text:
\section{Introduction}
While early studies of low surface brightness{} galaxies date back several decades \citep[e.g.][]{impey88,turner93,dalcanton97}, the discovery of a substantial number of very large diffuse galaxies in local clusters has drawn significant attention in recent years. \citet{vandokkum15} identified a significant population of large ($r_{\mathrm{eff}} \geq 1.5$ kpc) galaxies with low central surface
brightness ($\mu(g,0)=24-26\,\mathrm{mag\,arcsec^{-2}}$) in the Coma cluster, and dubbed them ultra-diffuse galaxies (UDGs).

\bigskip
In the two years since, several studies reported populations of UDGs in different galaxy cluster samples. Some of the studies focused on the nearby and well-studied Fornax \citep{munoz15}, Coma \citep{koda15, kadowaki17} and Virgo \citep{mihos15, davies16} clusters. Other works studied more distant clusters \citep{vdB16, roman16a}, including a Frontier-fields cluster at $z=0.3$ \citep{janssens17}. It is clear that UDGs are a common phenomenon in galaxy clusters, and their ubiquity has to be understood in the context of existing galaxy formation models.

Several explanations have been proposed in the literature to explain this class of galaxies. \citet{vandokkum15} suggested that UDGs are some sort of ``failed'' galaxy, with a relatively massive dark matter halo like the one expected for an $L^{*}$ galaxy. Such a high mass would make them capable of surviving very close \citep[down to only $\sim$ 300 kpc, see][]{vdB16} to the cluster centres. They may have stopped forming stars due to feedback processes, which is again possibly related to a massively overdense environment. \citet{yozinbekki15} were indeed able to reproduce several observed properties of UDGs with a simple tidal disruption model that linked their presence exclusively to dense environments.

There are also less exotic explanations. \citet{amorisco16} claimed that UDGs can be readily explained by a standard model of disk formation, and that UDGs are simply the extremes of a continuous distribution in size and luminosity. This would mean that they also exist in lower-density environments, albeit with possibly more disk-like morphologies. \citet{rong17} reached a similar conclusion based on semi-analytic galaxy formation models. \citet{dicintio17} used the NIHAO simulation to study the formation of UDG-type galaxies, and they also suggested that internal processes, particularly outflow-driven feedback, may be responsible for their formation. Feedback from active galactic nuclei in dwarf galaxies may also contribute to their diversity and range in morphologies \citep{silk17}.

Such opposing formation scenarios make different, observationally testable, predictions. One way forward is to investigate the failed-galaxy scenario by measuring the halo masses of UDGs. \citet{beasley16} estimated the mass of a UDG in the Virgo cluster based on the velocity dispersion of globular clusters that are associated with this galaxy, finding that they are overly massive for their stellar mass. Several other studies also based their mass estimates on the specific frequency, or number, of globular clusters in UDGs, reaching contradictory conclusions \citep{amorisco16b,penglim16,beasley16b}. \citet{dokkum16} measured a velocity dispersion of UDG DF44, which, upon extrapolation to $M_{200}$, is consistent with a halo mass of $M_{200}\sim 10^{12}\, \mathrm{M_{\odot}}$. In contrast, \citet{sifon17} measured the average mass associated with UDGs directly using weak gravitational lensing, finding that this is too low for them all to be ``failed'' massive galaxies. It is likely that the range of mass measurements of individual UDGs is real, and not (only) due to different assumptions in the analyses of different studies. This may hint at a range of different formation scenarios for UDGs \citep[e.g.][]{zaritsky17}.

Another critical observational study to constrain the formation scenario of UDGs is to measure their abundance in different environments. \citet[][hereafter vdB16]{vdB16} studied eight clusters that span an order of magnitude in halo mass, and found that the number of UDGs per cluster scales as $N_{\mathrm{UDG}}\propto M_{200}^{0.93\pm0.16}$, where $M_{200}$ is the virial mass of the host halo.
At even lower halo masses, \citet{roman16b} were able to identify UDGs in three nearby groups by making use of three-band photometric data from the SDSS deep Stripe 82. Their work shows that UDGs are also present in galaxy groups, with an abundance that is close to what is expected from extending the vdB16 relation down to lower masses. We note, however, that the sample they studied are the three Hickson Compact Groups \citep[][]{hickson82} that overlap with the SDSS Stripe 82, and this may not be fully representative of the underlying population of groups.

In this work we perform a systematic search for UDGs in galaxy groups, considering the 197 deg$^2$ overlap region between the optical-imaging Kilo-Degree Survey (KiDS) with the groups identified with the Galaxy And Mass Assembly (GAMA) spectroscopic survey. The groups are selected completely independently of their possible UDG content. By performing an analysis that is close to the one presented in vdB16, this allows us to study the abundance of UDGs in a consistent way over a wide halo mass range. A previous search for galaxies with low surface brightness was performed using SDSS imaging of the same GAMA region \citep{williams16}. Based on the locations of these galaxies, it was shown that they are plausibly associated with the $z<0.1$ large-scale structure \citep[Fig.~6 in][]{williams16}. In this paper we quantify this statement, focusing on the GAMA groups, based on the standardised galaxy selection using the UDG criteria on the KiDS imaging.

The paper is organised as follows. Section~\ref{sec:dataoverview} provides an overview of the data products from the KiDS and GAMA surveys we used. Section~\ref{sec:analysis} describes the selection of our sample of UDGs, and how we associated them with the galaxy
groups in a statistical sense. Section~\ref{sec:results} presents our main results: the abundance of UDGs as a function of halo mass, and the size distribution of the UDG sample. We discuss our findings in Sect.~\ref{sec:discussion}, compare them to previous work and to a measurement of the mass-richness relation for the same GAMA groups to help to interpret our results. We summarise in Sect.~\ref{sec:conclusion}. Appendix \ref{sec:robustness} lists several robustness tests that we performed to increase our confidence in the presented results.

All magnitudes we quote are in the AB magnitudes system, and we adopt angular-diameter and luminosity distances corresponding to $\Lambda$CDM cosmology with $\Omega_{\mathrm{m}}=0.3$, $\Omega_{\Lambda}=0.7$ and $\mathrm{H_0=70\, km\, s^{-1}\,  Mpc^{-1}}$. Unless stated otherwise, error bars denote 1$\sigma$ uncertainties, or 68\% confidence intervals.

\section{Data}\label{sec:dataoverview}
Our UDG candidates are identified using photometric data from the Kilo-Degree Survey \citep[KiDS,][]{kidsdejong15}, and the environments (in particular group properties) are characterised using the Galaxy And Mass Assembly \citep[GAMA,][]{drivergama11} group catalogue. In this study, we focus on the overlap region between the KiDS and GAMA surveys, totalling an effective (unmasked) area of 170 square degrees, and the surveys are summarised in turn below.

\begin{figure*}
\begin{center}
\resizebox{0.9\hsize}{!}{\includegraphics{allfields_show}}
\caption{Three KiDS+GAMA fields. The combined KiDS area is shown in white, with regions masking bright stars, reflective haloes, and image artefacts shown in grey. Circles indicate the actual angular sizes of $R_{200}$ disks of GAMA groups up to $z\leq 0.10$, with colours referring to their redshifts (cf.\ colour scale on the right). Small red dots show the positions of all UDG candidates that satisfy our final selection criteria.}
\label{fig:gama_allfields_overview}
\end{center}
\end{figure*}

\begin{table
\caption{Rough overview of the three KiDS+GAMA fields studied here. Each KiDS tile covers roughly 1 deg$^2$. The effective areas are reduced by $\sim 15\%$ due to the masking, see Fig.~\ref{fig:gama_allfields_overview} for a graphical representation.}
\label{tab:dataoverview}
\begin{center}
\begin{tabular}{l r r r r}
\hline
\hline
Field & KiDS tiles & Eff area [deg$^2$] & UDG candidates & Groups \\
\hline
G09&          64&50.7&        4635&          83\\
G12&          64&56.2&        5815&         140\\
G15&          69&62.5&        6145&         102\\
\hline
Total&         197&169.5&       16595&         325\\
\hline
\end{tabular}
\end{center}
\end{table}

\subsection{Kilo-Degree Survey}
 KiDS is an on-going ESO wide-area imaging survey in the $ugri$
bands that will cover 1500 deg$^2$, and which is designed to map the matter distribution in the Universe using weak gravitational lensing. Here we used the `KiDS-450' data release, which covers approximately 450 deg$^2$ \citep{hildebrandt17, dejong17}.

Our starting point is the KiDS $r$-band imaging, reduced and stacked with the THELI \citep{erben13} pipeline. The data are taken under dark conditions with minimum atmospheric extinction. To facilitate weak-lensing measurements in the $r$ band, they are taken in conditions with excellent seeing (PSF FWHM$<0.8''$). Stacks are composed of five dithers with a combined integration time of 1800s.

The stacks are designed for optimal weak-lensing shape measurements, and there are still residual background patterns that we have to account for in this study. We subtracted these residual background effects using a mesh size of 150 pixels ($\sim32''$), smoothed with a median filter of size 5$\times$5 meshes. This corrects background structures on a scale that is much larger than the sizes of the galaxies we study.

We used the conservative masks designed by the KiDS team, which mask $\sim15\%$ of the total survey area. This includes bright stars, their reflective haloes, and image artefacts \citep[for more details see][]{kuijken15}. Figure~\ref{fig:gama_allfields_overview} shows the unmasked area of the three KiDS fields we consider in this study in white.

\subsection{Galaxy And Mass Assembly (GAMA)}
GAMA \citep{drivergama11,liske15} is a spectroscopic survey that is complete down to an $r$-band magnitude limit of 19.8. Each GAMA field is visited several times to overcome sampling problems that are due to close galaxy pairs, which are common in groups and clusters of galaxies. Using a friends-of-friends (FoF) algorithm on the spectroscopic catalogue, \citet{robotham11} have constructed a group catalogue for the GAMA survey regions that links $\sim 14\%$ of the GAMA galaxies to a group with at least five spectroscopic members. Since this first version of the catalogue (G$^3$Cv1), the linking scheme has been updated several times. Including the current version of the spectroscopic catalogue, G$^3$Cv9 is now the most recent version of the group catalogue. While it covers a total of five survey fields, we focus on the three Equatorial GAMA fields G09, G12, and G15, which overlap with KiDS (cf.~Table~\ref{tab:dataoverview}). We considered all groups from G$^3$Cv9 in these survey fields that have at least five FoF members and are in the redshift range $0.01 \leq z\leq 0.10$. This selection resulted in a total of 325 groups, see Table~\ref{tab:dataoverview}.

Several group mass definitions are in use within the GAMA collaboration. Originally, group masses were measured dynamically, using the spectroscopic group members as tracers of the potential. While this dynamical method is expected to work reasonably well for groups with a large number of members, these masses are unreliable in the low-mass regime, however, where there are few members \citep{robotham11}. Instead, the masses used in this work are based on the total group luminosity in the $r$ band.
\citet{viola15} have performed a weak-lensing study of the overlap region between KiDS and GAMA to measure the total mass associated with groups, in bins of their total $r$-band luminosity. We used the almost-linear relation they found between total luminosity and weak-lensing mass to estimate $M_{200}$ for each group. The $R_{200}$ is defined, as usual, as the radius corresponding to a sphere that contains this mass. The group centres are defined as their $r$-band luminosity-weighted centre of mass.

Figure \ref{fig:gama_allfields_overview} gives an overview of the three GAMA regions studied here. The white area is the (unmasked) KiDS area considered here. The circles mark the locations of groups, with colours indicating their redshifts, and their sizes are the projected angular size of their $R_{200}$ disks.
An additional criterion to define our group sample is that less than 50\% of their $R_{200}$ disk is masked in the KiDS images (which removes 17 groups from the group sample, bringing the total to 325).

\section{Analysis}\label{sec:analysis}
Ultra-diffuse galaxies are defined in physical units \citep[$r_\mathrm{{eff}}\geq$ 1.5 kpc,][]{vandokkum15} and have a maximum size of about $r_\mathrm{{eff}}\sim7.0$ kpc (vdB16). Since we here consider GAMA groups over a wide redshift range, these physical sizes correspond to a range of different angular sizes. For example, at $z=0.01$, we would have to consider galaxies with angular sizes between $7\farcs3$ and 34$''$, while at $z=0.10,$ this would correspond to sizes between $0\farcs8$ and $3\farcs8$.
From an observational point of view, we have to work in angular units, which initially leads to a redshift-dependent selection. However, when we exploit the wide range of redshifts of the GAMA groups, we can make definitive statements about their UDG content, as outlined below.

\subsection{Large galaxies with low surface brightness in KiDS}\label{sec:selectioncriteria}
Although we work in angular units, we performed an analysis that is as close as possible to the analysis presented in vdB16, where data from the Canada-France-Hawaii Telescope (CFHT) were analysed.
As a first step, we used \texttt{SExtractor} \citep{bertinarnouts96} on the $r$-band stack to detect sources, using a Gaussian filter with a FWHM of 5 pixels ($\sim 1''$ with our detector) to improve our sensitivity towards faint extended galaxies. The KiDS pixel scale is slightly coarser than CFHT MegaCam (0.214$''$/pix versus to 0.185$''$/pix), so we require sources to have at least 15 adjacent pixels that are at least 0.90 sigma above the background. These choices keep the purity of detecting real objects (and not noise fluctuations) high (cf.~vdB16). We detected a total of $12\,243\,224$ sources in the KiDS fields.

We applied liberal selection criteria to these \texttt{SExtractor} output catalogues to perform a first selection of large extended galaxies (and thus potential UDGs at the redshifts of the GAMA groups). We selected against stars and other compact objects, requiring that $r_{2''} > 0.9 + r_{7''}$, where $r_{x''}$ is the $r$-band magnitude within a circular diameter of $x$ arcsec. This pre-selection was merely intended to speed up the overall analysis. We performed several tests to ensure that this cut did not affect the final selection of galaxies that we aimed to study, even if a fraction of UDGs is nucleated \citep[cf.][]{yagi16}. After applying the KiDS survey mask, we were left with $873\,702$ sources in the considered KiDS area.

These sources were our input to \texttt{GALFIT} \citep{peng02}, which we used to estimate morphological parameters while masking any pixels belonging to neighbouring sources. We then selected galaxies with best-fit S\'ersic parameters in the range $24.0 \leq \langle\mu(r,r_\mathrm{eff})\rangle \leq 26.0 \, \mathrm{mag\, arcsec^{-2}}$, circularised effective radii $1\farcs5\leq r_\mathrm{eff}\leq8\farcs0$ that are fit within 7 pixels from the \texttt{SExtractor} position, and have a S\'ersic index $n\leq$4. We note that the surface brightness criterion is somewhat shallower (26.0 versus 26.5) than the criterion applied in vdB16 because we work with shallower
data here. Some examples of selected galaxies are shown in Appendix~\ref{sec:examples}.

We find that \texttt{GALFIT} provides reasonable fits to the data in most cases; we therefore trust the \texttt{GALFIT} parameters, and the selection can be considered as being quite pure. We did not make any additional subjective or by-eye checks for individual sources, so that this study remains reproducible. We assume that any residual impurity does not correlate with the locations of groups and is eventually removed (in a statistical sense) by our background subtraction. Given that KiDS is untargeted, this is a reasonable assumption. To test the performance of our procedure and to gauge the completeness of our final UDG candidate selection, we also performed all processing steps on a set of tailored image simulations, as described next.

\subsection{Image simulations}\label{sec:simulations}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{aftergalfit_KiDS_new}}
\caption{Recovered fraction of sources that were injected in the KiDS $r$-band stacks as a function of circularised effective radius, and the mean surface brightness within the effective radius. The dotted region defines the selection criteria used in this study. Arrows show the negligible bias in the morphological parameters recovered with \texttt{GALFIT} in six corners of parameter space.}
\label{fig:completeness}
\end{figure}

Following vdB16, we injected objects with S\'ersic profiles with $n$=1 \citep[typical values for UDGs, cf.][]{koda15,vdB16} at random positions in the KiDS $r$-band image stacks. Effective radii ($r_\mathrm{eff}$) were drawn uniformly between 1$''$ and 10$''$ (we worked in angular units here). Central surface brightnesses were drawn uniformly from the range $22.7 < \mu(r,0)/[\mathrm{mag\, arcsec^{-2}}] < 26.0$, and the ellipticity parameter $\epsilon=(1-q)/(1+q)$, where $q$ is the ratio of minor/major axis, was drawn uniformly between 0 and 0.2. Injecting 1500 sources per realisation, with 10 realisations per KiDS tile, we simulated several million sources.

In the following we work in units of the mean surface brightness within the effective radius (following vdB16). This parameter is exactly 1.12 mag arcsec$^{-2}$ larger than the \textit{central} surface brightness for S\'ersic-profiles with $n$=1, and is more closely related to the detectability of a source. In particular, as noted in vdB16, the recovery rate of simulated sources with slightly different S\'ersic-indices ($n$=0.5 or $n$=1.5) is similar when expressed in these units. It is also noted there that the detectability of a source of a given effective (circularized) radius does not sensitively depend on its ellipticity.

Figure~\ref{fig:completeness} shows the completeness of the simulated sources. The completeness is defined as the fraction of injected sources, in bins of size and surface brightness, that are 1) detected with \texttt{SExtractor}, 2) pass the pre-selection based on the \texttt{SExtractor} output, and 3) have best-fit morphological parameters corresponding to the selection criteria we outlined above. These simulated sources were thus processed through exactly the same pipeline as the ``real'' sources. While only 2\% of ``real'' sources that were fed to \texttt{GALFIT} satisfy these criteria, the recovery rate of simulated sources is $\sim$66\%. Most ``real'' sources that did not match the final selection criteria are either too small (the pre-selection criteria on the \texttt{SExtractor} output catalogues were very inclusive, as outlined above) or too bright (there was no pre-selection on brightness).

With our final selection criteria, we are left with 16\,595 large and faint galaxies in an unmasked KiDS area of 169.5 deg$^2$. We ensured that we did not count the same galaxies several times because of the slightly overlapping KiDS tiles.
As a sanity check, we applied exactly the same selection criteria to the sources detected in the CFHTLS Deep fields (with noise added to resemble the cluster imaging used in vdB16). We found 343 sources that satisfy these criteria (in angular units) in a combined effective area of 3.52 deg$^2$. This means that the source densities are remarkably similar between the two surveys, providing further confidence that we can compare results from these different instrumental setups and depths.

\subsection{Associating UDGs with GAMA groups}\label{sec:analysis_correctionfactors}
We now link the selected sample of galaxies to UDGs at the redshifts of the GAMA groups. As in vdB16, we considered UDGs with physical sizes between 1.5 kpc and 7.0 kpc. We note that for galaxy groups with redshift $z>0.051$, we did not probe the smallest UDGs (because they would have an angular size smaller than $1\farcs5$). Correspondingly, for galaxy groups with redshift $z<0.044$, we did not probe the largest UDGs (because they have an angular size larger than $8\farcs0$). As described below, we combined information from all GAMA groups, exploiting the entire redshift range of the GAMA groups ($0.01\leq z\leq 0.10$), to turn our samples of galaxies with sizes between $1\farcs5$ and $8\farcs0$ into physical sizes (and thus UDGs, after accounting for fore- and background interlopers). Given that only 10\% of the current age of the Universe has passed since $z=0.10$, we assume that the UDG content of the galaxy groups does not evolve over this redshift range. We proceeded as follows:

\begin{itemize}
\item For each GAMA group, we counted the number of large galaxies
with low surface brightness that satisfy our \textit{angular} selection criteria (cf.~Fig.~\ref{fig:completeness}), the \textit{physical} size criteria corresponding to the group redshift, and that are projected within $R_{200}$ from the group centre. We estimated the background value on this number, namely the counts expected from the average number density of the particular KiDS field studied here (i.e.~G09, G12, or G15)\footnote{In principle, to properly estimate the background, structures that are physically associated with the group in question should be masked out before estimating the background. However, given the very low overdensities of UDGs in these groups together with the large statistical uncertainties in this work, we find that even in an extreme case, namely when we mask all \textup{}groups up to $z\leq 0.10$, this has a negligible effect, namely at most 10\% of the statistical uncertainty, on the final results (cf.~Appendix~\ref{sec:robustness}). The same is true when we estimate the background for groups using the average over all the fields. In the following we therefore use a simple average number density per field to estimate the background.}. The raw counts minus the background expectation and associated Poisson errors are our main measurements. The total overdensity of measured UDGs in all GAMA groups is 29\% compared to the background.

\item We corrected this number for the masked fraction of the $R_{200}$ disk by assuming that the UDG density is homogeneous. The average correction factor for all groups is 17\% to the number counts, which is close to the masked fraction of the entire KiDS survey area. We recall that we did not consider groups in which more than half the $R_{200}$ disk is masked.

\item We corrected this number for incompleteness, using our extensive set of image simulations (as described in Sect.~\ref{sec:simulations}). We counted the number of simulated galaxies with \textit{intrinsic} parameters in our selection region, and divided this by the number of simulated galaxies with \textit{measured} parameters in our selection region. Key here is that a larger region of parameter space is simulated than we considered for the analysis. This means that scatter between intrinsic and measured parameters can be taken into account. For this step we had to assume an underlying intrinsic distribution in morphological parameters. We started with the values measured for the cluster population of UDGs in vdB16; a steep size distribution given by $n \mathrm{[dex^{-1}]}\propto r_\mathrm{eff}^{-3.4}$, and a homogeneous distribution in surface brightness (in this region of parameter space). This led to correction factors ranging from 0.57 at the lowest redshift to 1.50 at $z=0.10$.

\item Since we only considered sources with sizes between $1\farcs5$ and $8\farcs0$ arcsec, we estimated which fraction of UDGs with physical sizes between 1.5 and 7.0 kpc we missed in this count. For this we again used the size distribution. There is no correction (correction factor=1) in the range $0.044<z<0.051$, since all physical sizes are represented in that redshift range. It increases only slightly towards lower redshift (since large UDGs are rare), but peaks at 8.3 for groups at $z=0.10$. This correction factor depends on the assumed UDG size distribution, which we iterated over in Sect.~\ref{sec:sizedist}. In Appendix~\ref{sec:robustnesssize} we show that our final results are not significantly affected by this assumed size distribution.

\item Since vdB16 were able to consider a slightly broader range in mean surface brightness within the effective radius ($24.0 \leq \langle\mu(r,r_\mathrm{eff})\rangle \leq 26.5 \, \mathrm{mag\, arcsec^{-2}}$, instead of up to 26.0), we increase our counts by 25\%, assuming that the distribution is homogeneous in surface brightness (vdB16, and supported by the current analysis, see Fig.~\ref{fig:sizedist}). We note that this correction was
made only for the purpose of comparing our results to those from vdB16 and other literature studies.
\end{itemize}

\section{Results}\label{sec:results}
The next two subsections present our main results, namely the abundance of UDGs as a function of halo mass, and their size distribution in the GAMA groups. They are measured iteratively, since the result of each of the two measurements potentially influences the other. We explain the reason for this interdependence in Appendix~\ref{sec:robustness}, and conclude there that each measurement is robust and only mildly influenced by the other (i.e.~to a degree that is smaller than the size of the statistical uncertainties).

\subsection{Abundance of UDGs in galaxy groups}\label{sec:abundancehalomass}

\begin{table
\caption{Average number of UDGs per group in bins of halo mass. Bins have a width of 0.4dex in halo mass, starting at $10^{12}\,\mathrm{M_{\odot}}$. These are the data points shown in Fig.~\ref{fig:abundance1}. Two types of uncertainty are given: first the Poisson uncertainty on the average value in each bin, second the bootstrap uncertainty from resampling groups in each bin (drawing with replacement). The latter uncertainty also includes the Poisson uncertainty on individual groups, and is therefore never smaller than the former (and they are equal when there is only one group in a bin). The last column lists as a comparison the richness of bright galaxies in the same groups and bins.}
\label{tab:abundance}
\begin{center}
\setlength{\extrarowheight}{0.05cm}
\begin{tabular}{c r r r}
\hline
\hline
Binmean & Ngroups & UDGs per group & Richness\\
log$_{10}[M_{200}/\mathrm{M_{\odot}}]$&&&$M_{r} < M_{r}^{*} +2.5$\\
\hline
12.23&8&$-0.21^{+0.51+0.52}_{-0.45-0.61}$&$1.80^{+0.50+0.40}_{-0.44-0.78}$\\
12.62&38&$0.32^{+0.62+1.08}_{-0.55-0.93}$&$2.93^{+0.30+0.39}_{-0.27-0.40}$\\
13.03&139&$1.38^{+0.56+1.26}_{-0.49-1.04}$&$4.69^{+0.20+0.30}_{-0.18-0.26}$\\
13.36&107&$3.31^{+0.89+1.51}_{-0.78-1.62}$&$7.98^{+0.31+0.45}_{-0.27-0.48}$\\
13.77&27&$7.54^{+2.52+5.22}_{-2.21-3.97}$&$19.31^{+0.94+1.97}_{-0.83-1.59}$\\
14.04&3&$44.93^{+13.95+21.54}_{-12.23-38.98}$&$39.60^{+3.93+5.22}_{-3.44-7.79}$\\
14.52&1&$64.29^{+18.16+18.16}_{-15.92-15.92}$&$81.84^{+9.66+9.66}_{-8.47-8.47}$\\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{abundance_plot1}}
\caption{Abundance of UDGs as a function of halo mass. \textit{Thick black points:} averages in bins of halo mass. Two types of error are shown: thick errors with hats are Poisson uncertainties, while thin errors show bootstrap errors after resampling the groups in each bin. \textit{Thick red points:} values measured for eight individual clusters in vdB16. \textit{Black lines:} best-fitting power-law relation including all data points shown here. \textit{Dashes:} relations corresponding to the $\pm 1$-$\sigma$ uncertainties on the power-law index. The numbers on the top axis indicate the number of groups in each bin.}
\label{fig:abundance1}
\end{figure}

Figure~\ref{fig:abundance1} shows the abundance of UDGs as a function of halo mass for the GAMA groups. Group masses are based on a weak-lensing calibration in bins of their total $r$-band luminosity \citep{viola15}. Two types of error bars are shown, namely Poisson uncertainties and bootstrap errors where groups in each bin have been resampled (drawing with replacement). Data points are also presented in Table~\ref{tab:abundance}. We note that the statistical background subtraction may in principle yield negative values for the abundance of UDGs in groups; in this case, for the bin with few low-mass systems that have a low overdensity compared to the background. In Fig.~\ref{fig:abundance1} we also show the measurements from vdB16, which were obtained from a similar analysis, but are based on individual clusters. We note that halo masses are determined differently between vdB16 and the current study. vdB16 measured cluster masses dynamically \citep[cf.][]{sifon15}, while our group masses are based on their total $r$-band luminosity. However, the group masses are calibrated directly using weak gravitational lensing \citep{viola15}, while the dynamical masses of the clusters studied in vdB16 are tightly correlated with lensing mass (Herbonnet et al., in prep.). In Appendix~\ref{sec:robustness} we perform a conservative test to ensure that residual uncertainties in mass calibration between the two studies do not affect our results.

We fitted\footnote{We used a $\chi^2$ fitting recipe on data with errors in both coordinates, as described in Sect.~15.3 of \citet{numericalrecipesinc}.} a power-law relation between the halo mass and UDG abundance, combining both the cluster measurements from vdB16, and the binned group abundances presented here. For the clusters we took errors on individual halo masses into account, while for the KiDS+GAMA points we used the Poisson uncertainties, and we assumed an uncertainty of 0.1 dex in mean halo mass for each bin while performing the fit. A power law of the form
\begin{equation}\label{eq:udgs}
N_{\mathrm{UDG}}=(19\pm2) \cdot \left[ \frac{M_{200}}{10^{14}\,\mathrm{M_{\odot}}}\right]^{1.11\pm0.07}
\end{equation}
provides a good description of the data points, with $\chi ^2/\mathrm{d.o.f.}=16.0/13$.

We note that as is customary when measuring the number of galaxies in a halo (e.g.~richness), the numbers we quote are measured in a cylinder with radius $R_{200}$. For the number of galaxies within a sperical volume with radius $r_{200}$, there is a deprojection factor to be considered. Assuming that galaxies to first order \citep[but see][]{vdB15} trace a Navarro-Frenk-White (NFW) profile \citep{NFW}, the deprojection factor depends on the concentration of the halo. The typical concentration is a function of halo mass \citep[e.g.][]{duffy08,dutton14}, so that the deprojection factor ranges from about $\sim$0.80 for groups (assuming $c_{200}$=6), to $\sim$0.75 for clusters (assuming $c_{200}$=3). If we were
to deproject all values and repeat the fit, it would only change the best-fit slope from 1.11 to 1.10. Since we are primarily interested in the relative abundance as a function of halo mass and not in the absolute normalisation, we did not apply a deprojection in this work.

\subsection{Size distribution of UDGs}\label{sec:sizedist}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{sizedist}}
\caption{Measured size distribution of galaxies in the central projected $R_{200}$ for the GAMA groups. \textit{Grey pluses:} average raw counts per group. \textit{Grey note:} average values expected from fore- and background interlopers. \textit{Grey $ diamond$:} background-subtracted values. \textit{Black data points:} background-subtracted values, corrected for incompleteness, including bootstrap resampling errors. \textit{Blue and red points:} measured sizes for galaxies split over two bins of mean effective surface brightness, also after correction. In this size range, the data are well described by a power-law index of $-2.71\pm0.33$ (in bins of equal logarithmic size). \textit{Grey dashed line:} size distribution with power-law index of $-3.40$, which we found for UDGs in clusters (vdB16), arbitrarily normalised.}
\label{fig:sizedist}
\end{figure}

A related measurement to the abundance as a function of halo mass is the size distribution of UDGs in groups. As mentioned before (and explained in more detail in Appendix~\ref{sec:robustness}), both measurements are slightly interdependent and have to be performed iteratively. Figure \ref{fig:sizedist} shows the size distribution of UDGs that passed our main selection criteria, but have a stricter selection in surface brightness of $24.0 \leq \langle\mu(r,r_\mathrm{eff})\rangle \leq 25.6 \, \mathrm{mag\, arcsec^{-2}}$, since we are highly complete in this region of parameter space (cf.~Fig.~\ref{fig:completeness}). Each of the 325 groups contributes to the estimated size distribution, but the physical range that is covered depends on the redshift of
each group. Since more massive groups are generally found at higher redshift (where the volume probed is larger), we accounted for the fact that they contain more UDGs within their $R_{200}$ disk than lower-mass groups (cf.~Fig.~\ref{fig:abundance1}). We therefore scaled numbers by $M_{200}^{1.1}$, that is, by the relation found in Sect.~\ref{sec:abundancehalomass}.

The drop in the background-subtracted counts (diamonds) at small physical sizes arises because groups at higher redshift do not contribute to these bins, and given that these are massive and therefore contain many UDGs, there is a large correction of up to a factor $\sim$5 for the smallest-size bin. After correcting the numbers and fitting a power-law distribution, we obtained a best fit $n \mathrm{[dex^{-1}]}\propto r_\mathrm{eff}^{-2.71\pm0.33}$ with $\chi^2/\mathrm{d.o.f.}=0.56$. We performed the same analysis on two subsamples with mean effective surface brightnesses of $24.0 \leq \langle\mu(r,r_\mathrm{eff})\rangle \leq 24.8 \, \mathrm{mag\, arcsec^{-2}}$ (shown in red) and $24.8 \leq \langle\mu(r,r_\mathrm{eff})\rangle \leq 25.6 \, \mathrm{mag\, arcsec^{-2}}$ (shown in blue), finding power-law indices that are identical within errors. The normalisation of each subsample is roughly 50\% of the full sample, indicating that the UDG distribution is roughly uniform as a function of mean effective surface brightness in this part of parameter space (similar to what was found by vdB16).

\section{Discussion}\label{sec:discussion}
The measured size distribution of UDGs places powerful constraints on models that aim to explain the formation of UDGs \citep[e.g.][]{amorisco16}. The size distribution we find ($n \mathrm{[dex^{-1}]}\propto r_\mathrm{eff}^{-2.71\pm0.33}$) is very steep, indicating that the largest UDGs are rare. Thanks to the strong overdensity of the clusters studied in vdB16, the authors were able to measure the size distribution more precisely to be $-3.40\pm 0.19$ (see their Fig.~7). While the UDG size distribution is certainly steep, there is evidence (at the $2\sigma$ level) that the distribution depends on environment. It would be worthwhile to measure the size distribution specifically only in the lowest-mass groups, but we find that this results in such a low overdensity of UDG candidates with respect to the field that we cannot place tighter constraints on a possible environment dependence of the UDG size distribution in this work.

\subsection{Abundance of UDGs as a function of halo mass}\label{sec:discussionabundance}
The measured abundance of UDGs as a function of halo mass is perhaps even more important in constraining formation scenarios, as it directly addresses the question of \textit{\textup{where}} UDGs may have formed. We have measured the abundance now in a consistent manner over three orders of magnitude in halo mass. For the lowest-mass groups that enter into the analysis, we note that they contain on average fewer than one UDG per group. A power law of the form $N \propto M_{200}^{1.11 \pm 0.07}$ provides a good description of the abundance of UDGs as a function of their host halo mass. In Appendix~\ref{sec:robustness} we show that this measurement is robust with respect to the main assumptions that enter into the analysis. We note that the measurement presented here is more precise than the measurement performed by vdB16, where our measurement on a cluster sample spans only one order of magnitude in halo mass. vdB16 found a power-law exponent of $0.93\pm0.16$, which is consistent within $\sim 1 \sigma$ with this measurement.

According to our best-fit relation, Eq.~\ref{eq:udgs}, roughly one in five systems like the Local Group, which has a combined mass of $M_{200}\approx 2\times 10^{12}\,\mathrm{M_{\odot}}$ \citep{gonzalezkravtsov14}, is expected to host a UDG within our selection boundaries. This rises to a substantial population of $\sim 200$ UDGs at halo mass scales of $M_{200}=10^{15}\,\mathrm{M_{\odot}}$.

\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{abundance_plot2}}
\caption{Same as Fig.~\ref{fig:abundance1}, but now expanding the plotted range, and showing literature measurements (as described in the text). Uncertainties shown here are the bootstrap error bars, from resampling groups within each bin.}
\label{fig:abundance2}
\end{figure}

We note that \citet{roman16b} measured the abundance of UDGs in three groups and found UDGs to be more abundant per unit halo mass in groups than in higher-mass systems (see Fig.~\ref{fig:abundance2} for a comparison), a conclusion that is different from ours. Low-number statistics may largely explain this apparent discrepancy. We note that many of the (especially low-mass) groups we studied in this work do not contain any UDGs, thus lowering the average to fewer than 1 UDG per group in the low-mass regime. Including Poisson scatter, the groups studied in \citet{roman16b} are fully consistent with being drawn from a sample comparable to ours.

There is another possible interpretation, related to the group properties themselves. The groups studied by \citet{roman16b} are the three Hickson Compact Groups \citep[HCGs,][]{hickson82},
which overlap with the SDSS Stripe 82. The HCG selection does not include loose groups, and is thus not representative for the diverse population of groups. In particular, constituent galaxies of HCGs have likely experienced only few merging events in their history \citep{proctor04,mendes05}. This may also have positively affected the survivability of UDGs in such groups.

In Fig.~\ref{fig:abundance2} we also show the result from \citet{janssens17}, who measured the abundance of UDGs in a very massive cluster from the Hubble Frontier Fields. In Fig.~4 of \citet{janssens17} and in the accompanying text \citep[see also Fig.~6 in][]{roman16b}, a comparison is made with other measurements of the UDG abundance in different systems that have recently been presented in the literature. We include them here for reference, noting that most of them are consistent within 1$\sigma$ with the extrapolation of our best-fit relation, even at higher masses. Several UDGs have also been found to be possibly associated with a local galaxy group around NGC 5473 \citep{merritt16}. However, the surface brightnesses of these UDGs are too faint to be detectable with the depth of the KiDS imaging. We therefore did not include these results in Fig.~\ref{fig:abundance2}. The difference in surface brightness limits is a general caveat when comparing results from different studies. For instance, \citet{mihos15} studied three UDGs with surface brightnesses that are $\sim$2 magnitudes fainter than we can detect in the KiDS imaging. Figure~\ref{fig:abundance2} \citep{roman16b,janssens17} only includes studies with detection limits comparable to ours.

The power law we find here suggests that the number of UDGs per unit of host halo mass is increasing towards higher-mass haloes. It is perhaps even more insightful to compare the abundance of UDGs to the abundance of more massive galaxies in group- and cluster-sized haloes. Measuring the so-called mass-richness relation has been the subject of multiple studies in the past decades \citep{marinoni02,lin04,rines04,muzzin07b,andreon10b}. With very few exceptions \citep{kochanek03,uitert16}, they all consistently measured a relation $N\propto M^{\alpha}$, where $\alpha < 1.0$. This is also consistent with the notion that the stellar mass fraction increases from clusters to group-mass haloes \citep{gonzalez13,lagana13,budzynski14,vdB14}. In particular, \citet{viola15} measured the relation between halo mass and apparent richness for the GAMA groups. They found $M_{200}\propto N_{\mathrm{FoF}}^{1.09\pm0.18}$, which, after inverting $M$ and $N$, is consistent with the general picture, namely that lower-mass haloes  per unit halo mass host more bright satellites than higher-mass haloes do. We note, however, that this is measured using all FoF members associated with the group. This is quite different from the way we measured the UDG abundances. We proceed by making a direct comparison between the UDG abundance and the richness of GAMA groups in the same mass bins.

\subsection{Mass-richness relation of GAMA groups}\label{sec:richness}
To help place the measured abundances of UDGs into context, we measured the abundance of more massive galaxies (i.e.~richness) in the same groups. To make a direct comparison, we closely followed the analysis performed for the UDGs; we measured the richness within $R_{200}$, applied the same KiDS mask, and corrected the counts by assuming that the spatial distribution is uniform within $R_{200}$. To reduce the background correction compared to a purely photometric richness measurement, we made use of the full GAMA catalogue of all spectroscopically targeted galaxies. This is a valid approach given that the GAMA catalogue is essentially 100\% complete down to an $r$-band magnitude limit of 19.8, and does not suffer from sampling problems of close galaxy pairs thanks to multiple visits of each field.

We measured richnesses down to a fixed magnitude past the characteristic magnitude in the $r$ band, which is $M^{*}_{r}=-21.21$ at $z=0.10$ \citep{blanton03}. The spectroscopic completeness limit of 19.8 in the $r$ band allowed us to measure the richness down to $M^{*}_{r} +2.5$ at this redshift. We k-corrected this limit based on a passively evolving stellar population with an age of 10 Gyr, and measured the richness consistently down to the same limit at all redshifts.

\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{abundance_plot3}}
\caption{Abundance of UDGs as a function of halo mass. \textit{Thick black points with errors:} same as shown in Fig.~\ref{fig:abundance1}, and these include the cluster points from vdB16. \textit{Black lines:} best-fitting power-law relation including all black data points shown here. \textit{Blue squares:} richnesses (i.e.~the abundance of massive galaxies) in the same GAMA groups, with error bars defined in a similar way. \textit{Blue lines:} best-fitting power-law relation to the richness points.}
\label{fig:abundance3}
\end{figure}

We estimated the velocity dispersion $\sigma_{\mathrm{LOS}}$ of each group based on its $M_{200}$, assuming the scaling of \citet{evrard08}. Then we selected galaxies within 5$\sigma_{\mathrm{LOS}}$ from the group redshift, which is defined as the median redshift of all FoF members. We subtracted the background (i.e.~galaxies that are interlopers due to their redshift) statistically, measuring the density of galaxies in the same redshift slice (corresponding to $\pm5\sigma_{\mathrm{LOS}}$) in the same GAMA field (G09, G12, G15), but outside the group $R_{200}$.

The data points are presented in Table~\ref{tab:abundance} and are shown in Fig.~\ref{fig:abundance3}. We fitted a power-law relation to these binned data points in the same way as was done for the UDG abundance measurements, finding a best-fit relation of
\begin{equation}
N_{\mathrm{Bright}}=(31\pm3) \cdot \left[ \frac{M_{200}}{10^{14}\,\mathrm{M_{\odot}}}\right]^{0.78\pm0.05},
\end{equation}
with $\chi ^2/\mathrm{d.o.f.}=4.2/5$.

The 5$\sigma$ cut is large enough to include all galaxies that are physically associated with the group, while it is small enough to not include substantial numbers of interlopers. A cut of $\pm3\sigma_{\mathrm{LOS}}$ is too strict to allow for all associated galaxies, especially for groups with low halo masses; in this case, we find the same best- fitting normalisation of $31\pm4$, but a slightly steeper slope of $0.86\pm0.06$. At the other extreme, we tested a fixed redshift cut of $\Delta z=$0.01. This conservative selection of group galaxies gives the same best-fit normalisation as the $\pm5\sigma$ cut and a similar slope of $0.79\pm0.05$. This shows that the richness measurement is robust with respect to the exact velocity cut along the line of sight. The slope is fully consistent with almost all relations measured by the literature studies that we listed above.

The difference in trends between UDGs and more massive satellites (mass-richness relation), indicates that UDGs are not merely a fixed fraction of the total galaxy population that is independent of the environment. It is striking that in cluster-sized haloes the number of UDGs is approximately equal to the number of galaxies down to $M_{r} < M_{r}^{*} +2.5$. Conversely, for groups of $M_{200}\simeq 10^{12}\,\mathrm{M_{\odot}}$, the bright galaxies are a factor $\sim$10 more abundant than UDGs. Although the number of UDGs is comparable to these brighter galaxies in massive groups and clusters, we note that they still constitute only a small fraction of the total $r$-band luminosity (or stellar mass) even for cluster-sized systems. The total stellar mass contained in UDGs is about 1.0\% of the total stellar mass in galaxies with $M_{r} < M_{r}^{*} +2.5$ for clusters. This fraction drops further when compared to the total stellar mass (due to the contribution from fainter galaxies, and the intra-group and intra-cluster light), and is lower for groups.

Ultra-diffuse galaxies therefore seem to either form, or survive, more easily in the most massive haloes. However, they may still simply be a fixed fraction of the general population of dwarf galaxies that is independent of the environment. There are claims that the luminosity function of cluster galaxies has an upturn at low luminosities compared to the field \citep{popesso05}, although recent measurements do not reproduce this upturn \citep{depropris13,agulli14}. If the overall dwarf galaxy population is relatively more abundant in cluster-sized haloes, this could mean that UDGs simply follow the same trend. An interesting approach would be to measure the total dwarf abundance in the same GAMA groups for a direct comparison, but the background correction for small and faint galaxies prevents this with the current data set (see the discussion in vdB16).

Galaxies that are currently in massive galaxy clusters have on average spent their life in more overdense environments than galaxies that are currently in lower-mass systems. Whether a satellite galaxy can survive in a massive halo depends on their subhalo mass, concentration, orbit, and mass distribution of the host halo. While tidal forces are weaker in group-sized haloes than in clusters, mergers with other satellite galaxies are typically more common because relative velocities are lower in groups. It is hard to predict which physical process (tidal stripping or mergers with other galaxies) would be dominant in destroying UDGs in massive haloes, but it is plausible that elliptic orbits in clusters that avoid their centres are the orbits that make the UDGs survive longest. This may explain why UDGs are relatively more abundant in clusters than in groups.

A detailed comparison of the morphological properties of UDGs in different environments may also provide clues as to the possible formation mechanisms for UDGs. The median S\'ersic index, an often-used morphological indicator of the UDG candidates in the clusters studied by vdB16, is $n_\mathrm{S\acute{e}rsic}=1.4$ (this median value is unchanged when we only consider UDG candidates brighter than the surface brightness limits reached in the present study). Interestingly, after accounting for background UDG candidates, we find that the median S\'ersic index of UDGs in the GAMA groups is $n_\mathrm{S\acute{e}rsic}=2.2$, and only about 33\% have a S\'ersic index consistent with 1 (within uncertainties). This means that if we had selected UDGs in groups with a similar morphology as the UDGs found in clusters, the abundance would rise even more steeply towards higher-mass systems. This suggests that the formation mechanisms for UDGs, or their subsequent evolution, depends on their environment.

\section{Summary and outlook}\label{sec:conclusion}
We have expanded the work of vdB16 and measured the abundance of UDGs in group-sized haloes in the KiDS and GAMA surveys. The KiDS images are only several tenths of a magnitude shallower than the CFHT imaging stacks used in vdB16, which ensured a consistent comparison (after applying the corrections we described). The main conclusions of this study can be summarised as follows.

\begin{itemize}
\item The size distribution at a given surface brightness is steep, with a best-fitting power law of $n\, \mathrm{[dex^{-1}]}\propto r_\mathrm{{eff}}^{-2.7\pm 0.3}$. This is slightly shallower, but still roughly consistent (within $\sim 2\sigma$) to what has been found for UDGs in clusters by vdB16.
\item We were able to constrain the abundance of UDGs down to halo masses of $M_{200}\sim 10^{12}\,\mathrm{M_{\odot}}$. In such low-mass haloes, UDGs are rare; only about one  in ten of these groups contains a UDG that we can detect. The relation between the number of UDGs and the host halo masses, which now spans three orders of magnitude in halo mass and is well constrained,
is $N_{\mathrm{UDGs}} (\mathrm{R} < R_{200})\propto M_{200}^{1.11\pm0.07}$. Based on this relation, we expect about one in five systems like the Local Group to contain a UDG that we can detect.
\item We also measured the mass-richness relation of more massive galaxies in a similar way, finding a significantly shallower relation: $N_{\mathrm{Bright}} (\mathrm{R} < R_{200})\propto M_{200}^{0.78\pm0.05}$. Interestingly, in cluster-sized haloes, the number of UDGs is approximately equal to the number of galaxies down to $M_{r} < M_{r}^{*} +2.5$. In contrast, for groups of $M_{200}\simeq 10^{12}\,\mathrm{M_{\odot}}$, the bright galaxies are a factor $\sim$10 more abundant than UDGs.
\item We find a median S\'ersic index of $n_\mathrm{S\acute{e}rsic}=2.2$ for UDGs in groups, versus $n_\mathrm{S\acute{e}rsic}=1.4$ for UDGs in clusters (vdB16). This is an interesting constraint on theoretical models that aim to explain the formation and survival of UDGs, as it may hint at different formation mechanisms, or a different subsequent evolution, in different environments.
\end{itemize}

The different relations between the UDG abundance and the richness of more massive galaxies versus halo mass suggest that UDGs are either destroyed more efficiently in group-sized haloes than in cluster-sized haloes, or that galaxy clusters may play a positive role in their creation. It may also be that the overall dwarf galaxy population is more abundant in clusters than in lower-mass systems, and that the UDGs constitute a fixed fraction of this general dwarf population.
More detailed modelling is required to better interpret the differences measured here.

On the observational side, progress can be made by performing detailed comparisons between the properties of UDGs in different environments (beyond comparing a median S\'ersic index).
Unfortunately, the photometric data set we used is not ideal to investigate this; except for the $r$ band, which we used in this analysis, the imaging in the other filters is too shallow to measure meaningful colours. This prevents us from increasing the contrast of UDGs against the background, and we reach an overdensity of only 29\%. The Hyper Suprime-Cam survey, which recently released imaging over an overlapping sky region \citep{HSC17}, may be the way forward to study UDGs with deeper imaging in more detail.

\begin{acknowledgements}
We thank Annie Robin, Emanuele Daddi, and Ryan Johnson for insightful discussions, and an anonymous referee for a constructive report with valuable points to improve and clarify this manuscript. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n$^{\circ}$ 340519. H.Hoekstra acknowledges support from the European Research Council under FP7 grant number 279396. C.H. acknowledges support from the European Research Council under grant number 647112. H.Hildebrandt is supported by an Emmy Noether grant (No. Hi 1495/2-1) of the Deutsche Forschungsgemeinschaft. D.K. is supported by the Deutsche Forschungsgemeinschaft in the framework of the TR33 `The Dark Universe'. K.K. acknowledges support by the Alexander von Humboldt Foundation. R.N. acknowledges support from the German Federal Ministry for Economic Affairs and Energy (BMWi) provided via DLR under project no. 50QE1103.

Based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017 and 177.A-3018, and on data products produced by Target/OmegaCEN, INAF-OACN, INAF-OAPD and the KiDS production team, on behalf of the KiDS consortium. OmegaCEN and the KiDS production team acknowledge support by NOVA and NWO-M grants. Members of INAF-OAPD and INAF-OACN also acknowledge the support from the Department of Physics \& Astronomy of the University of Padova, and of the Department of Physics of Univ. Federico II (Naples).

GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programmes including GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA website is http://www.gama-survey.org/.
\end{acknowledgements}

\bibliographystyle{aa}

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Luis Cordones
CEO MD
cordonesluis@gmail.com
Previous Career: Aviation Mechanic
"Life isn't about finding yourself. Life's about creating yourself." - George Bernard Shaw
Luis loves chicken!
"Life is so short and simple, we are as small as a grain of sand and yet we are as big as we want to be." Luis came to the United States when he was 10 years old from the Dominican Republic Raised by his single mother who worked hard day and night to raise her two sons. While living in the Dominican Republic as a child, Luis developed a dream to become an Aviation Mechanic. Growing up Luis would watch his mom struggle to make ends meet but couldn't quite understand why she struggled if America was the land of freedom and opportunity. He believed that he was so fortunate to have come to a land of success, opportunity and where dreams can come true. He knew his life would be different, and did everything he could to go to school for Aviation Mechanics. Luis attended City Tech College after high school to make his dreams come true but little did he know that while on the path to making his dreams happen he was headed down a different road of success. While working at Panasonic Luis was introduced to World Financial Group (WFG) and once he learned about the WFG mission he decided to build a business where he would be able to help people pursue financial independence.
WHY DID YOU GET STARTED WITH WFG?
Luis came to WFG's corporate overview with an open mind because he knew at heart he was an entrepreneur and that he needed a change of environment. His first introduction to WFG was by becoming a client and getting his own finances in order. Luis clearly internalized WFG's mission and message. After reaping the benefits of becoming financially educated he realized that he needed to be a part of this movement. He needed to be the voice of change for his family and his community. Six months after the initial invite Luis decided to change his life not only by changing his income but by educating others about getting their finances in order and changing their lives. Luis has grown and developed into a leader; a professional and a coach for everyone he comes into contact with. WFG has been the foundation for Luis's success. "WFG is the vehicle to my destination".
WHAT DO YOU ENJOY MOST ABOUT BEING PART OF WORLD FINANCIAL GROUP?
Luis enjoys the ability to personally grow and change his life. He considers himself a continuous learner who loves to discover himself through personally developing.
WHAT ARE SOME OF YOUR ACCOMPLISHMENTS IN THE BUSINESS?
Luis has achieved the ability to develop and build strong relationships with many people in his community. These relationships have helped him grow a business and develop future leaders.
WHAT ARE SOME OF THE THINGS YOU ARE LOOKING FORWARD TO IN THE FUTURE?
I look forward to growing my business in different locations by opening new offices and bringing the mission of WFG to people throughout the country. Through helping people discover their potential utilizing the WFG system, I envision becoming a millionaire, being recognized as one of the top CEO's and becoming the number one first generation Dominican to be successful in WFG.
Many people have experienced various levels of success with World Financial Group. However, individual member experiences may vary. This statement is not intended to nor does it represent that any current member's individual results are representative of what all participants achieve when following the WFG system.

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More evidence on the drug is being collected, until December 2018. After this, NICE will decide whether or not to recommend it for use on the NHS and update the guidance. It will be available through the Cancer Drugs Fund until then.


Free agent tight end Tyler Kroft has officially signed with the Buffalo Bills to a three-year contract, reportedly worth $21 million.
Kroft, 26, has spent the last four seasons with the Cincinnati Bengals after being taken in the third round (85th overall) in the 2015 NFL Draft out of Rutgers. His most productive season came in 2017, when he started all 16 games with 42 catches for 404 yards and seven touchdowns. Kroft will be a big target for the Bills, coming in at 6' 6" and 240-pounds.
Kroft has spent plenty of time backing up starting tight end Tyler Eifert, which has led to many starts due to Eifert's injury problems. Last season, Kroft himself was injured, missing 11 games with a foot injury.
The Bills only had Jason Croom under contract to start the 2019 season, and the team may not be done adding at the tight end position.


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Home > Television > NCIS
Source: Twitter/@RealDuaneHenry
Why Did Duane Henry Leave 'NCIS'? The Real Reason Is Quite Sad
By Stephanie Harper
Jul. 11 2022, Published 3:38 p.m. ET
The reason NCIS has been so successful since 2003 is that the show sheds light on what life is like for law enforcement officials as they try to solve crimes. After 21 seasons, NCIS has a dedicated fan base that's devoted to finding out how things are going to play out with each character.
For this reason, tons of die-hard NCIS fans are confused about Duane Henry's 2018 exit from the show. Why exactly did he leave NCIS? Here's some insight.
Why did Duane Henry leave 'NCIS'? Fans of the show are curious to know the truth.
While Duane was part of NCIS, he played none other than MI6 agent Clayton Reeves. Although the character was only around for two short seasons, viewers were instantly enchanted by him and wanted to see more.
It was quite a shock learning that Reeves — who died a hero when he sacrificed himself for Abby — wouldn't be continuing on with his storyline past Season 15. What exactly happened behind the scenes?
Love u bloody guys 💜 thank you for the love. #NCIS pic.twitter.com/Jx97HyQQkC
— Duane Henry (@RealDuaneHenry) October 25, 2017
Reeves was written into NCIS in the first place thanks to showrunner Gary Glasberg. Gary had plans for Reeves' future, but when Gary passed away in 2016, it seems his plans for the character died with him. According to Duane, the producers who were left behind had no idea how to move forward with Reeves' storyline. It appears that ultimately, they figured the easiest thing to do would be to write him out of the show entirely.
"Clayton was Gary's brainchild; he created this character," Duane told TVLine in 2018. "So when Gary passed, I was feeling a bit of panic in people's faces. I could tell that they were not really sure [what the future held]. I kind of got my bearings together and thought, 'What's going to happen now with the character?' I live in a world called You Never Know, so I'm always prepared for everything."
Source: CBS
What other projects has Duane Henry been part of aside from 'NCIS'?
Duane's role as agent Clayton Reeves isn't the only place viewers easily recognize him from. After being written out of NCIS, he didn't back down from his career as an actor. After the show, he went on to appear in Captain Marvel as well as Mistletoe in Montana, What Just Happened??!, and A Gingerbread Romance.
Some of his previous projects include Jericho, Doctor Who, and The Dark Knight Rises.
Anyone who wants to keep up with Duane on social media will have to request to follow him on his private Instagram account where he already has more than 29K followers. His Twitter account, on the other hand, is totally open to the public. He posts photos of his life in Beverly Hills as well as promotional ads for upcoming projects.
Abby Was an Iconic Member of Team Gibbs on 'NCIS' — What Happened to Her on the Show?
Ducky Mallard Has Been an Invaluable Member of 'NCIS' From the Start — How Old Is He?
Don't Worry, Wilmer Valderrama Aka Special Agent Nick Torres Is Not Leaving 'NCIS'
Latest NCIS News and Updates


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Joan Clark takes a personal and individual approach to everything she does. That includes her interactions with you.
You see, it isn't enough for Joan to have her clients follow a generic "roadmap to success." Why? Because no two people are alike. And because businesses are made up of people, no two organizations and no two cultures are alike either.
What is consistent is Joan's desire to help people become empowered to change their own lives (or careers). Because even though the path to freedom may be foggy, Joan can help you improve your vision, so you can find the way to your extraordinary.
"Success – what is it?


Samir was asked to leave. Transforming a collection of words into a well-formatted sentence is a quite an achievement, especially in English. He usually uses credit cards instead of cash. His family has to live on his small income. How much does this weigh? She quickly found a new boyfriend. For me this is a rumor. Blayne has been teaching us French. Her beauty attracted everyone in the room. He has sore eyes.
I was sworn to secrecy. How was that overlooked? Umm, I have a request... I'll go get my car. It's raining today. Where is my umbrella? "Doc" is an abbreviation of "doctor". He paid at most ten thousand dollars. And that is where the difference lies. Why is there no mobile version of Tatoteba? Hold still a moment while I fix your tie.
I'm currently working at the school as a teacher. Have you called Marc yet? Mayo went straight home after work. I asked my teacher what I should do next. How much will it cost? They believed Roberta. The airport is in Osaka Bay. "How old is she?" "She is twelve years old." Can you see that flickering light? We can't take that chance, can we?
Classes begin next Tuesday. Stay here! I'm not making changes. Rudolph is a reasonable person. What tempted him to propose to her? This is insanity. My tooth ached when I had ice cream, so I might have a cavity. Try not to cry. The trees have exuberant foliage. We went camping near the river.
This may not be coincidental. Love undeniably has wings to fly away from love, but it's also undeniable that it also has wings to fly back. I've packed gambling in. I've been loosing too much money. Don't squander your newly earned money! Don't cry over spilled milk. I have so many stories to tell. What train you are going to take? I thought it was a bottle. There were some men eating their lunches under the trees in front of the library. She was racked with pain.
I like showy clothes. Include these points in a 3-4 minute talk about your book. Knapper struggled to sit up. The movie is two and a half hours long. It's past midnight. This is a movie for men. I can't abide that fellow. Should I take Kanthan? I thought you were in Boston with Bonnie. We haven't been able to achieve that.
It was just for fun. Mann doesn't have to apologize to me. For the best fried chicken in Gulfport, go to Grandma's Kitchen. Srinivas never uses profanity. We elected him chairman. Merat asked us to be silent. He stood ankle-deep in the water. The men aren't shocked, but the women are. I was born in Boston. Can Vince be counted upon?
If there was a computer that didn't tire out my eyes, I'd definitely want to buy it. Fluency in English is a very marketable skill today. Why are you waiting here? Leon has breakfast at the local coffee shop every day. Pria said he knew you. Mr Murata, this is my friend Lisa. The portions at that restaurant are generous. I knew you'd want to go to the dance with Tomas. I'll see what happens. That's what I had thought at first.
Embrace your weirdness Norm visited us this past weekend. Have you ever picked your nose in the presence of the teacher? Rabin refused to talk to me. She was very nearly run over by a truck. It's not unusual that entire galaxies disappear all of the sudden. It would be nice if you could volunteer the information without being prompted. She is said to be very mean. When did you begin learning English? It's going to happen again.
I don't like being bossed around. I had a man build me a house. My brother makes it a rule to look over the newspaper before going to his office. I've almost finished this report. He seems to be in a dark humor. You couldn't live without water. No two snowflakes are exactly alike. She fairly jumped for joy. We aren't like that. Come on, hurry up. You'll miss the bus.


text:
The European Physical Journal A - Hadrons and Nuclei
August 2005 , Volume 25, Issue 2, pp 229–240 | Cite as
A new determination of the γπ → ππ anomalous amplitude via πe- → π-e-π0 data
I. Giller
A. Ocherashvili
T. Ebertshäuser
M. A. Moinester
S. Scherer
First Online: 05 September 2005
We discuss the reaction π-e- → π-e-π0 with the purpose of obtaining information on the γπ → ππ anomalous amplitude \( \cal {F}\)3π. We compare a full calculation at \( \cal {O}\)(p6) in chiral perturbation theory and various phenomenological predictions with the existing data of Amendolia et al. By integrating our theory results using Monte Carlo techniques, we obtain σ = 2.05 nb at \( \cal {O}\)(p6) and σ = 2.17 nb after including the dominant electromagnetic correction. Both results are in good agreement with the experimental cross-section of σ = (2.11±0.47) nb. On the basis of the ChPT results one would extract from the experimental cross-section as amplitudes \( \cal {F}\)3π(0)extr = (9.9±1.1) GeV-3 and \( \cal {F}\)3π(0)extr = (9.6±1.1) GeV-3, respectively, which have to be compared with the low-energy theorem \( \cal {F}\)3π = e/(4π2Fπ3) = 9.72GeV-3. We emphasize the need for new data to allow for a comparison of experimental and theoretical distributions and to obtain \( \cal {F}\)3π with smaller uncertainty.
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© Società Italiana di Fisica and Springer-Verlag 2005
1.School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
2.Institut für KernphysikJohannes Gutenberg-UniversitätMainzGermany
Giller, I., Ocherashvili, A., Ebertshäuser, T. et al. Eur. Phys. J. A (2005) 25: 229. https://doi.org/10.1140/epja/i2005-10116-1
Accepted 05 July 2005
First Online 05 September 2005
DOI https://doi.org/10.1140/epja/i2005-10116-1
Publisher Name Società Italiana di Fisica


Million thanks to Miyako for bringing us her first Chinese subtitled 'King's man' viewing update.
As 柳戲雨 said... 這部戲從一開始聽不懂韓文也沒有字幕時,我們就是被演員的演技所吸引. 不什明時才會給我們更多想像空間喔,難道不是嗎? :p The most important for me is whatever 感動 us initially remained.
We are all different (life experiences, family, education and maybe even cultural background), so each of our own individual interpretation of their interactions might also be different. I am not surprised to know that someone among us might watch this movie over 30 times and each time puts him- or her-self in different character in order to weave a fuller picture.
Thanks Miyako for answering my question. It may sound very philosophical, but searching for the complementary half is one of our goal (mission?) in this lifetime, is it not?
Also thanks 芭樂貓 for explaining those terms. I was like Brenda, never heard of any of them before.


Is Trading Health Coverage for a Higher Base Salary "Pension Spiking"?
SACRAMENTO – Legislation authored by State Senator Joe Simitian (D-Palo Alto) to curb "pension spiking" in the state´s two largest pension systems has passed the Legislature. Senate Bill 1425, which pertains to the California Public Employees´ Retirement System and the California State Teachers´ Retirement System, now moves to the Governor´s desk.
Pension spiking is the practice of boosting an employee´s final salary before retirement by cashing out on vacation time or administrative leave, among other methods.
Because pensions are often based on final-year compensation, inflating end-of-career incomes artificially boosts pensions throughout retirement. "For the most part," said Simitian, "the process is entirely legal. Simply put, the rules now in place encourage employees to game the system."
A 2007 study by San Francisco´s non-profit Pacific Research Institute indicates that pension spiking costs California taxpayers roughly $100 million each year.
"Prior pension legislation was never intended to authorize the pension spiking we are seeing," said the bill´s joint author, State Senator Lou Correa (D-Orange County). "It´s an abusive practice that robs Californians of vital services and precious dollars."
Senate Bill 1425 requires that an individual´s pension be based on a set of criteria that prevent employees from padding final salaries with one time bonuses, end-of-career promotions, and accrued vacation time.
"Pension spiking does a disservice to the public, who ultimately foots the bill; and it does a disservice to other public employees who rely on the resources and solvency of the system for a secure retirement," said Simitian.


Today, air travel is just another form of mass transit. Is there any going back to the glamorous days of yore?
Ah, the lost world of airline glamour. We seem to hear about it every time an airline introduces new uniforms, updates its airport lounges, or adds first-class amenities. Each new luxury or touch of style supposedly recalls the golden age of flying, before price competition, security checks, and slobs in sweatpants ruined everything. Yet despite the wardrobe tweaks and gourmet meals, the magic never returns. Airline glamour is an oxymoron," says a bicoastal friend.
In fact, bringing the glamour back to air travel is much harder than adding amenities. While luxury may provide an aura of glamour, the two are not the same thing. The glamour of air travel—its aspirational meaning in the public imagination—disappeared before its luxury did, dissipating as flying gradually became commonplace. Today it is mass transit, as much defined by crowds as a rush-hour New York subway ride. In 1956, when Life magazine devoted a special issue to the "Air Age," U.S. airlines carried 46 million passengers. In 2005, they carried nearly 739 million. (Each time you take a flight, you count as a passenger.) On a recent October afternoon, the passengers waiting to board a flight from Los Angeles to London filled every chair in the gate area, with some sitting on the floor. No wonder they wore jeans.
You can still find exclusivity in the sky, of course, as long as you're willing to pay. Today's international airlines sell their upscale cabins as mile-high oases of pampered privacy. Could these cabins offer a re-creation of old-fashioned, '50s-style glamour? To find out, I booked a flight from Los Angeles to London in Virgin Atlantic's "Upper Class" cabin, which the airline designed to reflect "an air of natural glamour." In economy seats, Virgin Atlantic passengers have to fight their neighbors for the armrest. But in Upper Class, which Forbes ranks as the best international business class, I enjoyed a private "suite"—a seat and ottoman that fold into a flat bed. The sides block your view of other passengers, creating a cozy sense of personal space. When you're ready to turn in, you change into a complimentary black jersey "sleep suit," and the flight attendant makes your bed. These amenities are all part of an international game of premium-class one-upmanship: ever-larger seats, more choices of food and entertainment, bigger work spaces with better computer hookups, increased privacy. Singapore Airlines recently announced a first-class flat bed thirty-five inches wide—or, as mischievous news reports noted, big enough for two.
On the way to London, I found plenty of luxury, from attentive meal service to a "hot hands" exfoliation-and-moisturizing treatment. But I didn't find glamour—because airline glamour was never on the planes themselves. It was in the imagination, especially the imagination of people who could only dream of flying. Airline glamour never promised anything as mundane as elbow room, much less a flat bed, a massage, or an arugula salad. It promised a better world. Service and dress reflected the more formal era, but no one expected air travel to be comfortable. It was amazing just to have hot food above the clouds. As a rookie Pan Am stewardess in the mid-1960s, Aimée Bratt was struck by "how crowded it was on an airplane, no place to put anything, lines for the lavatories, no place to sit or stand … Passengers got their food trays, there was no choice of meals, drinks were served from a hand tray, six at a time, pillows and blankets were overhead, and there were no extra amenities like headsets or hot towels." But, she writes in her memoir Glamour and Turbulence, "in those days nobody ever complained." Travel itself was privilege enough. Airline glamour was not about the actual experience of flying but about the idea of air travel—and the ideals and identity it represented.
Today's airlines can't recapture this jet-age glamour, not just because planes are crowded and less-than-luxurious but because society has changed. Home and work, labor and leisure, are not as separate as they once were. And jet-age glamour assumed a world run by an easily identifiable elite. People who flew, especially internationally, seemed like a special circle: the best and the brightest, the international playboys, the tennis pros, the carefree college students with time to see the world. Spy stories, from the James Bond novels to I Spy, disguised their heroes as members of this lucky leisure class. On the ground, the briefcase brigade dreamed of being like them.
Jet-age glamour depended on a particular worldview, as well: the innocent internationalism expressed in the opening ceremonies of the Olympics or the Miss Universe pageant, with their colorful costumes and shared aspirations; the sunny faith of Star Trek and trick-or-treating for UNICEF. The jet age's glamorous heroes traveled a world of superficial exoticism and essential sameness. In the 1966 film The Endless Summer, two young surfers fly around the world in search of the perfect wave. Everywhere, from Africa to Australia, they find friendly people who love the beach.
Contrary to popular belief, airline glamour didn't disappear with the coming of U.S. airline deregulation, in 1978. The magic had slipped away at least a decade earlier, as a new, negative idea of air travel seized the popular imagination. Terrorist hijackers turned glamour into horror, killing passengers, blowing up planes, and puncturing internationalist ideals. Airports replaced the free-flowing, glass-ringed spaces of their terminals with security checkpoints and fortified blank walls. Arthur Hailey's 1968 novel, Airport, and its movie spin-offs, portrayed flying as stressful, sordid, and dangerous. "In the popular imagination," writes architectural historian Alastair Gordon in Naked Airport, "air travel was now equated with boredom and disaster." The loss of glamour helped make deregulation possible. If airlines were just another business, why shouldn't they compete in an open marketplace?
Today air travel is just a more or less enjoyable way to get from place to place, not an emotionally resonant symbol of something greater than itself. We frequent flyers forget how unnatural it is to zoom through the air in a metal tube, and we imagine that airline glamour was all about real silverware and perfectly coiffed stewardesses—the experience on the plane. But glamour is always an illusion, an imaginative picture with the blemishes removed. With experience comes disillusionment, no matter how luxurious the reality may be. The glamour of twentieth-century air travel helped to persuade once-fearful travelers to take to the skies and encouraged parochial Americans to go out and see the world. The crowds on today's planes may have destroyed that glamour. But they also demonstrate its power.


I love Curvy Kate bras. They are elegant, supportive and made of good quality material. I own a few Curvy Kate bras in different styles and I really enjoy wearing them. However, the road to getting them to fit was quite arduous. It was confusing, frustrating and a time consuming process to find the right fit in Curvy Kate bras. After some extensive online research coupled with my own experience, I thought I'd share my findings. Please keep in mind that every woman's shape is different and what works for me might not work for you. I'm only sharing my opinion based on my experiences with the CK bras that I own.
Every brand of bras has it's pros and cons. I think Curvy Kate bras excel in the 'appeal' department. They really make very beautiful bras. Out of the hundreds of CK bras I've seen, I may have disliked only one bra because the colour did not suit my taste. Otherwise, CK bras are very pretty and feminine (always a bonus). Don't get me wrong though, I love industrial bras too (referred to as "ugly bras" by my husband) but they have a time and place. I do not think of CK bras as industrial. They are supportive enough for everyday wear yet elegant enough for a special occasion. They are also quite affordable. Curvy Kate bras are not as cheap as department store bras (which are very low quality in my opinion and in this case you really get what you pay for) but they won't break the bank either. More often than not, I find them on sale in many online stores, sometimes even up to 70% off on fashion colours, which makes them a steal at that point.
I've heard from many women as well as store owners that Curvy Kate bras are not very supportive. I personally think that this is untrue but I believe I understand why some might see CK bras that way. Over the course of the last year I've tried on several Curvy Kate bras through online and local stores. I found that Curvy Kate bras are supportive when they are in the correct size. They work fine and will even look okay initially but they will not offer enough support if not in the proper size, which for me felt very much like a game of Where's Waldo. Not only that, but also it took me a while to figure out that I am wearing the wrong size CK bra because with many CK bras, the tell-tale signs of bad fit are not very visible. Instead, CK bras tend to give an odd shape to the breast. I'll talk about that in a bit. Upon discovering this, and to prove the theory to myself, I tried one specific bra in multiple sizes. Sure enough, I did not get quadraboob (at least not initially), and I did not get any side spilling. However, the shape was all wrong.
Let's talk about shape. Different bras give different shapes. What may be ideal for one woman, may be undesirable for another. Not every woman thinks of shapes the same way. Some women think that a round shape is the ideal one for them. Some prefer a more natural shape. And some prefer a pointy shape. Of course, these are my own interpretations of the shapes down to the names. I am not sure if others even refer to them that way but the best way to demonstrate what those shapes are, from my perspective, is through pictures. Observe..
In general, Curvy Kate bras do not give me a pointy shape. In fact, I have yet to come across one that does. I find that CK bras give me a very natural shape. They do not lift up my breasts as high as some of my other bras, such as Freya, Fantasie and Panache (which actually give me a very uplifted round shape) but CK bras are supportive when I get fitted in the right size. And herein lies the problem. My biggest beef with Curvy Kate bras is the inconsistency in sizing between the different models. Sometimes even the same bra in a different colour will fit differently. It's borderline maddening!
When I wear a CK bra in the right size, I almost always end up with a natural shape which is the desired effect. It's also hard to get comfortable in some of the bras because I find that CK suits women with wider shoulders. I have narrow shoulders and when I wear a brand new CK bra I always get the dreaded chafing at the shoulder, where the strap meets the cup. This is partly due to my shape, I guess, as I'm narrow and boney at the shoulders. I don't have much meat around that area though I'm working on building muscle there so I can lift grocery bags without tipping over. But in all fairness that is a design issue which I can't really fault Curvy Kate for. I just have tiny shoulders. Thankfully, after a few washes the bra softens and the chafing goes away.
The first Curvy Kate bra I bought was the Emily. I fell in love with it because it looked so elegant but I could not get it to fit right no matter how hard I tried. It started to slowly slide down my torso into the smaller part of my waist and so after wearing the bra for an hour I would end up with a very odd shape, almost like a square/triangle/trapezoid. I didn't really get quadraboob, rather a slight bubble at the top edge of the cup. It looked like I am about to spill out yet not really spilling. It turns out that the band was too big.
I realize that in most cases a band that is too big will ride up, but I am certain that there are other women out there who are built like me with narrow shoulders and a narrow ribcage that kind of curves in towards the waist so that bras sometimes will slide down with the weight of the breasts. Basically the band slides down and so my breasts do not fill the cup entirely and I get spillage at the top of the cup. I have to thank Claire of Butterfly Collection for this discovery. I didn't realize that I was not filling the cup until I talked to her. She brought my attention to the fact that just because I'm spilling out of the cup does not necessarily mean that the bra is too small and that I shouldn't discard a bra until I make sure that I am filling the cup all the way. It's when I looked into why I am not filling the cup that I accidentally discovered why Curvy Kate gives such a weird shape.
The first time I tried the Emily I opted for my usual size of 30G. It looked fine initially but then 20 minutes later I got this square like shape you see above. I didn't know it at the time but it turns out this is what Curvy Kate looks like when it's too small. So I went back to the store and I tried the 30GG. I got the shape you see in the middle and it was huge. I had wrinkling at the top and the straps would not stay up even though I had them pretty tight. I left the store thinking that the Emily is just not for me. But I couldn't get over the colours. I just loved the bra and couldn't let it go. It's not until I came across this post by Bras I Hate that I realized how the sizing works in Curvy Kate. Thanks to her, I was able to go back and keep trying different sizes until I discovered that the Emily looks absolutely PERFECT on me in 28H. However, I discovered that 28H doesn't work for me in other CK styles. When I tried a black CK Princess in 28H for example, I couldn't even hook it. I had to wear it with an extender on the loosest setting and I still got a bit of that square shape. In 30H, the black Princess fits perfectly. Yet the Princess in other colours like pink or wine is too big in 30H and I end up with no support at all. Instead, the 28GG in the pink or wine fits perfectly.
I realize that 28GG, 30G, 28H, 30GG, and 32G are sister sizes and that many women can get away with that in some bras. In my case, I cannot. And definitely not with Curvy Kate bras. I have to be spot on with the sizing otherwise I don't get that natural shape and instead end up with either a square trapezoid shape or a droopy triangle shape. I also find the inconsistency between colours to be quite frustrating. It's especially annoying when ordering online because I can't just order a bra I like in a different colour. There is always a good chance that it will fit differently and returns are always a hassle especially for international orders. Case in point; the blue Portia fits good in 32G; the purple Portia fits good in 30G. Yet, in bras like the Emily and Tease Me, the size is consistent across colours respectively. Go figure!
In summary, I think Curvy Kate makes some stunning bras and I really hope that they can make their sizing more consistent in the future. As it stands now, any time I see a Curvy Kate bra that I like, it's always a gamble as to whether it will fit. Despite all this madness, I think Curvy Kate bras are beautiful enough that I would go through the trouble of trying to find the size that fits me best. The bras are also affordable and so a little bit of effort goes a long way. Okay, maybe a lot of effort but still, I think CK bras are too pretty to give up on them. Even though I have issues with the sizing, I'd say that Curvy Kate bras are among my favourites. I find myself reaching for one at least twice a week. Simply put, Curvy Kate bras are pretty, supportive and fairly comfortable. And when they fit properly, they look gorgeous.
To learn more about the latest styles of Curvy Kate bras you can visit them here.