text:
\section{Introduction}
Different types of objects are detected in series of CCD-frames during observations: solar system minor bodies (SSOs); stars and large-scale diffuse sources (non-SSOs); charge transfer tails from bright stars, bright streaks from satellites, and noise sources amongst others. The difference between the detected SSOs and non-SSOs is that the non-SSOs have a zero velocity apparent motion on a set of frames, while the SSOs have a non-zero one. Wherein, a rapid detection of the objects with a near-zero velocity apparent motion both from the main belt of asteroids and beyond the Jupiter's orbit is very important for the asteroid-comet hazard problem as well as for the earliest recording new SSOs.

Over the past few decades, several powerful software tools and methods had been developed, allowing discovery and cataloging of thousands of SSOs (asteroids, comets, trans-Neptunians, Centaurs, etc.).
First of all, it was the Lincoln Near-Earth Asteroid Research (LINEAR) project \citep[][]{stokes}, which outperformed all asteroid search programs acted until 1998. This project brought the number of discovered SSOs to over 230~000, including 2~423 near-Earth objects (NEOs) and 279 comets \citep[][]{stokes1}.
The second biggest asteroid survey, the Catalina Sky Survey \citep[][]{CSS}, started in 2005 as a search program for any potentially hazardous minor planets and allowed to discover more than 6~500 NEOs. The same program in the southern hemisphere, the Siding Spring Survey (SSS), was closed in 2013.

A successful operation of these programs has stimulated new instruments and advanced CCD-cameras manufacturing as well as the development of new methods and algorithms for image processing and detecting faint SSOs. These methods of the automated search for very faint objects in a CCD-frame series were based, mostly, on the matched filter or the combined multiple frames along the typical SSO's motion \citep[][]{yanagisawa}. For example, the implementation of a multi-hypothesis velocity matched filter for LINEAR archive of images has produced about 25 \% new detections (mostly of faint SSOs) that were missed at the stage of a primary processing of observations \citep[][]{shucker}. Another algorithm, the interacting multiple model (IMM), was introduced as a modification of matched filter and provided a new structure for effective management of multiple filter models, while the selected parameters must be considered for the IMM optimizing \citep[][]{genovese}.

The Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) for surveying the sky for moving objects on a continual basis was designed as an array of four telescopes. The first telescope, PS1, is in a full operation since 2010 and is able to observe objects down to 22.5${^m}$ apparent magnitude. With the help of PS1 more than 2~860 NEOs and many comets have already been discovered \citep[see][]{henry}. PS1 uses the Moving Object Pipeline System, MOPS \citep{heasley}, which includes some methods and techniques for searching for the extremely faint and distant Sedna-like objects \citep{jedicke}, such as for example the modified intra-nightly linking algorithm, which includes a partial Hough transform method for quickly identifying of the multiple detections and post-processing step for intra-nightly linking (see \citet{parker}, \citet{myers}).

These methods were successfully tested for simulations of processing the moving objects with MOPS on the Pan-STARRS and the Large Synoptic Survey Telescope, LSST \citep[][]{barnard}, the latter will be provided by the same pipeline system as on the Pan-STARRS \citep[][]{myers}. It is important that PS1 is a highly effective for discovering objects that could actually impact the Earth next 100 years \citep[][]{jedicke} and was complemented with the infrared data of the former WISE orbital telescope \citep[][]{dailey}.

In 2009, the authors of this paper developed the CoLiTec (Collection Light Technology) software for the automated detection of the solar system minor bodies in CCD-frames series (see, in detail, \url{http://www.neoastrosoft.com}
and papers by \citet{savanevych2,savanevych3,savanevych4,savanevych,vavilova1,vavilova,vavilova2,vavilova3,pohorelov}. Since 2009 it has been installed at several observatories: Andrushivka Astronomical Observatory (A50, Ukraine) \citep[][]{ivashchenko}, ISON-NM Observatory (H15, the US) \citep[][]{elenin}, ISON-Kislovodsk Observatory (D00, Russia) \citep[][]{kislovodsk}, ISON-Ussuriysk Observatory (C15, Russia) \citep[][]{elenin1}, Odessa-Mayaki (583, Ukraine) \citep[][]{troianskyi}, Vihorlat Observatory (Slovakia) \citep[][]{dubovsky}.

The preliminary object's detection with CoLiTec software is based on the accumulation of the energy of signals along possible object tracks in a series of CCD-frames. Such accumulation is reached by the method of the multivalued transformation of the object coordinates that is equivalent to the Hough transformation \citep{savanevych3,savanevych4}. In general, CoLiTec software allows detecting of the objects with different velocities of the apparent motion by individual plugins for fast and slow objects, and objects with the near-zero apparent motion. CoLiTec software is widely used in a number of observatories. In total, four comets (C/2011 X1 (Elenin), P/2011 NO1 (Elenin), C/2012 S1 (ISON) and P/2013 V3 (Nevski)) and more than 1560 asteroids including 5 NEOs, 21 Trojan Jupiter asteroids and one Centaur were discovered using CoLiTec software as well as more than 700 000 positional CCD-measurements were sent to the Minor Planet Center \citep{ivashchenko,elenin,elenin1,savanevych}. Our comparison of statistical characteristics of positional CCD-measurements with CoLiTec and Astrometrica \citep[http://www.astrometrica.at;][]{miller,raab} software in the same set of test CCD-frames has demonstrated that the limits for reliable positional CCD-measurements with CoLiTec software are wider than those with Astrometrica one, in particular, for the area of extremely low signal-to-noise ratio (S/N) \citep[][]{savanevych}.

Besides the requirement of large computational effort \citep[see][]{shucker}, the main disadvantage of all the above mentioned methods implemented into software is a neglecting of near-zero apparent motion of objects in CCD-frames that has yet to be described and tested. So, the aim of this paper is to introduce a new computational method for detection of SSOs with a near-zero velocity of apparent motion in a series of CCD-frames. We propose considering these SSOs as a separate subclass, which includes objects whose inter-frame shifts during the observational session are commensurate with the errors in measuring their positions. We call the maximum permissible velocity of a near-zero apparent motion as $\varepsilonup$-velocity. Then, a subclass of SSOs with a near-zero apparent motion includes such SSOs, which have velocities of apparent motion between CCD-frames that are not exceed three RMS errors, $3\sigma$, of measurements of their positions ($\varepsilonup=3\sigma$). We will also use the notation of $3\sigma$-velocity instead of $\varepsilonup$-velocity to describe a near-zero apparent motion of SSOs.

The economy in the observational search resource leads to a reduction in the time between CCD-frames. This, in turn, leads to the fact that a significant part of SSOs will have an $\varepsilonup$-velocity apparent motion, in other words, have a shift, which is commensurate with the errors in estimating of their position. In general, there are about 15\% of SSOs with $\varepsilonup$-velocity motion. They are the objects beyond the Jupiter's orbit as well as asteroids moving to the observer along the view axis (heading straight to the Earth). Of course, when such an object is close enough, a parallax from the Earth's rotation will introduce a significant transverse motion that can be detectable. The proposed method allows us to locate objects with a near-zero apparent motion, including the potentially dangerous objects, at larger distances from the Earth than trivial methods. It gives more time to study such objects and to warn about their approach to the Earth in case of their hazardous behavior.

The structure of our paper is as follows. We describe a problem statement, a model of the apparent motion and hypothesis verification in Chapter 2. The task solution and new method are described in Chapter 3. Analysis of quality indicators of near-zero motion detection is provided in Chapter 4. Concluding remarks and discussion are given in Chapter 5. A mathematical rationale of the method is described in Appendices A-C.

\section{Problem statement}
The apparent motion of any object may be represented as the projection of its trajectory on the focal plane of a telescope. It is described by the model of rectilinear and uniform motion of an object along each coordinate independently during the tracking and formation of the series of its CCD-measurements (see Appendix A).

Objects with significant apparent motion are easily detected by any methods of the trajectory determination, for example, the methods for inter-frame processing \citep{garcia,gong,vavilova}. The problem arises when we would like to detect an object with a near-zero apparent motion in CCD-frame series. Such an object can be falsely identified as the object with a $3\sigma$-velocity.

The first step for solving this problem is a formation of the set of measurements $\Omega_{set}$ (A.5) (no more than one measurement per frame) for the object, which was preliminarily assigned to the objects with $3\sigma$-velocities. In its turn, such objects should be registered in the internal catalog of objects that are motionless in the series of CCD-frames \citep[][]{vavilova1}. This catalog is also helpful to reduce the number of false SSO detections in the software for automatic CCD-frame processing of asteroid surveys \citep[][]{pohorelov}.

In other words, the hypothesis $H_{0}$ that a certain set $\Omega_{set}$ (A.5) of measurements complies to the objects with a $3\sigma$-velocity is as follows:
\begin{equation}
        \label{eq6}
        H_{0}:\sqrt{V_{x}^2+V_{y}^2}=0,
\end{equation}
where $V_{x}$, $V_{y}$ are the apparent velocities of object along each coordinate.

Then the more complicative alternative $H_{1}$ that the object with the set of measurements $\Omega_{set}$ (A.5) has a $3\sigma$-velocity will be written as:
\begin{equation}
        \label{eq7}
        H_{1}:\sqrt{V_{x}^2+V_{y}^2}>0.
\end{equation}
The false detection of the near-zero apparent motion of the object is an error of the first kind $\alpha$ assuming the validity of $H_{0}$ hypothesis (1). The skipping of the object with a $3\sigma$-velocity is an error of the second kind $\betaup$ under condition that the alternative hypothesis $H_{1}$ (2) is true. It is accepted in the community that the conditional probabilities of errors of the first $\alpha$ kind (conditional probability of the false detection, CPFD) and the second $\betaup$ kind (skipping of the object) are the indicators of a good quality detection \citep[][]{kuzmyn}. We also used the conditional probability of the true detection (CPTD) as a complement to the conditional probability of an error of the second $\betaup$ kind to unity ($1-\betaup$).

So, the task solution may be formulated as follows: 1) it is necessary to develop computational methods for detecting the near-zero apparent motion of the object based on the analysis of a set $\Omega_{set}$ of measurements (A.5) obtained from a series of CCD-frames; 2) computational methods have to check the competing hypotheses of zero $H_{0}$ (1) and near-zero $H_{1}$ (2) apparent motion of the object.

\textbf{Maximum likelihood criterion.} Usually, hypotheses such as $H_{0}$ (6) and $H_{1}$ (7) are tested according to a maximum likelihood criterion \citep[][]{masson}\citep[][]{myung}, \citep[][]{miura}, \citep[][]{sandersreed} or any other criterion of the Bayesian group \citep[][]{lee}. The sufficient statistic for all the criteria is the likelihood ratio (LR), which is compared with critical values that are selected according to the specific criteria \citep[][]{morey}. If there are no opportunities to justify the a priori probabilities of hypotheses and losses related to wrong decisions, the developer can use either a maximum likelihood criteria or Neyman-Pearson approach \citep[][]{lee}. The unknown parameters of the likelihood function are evaluated by the same sample in which the hypotheses are tested. In mathematical statistics, such rules are called "substitutional rules for hypothesis testing" \citep[][]{lehman,morey}. In the technical literature, such rules are called "detection-measurement" \citep[][]{morey}.

The "detection" procedure precedes the "measurement" procedure for the substitutional decision rule. And this is a general principle for solving the problem of mixed optimization with discrete and continuous parameters \citep[][]{arora}. The decision statistics of hypotheses that correspond to different values of discrete parameters are compared with each other after the optimization of conditional likelihood functions for the value of their continuous parameters.
The software developers use the substitution rule of maximum likelihood despite the fact that the evidence is not proved mathematically. It should be compared with any new methods of hypothesis testing with a priori parametric uncertainty \citep[][]{gunawan}. The quality indicators of hypothesis testing can be examined only by statistical modeling or on the training samples of large experimental datasets.

A likelihood function for detection of a near-zero apparent motion can be defined as the common density distribution of measurements of the object positions in a set of measurements (see Appendix B). Ordinary least square (OLS) evaluation of the parameters of the object's apparent motion as well as the variance of the object's positional estimates in a set of measurements are described in Appendix C. Using these parameters, one can obtain the maximum allowable (critical) value of the LR estimate for the detection of a near-zero apparent motion for the substitutional methods (C.11 - C.13).

\section{Task solution}
\textbf{Conversion of testing the hypothesis $H_{1}$ to the problem of validation of the statistical significance factor of the apparent motion.} One of the disadvantages of substitutional methods based on maximum likelihood criteria \citep{masson,myung} is the insufficient justification of their application when some parameters of likelihood function are unknown. The second one leads to the necessity of selecting the value of boundary decisive statistics \citep{miura,sandersreed}. Moreover, in our case, the substitutional methods are inefficient when the object's apparent motion is near-zero.

Models (A.1) and (A.2) of the independent apparent motion along each coordinate are the classical models of linear regression with two parameters (start position and the velocity along each coordinate). Thus, in our case, the alternative $H_{1}$ hypothesis (2) about the object to be the SSO with a near-zero apparent motion is identical to the hypothesis about the statistical significance of the apparent motion.
We propose to check the statistical significance of the entire velocity for detection of a $3\sigma$-velocity, which is equivalent to check the hypothesis $H_{1}$.

\textbf{A method for detection of the near-zero apparent motion using Fisher f-criterion.} We propose to check the statistical significance of the entire velocity of the apparent motion of the object using f-criterion. F-test should be applied, when variances of the positions in a set of measurements are unknown. It is based on the fact that the f-distribution does not depend on the distribution of positional errors in a set of measurements \citep[][]{phillips,johnson}. Furthermore, there are also tabulated values of the Fisher distribution statistics \citep[][]{burden,melard}.

The f-criterion to check the statistical significance of the entire velocity of the apparent motion is represented as \citep[][]{phillips}:
\begin{equation}
        \label{eq23}
        f(\Omega_{set}) = \frac{R_{0}^2 - R_{1}^2}{R_{1}^2} \frac{N_{mea} - r}{w},
\end{equation}
where $w=1$ is the number of factors of the linear regression model that are verified by the hypothesis. In our case, the factor is the velocity of the apparent motion;

$r$ is a rank of the plan matrix $F_{x}$ \citep{burden} ($rang(F_{x} = r \leq min(m, N_{mea}))$);
\begin{equation}
        \label{eq24}
        F_{x}=\left\|
        \begin{array}{cc}
                1 & \Delta_{\tau 1}=(\tau_{1}-\tau_{0}) \\
                ... & ...\\
                1 & \Delta_{\tau k}=(\tau_{k}-\tau_{0})\\
                ... & ...\\
                1 & \Delta_{\tau Nmea}=(\tau_{Nmea}-\tau_{0})
        \end{array}
        \right\|.
\end{equation}

The rank of the $F_{x}$ matrix defined by (4) is equal to two for the linear model of the motion along one coordinate because a number $m$ of the estimated parameters of the motion is equal to two. As the apparent motion occurs along two coordinates, the number $m$ of its estimated parameters is equal to four. Accordingly, the rank $r$ of the $F_{x}$ matrix is four because $r = m$.

The statistic (3) has a Fisher probability distribution with ($w$, $N_{mea} - r$) degrees of freedom \citep[][]{phillips}. Its distribution corresponds to the distribution of the ratio of two independent random variables with a chi-square distribution \citep[][]{park}, degrees of freedom $w,$ and $N_{mea} - r$. For example, let the number $N_{fr}$ of CCD-frames in a series of frames to be $N_{fr} = 4$, and each frame contains the measurement of the object's position. Hence, for two coordinates the number of measurements is $2N_{mea} = 8$, $w = 1$, and the rank $r$ of the matrix $F_{x}$ (4) is $r = 4$. Therefore, statistic (3) has a Fisher probability distribution with ($1$, $4$) degrees of freedom.

To determine the maximum allowable (critical) tabulated value of the Fisher distribution statistics, we have to use the predefined significance level $\alpha$. Its value is the conditional probability of the false detection, CPFD, of the near-zero apparent motion. For example, if $\alpha=10^{-3}$, the maximum allowable $f_{cr}$ value of the Fisher distribution statistics with ($1$, $4$) degrees of freedom is $f_{cr} = 74.13$ \citep[][]{melard}.

After transformation, the method for detection of the near-zero apparent motion using Fisher f-criterion is represented as:
\begin{equation}
        \label{eq25}
        \frac{R_{0}^2 - R_{1}^2}{R_{1}^2} \geq \frac{w f_{cr}}{N_{mea} - r}.
\end{equation}

\section{Indicators of quality of the near-zero apparent motion detection}
\textbf{Number of experiments for statistical modeling.} Errors in statistical modeling are defined by estimates of conditional probabilities of the false detection $\gammaup_{0}$ (validity of the $H_{0}$ hypothesis ) and true detection $\gammaup_{1}$ (validity of the alternative $H_{1}$ using the critical values of the decision statistics after modeling the $H_{0}$ hypothesis).

In our research we assumed that the reasonable values of errors of experimental frequencies are equal to $\gammaup_{0accept}=\alpha/10$, $\gammaup_{1accept}=10^{-3}$. Their dependence on the number of experiments for the statistical modeling (under the condition of a validity of the hypothesis $H_{0}$ and the alternative $H_{1}$) is determined by the empirical formulas:
\begin{equation}
        \label{eq26}
        N_{0exp} = 10^2 / \gammaup_{0accept};
\end{equation}
\begin{equation}
        \label{eq27}
        N_{1exp} = 10^2 / \gammaup_{1accept} = 10^{-6}.
\end{equation}

\textbf{Preconditions and constants for the methods of the statistical and in situ modeling.} To study the indicators of quality of the near-zero apparent motion detection using substitutional methods (see, Appendix C and formulas C.11 - C.13) in maximum likelihood approach, the appropriate maximum allowable values $\lambda_{cr}$ should be applied. These values are determined in accordance with the predefined level of significance $\alpha$ in the modeling of the hypothesis $H_{0}$ ($V = 0$).

For the statistical and in situ modeling, where the method (5) was used, we applied the tabulated value $f_{cr}$ of the Fisher distribution statistics with ($w$, $N_{mea} - r$) degrees of freedom \citep[][]{phillips}. As an alternative, the critical value $f_{cr}$ is determined according to the predefined level of significance $\alpha$ in the modeling of the hypothesis $H_{0}$ ($V = 0$).
 Normally distributed random variables were modeled using the Ziggurat method \citep[][]{marsaglia}. All the methods for detection of the near-zero apparent motion were analyzed on the same data set.

The following values of constants were used: the significance level is taken as $\alpha = 10^{-3}$ and $\alpha = 10^{-4}$; the number $N_{fr}$ of frames in a series is equal to $N_{fr} = (4, 6, 8, 10, 15)$. For modeling $H_{1}$ ($V > 0$) hypothesis the velocity module $V$ of the apparent motion was defined in relative terms, namely, RMS error of measurement deviations of the object's position ($V = k\sigma$).

Here the coefficient is equal to $k = (0, 0.5, 1, 1.25, 1.5, 1.75, 2, 3, 4, 5, 10)$. Mathematical expectation of external estimation of positional RMS error is $m(\hat{\sigma}_{out}) = 0$ and its RMS error is $\sigma(\hat{\sigma}_{out}) = (0.15, 0.25)$. If $\alpha = 10^{-3}$, the maximum allowable tabulated value of the Fisher distribution statistics with ($1$, $4$) degrees of freedom is equal to $f_{cr} = 74.13$ and if $\alpha = 10^{-4}$, it is $f_{cr} = 241.62$ \citep[][]{melard}.

\textbf{A method of statistical modeling for analysis of indicators of quality of the near-zero apparent motion detection in a series of CCD-frames.} Conditional probability of the true detection (CPTD) is calculated in terms of the frequency of LR estimates $\hat{\lambda}(\Omega_{set})$, or $f(\Omega_{set})$ exceeding the maximum allowable values $\lambda_{cr}$, or $f_{cr}$ for all methods of near-zero apparent motion detection:
\begin{equation}
        \label{eq28}
        D_{true} = N_{exc} / N_{1exp},
\end{equation}
where $N_{exc}$ is the number of exceedings of the critical value $\lambda_{cr}$ for the substitutional methods of maximum likelihood or $f_{cr}$ for the method with f-criterion. CPTD estimation is determined for the various number of frames $N_{fr}$ and various values of the apparent motion velocity module $V$.

Figure 1 ($\alpha = 10^{-3}$) shows the curves of near-zero apparent motion detected by different methods: the Fisher f-criterion (5) method (curve 1); substitutional method for maximum likelihood detection using the known variance of the position measurements (C.12) (curve 2); and substitutional method for maximum likelihood detection using external estimation of RMS error (C.13) $\hat{\sigma}_{out} = 0.15$ (curve 3) and $\hat{\sigma}_{out} = 0.25$ (curve 4).
\begin{figure}
        \label{pic1_1}
        \centering

        \begin{tikzpicture}[scale=0.8]
                \label{pic1a}
                \small

                \draw (0,0) -- coordinate (x axis mid) (8,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.012) (1,0.03) (2,0.1) (3,0.17) (4,0.28) (5,0.38) (6,0.63) (7,1.77) (8,3.17)};
                \draw[<-,thick] (6.8,1.5) -- (7.3,1) node[anchor=west] {1};
                \draw[thick,dash pattern=on 1pt off 3pt on 3pt off 3pt, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.05) (2,0.52) (3,1.18) (4,2.16) (5,3.23) (6,4.1) (7,4.89) (8,5)};
                \draw[<-,thick] (4.5,2.8) -- (3.8,3) node[anchor=east] {2};
                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.042) (2,0.43) (3,0.98) (4,1.86) (5,2.88) (6,3.82) (7,4.89) (8,5)};
                \draw[<-,thick] (4.4,2.2) -- (4.9,1.8) node[anchor=west] {3};
                \draw[thick,dotted, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.013) (2,0.059) (3,0.11) (4,0.21) (5,0.32) (6,0.61) (7,2.37) (8,4.25)};
                \draw[<-,thick] (7.35,3) -- (6.7,3.1) node[anchor=east] {4};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} a) $N_{fr}=4$ \end{tabular}
        \begin{tikzpicture}[scale=0.8]
                \label{pic1b}
                \small

                \draw (0,0) -- coordinate (x axis mid) (8,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.015) (1,0.24) (2,1.79) (3,3.07) (4,4.1) (5,4.7) (6,4.9) (7,5) (8,5)};
                \draw[<-,thick] (3.3,3.4) -- (3.9,3) node[anchor=west] {1};
                \draw[thick,dash pattern=on 1pt off 3pt on 3pt off 3pt, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.39) (2,3.67) (3,4.75) (4,4.98) (5,4.99) (6,5) (7,5) (8,5)};
                \draw[<-,thick] (2.4,4.3) -- (1.8,4.5) node[anchor=east] {2};
                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.33) (2,3.36) (3,4.62) (4,4.96) (5,4.99) (6,5) (7,5) (8,5)};
                \draw[<-,thick] (2.4,4) -- (2.9,3.7) node[anchor=west] {3};
                \draw[thick,dotted, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.048) (2,0.47) (3,1.05) (4,1.91) (5,2.93) (6,3.83) (7,4.88) (8,5)};
                \draw[<-,thick] (4.3,2.2) -- (4.9,1.8) node[anchor=west] {4};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} b) $N_{fr}=6$ \end{tabular}
        \begin{tikzpicture}[scale=0.8]
                \label{pic1c}
                \small

                \draw (0,0) -- coordinate (x axis mid) (8,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.019) (1,3.58) (2,4.8) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
                \draw[<-,thick] (1.5,4.3) -- (1.4,3.6) node[anchor=north] {1};
                \draw[thick,dash pattern=on 1pt off 3pt on 3pt off 3pt, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,4.19) (2,4.9) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
                \draw[<-,thick] (1.25,4.3) -- (0.6,4.4) node[anchor=east] {2};
                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,3.9) (2,4.9) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
                \draw[<-,thick] (1.4,4.65) -- (0.8,4.9) node[anchor=east] {3};
                \draw[thick,dotted, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.63) (2,4.31) (3,4.93) (4,4.99) (5,5) (6,5) (7,5) (8,5)};
                \draw[<-,thick] (1.4,2) -- (2,1.5) node[anchor=west] {4};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} c) $N_{fr}=10$ \end{tabular}
        \begin{tikzpicture}[scale=0.8]
                \label{pic1d}
                \small

                \draw (0,0) -- coordinate (x axis mid) (8,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.05) (1,4.45) (2,5) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
                \draw[<-,thick] (0.6,3) -- (0.3,3.6) node[anchor=south] {1};
                \draw[thick,dash pattern=on 1pt off 3pt on 3pt off 3pt, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,4.45) (2,5) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
                \draw[<-,thick] (1.3,4.8) -- (0.6,4.7) node[anchor=east] {2};
                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,4.25) (2,5) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
                \draw[<-,thick] (1.3,4.6) -- (1.8,4.5) node[anchor=west] {3};
                \draw[thick,dotted, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,3.9) (2,4.9) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
                \draw[<-,thick] (1.2,4.2) -- (1.8,3.8) node[anchor=west] {4};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} d) $N_{fr}=15$ \end{tabular}
        \caption{Curves of the near-zero apparent motion detection obtained by the method using Fisher f-criterion (1), substitutional methods with the known variance (2), with external estimations of RMS error 0.15 (3) and RMS error 0.25 (4)}
\end{figure}

Figure 2 ($\alpha = 10^{-3}$) shows the curves of near-zero apparent motion detection obtained by the Fisher f-criterion method (5) with the critical tabulated value $f_{cr}$ of the Fisher distribution statistics with ($w$, $N_{mea} - r$) degrees of freedom \citep[][]{phillips} and the critical value $f_{cr}$ according to the predefined significance level $\alpha$.
\begin{figure}
        \label{pic1_3}
        \centering

        \begin{tikzpicture}[scale=0.8]
                \label{pic2a}
                \small

                \draw (0,0) -- coordinate (x axis mid) (8,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};
                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.012) (2,0.04) (3,0.07) (4,0.12) (5,0.19) (6,0.28) (7,0.92) (8,1.94)};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.012) (1,0.03) (2,0.1) (3,0.17) (4,0.28) (5,0.38) (6,0.63) (7,1.77) (8,3.17)};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} a) $N_{fr}=4$ \end{tabular}
        \begin{tikzpicture}[scale=0.8]
                \label{pic2b}
                \small

                \draw (0,0) -- coordinate (x axis mid) (8,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};
                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.087) (2,0.93) (3,1.9) (4,3) (5,3.99) (6,4.59) (7,4.99) (8,5)};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.015) (1,0.24) (2,1.79) (3,3.07) (4,4.1) (5,4.7) (6,4.9) (7,5) (8,5)};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} b) $N_{fr}=6$ \end{tabular}
        \begin{tikzpicture}[scale=0.8]
                \label{pic2c}
                \small

                \draw (0,0) -- coordinate (x axis mid) (8,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};
                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,2.57) (2,4.7) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.019) (1,3.58) (2,4.8) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} c) $N_{fr}=10$ \end{tabular}
        \begin{tikzpicture}[scale=0.8]
                \label{pic2d}
                \small

                \draw (0,0) -- coordinate (x axis mid) (8,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};
                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,4) (2,4.9) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.05) (1,4.45) (2,5) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5)};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} d) $N_{fr}=15$ \end{tabular}
        \caption{The curves of the near-zero apparent motion detection obtained by the Fisher f-criterion method with the critical tabulated value (solid line) and the critical value according to the predefined significance level $\alpha$ (dashed line)}
\end{figure}

\textbf{A method of in situ modeling for analysis of indicators of quality of the near-zero apparent motion detection on a series of CCD-frames.} In this case, it is impossible to restore the real law of the errors' distribution completely. The method of in situ modeling is, therefore, more appropriate \citep[][]{kuzmyn}.

We compiled the set of objects with practically zero apparent motion in the framework of the CoLiTec project \citep{savanevych,savanevych1} and used it as the internal catalog (IC) of motionless objects in a series of frames \citep[][]{vavilova1}.

It is important to note that the objects exactly from the internal catalog were selected as in situ data. Because the positions of objects from this catalog are fixed, so deviations of their estimated positions from their average value can be regarded as evaluations of their errors. These values can be used in the in situ modeling.

Further, these deviations should be added to the determined values of the object's displacements according to their velocities of the apparent motion. Thereby, it is possible to use the real laws of the positional errors distribution in the study of their motion by the in situ modeling method.

\textbf{In situ data.} Series of CCD-frames from observatories ISON-NM (MPC code - "H15") \citep[][]{molotov} and ISON-Kislovodsk (MPC code - "D00") \citep[][]{kislovodsk} were selected as the in situ data. The ISON-NM observatory is equipped with a 40 cm telescope SANTEL-400AN with CCD-camera FLI ML09000-65 (3056 x 3056 pixels, the pixel size is 12 microns). Exposure time was 150 seconds.

The ISON-Kislovodsk observatory is equipped with a 19.2 cm wide-field telescope GENON (VT-78) with CCD-camera FLI ML09000-65 (4008 x 2672 pixels,the pixel size is 9 microns). Exposure time was 180 seconds. Figures 3 and 4 show the curves of the near-zero apparent motion detection obtained by the Fisher f-criterion (5) and by the substitutional method of maximum likelihood with an external estimation of RMS error (C.13) for two sources of in situ data.
\begin{figure}[H]

        \centering

        \begin{tikzpicture}[scale=0.8]
                \label{pic5a}
                \small

                \draw (0,0) -- coordinate (x axis mid) (10,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$,9/$5\sigma$,10/$10\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};
                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.008) (2,0.01) (3,0.015) (4,0.019) (5,0.026) (6,0.038) (7,0.26) (8,2) (9,4.7) (10,5)};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.01) (1,0.6) (2,1.8) (3,2.2) (4,2.5) (5,2.8) (6,3) (7,3.6) (8,4) (9,4.3) (10,5)};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} a) $\alpha = 10^{-3}$ \end{tabular}
        \begin{tikzpicture}[scale=0.8]
                \label{pic5b}
                \small

                \draw (0,0) -- coordinate (x axis mid) (10,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$,9/$5\sigma$,10/$10\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};
                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.008) (2,0.01) (3,0.015) (4,0.019) (5,0.023) (6,0.025) (7,0.03) (8,0.23) (9,1.25) (10,5)};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.01) (1,0.15) (2,0.7) (3,1) (4,1.4) (5,1.6) (6,1.9) (7,2.7) (8,3.1) (9,3.5) (10,4.7)};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} b) $\alpha = 10^{-4}$ \end{tabular}
        \caption{Curves of the near-zero apparent motion detection with the SANTEL-400AN telescope obtained by the Fisher f-criterion method (solid line) and by the substitutional method with external estimation of RMS error 0.15 (dashed line)}
\end{figure}
\begin{figure}[H]

        \centering

        \begin{tikzpicture}[scale=0.8]
                \label{pic6a}
                \small

                \draw (0,0) -- coordinate (x axis mid) (10,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$,9/$5\sigma$,10/$10\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};

                \draw[thick, dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.01) (2,0.03) (3,0.07) (4,0.1) (5,0.2) (6,0.7) (7,4.2) (8,4.8) (9,4.9) (10,5)};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.01) (1,1.1) (2,1.8) (3,2.0) (4,2.3) (5,2.5) (6,2.7) (7,3.3) (8,3.8) (9,4.2) (10,5)};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} a) $\alpha = 10^{-3}$ \end{tabular}
        \begin{tikzpicture}[scale=0.8]
                \label{pic6b}
                \small

                \draw (0,0) -- coordinate (x axis mid) (10,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$,9/$5\sigma$,10/$10\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};

                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.001) (1,0.002) (2,0.008) (3,0.017) (4,0.03) (5,0.06) (6,0.31) (7,1.37) (8,4.5) (9,4.9) (10,5)};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.002) (1,0.74) (2,1.2) (3,1.4) (4,1.6) (5,1.7) (6,1.9) (7,2.4) (8,2.8) (9,3.6) (10,4.7)};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} b) $\alpha = 10^{-4}$ \end{tabular}

        \caption{Curves of the near-zero apparent motion detection with the GENON (VT-78) telescope obtained by the Fisher f-criterion method (solid line) and by the substitutional method with external estimation of RMS error 0.15 (dashed line)}
\end{figure}

\textbf{Analysis of indicators of quality of the near-zero apparent motion detection in a series of CCD-frames by the method of statistical modeling.} Analyzing different approaches, we can note that the substitutional methods of maximum likelihood detection with known variance of the object's position (C.12) depicted by the curve 2 in Fig. 1, and the methods with external estimation of RMS errors $\hat{\sigma}_{out} = 0.15$ (C.13) represented by the curve 3 in the same figure are the most sensitive to the object velocity changes. For example, CPTD of the near-zero apparent motion for these methods increases in the series consisting of four frames and having $V = 0.5\sigma$. Here, $\sigma$ is an RMS error of the errors of estimated positions. For other methods the velocity module of the apparent motion is not less than $V = 1.25\sigma$, and if $N_{fr} = 6$, not less than $V = \sigma$.

The curve 1 in Fig. 1 demonstrates that the near-zero apparent motion detection method with Fisher f-criterion (5) is not effective enough with the data of statistical modeling, when the number of frames $N_{fr}$ is small. But if $N_{fr}$ is not less than eight, this method is not inferior to other ones by CPTD. In own turn, the substitutional method of maximum likelihood with the known variance of the object's position (C.12) exists only in theory and can not be applied in practice.

Hereby, the substitutional method of maximum likelihood with external estimation of RMS error (C.13) described by curve 3 in Fig. 1 is the most effective and flexible. We remember that the external estimation can be obtained from measurements of the other objects in CCD-frame.

On the other hand, the determination of critical values for all substitutional methods encounters formidable obstacles. First of all, it is not clear how to separate a set of stars (objects with a zero rate motion) from the objects with a near-zero apparent motion to determine them. Also, this process is very time- and resource-consuming and difficult to apply in rapidly changing conditions of observations in modern asteroid surveys.

In statistical modeling, the critical values $f_{cr}$ of the f-criterion determined according to the predefined significance levels are almost equal to the tabulated critical values of Fisher distribution statistics with ($w$, $N_{mea} - r$) degrees of freedom \citep[][]{phillips,melard} of the method (5). It is obviously seen in Fig. 2. Moreover, these figures demonstrate that the similarity of these critical values of decisive statistic does not depend on the number of frames in the series.

Hence, it is not necessary to determine them for the different number of frames $N_{fr}$ and observation conditions. It is enough to use the maximum allowable tabulated value \citep[][]{melard}.

Following from our statistical experiments, we can note that the method for the near-zero apparent motion detection with Fisher f-criterion (5) is more effective for the large number of CCD-frames and the velocity module of the apparent motion $V = 0.5\sigma$ as it's seen in Fig. 2.

\textbf{Analysis of indicators of quality of the near-zero apparent motion detection in a series of CCD-frames by the method of in situ modeling.} It is found that the method for detection of the object's near-zero apparent motion using Fisher f-criterion (5) is the most sensitive to changes in the object's velocity (Fig. 3, 4). As shown earlier, CPTD for this method increases when series includes four frames or more and when $V = 0.5\sigma$. For other methods the velocity module of the apparent motion should be not less than $V = 1.25\sigma$.

In addition, the method of the near-zero apparent motion detection using Fisher f-criterion (5) is stable and does not depend on the kind of telescope (Fig. 5a). Therefore, there is no need to undertake additional steps for determining the critical value of the decisive statistic after the equipment replacement or observational conditions change. Other methods of the apparent motion detection encounter problems when determining the critical values as it is obvious from Fig. 5b.
\begin{figure}

        \centering

        \begin{tikzpicture}[scale=0.8]
                \label{pic7a}
                \small

                \draw (0,0) -- coordinate (x axis mid) (10,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$,9/$5\sigma$,10/$10\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};

                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.01) (1,0.6) (2,1.8) (3,2.2) (4,2.5) (5,2.8) (6,3) (7,3.6) (8,4) (9,4.3) (10,5)};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.01) (1,1.1) (2,1.8) (3,2.0) (4,2.3) (5,2.5) (6,2.7) (7,3.3) (8,3.8) (9,4.2) (10,5)};
        \end{tikzpicture}
        \begin{tabular}{C{5cm}} a) \end{tabular}
        \begin{tikzpicture}[scale=0.8]
                \label{pic7b}
                \small

                \draw (0,0) -- coordinate (x axis mid) (10,0) node[right] {$V$};
                \draw (0,0) -- coordinate (y axis mid) (0,5) node[above=0.1cm] {$D_{true}$};

                \foreach \pos/\label in {0,1/$0.5\sigma$,2/$1\sigma$,3/$1.25\sigma$,4/$1.5\sigma$,5/$1.75\sigma$,
                                                                                6/$2\sigma$,7/$3\sigma$,8/$4\sigma$,9/$5\sigma$,10/$10\sigma$}
                        \draw (\pos,0pt) -- (\pos,-3pt) node [anchor=north]  {\label};
                \foreach \pos/\label in {0,1/0.2,2/0.4,3/0.6,4/0.8,5/1}
                        \draw (0pt,\pos) -- (-3pt,\pos) node [anchor=east]  {\label};

                \draw[thick,dashed, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.008) (2,0.01) (3,0.015) (4,0.019) (5,0.026) (6,0.038) (7,0.26) (8,2) (9,4.7) (10,5)};
                \draw[thick, mark=*] plot [smooth,tension=0.4] coordinates{(0,0.005) (1,0.01) (2,0.03) (3,0.07) (4,0.1) (5,0.2) (6,0.7) (7,4.2) (8,4.8) (9,4.9) (10,5)};

        \end{tikzpicture}
        \begin{tabular}{C{5cm}} b) \end{tabular}

        \caption{Curves of the near-zero apparent motion detection with the GENON (VT-78) (solid line) and SANTEL-400AN (dashed line) telescopes ($\alpha = 10^{-3}$) obtained by the Fisher f-criterion method (a), substitutional method for maximum likelihood detection with external estimation of RMS error (b)}
\end{figure}

\textbf{Examples of objects discovered by the method of near-zero apparent motion detection in a series of CCD-frames using significance criteria of the apparent motion.} There are many of objects with near-zero apparent motion that were detected by the CoLiTec software for automated asteroids and comets discoveries \citep[][]{savanevych1}. The plugin implements the method of detection using the Fisher f-criterion (5). Table 1 gives information about several observatories at which the CoLiTec software is installed.
\begin{table}[H]
        \label{table1}
        \caption{Information about observatories and telescopes at which the CoLiTec software is installed}
        \centering
        \scriptsize
        \begin{tabular}{| L{1.5cm} | C{1.3cm} | C{1.3cm} | C{1.4cm} | C{1.4cm} |}
                \hline
                Observatory & \multicolumn{2}{C{2.6cm} |}{ISON-Uzhgorod Observatory} & Cerro Tololo Inter-American Observatory (CTIO) & ISON-Kislovodsk Observatory \\ \hline
                MPC code & K99 & - & - & D00 \\ \hline
                Telescope & ChV-400 & BRC-250M & Promt8 & Santel-400AN \\ \hline
                Aperture, cm & 40 & 25 & 61 & 40 \\ \hline
                CCD-camera & FLI PL09000 & Apogee \newline Alta U9 & Apogee F42 & FLI ML09000-65 \\ \hline
                Resolution, pix & 3056 x 3056 & 3072 x 2048 & 2048 x 2048 & 3056 x 3056 \\ \hline
                Pixel size, $\mu$m & 12 & 9 & 13.5 & 12 \\ \hline
                Scale, " & 1.42 & 1.46 & 0.66 & 2.06 \\ \hline
        \end{tabular}
\end{table}
The real-life examples of detection of asteroids 1917, 6063, 242211, 3288 and 1980, 20460, 138846, 166 with a near-zero apparent motion are described in Tables 2 and 3 respectively.

The observations were conducted in 2017 in the period from 3 to 19 July with different small telescopes and confirmed an efficiency of the method even in bad conditions (strong backlight from the full Moon).

Tables 2 and 3 contain the following apparent motion parameters of the aforementioned asteroids:
date of observations;
name of telescope;
exposure time during the observation;
apparent velocities of object along each coordinate $\hat{V}_{x}$ and $\hat{V}_{y}$ in the rectangular coordinate system (CS) (see, Appendix C, formulas (C.1), (C.2);
apparent velocities of objects $\hat{V}_{RA}$ and $\hat{V}_{DE}$ in the equatorial CS determined from the observational data;
apparent velocities of object $\hat{V}_{RAcat}$ and $\hat{V}_{DEcat}$ in the equatorial CS determined from the Horizons system \citep[][]{giorgini} for the same times of observation;
velocity module $\hat{V}$ of the apparent motion of object determined from the observational data ($\hat{V}=\sqrt{\hat{V}_{x}^2+\hat{V}_{y}^2}$);
velocity module $\hat{V}_{cat}$ of the apparent motion of object determined from the Horizons system;
average FWHM of object in five frames;
average SNR of object in five frames;
RMS error of stars positional estimates $\hat{\sigma}_{0}$ (C.7) from UCAC4 catalog \citep[][]{zacharias} with SNR approximately equal to the object's SNR;
brightness ${Mag}_{cat}$ of the object determined from the Horizons system;
angular distance between the observed asteroid and the Moon;
phase of the Moon, percentage illumination by the Sun;
coefficient of the velocity module $\hat{V}_{cat}$ of the apparent motion of object determined in relative terms, in other words, RMS error of measurement deviations of the object's position ($k = \hat{V}/\hat{\sigma}_{0}$).

\begin{table}[H]
        \label{table2a}
        \caption{Examples of asteroids 1917, 6063, 242211, 3288 with a near-zero apparent motion that were detected by the proposed method using Fisher f-criterion (5)}
        \centering
        \scriptsize
        \begin{tabular}{| L{2.2cm} | C{1.2cm} | C{1.2cm} | C{1.2cm} | C{1.2cm}  |}
                \hline
                Parameters $\diagdown$ Objects
                & 1917            & 6063          & 242211        & 3288         \\ \hline
                Date of observation
                & 2017-07-11      & 2017-07-11     & 2017-07-13    & 2017-07-19     \\ \hline
                Telescope
                & Promt8         & Promt8        & Promt8        & Promt8         \\ \hline
                Exposure, s
                & 80             & 40            & 40            & 20              \\ \hline
                $\hat{V}_{x}$, pix/fr
                & 0.47           & 0.94          & -0.56         & 0.01           \\ \hline
                $\hat{V}_{y}$, pix/fr
                & -0.47          & 0.73          & 0.36          & -0.47           \\ \hline
                $\hat{V}_{RA}$, "/fr
                & -0.49          & 0.66          & -0.30         & -0.22           \\ \hline
                $\hat{V}_{DE}$, "/fr
                & -0.25          & 0.65          & -0.39         & -0.02            \\ \hline
                $\hat{V}_{RAcat}$, "/fr
                & -0.32          & 0.66          & -0.22         & -0.31           \\ \hline
                $\hat{V}_{DEcat}$, "/fr
                & -0.34          & 0.65          & -0.37         & -0.04           \\ \hline
                $\hat{V}$, pix/fr
                & \textbf{0.66}  & \textbf{1.19} & \textbf{0.67} & \textbf{0.50}    \\ \hline
                $\hat{V}$, "/fr
                & 0.55           & 0.93          & 0.49          & 0.22             \\ \hline
                ${V}_{cat}$, "/fr
                & 0.47           & 0.93          & 0.43          & 0.31              \\ \hline
                Average FWHM, pix
                & 3.48            & 3.68          & 4.62          & 5.70             \\ \hline
                Average SNR, "/fr
                & 6.86            & 10.04         & 12.83         & 11.86             \\ \hline
                $\hat{\sigma}_{0}$, pix (UCAC4)
                & \textbf{0.40}   & \textbf{0.45} & \textbf{0.41} & \textbf{0.30}      \\ \hline
                $\hat{\sigma}_{0}$, "
                & 0.30            & 0.19          & 0.28          & 0.20               \\ \hline
                ${Mag}_{cat}$, $^{m}$
                & 18.2            & 17.38         & 17.17         & 18.24              \\ \hline
                Asteroid-Moon dist., deg
                & 97              & 82.5          & 68            & 91.5               \\ \hline
                Moon phase \%
                & 91              & 91            & 76            & 14                  \\ \hline
                $k=\hat{V} / \hat{\sigma}_{0}$
                & \textbf{1.65}   & \textbf{2.64} & \textbf{1.63} & \textbf{1.67}           \\ \hline
        \end{tabular}
\end{table}

\begin{table}[H]
\label{table2b}
\caption{Examples of asteroids 1980, 20460, 138846, 166 with a near-zero apparent motion that were detected by the proposed method using Fisher f-criterion (5)}
\centering
\scriptsize
\begin{tabular}{| L{2.2cm} | C{1.2cm} | C{1.2cm} | C{1.2cm} | C{1.2cm} |}
        \hline
        Parameters $\diagdown$ Objects
        & 1980                          & 20460           & 138846        & 166\\ \hline
        Date of observation
        & 2017-07-09                    & 2017-07-03       & 2017-07-13    & 2017-07-19 \\ \hline
        Telescope
        & BRC-250M                      & ChV-400         & ChV-400       & ChV-400 \\ \hline
        Exposure, s
        & 30                            & 30              & 60            & 60\\ \hline
        $\hat{V}_{x}$, pix/fr
        & 0.06                          & 0.72            & -0.06         & -0.11 \\ \hline
        $\hat{V}_{y}$, pix/fr
        & 0.37                          & 0.51            & 0.58          & -0.21 \\ \hline
        $\hat{V}_{RA}$, "/fr
        & -0.11                         & -1.09           & 0.07          & 0.19 \\ \hline
        $\hat{V}_{DE}$, "/fr
        & -0.61                         & 0.76            & 1.34          & -0.32 \\ \hline
        $\hat{V}_{RAcat}$, "/fr
        & 0.09                          & -1.06           & 0.13          & 0.14 \\ \hline
        $\hat{V}_{DEcat}$, "/fr
        & -0.52                         & 0.88            & 0.83          & -0.28 \\ \hline
        $\hat{V}$, pix/fr
        & \textbf{0.37}                  & \textbf{0.88}   & \textbf{0.59} & \textbf{0.24} \\ \hline
        $\hat{V}$, "/fr
        & 0.62                          & 1.33            & 1.35          & 0.31 \\ \hline
        ${V}_{cat}$, "/fr
        & 0.53                          & 1.38            & 0.84          & 0.38 \\ \hline
        Average FWHM, pix
        & 3.35                          & 4.59            & 5.12          & 4.92 \\ \hline
        Average SNR, "/fr
        & 10.31                         & 7.76            & 7.26          & 42.14\\ \hline
        $\hat{\sigma}_{0}$, pix (UCAC4)
        & \textbf{0.38}                  & \textbf{0.39}   & \textbf{0.39} & \textbf{0.26} \\ \hline
        $\hat{\sigma}_{0}$, "
        & 0.54                          & 0.62            & 0.57          & 0.36 \\ \hline
        ${Mag}_{cat}$, $^{m}$
        & 15.32                         & 15.91           & 16.56         & 13.71 \\ \hline
        Asteroid-Moon dist., deg
        & 67.5                          & 79.5            & 83.5          & 84 \\ \hline
        Moon phase \%
        & 99                             & 79              & 76            & 14 \\ \hline
        $k=\hat{V} / \hat{\sigma}_{0}$
        & \textbf{0.97}                  & \textbf{2.26}   & \textbf{1.51} & \textbf{0.92} \\ \hline
\end{tabular}
\end{table}

\textbf{Discovery of the sungrazing comet C/2012 S1 (ISON).} On September 21, 2012 the sungrazing comet C/2012 S1 (ISON) was discovered (Fig. 6) at the ISON-Kislovodsk Observatory \citep[][]{kislovodsk} of the International Scientific Optical Network (ISON) project \citep[][]{molotov}, \citep[][]{mpc}. Information about observatory and telescope is available in the Table 1.
\begin{figure}
        \label{pic5}
        \centering
        \includegraphics[width=9cm]{pic1.eps}
        \small
        \begin{tabular}{R{2.2cm} C{3.4cm} L{3cm}}
                a) & & b)
        \end{tabular}
        \caption{Sungrazing comet C/2012 S1 (ISON) at the moment of discovery in the center of crop of CCD-frame with field of view 20 x 20 arcminutes (a), 8 x 8 arcminutes (b)}
\end{figure}
At the moment of discovery, the magnitude of the comet was equal to 18.8${^m}$, and its coma had 10 arc seconds in diameter that corresponds to 50 000 km at a heliocentric distance of 6.75 au. Its apparent motion velocity at the moment of discovery was equal to 0.8 pixels per frame. The size of the comet image in the frame was about five pixels. In Fig. 7a the cell size corresponds to the size of the pixel and is equal to 2~arc seconds. Within 26 minutes of the observation, the image of the comet has been moved by three pixels in the series of 4 CCD-frames (Fig. 7b).

\begin{figure}[H]
        \label{pic6}
        \centering
        \includegraphics[width=9cm]{pic2.eps}
        \small
        \begin{tabular}{R{2.2cm} C{3.4cm} L{3cm}}
                a) & & b)
        \end{tabular}
        \caption{a) Images of C/2012 S1 (ISON) comet on CCD-frames: the image size is five pixels (a), the shift of comet image between the first and the fourth CCD-frames of series is three pixels (b)}
\end{figure}

C/2012 S1 (ISON) comet (Fig. 8) was detected using the CoLiTec software for automated asteroids and comets discoveries \citep[][]{savanevych1} with the implemented method of detection using Fisher f-criterion (5).
\begin{figure}[H]
        \label{pic7}
        \centering
        \includegraphics[width=9cm]{pic3.eps}
        \caption{Sungrazing comet C/2012 S1 (ISON) in a series of four CCD-frames}
\end{figure}
C/2012 S1 (ISON) comet was disintegrated at an extremely small perihelion distance of about 1 million km on the day of perihelion passage, on November 28, 2013. Its disintegration was caused by the Sun's tidal forces and the significant mass loss due to the alterations in the moments of inertia of its nucleus. Despite having a short visible life time for our observations, this comet supplemented our knowledge of cometary astronomy.

\section{Conclusions}
We proposed a computational method for the detection of objects with the near-zero apparent motion on a series of CCD-frames, which is based on the Fisher f-criterion \citep[][]{phillips} instead of using the traditional decision rules that based on the maximum likelihood criterion \citep[][]{myung}.

For the analysis of the indicators of quality of the apparent motion detection, we applied statistical and in situ modeling methods and determined their conditional probabilities of true detection (CPTD) of the near-zero motion on a series of CCD-frames.

The statistical modeling showed that the most effective and adaptive method for the apparent motion detection is the substitutional method of maximum likelihood using the external estimation of RMS errors (C.13) (Fig. 1). But the process of determining the critical values of decisive statistics is very time- and resource-consuming in the rapidly changing observational conditions. By this reason, we recommended to apply the method of the near-zero apparent motion detection for the subclass of objects with $3\sigma$-velocity using Fisher f-criterion (5) for series with the number of frames $N_{fr} = 4$ or more (Fig. 1). The condition of a large number of frames in the series also makes the proposed method not inferior to other methods of apparent motion detection by CPTD.

When studying the indicators of quality of near-zero apparent motion detection by the in situ modeling method the objects from the internal catalog fixed on a series of CCD-frames were used as in situ data. It was found that in the case when the velocity does not exceed 3 RMS errors in object position per frame, the most effective method for near-zero apparent motion detection is the method which uses Fisher f-criterion (Fig. 3, 4). When compared with other methods, this method is stable at the equipment replacement (Fig. 5).

The proposed method for detection of the objects with $3\sigma$-velocity apparent motion using Fisher f-criterion was verified by authors and implemented in the embedded plugin developed in the CoLiTec software for automated discovery of asteroids and comets \citep[][]{savanevych1}.

Among the other objects detected and discovered with this plugin, there was the sungrazing comet C/2012 S1 (ISON) \citep[][]{mpc}. The velocity of the comet apparent motion at the moment of discovery was equal to 0.8 pixels per CCD-frame. Image size of the comet on the frame was about five pixels (Fig. 7a). Within 26 minutes of the observation, the image of the comet had moved by three pixels in the series of four CCD-frames (Fig. 7b). So, it was considered to belong to the subclass of SSOs that have a velocity of apparent motion between CCD-frames not exceeding three RMS errors $\sigma$ of measurements of its position ($\varepsilonup=3\sigma$). In total, about 15\% of SSO objects with $\varepsilonup$-velocity apparent motion in the CCD-frames. These are the objects beyond the Jupiter's orbit as well as asteroids heading straight to the Earth.

\section{Acknowledgments}
The authors thank observatories that have implemented CoLiTec software for observations. We especially thank Vitaly Nevski and Artyom Novichonok for their discovery of ISON comet and others SSOs. We are grateful to the reviewer for their helpful remarks that improved our paper and, in particular, for the suggestion "to add a few real-life examples, where the method provides a detection of motion for an object that would otherwise be difficult to detect". We express our gratitude to Mr. W. Thuillot, coordinator of the Gaia-FUN-SSO network \citep[][]{thuillot}, for the approval of CoLiTec as a well-adapted software to the Gaia-FUN-SSO conditions of observation (\url{https://gaiafunsso.imcce.fr}).

Research is supported by the APVV-15-0458 grant and the VVGS-2016-72608 internal grant of the Faculty of Science, P. J. Safarik University in Kosice (Slovakia).

The CoLiTec software is available on \url{http://neoastrosoft.com}.

redpajama_set_name: RedPajamaArXiv


The white airplane called Roc, which has a wingspan the length of an American football field and is powered by six engines on a twin fuselage, took to the air shortly before 7 a.m. Pacific time.
\"They say they wanted to hit some terrorists. What terrorists can you see here?\" said 62-year-old Nooran Shah, a resident of Jaba village, near the northeastern town of Balakot in Pakistan's Khyber Pakhtunkhwa province.
The bitter truth for Islamabad is that it simply cannot afford to take on the might of the Indian armed forces.
Indian Air Force has carried out surprise air strikes on militant camps in Pakistan-occupied Kashmir. Twelve Mirage 2000 jets struck Jaish-e-Muhammed (JeM) terror launch pads at around 3:30 am early on Tuesday.


In the History of Arts from Romania, Late Antiquity is absent and this is not justified. We are trying to find an individual place for this period of transition, for it is wellrepresented, both regarding its architecture and its artifacts (sculpture, painting, mosaic, minor arts), they all record the passing from the ancient vision to the Paleochristian one. One of the aspects of the architecture of this period is treated by the present research which is focused on military architecture from the two provinces on the territory of Romania: Dacia Traiana and Scythia Minor (today's Dobrogea).
Focuses on urban landscape painting in California. Influence of the 1930 Great Depression on the style of painting; Impact of cars on the picture of urban scene; Depictions of roadway culture; Importance of improvisation in expressive abstract painting.
An introduction to the reviews section of the issue is presented in which the reviews editor discusses articles in the section on topics related to art history as a discipline since 1913 and a publication on Dutch painting.
Discusses the themes that have influenced artistic works from ancient China. Influence of the Song Dynasty on the artistic works of ancient China; Shift from figure to landscape painting during the Northern Song rule; Interest of the last Northern Song emperor Huizong in arts.
Describes a mud glyph cave art site discovered in northern Alabama. Formation processes; Artifacts and chronology; Significance of the discovery of mud glyphs in 19th Unnamed Cave in Alabama.
The author writes about his experiences in this personal narrative about metal detecting, the artefacts he found, including thimbles, buttons, badges, jar lids, bronze tokens, and pennies.


Only hours sooner, a leak from Indonesia tipped that the Samsung Galaxy S10 series will go up for pre-orders the exceptionally following day after the launch i.e., on February 21. Presently, photographs of the Samsung Galaxy S10 series have leaked once more, corroborating past leaks, with supposed sham units appearing to concrete final plan details. Besides, the perplexity surrounding the less expensive variant of the Samsung Galaxy S10 has purportedly been finally put to rest, and the model is currently said to be called the Samsung Galaxy S10e. The naming was tipped by leaked images and another NBTC certification in Thailand. In conclusion, the camera details of all the three variant have likewise been tipped.
Starting with the leaked photographs Samsung Galaxy S10 and Samsung Galaxy S10+ asserted sham units of the couple have been snapped and are presently on SlashLeaks. These photographs are in line with every single past leak showing off the Infinity-O Display panel, the triple rear camera set up at the back, the USB Type-C port, 3.5mm sound jack, and the speaker grille at the bottom edge. The Samsung Galaxy S10+ sports a somewhat bigger display and accompanies dual front camera sensors too.
Independently, tipster Evleaks has likewise distributed another photograph showing off a part of the Samsung Galaxy S10. The triple rear camera setup can be seen, and the picture likewise uncovers that the phone will arrive in a Glossy White finish too.
Photographs of the foreseen less expensive model of the Samsung Galaxy S10 have additionally been leaked on Weibo. The gadget is booted up, and the words Samsung Galaxy S10e are seen flashing on the screen, tipping its name. This recommends the variant will be called the Samsung Galaxy S10e and not Samsung Galaxy S10 Lite or Samsung Galaxy S10E. The photographs additionally show how the screen will look with the Infinity-O Display and the selfie camera set inside a cut-out on the top right edge of the screen.
Independently, NBTC certification office has likewise confirmed the SM-G970F/DS, SM-G973F/DS, and SM-G975F/DS models and is calling it the Samsung Galaxy S10e, Samsung Galaxy S10, and Samsung Galaxy S10+ separately. This further bonds the moniker for the least expensive variant, finally putting a conclusion to all the disarray. These listings were first spotted by MySmartPrice.
There's additionally a different report from Chinese media new.qq.com that recommends that the Samsung Galaxy S10e will have same front and back camera specs as the Samsung Galaxy Note 9. In any case, Samsung is said to upgrade its computational photography calculations on the Galaxy S10e. The report likewise expresses that the two premium variants will accompany front camera support for 4K video and optical picture adjustment (OIS).
The Samsung Galaxy S10 and Samsung Galaxy S10+ are likewise tipped to sport a triple camera setup, and the report recommends that the specs will be - one 12-megapixel main sensor, another 16-megapixel wide-edge sensor, and a last 13-megapixel fax sensor. At the back, the two phones will support GoPro-like electronic picture adjustment (EIS).
The Samsung Galaxy S10e is required to sport a 5.8-inch display, a dual rear camera setup, and a side-mounted fingerprint scanner. The vanilla Samsung Galaxy S10 variant is relied upon to sport a 6.1-inch display and the Galaxy S10+ is required to sport a huge 6.4-inch display individually. It's solitary a matter of few days before Samsung uncovers all the official details, and we'll be at the event offering detailed inclusion. Stay tuned.


I'm praying for each of you this week as you are spending daily time, with God .
I hope each of you have taken so much treasure from our Scripture reading this week. Many of you may have come to salvation by reading Romans. Many times it is said to be the "Romans Road", because it tells us simply, how to accept Jesus as our Savior through faith. Romans is Paul's letter to Rome, sharing with them the Good News of Christ. Today, I simply want to share the verses that stood out to me.
Chapter four shares how Abraham, the founder of the Jewish nation, was counted as righteous by God all because of his faith, not his good deeds. Abraham was a sinner, too. Go back to Genesis and read over how many mistakes he made. Yet, because of His faith, God counted him as righteous.
Chapter 5 reminds us of the hope we have, now that we have been made right with God through our faith in Jesus Christ. Because of our faith, we can rejoice in trials. Our trials will develop endurance, our endurance will develop our character, and our character strengthens our hope of salvation.
What verses spoke straight to your heart this week?
What is God opening your eyes, heart and mind to?
I'm praying for each of you, as we continue our journey through Scripture.
Today we begin our Fall Bible Study. I'm looking forward to digging into God's Word with each of you! The purpose of our study here at Redeeming Elegance, is to encourage and support Christian women in their daily walk with Christ. Specifically, reading the Bible every day. I am a wife and a homeschool mother of (almost) six children. I understand how busy our lives can be and how hard it is to find time to read Scripture and be still before God. That is why Redeeming Elegance exists.
Our Bible studies are easy to follow. We read through the Bible, one book at a time, one chapter per day. Each week I will post a reading plan of 5 chapters, leaving the weekends open for you to catch up if needed. I hope to post through the week verses and things that stick out to me and I would love for each of you to comment on how God is speaking to you through His Word. Although our studies here are intended for individual reading, it is also very easy to adapt to your Ladies Small Group if you want to read through the Bible, one book at a time.
I am praying for each of you this week as you commit yourself to reading the Bible each day.
1. Be more supportive and encouraging to my Hubby.
2. Be more intentional and encouraging to my children.
3. Be encouraging to other wives and moms.
I wish I could say that I have reached each of these goals 100%. But sadly, I'm a work in progress and this is something I will forever be working on. Life is hard for everyone. Most of us take on way more than we should, some of us have been handed with unexpected trials, all of us have been hurt by others, and everyone of us have the every day "musts" that we have to contend with. Wouldn't a little bit of encouragement help ease our worries, stresses and anxieties?
Like I said, I'm still working on this. I try to remind myself everyday to be more encouraging. I don't always succeed, but I am trying. God gives us many verses on encouragement. I wanted to share a few that really stood out to me.
I'm going to stick with my goals. I'm going to try harder. I want to always remember, I know I will slip up, but my prayer is that God will help me to become an "Encourager".
*I want to build others up.
*I want to put others first and think about their interests instead of my own.
*I want to be a friend who supports, encourages and sharpens my friends.
*I want my words to be good and helpful to all those who hear them.
*In any way I can, I want to share other's burdens. I want to pray for them and help them in whatever need they may have.
*I want to be a motivator.
*I want to make time for others, starting with my husband and children and those God has placed in my life.
*I want to help others stand firm, when they feel like their feet are being taken out from under them.
This is my prayer. However, God wants to use me, I am His to use.
Lord, help me to be encouraging to my Sweet Hubby, my Precious Children, and my family and friends who You have placed in my life. Let my words, my actions, my reactions and my attitude always reflect Your love to those who need a bit of encouragement. And when I fail, pick me up and give me the strength to try again. Make me an Encourager! In Jesus' name I pray, amen.
Reach out and encourage your people, today!
Today we are supposed to begin our Fall Study, but due to a personal time constraint, I am pushing it off one more week. So, our Fall Bible Study through the books of Romans and Exodus, will begin next Monday, September 18, 2017. This is a great time of year to start a new study as we get more established in our fall school, work and holiday schedules. Please share this with all the ladies you know and also feel free to use this study in your own women's small group.
We are in the midst of our first few weeks of our homeschool year over here, so my days have been full of teaching, training and all the normal "keep the household together" routines. I look forward to studying God's Word with you over the next few months!


text:
Q: Ridge Regression in R: MASS implementation vs User Defined returning different coefficients I am currently having trouble with the results of a user defined ridge regression function (one that I have created) against that of the MASS::lm.ridge() function. Below is what is defined as my data:
x <- seq(-0.5, 0.5, 0.1)
b <- sin(2 * pi * x) - 0.1 + 0.2 * runif(11)
xb <- dataframe(x,b)

order <- 9
orders <- 1:order
X9 <- t(t(replicate(order,x))^orders)

The code above is describing my data, with x as the independent variable, b as the dependent, and X9 as my data used as ninth order polynomial matrix.
#MODEL USING MASS
lm_regularized_coef <- coef(lm.ridge(formula = b ~ poly(scale(x), degree = 9, raw = T), data = xb,  lambda = 0.001))
coef_fit <- as.matrix((lm_regularized_coef))

# USER DEFINED FUNCTION
ridge_line_fit <- function (x, b, lambda) {
    X = as.matrix(cbind(1,x))
    inv = solve((t(X) %*% X) + (lambda * diag(dim(t(X) %*% X)[2])))  #(X^T * X + lambda*I)^-1
    weights = inv  %*% t(X) %*% as.matrix(b) #(X^T * X + lambda*I)^-1 * X^T * y
    return(weights)
}

xs <- scale(X9) #For Scaling
rlf <- ridge_line_fit(x = xs, b = data.matrix(b), lambda = 0.001)

cbind(coef_fit, rlf)

which gives me the results below:
                                            [,1]         [,2]
                                     -0.02513236 -0.004314105
poly(scale(x), degree = 9, raw = T)1  2.00391977  1.999360461
poly(scale(x), degree = 9, raw = T)2  0.02474866  0.017679371
poly(scale(x), degree = 9, raw = T)3 -1.19300070 -2.073829478
poly(scale(x), degree = 9, raw = T)4 -0.11014914 -0.200522427
poly(scale(x), degree = 9, raw = T)5  0.02744687  0.086002817
poly(scale(x), degree = 9, raw = T)6  0.09376673  0.396663975
poly(scale(x), degree = 9, raw = T)7  0.08204332  0.633710744
poly(scale(x), degree = 9, raw = T)8 -0.01781305 -0.170140701
poly(scale(x), degree = 9, raw = T)9 -0.01465847 -0.237401993

The left side are the coefficients from the MASS::lm.ridge() function, and the coefficients on the right are from the user defined function.  I really cannot see why the two would be off. Wondering if someone could see if I may have an underlying issue that I may have missed.
Thanks

redpajama_set_name: RedPajamaStackExchange


Rent any motorhome and pick up between September 14 and October 31, 2016 for four nights or more and get 15% off the nightly rate. This also includes 50 free miles a night!
Wherever your imagination can take you, you can be there! There's no shortage of destinations, from the myriad music festivals to the Albuquerque International Balloon Festival from October 1 - 9! And, now is a perfect time to plan a leaf-peeping trip!
Use code AUTUMN when booking on our website, or call a reservation specialist and refer to this code.
But don't wait... availability will go fast!
El Monte RV wishes you and your family a very happy Labor Day! Get away and have some fun. Make time to take a long weekend and spend some time at a favorite beach, fishing hole, or mountain retreat.
Football Season Kicks Off - Make Plans Now for Your Great Tailgating Party!
Grab your friends, cheer for your favorite teams and make memories that will last a lifetime.
We make having fun easy with nationwide locations and a wide variety of motorhomes from which to choose. Don't wait to make your plans!
Here are the NASCAR races in September you won't want to miss!
The Bojangles Southern 500 NASCAR race at the Darlington Raceway on September 4!
The Federated Auto Parts 400 NASCAR race at the Richmond Int'l Raceway on September 10!
The September Chicagoland NASCAR race at the Chicagoland Speedway on September 18!
The New England 300 NASCAR race at the New Hampshire Motor Speedway on September 28!
Arrowhead RV Resort is an incredible camping retreat located on 188 beautifully wooded acres just minutes from the heart of the Wisconsin Dells. They are the perfect home base if you're looking to splash your way through the many water parks that have made the Dells famous.
At this Wisconsin Dells RV resort they're close to hiking trails around Devil's Lake, multiple golf courses and lots of shopping!
Spacious sites easily accommodate RVs and each have a picnic table and fire ring.
Amenities include swimming pool, mini golf, horseshoes, basketball, tennis court, billiards, volleyball, and more! Check out this Youtube video on the campground.
The friendly staff at Arrowhead RV Campground are ready to help you in any way they can. So check out this beautiful RV resort in Wisconsin, and have the trip of a lifetime! You'll be glad you did!
Taking an excursion to Apostle Islands National Lakeshore is worth the trip from your stay at Arrowhead RV Campground.
For the adventurous of heart, these islands in Lake Superior, off the coast of Wisconsin are covered with sea caves and lighthouses that beckon the explorer in you. Apostle Islands Cruises has several different tours and shuttles to help you quench your curiosity.
Check out the nps.gov page on this national lakeshore and make your plans now to visit before winter sets in! We have a rental location in Chicago that makes planning for this trip a breeze.
~ Mix together the butter, flour, oats, brown sugar, baking powder and salt.
~ Press half the oat mixture into the prepared pan.
~ Spread with the strawberry preserves.
~ Sprinkle the other half of the oat mixture over the top and pat lightly.
~ Bake until light brown, 30 to 40 minutes.
~ Let cool completely, and then cut into squares.
This best selling Class C model sleeps up to six, comes with stove, refrigerator/freezer, microwave, generator and so much more. Looks like new! Includes warranty.


Born in Jerusalem on 4 June 1948; received a BA in Commerce from Cairo University (1971) and an MA in Association Management (2003), American Society of Association Management, Washington DC (2003) and Faculty member of the International School of Association Management; owner and manager of Technical Supplies Company in Ramallah from 1971-83; Commercial Manager, Al-Bireh Trading Co., 1975-80; Director of the Tenders and Supplies Dept. at Birzeit University from 1982-96; representative of 32 main international publishers in Palestine and Jordan and promoting academic textbooks and journals, 1996-1999; Director of the Arab Chamber of Commerce and Industry since Jan 1999; contributes in activating cultural life in East Jerusalem through studies and working with various organizations, incl. the Palestinian National Theater, the Arab Thought Forum, the Jerusalem Society for Welfare and Development, and the Jerusalem Center for Social & Economic Rights; works as a columnist, contributing to Al-Quds newspaper and other Palestinian magazines; his work as playwright includes two TV series Khirbet Mabroke and Bab El-Silseleh, Palestine TV (awaiting for production finance since 2004); publishes satirical writings related to the daily life of the Palestinians, incl. Mish Heik Ya Balad (Arabic, 1999) and Tar El-Hamam (Pigeons Flew) (Arabic, 2004).


Inbound Marketing is method of promoting your business that has dramatically transformed the way people consume information in the market place. With an inbound approach, the idea is to provide useful content that draws people in.
Inbound Marketing is broken out into four stages: Attract, Convert, Close, and Delight.
To attract new visitors you need to focus on developing compelling content that will attract the people that are most likely to benefit from what you have to offer. Once people start to find your site you can trade premium content in the form of helpful tools, spreadsheets, code samples, or eBooks, so your visitors are benefiting from something you worked hard to create. The more helpful the content you provide, the more likely the people you want to attract to your business will find you.
Once you've attracted people to your content and demonstrated value, the next step is to convert that mysterious visitor into an identifiable person (a Contact or a Lead). To do this, you must create compelling content that your unnamed visitor is willing to lift the veil of anonymity and trade you a precious piece of personal data (like a name or email address) to get whatever it is you're offering up.
As you get to know each other better, your contact becomes more and more comfortable trading more of their personal details and you're better able to put more relevant content in front of your visitor, and everyone benefits from the relationship.
Once you've attracted people to your content, and converted them into a lead, the next stage is to use all of the information you've collected to know when it's time to get the right product in front of the right person so you can close the deal and make the sale.
Inbound does not end when you've made the sale. The final stage is to delight your customer base and turn them into ambassadors for your products and services. You want to get your customers to openly share their stories and spread the word of how amazing you are to any of their friends and colleagues that would be interested in what you have to offer.
There are many tools, platforms, and software packages you can use to help move your marketing to a more inbound approach, but whatever technology stack you choose, it's important to keep in mind that you should be using the inbound marketing methodology to gently guide your visitors through the buying process. You should do this in a way that provides them with value at every step. Tha's the heart of Inbound Marketing.


Plasmonic nanostructures locally increase the electric field and are also themselves highly optically nonlinear, so that processes such as the generation of the second (SHG) or third harmonic (THG) are very efficient. The interaction of fundamental field distribution, plasmonic resonance, nonlinear polarization and finally nonlinear emission is quite complex. In special cases, this can be observed despite the limited resolution of a far-field microscope (D. Wolf et al., Nat.Comm 2016). Within the framework of a DFG project, however, we are looking for structures that enable well-controlled plasmonic near-fields, for example, in order to excite molecules at specific positions on the nanometer scale. Therefore, we want to image non-linear plasmonic near-fields in a near-field microscope consisting essentially of a scattering AFM tip as a probe. In an iterative process of simulation, production and measurement, a structured light source is to be developed that works below the diffraction limit.


text:
Population exposure to smoking and tobacco branding in the UK reality show 'Love Island'
Alexander B Barker1,
Magdalena Opazo Breton1,
Jo Cranwell2,
John Britton1,
http://orcid.org/0000-0001-5477-2557Rachael L Murray1
1 Division of Epidemiology and Public Health, UK Centre for Tobacco and Alcohol Studies, Clinical Sciences Building, City Hospital, University of Nottingham, Nottingham, UK
2 Department for Health, UK Centre for Tobacco and Alcohol Studies, University of Bath, Bath, UK
Correspondence to Dr Rachael L Murray, Division of Epidemiology and Public Health, UK Centre for Tobacco and Alcohol Studies, University of Nottingham, Clinical Sciences Building, City Hospital, Nottingham NG5 1PB, UK; rachael.murray{at}nottingham.ac.uk
Background Reality television shows are popular with children and young adults; inclusion of tobacco imagery in these programmes is likely to cause smoking in these groups. Series 3 of the UK reality show Love Island, broadcast in 2017, attracted widespread media criticism for high levels of smoking depicted. We have quantified this tobacco content and estimated the UK population exposure to generic and branded tobacco imagery generated by the show.
Methods We used 1-min interval coding to quantify actual or implied tobacco use, tobacco paraphernalia or branding, in alternate episodes of series 3 of Love Island, and Census data and viewing figures from Kantar Media to estimate gross and per capita tobacco impressions.
Results We coded 21 episodes comprising 1001 min of content. Tobacco imagery occurred in 204 (20%) intervals; the frequency of appearances fell significantly after media criticism. An identifiable cigarette brand, Lucky Strike Double Click, appeared in 16 intervals. The 21 episodes delivered an estimated 559 million gross tobacco impressions to the UK population, predominantly to women, including 47 million to children aged <16 and 44 million gross impressions of Lucky Strike branding, including 4 million to children <16.
Conclusion Despite advertising legislation and broadcasting regulations intended to protect children from smoking imagery in UK television, series 3 of Love Island delivered millions of general and branded tobacco impressions both to children and adults in the UK. More stringent controls on tobacco content in television programmes are urgently needed.
priority/special populations
http://dx.doi.org/10.1136/tobaccocontrol-2017-054125
Contributors ABB led coding of data, supported the analysis of data and contributed to drafting the initial manuscript. MOB led the analysis of data. JC and JB contributed to drafting the manuscript. RM conceived the study and contributed to drafting the initial manuscript. All authors read and approved the final manuscript.
Funding This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.
Data sharing statement Data are available from the authors on request.
Correction notice This article has been corrected since it published Online First. The author name 'Joanne Cranwell' has been corrected to 'Jo Cranwell'.

redpajama_set_name: RedPajamaCommonCrawl


When I read Al Gore's latest speech on global warming, my reaction was much like my initial response to that New Yorker cover (see previous post): What am I supposed to make of this?
The call to produce "100 percent of our electricity from renewable energy and truly clean carbon-free sources within 10 years" is off the charts. To blandly claim that this is both "achievable" and "affordable" is a typical Gore touch--as is the hyperbole about the end of life as we know it if we fail to do as he advises. Gore says, "The leading experts predict that we have less than 10 years to make dramatic changes in our global warming pollution lest we lose our ability to ever recover from this environmental crisis." Well, among other things, that depends what you mean by "dramatic"; so far as am I aware, nobody else is saying, "eliminate carbon from the US electricity supply by 2018 or we are doomed."
Gore is right, however, that meeting his target would be "transformative". That is why the inevitable invocation of Kennedy's moon-shot commitment is ill-conceived. Putting a man on the moon within nine years of getting the first American into space was self-evidently a staggering accomplishment. But unlike what Gore is calling for, it did not represent "a challenge to all Americans - in every walk of life: to our political leaders, entrepreneurs, innovators, engineers, and to every citizen."
I agree with Gore about some things. I agree with his preference for a carbon tax over other carbon control regimes. And history has been on the side of technology optimists. At today's energy prices, progress on new technologies for conservation and renewables will probably happen much faster than we think. (See this for instance.) But eliminating carbon from electricity within 10 years? Does he even mean it? "I see my role as enlarging the political space in which Senator Obama or Senator McCain can confront this issue as president next year," he says. Translation: I advocate the impossible so that the possible becomes more probable. Fair enough, one might say. But propaganda in a good cause is still propaganda, isn't it?


Whether you are a connoisseur of fine wine and food or you just looking for a good day out with your friends or partner, The Wine Show Jo'burg should be greenlit for your diary.
Patrons get the chance to sample and buy wine from over 130 different estates and the event is also is there to heighten patron's knowledge of wines. There will also be delectable food on sale.
South Africa's marquee wine even has returned much to the delight of many. Last year's event had a turnout of approximately 9000 patrons and sales volumes exceeded R2, 7million. If that is anything to go by, this year's event is going to be a cracker.
Tickets for The Wine Show Jo'burg 2012 are available from Computicket at R100 per person and an additional R20 is payable at the door for access to the WineExtra and IWSC Gold Lounge. Weekend passes cost R180 per person and special group rates of R90 per person apply for bookings of ten or more people. For more information call (021) 888 8803.