Input: Does DUO contain more instances per image than COCO?

Context:
\section{Introduction}
Underwater robot picking is to use the robot to automatically capture sea creatures like holothurian, echinus, scallop, or starfish in an open-sea farm where underwater object detection is the key technology for locating creatures. Until now, the datasets used in this community are released by the Underwater Robot Professional Contest (URPC$\protect\footnote{Underwater Robot Professional Contest: {\bf http://en.cnurpc.org}.}$) beginning from 2017, in which URPC2017 and URPC2018 are most often used for research. Unfortunately, as the information listed in Table \ref{Info}, URPC series datasets do not provide the annotation file of the test set and cannot be downloaded after the contest.
Therefore, researchers \cite{2020arXiv200511552C,2019arXiv191103029L} first have to divide the training data into two subsets, including a new subset of training data and a new subset of testing data, and then train their proposed method and other \emph{SOTA} methods. On the one hand, training other methods results in a significant increase in workload. On the other hand, different researchers divide different datasets in different ways,
\begin{table}[t]
\renewcommand\tabcolsep{3.5pt}
\caption{Information about all the collected datasets. * denotes the test set's annotations are not available. \emph{3} in Class means three types of creatures are labeled, \emph{i.e.,} holothurian, echinus, and scallop. \emph{4} means four types of creatures are labeled (starfish added). Retention represents the proportion of images that retain after similar images have been removed.}
\centering
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Dataset&Train&Test&Class&Retention&Year \\
\hline
URPC2017&17,655&985*&3&15\%&2017 \\
\hline
URPC2018&2,901&800*&4&99\%&2018 \\
\hline
URPC2019&4,757&1,029*&4&86\%&2019 \\
\hline
URPC2020$_{ZJ}$&5,543&2,000*&4&82\%&2020 \\
\hline
URPC2020$_{DL}$&6,575&2,400*&4&80\%&2020 \\
\hline
UDD&1,827&400&3&84\%&2020 \\
\hline

\end{tabular}
\label{Info}
\end{table}
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=1\linewidth]{example.pdf}
\end{center}
   \caption{Examples in DUO, which show a variety of scenarios in underwater environments.}
\label{exam}
\end{figure*}
causing there is no unified benchmark to compare the performance of different algorithms.
In terms of the content of the dataset images, there are a large number of similar or duplicate images in the URPC datasets. URPC2017 only retains 15\% images after removing similar images compared to other datasets. Thus the detector trained on URPC2017 is easy to overfit and cannot reflect the real performance.
For other URPC datasets, the latter also includes images from the former, \emph{e.g.}, URPC2019 adds 2,000 new images compared to URPC2018; compared with URPC2019, URPC2020$_{ZJ}$ adds 800 new images. The URPC2020$_{DL}$ adds 1,000 new images compared to the URPC2020$_{ZJ}$. It is worth mentioning that the annotation of all datasets is incomplete; some datasets lack the starfish labels and it is easy to find error or missing labels. \cite{DBLP:conf/iclr/ZhangBHRV17} pointed out that although the CNN model has a strong fitting ability for any dataset, the existence of dirty data will significantly weaken its robustness.
Therefore, a reasonable dataset (containing a small number of similar images as well as an accurate annotation) and a corresponding recognized benchmark are urgently needed to promote community development.

To address these issues, we introduce a dataset called Detecting Underwater Objects (DUO) by collecting and re-annotating all the available underwater datasets. It contains 7,782 underwater images after deleting overly similar images and has a more accurate annotation with four types of classes (\emph{i.e.,} holothurian, echinus, scallop, and starfish).
Besides, based on the MMDetection$\protect\footnote{MMDetection is an open source object detection toolbox based on PyTorch. {\bf https://github.com/open-mmlab/mmdetection}}$ \cite{chen2019mmdetection} framework, we also provide a \emph{SOTA} detector benchmark containing efficiency and accuracy indicators, providing a reference for both academic research and industrial applications. It is worth noting that JETSON AGX XAVIER$\protect\footnote{JETSON AGX XAVIER is an embedded development board produced by NVIDIA which could be deployed in an underwater robot. Please refer {\bf https://developer.nvidia.com/embedded/jetson-agx-xavier-developer-kit} for more information.}$ was used to assess all the detectors in the efficiency test in order to simulate robot-embedded environment. DUO will be released in https://github.com/chongweiliu soon.

In summary, the contributions of this paper can be listed as follows.

  $\bullet$ By collecting and re-annotating all relevant datasets, we introduce a dataset called DUO with more reasonable annotations as well as a variety of underwater scenes.

  $\bullet$ We provide a corresponding benchmark of \emph{SOTA} detectors on DUO including efficiency and accuracy indicators which could be a reference for both academic research and industrial applications.

\pagestyle{empty}
\section{Background}
In the year of 2017, underwater object detection for open-sea farming is first proposed in the target recognition track of Underwater Robot Picking Contest 2017$\protect\footnote{From 2020, the name has been changed into Underwater Robot Professional Contest which is also short for URPC.}$ (URPC2017) which aims to promote the development of theory, technology, and industry of the underwater agile robot and fill the blank of the grabbing task of the underwater agile robot. The competition sets up a target recognition track, a fixed-point grasping track, and an autonomous grasping track. The target recognition track concentrates on finding the {\bf high accuracy and efficiency} algorithm which could be used in an underwater robot for automatically grasping.

The datasets we used to generate the DUO are listed below. The detailed information has been shown in Table \ref{Info}.

  {\bf URPC2017}: It contains 17,655 images for training and 985 images for testing and the resolution of all the images is 720$\times$405. All the images are taken from 6 videos at an interval of 10 frames. However, all the videos were filmed in an artificial simulated environment and pictures from the same video look almost identical.

   {\bf URPC2018}: It contains 2,901 images for training and 800 images for testing and the resolutions of the images are 586$\times$480, 704$\times$576, 720$\times$405, and 1,920$\times$1,080. The test set's annotations are not available. Besides, some images were also collected from an artificial underwater environment.

  {\bf URPC2019}: It contains 4,757 images for training and 1029 images for testing and the highest resolution of the images is 3,840$\times$2,160 captured by a GOPro camera. The test set's annotations are also not available and it contains images from the former contests.

  {\bf URPC2020$_{ZJ}$}: From 2020, the URPC will be held twice a year. It was held first in Zhanjiang, China, in April and then in Dalian, China, in August. URPC2020$_{ZJ}$ means the dataset released in the first URPC2020 and URPC2020$_{DL}$ means the dataset released in the second URPC2020. This dataset contains 5,543 images for training and 2,000 images for testing and the highest resolution of the images is 3,840$\times$2,160. The test set's annotations are also not available.

  {\bf URPC2020$_{DL}$}: This dataset contains 6,575 images for training and 2,400 images for testing and the highest resolution of the images is 3,840$\times$2,160. The test set's annotations are also not available.

  {\bf UDD \cite{2020arXiv200301446W}}: This dataset contains 1,827 images for training and 400 images for testing and the highest resolution of the images is 3,840$\times$2,160. All the images are captured by a diver and a robot in a real open-sea farm.

\begin{figure}[t]
\begin{center}
\includegraphics[width=1\linewidth]{pie.pdf}
\end{center}
   \caption{The proportion distribution of the objects in DUO.}
\label{pie}
\end{figure}

\begin{figure*}
  \centering
  \subfigure[]{\includegraphics[width=3.45in]{imagesize.pdf}}
  \subfigure[]{\includegraphics[width=3.45in]{numInstance.pdf}}
  \caption{(a) The distribution of instance sizes for DUO; (b) The number of categories per image.}
  \label{sum}
\end{figure*}
\section{Proposed Dataset}

\subsection{Image Deduplicating}
As we explained in Section 1, there are a large number of similar or repeated images in the series of URPC datasets. Therefore, it is important to delete duplicate or overly similar images and keep a variety of underwater scenarios when we merge these datasets together. Here we employ the Perceptual Hash algorithm (PHash) to remove those images. PHash has the special property that the hash value is dependent on the image content, and it remains approximately the same if the content is not significantly modified. Thus we can easily distinguish different scenarios and delete duplicate images within one scenario.

After deduplicating, we obtain 7,782 images (6,671 images for training; 1,111 for testing). The retention rate of the new dataset is 95\%, which means that there are only a few similar images in the new dataset. Figure \ref{exam} shows that our dataset also retains various underwater scenes.

\subsection{Image Re-annotation}
Due to the small size of objects and the blur underwater environment, there are always missing or wrong labels in the existing annotation files. In addition, some test sets' annotation files are not available and some datasets do not have the starfish annotation. In order to address these issues, we follow the next process which combines a CNN model and manual annotation to re-annotate these images. Specifically, we first train a detector (\emph{i.e.,} GFL \cite{li2020generalized}) with the originally labeled images. After that, the trained detector predicts all the 7,782 images. We treat the prediction as the groundtruth and use it to train the GFL again. We get the final GFL prediction called {\bf the coarse annotation}. Next, we use manual correction to get the final annotation called {\bf the fine annotation}. Notably, we adopt the COCO \cite{Belongie2014} annotation form as the final format.
\subsection{Dataset Statistics}
{\bf The proportion of classes}: The total number of objects is 74,515. Holothurian, echinus, scallop, and starfish are 7,887, 50,156, 1,924, and 14,548, respectively. Figure \ref{pie} shows the proportion of each creatures where echinus accounts for 67.3\% of the total. The whole data distribution shows an obvious long-tail distribution because the different economic benefits of different seafoods determine the different breed quantities.

{\bf The distribution of instance sizes}: Figure \ref{sum}(a) shows an instance size distribution of DUO. \emph{Percent of image size} represents the ratio of object area to image area, and \emph{Percent of instance} represents the ratio of the corresponding number of objects to the total number of objects. Because of these small creatures and high-resolution images, the vast majority of objects occupy 0.3\% to 1.5\% of the image area.

{\bf The instance number per image}: Figure \ref{sum}(b) illustrates the number of categories per image for DUO. \emph{Number of instances} represents the number of objects one image has, and \emph{ Percentage of images} represents the ratio of the corresponding number of images to the total number of images. Most images contain between 5 and 15 instances, with an average of 9.57 instances per image.

{\bf Summary}:
In general, smaller objects are harder to detect. For PASCAL VOC \cite{Everingham2007The} or COCO \cite{Belongie2014}, roughly 50\% of all objects occupy no more than 10\% of the image itself, and others evenly occupy from 10\% to 100\%.
In the aspect of instances number per image, COCO contains 7.7 instances per image and VOC contains 3. In comparison, DUO has 9.57 instances per image and most instances less than 1.5\% of the image size.
Therefore, DUO contains almost exclusively massive small instances and has the long-tail distribution at the same time, which means it is promising to design a detector to deal with massive small objects and stay high efficiency at the same time for underwater robot picking.

\section{Benchmark}
Because the aim of underwater object detection for robot picking is to find {\bf the high accuracy and efficiency} algorithm, we consider both the accuracy and efficiency evaluations in the benchmark as shown in Table \ref{ben}.

\subsection{Evaluation Metrics}
Here we adopt the standard COCO metrics (mean average precision, \emph{i.e.,} mAP) for the accuracy evaluation and also provide the mAP of each class due to the long-tail distribution.

{\bf AP} -- mAP at IoU=0.50:0.05:0.95.

{\bf AP$_{50}$} --  mAP at IoU=0.50.

{\bf AP$_{75}$} --  mAP at IoU=0.75.

{\bf AP$_{S}$} --   {\bf AP} for small objects of area smaller than 32$^{2}$.

{\bf AP$_{M}$} --   {\bf AP} for objects of area between 32$^{2}$ and 96$^{2}$.

{\bf AP$_{L}$} --   {\bf AP} for large objects of area bigger than 96$^{2}$.

{\bf AP$_{Ho}$} --  {\bf AP} in holothurian.

{\bf AP$_{Ec}$} --  {\bf AP} in echinus.

{\bf AP$_{Sc}$} --  {\bf AP} in scallop.

{\bf AP$_{St}$} --  {\bf AP} in starfish.

For the efficiency evaluation, we provide three metrics:

{\bf Param.} --  The parameters of a detector.

{\bf FLOPs} --  Floating-point operations per second.

{\bf FPS} --  Frames per second.

Notably, {\bf FLOPs} is calculated under the 512$\times$512 input image size and {\bf FPS} is tested on a JETSON AGX XAVIER under MODE$\_$30W$\_$ALL.

\subsection{Standard Training Configuration}
We follow a widely used open-source toolbox, \emph{i.e.,} MMDetection (V2.5.0) to produce up our benchmark. During the training, the standard configurations are as follows:

  $\bullet$ We initialize the backbone models (\emph{e.g.,} ResNet50) with pre-trained parameters on ImageNet \cite{Deng2009ImageNet}.

  $\bullet$ We resize each image into 512 $\times$ 512 pixels both in training and testing. Each image is flipped horizontally with 0.5 probability during training.

  $\bullet$ We normalize RGB channels by subtracting 123.675, 116.28, 103.53 and dividing by 58.395, 57.12, 57.375, respectively.

  $\bullet$ SGD method is adopted to optimize the model. The initial learning rate is set to be 0.005 in a single GTX 1080Ti with batchsize 4 and is decreased by 0.1 at the 8th and 11th epoch, respectively. WarmUp \cite{2019arXiv190307071L} is also employed in the first 500 iterations. Totally there are 12 training epochs.

  $\bullet$ Testing time augmentation (\emph{i.e.,} flipping test or multi-scale testing) is not employed.

\subsection{Benchmark Analysis}
Table \ref{ben} shows the benchmark for the \emph{SOTA} methods. Multi- and one- stage detectors with three kinds of backbones (\emph{i.e.,} ResNet18, 50, 101) give a comprehensive assessment on DUO. We also deploy all the methods to AGX to assess efficiency.

In general, the multi-stage (Cascade R-CNN) detectors have high accuracy and low efficiency, while the one-stage (RetinaNet) detectors have low accuracy and high efficiency. However, due to recent studies \cite{zhang2019bridging} on the allocation of more reasonable positive and negative samples in training, one-stage detectors (ATSS or GFL) can achieve both high accuracy and high efficiency.

\begin{table*}[htbp]
\renewcommand\tabcolsep{3.0pt}

\begin{center}
\caption{Benchmark of \emph{SOTA} detectors (single-model and single-scale results) on DUO. FPS is measured on the same machine with a JETSON AGX XAVIER under the same MMDetection framework, using a batch size of 1 whenever possible. R: ResNet.}
\label{ben}
\begin{tabular}{|l|l|c|c|c|ccc|ccc|cccc|}
\hline
Method&Backbone&Param.&FLOPs&FPS&AP&AP$_{50}$&AP$_{75}$&AP$_{S}$&AP$_{M}$&AP$_{L}$&AP$_{Ho}$&AP$_{Ec}$&AP$_{Sc}$&AP$_{St}$ \\
\hline
\emph{multi-stage:} &&&&&&&&&&&&&& \\

\multirow{3}{*}{Faster R-CNN \cite{Ren2015Faster}}
&R-18&28.14M&49.75G&5.7&50.1&72.6&57.8&42.9&51.9&48.7&49.1&60.1&31.6&59.7\\
&R-50&41.14M&63.26G&4.7&54.8&75.9&63.1&53.0&56.2&53.8&55.5&62.4&38.7&62.5\\
&R-101&60.13M&82.74G&3.7&53.8&75.4&61.6&39.0&55.2&52.8&54.3&62.0&38.5&60.4\\
\hline

\multirow{3}{*}{Cascade R-CNN \cite{Cai_2019}}
&R-18&55.93M&77.54G&3.4&52.7&73.4&60.3&\bf 49.0&54.7&50.9&51.4&62.3&34.9&62.3\\
&R-50&68.94M&91.06G&3.0&55.6&75.5&63.8&44.9&57.4&54.4&56.8&63.6&38.7&63.5\\
&R-101&87.93M&110.53G&2.6&56.0&76.1&63.6&51.2&57.5&54.7&56.2&63.9&41.3&62.6\\
\hline

\multirow{3}{*}{Grid R-CNN \cite{lu2019grid}}
&R-18&51.24M&163.15G&3.9&51.9&72.1&59.2&40.4&54.2&50.1&50.7&61.8&33.3&61.9\\
&R-50&64.24M&176.67G&3.4&55.9&75.8&64.3&40.9&57.5&54.8&56.7&62.9&39.5&64.4\\
&R-101&83.24M&196.14G&2.8&55.6&75.6&62.9&45.6&57.1&54.5&55.5&62.9&41.0&62.9\\
\hline

\multirow{3}{*}{RepPoints \cite{yang2019reppoints}}
&R-18&20.11M&\bf 35.60G&5.6&51.7&76.9&57.8&43.8&54.0&49.7&50.8&63.3&33.6&59.2\\
&R-50&36.60M&48.54G&4.8&56.0&80.2&63.1&40.8&58.5&53.7&56.7&65.7&39.3&62.3\\
&R-101&55.60M&68.02G&3.8&55.4&79.0&62.6&42.2&57.3&53.9&56.0&65.8&39.0&60.9\\
\hline
\hline
\emph{one-stage:} &&&&&&&&&&&&&& \\
\multirow{3}{*}{RetinaNet \cite{Lin2017Focal}}
&R-18&19.68M&39.68G&7.1&44.7&66.3&50.7&29.3&47.6&42.5&46.9&54.2&23.9&53.8\\
&R-50&36.17M&52.62G&5.9&49.3&70.3&55.4&36.5&51.9&47.6&54.4&56.6&27.8&58.3\\
&R-101&55.16M&72.10G&4.5&50.4&71.7&57.3&34.6&52.8&49.0&54.6&57.0&33.7&56.3\\
\hline

\multirow{3}{*}{FreeAnchor \cite{2019arXiv190902466Z}}
&R-18&19.68M&39.68G&6.8&49.0&71.9&55.3&38.6&51.7&46.7&47.2&62.8&28.6&57.6\\
&R-50&36.17M&52.62G&5.8&54.4&76.6&62.5&38.1&55.7&53.4&55.3&65.2&35.3&61.8\\
&R-101&55.16M&72.10G&4.4&54.6&76.9&62.9&36.5&56.5&52.9&54.0&65.1&38.4&60.7\\
\hline

\multirow{3}{*}{FoveaBox \cite{DBLP:journals/corr/abs-1904-03797}}
&R-18&21.20M&44.75G&6.7&51.6&74.9&57.4&40.0&53.6&49.8&51.0&61.9&34.6&59.1\\
&R-50&37.69M&57.69G&5.5&55.3&77.8&62.3&44.7&57.4&53.4&57.9&64.2&36.4&62.8\\
&R-101&56.68M&77.16G&4.2&54.7&77.3&62.3&37.7&57.1&52.4&55.3&63.6&38.9&60.8\\
\hline

\multirow{3}{*}{PAA \cite{2020arXiv200708103K}}
&R-18&\bf 18.94M&38.84G&3.0&52.6&75.3&58.8&41.3&55.1&50.2&49.9&64.6&35.6&60.5\\
&R-50&31.89M&51.55G&2.9&56.8&79.0&63.8&38.9&58.9&54.9&56.5&66.9&39.9&64.0\\
&R-101&50.89M&71.03G&2.4&56.5&78.5&63.7&40.9&58.7&54.5&55.8&66.5&42.0&61.6\\
\hline

\multirow{3}{*}{FSAF \cite{zhu2019feature}}
&R-18&19.53M&38.88G&\bf 7.4&49.6&74.3&55.1&43.4&51.8&47.5&45.5&63.5&30.3&58.9\\
&R-50&36.02M&51.82G&6.0&54.9&79.3&62.1&46.2&56.7&53.3&53.7&66.4&36.8&62.5\\
&R-101&55.01M&55.01G&4.5&54.6&78.7&61.9&46.0&57.1&52.2&53.0&66.3&38.2&61.1\\
\hline

\multirow{3}{*}{FCOS \cite{DBLP:journals/corr/abs-1904-01355}}
&R-18&\bf 18.94M&38.84G&6.5&48.4&72.8&53.7&30.7&50.9&46.3&46.5&61.5&29.1&56.6\\
&R-50&31.84M&50.34G&5.4&53.0&77.1&59.9&39.7&55.6&50.5&52.3&64.5&35.2&60.0\\
&R-101&50.78M&69.81G&4.2&53.2&77.3&60.1&43.4&55.4&51.2&51.7&64.1&38.5&58.5\\
\hline

\multirow{3}{*}{ATSS \cite{zhang2019bridging}}
&R-18&\bf 18.94M&38.84G&6.0&54.0&76.5&60.9&44.1&56.6&51.4&52.6&65.5&35.8&61.9\\
&R-50&31.89M&51.55G&5.2&58.2&\bf 80.1&66.5&43.9&60.6&55.9&\bf 58.6&67.6&41.8&64.6\\
&R-101&50.89M&71.03G&3.8&57.6&79.4&65.3&46.5&60.3&55.0&57.7&67.2&42.6&62.9\\
\hline

\multirow{3}{*}{GFL \cite{li2020generalized}}
&R-18&19.09M&39.63G&6.3&54.4&75.5&61.9&35.0&57.1&51.8&51.8&66.9&36.5&62.5\\
&R-50&32.04M&52.35G&5.5&\bf 58.6&79.3&\bf 66.7&46.5&\bf 61.6&55.6&\bf 58.6&\bf 69.1&41.3&\bf 65.3\\
&R-101&51.03M&71.82G&4.1&58.3&79.3&65.5&45.1&60.5&\bf 56.3&57.0&\bf 69.1&\bf 43.0&64.0\\

\hline
\end{tabular}
\end{center}
\end{table*}
Therefore, in terms of accuracy, the accuracy difference between the multi- and the one- stage methods in AP is not obvious, and the AP$_{S}$ of different methods is always the lowest among the three size AP. For class AP, AP$_{Sc}$ lags significantly behind the other three classes because it has the smallest number of instances. In terms of efficiency, large parameters and FLOPs result in low FPS on AGX, with a maximum FPS of 7.4, which is hardly deployable on underwater robot. Finally, we also found that ResNet101 was not significantly improved over ResNet50, which means that a very deep network may not be useful for detecting small creatures in underwater scenarios.

Consequently, the design of high accuracy and high efficiency detector is still the main direction in this field and there is still large space to improve the performance.
In order to achieve this goal, a shallow backbone with strong multi-scale feature fusion ability can be proposed to extract the discriminant features of small scale aquatic organisms; a specially designed training strategy may overcome the DUO's long-tail distribution, such as a more reasonable positive/negative label sampling mechanism or a class-balanced image allocation strategy within a training batch.

\section{Conclusion}
In this paper, we introduce a dataset (DUO) and a corresponding benchmark to fill in the gaps in the community. DUO contains a variety of underwater scenes and more reasonable annotations. Benchmark includes efficiency and accuracy indicators to conduct a comprehensive evaluation of the \emph{SOTA} decoders. The two contributions could serve as a reference for academic research and industrial applications, as well as promote community development.

Answer: Yes, DUO has 9.57 instances per image while COCO contains 7.7.


Input: When did the Tevatron Collider Run II start and when is it expected to end?

Context:
\section{INTRODUCTION}
The Tevatron Collider Run II started in March 2002 and is expected
to continue until the end of this decade. The Tevatron and the
two detectors, CDF and D\O, have been performing  well in 2004,
each experiment is collecting data at the rate
of $\approx$10 pb$^{-1}$ per week.
The total  luminosity accumulated by August 2004 is $\approx$500 pb$^{-1}$
per detector.
The rich physics program includes the
production and precision measurement of properties of  standard model (SM)
objects, as well as searches for phenomena beyond standard model.
In this brief review we focus on areas of most interest
to the lattice community. We present
new results on the top quark mass
and their implication for the mass of the SM Higgs boson,
on searches for the SM Higgs boson, on evidence for the $X(3872)$ state,
on searches for pentaquarks, and on $b$ hadron properties.
All Run II results presented here are preliminary.

\section{TOP QUARK MASS}

The experiments CDF and D\O\ published several direct  measurements of
the top quark pole mass, $\ensuremath{M_{\mathrm{top}}}$,
based on Run I data (1992-1996).
The ``lepton $+$ jets'' channel yields the most precise determination of
$\ensuremath{M_{\mathrm{top}}}$. Recently, the
D\O\ collaboration published a new measurement~\cite{Mtop1-D0-l+j-new},
based on a powerful analysis technique yielding  greatly improved precision.
The differential probability
that the measured variables in any event correspond to the signal
is calculated as a function of $\ensuremath{M_{\mathrm{top}}}$.
The maximum in the product of the individual event probabilities
provides the best estimate of $\ensuremath{M_{\mathrm{top}}}$.
The critical differences from previous analyses
in the lepton $+$ jets decay channel lie in
the assignment of more
weight to events that are well measured or more likely to correspond to
$t \bar t$ signal,
and  the handling of the combinations of final-state objects
(lepton, jets, and imbalance in transverse momentum)
and their identification with
top-quark decay products in an event.
The new combined value for the top-quark mass from Run I is
$\ensuremath{M_{\mathrm{top}}}  =  178.0\pm4.3~\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$.

In Run II, both collaborations  have been exploring several different techniques
for $\ensuremath{M_{\mathrm{top}}}$
measurements. The best single CDF result comes from a dynamic likelihood method
(DLM). The method is similar to
the technique used in Ref.~\cite{Mtop1-D0-l+j-new}.
The result is $\ensuremath{M_{\mathrm{top}}} = 177.8^{+4.5}_{-5.0} (stat) \pm  6.2 (syst) ~\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$.
The joint likelihood of the selected events is shown in Fig. ~\ref{fig:cdf_tml}.
The Run II goal is a 1\% uncertainty on $\ensuremath{M_{\mathrm{top}}}$.

\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=5.8cm,width=8.1cm]  {data_22ev_likelihood.eps}
\vspace*{-1.2cm}
\caption{The joint likelihood of top candidates(CDF).}
\label{fig:cdf_tml}
\end{figure}

\section{SEARCH FOR SM HIGGS BOSON}

The constraints on the SM Higgs ($H$)  boson  mass from
published  measurements, updated to include the new D\O\ top mass
measurement~\cite{Mtop1-D0-l+j-new}, are
$M_H = 117 ^{+67}_{-45}~\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$, $M_H < 251~\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$ at 95\% C.L.
The  new most likely  value of $M_H$
is above the experimentally excluded range,
and sufficiently low for $H$ to be observed at the Tevatron.

\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=7.5cm,width=7.8cm]  {d0_wbb_fig_3_err.eps}
\vspace*{-1.1cm}
\caption{Distribution of the dijet
invariant mass for $W+2 b$-tagged jets  events,
compared to the expectation (D\O).
}
\label{fig:d0_wbb_2tag}
\end{figure}

D\O\  has conducted a search for $H$ at $M_H < 140~\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$
in the production channel
$p \bar{p} \rightarrow WH \rightarrow  e \nu b \bar{b}$.
The experimental signature of  $WH \rightarrow e \nu b \bar{b}$
is a final state with
one high $p_T$ electron, two  $b$ jets, and
large missing transverse energy  resulting from
the undetected neutrino.
The dominant backgrounds to $WH$ production
are  $W b \bar{b}$, $t \bar{t}$ and single-top production.
The distribution
of the dijet mass for events with two $b$-tagged jets is shown in
Fig.~\ref{fig:d0_wbb_2tag}.
Also shown is the  expected contribution ($0.06$ events)
from the $b \bar{b}$ decay of a
SM Higgs boson with $M_H =$ 115 $\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$.
No events are observed in the  dijet mass window of 85--135  $\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$.
D\O\ sets a limit on the cross section
for $\sigma( p\bar{p} \rightarrow WH) \times B(H \rightarrow b \bar{b}) $
of 9.0 pb at the 95\% C.L.,  for a 115  $\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$ Higgs boson.
The results for mass points 105, 125, and 135 $\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$
 are 11.0, 9.1 and 12.2 pb,  respectively.

\begin{figure}[htb]
\vspace*{-1.2cm}
\includegraphics[height=0.33\textheight,width=8.0cm]{whww_aps04_bw.eps}

\vspace*{-1.2cm}
\caption{95\% limits on the $H$ production (CDF).}
\label{fig:cdf_whww}
\end{figure}

CDF  has done  a similar search, allowing either an  electron or a muon
in the final state.  Both groups have also searched for $H$ produced in
gluon-gluon fusion, with subsequent decay to a pair of $W$ bosons.
The CDF results for both channels  are shown in Fig.~\ref{fig:cdf_whww}.

\section{THE STATE X(3872)}

\begin{figure}[htb]

\includegraphics[height=8.0cm,width=7.5cm]  {X3872cdfPRL1FullM.eps}
\vspace*{-1cm}
\caption{The $X(3872)$ signal (CDF).}
\label{fig:cdf_x}
\end{figure}

 The existence of the $X(3872)$ state discovered by
the Belle Collaboration~\cite{Belle-X}
 has been confirmed
 in $p \bar{p}$ collisions by  CDF~\cite{cdf-X} (see Fig.~\ref{fig:cdf_x})
and D\O~\cite{d0-X}.
 It is still unclear whether this particle is a $c\bar{c}$ state,
 or a more complex object.  When the data are separated according to
production and decay variables, D\O\  finds no significant
differences between the $X(3872)$ and
the $c \bar{c}$ state $\psi(2S)$.
CDF has analysed the ``lifetime'' distribution of the $X(3872)$ events in order to
quantify what fraction of this state arises from decay of $B$ hadrons, as opposed to
those produced promptly. The authors find that for the selected samples
28.3$\pm$1.0$(stat)\pm$0.7$(syst)$\% of $\psi(2S)$ candidates are from $b$ decays,
whereas 16.1$\pm$4.9$(stat)\pm$2.0$(syst)$\% of $X$ mesons arise from such decays.

\section{SEARCH FOR PENTAQUARKS}

\begin{figure}[htb]

\includegraphics[height=0.27\textheight,width=7.6cm]  {mpks_1stminbias.eps}
\vspace*{-1.2cm}

\caption{Invariant mass distribution of an identified proton and a $K^0_s$ candidate. (CDF)
}
\label{fig:pqtheta}
\end{figure}

\begin{figure}[htb]

\vspace*{-0.9cm}
\includegraphics[height=0.25\textheight,width=8.0cm]  {CM_xicst_cc_1.eps}
\vspace*{-1.2cm}
\caption{Invariant mass distribution of the $(\Xi^-,\pi^+)$ system. (CDF)
}
\label{fig:pqxi}
\end{figure}

\begin{figure}[htb]
\vspace*{-0.9cm}

\includegraphics[height=0.25\textheight,width=7.6cm]  {theta_note_dstp_dedx_pt.eps}
\vspace*{-1.2cm}
\caption{Mass of the ($D^{*+}\bar p$) system. The arrow indicates the position of
the $\Theta_c$ state (CDF).}
\label{fig:pqthetac}
\end{figure}

Following reports of evidence for exotic
baryons containing five quarks (pentaquarks), CDF has analysed
its data for evidence of the following pentaquarks:
$\Theta^+$ ($uud\bar d \bar s$), doubly strange states
$\Xi_{3/2}$, charmed states $\Theta_c$, and, most recently,
a state $(udus\bar b)$, dubbed $R^+_s$, through its weak decay to $(J/\psi, p)$.
With its excellent particle indentification and mass resolution,
CDF has a unique capability to search for  pentaquark states.
The signals of known states: $\phi$, $\Lambda$,
$\Lambda(1520)$, $K^*$, $\Xi$,
compare favorably with those provided
by the authors of  the pentaquark evidence.
The group finds no evidence for pentaquark states, see Figs
~\ref{fig:pqtheta},{\ref{fig:pqxi},\ref{fig:pqthetac}.
This can be interpreted as an indication that the pentaquark production
in $p \bar p$ collisions is heavily suppressed compared to the conventional
hadron production, or as an evidence against the existence of pentaquarks.

\clearpage

\section{RECENT B PHYSICS RESULTS}

\subsection{Spectroscopy}

CDF has measured the mass of $b$ hadrons in exclusive $J/\psi$ channels.
The measurements of the $B_s$ and $\Lambda_b$ (Fig. \ref{fig:masslb})
masses are the current world's best.\\

$m(B^+)$ = 5279.10$\pm$0.41$(stat)\pm$0.36$(syst)$,

$m(B^0)$ = 5279.63$\pm$0.53$(stat)\pm$0.33$(syst)$,

$m(B_s)$ = 5366.01$\pm$0.73$(stat)\pm$0.33$(syst)$,

$m(\Lambda_b)$ = 5619.7$\pm$1.2$(stat)\pm$1.2$(syst)$ MeV/$c^2$.\\

\begin{figure}[htb]
\vspace*{-1mm}
\includegraphics[height=0.30\textheight,width=7.5cm]  {lambdav1c.eps}
\vspace*{-1cm}

\caption{The mass spectrum of $\Lambda_b$ candidates (CDF).}
\label{fig:masslb}
\end{figure}

D\O\ reports the first observation of the excited $B$ mesons
$B_1$ and $B^*_2$ as two separate states in fully reconstructed
decays to $B^{(*)}\pi$. The mass of $B_1$ is measured to be
5724$\pm$4$\pm$7 MeV/c$^2$, and the mass difference $\Delta M$ between
$B^*_2$ and $B_1$ is 23.6$\pm$7.7$\pm$3.9 MeV/c$^2$
(Fig.  \ref{fig:d0_bexc}).

D\O\ observes semileptonic $B$ decays to narrow $D^{**}$ states,
the orbitally excited states  of the $D$ meson
seen as resonances in the $D^{*+}\pi^-$ invariant mass spectrum.
The $D^*$ mesons are reconstructed through the decay sequence
$D^{*+} \rightarrow D^0\pi^+$, $D^0\rightarrow K^-\pi^+$.
The invariant mass  of oppositely charged $(D^*,\pi)$ pairs
is shown in Fig.  \ref{fig:d0_dstst}.
The mass peak between 2.4 and 2.5 GeV/$c^2$ can be interpreted as two merged
narrow $D^{**}$ states, $D^0_1(2420)$ and $D^0_2(2460)$.
The combined branching fraction is
$ {\cal B}(B\rightarrow D^0_1,D^0_2)\cdot {\cal B}(D^0_1,D^0_2\rightarrow D^{*+}\pi^-)=(0.280\pm0.021(stat)\pm0.088(syst)$\%. The systematic error includes the unknown phase between the
two resonances. Work is in progress on extracting the two Breit-Wigner
amplitudes.

\begin{figure}[htb]
\vspace*{-2mm}
\hspace*{-3mm}
\includegraphics[height=0.28\textheight,width=8.3cm]  {B08F02.eps}

\vspace*{-1cm}
\caption{Mass difference $\Delta M = M(B\pi)-M(B)$ for exclusive $B$ decays.
The background-subtracted signal is a sum of
$B^*_1 \rightarrow B^* \pi$, $B^* \rightarrow B \gamma $ (open area)
and $B^*_2 \rightarrow B^*\pi$ $B^*\rightarrow B \gamma$ (lower peak in the shaded area)
and $B^*_2 \rightarrow B \pi$ (upper peak in the shaded area)
(D\O).}
\label{fig:d0_bexc}
\end{figure}

\begin{figure}[htb]
\includegraphics[height=0.25\textheight,width=7.5cm]  {B05F03.eps}

\vspace*{-1cm}
\caption{The invariant mass distribution of
$(D^*,\pi)$ pairs, opposite sign (points) and same-sign (solid histogram).}
\label{fig:d0_dstst}
\end{figure}

\subsection{Lifetimes}

CDF and D\O\ have measured  lifetimes of $b$ hadrons through the exclusively
reconstructed decays $B^+ \rightarrow J/\psi K^+$, $B^0 \rightarrow J/\psi K^{*0}$,
$B_s \rightarrow J/\psi \phi$,
and $\Lambda_b \rightarrow J/\psi \Lambda$
(Fig. \ref{fig:d0_lbctau}).
The latest results are:  \\

 $\tau(B^+)$=1.65 $\pm$ 0.08 $^{+0.096}_{-0.123}$  ps ~(D\O\ 2003),

 $\tau(B^+)$=1.662 $\pm$ 0.033  $\pm$ 0.008  ps ~(CDF),

 $\tau(B^0_d)$=1.473  $^{+0.052}_{-0.050}$ $\pm$ 0.023    ps ~(D\O).

 $\tau(B^0_d)$=1.539 $\pm$ 0.051  $\pm$ 0.008  ps ~(CDF),

 $\tau(B^0_s)$=1.444   $^{+0.098}_{-0.090}$ $\pm$ 0.020   ps ~(D\O),

 $\tau(B^0_s)$=1.369 $\pm$ 0.100 $\pm$ $^{+0.008}_{0.010}$  ps ~(CDF),

 $\tau(\Lambda_b)$=1.221 $^{+0.217}_{-0.179}$ $\pm$ 0.043  ps ~(D\O),

 $\tau(\Lambda_b)$=1.25 $\pm$ 0.26 $\pm$ 0.10  ps ~(CDF 2003).\\

The measured lifetimes correspond to the following lifetime ratios:\\

$\tau(B^+)/\tau(B^0_d)$   =  1.080$\pm$0.042     ~(CDF),

$\tau(B^0_s)/\tau(B^0_d)$ =  0.890$\pm$0.072    ~(CDF),

$\tau(B^0_s)/\tau(B^0_d)$ = 0.980$ ^{+0.075}_{-0.070}   \pm$0.003    ~(D\O),

$\tau(\Lambda_b)/\tau(B^0_d)$ =  0.874$ ^{+0.169}_{-0.142}   \pm$0.028    ~(D\O).\\

\begin{figure}[htb]
\includegraphics[height=0.3\textheight,width=8.2cm]  {d0_lbctau_B11F02.eps}
\vspace*{-1cm}

\caption{ Fit projection on  $c\tau$ for the $\Lambda_b$ candidates.  (D\O)}
\label{fig:d0_lbctau}
\end{figure}

The $B_s$ lifetime measurements listed above are results of
a single-lifetime fit to data, integrated over the decay angles.
Because  of the presence of  final
states  common to ${B_s^0}$\ and its charge conjugate ${\overline{B}_s^0}$,
the two meson states   are expected
to mix in such a way that the two CP  eigenstates may have a relatively
large lifetime difference.
It is possible to
separate the two CP components of ${B_s^0 \rightarrow J/\psi \phi}$\ and thus to measure the
lifetime difference by studying the time evolution of the
polarization states of the vector mesons in the final state.
CDF has carried out a combined analysis of $B_s$ lifetimes
and polarization amplitudes. The results for the lifetimes of the
low mass (CP even) and high mass (CP odd) eigenstates, and the relative
width difference are:\\

 $\tau_L = 1.05 ^{+0.16}_{-0.13} \pm 0.02$ ~ps,

 $\tau_H = 2.07 ^{+0.58}_{-0.46} \pm 0.03$ ~ps,

 $\Delta \Gamma /\overline \Gamma   = 0.65 ^{+0.25}_{-0.33} \pm 0.01$.\\

Figure \ref{fig:cdf_dg} shows  the scan of the likelihood function
for $\Delta \Gamma /\overline \Gamma$.
Pseudoexperiments tossed with $\Delta \Gamma /\overline \Gamma =0$
yield the betting odds for observing the above results at
1/315. For $\Delta \Gamma /\overline \Gamma = 0.12$ (SM prediction,
which has recently been updated to 0.14$\pm$0.05~\cite{dg_un}) the betting odds are
1/84.

\begin{figure}[htb]
\vspace*{-1mm}
\includegraphics[height=0.3\textheight,width=8.2cm]  {cdf_scan-dg-un.eps}

\vspace*{-1cm}
\caption{Scan of the likelihood function
for $\Delta \Gamma /\overline \Gamma$ (CDF).
}
\label{fig:cdf_dg}
\end{figure}

D\O\ has used a novel technique to  measure the lifetime ratio
of the charged and neutral $B$ mesons, exploiting the large
semileptonic sample. $B$ hadrons were reconstructed in the channels
$B\rightarrow \mu^+ \nu D^*(2010)^-X$, which are dominated by $B^0$ decays,
and  $B\rightarrow \mu^+ \nu D^0X$, which are dominated by $B^+$ decays.
The lifetime ratio was
obtained from the variation of the ratio of the number of events in these two
processes at different decay lengths.
The result is \\

$\tau(B^+)/\tau(B^0_d)$   =  1.093$\pm$0.021$\pm$0.022.      ~(D\O)

\subsection{Towards $B_s$ mixing}

Measurement of the $B_s$ oscillation frequency via ${B_s^0}$ -${\overline{B}_s^0}$ ~mixing
will provide an important constraint on the CKM matrix. The oscillation
frequency is proportional to the mass difference between the mass eigenstates,
$\Delta m_s$, and is related to the CKM matrix through
$\Delta m_s \propto |V_{tb}V_{ts}|$. When combined with the
$B_d$ mass difference, $\Delta m_d$ it helps in extraction of $|V_{td}|$,
and thereby the CP violating phase.

As a benchmark for future $B_s$ oscillation measurement, both groups
study  $B_d$ mixing, gaining an understanding of the different components
of a $B$ mixing analysis (sample composition, flavor tagging, vertexing,
asymmetry fitting). For a sample of partially reconstructed decays
$B\rightarrow D^*(2010)^+\mu^-X$, D\O\ obtains
$\Delta m_d = 0.506 \pm 0.055 (stat) \pm  0.049 (syst))$ ps$^{-1}$ and
$\Delta m_d = 0.488 \pm 0.066 (stat) \pm  0.044 (syst))$ ps$^{-1}$
when employing  opposite side muon tagging and the same side tagging,
respectively.

The CDF result for semileptonic channels is
$\Delta m_d = 0.536 \pm 0.037 (stat) \pm  0.009 (s.c.) \pm 0.015 (syst)$ ps$^{-1}$.
CDF also reports a result on $B$ oscillations using fully reconstructed
decays:
$\Delta m_d = 0.526 \pm 0.056 (stat) \pm  0.005 (syst))$ ps$^{-1}$.

Reconstructing $B_s$ decays into different final states is another
important
 step in the ${B_s^0}$ -${\overline{B}_s^0}$ ~mixing analysis.
Thanks to the  large muon and tracking coverage,   D\O\ is accumulating
a  high statistics sample of semileptonic $B_s$ decays.
D\O\ reconstructs the $B_s \rightarrow D^+_s \mu^- X$ decays, with
$D^+_s \rightarrow \phi \pi^+ $ and
$D^+_s \rightarrow K^* K^+ $,
at a rate of $\approx$ 40(25) events per pb$^{-1}$,  respectively.
Figure \ref{fig:d0_bsdsphipi} shows the mass distribution of the
$D^+_s \rightarrow \phi \pi$ candidates.

\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=0.3\textheight,width=8.0cm]  {blds-250.eps}
\vspace*{-1.2cm}
\caption{  $D^+_s \rightarrow \phi \pi^+$  signal. (D\O)}
\label{fig:d0_bsdsphipi}
\end{figure}

\begin{figure}[htb]
\vspace*{-10mm}
\hspace*{-4mm}
\includegraphics[height=0.35\textheight,width=7.9cm]  {cdf_Bs-DsPi-PhiPi.eps}

\vspace*{-1.0cm}
\caption{ $B_s \rightarrow D_s \pi$, $D_s \rightarrow \phi \pi$  signal.  (CDF)}
\label{fig:cdf_bsdsphipi}
\end{figure}

CDF has clean signals for fully hadronic, flavor-specific  $B_s$ decays,
providing the best sensitivity to $B_s$ oscillations at high
$\Delta m_s$. Figure \ref{fig:cdf_bsdsphipi} shows the signal for
the best channel, $B_s \rightarrow D_s \pi$, $D_s \rightarrow \phi \pi$.

\clearpage

\subsection{Rare decays}

The purely leptonic decays $B_{d,s}^0 \rightarrow \mu^+
\mu^-$ are flavor-changing neutral current (FCNC) processes.
In the standard model, these decays are forbidden at the tree level and
proceed at a very low rate through higher-order diagrams.
The latest SM prediction~\cite{sm_ref3}
is ${\cal B}(B^0_s \rightarrow \mu^+ \mu^-)=(3.42\pm 0.54)\times
10^{-9}$, where the error is dominated by non-perturbative uncertainties. The
leptonic branching fraction of the $B_d^0$ decay is suppressed by CKM matrix elements $|V_{td}/V_{ts}|^2$
leading to a predicted SM branching fraction of $(1.00\pm0.14)\times 10^{-10}$.
The best published experimental bound (Fig.~\ref{fig:cdf_bsmumu})
 for the branching fraction
of $B^0_s$ $(B^0_d)$ is presently
${\cal B}(B^0_s \, (B^0_d) \rightarrow \mu^+\mu^-)<7.5\times 10^{-7}\,
(1.9\times 10^{-7})$ at the 95\% C.L.~\cite{cdfII}.
The decay amplitude of $B^0_{d,s} \rightarrow \mu^+ \mu^-$ can be
significantly enhanced in some extensions of the SM.

\begin{figure}[htb]
\includegraphics[height=8.3cm,width=7.9cm]  {cdfbsmumu_results_prl.eps}

\vspace*{-1cm}
\caption{Invariant mass for the events passing all requirements. (CDF)}
\label{fig:cdf_bsmumu}
\end{figure}

Assuming no contributions
from the decay $B^0_d\rightarrow \mu^+\mu^-$ in the signal region,
D\O\  finds the conservative upper limit on the branching fraction
to be ${\cal B}(B^0_s \rightarrow \mu^+ \mu^-) \leq 4.6\times 10^{-7}$
at the 95\% C.L. (Fig.~\ref{fig:d0_bsmumu}).

\begin{figure}[htb]
\includegraphics[height=5.0cm,width=8.0cm]  {B06F03.eps}
\vspace*{-1cm}
\caption{Invariant mass for the events  passing all requirements. (D\O)}
\label{fig:d0_bsmumu}
\end{figure}

Answer: The Tevatron Collider Run II started in March 2002 and is expected to continue until the end of this decade.


Input: What molecule was the focus of the study?

Context:
\section{Introduction}

Spectral line surveys have revealed that high-mass star-forming
regions are rich reservoirs of molecules from simple diatomic species
to complex and larger molecules (e.g.,
\citealt{schilke1997b,hatchell1998b,comito2005,bisschop2007}).
However, there have been rarely studies undertaken to investigate the
chemical evolution during massive star formation from the earliest
evolutionary stages, i.e., from High-Mass Starless Cores (HMSCs) and
High-Mass Cores with embedded low- to intermediate-mass protostars
destined to become massive stars, via High-Mass Protostellar Objects
(HMPOs) to the final stars that are able to produce Ultracompact H{\sc
  ii} regions (UCH{\sc ii}s, see \citealt{beuther2006b} for a recent
description of the evolutionary sequence). The first two evolutionary
stages are found within so-called Infrared Dark Clouds (IRDCs).  While
for low-mass stars the chemical evolution from early molecular
freeze-out to more evolved protostellar cores is well studied (e.g.,
\citealt{bergin1997,dutrey1997,pavlyuchenkov2006,joergensen2007}),
it is far from clear whether similar evolutionary patterns are present
during massive star formation.

To better understand the chemical evolution of high-mass star-forming
regions we initiated a program to investigate the chemical properties
from IRDCs to UCH{\sc ii}s from an observational and theoretical
perspective. We start with single-dish line surveys toward a large
sample obtaining their basic characteristics, and then perform
detailed studies of selected sources using interferometers on smaller
scales. These observations are accompanied by theoretical modeling of
the chemical processes.  Long-term goals are the chemical
characterization of the evolutionary sequence in massive star
formation, the development of chemical clocks, and the identification
of molecules as astrophysical tools to study the physical processes
during different evolutionary stages. Here, we present an initial
study of the reactive radical ethynyl (C$_2$H) combining single-dish
and interferometer observations with chemical modeling.  Although
C$_2$H was previously observed in low-mass cores and Photon Dominated
Regions (e.g., \citealt{millar1984,jansen1995}), so far it was not
systematically investigated in the framework of high-mass star
formation.

\section{Observations}
\label{obs}

The 21 massive star-forming regions were observed with the Atacama
Pathfinder Experiment (APEX) in the 875\,$\mu$m window in fall 2006.
We observed 1\,GHz from 338 to 339\,GHz and 1\,GHz in the image
sideband from 349 to 350\,GHz.  The spectral resolution was
0.1\,km\,s$^{-1}$, but we smoothed the data to
$\sim$0.9\,km\,s$^{-1}$. The average system temperatures were around
200\,K, each source had on-source integration times between 5 and 16
min. The data were converted to main-beam temperatures with forward
and beam efficiencies of 0.97 and 0.73, respectively
\citep{belloche2006}. The average $1\sigma$ rms was 0.4\,K.  The main
spectral features of interest are the C$_2$H lines around 349.4\,GHz
with upper level excitation energies $E_u/k$ of 42\,K (line blends of
C$_2$H$(4_{5,5}-3_{4,4})$ \& C$_2$H$(4_{5,4}-3_{4,3})$ at
349.338\,GHz, and C$_2$H$(4_{4,4}-3_{3,3})$ \&
C$_2$H$(4_{4,3}-3_{3,2})$ at 349.399\,GHz). The beam size was $\sim
18''$.

The original Submillimeter Array (SMA) C$_2$H data toward the
HMPO\,18089-1732 were first presented in \citet{beuther2005c}. There
we used the compact and extended configurations resulting in good
images for all spectral lines except of C$_2$H. For this project, we
re-worked on these data only using the compact configuration. Because
the C$_2$H emission is distributed on larger scales (see
\S\ref{results}), we were now able to derive a C$_2$H image. The
integration range was from 32 to 35\,km\,s$^{-1}$, and the achieved
$1\sigma$ rms of the C$_2$H image was 450\,mJy\,beam$^{-1}$.  For more
details on these observations see \citet{beuther2005c}.

\section{Results}
\label{results}

The sources were selected to cover all evolutionary stages from IRDCs
via HMPOs to UCH{\sc ii}s. We derived our target list from the samples
of \citet{klein2005,fontani2005,hill2005,beltran2006}.  Table
\ref{sample} lists the observed sources, their coordinates, distances,
luminosities and a first order classification into the evolutionary
sub-groups IRDCs, HMPOs and UCH{\sc ii}s based on the previously
available data. Although this classification is only based on a
limited set of data, here we are just interested in general
evolutionary trends. Hence, the division into the three main classes
is sufficient.

Figure \ref{spectra} presents sample spectra toward one source of each
evolutionary group. While we see several CH$_3$OH lines as well as
SO$_2$ and H$_2$CS toward some of the HMPOs and UCH{\sc ii}s but not
toward the IRDCs, the surprising result of this comparison is the
presence of the C$_2$H lines around 349.4\,GHz toward all source types
from young IRDCs via the HMPOs to evolved UCH{\sc ii}s.  Table
\ref{sample} lists the peak brightness temperatures, the integrated
intensities and the FWHM line-widths of the C$_2$H line blend at
349.399\,GHz. The separation of the two lines of 1.375\,MHz already
corresponds to a line-width of 1.2\,km\,s$^{-1}$. We have three C$_2$H
non-detections (2 IRDCs and 1 HMPO), however, with no clear trend with
respect to the distances or the luminosities (the latter comparison is
only possible for the HMPOs). While IRDCs are on average colder than
more evolved sources, and have lower brightness temperatures, the
non-detections are more probable due to the relatively low sensitivity
of the short observations (\S\ref{obs}). Hence, the data indicate
that the C$_2$H lines are detected independent of the evolutionary
stage of the sources in contrast to the situation with other
molecules.  When comparing the line-widths between the different
sub-groups, one finds only a marginal difference between the IRDCs and
the HMPOs (the average $\Delta v$ of the two groups are 2.8 and
3.1\,km\,s$^{-1}$).  However, the UCH{\sc ii}s exhibit significantly
broader line-widths with an average value of 5.5\,km\,s$^{-1}$.

Intrigued by this finding, we wanted to understand the C$_2$H spatial
structure during the different evolutionary stages.  Therefore, we
went back to a dataset obtained with the Submillimeter Array toward
the hypercompact H{\sc ii} region IRAS\,18089-1732 with a much higher
spatial resolution of $\sim 1''$ \citep{beuther2005c}.  Albeit this
hypercompact H{\sc ii} region belongs to the class of HMPOs, it is
already in a relatively evolved stage and has formed a hot core with a
rich molecular spectrum.  \citet{beuther2005c} showed the spectral
detection of the C$_2$H lines toward this source, but they did not
present any spatially resolved images. To recover large-scale
structure, we restricted the data to those from the compact SMA
configuration (\S\ref{obs}). With this refinement, we were able to
produce a spatially resolved C$_2$H map of the line blend at
349.338\,GHz with an angular resolution of $2.9''\times 1.4''$
(corresponding to an average linear resolution of 7700\,AU at the
given distance of 3.6\,kpc). Figure \ref{18089} presents the
integrated C$_2$H emission with a contour overlay of the 860\,$\mu$m
continuum source outlining the position of the massive protostar. In
contrast to almost all other molecular lines that peak along with the
dust continuum \citep{beuther2005c}, the C$_2$H emission surrounds the
continuum peak in a shell-like fashion.

\section{Discussion and Conclusions}

To understand the observations, we conducted a simple chemical
modeling of massive star-forming regions. A 1D cloud model with a mass
of 1200\,M$_\sun$, an outer radius of 0.36\,pc and a power-law density
profile ($\rho\propto r^p$ with $p=-1.5$) is the initially assumed
configuration. Three cases are studied: (1) a cold isothermal cloud
with $T=10$\,K, (2) $T=50$\,K, and (3) a warm model with a temperature
profile $T\propto r^q$ with $q=-0.4$ and a temperature at the outer
radius of 44\,K. The cloud is illuminated by the interstellar UV
radiation field (IRSF, \citealt{draine1978}) and by cosmic ray
particles (CRP). The ISRF attenuation by single-sized $0.1\mu$m
silicate grains at a given radius is calculated in a plane-parallel
geometry following \citet{vandishoeck1988}. The CRP ionization rate is
assumed to be $1.3\times 10^{-17}$~s$^{-1}$ \citep{spitzer1968}. The
gas-grain chemical model by \citet{vasyunin2008} with the desorption
energies and surface reactions from \citet{garrod2006} is used.
Gas-phase reaction rates are taken from RATE\,06 \citep{woodall2007},
initial abundances, were adopted from the ``low metal'' set of
\citet{lee1998}.

Figure \ref{model} presents the C$_2$H abundances for the three models
at two different time steps: (a) 100\,yr, and (b) in a more evolved
stage after $5\times10^4$\,yr. The C$_2$H abundance is high toward the
core center right from the beginning of the evolution, similar to
previous models (e.g., \citealt{millar1985,herbst1986,turner1999}).
During the evolution, the C$_2$H abundance stays approximately
constant at the outer core edges, whereas it decreases by more than
three orders of magnitude in the center, except for the cold $T=10$~K
model.  The C$_2$H abundance profiles for all three models show
similar behavior.

The chemical evolution of ethynyl is determined by relative removal
rates of carbon and oxygen atoms or ions into molecules like CO, OH,
H$_2$O. Light ionized hydrocarbons CH$^+_{\rm n}$ (n=2..5) are quickly
formed by radiative association of C$^+$ with H$_2$ and hydrogen
addition reactions: C$^+$ $\rightarrow$ CH$_2^+$ $\rightarrow$
CH$_3^+$ $\rightarrow$ CH$_5^+$.  The protonated methane reacts with
electrons, CO, C, OH, and more complex species at later stage and
forms methane.  The CH$_4$ molecules undergo reactive collisions with
C$^+$, producing C$_2$H$_2^+$ and C$_2$H$_3^+$. An alternative way to
produce C$_2$H$_2^+$ is the dissociative recombination of CH$_5^+$
into CH$_3$ followed by reactions with C$^+$.  Finally, C$_2$H$_2^+$
and C$_2$H$_3^+$ dissociatively recombine into CH, C$_2$H, and
C$_2$H$_2$. The major removal for C$_2$H is either the direct
neutral-neutral reaction with O that forms CO, or the same reaction
but with heavier carbon chain ions that are formed from C$_2$H by
subsequent insertion of carbon. At later times, depletion and
gas-phase reactions with more complex species may enter into this
cycle.  At the cloud edge the interstellar UV radiation
instantaneously dissociates CO despite its self-shielding,
re-enriching the gas with elemental carbon.

The transformation of C$_2$H into CO and other species proceeds
efficiently in dense regions, in particular in the ``warm'' model
where endothermic reactions result in rich molecular complexity of the
gas (see Fig.~\ref{model}).  In contrast, in the ``cold'' 10\,K model
gas-grain interactions and surface reactions become important.  As a
result, a large fraction of oxygen is locked in water ice that is hard
to desorb ($E_{\rm des} \sim 5500$~K), while half of the elemental
carbon goes to volatile methane ice ($E_{\rm des} \sim 1300$~K). Upon
CRP heating of dust grains, this leads to much higher gas-phase
abundance of C$_2$H in the cloud core for the cold model compared to
the warm model. The effect is not that strong for less dense regions
at larger radii from the center.

Since the C$_2$H emission is anti-correlated with the dust continuum
emission in the case of IRAS\,18089-1732 (Fig.\,\ref{18089}), we do
not have the H$_2$ column densities to quantitatively compare the
abundance profiles of IRAS\,18089-1732 with our model. However, data
and model allow a qualitative comparison of the spatial structures.
Estimating an exact evolutionary time for IRAS\,18089-1732 is hardly
possible, but based on the strong molecular line emission, its high
central gas temperatures and the observed outflow-disk system
\citep{beuther2004a,beuther2004b,beuther2005c}, an approximate age of
$5\times10^4$\,yr appears reasonable.  Although dynamical and chemical
times are not necessarily exactly the same, in high-mass star
formation they should not differ to much: Following the models by
\citet{mckee2003} or \citet{krumholz2006b}, the luminosity rises
strongly right from the onset of collapse which can be considered as a
starting point for the chemical evolution. At the same time disks and
outflows evolve, which should hence have similar time-scales.  The
diameter of the shell-like C$_2$H structure in IRAS\,18089-1732 is
$\sim 5''$ (Fig.\,\ref{18089}), or $\sim$9000\,AU in radius at the
given distance of 3.6\,kpc.  This value is well matched by the modeled
region with decreased C$_2$H abundance (Fig.\,\ref{model}).  Although
in principle optical depths and/or excitation effects could mimic the
C$_2$H morphology, we consider this as unlikely because the other
observed molecules with many different transitions all peak toward the
central submm continuum emission in IRAS\,18089-1732
\citep{beuther2005c}. Since C$_2$H is the only exception in that rich
dataset, chemical effects appear the more plausible explanation.

The fact that we see C$_2$H at the earliest and the later evolutionary
stages can be explained by the reactive nature of C$_2$H: it is
produced quickly early on and gets replenished at the core edges by
the UV photodissociation of CO. The inner ``chemical'' hole observed
toward IRAS\,18089-1732 can be explained by C$_2$H being consumed in
the chemical network forming CO and more complex molecules like larger
carbon-hydrogen complexes and/or depletion.

The data show that C$_2$H is not suited to investigate the central gas
cores in more evolved sources, however, our analysis indicates that
C$_2$H may be a suitable tracer of the earliest stages of (massive)
star formation, like N$_2$H$^+$ or NH$_3$ (e.g.,
\citealt{bergin2002,tafalla2004,beuther2005a,pillai2006}). While a
spatial analysis of the line emission will give insights into the
kinematics of the gas and also the evolutionary stage from chemical
models, multiple C$_2$H lines will even allow a temperature
characterization. With its lowest $J=1-0$ transitions around 87\,GHz,
C$_2$H has easily accessible spectral lines in several bands between
the 3\,mm and 850\,$\mu$m.  Furthermore, even the 349\,GHz lines
presented here have still relatively low upper level excitation
energies ($E_u/k\sim42$\,K), hence allowing to study cold cores even
at sub-millimeter wavelengths.  This prediction can further be proved
via high spectral and spatial resolution observations of different
C$_2$H lines toward young IRDCs.

\acknowledgments{H.B. acknowledges financial support
  by the Emmy-Noether-Programm of the Deutsche Forschungsgemeinschaft
  (DFG, grant BE2578). }

Answer: The focus of the study was on the reactive radical ethynyl (C$_2$H).


Input: What were the vaccines trialed against?

Context:
A special tribute to Del Bigtree (pictured) and his team at ICAN for his stunning 88 page letter to the HHS regarding vaccine safety. As Del reported - in the latest edition of Highwire - the letter, in response to an earlier reply from the then acting Director National Vaccine Program Office, Melinda Wharton, took virtually a year to compile, and is a meticulous piece of research. Most sensationally they researched the HHS claim through US government archives that at least some pediatric vaccines had been trialed against genuine placebo, and came to a negative conclusion. Not only that, they established that none of the vaccines those vaccines had been trialed against had ever been trialed against genuine placebo either. At the end of the line the toxic products were only being compared with other toxic products, rather than against saline.
Leave aside the sceptics, for any believer in the vaccine program as a necessary intervention in public health, this should be a devastating finding. Fundamentally, the research into the safety of any of the products before marketing was simply not there. The manufacturers apparently had no faith that their proto-products could withstand this scrutiny, and for the rest they just did not care: under the alleged imperative of protecting the population it seems anything went. So even before all the sham monitoring procedures and reviews which Del and his team dismantle in forensic detail we are left with the proposition that none of the present products being given to US children – and frequently other children across most of the developed world – have any meaningful pre-marketing safety data all. If you are believer in the program you have been let down: if you wanted a program with any pretensions to safety - supposing such a thing to be possible - it looks like you would have to start from scratch. The manufacturers did this: the governments, the politicians and the regulators (internationally) let it happen.
This damning document is published simultaneously with a demand in the UK from the Royal Society for Public Health (which I had never heard of) to shut down comment about vaccines on the web. It echoes calls from Seth Berkley of GAVI, Heidi Larson of the Vaccine Confidence Project and the European Parliament. The pamphlet airily dismisses concerns that vaccines have side effects or that you could possibly have too many. It is pure public relations, and if the RSPH claims to be "independent" it also admits that the publication was paid for by Merck, a detail which was reported by British Medical Journal and the Guardian, but not true to form by the BBC. We have, in truth, been building to this moment for two decades: as the evidence piles up that every single aspect of the program lacks integrity or is simply rotten to the core all the perpetrators can do is call for the silencing of their critics, and maintain the products are safe because they say so.
Please help give the ICAN letter the widest possible distribution, particularly to politicians.
"The outcome of disease always depends both on the virulence of the pathogen and the health of the individual immune system."
Nope. This makes no sense. Lots of people who seemed vibrant will get a very severe case of the same illness that a vulnerable baby overcomes in a day.
And under the germ theory it doesn't matter how strong your immune system *was*. Once it's been overcome by the pathogen it is every bit as weak as anybody else's with that pathogen.
What you say makes no sense. There's no reason for me to reply to you again.
"Why do you think that within a few years (not many) of the introduction of the vaccines for them, pertussis, measles, mumps, rubella, tetanus, diphtheria, Hib disease, and chickenpox (and others) almost entirely disappeared?"
Why do you keep asking this question when I've already provided the answer hundreds of times? Why are you so desperate to believe the people who you already recognize are harming our children?
Why would Walter Reed be any more trustworthy than Paul Offit or Senator Pan? Why would Jenner or Pasteur?
And you went no way to explaining my arguments against germ theory. If we are attacked by billions of viruses every day then if even a tiny fraction of them are pathogenic then we couldn't possibly survive. And even if we could, we would already be immune rendering every vaccine pointless. Once we had survived our first few days on earth, then we could never get sick again.
If that's wrong then we must conclude that precisely 0% of germs are pathogenic.
Plus your comment about the immune system completely misunderstood my point. The immune system does not allow us to overcome our math problem. In fact, it makes it worse.
You did provide one solitary example of a patient with what are presumably yellow fever symptoms but you didn't say whether they had been given any toxic medical treatments.
And like I said before, the whole "incubation period" is more than a little suspicious. Clearly they never found what they thought they would and just rigged the results to tell them what they want to hear.
Like every other germ theorist/vaccine promoter in history.
Many kinds of bacteria are constantly evolving and changing, like flu viruses. Others are more stable over time, like the yellow fever virus. Those that change develop new ways of infiltrating the cells of the organism being attacked (from our point of view, from its unconscious point of view, it's just carrying out its need to replicate, which it can only do inside the cells of its host). The changes which allow it to better infiltrate are more successful and result in more viruses with those traits.
Our immune system is designed to detect and destroy potentially dangerous invading pathogens. Many bacteria are usually harmless and absolutely necessary. The minority are dangerous, and most people's immune systems do a good job of analyzing them and killing them, often with no signs of disease. Others experience a clinical infection, and the immune system usually mounts a successful attack on them.
The outcome of disease always depends both on the virulence of the pathogen and the health of the individual immune system. Vaccines are usually effective in giving immunity to the targeted diseases. They also have many dangers which everyone should be aware of, and vaccines should be avoided whenever possible. But in the case of the most dangerous diseases, everyone should learn about them and think about what he wants to do to protect himself and his children from them, considering all the factors involved. And no one can have 100% certainty that he has made the right decision, but that's life. But if you live in the Congo and many people around you are currently dying of yellow fever, then that means that you yourself are at risk of being bitten by a loaded mosquito and getting, often dying, of yellow fever. The yellow fever vaccine is very effective at preventing yellow fever. From there, each person must make a choice.
At the end of this stage there is a remission of two or three days. About 80% of those with clinical disease recover at this point, with permanent immunity. The other 20% enter the toxic stage, with a return of the fever, black vomit (coffee-ground emesis), diarrhea, a slowing of the pulse (Faget's sign), jaundice, yellow eyes, yellow skin, and failure of the kidneys, liver, and heart. The patient gets a strange hiccup (like with Ebola, a related disease), falls into a coma, and dies. About half of those patients who enter the toxic stage dies, even now, even with the best of hospital care. The Faget's sign can also occur at the end of the first stage.
You asked specifically about the symptoms of the Americans on Dr. Reed's team who got yellow fever in Cuba in 1900. I'll give the passage from The American Plague (162-5), which describes the course of Jesse Lazear's illness. "In his logbook, Lazear wrote an unusual entry on September 13. In all cases before those, page after page of records, Lazear had used the soldier's name and simply the date he was bitten, with no other attention to the mosquito. A one-line entry with a name and a date. On that day, however, in his elegant hand, Lazear did not write the soldier's name, but instead wrote 'Guinea Pig No. 1.' He went on to write that this guinea pig had been bitten by a mosquito that developed from an egg laid by a mosquito that developed from an egg laid by a mosquito that fed on a number of yellow fever cases: Suarez, Hernández, De Long, Ferández. It was a precise, detailed history that proved beyond doubt that the mosquito was loaded with the virus when it bit a healthy soldier...(If he had entered his name, then his death would have been considered medical suicide by the insurance company, and his wife and two children would not have gotten any payment.) For the next few days, Lazear's life continued much as it had over the last few months in Cuba. He fed and cared for the mosquitoes in the lab. ..Then he began to lose his appetite. He skipped a few meals in the mess hall. He didn't mention it to anyone, nor did he ask to see one of the yellow fever doctors; instead, he worked hard in the lab trying to ignore the oncoming headache.
"On September 18, he complained of feeling 'out of sorts,' and stayed in his officer's quarters. His head pounded and L. decided to write a letter. ..(he wrote to his mother, and referred to his one-year old son Houston and the baby his wife Mabel was about to have: they were staying with his mother in the US). ..That night, L. started to feel chilled as the fever came on. He never went to sleep but worked at his desk all through the night, trying to get all the information about the mosquitoes organized. By morning, he showed all the signs of a severe attack of yellow fever. The camp doctors made the diagnosis, and L. agreed to go to the yellow fever ward. ..L. was carried by litter out of the two-room, white pine board house in which he had lived since he and Mabel first arrived in Cuba. ..(In the yellow fever ward, in a separate one-room building), Lena Warner (the immune nurse who had survived the yellow fever in 1878, when she was nine, and was found in her boarded-up house by a former slave who first thought she was dead, and carried her to safety) nursed J.L., recording his vitals. (I put up a link to his case record and vital signs last week. The surgeon general required that this record be made for every yellow fever patient.)... (On September 25,) Lena Warner braced L's arms with all of her weight, shouting for help. Still he bolted from the bed, darting around the small frame-wood room as wildly as a trapped insect beating against glass. Two soldiers ran into the ward, pinning L to his bed, tying restraints around his wrists and elbows. ..Warner sponged his body with iced whiskey and water. She recorded his temperature, which had held at 104 degrees for days, on the chart beside his bed. ..(Warner watched him sleep.) But the quiet did not last. L's body began to lurch, and black vomit rolled from his mouth; through the bar hanging above his hospital cot. He writhed in the bed, and his skin grew deep yellow. His 104 temperature slowly fell, leveling out 99 degrees, and JL died at 8:45 p.m. at the age of thirty-four."
As is obvious, there are many problems with vaccines. But, that being said, most of them usually work for a period of time to prevent the targeted diseases. The basic science behind vaccines is correct. Why do you think that within a few years (not many) of the introduction of the vaccines for them, pertussis, measles, mumps, rubella, tetanus, diphtheria, Hib disease, and chickenpox (and others) almost entirely disappeared? In the case of the routine childhood diseases, this was a bad thing, but it is a true thing.
Vaccines usually don't cause any obvious reactions. While they usually prevent the diseases, and that's why people continue to get them. With the increasing vaccination schedule, more and more are severely and permanently damaged, and it is immoral to mandate any vaccine for anyone for this reason. But it would also be immoral to prohibit vaccines for those who want them enough to take the risk.
Your article said as though it had any probative value that 90% of those who get pertussis had been vaxxed. The old DPT vaccine was MUCH more effective at preventing pertussis, but it was so dangerous (again, not to most, but to many), that developed countries replaced it with the acellular version, DTaP. From the beginning about twenty years ago, it was clear that it was not very effective and that huge numbers of vaxxed people got pertussis anyway, including my daughter who got pertussis at eight month old after having gotten three DTaPs. The pertussis vaccine continues to be very dangerous, and I do not recommend that anyone get it. It used to be a killer disease, but evolved to become much milder, to the extent that the disease is very rarely dangerous (usually only to newborns under three months old), while the vaccine is very dangerous. And they're trying to see how they can go back to the old DPT. This does not show that vaccine science has collapsed, but rather that the vaccine they developed to replace the DPT turned out to be much less effective than they first thought, while continuing to be much more dangerous than they first thought.
Your article extrapolated from that that modern medical science in general has collapsed, but that, again, is going too far. A older woman in Mexico City who is like my mother to me had a pacemaker inserted about two months ago to aid her failing heart, and it has restored her to optimism and energy, when she was despondent, weak, and close to death. I took my daughter to the dentist yesterday, who said she has three wisdom teeth coming in and that she said that the lower right one was sore. So, although I am cautious about X-rays, I made an appointment for a panoramic X-ray in a month to assess the wisdom teeth, and, if it seems appropriate, I'll take her to an oral surgeon to have one or more extracted under IV sedation, in his office, if possible (the dentist thought that it would be). And I am confident that there will be no serious problems, but this is thanks to technology and training in modern medicine that haven't been available for that long.
I think that everyone should inform himself on all medical procedures before agreeing to anything, but I also think that he should have access to any medical procedure which is reasonable (and opinions can differ as to that).
One problem is that you have not said how you think people should protect themselves against tetanus, bacterial meningitis, and yellow fever in the relevant cases, for example. These are diseases which healthy, well-nourished people used to die from very readily.
If most people stopped vaxxing and the mortality from these diseases rose to something like pre-vaccine levels, do you think they should just accept dying from them?
I put that in a separate paragraph because it is the crucial issue.
balinaheuchter Air Traffic Control You Tube - Colin Campbell example of - How to "Fudge a Nudge" -"Deal" or "No Deal" "Not in a month of Sundays" "No exceptions/no compromise?" -make a trade off -do an exception- everyone get's a good deal /good outcome!
Hans, you are right that we are looking at one of the biggest crimes in all history. When I read the story of that poor girl who was so healthy and is now confined to a wheelchair after getting her third Gardasil shot I could not believe that Merck could produce such a toxic vaccine and give it out to girls like it was something they absolutely had to have only to be mislead and made into cripples. Merck should be prosecuted for the damage they have done to so many girls who got the Gardasil vaccine and were physically debilitated for life. There is a place for the people who perpetrated this crime on young girls and women and it is called hell. They have destroyed people's lives and gotten away with it. My heart goes out to those who have suffered this damage for no damn good reason except to help make huge profits for Merck!
Here is the reason that the germ theory is nonsense.
1) Everyday we are bombarded with billions of germs. Presumably at least some of them are of the kind that germ theorists believe are dangerous (otherwise we would have to conclude that none of them are dangerous). So how do we survive?
2) Let's just say that we ignore 1 and imagine that, by way of magic, none of the billions of viruses we get bombarded with are pathogenic but all those that are are tucked away somewhere. Ok. But presumably they reside in sick people right? So where are there lots of sick people? Doctor offices and hospitals! So everybody must be dying the moment they enter these places right?
3) I love this one because I have never seen anybody else ever raise it. Under the germ theory there are no negative feedbacks. This makes a stable biological system by definition impossible. The immune system is *not* a negative feedback it is the opposite. It actually reinforces our math problem because the immune system will weaken as the number of pathogens increase.
There is no way of resolving this problem without a discontinuity. A Deus ex Machina as The Almighty Pill so beautifully put it. So the germ theory is quite literally, mathematically impossible.
There is as much chance of it being true as 2+2 = 5.
There are plenty of other massive problems with germ theory such as why did things like SARS and bird flu magically disappear? Why do we have the symptoms that we do? Is our body controlling the symptoms to help fight the germs and if so, why would suppressing the symptoms with antibiotics or Tamiflu be considered a good idea? If the virus is causing the symptoms then why would it cause these kinds of things?

Answer: Other toxic products.


Input: What types of data did the authors use in their experiments?

Context:
\section{Introduction}
\label{sec:introduction}

Probabilistic models have proven to be very useful in a lot of applications in signal processing where signal estimation is needed \cite{rabiner1989tutorial,arulampalam2002tutorial,ji2008bayesian}. Some of their advantages are that 1) they force the designer to specify all the assumptions of the model, 2) they provide a clear separation between the model and the algorithm used to solve it, and 3) they usually provide some measure of uncertainty about the estimation.

On the other hand, adaptive filtering is a standard approach in estimation problems when the input is received as a stream of data that is potentially non-stationary. This approach is widely understood and applied to several problems such as echo cancellation \cite{gilloire1992adaptive}, noise cancellation \cite{nelson1991active}, and channel equalization \cite{falconer2002frequency}.

Although these two approaches share some underlying relations, there are very few connections in the literature. The first important attempt in the signal processing community to relate these two fields was the connection between a linear Gaussian state-space model (i.e. Kalman filter) and the RLS filter, by Sayed and Kailath \cite{sayed1994state} and then by Haykin \emph{et al.} \cite{haykin1997adaptive}. The RLS adaptive filtering algorithm emerges naturally when one defines a particular state-space model (SSM) and then performs exact inference in that model. This approach was later exploited in \cite{van2012kernel} to design a kernel RLS algorithm based on Gaussian processes.

A first attempt to approximate the LMS filter from a probabilistic perspective was presented in \cite{park2014probabilistic}, focusing on a kernel-based implementation. The algorithm of \cite{park2014probabilistic} makes use of a Maximum a Posteriori (MAP) estimate as an approximation for the predictive step. However, this approximation does not preserve the estimate of the uncertainty in each step, therefore degrading the performance of the algorithm.

In this work, we provide a similar connection between state-space models and least-mean-squares (LMS). Our approach is based on approximating the posterior distribution with an isotropic Gaussian distribution. We show how the computation of this approximated posterior leads to a linear-complexity algorithm, comparable to the standard LMS. Similar approaches have already been developed for a variety of problems such as channel equalization using recurrent RBF neural networks \cite{cid1994recurrent}, or Bayesian forecasting \cite{harrison1999bayesian}. Here, we show the usefulness of this probabilistic approach for adaptive filtering.

The probabilistic perspective we adopt throughout this work presents two main advantages. Firstly, a novel LMS algorithm with adaptable step size emerges naturally with this approach, making it suitable for both stationary and non-stationary environments. The proposed algorithm has less free parameters than previous LMS algorithms with variable step size \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}, and its parameters are easier to be tuned w.r.t. these algorithms and standard LMS. Secondly, the use of a probabilistic model provides us with an estimate of the error variance, which is useful in many applications.

Experiments with simulated and real data show the advantages of the presented approach with respect to previous works. However, we remark that the main contribution of this paper is that it opens the door to introduce more Bayesian machine learning techniques, such as variational inference and Monte Carlo sampling methods \cite{barber2012bayesian}, to adaptive filtering.\\

\section{Probabilistic Model}

Throughout this work, we assume the observation model to be linear-Gaussian with the following distribution,

\begin{equation}
p(y_k|{\bf w}_k) = \mathcal{N}(y_k;{\bf x}_k^T {\bf w}_k , \sigma_n^2),
\label{eq:mess_eq}
\end{equation}
where  $\sigma_n^2$ is the variance of the observation noise, ${\bf x}_k$ is the regression vector and ${\bf w}_k$ is the parameter vector to be sequentially estimated, both $M$-dimensional column vectors.

In a non-stationary scenario, ${\bf w}_k$ follows a dynamic process. In particular, we consider a diffusion process (random-walk model) with variance $\sigma_d^2$ for this parameter vector:

\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;{\bf w}_{k-1}, \sigma_d^2 {\bf I}),
\label{eq:trans_eq}
\end{equation}
where $\bf I$ denotes the identity matrix. In order to initiate the recursion, we assume the following prior distribution on ${\bf w}_k$

\begin{equation}
p({\bf w}_0)= \mathcal{N}({\bf w}_0;0, \sigma_d^2{\bf I}).\nonumber
\end{equation}

\section{Exact inference in this model: Revisiting the RLS filter}

Given the described probabilistic SSM, we would like to infer the posterior probability distribution $p({\bf w}_k|y_{1:k})$.
Since all involved distributions are Gaussian, one can perform exact inference, leveraging the probability rules in a straightforward manner. The resulting probability distribution is
\begin{equation}
p({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k}, \boldsymbol\Sigma_{k}), \nonumber
\end{equation}
in which the mean vector ${\bf\boldsymbol\mu}_{k}$ is given by
\begin{equation}
{\bf\boldsymbol\mu}_k = {\bf\boldsymbol\mu}_{k-1} + {\bf K}_k (y_k - {\bf x}_k^T {\bf\boldsymbol\mu}_{k-1}){\bf x}_k, \nonumber
\end{equation}
where we have introduced the auxiliary variable
\begin{equation}
{\bf K}_k = \frac{ \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)}{{\bf x}_k^T  \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)  {\bf x}_k + \sigma_n^2}, \nonumber
\end{equation}
and the covariance matrix $\boldsymbol\Sigma_k$ is obtained as
\begin{equation}
\boldsymbol\Sigma_k = \left( {\bf I} -  {\bf K}_k{\bf x}_k {\bf x}_k^T \right) ( \boldsymbol\Sigma_{k-1} +\sigma_d^2), \nonumber
\end{equation}
Note that the mode of $p({\bf w}_k|y_{1:k})$, i.e. the maximum-a-posteriori estimate (MAP), coincides with the RLS adaptive rule
\begin{equation}
{{\bf w}}_k^{(RLS)} = {{\bf w}}_{k-1}^{(RLS)} + {\bf K}_k (y_k - {\bf x}_k^T {{\bf w}}_{k-1}^{(RLS)}){\bf x}_k .
\label{eq:prob_rls}
\end{equation}
This rule is similar to the one introduced in \cite{haykin1997adaptive}.

Finally, note that the covariance matrix $\boldsymbol\Sigma_k$ is a measure of the uncertainty of the estimate ${\bf w}_k$ conditioned on the observed data $y_{1:k}$. Nevertheless, for many applications a single scalar summarizing the variance of the estimate could prove to be sufficiently useful. In the next section, we show how such a scalar is obtained naturally when $p({\bf w}_k|y_{1:k})$ is approximated with an isotropic Gaussian distribution. We also show that this approximation leads to an LMS-like estimation.

\section{Approximating the posterior distribution: LMS filter }

The proposed approach consists in approximating the posterior distribution $p({\bf w}_k|y_{1:k})$, in general a multivariate Gaussian distribution with a full covariance matrix, by an isotropic spherical Gaussian distribution

\begin{equation}
\label{eq:aprox_post}
\hat{p}({\bf w}_{k}|y_{1:k})=\mathcal{N}({\bf w}_{k};{\bf \hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_{k}^2 {\bf I} ).
\end{equation}

In order to estimate the mean and covariance of the approximate distribution $\hat{p}({\bf w}_{k}|y_{1:k})$, we propose to select those that minimize the Kullback-Leibler divergence with respect to the original distribution, i.e.,

\begin{equation}
\{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k\}=\arg \displaystyle{  \min_{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k}} \{ D_{KL}\left(p({\bf w}_{k}|y_{1:k}))\| \hat{p}({\bf w}_{k}|y_{1:k})\right) \}. \nonumber
\end{equation}

The derivation of the corresponding minimization problem can be found in Appendix A. In particular, the optimal mean and the covariance are found as
\begin{equation}
{\hat{\boldsymbol\mu}}_{k} = {\boldsymbol\mu}_{k};~~~~~~ \hat{\sigma}_{k}^2 = \frac{{\sf Tr}\{ \boldsymbol\Sigma_k\} }{M}.
\label{eq:sigma_hat}
\end{equation}

We now show that by using \eqref{eq:aprox_post} in the recursive predictive and filtering expressions we obtain an LMS-like adaptive rule. First, let us assume that we have an approximate posterior distribution at $k-1$, $\hat{p}({\bf w}_{k-1}|y_{1:k-1}) =  \mathcal{N}({\bf w}_{k-1};\hat{\bf\boldsymbol\mu}_{k-1}, \hat{\sigma}_{k-1}^2 {\bf I} )$. Since all involved distributions are Gaussian, the predictive distribution
is obtained as %
\begin{eqnarray}
\hat{p}({\bf w}_k|y_{1:k-1}) &=& \int p({\bf w}_k|{\bf w}_{k-1}) \hat{p}({\bf w}_{k-1}|y_{1:k-1}) d{\bf w}_{k-1} \nonumber\\
&=& \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k|k-1}, \boldsymbol\Sigma_{k|k-1}),
\label{eq:approx_pred}
\end{eqnarray}
where the mean vector and covariance matrix are given by
\begin{eqnarray}
\hat{\bf\boldsymbol\mu}_{k|k-1} &=& \hat{\bf\boldsymbol\mu}_{k-1} \nonumber \\
\hat{\boldsymbol\Sigma}_{k|k-1} &=& (\hat{\sigma}_{k-1}^2 + \sigma_d^2 ){\bf I}\nonumber.
\end{eqnarray}

From \eqref{eq:approx_pred}, the posterior distribution at time $k$ can be computed using Bayes' Theorem and standard Gaussian manipulations (see for instance \cite[Ch. 4]{murphy2012machine}). Then, we approximate the posterior $p({\bf w}_k|y_{1:k})$ with an isotropic Gaussian,
\begin{equation}
\hat{p}({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k ; {\hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_k^2 {\bf I} ),\nonumber
\end{equation}
where
\begin{eqnarray}
{\hat{\boldsymbol\mu}}_{k} &= & {\hat{\boldsymbol\mu}}_{k-1}+ \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k  \nonumber  \\
&=& {\hat{\boldsymbol\mu}}_{k-1}+ \eta_k (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k .
\label{eq:prob_lms}
\end{eqnarray}
Note that, instead of a gain matrix ${\bf K}_k$ as in Eq.~\eqref{eq:prob_rls}, we now have a scalar gain $\eta_k$ that operates as a variable step size.

Finally, to obtain the posterior variance, which is our measure of uncertainty, we apply \eqref{eq:sigma_hat} and the trick ${\sf Tr}\{{\bf x}_k{\bf x}_k^T\}= {\bf x}_k^T{\bf x}_k= \|{\bf x}_k \|^2$,

\begin{eqnarray}
\hat{\sigma}_k^2 &=& \frac{{\sf Tr}(\boldsymbol\Sigma_k)}{M} \\
&=& \frac{1}{M}{\sf Tr}\left\{ \left( {\bf I} -  \eta_k {\bf x}_k {\bf x}_k^T \right) (\hat{\sigma}_{k-1}^2 +\sigma_d^2)\right\} \\
&=& \left(1 - \frac{\eta_k \|{\bf x}_k\|^2}{M}\right)(\hat{\sigma}_{k-1}^2 +\sigma_d^2).
\label{eq:sig_k}
\end{eqnarray}
If MAP estimation is performed, we obtain  an adaptable step-size LMS estimation

\begin{equation}
{\bf w}_{k}^{(LMS)} = {\bf w}_{k-1}^{(LMS)} + \eta_k (y_k - {\bf x}_k^T {\bf w}_{k-1}^{(LMS)}){\bf x}_k,
\label{eq:lms}
\end{equation}
with
\begin{equation}
\eta_k = \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2}.\nonumber
\end{equation}
At this point, several interesting remarks can be made:

\begin{itemize}

\item The adaptive rule \eqref{eq:lms} has linear complexity since it does not require us to compute the full matrix $\boldsymbol\Sigma_k$.

\item For a stationary model, we have $\sigma_d^2=0$ in \eqref{eq:prob_lms} and \eqref{eq:sig_k}. In this case, the algorithm remains valid and both the step size and the error variance, $\hat{\sigma}_{k}$, vanish over time $k$.

\item Finally, the proposed adaptable step-size LMS has only two parameters, $\sigma_d^2$ and $\sigma_n^2$, (and only one, $\sigma_n^2$, in stationary scenarios) in contrast to other variable step-size algorithms \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}. More interestingly, both $\sigma_d^2$ and $\sigma_n^2$ have a clear underlying physical meaning, and they can be estimated in many cases. We will comment more about this in the next section.
\end{itemize}

\section{Experiments}
\label{sec:experiments}

We evaluate the performance of the proposed algorithm in both stationary and tracking experiments. In the first experiment, we estimate a fixed vector ${\bf w}^{o}$ of dimension $M=50$. The entries of the vector are independently and uniformly chosen in the range $[-1,1]$. Then, the vector is normalized so that $\|{\bf w}^o\|=1$. Regressors $\boldsymbol{x}_{k}$ are zero-mean Gaussian vectors with identity covariance matrix. The additive noise variance is such that the SNR is $20$ dB. We compare our algorithm with standard RLS and three other LMS-based algorithms: LMS, NLMS \cite{sayed2008adaptive}, VSS-LMS \cite{shin2004variable}.\footnote{The used parameters for each algorithm are: for RLS $\lambda=1$, $\epsilon^{-1}=0.01$; for LMS $\mu=0.01$; for NLMS $\mu=0.5$; and for VSS-LMS $\mu_{max}=1$, $\alpha=0.95$, $C=1e-4$.} The probabilistic LMS algorithm in \cite{park2014probabilistic} is not simulated because it is not suitable for stationary environments.

In stationary environments, the proposed algorithm has only one parameter, $\sigma^2_n$. We simulate both the scenario where we have perfectly knowledge of the amount of noise (probLMS1) and the case where the value $\sigma^2_n$ is $100$ times smaller than the actual value (probLMS2). The Mean-Square Deviation (${\sf MSD} = {\mathbb E} \| {\bf w}_0 - {\bf w}_k \|^2$), averaged out over $50$ independent simulations, is presented in Fig. \ref{fig:msd_statationary}.

\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{results_stationary_MSD}}
\end{minipage}
\caption{Performance in terms of MSD of probabilistic LMS with both optimal (probLMS1) and suboptimal (probLMS2) compared to LMS, NLMS, VS-LMS, and RLS.}
\label{fig:msd_statationary}
\end{figure}

The performance of probabilistic LMS is close to RLS (obviously at a much lower computational cost) and largely outperforms previous variable step-size LMS algorithms proposed in the literature. Note that, when the model is stationary, i.e. $\sigma^2_d=0$ in \eqref{eq:trans_eq},  both the uncertainty $\hat{\sigma}^2_k$, and the adaptive step size $\eta_k$, vanish over time. This implies that the error tends to zero when $k$ goes to infinity. Fig. \ref{fig:msd_statationary} also shows that the proposed approach is not very sensitive to a bad choice of its only parameter, as demonstrated by the good results of probLMS2, which uses a $\sigma^2_n$ that is $100$ times smaller than the optimal value.

\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{fig2_final}}
\end{minipage}
\caption{Real part of one coefficient of the measured and estimated channel in experiment two. The shaded area represents two standard deviations from the prediction {(the mean of the posterior distribution)}.}
\label{fig_2}
\end{figure}

\begin{table}[ht]
\begin{footnotesize}
\setlength{\tabcolsep}{2pt}
\def1.5mm{1.5mm}
\begin{center}
\begin{tabular}{|l@{\hspace{1.5mm}}|c@{\hspace{1.5mm}}|c@{\hspace{1.5mm}}|c@{\hspace{1.5mm}}|c@{\hspace{1.5mm}}|c@{\hspace{1.5mm}}|c@{\hspace{1.5mm}}|}
\hline
Method &  LMS &  NLMS & LMS-2013 & VSSNLMS & probLMS & RLS \\
\hline
\hline
MSD (dB) &-28.45 &-21.07 &-14.36 &-26.90 &-28.36 &-25.97\\
\hline
\end{tabular}
\end{center}
\caption{Steady-state MSD of the different algorithms for the tracking of a real MISO channel.}
\label{tab:table_MSD}
\end{footnotesize}

\end{table}
\newpage
In a second experiment, we test the tracking capabilities of the proposed algorithm with {real} data of a wireless MISO channel acquired in a realistic indoor scenario. More details on the setup can be found in \cite{gutierrez2011frequency}. Fig. \ref{fig_2} shows the real part of one of the channels, and the estimate of the proposed algorithm. The shaded area represents the estimated uncertainty for each prediction, i.e. $\hat{\mu}_k\pm2\hat{\sigma}_k$. Since the experimental setup does not allow us to obtain the optimal values for the parameters, we fix these parameters to their values that optimize the steady-state mean square deviation (MSD). \hbox{Table \ref{tab:table_MSD}} shows this steady-state MSD of the estimate of the MISO channel with different methods. As can be seen, the best tracking performance is obtained by standard LMS and the proposed method.

\section{Conclusions and Opened Extensions}
\label{sec:conclusions}

{We have presented a probabilistic interpretation of the least-mean-square filter. The resulting algorithm is an adaptable step-size LMS that performs well both in stationary and tracking scenarios. Moreover, it has fewer free parameters than previous approaches and these parameters have a clear physical meaning. Finally, as stated in the introduction, one of the advantages of having a probabilistic model is that it is easily extensible:}

\begin{itemize}
\item If, instead of using an isotropic Gaussian distribution in the approximation, we used a Gaussian with diagonal covariance matrix, we would obtain a similar algorithm with different step sizes and measures of uncertainty, for each component of ${\bf w}_k$. Although this model can be more descriptive, it needs more parameters to be tuned, and the parallelism with LMS vanishes.
\item Similarly, if we substitute the transition model of \eqref{eq:trans_eq} by an Ornstein-Uhlenbeck process,

\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;\lambda {\bf w}_{k-1}, \sigma_d^2), \nonumber
\label{eq:trans_eq_lambda}
\end{equation}
a similar algorithm is obtained but with a forgetting factor $\lambda$ multiplying ${\bf w}_{k-1}^{(LMS)}$ in \eqref{eq:lms}. This algorithm may have improved performance under such a kind of autoregresive dynamics of ${\bf w}_{k}$, though, again, the connection with standard LMS becomes dimmer.

\item As in \cite{park2014probabilistic}, the measurement model \eqref{eq:mess_eq} can be changed to obtain similar adaptive algorithms for classification, ordinal regression, and Dirichlet regression for compositional data.

\item A similar approximation technique could be applied to more complex dynamical models, i.e. switching dynamical models \cite{barber2010graphical}. The derivation of efficient adaptive algorithms that explicitly take into account a switch in the dynamics of the parameters of interest is a non-trivial and open problem, though the proposed approach could be useful.

\item Finally, like standard LMS, this algorithm can be kernelized for its application in estimation under non-linear scenarios.

\end{itemize}

\begin{appendices}

\section{KL divergence between a general gaussian distribution and an isotropic gaussian}
\label{sec:kl}

 We want to approximate  $p_{{\bf x}_1}(x) = \mathcal{N}({\bf x}; \boldsymbol\mu_1,\boldsymbol\Sigma_1)$ by $p_{{\bf x}_2}({\bf x}) = \mathcal{N}({\bf x}; \boldsymbol\mu_2,\sigma_2^2 {\bf I})$. In order to do so, we have to compute the parameters of $p_{{\bf x}_2}({\bf x})$, $\boldsymbol\mu_2$ and $\sigma_2^2$, that minimize the following Kullback-Leibler divergence,

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) &=&\int_{-\infty}^{\infty} p_{{\bf x}_1}({\bf x}) \ln{\frac{p_{{\bf x}_1}({\bf x})}{p_{{\bf x}_2}({\bf x})}}d{\bf x} \nonumber  \\
&= &  \frac{1}{2} \{ -M + {\sf Tr}(\sigma_2^{-2} {\bf I}\cdot \boldsymbol\Sigma_1^{-1})  \nonumber \\
  & &  + (\boldsymbol\mu_2 - \boldsymbol\mu_1 )^T \sigma^{-2}_2{\bf I} (\boldsymbol\mu_2 - \boldsymbol\mu_1 )  \nonumber \\
 & &   +  \ln \frac{{\sigma_2^2}^M}{\det\boldsymbol\Sigma_1} \}.
\label{eq:divergence}
\end{eqnarray}
Using symmetry arguments, we obtain
\begin{equation}
\boldsymbol\mu_2^{*} =\arg \displaystyle{  \min_{\boldsymbol\mu_2}} \{ D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) \} = \boldsymbol\mu_1.
\end{equation}
Then, \eqref{eq:divergence} gets simplified into

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) = \frac{1}{2}\lbrace { -M + {\sf Tr}(\frac{\boldsymbol\Sigma_1}{\sigma_2^{2}}) + \ln \frac{\sigma_2^{2M}}{\det\boldsymbol\Sigma_1}}\rbrace.
\end{eqnarray}
The variance $\sigma_2^2$ is computed in order to minimize this Kullback-Leibler divergence as

\begin{eqnarray}
\sigma_2^{2*} &=& \arg\min_{\sigma_2^2} D_{KL}(P_{x_1}\| P_{x_2}) \nonumber \\
 &=& \arg\min_{\sigma_2^2}\{ \sigma_2^{-2}{\sf Tr}\{\boldsymbol\Sigma_1\} + M\ln \sigma_2^{2} \} .
\end{eqnarray}
Deriving and making it equal zero leads to

\begin{equation}
\frac{\partial}{\partial \sigma_2^2} \left[ \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{\sigma_2^{2}} + M \ln \sigma_2^{2} \right] = \left. {\frac{M}{\sigma_2^{2}}-\frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{(\sigma_2^{2})^2}}\right|_{\sigma_2^{2}=\sigma_2^{2*}}\left. =0 \right. .
\nonumber
\end{equation}
Finally, since the divergence has a single extremum in $R_+$,
\begin{equation}
\sigma_2^{2*} = \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{M}.
\end{equation}

\end{appendices}

\vfill
\clearpage

\bibliographystyle{IEEEbib}

Answer: The authors used simulated data and real data from a wireless MISO channel.