text:
\section{Introduction} \label{sec:intro}
The 21-cm hyperfine transition of neutral hydrogen (HI) is a prime tracer of the HI gas in galaxies as well as the inter-galactic medium at high redshifts. Observations at the radio frequencies of this line (at or below $1420$ MHz) are free from dust absorption and have high redshift resolutions due to standard digital processing techniques from radio telescopes. Given the abundance of HI gas in the Universe, observations with the 21-cm line can be a powerful probe of the structure and evolution of the Universe.
However, the inherent weakness of the 21-cm signal has restricted HI galaxy surveys to the nearby Universe (redshifts $\lesssim 0.1$; e.g. \citet{Jones_2018}). The upcoming generation of radio telescopes will push this to higher redshifts but only over small areas and large HI masses \citep{Maddox_2021}. The intensity mapping (IM) technique circumvents this sensitivity issue by averaging over the collective emission from many unresolved galaxies and producing low angular resolution maps of the HI intensity (albeit still with redshift resolutions comparable to galaxy spectroscopic surveys). The possibilities
opened by this new observational window promise a breakthrough in constraining cosmological theories in the very near future \citep{Bharadwaj_Sethi_2001, 2004MNRAS.355.1339B,2010ARA&A..48..127M, Bull_2015}.
However, the continuous mapping of the skies requires separating the 21-cm line from other signals at a given frequency.
This is particularly challenging due to the presence of strong radio foregrounds \citep{Wolz_2014, Olivari_2016, Cunnington_2019, Liu_2020}.
So far, measurements have been hampered by the extraordinary calibration challenges and detections have only been achieved through cross-correlations with galaxy surveys since the systematics in the  HI maps are not expected to correlate with data at other wavelengths. The first cross-correlation detection was obtained with the single-dish Green Bank Telescope (GBT)\citep{Chang_2010,Masui_2013,Wolz_2022}, followed by measurements with the Parkes telescope \citep{Anderson_2018}. More recently, the CHIME interferometer made a stacking detection using this technique over a wide area \citep{CHIME_2022} (but see also \citet{Chowdhury_2020}).

The recently built MeerKAT 64-dish telescope array in South Africa has also been proposed as an exquisite survey machine for HI IM \citep{Santos_2017} and acts as a precursor for the upcoming SKA Observatory \citep{Santos_2015,SKA_2018}.
For IM surveys, the array is used in single dish mode in order to access the large cosmological scales and pilot tests have been underway \citep{Wang_2021}. A detection of the HI power spectrum in cross-correlation with galaxies from the WiggleZ Dark Energy Survey has been made \citep{2023MNRAS.518.6262C}. However, the large primary beam limits the minimum accessible scales to tens of Mpc and interferometric IM observations can provide crucial information on smaller scales and the nonlinear regime
The core heavy dish distribution of the MeerKAT interferometer can measure scales up to $60$ Mpc at $z<0.65$ in the $L$-band frequency range (856 MHz $< \nu <$ 1712 MHz) and up to $165$ Mpc at $z<1.4$ in the $UHF$-band (580 MHz $< \nu <$ 1015 MHz).
{Forecasts in \citet{Paul_2021} predict that a statistical detection of the HI IM power spectrum is possible with $\sim 100$ hrs of observation with MeerKAT. Using surveys such as the MeerKAT International GHz Tiered Extragalactic Exploration (MIGHTEE) Survey \citep{MIGHTEE}, we can detect the HI power spectrum in narrow redshift bins across a range of redshifts, opening up a new window for probing the astrophysics of the HI galaxies \citep{Chen_2021}.}

In this work, we report the first detection of the HI intensity auto-power spectrum in the interferometric IM mode at frequencies of $986$ MHz ($z \approx 0.44$) and $1077.5$ MHz ($z \approx 0.32$) by analysing $\sim 96$ hrs of MeerKAT data. Observations were carried out with the $L$-band configuration and 4,000 frequency channels with a resolution of $209$ KHz, during the commissioning process of MeerKAT in 2018. The total 96 hrs of data spanned through 9 separate observing sessions all targeted one single pointing corresponding to a sky area of $\sim 2$ deg$^2$ at 1.0 GHz. The J2000 center position of this field at $\alpha=04^{\rm{h}}13^{\rm{m}}26.4^{\rm{s}}, \delta=\ang{-80;00;00}$ in the Southern Hemisphere was specifically selected to minimize artifacts caused by bright sources, making it a prime candidate for the IM technique. We processed the raw data from scratch, flagging radio frequency interference (RFI) and doing full polarization calibration using the processMeerKAT \citep{processmeerkat} software pipeline. We then focused our analysis on two RFI free frequency ranges centered at $986$ and $1077.5$ MHz with 220 frequency channels each, equalling a frequency width of $\approx 46$ MHz. The smaller bandwidth segments are chosen to minimize the effects of cosmological evolution ($\Delta z \sim 0.06$).

\section{Detection methodology} \label{sec:method}
{In this section we present the formalism to compute the HI power spectrum, which acts as a probe of the underlying dark matter distribution and the biased relation to the HI.}
We estimate the power spectrum using the `delay spectrum' approach \citep{Morales_2004, Mcquinn_2006, Parsons_2012a, Parsons_2014, Paul_2016} with our custom-built pipeline \citep{Paul_2021}: instead of producing images and HI cubes, we work directly with the interferometer visibilities, $V(\bm{b}, t, \nu)$, which records the spatial correlation of the electric field for each antenna-pair (denoted by the baseline vector $\bm{b}$). In the limit of small sky areas, $V(\bm{b}, t, \nu)$ measures the 2-d Fourier transform of the sky at each frequency $\nu$ and time $t$. The ``angular" Fourier mode is related to the baseline vector by $\bm{k_\perp} = \frac{2\pi \bm{b}}{\lambda x}$, where $\nu_{21}$ is the rest frame frequency of the 21-cm line, $H(z)$ is the Hubble parameter at redshift $z$ ($1+z=\nu_{21}/\nu$) and $x$ is the comoving distance to that redshift.
Since we are observing the same sky patch, visibilities at a given frequency with the same $\bm{k_\perp}$ should have the same value at any given time (except for noise and RFI). We, therefore, average them in a 2-d $\bm{k_\perp}$ grid, with a bin width chosen to avoid decorrelation given the primary beam size. {The primary beam convolves with the sky signal in Fourier space, mixing different $k_\perp$-modes. In order to minimise the effects of mode-mixing by the primary beam, we choose a large grid size, $\Delta u = 60 \lambda$, corresponding to $\Delta k_\perp \sim 0.3\,{\rm Mpc^{-1}}$, so that the width of the primary beam in $k_\perp$-space is negligible.} This bin size is kept constant across the frequency since, as mentioned above, the redshift evolution within each of the small bands analyzed is negligible. Finally, we Fourier transform the visibilities along the frequency axis (called the delay transform), resulting in the delay-space ($\tau$) visibility function $\tilde{V}(\bm{k_\perp}, \tau)$. In the limit of small bandwidths, this delay-space can be related to the line of sight Fourier mode, $k_\parallel$, through $k_\parallel = \frac{2\pi\tau\nu_{21}H(z)}{c(1+z)^2}$. This creates the 3-d Fourier cube $\bm{k}=(\bm{k_\perp}, k_\parallel)$ and the 3-d power spectrum \citep[`delay spectrum'--][]{Morales_2004, Mcquinn_2006, Parsons_2012a, Parsons_2014, Paul_2016}:
\begin{linenomath*}
\begin{equation}
P_{\rm{D}}(\bm{k_\perp}, k_\parallel) \equiv  \frac{x^2 y}{\Omega_{\rm ps}B} \left(\frac{\lambda^2}{2k_B}\right)^2 {\rm Re}\{\tilde{V}_1(\bm{b},\tau)\tilde{V}_2^{*}(\bm{b},\tau)\},
\label{PS_eq}
\end{equation}
\end{linenomath*}
with $\lambda$ denoting the wavelength at the band-center, $k_B$ the Boltzmann constant and $y$ the comoving depth corresponding to the bandwidth $B$; the visibility functions $\tilde{V}_1$ and $\tilde{V}_2$ correspond to different times and we are taking the real part (${\rm Re}\{\}$) of the product (such cross product will remove the noise bias on average). {$\Omega_{\rm ps}$ denotes the power-squared primary beam area, i.e. the solid angle integral of the primary beam squared \citep{2005ApJ...625..575S,Parsons_2014}},
\begin{equation}
    \Omega_{\rm ps} = \int {\rm d}l\,{\rm d}m\, A^2(l,m),
\end{equation}
where $A(l,m)$ is the primary beam attenuation.
The power-squared primary beam area is calculated using the \textsc{eidos} MeerKAT beam model \citep{2021MNRAS.502.2970A}.

\subsection{Cosmological parameters}
We use the flat $\Lambda$CDM cosmological parameters $[\Omega_m, \Omega_b, h, n_s, \sigma_8] = [0.311, 0.049, 0.677, 0.967, 0.8102]$ from Planck 2018 results \citep{Planck_2018} throughout this work.

\subsection{MeerKAT observations}
Situated in the Karoo region in South Africa, the MeerKAT telescope consists of 64 dish antennas of $13.5$ m diameter. The telescope array has a dense core of $48$ antennas placed within a $1$ km diameter, and another $16$ antennas are mounted at larger distances up to a 4 km radius from the center. This array layout leads to a high number of short baselines which leads to higher sensitivity at low $\bm{k_\perp}$, i.e. the larger cosmological scales.

The observed field J2000, located at $\alpha=04^{\rm{h}}13^{\rm{m}}26.4^{\rm{s}}, \delta=\ang{-80;00;00}$, contains a limited amount of bright radio foreground sources. It was previously studied to produce the deepest radio source count to date using the P(D) technique, revealing the bulk of the star formation history of the universe \citep{Mauch_2020} and therefore has been chosen for the analysis presented in this paper. The observations were carried out with the $L$-band configuration and the data used in this work is a subset of the publicly available data observed under the Proposal ID: SCI-20180426-TM-01 \citep{Mauch_2020}. The total 96 hrs of data spanned through 9 separate observing sessions and all of them utilize at least 58 antennas. Each scan on the target field lasted for 15 minutes followed by a 2 minutes observation of the secondary (e.g. phase) calibrator $\rm{PKS}\, \rm{J}0252{\text -}7104$. The primary flux and bandpass calibrator $\rm{PKS}\, \rm{B}1934{\text -}638$ was observed for 10 minutes at the beginning of each session and after that, in regular intervals of 3 hours. The flagging of unwanted radio frequency interference (RFI) from terrestrial sources and subsequent calibration of the uncalibrated data were performed with the processMeerKAT \citep{processmeerkat} software which is built for calibrating MeerKAT interferometer data. The processMeerKAT uses Common Astronomy Software Applications (CASA, \citealt{CASA}) package routines for RFI removal, calculations of phase and flux gains from the reference calibrator observations. In particular, it uses $\rm{PKS}\, \rm{B}1934{\text -}638$ to estimate the delay and bandpass solutions which are applied on $\rm{PKS}\, \rm{J}0252{\text -}7104$ data to calculate the time-dependent complex gains. The final gain corrections are applied to the target data to perform a full-polarization calibration. Following this, we perform three rounds of phase only self-calibration to 1 of the 9 observing sessions datasets with a solution interval of $60$s. The final resulting model is then used as the initial model to perform phase self-calibration to the rest of the 8 datasets. As the target field does not contain any dominant point sources, we do not perform any amplitude self-calibration. {We report that, for each observing sessions, the signal-to-noise ratio of the gain solutions is $\sim 10^{3}$. Assuming that the calibration errors scale down as we average over all observing sessions, the overall calibration error is $10^{-5}$, sufficient for containing foreground leakage \citep{2016MNRAS.461.3135B}}. After flagging the undesired data and performing the calibration steps, the data were split to a $952{\text -}1170$ MHz sub-band and $8$s time integration. For computing the delay power spectrum, we adopt a \textit{visibility}-based approach as described in the previous section. Therefore, no image cubes are used in our analysis pipeline. Moreover, our pipeline relies on foreground isolation and thus identifying individual foreground sources in the map domain is not required. However, for demonstration purposes, we present a frequency-averaged Stokes~I image of the field at $1077.5$ MHz from a single dataset of $12.75$ hrs tracking using the subband $952-1170$ MHz in \autoref{DEEP2}. As shown, the flux scale for the continuum foreground emission of the DEEP2 field is below 0.1mJy/beam, ideal for detecting the faint HI signal.
\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{DEEP2.png}
 \caption{Stokes I frequency-averaged image of the field at $1077.5$ MHz from 1 ($12.75$ hrs tracking) out of 9 datasets. (Image rendered with CARTA \citep{CARTA})}
\label{DEEP2}
\end{figure}

\subsection{Power spectrum estimation}
To calculate the power spectrum, we further split the visibility datasets into two smaller bands of $\approx 46$ MHz width centered at $986$ and $1077.5$ MHz respectively. The smaller bandwidth segments are chosen to minimize the effects of cosmological evolution. We compute the power spectrum for these two cases separately.

In a tracking observation, each baseline refers to a particular $(u,v)$ point on the $uv$ plane at a given time. The $(u,v)$ coordinate changes to a new value at every integration interval. We generate visibility cubes in the baseline ($uv$)-frequency ($\nu$) domain by gridding the $uv$ plane into discrete cells of size $\Delta u = \Delta v = 60\lambda$. The $(u,v)$ coordinates are calculated at the band center. {In each observing session, we split the baselines based on the parity of the scan id they belong to. We construct the ``even scan'' and ``odd scan'' visibility gridded cubes} from the 9 observing sessions, excluding the flagged baselines. We only consider $uv$ points for which the visibility data has at least $80$ percent unflagged channels {to mitigate potential weak wide-band RFI}. Moreover, the flagged channels (if any) are substituted with the nearest neighbour unflagged channel data. {This ensures even sampling across channels when we perform the Fast Fourier Transform (FFT) during the delay transformation.} {We checked that the results from unsubstituted gridded visibility are consistent with the nearest neighbour substituted data.} Assuming the sky signal to be the same within a $uv$ pixel, all visibility data within it are then averaged. Following this, we apply an FFT along the frequency axis for every $uv$ pixel, producing two $u$-$v$-delay ($\tau$) cubes. During this delay transformation, the averaged visibility data is multiplied with the Blackman-Harris spectral window function that suppresses the foreground power leakage to higher $k_\parallel$ modes. Next, we cross-correlate between different cubes, computing the cylindrical $(\bm{k_\perp}, k_\parallel)$ power spectrum using \autoref{PS_eq}. This cross-correlation removes the noise bias in our calculation, and it is also useful to minimize any time-dependent systematics present in the data.

In the case of statistical isotropy and homogeneity, the power spectrum should be only a function of $k=\sqrt{k_\perp^2 + k_\parallel^2}$. However, the instrumental noise contribution evolves as a function of $k_\perp$ due to the baseline distribution of the array. In our analysis, since we split the dataset into two time blocks and cross-correlate the visibilities, the noise should decorrelate between different times and therefore noise bias correction is not needed. The isotropy is also broken due to HI redshift space distortions, which at these small scales tend to reduce the amplitude of fluctuations for large $k_\parallel$ along the line of sight. Nevertheless, the attenuation along the line of sight for the HI signal is much weaker than the attenuation for the foregrounds. The foregrounds such as the Galactic synchrotron emission and radio galaxies are smooth in frequency and contribute power at short delays (small $k_\parallel$). This allows us to isolate the contaminants from the HI signal by using the foreground avoidance technique \citep{Datta_2010, Morales_2012, Parsons_2012a, Vedantham_2012, Pober_2013, Paul_2016,2023MNRAS.518.2971C}.

Here, we employ a conservative approach to compute the 1-d power spectrum and only include those $(\bm{k_\perp}, k_\parallel)$ pixels which are way above the \textit{horizon limit} of the foreground wedge. The horizon limit is defined as \citep{2014PhRvD..90b3018L}
\begin{linenomath*}
\begin{equation}
    k_\parallel = \frac{x H_0 E(z) \sin \theta_0}{c(1+z)} k_\perp,
    \label{wedge_eq}
\end{equation}
\end{linenomath*}
where $\theta_0$ is the angular extent of the MeerKAT dishes. Due to the small field-of-view and small sidelobes of the MeerKAT telescope, the horizon limit is close to $k_\parallel \sim 0.02 k_\perp$. However, it does not exclude modes that are susceptible to foreground leakage from the sidelobes and instrument systematics. Assuming the most extreme case that leakage of bright foreground sources from the side lobes extends to the entire sky, i.e. $\sin \theta_0 = 1$, the horizon limit becomes $k_\parallel \sim 0.26 k_\perp$. Therefore, we use a much stricter selection criterion $k_\parallel = 0.3 k_\perp$ to ensure the robustness of our results.

Apart from the foreground avoidance, we also use thermal noise simulations to flag outliers of the delay powers in $k$-space to mitigation residual systematics, which we discuss in details in \autoref{sec:sys}. To estimate the instrument noise contribution in the power spectrum variance, we generate a cylindrical thermal noise power spectrum where each calibrated visibility component is replaced with a simulated thermal noise visibility generated from a zero mean random process with standard deviation \citep{Taylor_1999}
\begin{linenomath*}
\begin{equation}
\sigma_{\rm{N}} = \frac{2k_{\rm{B}} T_{\rm{sys}}}{A_e\sqrt{\delta\nu \delta t}}
\label{sigma_N}
\end{equation}
\end{linenomath*}
where $T_{\rm{sys}}$ and $A_e$ are the system temperature and effective area of each antenna respectively, $k_{\rm{B}}$ is the Boltzmann constant, $\delta\nu = 208.984$ kHz is the channel width and $\delta t = 8$s is the time resolution. {The thermal noise amplitude depends on the natural sensitivity of the instrument $A_e/T_{\rm sys}$. We estimate the natural sensitivity at each frequency sub-band by calculating the variance of the Stokes V visibility data (see \autoref{sec:sys} for more details). We find $A_e/T_{\rm sys} = 6.65\,{\rm m^2/K}$ for $z=0.32$ and $6.48\,{\rm m^2/K}$ for $z=0.44$, consistent with the anticipated performance $A_e/T_{\rm sys} = 6.22\,{\rm m^2/K}$. We then generate $10000$ realizations of the noise power spectrum given the natural sensitivity $A_e/T_{\rm sys}$ to verify our calculations.} The Stokes~V estimate and the simulation outcome are presented in \autoref{2d_ps_TN} in \autoref{sec:sys}. We then assign the variance of the thermal noise power across the 10000 realisations at each $k-$pixel $\sigma^2(\bm{k_\perp}, k_\parallel)$ as the variance of the delay power spectrum. For a given $\bm{k_\perp}$, $\sigma$ is approximately constant across $k_\parallel$, only changing across $\bm{k_\perp}$ depending on the baseline density.
At this stage, we flag $\bm{k}$ pixels for which $|P(\bm{k_\perp}, k_\parallel)-P_{\rm TN}(\bm{k_\perp}, k_\parallel)| > 5\sigma(\bm{k_\perp}, k_\parallel)$ to exclude $k$-modes affected by any residual systematics. Note that the thermal noise simulations are performed for each gridded visibility data set separately, since different timeblocks and Stokes parameters at different frequency sub-bands have different $u$-$v$ sampling. The $5-\sigma$ flagging is done in the 3-d auto-power of the Stokes I visibility for each timeblock at each sub-band individually. If the 3-d $k$-pixel is flagged in either even or odd scans, it is excluded from the power spectrum estimation in the cross-power. We have verified that this selection criteria results in negligible signal loss with our simulations.

From the 3-d cylindrical power spectrum $P(\bm{k_\perp}, k_\parallel)$, we use inverse noise variance weighting ($1/\sigma^{2}(\bm{k_\perp}, k_\parallel)$) to calculate the optimal 2-d power spectrum $P(k_\perp, k_\parallel)$ with $k_\perp=|\bm{k_\perp}|$ and 1-d power spectrum $P(k)$ with $k=|(\bm{k_\perp}, k_\parallel)|$.

\begin{equation}
    \hat{P}_{\rm D}^{i} = \big(\sum_j P_{\rm D}^{ j} w_{ j} \big)/\big(\sum_j w_{ j}\big),
\label{eq:avg}
\end{equation}
where $j$ loops over all the $k$-points that fall into the $i^{\rm th}$ $k$-bin and $w_{ j} = 1/\sigma^2_j$ is the inverse covariance weight calculated from the thermal noise simulations mentioned above.
The errors are calculated from the sampling variance of the 3-D powers in each bin
\begin{equation}
    (\Delta P_{\rm D}^{\rm i})^2 = \big(\sum_j (P_{\rm D}^{\rm j}-\hat{P}_{\rm D}^{\rm i})^2 w_{\rm j}^2 \big) \Big/ \big(\sum_j w_{\rm j}\big)^2.
\label{eq:err}
\end{equation}

\section{Results} \label{sec:results}

\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{fig/p2dresult.png}
\caption{{\bf 2-d power spectrum from the analysis of $\mathbf{ 96}$-hrs MeerKAT interferometer data at $\mathbf{z=0.32}$ and $\mathbf{z=0.44}$.} {\bf a,} with Stokes I visibility data, cross-correlating the even and odd scan cubes for ${z=0.32}$. {\bf b,} for $z=0.44$. Negative values are left blank. The $(\bm{k_\perp}, k_\parallel)$ modes above the black dashed line $(k_\parallel > 0.3 k_\perp)$ are used to calculate the 1-d power spectrum.}
\label{2d_ps_1078MHz}
\end{figure}

In \autoref{2d_ps_1078MHz}, we show the 2-d power spectrum as a function of $(k_\perp, k_\parallel)$ at $z=0.32$ and $z=0.44$. The dashed line in \autoref{2d_ps_1078MHz} defines the wedge, $k_\parallel = 0.3 k_\perp$, above which we do not expect any foreground contamination (e.g. \citealt{Morales_2012}). In our case, the superb stability of MeerKAT and its primary beam size move the contaminants to a much smaller region. Nevertheless, we take a conservative approach and restrict our analysis to values above the dashed line. This foreground avoidance technique has the advantage of being robust to signal loss. Apart from the foreground-dominated modes at low $k_\parallel$, the rest of the cylindrical $k$-space is largely noise-dominated.

We further average the power spectrum into bins in $k$ to increase the statistical significance. For each redshift bin, we choose 7 logarithmic $k$-bins and average the 3-d powers into the 1-d power spectrum with inverse covariance weighting described in \autoref{eq:avg} and \autoref{eq:err}. The resulting 1-d HI power spectra are shown in \autoref{1d_ps}. For both redshifts, the 1-d power spectrum shows a clear detection, with the statistical significance being $8.0\sigma$ and $11.5\sigma$ respectively for $z=0.32$ and $z=0.44$. The measured power spectrum drops as $k$ increases which is expected. The amplitude of the signal at small scales is $\sim 0.1\,{\rm mK^2Mpc^3}$, consistent with the combination of the expected shot noise level of $\sim 1\,{\rm mK^2Mpc^3}$ (e.g. \citet{Chen_2021}) and heavy attenuation due to the Finger-of-God (FoG) effects at high $k_\parallel$. For both redshifts, second $k$-bins of the power spectrum is lower than expected. This is due to the systematics at short baselines $u\approx 0$ as we show in \autoref{sec:sys}. While the systematics are largely removed, some weak effects may not be picked up by the $5-\sigma$ flagging. It does not impact the robustness of our detection, because the expected amplitude of the power spectrum is still well within the $3-\sigma$ region of our measurements. Furthermore, since we use the sampling variance for calculating the measurement errors, the impact of the systematics is incorporated and results in larger error bars. As a result of the low power spectrum amplitude at the second $k$-bin, when we perform the model fitting in \autoref{modelfit}, the median fitting results are biased towards lower values for the first $k$-bin as seen in \autoref{contour1} and \autoref{contour2}.

To validate our results, we perform a null test by cross-correlating the data of the two frequency sub-bands used in the detection. The individual sub-bands have no overlap
and each of them is split into two time blocks of even and odd scans. When calculating the cross-power, only $k$-points that are above the $k_\parallel = 0.3 k_\perp$ wedge and not flagged by the $5-\sigma$ criterion are used. Therefore, the cross-power should give results consistent with zero.
As shown in \autoref{null_test}, we detect no significant correlation in these tests which shows that no obvious frequency correlated signals remain in the Fourier window used in the analysis. The null test suggests that the observation window we choose does not have sizeable foreground leakage, and residual systematics have been largely mitigated.

\begin{figure}[ht!]
\centering
\includegraphics[width=0.45\textwidth]{fig/p1d1.png}
\includegraphics[width=0.45\textwidth]{fig/p1d2.png}
\caption{{\bf 1-d power spectrum from the analysis of the MeerKAT DEEP2 data.} {\bf a,} at $z=0.32$ and {\bf b,} $z=0.44$. The measurement is denoted as ``MeerKAT DEEP2'' and the expected level of signal denoted as ``Model'' is calculated following the best fit results of the ``No Prior'' case in \autoref{modelfit}.}
\label{1d_ps}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{fig/null_test.png}
\caption{{\bf Null test--} cross power spectrum between the two redshift bins. As each redshift bin has been divided into two time bins of even and odd scans, we have four possible cross-correlations between the two redshift bins, shown as the dashed lines. The black data points shows the average of the four cases and we observe no significant correlation. The value of the $k$ bins and normalization of the power spectrum are calculated at $1032$ MHz which is approximately midway to the central frequencies of the two redshift bins. The error bars are calculated by taking the sampling variance of the four cases.
}
\label{null_test}
\end{figure}

\begin{table}
\centering
\begin{tabular}{| c | c c c | c | c c c |}
 \hline
 \multicolumn{4}{|c|}{$z=0.32$} & \multicolumn{4}{c|}{$z=0.44$} \\
 \hline

 $k$[Mpc$^{-1}$] & $P(k)$[mK$^2$Mpc$^3$] & $\sigma_{P}$[mK$^2$Mpc$^3$] & $P(k)/\sigma_{P}$ & $k$[Mpc$^{-1}$] & $P(k)$[mK$^2$Mpc$^3$] & $\sigma_{P}$[mK$^2$Mpc$^3$] & $P(k)/\sigma_{P}$\\
 \hline
 0.43 & 36.48 & 19.03 & 1.92 & 0.34 & 53.43 & 27.80 & 1.92 \\
 0.74 & 0.12 & 3.04 & 0.04 & 0.61 & 0.60 & 4.13 & 0.14 \\
 1.25 & 2.26 & 0.57 & 3.98 & 1.01 & 4.34 & 0.85 & 5.12\\
 2.04 & 0.96 & 0.19 & 4.96 & 1.68 & 1.51 & 0.27 & 5.62\\
 3.30 & 0.19 & 0.09 & 2.12 & 2.81 & 0.96 & 0.13 & 7.54\\
 5.19 & 0.33 & 0.09 & 3.80 & 4.31 & 0.56 & 0.15 & 3.80\\
 7.96 & 0.21 & 0.20 & 1.07 & 7.04 & 0.28 & 0.41 & 0.69\\
 \hline
\end{tabular}
\caption{{Table summarizing the HI power spectrum constraints at $z=0.32$ and $z=0.44$. From left to right, the columns show the centre of each $k$-bin, the value of the measured HI power spectrum in the units of mK$^2$Mpc$^3$, the measurement uncertainties in the units of mK$^2$Mpc$^3$, and the significance of the detection in each $k$-bin. }}
\label{ps_table}
\end{table}

To further quantify the amplitude of the HI fluctuations measured, we use the 3-d power from our data to compute the variance of the fluctuation at a given scale R
\begin{equation}
\sigma_{\rm HI}^2 = \frac{3}{4\pi R^3}\,\frac{\int_{k_{\rm window}}\frac{{\rm d}^3k} {(2\pi)^3}W^2(kR)P_{\rm D}(\bm{k})w(\bm{k})}{\int_{k_{\rm window}}\frac{{\rm d}^3k}{(2\pi)^3}W^2(kR)w(\bm{k})},
\label{eq:sigmahi}
\end{equation}
where the integration sums over all the $k$-pixels in the delay-transformed visibility data cube and $w(\bm{k})$ is the inverse covariance weight discussed before. $W(kR) = 3 j_1(kR)/(kR)$ is the Fourier transform of the spherical top-hat function with $j_1$ being the spherical Bessel function of the first kind.

As shown in \autoref{ps_table}, our measurements are most sensitive to scales around 1.0 Mpc. We use the bootstrapping method to estimate $\sigma_{\rm HI}$ at $R= 1.0$ Mpc, by running 8000 realizations with each realization selecting $10^4$ k-pixels that are not flagged. In each realization, a value of $\sigma_{\rm HI}$ is calculated according to the $k-$pixels selected. This allows us to constrain the rms of HI density to be $\sigma_{\rm HI} = (0.44\pm 0.04)\,{\rm mK}$ at $z\sim0.32$ and $\sigma_{\rm HI} = (0.63\pm 0.03)\,{\rm mK}$ at $z\sim0.44$, at the scale of 1.0 Mpc. The results for $\sigma_{\rm HI}$ coherently sum over all $k$-points used, and thus can be used as an indicator for the overall statistical significance of the measurements. Our results have high statistical significance of our detections at both redshifts, demonstrating the capability of MeerKAT observations for interferometric HI intensity mapping.

\section{Model fitting}
\label{modelfit}
In this section, we use the obtained measurement of the HI power spectrum to constrain the astrophysics of the HI galaxies and the HI mass function (HIMF) following \citet{Chen_2021}. The power spectrum is modelled as a combination of a 2-halo and 1-halo term as well as a scale-independent shot noise term. We further include redshift space distortions, e.g. the Kaiser term \citep{Kaiser_1987} which increases the amplitude of the power spectrum on large scales and the ``fingers-of-god" term \citep{Jackson_1971} which smooths out fluctuations on small scales. The theoretical power spectrum is averaged in the same way as the observed one in order to produce the 1-d power spectrum.

We model the HI power spectrum as a combination of a 2-halo, 1-halo, and shot noise term \citep{Cooray_2002,2010MNRAS.404..876W,Chen_2021}:
\begin{linenomath*}
\begin{equation}
    P_{\rm D}(k_\perp,k_\parallel) = P_{\rm 2h}(k_\perp,k_\parallel) + P_{\rm 1h}(k_\perp,k_\parallel) + P_{\rm SN}(k_\parallel).
\end{equation}
\end{linenomath*}
The 2-halo term can be modelled as
\begin{linenomath*}
\begin{equation}
\begin{split}
P_{\rm 2h}(k,k_\parallel) =& C_{\rm HI}^2 \bigg[\int{\rm d}n\,n(m) b(m)\langle M_{\rm HI}(m) \rangle u_{\rm HI}(k|m) \bigg]^2(1+\frac{fk_\parallel^2}{b_{\rm HI}^0k^2})^2 \frac{P_{m}(k)}{{1+(k_\parallel\sigma_p)^2}/2},
\end{split}
\end{equation}
\end{linenomath*}
where $m$ integrates over the halo mass range, $C_{\rm HI}$ is the conversion factor from HI mass density to temperature, $n(m)$ is the halo mass function \citep{Tinker_2008}, $b(m)$ is the linear halo bias \citep{Tinker_2010}, $\langle M_{\rm HI}(m) \rangle$ is the HI-halo mass relation, $u_{\rm HI}(k|m)$ is the normalised Fourier-transformed density profile, $k=\sqrt{k_\perp^2+k^2_\parallel}$ and $b_{\rm HI}^0$ is the linear HI bias. We include redshift space distortions in our analysis, taking into account both the Kaiser effect \citep{Kaiser_1987} as well as the FoG effect \citep{Jackson_1971} which affects the small line of sight scales (large $k_\parallel$) we are observing. For these terms, $f$ is the growth rate and $\sigma_p =\sigma_v(1+z)/(Hf) $ where $\sigma_v$ is the velocity dispersion and $H$ is the Hubble parameter. Here, we simplify the modelling of the FoG effects by assuming an averaged velocity dispersion $\sigma_v$, which can be related to the velocity width of the emission line profile of the stacked HI galaxies.
$P_{\rm m}(k)$ is the matter power spectrum calculated using halofit \citep{Takahashi_2012}.

 The linear HI bias is calculated as
 \begin{linenomath*}
\begin{equation}
    b_{\rm HI}^0 = \frac{\int{\rm d}n\,n(m) b(m)\langle M_{\rm HI}(m) \rangle}{\int{\rm d}n\,n(m) \langle M_{\rm HI}(m) \rangle},
\end{equation}
\end{linenomath*}
while the 1-halo term can be modelled as
\begin{linenomath*}
\begin{equation}
    P_{\rm 1h}(k,k_\parallel) =C_{\rm HI}^2 \int {d}n \,n(m)\langle M_{\rm HI}(m)\rangle ^2 u_{\rm HI}(k|m)\frac{1}{{1+(k_\parallel\sigma_p)^2}/2} .
\end{equation}
\end{linenomath*}

The shot noise term $P_{\rm SN}(k_\parallel)$ can be written as
\begin{linenomath*}
\begin{equation}
    P_{\rm SN}(k_\parallel) = \frac{P_{\rm SN}^0 }{1+(k_\parallel\sigma_p)^2/2}.
\end{equation}
\end{linenomath*}
Note that, the shot noise in the HI power spectrum is also attenuated by the FoG effect. This is due to the fact that the HI sources are not treated as point sources on the sky but as intensity maps. The emission line width, corresponding to the velocity dispersion of the sources, breaks the point-source assumption along the line of sight, smearing the overall power spectrum along the $k_\parallel$ direction.

We further parameterize:
\begin{linenomath*}
\begin{equation}
    \langle M_{\rm HI}(m)\rangle = A_{\rm HI}\big(\frac{m}{M_0}\big)^\beta,\, \rho_{\rm HI}(r|m) = \rho_0 \big(\frac{r}{r_0}\big)^\gamma {\rm exp}[-ar/r_0],
\end{equation}
\end{linenomath*}
where $M_0 = 10^{10} M_\odot {\rm h}^{-1}$, $r$ is the distance relative to the halo centre, and $r_0$ is the characteristic scale of the dark matter halo, calculated using the concentration-mass relation \citep{Maccio_2006}. The parameter $a$ is fixed by imposing that at $m=10^{12} M_\odot {\rm h}^{-1}$ the cut-off scale is $r_0/a = 1 {\rm Mpc \,h^{-1}}$ \citep{Spinelli_2019}. We further fix the power law index for the velocity dispersion as in \citep{Villaescusa-Navarro_2018}, which leaves us with a 5-parameter model:
\begin{linenomath*}
\begin{equation}
    \theta = \{A_{\rm HI}, \beta, \gamma, \sigma_v, P_{\rm SN}^0  \}.
\end{equation}
\end{linenomath*}

After choosing the halo model parameters, we compute the analytical power spectrum using \textsc{halomod} \citep{Murray_2020} and average across the k-range probed as shown in \autoref{2d_ps_1078MHz}. We construct a Gaussian likelihood from the results reported and use \textsc{emcee} \citep{Foreman-Mackey_2012} to fit for the model parameters with 50 random walkers and 8000 steps to ensure convergence. We discard the first 2000 steps. Flat priors are imposed so that all parameters are positive. For $\gamma$, we impose $1<\gamma<3$ since values outside this range lead to divergence. From the stacking of selected HI galaxies using MeerKAT observations at similar redshifts \citep{2022ApJ...935L..13S}, we impose a flat prior on the velocity dispersion $0<\sigma_v<500\,{\rm km\,s^{-1}}$. The HOD parameters $A_{\rm HI}$ and $\beta$ are specific to the parameterization, and their physical meanings can be hard to interpret. Therefore, in each sample in the posterior, we calculate $\Omega_{\rm HI}$ and the HI bias $b_{\rm HI}$ at 1.0 Mpc$^{-1}$ and present the results with these two parameters instead of $A_{\rm HI}$ and $\beta$.

As we show later, the parameter fitting does not converge for all parameters due to parameter degeneracy and lack of information on the $k_\parallel$ dependence of the power spectrum. We therefore consider another case where we impose a prior on $\Omega_{\rm HI}$ with $\Omega_{\rm HI} = (0.50\pm0.18) \times 10^{-3}$ at $z\sim 0.32$ from stacking \citep{Rhee_2018} and $\Omega_{\rm HI} = (0.77\pm0.26) \times 10^{-3}$ at $z\sim 0.44$ from Damped Lyman Alpha (DLA) systems \citep{Rao_2017}.

\begin{figure}
\centering
\includegraphics[width=\textwidth]{fig/result1.png}
\caption{{\bf Model fitting results for $z\sim0.32$.} The 2-d contour plots show posterior distribution of model parameters for power spectrum fitting. The darker region shows the $1\sigma$ confidence interval of the posterior and the lighter region shows the $2\sigma$ confidence interval. $\sigma_v$ is shown in the unit of km/s. $P_{\rm SN}$ is shown in the unit of $\rm mK^2 Mpc^3$. The upper lines of the titles of the 1-d histograms show the median and the $1\sigma$ confidence level for each parameter for the model fitting with the shot noise modelled by the HIMF (``HIMF''). The middle lines of the titles of the 1-d histograms show the results without the $\Omega_{\rm HI}$ prior (``NP''). The lower lines show the results with the $\Omega_{\rm HI}$ prior (``WP''). The 2-d posterior distribution is smoothed with a Gaussian kernel of 0.5 grid width for better visualization. In the upper right zoom-in panel, the fitting result of the power spectrum measurement with no prior is shown. The vertical dash line shows the lower and upper limit for each k-bin. }
\label{contour1}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=\textwidth]{fig/result2.png}
\caption{{\bf Model fitting results for $z\sim0.44$.} The 2-d contour plots show posterior distribution of model parameters for power spectrum fitting. The darker region shows the $1\sigma$ confidence interval of the posterior and the lighter region shows the $2\sigma$ confidence interval. $\sigma_v$ is shown in the unit of km/s. $P_{\rm SN}$ is shown in the unit of $\rm mK^2 Mpc^3$. The upper lines of the titles of the 1-d histograms show the median and the $1\sigma$ confidence level for each parameter for the model fitting without the $\Omega_{\rm HI}$ prior (``NP''). The lower lines show the results with the $\Omega_{\rm HI}$ prior (``WP''). The 2-d posterior distribution is smoothed with a Gaussian kernel of 0.5 grid width for better visualization. In the upper right zoom-in panel, the fitting result of the power spectrum measurement with no prior is shown. The vertical dash line shows the lower and upper limit for each k-bin.}
\label{contour2}
\end{figure}

The fitting results are shown in \autoref{contour1} for $z\sim0.32$ and \autoref{contour2} for $z\sim0.44$. The green regions show the posterior distribution for the model parameters without the $\Omega_{\rm HI}$ priors while the purple region shows the distribution with the prior. The blue regions show the results with the HIMF modelling which we discuss later. From the posterior, one can see that the small number of $k$-bins and the relatively low signal-to-noise ratio lead to divergence in the MCMC fitting, with most of the parameters unconstrained. The constraints on $\Omega_{\rm HI}$ are noticeably worst if no prior is imposed. This is expected, since the small inner-halo scales do not contain enough information to constrain the overall amplitude which is on the large scales. The density profile parameter $\gamma$ is also poorly constrained, because the small scale power spectrum is dominated by the effects of the shot noise and the velocity dispersion.

Comparing the results at $z\sim 0.32$ and $z\sim 0.44$, one can see that both redshift bins show tangible evidence that the velocity dispersion $\sigma_v$ and the HI shot noise can be jointly constrained. For $z\sim 0.32$, the posterior is reaching the flat prior $\sigma_v<500\,{\rm km/s}$ we impose. We verify that relaxing this prior simply leads to the posterior streching to higher, more unphysical values of $\sigma_v$. The large tail of the $\sigma_v$ distribution also drives the shot noise $P_{\rm SN}^0$ to be larger, since these two parameters are degenerate.
The $z\sim 0.44$ redshift bin has higher signal-to-noise ratio and the results give a reasonable constraint on the velocity dispersion $\sigma_v = 145.84^{+124.00}_{-52.34} \,{\rm km/s}$, although the problem of the degeneracy between $\sigma_v$ and $P_{\rm SN}^0$ persists. This is again expected for our preliminary result. The cylindrical power spectrum for the DEEP2 data is dominated by thermal noise, and therefore we need to average the $k$-pixels down to 1-d power spectrum in order to constrain our model. Binning the cylindrical power spectrum into 1-d power spectrum means that we lose the information along the $k_\parallel$ direction, which entails the effects of the $\sigma_v$ parameter. In the future with more data resulting in higher signal-to-noise ratio, binning the $k$-points into $k_\perp-k_\parallel$ grids will help preserve the $k_\parallel$ dependence of the FoG effect, breaking the degeneracy between $\sigma_v$ and the rest of the parameter set. The degeneracy can also be resolved by using the HIMF to model the shot noise instead of treating it as a free parameter, which we discuss later in this section.

Comparing the results with and without the $\Omega_{\rm HI}$ prior, we find that the fitting converges on the $\Omega_{\rm HI}$ parameter as expected, but shows no visible improvement for shot noise and velocity dispersion. For $z\sim 0.44$ with better signal-to-noise ratio, we start to be able to constrain the density profile parameter $\gamma$, which leads to better constraints on the HI bias at small scales. The results suggest that at 1-D power spectrum level, the signal at the scales of the measurement is dominated by the 1-halo term and the shot noise, which are less correlated with the overall HI amplitude comparing to the large scales dominated by the 2-halo term.

Our results show that using MeerKAT interferometric observations, we can constrain the velocity dispersion and the HI shot noise even with preliminary measurements with relatively low signal-to-noise ratio. In the future with more data from the MeerKAT observations, intensity mapping measurements can be used to constrain the halo model of HI, providing a detailed picture of the galaxy evolution from $z\sim 0.0$ to $z\sim 1.0$.

The measurements on the HI shot noise can also be used to constrain the HI mass function \citep{Chen_2021}.
The HIMF is parameterized with a Schechter function \citep{Schechter_1976}:
\begin{linenomath*}
\begin{equation}
    \phi_{\rm HI}(M_{\rm HI})\equiv \frac{{\rm d}n_{\rm HI}}{{\rm dlog}M_{\rm HI}} = {\rm ln}10\,\phi_* \big(\frac{M_{\rm HI}}{M_*}\big)^{\alpha +1}e^{-\frac{M_{\rm HI}}{M_*}}.
\end{equation}
\end{linenomath*}
This can be related to the number density, $n_{\rm HI}$, HI mass density and shot noise as:
\begin{equation}
    n_{\rm HI} = \int_{M_{\rm min}} {\rm dlog}M_{\rm HI}\,\phi_{\rm HI}(M_{\rm HI})
\label{eq:nhi}
\end{equation}
\begin{equation}
    \Omega_{\rm HI} = \int {\rm dlog}M_{\rm HI}\,\phi_{\rm HI}(M_{\rm HI}) M_{\rm HI}/\rho_c
\label{eq:omegahi}
\end{equation}
\begin{equation}
    P_{\rm SN}^0 = (C_{\rm HI}\rho_c \Omega_{\rm HI})^2 \frac{\int {\rm dlog}M_{\rm HI}\,\phi_{\rm HI}(M_{\rm HI})M_{\rm HI}^2}{\big(\int {\rm dlog}M_{\rm HI}\,\phi_{\rm HI}(M_{\rm HI})M_{\rm HI}\big)^2}
\label{eq:psn}
\end{equation}

For intensity mapping experiments, we have independent measurements for $\Omega_{\rm HI}$ and $P_{\rm SN}^0$. For $n_{\rm HI}$ we need external information since the 1-halo and 2-halo terms are only sensitive to the HI density distribution inside halos and not the number of HI galaxies itself. The ideal approach is to use HI stacking of the same field that we are observing. Here, however, we are only interested in a proof-of-concept to illustrate its feasibility on future MeerKAT data. We use the reported number density (165 galaxies in a 57' field) and average HI mass ($\langle M_{\rm HI}\rangle = (5.73\pm1.47)\times10^9M_\odot$ ) of the stacking sample in \cite{Rhee_2018}.
For $\Omega_{\rm HI}$ and $P_{\rm SN}$, intensity mapping observations integrate over all the HI mass and therefore the integration limit is chosen to be $10^7$ to $10^{13}$ solar masses for convergence. The number density, on the other hand, depends on the lower mass limit of the stacking sample. In order to incorporate the ignorance of this minimum mass and use the stacking result as a prior in our model fitting, we use the following routine in the MCMC:
\begin{itemize}
    \item
    In each step, a random set of halo model parameters as well as the HIMF parameters $\phi_*$ and $M_*$ is sampled.
    \item
    Based on the halo model parameter, the overall HI density $\Omega_{\rm HI}$ is calculated. Using \autoref{eq:omegahi}, we can calculate the value for the last HIMF parameter $\alpha$.
    \item
    The shot noise is calculated using \autoref{eq:psn}.
    \item
    Using the number density $ n_{\rm HI} = 2.987\times 10^{-3}\,{\rm Mpc^{-3}}$ corresponding to 165 galaxies in a 57' field, from \autoref{eq:nhi} we can find the lower mass limit of the stacking sample.
    \item
    From the lower mass limit, we randomly sample 165 galaxies from the HIMF distribution. The average HI mass of the samples $\bar{M}_{\rm HI}$ is compared against the stacking result $\langle M_{\rm HI}\rangle = (5.73\pm1.47)\times10^9 M_\odot$. A Gaussian prior can be calculated as part of the likelihood ${\rm log[p]} = -0.5\times(\bar{M}_{\rm HI}/10^9 M_\odot-5.73)^2/1.47^2$.
\end{itemize}

Incorporating the HIMF into our model fitting, the shot noise $P_{\rm SN}^0$ is no longer a free parameter. Instead, it is derived from the HIMF and the halo model parameters, making the fitting results on the model parameters different from the previous cases shown. The updated results using the measurements at $z\sim 0.32$ are shown as the blue regions in \autoref{contour1}. Comparing the results with previous fittings where shot noise is treated as a free parameter, we find that the shot noise is no longer biased towards larger values. This is due to the fact that, in order to get to the large values for the shot noise, the HIMF needs to be skewed towards the higher mass end. On the other hand, the parameter space where the HIMF is skewed towards the higher mass end is disfavoured by the MCMC fitting, because it would result in a much higher average HI mass when we sample the 165 HI galaxies. Comparing the fitting with the HIMF with a naive $\Omega_{\rm HI}$ prior, we can see that, apart from the shot noise no longer being biased, the posterior also shows anti-correlation between $\Omega_{\rm HI}$ and $b_{\rm HI}$. The anti-correlation suggests that the information from stacking provides extra constraints for the large scale, leading to the degeneracy between the HI density and the bias as they equally contribute to the power spectrum amplitude. In conclusion, using the HIMF to model the shot noise helps excluding part of the parameter space that is not physical, resulting in better constraints for the parameters.

We note that, however, the fitted values for $\sigma_v$ and $P_{\rm SN}^0$ are much lower than the results without the HIMF parameters. It is possible that the stacking sample we are using is not complete, making the HIMF lower than the underlying truth. As the HIMF is lower, the shot noise is also lower and to compensate the drop in the power spectrum amplitude, the velocity dispersion becomes smaller as well. This issue can be mitigated by using the MeerKAT DEEP2 data itself to perform HI galaxy stacking, which we leave for future work. Here, we are only interested in demonstrating the effects of using the HIMF in the model fitting to improve the constraints on HI, which is evident in the posterior. Our findings strongly advocates for combined constraints on HI galaxies from the same MeerKAT observations using both stacking and intensity mapping techniques. The intensity mapping measurements in turn help constrain the HIMF parameters, without the need to resolve individual galaxies or split galaxy samples, significantly extending the redshift range where the HIMF can be probed comparing to the conventional HI galaxy stacking technique.

In \autoref{himf}, we show the posterior distribution of the HIMF parameters as well as the fitted HIMF at $z\sim 0.32$. The HI power spectrum is more sensitive to the ``knee mass'' $M_*$ and the slope parameter $\alpha$ while being less sensitive to the overall abundance $\phi_*$, since the HI shot noise is weighted by the HI mass squared and therefore sensitive to the relative abundance of the HI-heavy galaxies. Comparing our results to the HIMF measured at the local Universe \citep{Jones_2018}, we find a lower ``knee mass'', suggesting that galaxies obtain more HI from $z\sim 0.32$ to $z\sim 0.0$. Our findings are also consistent with measurements of the HIMF at similar redshift \citep{2022ApJ...940L..10B}. It is also consistent with the MIGHTEE-HI measurements of the scaling relations \citep{2022ApJ...935L..13S,2022arXiv221004651P}. In the future, combining measurements of intensity mapping and HI galaxy surveys using MIGHTEE survey data \citep{Maddox_2021}, we will be able to probe the HIMF from $z\sim 0.0$ to $z\sim 0.5$, opening up a new window into studying the evolution of the star-forming properties of galaxies.

\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{fig/himf.png}
\caption{{\bf Measurement of HIMF parameters for power spectrum fitting at $z\sim 0.32$.} \textbf{a}, the darker region shows the $1\sigma$ confidence interval of the posterior and the lighter region shows the $2\sigma$ confidence interval. $\phi$ is shown in the unit of $\rm Mpc^{-3}dex^{-1}$. $M_*$ is shown in the unit of solar mass. The titles of the 1-d histograms show the median and the $1\sigma$ confidence level for each parameter. The 2-D posterior distribution is smoothed with a Gaussian kernel of 0.5 grid width for better visualization. \textbf{b}, the $1\sigma$ region of the fitted HIMF is shown in blue with the solid line being the mean. The result from HI galaxy survey in the local Universe \citep{Jones_2018} is shown in dotted line for comparison.}
\label{himf}
\end{figure}
\section{Conclusion}
In this letter, we use the MeerKAT observations of the DEEP2 field to make the first-ever detection of the HI autopower spectrum at $z\sim 0.32$ and $z\sim 0.44$. The deep observations allow accurate calibration of the data with errors $\sim 10^{-5}$. Splitting the frequency range into two sub-bands, each with 220 frequency channels centred around $z\sim 0.32$ and $z\sim 0.44$, we cross-correlate the visibility data from different timeblocks to minimise time-dependent systematics and remove noise bias. The resulting power spectrum measurements in the $k_\parallel > 0.3 k_\perp$ region are binned into 1-d $k$-bins. The HI power spectrum at both redshifts is measured with high statistical significance. Cross-correlating the timeblocks between the two frequency sub-bands as a null test, we find the cross-power spectrum fluctuates around zero with no indication of excess power, suggesting that our measurements are free of foreground leakage. Residual systematic effects are mitigated by $5$-$\sigma$ flagging in $k$-space using thermal noise simulations derived from the Stokes V data.

The resulting HI power spectrum is then used to perform parameter inference. Using halo model formalism to calculate the HI power spectrum and including the effects of redshift space distortion, we find that the scales probed by our measurements are dominated by the combination of the HI shot noise and the velocity dispersion. Due to the fact that the power spectrum is measured in 1-d $k$-bins, losing the information of the $k_\parallel$ dependence, the parameters are degenerate with each other, leading to a divergence in the fitting. Imposing a $\Omega_{\rm HI}$ prior based on existing constraints, we find that preliminary constraints on the HI shot noise, the velocity dispersion, and the HI density profile can be made. We also demonstrate that incorporating the HI mass function into the modelling can help mitigate divergence for the shot noise and the velocity parameters, yielding better constraints. The HI mass function itself can in turn be constrained.

Our results show that it is possible to isolate the HI signal from the strong foregrounds using the avoidance technique and obtain a high signal to noise detection, opening the window to observational cosmology on quasi-linear scales. These measurements can be used to extract astrophysical information on the HI galaxies, informing us on the accuracy of current hydrodynamic HI simulations \citep{Dave_2019}. MeerKAT surveys such as MIGHTEE \citep{MIGHTEE} and Laduma \citep{LADUMA} are currently undertaking deep observations on several fields which will allow to measure the small scale HI power spectrum to extreme precision and provide exquisite constraints on the HI mass function, the halo model and the nature of redshift space distortions on these scales.

\section*{Data availability}
The uncalibrated visibility data used in this work are publicly available under the Proposal ID: SCI-20180426-TM-01 in SARAO archive (\url{https://archive.sarao.ac.za}).

\section*{Code availability}
The initial processing and calibration of the MeerKAT data was carried out using the processMeerKAT pipeline developed at the IDIA and available at \url{https://idia-pipelines.github.io/docs/processMeerKAT}. The self~calibrations were performed with standard CASA tasks: \textit{tclean}, \textit{gaincal} and \textit{applycal} (\url{https://casa.nrao.edu/casa_obtaining.shtml}). Further codes are available on a reasonable request to the authors.

\section*{Acknowledgements}
M.G.S. and S.P. acknowledge support from the South African Radio Astronomy Observatory and National Research Foundation (Grant No. 84156).  LW is a UK Research and Innovation Future Leaders Fellow [grant MR/V026437/1]. We acknowledge the use of the Ilifu cloud computing facility - \url{www.ilifu.ac.za}, through the Inter-University Institute for Data Intensive Astronomy (IDIA). We thank Tom Mauch and Jordan D. Collier for useful discussions on the MeerKAT data used in this work. We also thank the developers of open-source Python libraries NumPy \citep{Numpy}, SciPy \citep{Scipy}, emcee \citep{Foreman-Mackey_2012}, corner \citep{corner}  and Matplotlib \citep{Matplotlib}.
The MeerKAT telescope is operated by the South African Radio Astronomy Observatory,
which is a facility of the National Research Foundation, an agency of the Department
of Science and Innovation.

redpajama_set_name: RedPajamaArXiv


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text:
The future of veterinary medicine is here. Are you ready?
Jessica Vogelsang, DVM
Youre not crazy. Change and innovation are happening faster everywhere, including veterinary practice. Heres your gut check so youre not left out of the glorious future of pet care.
(Shutterstock.com)In the Monday of the future, your receptionist picks up the phone and listens to Mrs. Jones request an appointment for her cat Fluffy because she suspects diabetes.
"Have you noticed any changes in her eating or drinking?" asks the receptionist.
"No," Mrs. Jones replies, "but her litterbox just told me she's lost a pound and has glucose in her urine." Your smart software handles the back-and-forth "does this day work for you?" logistics, freeing up your receptionist to check on Ranger, who's in because his collar reported a 25 percent decrease in activity over the last month. Instead of Googling a variety of incorrect differentials, Ranger's owners used your practice's convenient triage service and realized he needed to come in right away.
We live at a turning point in history when it comes to technology, and the future of veterinary medicine is overflowing with possibility and opportunity. There's only one thing standing in the way: veterinarians.
Digital evolution is crazy fast
The "drag me kicking and screaming only when there's no other option" approach has long been the default reaction of our profession when confronted with the changing face of technology, which wasn't a big issue when fundamental change took place over the course of decades and the only thing at stake was a phone with or without a cord. Now substantial evolutions happen in years or even months-and that changes everything.
Digital evolution is taking place not on a linear scale but an exponential one, a trend that didn't really take off until the turn of the millennium. Life in 1975 versus 1985 wasn't all that different, at least as far as available technology goes. In contrast, consider this: Just 10 years ago, smartphones didn't exist. (The iPhone debuted January 2007.) Every single way that mobile technology has changed our lives occurred in that time.
The fundamental nature of how clients obtain and process information has moved out of the clinic and onto the web. Who would have anticipated a decade ago that a client could broadcast your exam live on the internet and crowdsource the veracity of your diagnosis in real time using a phone? And if that's what's happened in the previous 10 years, what will 2027 look like for veterinarians?
Staying ahead of the game and anticipating the changes in store for us is a daunting prospect. Fortunately, Adam Little, DVM, has been thinking about it quite a bit. As director of veterinary innovation and entrepreneurship at Texas A&M University, it's his job and his passion to explore the future of the profession-and rather than finding it daunting, he's exuberant about what he sees in store.
'Most vet practices think they're innovating. They're not.'
In order to understand the types of changes Little speaks of when he says "disruptive innovation," one first needs to know what innovation truly means. Little points to the car company Tesla as an obvious example.
"If you talked to car companies five years ago, they'd all say they're innovative," he says. But while the big automakers were congratulating themselves on hybrids and Bluetooth integration, Tesla created the first all-electric vehicle that drove like a sports car, dissolved the traditional car dealership model and created self-driving technology that carries people over 1.5 million miles a day. But most importantly, says Little, Tesla created a model where cars can be upgraded and changed with the push of a button. That's disruption.
"Most vet practices think they're innovating," Little says. "They're not."
In other words, most of us are the Ford in this equation, sitting in bewilderment as our clients zip right on past us to the Tesla store.
(Shutterstock.com)3 biggies on the horizon
Despite the relatively static nature of veterinary practice as far as our day-to-day job functions, changes are happening all around us. Take Rover, for example, an on-demand pet-sitting company that recently raised $40 million in investments.
"The veterinary industry would never see that as being disruptive to their space," Little says. "[But] HBO didn't see Netflix as a disruptor."
Little is looking at technology and apps like Rover and extrapolating how they might apply to veterinary medicine in three distinct categories:
1. Clients want to gather data faster. Little sees big changes already happening in American homes. Such wearable technology as the Whistle and Voyce monitoring collars have already established themselves in the consumer space as the dog version of activity trackers, but the next level goes even further-into diagnostic samples.
"Where information is generated has shifted from the four clinic walls," Little says. Sure, a collar that tells you your dog's heart rate or hours spent sleeping is one thing, but what about that litterbox we dreamed up earlier that tells whether a cat is PU/PD or losing weight? Or a device paired with an app that allows owners to monitor a diabetic pet at home? They're coming.
"All of a sudden, tasks that were reserved to the domain of the clinic begin to shift," says Little. "An app similar to Rover that shows up overnight and offers the ability to manage diabetics at home changes things."
Consumers who visit that company's website enter a pet's name, signalment and a variety of other data points and out pops a personalized, pre-portioned homemade diet sent right to their doorstep.
2. Clients want genetically individualized veterinary care. Little's second area of interest is genomics, using a pet's individual genome to provide a perfectly individualized plan of care in everything from drug therapy choices to diet. While that technology is still on the horizon, Little points to a pet food company that delivers individually formulated pet foods directly to the pet owner's home as evidence that consumers are looking for this. Consumers who visit that company's website enter a pet's name, signalment and a variety of other data points and out pops a personalized, pre-portioned homemade diet sent right to their doorstep.
While this company caters to maintenance diets for now, Little predicts a future where owners and veterinarians in search of specialized diets can enter similar information and receive a prescription formulation via home delivery.
"We know prescription diet compliance is terrible after the first couple of bags," he says. "It gets difficult to manage that relationship." But in addition to individualized client choice, we can't underestimate their need and expectation for convenience-and how that in turn generates profit.
"Amazon Prime customers generate three times the revenue of regular customers," Little says. As veterinarians begin to incorporate genetic information into diet and therapy recommendations, highly personalized services like this will continue to grow.
3. Clients want telemedicine. You only need to look to your own doctor's office to see which way the winds are blowing. The number of medical video consultations is expected to grow fourfold in the next four years, according to research company Tractica. And if consumers want it for themselves, they'll want it for their pets too.
As things currently stand, the opportunity for veterinarians to practice medicine directly with new clients via mobile technology is limited by individual state practice acts, many of which (but not all) require an in-person examination to establish a valid veterinarian-client-patient relationship (VCPR). This has led to a confusing mishmash of on-demand websites and apps offering everything from advice posing as entertainment to second opinions and triage, skirting in and out of questionably legal territory.
Veterinarians may never agree to virtual physical exams the way human medicine has accepted them, but there are still huge opportunities for telemedicine to make a difference. While many practice acts forbid using telemedicine to establish a relationship with a new client, they say nothing about using it to provide improved care to current clients. Nor do they mention teleconsulting, in which referring DVMs do remote consults with specialists.
This is what has Little really excited, as it's a win for clients, DVMs and specialists alike. A pet owner who may or may not go through the process and expense of making a separate appointment at a different specialty facility may be more than happy to have their regular veterinarian talk to that specialist right there during their appointment.
"Specialists would love the opportunity to be a part of those decisions" in which clients would normally decline referrals, says Little. "We want to create a model where the collaboration between specialists and clinicians happens in real time on a micro scale."
'It's not about an app; it's a relationship business'
For those who cringe at the idea of keeping up on apps and technologies in the coming years, Little reminds us that the technology itself will always play a secondary role in the veterinary profession-and that's a good thing.
"It's not about an app, it's a relationship business," he emphasizes. "People trust us and really do enjoy those individual relationships. These are tools to help us build that.
"While the models will need to change, there are opportunities for veterinarians to be a part of that. I believe the goal of veterinarians is to provide the best care to the most number of patients. We want a better life for animals and the people who depend on them."
And that will never change.
Jessica Vogelsang, DVM, is a certified veterinary journalist, a regular contributing writer for a number of publications, author of the memoir All Dogs Go to Kevin, and creator of the popular blog Pawcurious.com.
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redpajama_set_name: RedPajamaCommonCrawl


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Juvenile Justice Ministry RSS Feed
Category: Juvenile Justice Ministry
Covid Positives
12/1/2020 in Category Juvenile Justice Ministry
By Claire Froehlich, Juvenile Justice Ministry Chaplain
"She has improved immensely and mentoring has been a key factor." – Court liaison for Caitlyn in a court post-trial conference.
A... More
Image Bearing Souls
By Natalie Baddour, Director of Mentoring and Therapeutic Services
Most of the young people within the Juvenile Justice System are battling extreme mental health disorders. We recently did... More
11/2/2020 in Category City Life, Juvenile Justice Ministry
By Preston Adams, Regional Director - Central Area
Jesus said, "Which of these three do you think was a neighbor to the man who fell into the hands of robbers?"
The expert in the law... More
You Just Never Know
When you first meet "Shay", it's impossible not to notice her big, boisterous personality and strong presence. With both parents... More
The Story is Not Over
By Tim Butler, Juvenile Justice Ministry Chaplain
"You were born a mistake. You are such a failure and always will be… you are a waste of tax dollars."
"You should be ashamed of... More


The Digifant 1 ecu uses closed loop lambda control for computing the injection amount at idle and part throttle (regulation to lambda 1). At full throttle (WOT, wide open throttle), the engine has to run richer than lambda 1 to produce more power / torque and to keep the temperature in the combustion chamer in a safe area. This WOT enrichment is achieved by deactivating the lambda control and computing the injected fuel out of the map data (open loop).
The firmware of the digifant ecu (program code) contains a "feature" which delays the reaction on pressing the WOT switch for 3-4 seconds. The closed loop lambda control is still active (and regulates to lambda 1) for the first 3-4 seconds on WOT. This "feature" is called "Digi-Lag". It leads to a poor reaction on flooring the gas pedal (gasoline engine produces it's maximum torque around lambda 0.85 - 0.9 and not at lamda 1!).
All factory chips and many tuning chips contain the Digi-Lag!
I changed the program code to get instant WOT enrichment and succeeded in completely removing the "digi-lag"!
Beyond that, I'm able to activate the WOT open-loop (WOT enrichment) on part throttle based on a boost threshold (i.e. activate WOT-enrichment if boost >= 0.8 bar / 11 psi).
Some nice graphs to get the point better.


I am loving ..carring..honest.. Easy going.. Easy to get on with!!
Im staying in Thailand quite a long time. Work in the Medical Proffesion, With people with Brain Difficulties and challenging behaviour. Im smart, funny, and very respectable and caring. I love Thailand and what the people stand for.
I am a very happy man seeking a nice lady.
Am a normal hard working desent bloke.Who would love too share my life with someone.


The first generation of the Morris Isis was produced between 1929 and 1935 and was the only pre-war iteration of the model. The Isis received a facelift for 1932 to resemble the model offered here. It's one of the more attractive Morris' I've ever seen.
The Isis was the six-cylinder Morris with a 2.5 liter engine making 17.7 horsepower – enough to propel the car to 65 mph. It has a steel body over an ash frame – very British. 3,467 facelifted Isis models were made, including the deluxe Twenty-Five version with the larger 3.5 liter engine.
The car offered here shows very well with it's attractive red coachwork. Mechanicals were "freshened" in 2006 and the interior is quite nice as well. Bonhams estimates this car between $15,000-$24,000, which sounds quite reasonable for something that looks this nice. For more on this car, click here and for more on Bonhams in Oxford, click here.
This entry was posted in Uncategorized and tagged 1933, auction, Bonhams, car, classic, classic car, coupe, isis, morris by classiccarweekly. Bookmark the permalink.


I attended the BRIT school of performing arts and Vocaltech where I studied extensively both vocal technique and composition. I have years of experience as a solo artist and as a backing singer, both in recorded sessions and live. I aim to bring out the natural voice and ability of my students, tailoring all of my lessons to the need of each individual so that each lesson is challenging and varied. In a friendly and relaxed environment, we will cover: breathing technique, posture and support, developing tone, extending your range, chest and head voice and working through any breaks in the voice, projection, increasing confidence, learning a repertoire of songs, and learning how to develop an overall stronger and healthier voice.
Say hello to Valentina, enquire about an activity, arrange a date.


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Olive: Spouses Should Be Introduced To Parents When A Relationship Is Solid Enough
Getting your significant other to meet with your parents can be a nervous feeling. It brings about questions of whether or not they will like your partner and how their opinions might affect or validate your relationship.
Today on The Fatboy Show, while discussing the right time to introduce your significant other to your parents, Olive Najjuma a presenter of the show opined that it should be done when both partners feel ready.
"When you introduce your partner too early to your family before the relationship is on solid ground, it won't work out well. You need to bring in third parties at the point where both of you want and feel the same way about the decision," she reasoned.
As a parent, she noted that her considerations for her future daughter/sons-in-law will have to do with which families they come from, and their level of education regardless of political party affiliations or tribe.
"I've seen parents that deter their children from marrying from a certain family because of its political or religious affiliations and most times the girl ends up unmarried. Meanwhile, the guy gets to marry and live a happy marriage while the lady can reach her 40's unmarried, bitter, and sometimes without a child," she said.
She, therefore, suggested that parents must overlook some issues and embrace the character of the person than their family's political or religious beliefs.
Also, Pacific, a contributor to the show reasoned that meeting the parents of one's significant other should be in a serious relationship one that is potentially aimed at marriage.
"If you're serious, the grace period should be between 6 months - 1 year. Then we expect the relationship to be intentional about marriage and not just the laissez-faire economy of trying out things to see how they'll go," he said.


Thank you for the suggestion, I'll give it a shot. Just concerned as this has not happened before, and really bogs down the production of getting edits done. Why would the retail version suddenly stop? If there are significant improvements, why not make a release to help eliminate issues like I'm having if that be the fix?


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Q: how to keep telegram bot users language? I created a telegram bot with Nodejs and node-telegram-bot-api package and Mongodb for the database. Now, I want to add multiple languages in the bot and I want to keep the user language selected, and back data with his language.
I know I can save user language on a field in Mongodb. but I wanna know if there is any best way to keep that for the best performance. because there are many responses to users, I must check the language users every second with this method.

redpajama_set_name: RedPajamaStackExchange


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George Ripley (1415 k.-1490)  Ágoston-rendi szerzetes, egyike a legfontosabb angol alkimistáknak.

Életútja
Valószínűleg a Yorkshire-ben fekvő bridlingtoni Szent Ágoston perjelség kanonoka volt. Franciaországban és Olaszországban tanult és egy ideig Rómában élt, ahol a pápától 1477-ben kamarási címet kapott. VIII. Ince pápa oldozta fel kanonoki kötelezettségei alól, hogy behatóbb tanulmányokat végezhessen. 1478-ban a transzmutáció titkának állítólagos ismeretében visszatért Angliába. 100 ezer fonttal támogatta a máltai lovagokat a törökök elleni harcukban, ezért azt feltételezték róla, hogy birtokában volt a bölcsek kövének.  Munkáját az apát és a többi kanonok nem nézte jó szemmel, ezért elhagyta rendjét és belépett a bostoni karmelitákhoz, ahol 1490-ben meghalt.

Cantilena című művében allegorikusan írja le a színek játékát, melyet olyan asszony bőrének elszíneződéséhez hasonlít, akinek a királyt kell megszülnie. A rothadás (putrefactio) művelete után azt mondja: "Igen sokszínűnek látszott az asszony bőre/ feketének és zöldnek és hamuszürkének,/ és a páva farkához hasonlónak/ vérvörösnek is, mint a vér, ha kiömlik az erekből".

Művei

A Zedler lexikon (1742/XXXI.) 15 művét sorolja fel. A legismertebbek:

Compound of Alchymie… divided into 12 gates. London, 1591
Opuscula chemica. Frankfurt, 1614
Chimische Schriften. Erfurt, 1624, Nürnberg, 1717
Opera Omnia. Kassel, 1649
Medulla Alchymiae. (1476) London, 1692

Jegyzetek

Források
George Ripley a The Alchemy web site-on

További információk
 Ripley művei

Kapcsolódó szócikkek
Bölcsek köve

Alkimisták
1490-ben elhunyt személyek

redpajama_set_name: RedPajamaWikipedia


A1 Envelope Dur-O-Tone Packing Brown Wrap: 100% recycled, kraft or grocery bag brown 3 5/8 x 5 1/8 envelope with smooth, printable surface. Paper has subtle inclusions for a unique appearance. Contemporary, rustic style and smooth surface make this envelope perfect for use as a response or note card envelope. Envelopes are made in America. The Dur-O-Tone collection is inspired by everyday utilitarian papers.


Vending machines like this one have been networked to create a wireless security blanket.
Currently on test in Osaka, the ' Robot Locator ' system from Network Security Japan (NSJ) is intended to showcase the company's location technology and its ability to provide a rapid response to dangerous situations faced by its customers.
NSJ's squadron of bicycle-pushing security staff patrol part of Osaka known as America Village waiting for an alert from their control room in response to a customer setting off an alarm in the form of a small IC tag they have been issued with.
Anyone who isn't an NSJ customer can also benefit from the service if they can find one of the networked machines in time and press the alert button mounted on the front of each one. As America Village is popular with tourists and school-trip parties, NSJ sees it as an ideal opportunity to put its services in the shop window.


This is a scalable context timeline. It contains events related to the event August 28, 2001: CIA Fails to Connect KSM, Bin Al-Shibh, and Moussaoui. You can narrow or broaden the context of this timeline by adjusting the zoom level. The lower the scale, the more relevant the items on average will be, while the higher the scale, the less relevant the items, on average, will be.
FBI agents are able to get prisoner Abu Zubaida to confess that Khalid Shaikh Mohammed (KSM) is the mastermind of the 9/11 attacks. Zubaida, captured in Pakistan in March 2002 (see March 28, 2002), is considered a high-ranking prisoner, although there is controversy over just how important he was to al-Qaeda (see Shortly After March 28, 2002). FBI agents Ali Soufan and Steve Gaudin have been interrogating Zubaida using traditional non-coercive interrogation methods, while he is being held in a secret CIA prison in Thailand (see Late March through Early June, 2002). A CIA interrogation team is expected but has not yet arrived. Soufan shows Zubaida a sheaf of pictures of high priority suspects; Zubaida points at KSM's photo and says, "That's Mukhtar… the one behind 9/11." Shortly before 9/11, US intelligence had learned that the alias "Mukhtar" was frequently used by KSM (see August 28, 2001), but this is important confirmation. The more stunning revelation is that KSM is "the one behind 9/11." [Mayer, 2008, pp. 155; New York Times, 4/22/2009; Newsweek, 4/25/2009] Zubaida proceeds to lay out more details of the 9/11 plot, although exactly what and how much he says is unknown. [Vanity Fair, 7/17/2007] It is unclear how much US intelligence knew about KSM's role in the 9/11 attacks prior to this, although at least some was known (see (December 2001)).
Abu Zubaida circa 2008. [Source: Defense Department.]In a speech defending the US treatment of high-level al-Qaeda prisoners, President Bush apparently makes some false claims about how valuable the intelligence from some prisoners was. He says that Abu Zubaida, who was captured in March 2002 (see March 28, 2002), revealed that 9/11 mastermind Khalid Shaikh Mohammed (KSM) used the alias 'Mukhtar.' "This was a vital piece of intelligence that helped our intelligence community pursue KSM." However, the 9/11 Commission's final report published in 2004 revealed that the fact that KSM had that alias was known to US intelligence before 9/11 (see August 28, 2001). Bush also claims that Zubaida's interrogation identified Ramzi bin al-Shibh as an accomplice in the 9/11 attacks. [New York Times, 9/8/2006] However, this was known months before Zubaida's capture, and reported in the US press as early as September 2001. A CBS News report from that time said bin al-Shibh was "believed to have provided logistics backup for the hijackers." [CBS News, 9/29/2001] Bush also describes the interrogation techniques used on the prisoners as "safe, lawful and effective," and he claims torture was not used. However, the New York Times notes that "the Bush administration has yet to make public the legal papers prepared by government lawyers that served as the basis for its determination that those procedures did not violate American or international law." [New York Times, 9/8/2006] Both the New York Times and Washington Post publish prominent stories pointing out the factual errors in Bush's statements, but this does not become a big political issue. [Washington Post, 9/7/2006; New York Times, 9/8/2006] Bush repeatedly exaggerated the importance of Zubaida in the months after his capture as well (see Shortly After March 28, 2002).


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Police: Man who shot 3 at Arizona complex felt bullied
Jacques Billeaud and Terry Tang
Glendale, Ariz. – A 20-year-old gunman who opened fire at a suburban Phoenix restaurant and retail complex, injuring three people, wanted to target people in his own age group in retaliation for bullying he had suffered, authorities said Thursday.
The shooting rampage occurred Wednesday night after suspect Armando Hernandez scoped out Westgate Entertainment District in Glendale, returned to his car to make a social media video and loaded three rifle magazines, police said.
"I'm going to be the shooter of Westgate 2020," Hernandez said in his Snapchat video, holding a beer in one hand. The footage also shows an AR-15-type rifle in the backseat of his car.
"Let's get this done, guys," he said.
Hernandez later surrendered, telling detectives that he intended to harm 10 people, though it's unclear why he chose that number.
"He wanted to gain some respect, and he felt that he had been bullied in his life," Glendale police Sgt. Randy Stewart said.
Police say Hernandez filmed the attack while holding a cellphone with his left hand and blasting away with the rifle in his right hand.
The first two victims were shot outside a restaurant. As he moved through the complex, police say Hernandez fired shots to intimidate people before shooting the third victim.
Hernandez surrendered without incident to one of the first officers who arrived at the complex adjacent to the stadium where the NFL's Arizona Cardinals play.
Authorities say Hernandez wounded a 19-year-old adult man, who remained hospitalized in critical condition. A 16-old-girl was taken to a hospital with non-life-threatening injuries. A third victim, a 30-year-old woman, did not require hospitalization.
Hernandez, who lives in the neighboring suburb of Peoria, could face more than a dozen felony charges. It was not immediately known if he had a lawyer who could speak on his behalf.
It's not yet clear how many shots were fired. Investigators say Hernandez moved around the complex during the attack, leaving spent shells in different spots.
"Armando stated multiple times he wanted to do this mass shooting for respect and spoke of interest in previous mass shootings that have occurred in recent years," police said in court records.
Bystander video posted on social media showed people running from the area and embracing after being reunited.
Eliana Rivera, a sales associate at a pottery painting store, posted on Twitter that she and a co-worker heard police running and helicopters overhead as they huddled in the back of the business, the Arizona Republic reported.
"It's just unreal," Rivera said. "You see it on the news. It happens and you never really think that you would be put in that position. You just think, 'What are the chances that that will happen?' – and it did."
State Sen. Martin Quezada was at the complex during the shooting and wrote on Twitter that he witnessed a person "with an AR-15 shoot up Westgate."
In a statement Thursday, the Democratic lawmaker said he was "beyond horrified what I saw with my own eyes." He then called for more gun control.
"Civilians should not have access to these types of weapons and be able to walk into a mall to destroy people's lives in a matter of seconds," Quezada said.
Associated Press writers Jonathan J. Cooper and Paul Davenport in Phoenix contributed to this report.


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The full story...
Millions wasted by low performing IVF clinics, says leading fertility expert
Norman Swan reported this story on Monday, May 11, 2015 08:10:00
Listen to MP3 of this story ( minutes)
| MP3 download
MICHAEL BRISSENDEN: A leading expert in in vitro fertilisation (IVF) claims that tens of millions of dollars of Medicare funds is wasted each year, and perhaps around $20 million in personal out of pocket expenses by couples because they're unknowingly attending low performing IVF Clinics.
And even though the performance information is available on each clinic, it's being kept from infertile couples by the Fertility Society of Australia.
Norman Swan of RN's Health Report has the story.
NORMAN SWAN: In 2012 there were 62,000 IVF cycles - that's an implantation of an embryo - performed on 33,000 women with 12,000 live babies being produced.
Australia has been a global leader in IVF and has kept detailed performance information on each IVF clinic for several decades. Clinics only know what their own take home baby rate is and consumers know nothing.
Yet, as Dr Richard Henshaw of Monash IVF argues, there's a huge gap between the highest performing clinics and the lowest ones.
RICHARD HENSHAW: In 2012 the top clinic had a live birth rate of 30.9 per cent and the bottom clinic had a live birth rate of only 4 per cent.
NORMAN SWAN: What does that mean for the patient?
RICHARD HENSHAW: Well that means that if you are a patient in the lowest performing clinic you have to undergo seven times more treatment cycles than if you're a patient attending the top clinic.
For Medicare, clinics in the top 25th percentile cost around $2 million of Medicare money to produce a 100 live births, whereas clinics in the bottom 25th percentile spend around $6 million to produce a 100 live babies.
NORMAN SWAN: And that's not to mention the out of pocket costs of the individual, because there's a gap for a lot of clinics.
RICHARD HENSHAW: There is. There's around about a $4000 gap paid by the patients.
NORMAN SWAN: So the Government's out of money and you can be out of money as a couple by anything up to $20,000 per live birth.
RICHARD HENSHAW: That's right.
NORMAN SWAN: And so if you gross that up, what does that come down to in terms of potentially wasted money by the Commonwealth?
RICHARD HENSHAW: The answer is an excess cost of around about $50 million, just in Commonwealth outlay.
NORMAN SWAN: Surely the public have a right to know so they can choose the best place to go.
Why does the fertility society refuse to make this available? What have they told you?
RICHARD HENSHAW: That it's not policy currently.
NORMAN SWAN: And is this lobbying by the low performing clinics or what?
RICHARD HENSHAW: Well of course the only people who really benefit from the data remaining covert are the low performing clinics. Patients don't benefit, Medicare doesn't benefit, and of course good quality providers don't benefit. That can only be the answer, that it is lobbying by the low performing clinics.
The thing that frightens me though is that here we are collecting these data and we're keeping them covert.
NORMAN SWAN: With me on the line is associate professor Mark Bowman of the University of Sydney, he's also president of the Fertility Society of Australia.
Mark Bowman, what do you think of what Richard Henshaw's been saying, that essentially we owe it to the Government as the payer, and we owe it above all to women and their partners to tell people what are high performing clinics and what are low performing clinics?
MARK BOWMAN: I think we would all agree that patients and society need the best results and the best information they can to help them about the decisions regarding treatment and about outcomes. The challenge is that it's not quite as simple as using very simple numerators and denominators because there's so many differences. Some infertility problems are more difficult to treat with IVF.
NORMAN SWAN: Richard Henshaw is alleging that basically the low performing clinics are jacking up and the fertility society's run for cover.
MARK BOWMAN: Don't get me wrong. I personally do believe that a reasonable amount of benchmarking is an appropriate process. I'm just simply saying that a simple numerator and denominator is not necessarily going to help our patients who might then make a choice on a small percentage between unit A or unit B.
NORMAN SWAN: But a 32 per cent chance for a success cycle compared to a 4.7 per cent is not a small percentage.
MARK BOWMAN: And I do believe that that warrants some analysis.
MICHAEL BRISSENDEN: Associate professor Mark Bowman from the Fertility Society of Australia ending that report by Norman Swan. And you can hear the full story on Radio National's Health Report.
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RN Health Report: Fertility Data Kept from Public
Around 500 Indigenous people fought in the First World War, and as many as 5,000 in the second. But many Aboriginal and Torres Strait Islander diggers who made it home received little or no recognition for their contribution. On Anzac Day, 2007, the first parade to commemorate their efforts and bravery was held in Sydney. Listen to our report from that day by Lindy Kerin.
From the archives: Indigenous soldiers get their own parade
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Jean II de Żagań  connu également sous le nom de  le Fou, le Sauvage ou le Cruel (; né le  – † ), fut duc de Żagań-Przewóz à partir de 1439, avec ses frères comme corégents jusqu'en 1449, à partir de 1449 duc de Przewóz avec son frère cadet comme corégent, entre 1459/1461 et 1467 et entre 1472 et 1476, duc de Żagań et entre 1476 et 1488 duc de la moitié   puis de la totalité de Głogów après la réunification du duché en 1480.

Biographie

Origine
Jean II est le quatrième et plus jeune fils du duc  Jean  de Żagań et de son épouse Scholastika, fille de l'Electeur de Saxe  Rodolphe III de Saxe.

Duc de  Żagań
À la mort de son père en 1439, Jean II est encore mineur et confié avec son frère Venceslas sous la garde de leurs frères ainés Balthasar et Rodolphe. la division du patrimoine familiale n'est effective qu'en 1449. Jean II et Wenceslas reçoivent conjointement Przewóz. Cependant comme à cette époque Venceslas apparait comme mentalement déficient, la totalité du pouvoir dans le duché est exercée par Jean II.

Le petit  Duché de Przewóz est insuffisant pour satisfaire les ambitions de  Jean II, qui réclame une répartition en part égale des domaines à ses autres frères. Le conflit se termine en 1453, après une médiation de l'Électeur Frédéric II de Saxe. Après la mort de  Rodolphe le  et en l'absence de Balthasar, qui est en Prusse, Jean II s'autoproclame  « Gouverneur du duché de Żagań ». Il contraint les États du duché à lui payer un tribut, une disposition qui est ensuite annulée lorsque Balthasar revient en Silésie en 1458.

Le conflit entre les frères éclate de nouveau en 1461, lorsque Jean II, avec l'appui de Georges de Poděbrady, roi de Bohême envahit Żagań et dépose son frère, qui est obligé de s'enfuir. La même année après que meurt leur mère de leur mère  Scholastique de Wettin, Jean II hérite de son douaire ou Oprawa wdowia constitué de la cité de Nowogród Bobrzański. Sept ans plus tard en 1468, Jean II perd Żagań, lorsque Balthasar reprend le contrôle du duché avec l'aide de leur cousin Henri XI de Głogów.

Jean II est déterminé à recouvrer le duché de Żagań. Avec l'aide du roi de Hongrie Matthias Corvin dont il reçoit  florins pour recruter une troupe afin de l'aider à combattre la couronne de Bohême, il envahit victorieusement le duché en 1472. Balthasar est capturé et emprisonné, d'abord dans le village de Witoszyn et ensuite dans la tour du château de Przewóz, où selon certaines sources Jean II le laisse mourir de faim († ).

Toutefois inexplicablement après avoir régné seulement quelques mois à Żagań, il vend son duché de Żagań incluant Przewóz, aux électeurs corégents de Saxe Ernest et Albert III, pour  florins hongrois (). Cet accord, confirmé par Matthias Corvin, garantit également une pension substantielle à son frère Venceslas († ). Depuis lors le duc Jean II est surnommé  « Jean Sans-Terre » (polonais: Janem Bez Ziemi). Malgré la vente de son duché, Jean II cherche à continuer à jouer un rôle actif dans les affaires politiques. En 1474, à la tête de troupes recrutées par Matthias Corvin, il organise une expédition réussie en Grande-Pologne, d'où il  rapporte un important butin ce qui est bien entendu néfaste à ses bonnes relations avec la Pologne.

Succession de Głogów
Le   Henri XI de Głogów meurt, sans doute empoisonné par des agents brandebourgeois. Selon sa volonté il laisse ses possessions à sa très jeune veuve de 12 ans Barbara, fille de Albert III Achille de Brandebourg avec éventuelle réversion de ses domaines à la famille de cette dernière. Comme héritier male de la lignée des Piast, Jean II refuse de reconnaitre les dernières volontés du défunt duc et réclame la succession de son cousin. Mais il doit faire face non seulement aux prétentions d'Albert III Achille de Brandebourg mais aussi au roi Casimir IV de Pologne et à Matthias Corvin de Hongrie !

Pendant la premier phase de la guerre qui s'ensuit, Jean II reçoit l'appui de  Matthias Corvin. Ses opposants de leurs côtés bénéficient du support du Brandebourg et de la Pologne.  Le mariage entre le roi  Vladislas II de Bohême, fils ainé de Casimir IV de Pologne et Barbara de Brandebourg est célébré le 20 aout 1476; cependant cette union ne sera jamais consommée et est donc invalide selon le droit canon. Jean II use de l'influence du roi de Bohême à Rome et tente d'obliger les habitants de Głogów à lui prêter un serment de fidélité, mais ils refusent et le Légat du Pape doit lancer une excommunication sur la cité Głogów, qui est forcée de se soumettre à Jean II. Jean II obtient un autre succès lorsque le , les États de Żagań lui rendent l'Hommage féodal. Le jour suivant le 8 décembre, Matthias Corvin déclare que les dernières volontés d'Henri XI sont nulles et non avenue prétextant que la teneur de son testament révélait son incapacité mentale. Au cours de la période 1476–1477 les troupes Bohémienne prennent Szprotawa et Kożuchów seul Krosno Odrzańskie demeure sous le contrôle de  l'Électeur Albert III Achilles. Au début de 1477 une trêve est proclamée entre les parties combattantes elle se poursuit jusqu'en . La guerre reprend au printemps 1477 et débouche sur des années de combats entre les troupes Bohémiennes et Hongroises.

Cette poursuite des hostilités affaiblit la position de Jean II et ses alliés hongrois, les habitants de Głogów lui maintiennent néanmoins leur fidélité. Les diverses parties tentent finalement de régler leurs différends par un accord diplomatique mais à l'automne 1477 la trêve est de nouveau rompue par Jean II, qui envahit les possessions du Brandenbourg et assiège même Berlin et Francfort-sur-l'Oder. La guerre se poursuit jusqu'en 1478 avec plusieurs batailles rangées importante. Pendant ce temps, l'ex-duc de Żagań renforce la position à Głogów.

À la mi-1479 une nouvelle trêve est conclue entre les belligérants, elle prévoit que l'électeur Albert III Achille abandonne ses prétentions sur le duché de Głogów contre le versement d'une somme de 50.000 florins. Les négociations entre Jean II et l'Électeur de Brandebourg se poursuivent jusqu'en septembre 1482. Jean II obtient Głogów sans le château et une partie de la cité dévolue initialement à la lignée de  Cieszyn qui était désormais intégrée à la Couronne de Bohême. La partie nord du duché avec Odrzańskie, Sulechów, Świebodzin et Lubsko) est annexée par le  Brandebourg. Avec l'accord de  Matthias Corvin, Jean II règne sur Głogów pendant le reste de sa vie mais après sa mort le duché revient au roi de Hongrie ou à ses descendants. En 1480 Jean II tente de réunifier le duché de Głogów qui avait été divisé en deux parties entre les dynastes Piast. Après un siège de sept semaines le  mai 1480 le château est conquis et le duché réunifié après 150 années de divisions.

Tous ces succès attisent l'ambition de Jean II et l'entraine à s'opposer à  Matthias Corvin. Le duc de Głogów devient alors l'allié de Henri  l'Ainé de Poděbrady, fils de Georges de Poděbrady duc d'Oleśnica; Leur alliance prévoit le mariage des trois fils d'Henri  avec trois filles de Jean II. L'union de l'ainée de filles célibataires de Jean II, Salomé, avec le fils ainé d'Henri Ier Albert est célébrée le  à Głogów.  Un an plus tard le , sont célébrés les fiançailles officielles des deux autres filles de Jean II, Hedwige et Anna, avec les deux autres fils d'Henri Ier, Georges  et Charles . Lors de la cérémonie, Jean II proclame son intention de léguer le duché de Głogów à ses trois gendres après sa mort.

Cette disposition déclenche une réaction immédiate de Matthias Corvin qui lui déclare la guerre. Jean II cherche coute que coute à obtenir de ses sujets qu'ils prêtent un serment de fidélité à leurs futurs souverains. Malheureusement il rencontre une forte résistance, même après avoir prononcé une sentence de mort contre sept membres du conseil de la cité. En mai 1488 le siège de Głogów commence, il se termine par la capitulation de Jean II en novembre de la même. Le duc déposé renonce à droits sur le duché de Głogów contre  20,000 florins. Matthias Corvin investit alors du duché son fils illégitime Jean Corvin.

Fin de vie
Au cours des années suivantes il tente sans succès  d'obtenir une parcelle des duchés de Silésie afin d'y passer le reste de sa turbulente existence. Il réclame Ścinawa ou même son retour à Głogów, sans aucun succès. En 1497 Henri  l'Ainé lui donne la cité de Wołów comme fief pour le reste de ses jours. Jean II meurt à Wołów le , avec lui s'éteint la lignée male de Żagań-Głogów. Il est inhumée dans l'église paroissiale locale. Figure controversée, Jean II est loué par quelques historiens pour ses  ambitions et critiqué par d'autres pour sa politique irréaliste et ses idées folles.

Union et postérité
En 1462 Jean II épouse Catherine (née 1443 –  † ), fille du duc Guillaume d'Opava. Ils ont cinq filles:
 Margarete (née 1465/1472 – †  après 1502), épouse le   Miklós Bánffy de Limbow et ensuite vers 1500 Johann Hampo.
 Salomé (née 1475/1476 – † ), épouse le  Albert  de Poděbrady, duc de Ziębice-Oleśnica (Münsterberg-Oels), et en secondes noces en 1512/1513 Johann IV de Kurzbach, Baron de Trachenberg-Militsch (†  ).
 Hedwige (née septembre/ – † Ziębice, ), épouse en   Georges Ier de Poděbrady, duc de Ziębice-Oleśnica (Münsterberg-Oels), puis le   Sigismund, Baron de Wartenberg.
 Anne de Żagań, épouse le   Charles Ier de Poděbrady, duc de Ziębice-Oleśnica (Münsterberg-Oels).
 Barbara (née vers 1480 – †  ), Abbesse en 1495 de Sainte-Claire de Strzelin.

Sources
 , édition du 30 aout 2014.
  &  Peter Truhart, Regents of Nations, K. G Saur Münich, 1984-1988 , Art. « Sagan (Pol. Zagan) »,  2.454.
  Europäische Stammtafeln Vittorio Klostermann, Gmbh, Francfort-sur-le-Main, 2004 ,  Die Herzoge von Glogau †1476, und Sagan †1504 des Stammes der Piasten  Volume III Tafel 13.
 Pour la chronologie cf.  Jan II Szalony (Okrutny, Dziki, ZŁy)

Piast
Duc de Silésie
Naissance en juin 1435
Décès en septembre 1504
Décès dans la voïvodie de Basse-Silésie
Décès à 69 ans
Duc de Głogów
Duc de Żagań


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