. Consequently, an electric bike costs more to run and maintain over a conventional bike.
Because e-Bikes use electricity rather than say petrol or diesel, your e-bike emits a lot less carbon than a car would. In 2011 the European Cyclists' Federation found that when comparing electric bikes to cars, the bikes emitted just 8.1% per km of CO2 that a car emits making a measurable impact on the environment. This Makes the Electric bike the most fuel efficient mode of transport for everyday use. The typical fuel consumption is 8-16 watt-hours per mile, or something like a tenth as much as a small motorcycle.
The electric bike is a very different experience to riding any other type of bike. It gives you the chance to sit back, relax and take in the surroundings you might not have noticed before on your journey. It's much less effort. You won't get as easily tired on your journey when the motor is on. Of course you can turn the motor on and off and you basically get two bikes in one. This means you can have a leisurely cycle to work and then turn the motor off for a good workout on your journey home.
Recent research shows that around 46% of conventional bikes are only used once or twice a week. With about another 30% being used once a fortnight or even less. And to this, that figures confirm that an electric bike typically gets used at least twice as often as a conventional machine. There is going to be an health benefit to riding an electric bike, after all you still have to pedal the bike. The motor provides up to half the effort, but more regular use means more exercise for the rider.


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Mom humiliated after Six Flags security 'body shames' her over 'too short shorts' and 'told her to buy new clothes or leave'
The interaction was captured on TikTok, where it quickly went viral
A Colorado mom says she's embarrassed and feeling body shamed after she was confronted and threatened with criminal charges by Oklahoma City theme park security guards who told her her shorts were "too short" and she would have to buy new attire.
According to The Daily Mail, body shamed Colorado mom Bailey Breedlove had traveled to Oklahoma City's Frontier City theme park on April with her young daughter for a day at the park.
Instead she said she was yelled at and disparaged by park security for not following orders after she was confronted over her attire, and was told her shorts were "too short" and she would need to purchase new attire to remain in the park.
"I committed no crime and proceeded to walk to my boyfriend as I am autistic and have a hard time talking to officers," she recounted. "[The officer] followed me yelling and calling for backup."
In her post, she claimed that her 11-year-old daughter was 'yelled at by a park police officer' for rolling down a hill on wheeled sneakers
Autistic mom claims Six Flags police officer 'body shamed' her and kicked her out of the park because her shorts were 'too short'
"Then your incompetent manager showed up and began body shaming me. I was told I needed to go buy new shorts, which I am not obligated to purchase anything I don't want to," the shamed and embarrassed Colorado mom wrote of the demeaning encounter, which she posted to TikTok and Facebook.
Breedlove said her 11-year-old child was brought to tears by the security officials' yelling.
According to The Daily Star, when she attempted to leave the park she was "threatened with criminal trespassing."
"And that's how I got into trouble. Apparently you can't walk away and you can't argue when they are body-shaming you."
Bailey claimed: "She [the officer] then started yelling at me about my shorts, at which point I tried to walk away because it sounded ridiculous and she was on a power trip.
Mum horrified as theme park staff 'confront her for wearing shorts that are too short'
The Daily Star
The Independent reported one of the officials told Breedlove they were "police" when asking to see her ID.
Despite the fact one of the security officials seen confronting her in the video is wearing a polo with the word POLICE, the body shamed Colorado mom said she was later told by the city police department that the person was not an actual law enforcement officer.
"We have spoken with the Oklahoma police department captain and he assured us this was not an Oklahoma city police officer," Breedlove wrote on her Facebook.
"He sends his apologies and is going to look into the situation."
"They wouldn't provide us with badge numbers and I was terrified I was about to go to jail over a pair of shorts," she said.
Tik Tok user shares video of police threatening her for wearing 'shorts that are too short'
The body shamed Colorado mom wrote she was later slapped with a ban from Frontier City after requesting information about the incident.
"I have a 5-year ban now but will not be returning to any of your parks if this is how you treat a woman and law abiding citizens and visitors. You should be ashamed with yourselves," Breedlove wrote.
Six Flags, which operates Frontier City, did not respond to multiple outlets' requests for comment.
MEAWW reported that Breedlove has said she's contacted an attorney about the situation.
"I will not be letting this go as this is 2021 and not middle school. The Oklahoma police department has been contacted as well as an attorney. I have a five-year ban now but will not be returning to any of your parks if this is how you treat [women] and law-abiding citizens and visitors. You should be ashamed with yourselves."
Autistic mom Bailey Breedlove says Six Flags cop 'body shamed' her for 'too short' pants as daughter, 11, cried
Meaww
Tobey Maguire just sold an empty lot in Brentwood for $10.7 million


Vector Madness is a game about guns. The player-controlled character can move left or right and needs to kill all enemies which enter from the sides of the screen. Surviving waves unlocks more accessories for the character and additional guns (over 200 which range from traditional pistols to fantasy guns like the nailgun) which can be purchased between waves. Killed enemies may leave guns, ammo, health or time (for activating bullet time mode) behind and the last reached waved can be chosen directly from the main menu.
Besides the campaign, there is also an experimental mode without money restrictions. Here the player can experiment with already unlocked guns and change the enemy type.


National-Anarchist Movement: Troy Southgate Records Vocals For Greek National-Anarchist Band, Mind Terrorist.
Troy Southgate Records Vocals For Greek National-Anarchist Band, Mind Terrorist.
Troy Southgate has just written and recorded some vocals for a track on a forthcoming album by the Greek National-Anarchist band, Mind Terrorist. We are looking forward to hearing the rest of this release when it becomes available, too.
For more information visit the Mind Terrorsit Facebook page here or the official website here.


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Bill White for Governor essay
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Bill White for Governor. Custom Bill White for Governor Essay Writing Service || Bill White for Governor Essay samples, help
Bill White is currently among the main Texas candidates running for the Governor's seat. San Antonio was his place of birth. The candidate attended Harvard on a scholarship in economics. He also studied law at Texas University at Austin. White joined Clinton administration in 1993 as a Deputy secretary of Energy. He was also a chairman of the Texas Democratic Party at the time and has been a mayor of Houston since January 2004 to January 2010.His background has various elements and qualities that can make him particularly a strong candidate for a Governor. First of all, he is a supporter of open government and civic access. White has sensible ideas about how to deal with Texas threatening budget shortfall of $20-plus billion and he is ready to make the tough decisions that will have to be made. He has verified that he knows how to solve disasters. He proved this during his six years leadership as a mayor of Houston after Hurricane Katrina as he was able to accommodate around 200,000 evacuees. This showed that he is a manager who does not only set the track that has become a representation for such crises but also well-known, accumulated and in some cases forced the people who could make red tape vanish so that the countless barriers to the huge transition could be overcome (Jones, pp 423).The mayor has sound economic qualifications which are perhaps expressed by his distinguished skills in energy which is one of the state's important industries. He is a ex- U.S energy deputy secretary that owned an oil company which was successful. Therefore he would be an effective winner for both the traditional oil and gas sector and the rising renewable energy sector.
White has an obligation to an improved economic future for the Texans that are yet to be born. This has been emphasized in his campaign where he insists the need for educational advancement that will make Texas more aggressive in the international economy. To enable that to happen, he has promised that he would hit the state's shocking dropout crisis by taking care of it as an urgent situation when students do refuse to go back to school. As a Houston mayor, he formed a program in which thousands of people, him and his wife included paid visits to young people's homes and were able to encourage almost 9,000 of them to return to school. The mayor also has an extraordinary degree of competence which can qualify him as a strong Governor. He describes himself as a workhorse and not a show horse (Jones, 422).Bill White has different positions on various issues. Among the issues is jobs and economy on which he claims that getting the state's financial system back on track requires economic revival for common Texas families. In order to make the state more viable in a global economy, he promises to work on lasting resolutions. He would therefore revive Texas' job growth which is very important to the state's economy (Jones, 427).He promises that he would also attract jobs with a future because he is an earlier businessman and he knows the cost of meeting a payroll and make tough selections. He sees the need as a Governor to magnetize new businesses to the state, permit small businesses to develop and ensure an experienced workforce through education and job training (Maxwell, 235).
On the same issue he promises to work on stopping the squeeze on savings created by insurance rates. He claims he will do this do this by selecting appointees who will serve Texans and will be a focus a competitive group of organizations offering more competitive pace. White also focuses on investing in critical transportation and other infrastructure by facilitating local transportation results that promote economic growth and respect personal property. Using tax dollars for enticements more wisely is also among Bill's obligations and he promises to work on it.Another issue is Education which is the most significant industry of the state as the savings made in students now will establish the economic future for the state. Bill promises to improve professional and technical education through working with school districts, neighborhood colleges and employers. He will also develop pre-K programs that are applicable, let tutors teach problem-solving abilities, writing and logic instead of teaching how to make a lowest score on an annual high-stakes multiple choice exams. He also intends to make college education more affordable for Texans (Maxwell, 231).White's stand on gay marriages is quite contracting. He claims the need for traditional marriage between a man and a woman an on the other hand he supports it. On the immigration issue he promises to effectively fight hard for the central resources required to protect the state's borders against gangs, drug and human traffickers without disturbing legal daily trade and travel. On the health care issue, his stand and position is to boost the number of covered Texans, reducing the healthcare costs for families and supporting firm, community based medical practices and clinics as the foundation of health care delivery. He also sees the need for Texans to pay taxes. He feels Texas needs to start collecting taxes for internet sales (Maxwell, 232).White has considered his ethics and morals plan by revealing on his own connections. For example he knows some things about tearing up public proceedings and appointing top donors to handle the assignments. On the state budget deficit issue Bill White accuses the fellow candidate Rick Perry of hiding from an anticipated $18 billion budget deficit for the state. He personally promises to be providing all the updates of the state's financial condition if he is elected as a governor (Maxwell, 233).The candidates can however be contrasted in different manners as each of them has his own character and campaign tactics that opposes the other's. For instance, they are referring to each other as a liar due to Bill White's position on cap and trade something White himself denies that he has never promoted. Texans therefore may in a way find it hard to judge the right candidate for the Governor who is more honest, reliable and dependable.
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To put down a learning and development strategy is a way to ensure that the efforts of competence development and performance improvements are taking place in a structured and targeted fashion. It will ensure that the organisation share a common perception of how the organisation learns and develops. In cooperation with management we can facilitate the development of the strategy or philosophy and define how to manage the process and the instruments to be applied in order to achieve the desired development culture.


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Federal elections complaint accuses Rick Scott of misusing political committee to prepare for Senate bid
by John Lucas | Apr 10, 2018
A complaint filed with the Federal Elections Commission charges Gov. Rick Scott with violating federal election laws. The complaint accuses Scott of using his political committee, New Republican PAC, to raise and spend unlimited amounts of campaign money to prepare for his bid for the U.S. Senate.
The complaint was filed following Scott's announcement Monday that he is a candidate for the U.S. Senate. It was filed by End Citizens United, an organization that tends to lean toward the Democrats.
It says that the "very same day that Scott declared his federal candidacy, the Committee revamped its website, suddenly transforming itself from a committee dedicated to supporting President Trump to one dedicated to supporting Scott. This timing cannot be mere coincidence.."
The complaint charges that "it appears as if [Scott] has been using the Committee to improperly pay for expenses associated with his exploratory activities and his candidacy for Senate."
The PAC was established last year with a mission of re-inventing the Republican Party to appeal to younger and hispanic voters and support the agenda of President Donald Trump.
According to the complaint, the New Republican PAC was a way for Scott to avoid federal restrictions on campaign fundraising while he put together his campaign to unseat Democratic incumbent Bill Nelson.