text:
\subsection{Deep Neural Networks}
\label{sec:dnn}

\begin{figure}[t]
  \centering
\includegraphics[width=0.7\textwidth,clip]{Plots/FC.pdf}
\caption{A schematic architecture of the fully-connected NN (FC or DNN) used in this paper.}
\label{fig:FC}
\end{figure}

A fully-connected layer (FC or DNN) is the basic type of neural networks where every neuron in each layer is connected to all neurons in two consecutive layers. An input layer is composed of the combination of four-momenta of reconstructed particles in Eqs.~(\ref{eq:vis_4momenta}-\ref{eq:inv}), or kinematic variables in Eqs.~(\ref{eq:11kin}-\ref{eq:21}). It is followed by 8 hidden layers with the decreasing number of neurons from 2400 to 300 as shown in Fig. \ref{fig:FC}, and ReLU (Rectified Linear Unit) function \cite{pmlr-v15-glorot11a} is used to activate each neuron.
The final hidden layer is connected to the output layer that contains two neurons with each representing a label of 1 for the signal and 0 for the backgrounds. A softmax activation function is used for the output layer.
We use Adam optimizer \cite{Kingma:2014vow} and a learning rate of $10^{-4}$ to minimize the binary cross entropy loss function.
To prevent overfitting, we add the $L_2$ regularization term to the loss function by setting {\tt weight\_decay}=$5 \times 10^{-4}$ in {\tt Pytorch} implementation. When training the DNN, we use a mini-batch size of 20 and the epochs of 30~\footnote{There are some common features used for all NNs in this paper. Except for CapsNets, we use the $L_2$ regularization for all NNs, which shifts the loss function, $L \rightarrow L +  \frac{1}{2} \lambda \|\textbf{W}\|^2$, where $\textbf{W}$ represents all weights, and the $\lambda$ denotes {\tt weight\_decay}. Throughout this paper, DNN hidden layers will appear repetitively in most of neural networks. For each layer, we apply the ReLU activation function.
Also the configuration of the output layer is the same for all other neural networks throughout this paper, unless otherwise mentioned.}. \begin{figure*}[tb]
\begin{center}
\includegraphics[width=0.47\textwidth,clip]{Plots/FC_bare.pdf}  \hspace*{0.1cm}
\includegraphics[width=0.47\textwidth,clip]{Plots/FC_Norm.pdf}  \hspace*{0.05cm}
\caption{Significance of observing double Higgs production at the HL-LHC with ${\cal L}$ = 3 ab$^{-1}$ for DNNs with bare inputs (left) and DNNs with normalized inputs (right).}
\label{fig:FCresults}
\end{center}
\end{figure*}

Fig. \ref{fig:FCresults} shows the final significance as a function of the number of signal events ($N_s$) with bare inputs (left) and normalized\footnote{
We perform the linear transformation, $x_i \to x_i^\prime = a x_i + b$ for each input $x_i$ such that $x_i^\prime \in [0, 1]$.} inputs (right) for various combinations of input features. First observation is that DNN with the normalized input features lead to a slightly higher significance than that with the bare input for most NNs. The variation in the significances for different input is slightly narrower with the normalized features.
Secondly, it is clear that when the DNN is trained with four-momenta of visible particles ($V^{(\text{vis})}_{ p_{\mu} }$), its performance does not stand out (see cyan-solid curves).
Even when additional four-momenta of reconstructed neutrinos ($V^{(\nu_{\text{H}})}_{ p_{\mu} }$) are supplemented, there is no clear impact on the significance (see magenta-dashed curves). This result indicates that the simple DNN is unable to efficiently identify features of the signal and backgrounds with the primitive input data, given a finite number of training samples and the depth of DNN.
On the other hands, addition of the human-engineered kinematic variables plays a very important role.
$V_{\text{11-kin}}$, $V_{\text{15-kin}}$ and
$V_{\text{21-kin}}$ are introduced in section \ref{sec:input}.
$V_{10-kin}^{\rm Ref.[62]}$ and $V_{8-kin}^{\rm Ref.[63]}$ are sets of kinematic variables used in Ref. \cite{Adhikary:2017jtu} and Ref. \cite{Sirunyan:2017guj}, respectively.
Note that the performance of DNN increases, when using more kinematic variables.
We find that when the DNN is trained with 11 kinematic variables ($V_{\text{11-kin}}$)
the significance increases up to
10\%-50\% compared to the results using the four-momenta for a wide range of signal number of events (see the green dotted curve in the left panel.).
Interestingly, 15 kinematic variables ($V_{\text{15-kin}}$), which include the kinematic variables using the momentum of the reconstructed neutrinos, provide an additional steady $\sim10\%$ improvement on the significance (see the red and green curves in the left panel).
It is worth noting that the 6 high-level variables ($V_{\text{6-kin}}$) adds the orthogonal set of information to $V_{\text{15-kin}}$, which enables the DNN to better disentangle the backgrounds from the signal and brings additional $\sim10\%$ improvement. Finally, as mentioned previously, while the relative improvement is diminished with the normalized input features (as shown in the right panel), the importance of the kinematic variables still remain.

\subsection{Convolutional Neural Networks}
\label{sec:cnn}

When the final state is fully represented by a set of images as in Eq.(\ref{eq:3images}-\ref{eq:5images}),
deep neural networks specialized for the image recognition provide useful handles.
One of the most commonly used algorithms is a convolutional neural network (CNN) as shown in Fig. \ref{fig:CNN}.
The input to the CNN is the 3D image of $V^{(C, N, \ell, \nu_{\text{H}}, \nu_{\text{T}})}_{\text{image}}$ ($V^{(C, N, \ell ) }_{\text{image}}$)
whose dimension is given by $ 5 \times 50 \times 50  $ ($ 3 \times 50 \times 50  $ ) where 5 (3) denotes a number of channels.
In order to exploit the spatial correlation among different channels, we first apply the 3D convolution using the kernel size of $5 \times 3 \times 3$ ($3 \times 3 \times 3$), the stride 1, the padding size 1, and 32 feature maps \footnote{For all 2D and 3D convolutional layers, the padding size is fixed to 1. After each convolutional layer, we apply the batch normalization and ReLU activation function.}.
Next, we apply the max-pooling using the kernel size of $2 \times 2 \times 2$ without the stride and padding, which subsequently reduces the image dimension down to $ 32 \times 25 \times 25 $.
\begin{figure*}[t!]
\centering
\begin{center}
\includegraphics[width=1.\textwidth,clip]{Plots/CNN2.pdf}
\caption{A schematic architecture of the convolutional neural network (CNN) used in this paper. The separate DNN chain in the right-upper corner is used only when the kinematic variables are included.}
\label{fig:CNN}
\end{center}
\end{figure*}

Since this is effectively the same dimension as the 2D image with 32 feature maps, in what follows,
we apply the 2D convolution using the kernel size of $3 \times 3$, the stride of 1, and 32 feature maps, followed by the max-pooling with the kernel size of $2 \times 2$. We repeat this procedure until the image dimension is reduced to $ 32 \times 6 \times 6 $.
Each of these neurons are connected to 3 hidden DNN layers with 600 neurons, and the final DNN layer is connected to the output layer.
To study the effect of the kinematic variables along with the image inputs, we slightly modify the NN structure, in which case these 3 hidden DNN layers are not used. Instead we construct the separate DNN consisting of 6 hidden layers with the decreasing number of neurons from 1200 to 600 (as shown in the right-upper corner in Fig. \ref{fig:CNN}). The last layer with 600 neurons for the kinematic variables are combined with the output of CNN of dimension $ 32 \times 6 \times 6 $.
When training the network, we use Adam optimizer, the learning rate of $10^{-4}$,  regularization term of {\tt weight\_decay}=$5 \times 10^{-4}$, mini-batch size of 20, and epochs of 24.

Fig. \ref{fig:CNN_ResNet} shows the performances of CNNs with various inputs.
First, we roughly reproduce results (green, solid) presented in Ref. \cite{Kim:2019wns}, taking $V^{(C,N)}_{\text{image}}$ as input. It is important to check this, as all event generations and simulations are performed completely independently.
Adding lepton images, the overall significance of $0.9\sim 1$ can be achieved with $V^{(C,N,\ell)}_{\text{image}}$ image information without kinematic variables (red, dotted), which is substantially larger than that with $V^{(C,N)}_{\text{image}}$.
Albeit not a substantial impact, addition of neutrino images ($V^{(C,N,\ell, \nu_H, \nu_T)}_{\text{image}}$) helps improve the significance up to $\sim 5\%$ (red, solid).
Finally, addition of $V_{\text{21-kin}}$ kinematic variables increases the significance substantially, making CNN with $V_{\text{21-kin}} + V^{(C,N,\ell, \nu_H, \nu_T)}_{\text{image}}$ as the best network in terms of the signal significance (blue, solid).

\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.6\textwidth,clip]{Plots/resNet_CNN.pdf}
\caption{Significance at the HL-LHC with ${\cal L}$ = 3 ab$^{-1}$ for CNNs and ResNets.}
\label{fig:CNN_ResNet}
\end{center}
\end{figure*}

\subsection{Residual Neural Networks}
\label{sec:resnet}
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.85\textwidth,clip]{Plots/ResNet.pdf}
\caption{A schematic architecture of the residual neural network (ResNet) used in this paper. The separate DNN chain in the right-bottom corner is used only when the kinematic variables are included.}
\label{fig:ResNet}
\end{center}
\end{figure*}

The image sets that we feed into CNNs contain only few activated pixels in each channels.
Given the sparse images, the performance of the CNNs could greatly diminish.
One of possibilities to ameliorate this problem is to design neural networks at much deeper level.
As the CNN goes deeper, however, its performance becomes saturated or even starts degrading rapidly~\cite{simonyan2014very}.
A residual neural network (ResNet)~\cite{he2016deep,he2016identity} is one of the alternatives to the CNN
which introduces the skip-connections that bypass some of the neural network layers as shown in Fig. \ref{fig:ResNet}.

The input to the ResNet is the 3D image of $V^{(C, N, \ell, \nu_{\text{H}}, \nu_{\text{T}})}_{\text{image}}$ ($V^{(C, N, \ell ) }_{\text{image}}$).
We apply the 3D convolution using the kernel size of $5 \times 3 \times 3$ ($3 \times 3 \times 3$), the stride of 1, and 32 feature maps. Note that we apply the batch normalization and ReLU activation function after each convolutional layer, but we do not use the max-pooling.
We introduce the three-pronged structure: i) three series of 2D convolutions using the kernel size of $3 \times 3$, the stride of 1, and 32 feature maps, ii) two series of 2D convolutions using the same configurations, and iii) the skip-connection.
All three paths are congregated into the single node, which enables the ResNet to learn various features of convoluted images, while keeping the image dimension unchanged.
One way to reduce the dimensionality of the image is to change the striding distance, and we consider the two-pronged structure: i) the 2D convolution using the stride of 2, and ii) the 2D convolution using the stride of 2 followed by another 2D convolution using the stride of 1. Both layers are congregated again into the single node.
These are basic building blocks of our ResNet, and repeated several times in hybrid ways,
until the image dimension is brought down to $ 32 \times 4 \times 4 $.
In parallel to the ResNet pipeline, we construct the DNN consisting of 6 hidden layers with the decreasing number of neurons from 1200 to 600. The inputs to the DNN are the  kinematic variables. This step is similar to what has been done in CNN when including the kinematic variables in addition to the image inputs.
The final neurons of two pipelines are congregated into the same layer, and subsequently connected to the output layer.
We use the learning rate of $10^{-4}$, the regularization term of {\tt weight\_decay}=$5 \times 10^{-4}$, the mini-batch size of 20, and the epochs of 11.

We obtain the overall significance of $\sim 1 -1.25$ can be achieved as shown in Fig. \ref{fig:CNN_ResNet}, which marks very high significance along with CNNs.
The impact of neutrino images turns out to be mild, when including kinematic variables. This is partially because the reconstructed momentum of neutrinos (and the corresponding images) are byproducts of the {\it Higgsness} and {\it Topness} variables, which are already included in the variables of $V_{\text{6-kin}}$. Additional neutrino images, therefore, are redundant information in the ResNets and CNNs.

\subsection{Capsule Neural Networks}
\label{sec:capsnet}

The max pooling method in the CNN selects the most active neuron in each region of the feature map, and passes it to the next layer. This accompanies the loss of spatial information about where things are,
so that the CNN is agnostic to the geometric correlations between the pixels at higher and lower levels.
A capsule neural network (CapsNet) ~\cite{NIPS2017_2cad8fa4,Diefenbacher:2019ezd} was proposed to address the potential problems of the CNN, and Fig. \ref{fig:Caps} shows the schematic architecture of the CapsNet used in our analysis.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=1\textwidth,clip]{Plots/CapsNet.pdf}
\caption{A schematic architecture of the capsule network model (CapsNet) used in this paper.}
\label{fig:Caps}
\end{center}
\end{figure*}

The input to the CapsNet is the 3D image of $V^{(C, N, \ell, \nu_{\text{H}}, \nu_{\text{T}})}_{\text{image}}$ ($V^{(C, N, \ell ) }_{\text{image}}$). We apply the 3D convolution using the kernel size of $5 \times 3 \times 3$ ($3 \times 3 \times 3$), the stride of 1, and 32 feature maps. Note that we do not use the max-pooling.
We apply the series of 2D convolution using the kernel size of $3 \times 3$, the stride of 1, and 256 feature maps, until the image dimension is reduced to $ 32 \times 42 \times 42 $.
The output neurons of 2D convolution are reshaped to get a bunch of 8-dimensional primary capsule vectors,
which contain the lower-level information of the input image. For each feature map, there are $40 \times 40$ arrays of primary capsules, and there are 32 feature maps in total. To sum up, there are $N_{caps} = 40 \times 40 \times 32 = 51200$ primary capsules formed in this way.
We denote primary capsule vectors as $\textbf{u}_{i}$ where the index $i$ runs from 1 to $N_{caps}$.
Each primary capsule is multiplied by a $16 \times 8$ weight matrix $\textbf{W}_{ij}$ to get a 16-dimensional vector $\hat{\textbf{u}}_{j|i}$ which predicts the high-level information of the image
\begin{eqnarray}
[\hat{\textbf{u}}_{j|i}]_{16 \times 1} = [\textbf{W}_{ij}]_{16 \times 8} [\textbf{u}_{i}]_{8 \times 1} \;, \label{eq:routing1}
\end{eqnarray}
where $j$ denotes a class label, 0 or 1.
To construct a digital capsule vector ($\textbf{v}_{j}$), we first take the linear combination of prediction vectors $\hat{\textbf{u}}_{j|i}$ from all capsules in the lower layer and define capsules $\textbf{s}_j$ in the higher-level,
\begin{eqnarray}
[\textbf{s}_{j}]_{16 \times 1} = \sum^{N_{caps}}_{i=1} c_{ij} [\hat{\textbf{u}}_{j|i}]_{16 \times 1} \;,
\end{eqnarray}
where the summation runs over the index $i$, and $c_{ij}$ denote routing weights
\begin{eqnarray}
c_{ij} = \frac{\exp b_{ij}}{\sum^1_{j=0} \exp b_{ij}} \;,
\end{eqnarray}
where all coefficients $b_{ij}$ are initialized to 0.
The digital capsule vector is defined by applying a squash activation function to $\textbf{s}_j$
\begin{eqnarray}
\textbf{v}_{j} = \frac{||\textbf{s}_{j}||^{2}}{1+||\textbf{s}_{j}||^{2}} \frac{\textbf{s}_j}{||\textbf{s}_{j}||} \;,
\label{eq:squash}
\end{eqnarray}
where $j$ again denotes a class label, 0 or 1.
The final length of each digital capsule vector $||\textbf{v}_{j}||$ represents
a probability of a given input image being identified as a class of $j$.
The CapsNet adjusts the routing weights $c_{ij}$ by updating the coefficients $b_{ij}$ such that
the prediction capsules $\hat{\textbf{u}}_{j|i}$ having larger inner products with the high-level capsules $\textbf{v}_{j}$ acquire larger weights
\begin{eqnarray}
b_{ij} \leftarrow b_{ij} + [\hat{\textbf{u}}_{ij}]_{1 \times 16} \cdot [\textbf{v}_{j}]_{16 \times 1} \;.
\label{eq:bij}
\end{eqnarray}
The procedure from Eq. (\ref{eq:routing1}) to Eq. (\ref{eq:bij}) is referred to as the routing by agreement algorithm \cite{NIPS2017_2cad8fa4}, which we repeat three times in total to adjust the routing weights $c_{ij}$.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.46\textwidth,clip]{Plots/CapsNet_MatrixCapsNet.pdf} \hspace*{0.3cm}
\includegraphics[width=0.46\textwidth,clip]{Plots/MPNN.pdf}
\caption{Significance at the HL-LHC with ${\cal L}$ = 3 ab$^{-1}$ for CapsNets/Matrix CapsNets (left) and MPNN (right).}
\label{fig:CapsNet_MPNN}
\end{center}
\end{figure*}
The output digital capsule vectors $\textbf{v}_{j}$ are used to define the margin loss function
\begin{eqnarray}
  L_{j} =
  && T_{j} \, \text{max} \Big (0,m^{+} - ||\textbf{v}_{j}|| \Big )^{2}
+ \lambda(1-T_{j}) \, \text{max} \Big (0,||\textbf{v}_{j}||-m^{-} \Big )^{2} \;,
\end{eqnarray}
where $T_{1}=1$ and $T_{0}=0$ for the signal and backgrounds respectively, $m^+ = 0.9$, $m^- = 0.1$, and $\lambda = 0.5$.

Another parallel routine, called a decoder, attempts to reconstruct the input image out of
the digital capsule vectors $\textbf{v}_{j}$.
The digital capsule vectors are fed into
3 DNN hidden layers with increasing number of neurons from 512 to 12500 (7500),
and reshaped into the input image size of $5 \times 50 \times 50$ ($3 \times 50 \times 50$).
The reconstruction loss function for the decoder is defined by the sum of squared differences in pixel intensities
between the reconstructed ($\mathcal{I}^{\text{(reco)}}_k$) and input ($\mathcal{I}^{\text{(input)}}_k$) images
\begin{eqnarray}
  L_{\text{deco}} = \frac{1}{N} \sum_{k=1} ( \mathcal{I}^{(reco)}_k - \mathcal{I}^{\text{(input)}}_k)^2 \;,
\end{eqnarray}
where the index $k$ runs from 1 to the total number of pixels in the image, and $N$ is a normalization factor
defined by the total number of pixels times the total number of training events.
In order for capsule vectors to learn features that are useful to reconstruct the original image,
we add the reconstruction loss into the total loss function
\begin{eqnarray}
  L =  L_{j}  + \alpha L_{\text{deco}} \, ,
\end{eqnarray}
which is modulated by the overall scaling factor of $\alpha$.
Following the choice of Ref. \cite{NIPS2017_2cad8fa4}, we set $\alpha = 5.0 \times 10^{-4}$.
When training the network, we used Adam optimizer, the learning rate of $10^{-4}$,  regularization term of {\tt weight\_decay}=$0$, mini-batch size of 20, and epochs of 11.

Fig. \ref{fig:CapsNet_MPNN} shows the performances of the CapsNet where the overall significance of $0.8 \sim 0.9$ can be achieved, which is slightly lower than CNN with the same image inputs but better than CNN with $V^{(C,N)}$ (see Fig. \ref{fig:CNN_ResNet}).
Albeit not a substantial impact, additional neutrino images help to improve the significance up to $\sim 5\%$.

\subsection{Matrix Capsule Networks}
\label{sec:matrix}
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=1\textwidth,clip]{Plots/Matrix_CapsNet.pdf}
\caption{A schematic architecture of the matrix capsule network model (Matrix CapsNet) used in this paper.}
\label{fig:Matrix_CapsNet}
\end{center}
\end{figure*}

Regardless of the progress made in the CapsNet, there are a number of deficiencies that need to be addressed.
First, it uses vectors of length $n$ to represent features of an input image, which introduces too many $n^2$ parameters for weight matrices.
Second, in its routing by agreement algorithm, the inner products are used to measure the accordance between two pose vectors ($cf.$ Eq.(\ref{eq:bij})).
This measure becomes insensitive when the angle between two vectors is small.
Third, it uses the ad hoc non-linear squash function to force the length of the digital vector to be smaller than 1 ($cf.$ Eq.(\ref{eq:squash})).

The aim of the Matrix CapsNet \cite{hinton2018matrix} is to overcome the above problems
by generalizing the concept of the capsules.
The major difference with the original CapsNet is that here each capsule is not a vector, but it is the entity which
contains a $n \times n$ pose matrix $M$ and an activation probability $a$.
The utility of the pose matrix is to recognize objects in various angles, from which they are viewed,
and it requires a less number of hyper-parameters in its transformation matrix.\footnote{The same number of components in the vector of length $n$ can be contained in the $\sqrt{n} \times \sqrt{n}$ matrix. The former requires the $n \times n$ weight matrix, but the later requires the $\sqrt{n} \times \sqrt{n}$ weight matrix giving rise to a significant reduction in the amount of hyper-parameters.}

Fig. \ref{fig:Matrix_CapsNet} shows the architecture of the Matrix CapsNet used in our analysis.
The input is the 3D image of $V^{(C, N, \ell, \nu_{\text{H}}, \nu_{\text{T}})}_{\text{image}}$ ($V^{(C, N, \ell ) }_{\text{image}}$). We apply the 3D convolution using the kernel size of $5 \times 3 \times 3$ ($3 \times 3 \times 3$), the stride of 1, and 64 feature maps.
It is followed by a series of 2D convolutions using the kernel size of $3 \times 3$, the stride of 1 or 2, and 64 feature maps, until the image dimension is reduced to $ 64 \times 22 \times 22 $.
Next, we apply the modified 2D convolution using the kernel size of $1 \times 1$ and the stride of 1 where
each stride of the $1 \times 1$ convolution transforms the 64 feature maps into 8 capsules.
As a result of this operation, we obtain the layer $L$ of capsules whose dimension is denoted as
$8 \times 22 \times 22$ in unit of capsules.

Each capsule $i$ in the layer $L$ encodes low-level features of the image,
and it is composed by the $2 \times 2$ pose matrix $M_i$ and the activation probability $a_i$.
To predict the capsule $j$ in the next layer $L+1$, which encodes high-level features,
we multiply the pose matrix $M_i$ by a $2 \times 2$ weight matrix $W_{ij}$
\begin{eqnarray}
  [V_{ij}]_{2 \times 2} = [M_i]_{2 \times 2} [W_{ij}]_{2 \times 2}  \;,
\label{eq:vote}
\end{eqnarray}
where $V_{ij}$ is the prediction for the pose matrix of the parent capsule $j$.
Since each capsule $i$ attempts to guess the parent capsule $j$,
we call $V_{ij}$ the vote matrix.

On the other hand, the $2 \times 2$ pose matrix of the parent capsule $j$ is modeled by 4 parameters of $\mu^h_j$ (with $h = 1, 2, 3, 4$),
which serve as the mean values of the 4 Gaussian distribution functions with standard deviations of $\sigma^h_j$.
In this model, the probability of $V_{ij}$ belonging to the capsule $j$ is computed by
\begin{eqnarray}
P^{h}_{i|j}=\frac{1}{\sigma^{h}_{j} \sqrt{2 \pi }} \exp \bigg(-\frac{(V^{h}_{ij}-\mu^{h}_{j})^{2}}{2(\sigma^{h}_{j})^{2}} \bigg) \;,
\label{eq:probability}
\end{eqnarray}
where $V^{h}_{ij}$ denotes the $h^{th}$ component of the vectorized vote matrix.
Using this measure, we define the amount of cost ${\mathcal{C}}$ to activate the parent capsule $j$
\begin{eqnarray}
\mathcal{C}^{h}_{ij} = -\ln (P^{h}_{i|j}) \, ,
\label{eq:cost}
\end{eqnarray}
so that the lower the cost, the more probable that the parent capsule $j$ in the layer $L +1$ will be activated by the capsule $i$ in the layer $L$.
We take the linear combination of the costs from all the capsules $i$ in the layer $L$
\begin{eqnarray}
\mathcal{C}^{h}_{j} = \sum_{i} r_{ij} \,  \mathcal{C}^{h}_{ij} \, ,
\label{eq:costsum}
\end{eqnarray}
where each cost is weighted by an assignment probability $r_{ij}$.
To determine the activation probability $a_j$ in the layer $L+1$, we use the following equation
\begin{eqnarray}
a_j = sigmoid \Big (\lambda (b_j -  \sum^4_{h=1} \mathcal{C}^h_j ) \Big ) \;,
\label{eq:activation}
\end{eqnarray}
where $b_j$ and $\lambda$ are a benefit and an inverse temperature parameters, respectively.

Hyper-parameters such as $W_{ij}$ and $b_j$ are learned through a back-propagation algorithm,
while $\mu^{h}_{j}$, $\sigma^{h}_{j}$, $r_{ij}$, and $a_j$ are determined iteratively by the Expectation-Maximization (EM) routing algorithm.\footnote{It is the algorithm based on the series of steps described in Eq.(\ref{eq:vote}-\ref{eq:activation}). The algorithm is repeated several times to determine the hyper-parameters iteratively.
More details can be found in Ref. \cite{hinton2018matrix}.}
The inverse temperature parameter $\lambda$, on the other hand, is fixed to $10^{-3}$.
After repeating the EM iteration 3 times, the last $a_j$ is the activation probability, and
the final parameters of $\mu^h_j$ (with $h = 1, 2, 3, 4$) are reshaped to form the $2 \times 2$ pose matrix of the layer $L+1$.

The above prescription of computing the capsules in the layer $L+1$ from the layer $L$ can be systematically
combined with the convolutional operation.
Recall that we ended up with the capsule layer with the dimension of $8 \times 22 \times 22$.
Here, we apply the convolutional capsule operation using the kernel size of $5 \times 5$, the stride of 2, and 16 feature maps.
This operation is similar to the regular CNN, except that it uses the EM routing algorithm to compute the pose matrices and the activation probability of the next layer.
It is followed by another convolutional capsule operation until the image dimension is brought to $16 \times 7 \times 7$ in unit of capsules.
These capsules in the last layer are connected to the class capsules, and it outputs one capsule per class.
The final activation value of the capsule of a class $j$ is interpreted as the likelihood
of a given input image being identified as a class $j$.

The final loss function is defined by
\begin{eqnarray}
L = \frac{\sum_{j \neq t}  (max(0,m-(a_{t}-a_{j})))^{2}}{N_{train}} - m^{2} \, ,
\end{eqnarray}
where $a_{t}$ denotes a true class label, and $a_{j}$ is the activation for class $j$, $N_{train}$ denotes a number of training events.
If the difference between the true label $a_{t}$ and the activation for the wrong class $j ( \neq t)$ is smaller than the threshold of $m$,
the loss receives a penalty term of $(m-(a_{t}-a_{j}))^{2}$ \cite{hinton2018matrix}.
The threshold value $m$ is initially set to 0.1, and it is linearly increased by $1.6 \times 10^{-2}$ per each epoch of training.
It stops growing when it reaches to 0.9 at the final epoch.
When training the network, we used Adam optimizer, the learning rate of $10^{-4}$,  regularization term of {\tt weight\_decay}=$5 \times 10^{-5}$, mini-batch size of 20, and epochs of 20.

Fig. \ref{fig:CapsNet_MPNN} shows the performances of the Matrix CapsNet
where the overall significance of 0.7-0.8 can be achieved. It is slightly lower than that for CapsNet.
We find that performance of CapsNet and Matrix CapsNet is comparable or slightly worse than those using CNN with the same image inputs (see Fig. \ref{fig:CNN_ResNet}).

\subsection{Graph Neural Networks}
\label{sec:mpnn}

Instead of using the image-based neural networks that could suffer from the sparsity of the image datasets, one could encode the particle information into the graphical structure which consists of nodes and edges.
Graph neural networks (GNNs) \cite{1555942,4700287} take into account topological relationships among the nodes and edges in order to learn graph structured information from the data. Each reconstructed object (in Eq.(\ref{eq:vis_4momenta})) including neutrinos obtained from the Higgsness (in Eq.(\ref{eq:Hneu_4momenta})) is represented as a single node.
Each node $i$ has an associated feature vector $\textbf{x}_i$ which collects the properties of a particle.
The angular correlation between two nodes $i$ and $j$ is encoded in an edge vector $\textbf{e}_{ij}$.

The first type of GNN architectures that we consider is a modified edge convolutional neural network (EdgeConv) \cite{DBLP:journals/corr/abs-1801-07829}, which efficiently exploits Lorentz symmetries, such as an invariant mass, from the data~\cite{Abdughani:2020xfo,Abdughani:2018wrw}. Its schematic architecture is shown in Fig. \ref{fig:EdgeConv}.
All nodes are connected with each other, and the node $i$ in the input layer is represented by a 4-momentum feature vector of $\textbf{x}^0_i = (p_x,\ p_y,\ p_z,\ E)_i$, while leaving the edge vectors $\textbf{e}_{ij}$ unspecified.
The EdgeConv predicts the features of the node in the next layer as a function of neighboring features.
Specifically, each feature vector of a node $i$ in the layer $n$ is defined by
\begin{eqnarray}
\textbf{x}_i^{n} = \sum_{j \in \mathcal{E}} \textbf{W}^n &\Big(& \textbf{x}_i^{n-1} \oplus  (\textbf{x}_i^{n-1} \circ  \textbf{x}_i^{n-1})
 \oplus   (\textbf{x}_i^{n-1} \circ  \textbf{x}_j^{n-1})   \oplus  ( \textbf{x}_j^{n-1} - \textbf{x}_i^{n-1} )  \Big) \;,
 \end{eqnarray}
where the symbol $\oplus$ denotes a direct sum and $\mathcal{E}$ stands for the a set of feature vectors $j$ in the layer $n-1$ that are connected to the node $i$. $\circ$ and $\textbf{W}^n$ denote the element-wise multiplication and a weight matrix, respectively.
\begin{figure*}[t]
\centering
\includegraphics[scale=0.57]{./Plots/EdgeConv.pdf}
\caption{
A schematic architecture of the edge convolutional neural network (EdgeConv) used in this paper.
}
\label{fig:EdgeConv}
\end{figure*}
We do not include the bias vector and the ReLu activation function.
This completes one round of the graph convolution, and we repeat it $N=3$ times.

In the output layer, all feature vectors are concatenated and multiplied by a weight matrix, leading to a vector of dimension 2
\begin{eqnarray}
\hat{\textbf{p}} = \textbf{W}^N \bigoplus_{i}    \textbf{x}_i^{N} \;.
\end{eqnarray}
We apply the sigmoid activation function on each component of $\hat{\textbf{p}}$
\begin{eqnarray}
p_{k} = \frac{1}{1+e^{-\hat{p}_{k}}} \;,
\end{eqnarray}
where $k$ stands for the class label, 0 or 1, and
$p_{k}$ represents a probability which is used to classify the signal and backgrounds.
We use the cross entropy as the loss function
\begin{eqnarray}
L_k = -y_k \log p_k - (1-y_k) \log (1-p_k) \;,
\end{eqnarray}
where $y_1 = 1$ and $y_0 = 0$ for the signal and backgrounds respectively.
When training the network, we adopted Adam optimizer with the learning rate of $9.20 \times 10^{-7}$ and the momentum parameters $\beta_1= 9.29 \times 10^{-1}$ and $\beta_2 = 9.91 \times 10^{-1}$. We used the regularization term with {\tt weight\_decay}=$3.21 \times 10^{-2}$, a multiplicative factor of the learning rate decay $\gamma = 1$, mini-batch size of 130, and epochs of 70.

The second type of GNN architectures is a message passing neural network (MPNN)  \cite{DBLP:journals/corr/GilmerSRVD17} as shown in Fig. \ref{fig:mpnn}.
The node $i$ in the input layer is represented by a feature vector of $\textbf{x}^0_i = (I_{\ell}, I_{b}, I_{\nu},\ m,\ p_T,\ E)_i$,
where $m$, $p_T$, and $E$ denote the invariant mass, transverse momentum, and energy of a particle, respectively.
The default values for $I_i$ are set to zero.
$I_{\ell} = 1$ for the hardest lepton and $I_{\ell} = -1$ for the second hardest lepton, $I_{b} = 1$ for the hardest $b$-jet and $I_{b} = -1$ for the second hardest $b$-jet, and $I_{\nu} = 1$ for the hardest neutrino and $I_{\nu} = -1$ for the second hardest neutrino (note that we are using reconstructed neutrino momenta from Higgsness).
All nodes are connected to each other, and a single component edge vector $\textbf{e}_{ij}$ is represented by the angular separation ($\Delta R_{\textbf{x}_i, \textbf{x}_j}$) between two particles in the node $i$ and $j$.
The MPNN preprocesses each input node $i$, multiplying the $\textbf{x}^0_i$ by a weight matrix $\textbf{W}^0$
\begin{eqnarray}
\textbf{m}^0_i =  \textbf{W}^0  \textbf{x}^0_i \;.
\end{eqnarray}
Each feature vector of a node $i$ in the layer $n$ is defined by
\begin{eqnarray}
\textbf{m}_i^{n} = \sum_{j \in \mathcal{E}} \textbf{W}^n  \Big( \textbf{m}_i^{n-1}\oplus \big(\textbf{W}^{'n}  ( \textbf{m}_j^{n-1}\oplus \textbf{e}_{ij}) \big) \Big)  ,
\end{eqnarray}
where $\textbf{W}^{n}$ and $\textbf{W}^{'n}$ denote weight matrices. This completes one round of the graph convolution, and we repeat it $N=3$ times.
\begin{figure*}[t]
\centering
\includegraphics[scale=0.57]{./Plots/MPNN_NN.pdf}
\caption{A schematic architecture of the message passing neural network (MPNN) used in this paper.
}
\label{fig:mpnn}
\end{figure*}

In the output layer, each feature vector of node $i$ is multiplied by a weight matrix to become the vector of length 2
\begin{eqnarray}
\hat{\textbf{p}}_i = \textbf{W}^N \textbf{m}_i^{N} \;.
\end{eqnarray}
We apply the sigmoid activation function on the each component of $\hat{\textbf{p}}_i$
\begin{eqnarray}
p_{ik} = \frac{1}{1+e^{-\hat{p}_{ik}}} \;,
\end{eqnarray}
where $k$ stands for the class label, and
$p_{i1}$ represents a probability of a node $i$ being identified as the signal.
To identify the signal, we require the graph to pass the cut
\begin{eqnarray}
\sum_i p_{i1} >  p^{cut}_{1}   \;,
\end{eqnarray}
where $p^{cut}_{1}$ is optimized to yield a best significance.
We use the cross entropy as the loss function
\begin{eqnarray}
L_k = \sum_i  \Big( -y_k \log p_{ik} - (1-y_k) \log (1-p_{ik})  \Big),
\end{eqnarray}
When training the network, we used Adam optimizer, the learning rate of $1.44 \times 10^{-5}$,  regularization term of {\tt weight\_decay}=$9.67 \times 10^{-2}$, $\beta_1= 9.16 \times 10^{-1}$, $\beta_2 = 9.92 \times 10^{-1}$, $\gamma = 9.81 \times 10^{-1}$, mini-batch size of 164, and epochs of 21.

We find that the EdgeConv and MPNN show a very good performance with the basic momentum input, and show their signal significance in the right panel of Fig. \ref{fig:CapsNet_MPNN}, which clearly surpasses the performance of FC with the same input features (shown in Fig. \ref{fig:FCresults}).
Moreover, before combining kinematic variables and image-based input features, the performance of the EdgeConv and MPNN is comparable to or slightly better than those based on NN with image only such as CapsNet, Matrix CapsNet, or CNN (see Fig. \ref{fig:CNN_ResNet} and the left panel in Fig. \ref{fig:CapsNet_MPNN}.).
This comparison illustrates a couple of important points. First, one can further improve vanilla DNN with basic four momenta input introducing evolution of nodes and edges in MPNN/EdgeConv. Second, to bring additional improvement in the signal significance, it is crucial to consider different types of inputs such as image-based features and high kinematic variables in addition to basic four momenta, and develop NN architecture suitable for the corresponding input features.

\section{A brief review on kinematic variables}
\label{appen:variable}

In this appendix, we provide a short review on kinematic variables used in this paper. For more details, we refer to Refs. \cite{Barr:2010zj,Barr:2011xt}.
We consider the following 21 kinematic variables in total (Eqs. (\ref{eq:11_kin})-(\ref{eq:15_kin})), in addition to four momentum of all visible and invisible particles (Eqs. (\ref{eq:vis_4momenta})-(\ref{eq:Hneu_4momenta})),
\begin{eqnarray}
V_{\text{21-kin}} = &\{& p_T(\ell_1), p_T(\ell_2), p_{Tbb}, p_{T\ell\ell}, \mpt, \Delta R _{bb}, \Delta R_{\ell\ell}, \Delta\phi_{bb,\ell\ell}, m_{\ell\ell}, m_{bb}, \label{eq:V21} \\
      & & min[\Delta R_{b\ell}], \Delta R^{\text{H}}_{\nu \nu}, m^{\text{H}}_{\nu \nu},  \Delta R^{\text{T}}_{\nu \nu}, m^{\text{T}}_{\nu \nu},  \sqrt{\hat{s}}_{min}^{(bb\ell \ell)},  \sqrt{\hat{s}}_{min}^{(\ell \ell)}, M_{T2}^{(b)}, M_{T2}^{(\ell)}, {\rm H}, {\rm T} \,  \} \, . \nonumber
\end{eqnarray}
The distributions of 16 variables (out of 21 except for $min[\Delta R_{b\ell}], \Delta R^{\text{H}}_{\nu \nu}, m^{\text{H}}_{\nu \nu},  \Delta R^{\text{T}}_{\nu \nu}, m^{\text{T}}_{\nu \nu}$) in Eq. \ref{eq:V21} are shown in Refs. \cite{Kim:2019wns,Kim:2018cxf} and we obtained similar results.

\begin{itemize}
\item $p_T(\ell_1)$ and $p_T(\ell_2)$ are the transverse momentum of the hardest and the next hardest leptons, respectively.
\item $p_{T\ell\ell}$ and $p_{Tbb}$ are the transverse momentum of the two-lepton system and the two-$b$-tagged jets, respectively.
\item $\mpt$ is the missing transverse momentum.
\item $\Delta R_{\ell\ell}$ and $\Delta R _{bb}$ are the angular separation between two leptons, between two $b$-tagged jets, respectively.
The angular distance is defined by
\begin{eqnarray}
\Delta R_{ij} = \sqrt{(\Delta\phi_{ij})^2+(\Delta \eta_{ij})^2}, \label{deltaR}
\end{eqnarray}
where $\Delta\phi_{ij} = \phi_i-\phi_j$ and $\Delta\eta_{ij}=\eta_i-\eta_j$ denote the differences in the azimuthal angles and rapidities respectively between particles $i$ and $j$.
\item $\Delta\phi_{bb,\ell\ell}$ is the angular separation between the $b\bar b$ system and the two lepton system.
\item $m_{\ell\ell}$ and $m_{bb}$ are the invariant mass of two leptons and two $b$-tagged jets, respectively.
\item $min[\Delta R_{b\ell}]$ is the smallest angular distance between a $b$-jet and a lepton out of 4 possible combinations, which is shown in the left-upper corner in Fig. \ref{fig:5dists}.
\item $\sqrt{\hat{s}}_{min}^{(bb\ell \ell)}$ and $\sqrt{\hat{s}}_{min}^{(\ell \ell)}$ are the minimum $\sqrt{\hat s}$ for the $(bb\ell \ell)$ subsystem and the $(\ell \ell)$ subsystem, respectively. The $\hat{s}_{min}$ variable \cite{Konar:2008ei,Konar:2010ma} is defined as
\begin{equation}
 \hat{s}_{min}^{({\rm v})} = m_{{\rm v}}^2 + 2 \left ( \sqrt{ |\vec P_{T}^{\rm v} |^2 + m_{\rm v}^2 }\ |{\;/\!\!\!\! \vec{P}_T}| - \vec P_{T}^{\rm v} \cdot {\;/\!\!\!\! \vec{P}_T} \right ) \, ,
 \label{smindef}
\end{equation}
where $({\rm v})$ represents a system of visible particles, and $m_{\rm v}$ ($\vec P_{T}^{\rm v}$) is corresponding invariant mass (transverse momentum). It provides a way to approximate the Mandelstam variable $\hat s$ of the system $({\rm v})$ in the presence of the missing momenta.
Here we consider two systems, ${\rm v}=\{bb\ell\ell\}$ and ${\rm v}=\{\ell\ell\}$. First, $\sqrt{\hat s}^{(bb\ell\ell)}_{min}$ represents a minimum energy required to produce two Higgs bosons (two top quarks) in the signal ($t\bar{t}$) events. Second, $\sqrt{\hat s}^{(\ell\ell)}_{min}$ represents a minimum energy required to produce two $W$ bosons. In case of the $t\bar{t}$ background, the peak of the distribution appears near $2m_W$. In case of the signal, the peak appears around the Higgs mass. The distributions of the variables $\hat{s}_{min}^{(bb\ell\ell)}$ and $\hat{s}_{min}^{(\ell\ell)}$ are shown in Refs. \cite{Kim:2019wns,Kim:2018cxf}.

\item $M_{T2}^{(b)}$ and $M_{T2}^{(\ell)}$ are the stransverse of ($b\bar b$) and $(\ell^+\ell^-)$ subsystem, respectively.
The $M_{T2}$ \cite{Lester:1999tx} defined as
\begin{equation}
M_{T2} (\tilde m) \equiv \min\left\{\max\left[M_{TP_1}(\vec{p}_{T \nu },\tilde m),\;M_{TP_2} (\vec{p}_{T \bar\nu },\tilde m)\right] \right\} \, ,
\label{MT2def}
\end{equation}
where $\tilde m$ is the test mass for the daughter particle, and the minimization over the transverse masses of the parent particles $M_{TP_i}$ ($i=1, 2$) is performed by scanning the unknown transverse neutrino momenta $\vec{p}_{\nu T}$ and $\vec{p}_{\bar\nu T}$, under the ${\;/\!\!\!\! \vec{P}_T} $ constraint. See Refs.~\cite{Barr:2011xt,Kim:2017awi,Cho:2014naa,Konar:2009wn,Konar:2009qr,Baringer:2011nh,Kim:2015uea,Goncalves:2018agy,Debnath:2017ktz} for more information and other variants of $M_{T2}$.

When $M_{T2}$ is applied to the $b \bar b$ visible system we abbreviate it as $M_{T2}^{(b)}$. In this case, the daughter particles are the $W$ bosons, and hence we set $\tilde m = m_W = 80$ GeV. By construction, $M_{T2}^{(b)}$ is bounded by the mass of the corresponding parent particle, so that its distribution exhibits a sharp drop at around $M_{T2}^{(b)} = m_t$, for $t\bar{t}$ events. Double Higgs production, on the other hand, does not obey this bound, which in turn can be used to disentangle two different samples.

When $M_{T2}$ is applied to the $\ell^+\ell^-$ visible system, we abbreviate it as $M_{T2}^{(\ell)}$. In this case, we set $\tilde m = m_\nu = 0$. The $M_{T2}^{(\ell)}$ distribution for the di-leptonic $\tau \tau b b$ events drops at around $\sim m_{\tau}$. This suggests that $M_{T2}^{(\ell)}$ can be used to discriminate the $\tau \tau b b$ background.

\item Topness measures a degree of consistency of a given event with $t\bar{t}$ production.
Topness itself is a minimized chi-square value constructed by using four on-shell constraints, $m_t$, $m_{\bar t}$, $m_{W^+}$ and $m_{W^-}$, and 6 unknowns (the three-momenta of the two neutrinos, $\vec p_{\nu}$ and $\vec p_{\bar\nu}$)
\begin{eqnarray}
\chi^2_{ij} &\equiv& \min_{\tiny {\;/\!\!\!\! \vec{P}_T} = \vec p_{\nu T} + \vec p_{ \bar\nu T}}  \left [
\frac{\left ( m^2_{b_i \ell^+ \nu} - m^2_t \right )^2}{\sigma_t^4} \,   +
\frac{\left ( m^2_{\ell^+ \nu} - m^2_W \right )^2}{\sigma_W^4}  \, \right.  \label{eq:tt} \nonumber \\
&& \hspace*{1.9cm}\left . + \frac{\left ( m^2_{b_j \ell^- \bar \nu} - m^2_t \right )^2}{\sigma_t^4} \, +
\frac{\left ( m^2_{\ell^- \bar\nu} - m^2_W \right )^2}{\sigma_W^4}   \right ]  \; ,
\label{eq:Tness1}
\end{eqnarray}
subject to the constraint, $ {\;/\!\!\!\! \vec{P}_T} = \vec p_{\nu T} + \vec p_{ \bar\nu T}$.
Due to the twofold ambiguity in paring a $b$-jet and a lepton, we define Topness as the smallest chi-square value
\begin{eqnarray}
T &\equiv&  { \min} \left ( \chi^2_{12} \, , \, \chi^2_{21} \right ) \, .
\label{eq:Tness2}
\end{eqnarray}

\item Similarly, Higgsness measures a degree of consistency of a given event with double Higgs production defined by
\begin{eqnarray}\nonumber
H &=&
\min_{\tiny {\;/\!\!\!\! \vec{P}_T} = \vec p_{\nu T} + \vec p_{ \bar\nu T}}  \left [
 \frac{\left ( m^2_{\ell^+\ell^-\nu \bar\nu} - m^2_h \right )^2}{\sigma_{h_\ell}^4}    \right.
 + \frac{ \left ( m_{\nu  \bar\nu}^2 -  m_{\nu\bar\nu, peak}^2 \right )^2}{ \sigma^4_{\nu}}
   \nonumber \\
 && \hspace*{1.1cm}   + {\min} \left (
\frac{\left ( m^2_{\ell^+ \nu } - m^2_W \right )^2}{\sigma_W^4} +
\frac{\left ( m^2_{\ell^- \bar \nu} - m^2_{W^*, peak} \right )^2}{\sigma_{W_*}^4}  \, , \right. \label{eq:hww}  \\ \nonumber
 && \hspace*{2.3cm}    \left.  \left .
\frac{\left ( m^2_{\ell^- \bar \nu} - m^2_W \right )^2}{\sigma_W^4} +
\frac{\left ( m^2_{\ell^+ \nu} - m^2_{W^*, peak} \right )^2}{\sigma_{W_*}^4}
\right )  \right ]   \, ,
\label{eq:Hness}
\end{eqnarray}
where $m_{W^*}$ is bounded above $m_{W^*} \leq m_h- m_W$ and its location of the peak can be estimated by
\begin{eqnarray}
m_{W^*}^{peak} &=& \frac{1}{\sqrt{3}} \sqrt{ 2 \left ( m_h^2 + m_W^2 \right ) - \sqrt{m_h^4 + 14 m_h^2 m_W^2 + m_W^4}} \approx 40 \text{ GeV} \, .
\end{eqnarray}
The location of the peak in the invariant mass $m_{\nu\bar\nu}$  distribution of two neutrinos appears at $m_{\nu\bar\nu}^{peak} \approx 30$ GeV.
The definitions of Topness and Higgsness involve $\sigma$ hyperparameters which stand for experimental uncertainties and particle widths. In our numerical study, we use $\sigma_t=5$ GeV, $\sigma_W=5$ GeV, $\sigma_{W^*}=5$ GeV, $\sigma_{h_\ell}=2$ GeV, and $\sigma_\nu = 10 $ GeV. Our results are not sensitive to these numerical values. The Topness and Higgsness are shown in Fig. \ref{fig:TH}. See Ref. \cite{Alves:2022gnw} for the heavy Higgs decaying to two $W$ bosons.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.54\textwidth,clip]{Plots/Topness_Higgsness_signal2.pdf}  \hspace*{-1.5cm}
\includegraphics[width=0.54\textwidth,clip]{Plots/Topness_Higgsness_backgrounds2.pdf}
\caption{$\log T$ vs $\log H$ for signal (left) and all backgrounds (right).
\label{fig:TH}}
\end{center}
\end{figure*}

\item $\Delta R^{\text{H}}_{\nu \nu}$ (left-middle panel in Fig. \ref{fig:5dists}) and $ \Delta R^{\text{T}}_{\nu \nu}$ (right-top panel in Fig. \ref{fig:5dists}) are the angular separation of the two neutrinos which are reconstructed using Higgsness and Topness, respectively. It is expected that they resemble $\Delta R_{\ell\ell}$.
\item $m^{\text{H}}_{\nu \nu}$ (left-bottom panel in Fig. \ref{fig:5dists}) and $m^{\text{T}}_{\nu \nu}$ (right-middle panel in Fig. \ref{fig:5dists}) are the invariant mass of the two neutrinos which are reconstructed using Higgsness and Topness, respectively. It is expected that they resemble $m_{\ell\ell}$.
\end{itemize}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.51\textwidth,clip]{Plots/min_R_bl_stack2.pdf} \hspace*{-0.7cm}
\includegraphics[width=0.51\textwidth,clip]{Plots/R_T_vv_stack2.pdf} \\
\includegraphics[width=0.51\textwidth,clip]{Plots/R_H_vv_stack2.pdf}  \hspace*{-0.7cm}
\includegraphics[width=0.51\textwidth,clip]{Plots/m_T_vv_stack2.pdf} \\
\includegraphics[width=0.51\textwidth,clip]{Plots/m_H_vv_stack2.pdf} \hspace*{-0.7cm}
\includegraphics[width=0.51\textwidth,clip]{Plots/m_bbllvv_stack.pdf} \\
\caption{$min[\Delta R_{b\ell}]$ (left, top), $\Delta R^{\text{T}}_{\nu \nu}$ (right, top), $\Delta R^{\text{H}}_{\nu \nu}$ (left, middle), $m^{\text{T}}_{\nu \nu}$ (right, middle), $m^{\text{H}}_{\nu \nu}$ (left, bottom), $m_{b\bar b \ell\ell\nu\bar\nu}$  with reconstructed neutrino momenta using Higgsness (right, bottom).
\label{fig:5dists}}
\end{center}
\end{figure*}

Finally we would like to comment on the use of the reconstructed neutrinos.
Some of the 21 kinematic variables which are defined in the laboratory frame show the global properties of the signal and background processes. For example, distributions of invariant masses ($m_{\ell\ell}$ and $m_{bb}$ and angular variables ($\Delta R_{\ell\ell}$, $\Delta\phi_{bb,\ell\ell}$, $min[\Delta R_{b\ell}]$, etc) show that the pencil-like production of $hh$ and isotropic production of $t \bar t$.
These are better measured via shape variables in the CM frame.
Now that we have obtained approximate momenta of the missing neutrinos, we are able to Lorentz-boost to the CM frame of the production event by event. Among many shape variables, we consider the sphericity and thrust variables.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.325\textwidth,clip]{Plots/Thrust_LAB.png}\hspace*{-0.1cm}
\includegraphics[width=0.325\textwidth,clip]{Plots/Sphericity_LAB.png}\hspace*{-0.1cm}
\includegraphics[width=0.325\textwidth,clip]{Plots/T_vs_S_LAB.png} \\
\includegraphics[width=0.325\textwidth,clip]{Plots/Thrust_CM.png}\hspace*{-0.1cm}
\includegraphics[width=0.325\textwidth,clip]{Plots/Sphericity_CM.png}\hspace*{-0.1cm}
\includegraphics[width=0.325\textwidth,clip]{Plots/T_vs_S_CM.png}
\caption{Thrust (left), sphericity (middle) and thrust vs. sphericity (right) in the lab frame (top) and CM frame (bottom) for the $hh$ (red) and $t\bar t$ (blue) productions.
The quantities with subscript `H' (`T') are obtained using Higgsness (Topness).
\label{fig:shapes} }
\end{center}
\end{figure*}

The thrust ($T$) is defined as
\begin{eqnarray}
T &=& \max_{ \vec n} \left ( \frac{\sum_i \Big | \vec p_i \cdot \vec n \Big | }{ \sum_i \Big | \vec p_i \Big |} \right ) \, ,
\end{eqnarray}
with respect to the direction of the unit vector $ \vec{n} $ which maximizes $ T $, identified as the thrust axis $\vec n_T$.
This definition implies that for $T = 1$ the event is perfectly back-to-back while for $T = 1/2$ the event is spherically symmetric.
The sphericity ($S$) provides global information about the full momentum tensor ($M$) of the event via its eigenvalues:
\begin{eqnarray}
    M_{i j} &=& \frac{\sum_{a=1}^{n_j} p_{i a} p_{j a}}{\sum_{a=1}^{n_j} |\vec p_a|^2} \, ,
\end{eqnarray}
where $i,j$ are the spatial indices and the sum runs over all jets. The ordered eigenvalues $\lambda_i$ ($\lambda_1 > \lambda_2 > \lambda_3$) with the normalization condition $\sum_i \lambda_i = 1$ defines the sphericity
\begin{eqnarray}
S   &=& \frac{3}{2} \Big ( \lambda_2 + \lambda_3 \Big )\, .
\end{eqnarray}
The sphericity axis $\vec n_S$ is defined along the direction of the eigenvector of $\lambda_1$.
The sphericity measures the total transverse momentum with respect to the sphericity axis defined by the four momenta in the event. In other words, the sphericity of an event is a measure of how close the spread of energy in the event is to a sphere in shape. The allowed range for $S$ is $0 \leq  S < 1$. For a pencil-like production, $S\approx0$, while $S\approx 1$ for isotropic production.

We obtain neutrino momenta using Higgsness and Topness, which in fact show similar distributions statistically, as shown in the neutrino images. Fig. \ref{fig:shapes} shows the thrust (left), sphericity (middle) and thrust vs. sphericity (right) in the lab frame (top) and CM frame (bottom) for the $hh$ (red) and $t\bar t$ (blue) productions.
The numerical values of ($T_H^{LAB}$, $S_H^{LAB}$, $T_H^{CM}$, $S_H^{CM}$) are obtained using Higgsness,
while ($T_T^{LAB}$, $S_T^{LAB}$, $T_T^{CM}$, $S_T^{CM}$) are obtained using Topness.
Two plots in the right panel show the correlation of the thrust and sphericity, taking Topness minimization for the neutrino momenta. We do not show results with Higgsness, as they are very similar.

\section{Introduction}
\label{sec:intro}
\input{1_intro.tex}

\section{Theoretical setup and simulation}
\label{sec:eventgen}
\input{2_EventGen.tex}

\section{Pre-processing input data}
\label{sec:input}
\input{3_InputData.tex}

\section{Performance of machine learning algorithms}
\label{sec:analysis}
\input{4_analysis.tex}

\section{Comparison of different networks}
\label{sec:results}
\input{5_results.tex}

\section{Discussion and outlook}
\label{sec:outlook}
\input{6_conclusion.tex}

\bigskip\bigskip
\emph{Acknowledgements:}
We thank Minho Kim, Myeonghun Park, and Konstantin Matchev for useful discussion and collaboration at the early stage of the work. This work is supported by Chungbuk National University Korea National University Development Project (2020).
KK acknowledges support from the US DOE, Office of Science under contract DE-SC0021447.
LH is supported by the Fundamental Research Funds for the Central Universities, and the Bureau of International Cooperation, Chinese Academy of Sciences.
\bigskip

redpajama_set_name: RedPajamaArXiv


So one of my favorite blogs is styleberry BLOG. Last year she came up with Project 52. She picks a theme for the year and each Friday she posts a picture and writes a post to go along with the theme. Then, at the end of the year she got her book printed. Her theme for 2011 was "Motherhood Is..." and her theme for this year is "I want them to know". She is posting different lessons she wants her kids to know.
I've decided to join her this year.
Y'all obviously being the twins.
I've already realized that there is going to be so many new things this year. So many ways to grow. And so many things to learn.
And all of it is going to be a blur.
So, if I can pull one moment of joy or one lesson learned every week of my first year with the boys, I think I will be able to look back and see how much motherhood has allowed me to grow and change.
Tomorrow morning will be my first post! I know I missed week one, so I'll be catching up at some point.


Is a Happy Marriage Between Gemini and Taurus Possible? Find Out.
One talks a mile a minute, the other just wants a margarita in the sunshine – Gemini and Taurus are two sunny zodiac pals indeed.
Yet in love, their differences can prove as inspiring as they are challenging, and they're going to have to learn to work with one another's quirks to get by.
This is a relationship where zippy ideas zing off solid as a rock dependability, and where two social butterflies of very different sorts can combine their talents.
Yet doing so means getting over some pretty big differences in outlook, and to do that, you'll need this introduction to Gemini and Taurus compatibility – the definitive guide to understanding this partnership from top to bottom.
If you look at the calendar and tot up where in the year the birthdays for Gemini and Taurus fall, you'll find they're not far apart at all – a few weeks at best.
Gemini and Taurus are what are sometimes called astrological neighbours or adjacent signs.
This means, essentially, that their star signs are directly sequential in the zodiac, with one following the other.
Astrologers often find that relationships between neighbouring star signs prove oh so intriguing, although for a mix of good and bad reasons!
To understand why, it's important to understand the individual qualities of each star sign. That's the best way of identifying the complementary – and less readily mixed – things to expect in a Gemini and Taurus relationship.
Beginning with Gemini, it's a good idea to get to know the symbolism behind them – the twins.
Gemini people have a unique gift in all of astrology, for they can view situations from countless angles simultaneously and internally debate them, just as two lifelong siblings might.
Yet the symbolism of the twins explains not only why Gemini people seem to adopt multiple personality quirks they shuffle through on a whim.
It also represents how no single one of those personas is dominant – they are all fully realised, just as two twins are individuals as much as identical.
Gemini people go through life internally mulling countless ideas concurrently, and this is partly why they find it difficult making and sticking to decisions.
Seeing something from every angle at once, then condensing it down to one reality, is a challenge few of us can relate to.
That complexity is a far cry from the world of the Taurus soul. Taurus is represented by the bull in astrology – a slow paced and docile animal, but also fierce if provoked and certainly not to be trifled with.
And just as bulls helped pull ploughs in olden times, Taurus people are always keen to lend a hand.
These are generous and giving people, but also as territorial and stubborn as a bull too in many respects.
A Taurus host might give you the best dinner he can provide, for instance, but certainly would draw the line at letting you run free in his larder. He earned that feast for himself.
But of course, chief among the concerns of Taurus people – not that these laid back individuals have many of those – is merely the idea of a simple, enjoyable existence.
While Taurus folks can certainly be ambitious, they take life slow and steady, seeing no reason to stress.
And of course, that simplicity is a million miles from Gemini's own multifaceted view of everything. Similarly, the placid pace of life that Taurus prefers will feel like plodding through treacle for Gemini.
These are some of the differences these star signs face – they're astrological neighbours, but by no means similar.
The electric vitality that emanates from the Gemini woman can't help but entice the Taurus man.
He's someone naturally drawn to life's pleasures, beautiful women being among them – and the Gemini woman is often a graceful individual with remarkably good looks.
Of course, the Taurus man is often a handsome devil in his own right, with an earthy charm and a winning smile – after all, George Clooney was born under this very star sign.
However he looks, he likely notices the Gemini woman fast, and she him.
Of course, they're unlikely to get together quite as swiftly as they feel butterflies for one another, for a few key reasons.
In the case of the Taurus man, it's simply because he advances slowly but assuredly to what he wants. In the case of the Gemini woman, she's likely to be super scattered.
Not in the sense of being scatterbrained – although there can be a touch of that in Gemini folks – as much as being scattered among dozens of activities and pursuits with various people.
She's a popular person who's always in demand, and might not have time to commit to dating right away.
A lively energy backed up by wit and flirtation will eventually pick up though, and the dates that the Gemini woman and Taurus man in love go on often mix the intellectual and the romantic to superb effect.
However, the Taurus man will notice as time goes on that entering a relationship does little to slow the Gemini woman down.
She will still be flitting to and fro among countless friends and family members, and always has places to go and people to see.
Taurus takes life slow, and doesn't enjoy the idea of being dragged around countless places to talk over nothing in particular with countless faces.
He's a social creature too, but at a much different pace, and will soon elect to stay home or do his own thing.
The relationship gives both partners lots of time to be their own person, but the Taurus man has an occasionally possessive side that the Gemini woman won't like.
She's born to be free as a bird, and will find his insistences on her spending more time with him cloying.
Add to that the fact that the Taurus man has a jealous streak from time to time, and the Gemini woman's naturally flirtatious ways are likely to ruffle some feathers.
Yet her detached air and refusal to keep still every so often to indulge his romantic overtures does more harm than help too.
One's a fly by night smart alec, and the other's a relaxed, earthy and sensual nature lover who's keen to see some loyalty and stability in life.
The Gemini man, being the natural actor that he is, can certainly play the role here – but can he play for keeps?
There's very little that the Gemini man can't talk his way into or out of. His is a superb gift of the gab, with a natural affinity for wordplay and fast comebacks that allow him to effortlessly slide into or out of as many social circles as he pleases. He's fun, popular and widely beloved.
He's also something of a flirt, which the romantically inclined Taurus woman, all sensuality and snuggles, can't help but notice.
She's always keen to play the game of love, but also in no hurry to do so – she loves to enjoy the moment, so why rush things?
She's also patient enough to realise early on that the Gemini man needs to have a lot of things to see, do and get stuck into to feel engaged, so she won't rush him or insist upon his attention early on. He'll likely gravitate towards her more as she intrigues him anyway.
The Taurus woman enjoys feeling a dependable rhythm in life, and it might be difficult establishing that with the Gemini man – his plans change on the fly.
Having said that, he's intelligent and fun, and the pair can sit up all hours talking about life's big questions.
The Gemini man is touched by the loyalty and reliability of the generous Taurus woman, but he won't really let that stop him doing his own thing either.
This is mostly harmless, albeit with a chance to leave the Taurus woman in the lurch sometimes – but he should still be careful.
The reason the Gemini man should be careful when dating a Taurus woman is that he likes to look clever and lets his mouth run away sometimes.
In the company of a good looking lady, or if among male friends eyeing up a good looking woman across the room, he might say something the Taurus woman dislikes.
She won't flare up over jealousy instantly, but eventually she'll bubble over within in the slow way a Taurus temper rises – and once boiling point is reached, the Gemini man's ability to move fast will come in very handy. In short, he'll want to run away from her terrible wrath!
However, the Taurus woman also has an outlook on life that can sometimes cause her to stop making effort in a relationship once it's established.
The Gemini man needs to feel stimulated in the mind to want to stick at anything, but the partnership could get samey for him fast.
Both Gemini and Taurus have personalities that are often appealing to those around them.
Gemini's intelligence and fast humour always wins plenty of praise, and the way Taurus tends to relax and slow down the frayed nerves of people they connect with is one of their greatest gifts.
It therefore makes sense that both of these people make and keep friends easily, and because of that Gemini and Taurus friendships also tend to blossom wonderfully too.
Here, their differences and complementary values alike can mesh together without any high strung emotions.
The Gemini friend will be tricky to get hold of from time to time though, simply because their life is so full and socially active.
Gemini people bounce from place to place at an almost impossible speed, and they may duck out of plans last minute as new and exciting opportunities arise.
This tangles the thread of the more laid back Taurus friend's lifestyle somewhat, but it's unlikely to cause any real friction.
While Taurus folks have their grumpy moments, for the most part they are very live and let live.
They're also fond of good food and delicious drinks, so social occasions between Gemini and Taurus friends are likely to revolve around restaurants and bars if Taurus is given the choice of venue.
Gemini will go for any number of choices if left in charge, from night clubs to art galleries.
This variation can prove inspiring to Taurus, and through Gemini they're able to learn about dozens of new perspectives and fresh ideas.
Meanwhile, Taurus people themselves have a dependability that few others can match, which ground the Gemini friend when their lofty ways get too much even for them.
Taurus can remind electric Gemini to relax and not take on too much, and likewise Gemini can incentivise Taurus to get up and live a little if they get a bit stuck in their ways.
It's a wonderful match of complementary energies here, that brings warmth and comfort to the lives of each of them.
There's a bit of a difference in attitude towards marriage between Gemini and Taurus that it's important to keep in mind.
It's nothing irreconcilable, but having foreknowledge of how it could affect the big picture could calm troubled waters for both partners.
Essentially, it's once again a question of the speed at which life is taken, and what is given to the time we have as individuals, that defines who approaches married life in which way.
Naturally, anyone who gets to know Taurus likely isn't surprised that they're happy to settle down long-term.
Taurus won't rush into things though, as recklessness is not their forte, and neither is committing to anyone who isn't entirely on the level.
However, fidelity and loyalty are the strong suits of the bull of the zodiac, and they make committed, reliable and giving spouses.
Gemini has a little more complex a relationship with marriage, thank to the many fragments of personality that make up their whole.
It's like half of them loves it and half of them is terrified by the idea, and it all comes down to whether or not the marriage will clip their wings of freedom.
Many Gemini folks like to play the field too, and it's important they've got this out of their system before wedded life begins to save a lot of skulking around and potential heartache for Taurus.
The sunny bull expects the same loyalty they give, and can be baffled if Gemini doesn't want to do so.
The marriage is likely to be a tug of war between these notions of personal freedom versus devotion and staying centred together.
The Taurus and Gemini couple won't argue much, but that's because they both prefer avoiding drama – meaning a lot of unspoken grievances that could snowball.
However, with a good mix of travel, laughter, togetherness and independent lifestyles, there's every reason for this marriage to go the distance.
Stubborn Taurus won't always want to meet in the middle on issues like adaptive Gemini, but both will need to compromise to help the marriage last.
The differences in worldview held by both the Gemini and Taurus partners might devolve into some resentment over time.
Gemini's flighty mannerisms and constant charm to everyone – sometimes contrasted with the less glowing ways they discuss those same people with others – can strike Taurus as superficial.
Yet Gemini views Taurus as a slow-moving, slow reacting, unflappable kind of individual to whom life never seems to mean anything. It does – it's just that emotions are very internal to Taurus, and are rarely displayed on their features too much.
Besides which, excessive emotions make Gemini uncomfortable, and big displays of affection from Taurus might indeed ignite that fearful spark and send them fluttering away in a tizzy.
Taurus is hurt and bemused by that reaction and moves to hold on tighter, which just makes things worse.
However, Taurus is also quite an indulgent partner, and might be content to eat too much, drink too much, spend too much and sleep too much from time to time.
Taurus folks are powerfully connected to the power of rest and relaxation, but can take it to excess.
Yet Gemini sees this as out and out lazy, but they have plenty of ways to tick off Taurus too. For one thing, the predictable way Taurus likes their life to go is upended by Gemini's changing of the mind, inability to make a decision and way of almost arguing with themselves over things.
Plans went awry and a slowness to reach consensus on even the simplest of choices could well be the reality for a Gemini and Taurus couple from time to time, and they way they each approach that issue can all but madden the other.
Relationships that take place between those star signs who are next to one another in our calendar – and in the zodiac – never cease to surprise.
Many astrologers advise against it, as the differences between neighbouring signs is often pretty pronounced, but these alliances can work out too.
Interestingly, even the ruling planets of both Gemini and Taurus are neighbours in the night sky too. Mercury, the planet of communication and thought, rules Gemini – and loving, gentle, tender Venus rules Taurus. Again, neighbouring planets, but hugely different symbolism and personality types.
The gift with words that Gemini has, coupled with their charm, means they can flirt just as well as Taurus likes, but over time there may be a sense that their words don't always translate into actions.
Meanwhile, Taurus gives their word as their bond, and expects results!
Gemini's fleeting and freewheeling ways will cause issues for Taurus, but only if the proud bull refuses to relinquish control.
Letting Gemini flutter free is simply the best way to let them be their best selves, and that's what love is all about.
Of course, this doesn't give Gemini carte blanche to come and go as they please without shoeing some responsibility.
Through love, they can learn that having ties to someone needn't tie them down, as such – and as lovers go, Taurus is among the most easygoing around.
If the pair can adapt to one another and share their hang-ups without fear, there's every reason for this love match to succeed – but both Taurus and Gemini need to give some ground to help that happen.


To help you enhance your relationship and deepen your value proposition with Plan Sponsors Invesco has developed a series of public approved materials focused on educating participants about retirement. Check out the tabs below for more information.
Dividends are the only way investors can collect cash from their stock investments without having to sell them. Dividends are also one of the oldest gauges investors can use to help evaluate the financial health of a company.
1 Ned Davis Research Inc. Dec. 31, 1929, through Dec. 31, 2014.
2 Source: All data is sourced from Ned Davis Research Inc. as of Dec. 31, 2014. Further distribution prohibited without prior permission. Copyright© 2015 Ned Davis Research, Inc.
All rights reserved. S&P 500 Index past performance is not indicative of future results. Index performance is shown for illustrative purposes only and is not intended to represent historical or to predict future performance of any specific investment. Indexes are unmanaged and an investor cannot invest in an index. They do not include any fees or charges. Returns based on monthly equal-weighted geometric average of total returns of S&P 500 component stocks, with components reconstituted monthly. "Standard & Poor's" and "S&P" are trademarks of The McGraw-Hill Companies, Inc.
Essentially, dividends are payouts that some companies make to shareholders as a return on their investment. The money paid comes from a company's earnings, and it is generally seen as a reflection of the company's past performance as well as its potential for the future. Older, more established companies are more likely to pay dividends than newer firms. Dividends are often paid quarterly at a pre-announced rate that gives shareholders a predictable payout for that quarter, regardless of daily share price fluctuations. Dividends are taxed at your individual tax rate.
Keep in mind, however, that the payment of stock dividends is never guaranteed and may vary over time. This material is for educational purposes only and does not contend to address the financial objectives, situation or specific needs of any individual investor. It is not a solicitation or an offer to buy or sell any security or investment product.
Making the most of your retirement savings: Traditional or Roth: Which is right for you?
Helps investors understand the tax-deductible benefit of traditional IRAs and the tax-free earnings growth of Roth IRAs.
Divides comprehensive financial planning into six parts and includes a checklist of financial issues for considerations.
As stock prices go up, some investors may feel more optimistic — and more willing to buy stocks. Likewise, as stock prices go down, investors may feel more pessimistic and perhaps less inclined to buy stocks.
Source: This illustration is based on the Cycle of Market Emotions chart created by Christianna Wood at Westcore Funds/Denver Investment Advisors LLC, 1998.
Understanding the importance of allocation.
Exposes the myth perpetuated by the financial industry that it's more important to avoid missing the best days without consideration to the impact of missing the worst days.
A growing global middle class equals opportunities for US investors.
Explore the option of diversifying your portfolio to include international markets.
Introduces a better way to view asset class performance — by economic environment.
Once they made it to retirement, Bob and Julie had to change their game plans. A retiree's goals are more focused on preserving hard-earned savings while also generating enough income to fund a comfortable retirement. And once you begin making withdrawals from your savings, the sequence of returns can make a big difference.
Julie Blue's retirement portfolio started strong. Her returns, shown below, follow the actual returns of the S&P 500 Index from 1990 through 2010, taking into account an annual $50,000 withdrawal (indexed for inflation). As you may remember, the returns in the early '90s were stronger than returns since 2008. But even though Julie's portfolio took some major hits in 2002 and then again in 2008, strong early returns sustained her portfolio so that she ended up with more than twice what she started with — all while taking out money to live on each year.
During Bob Green's retirement, the market finished much stronger than it started. His returns are also based on the S&P 500 Index — but in the reverse order of Julie's. So Bob experienced early setbacks, not early gains. Therefore, even with strong market returns in later years, his portfolio never fully recovered.
Source: Lipper. Data as of Dec. 31, 2015.
For illustrative purposes only. Past performance does not guarantee future results. The S&P 500 is an unmanaged index considered representative of the US stock market. An investment cannot be made directly in an index. Each portfolio assumes a first year withdrawl that was subsequently adjusted for actual inflation, represented by changes to the historical Consumer Price Index (CPI). The table assumes annual withdrawls are taken at the end of each year. The (CPI) is a measure of change in consumer prices as determined by the US Labor of Statistics.
This is a hypothetical example with fictional characters. They do not necessarily represent the experiences of actual clients, nor do they indicate future performance.
The sequence of returns does matter when clients begin making withdrawls from their retirement accounts.
Discover the truth behind five common investment myths and five actions you can take toward building an intentional portfolio.


MINSK, 21 February (BelTA) – Belarus' accession to the World Trade Organization does not entail serious risks for the domestic manufacturing industry, Industry Minister Pavel Utyupin told reporters prior to a session of the ministry's board on 21 February, BelTA has learned.
"We do not expect any other major risks. We are already working to the WTO rules. All the risks are already behind," Pavel Utyupin said.
He meant Belarus' arrangements and commitments within the framework of the Eurasian Economic Union. "Respectively, de facto we are already in the WTO, while it has not been finalized de jure," the minister said.
He believes that the advantages of the WTO accession outnumber and outweigh the risks associated with it, because the WTO accession will give more legal instruments to protect the interests of Belarusian manufacturers when their products are not allowed to foreign markets.
As BelTA reported earlier, Belarus expects to join the WTO in 2020.


How we have helped others.
Below are few of the numerous success stories from successful satisfied clients, past and present.
A key component of the success of a small business is its branding and image package. The small business owner is faced with a daunting dilemma. How you project the uniqueness of your company within the constraints of a limited budget.
My solution was to hire Dave Pedersen of Pedersen Design, LLC. I had interviewed several of the local graphic art and design shops. Their offerings appeared to me to be flat and mostly looked the same.
Dave is a great listener. He combines this great gift with a refined aesthetic. After some probing conversation, he can usually produce a quite startling and spot-on representation of your business vision. It's truly a great moment when he brings your ideas to life.
Dave has won my continuing confidence to produce refined, timely and unique images to present my business in the best light. I am pleased to recommend Dave Pedersen for graphic art and design work.
I have been working with Pedersen Design for 6+ years. Dave and his team have created numerous marketing materials ranging from business cards, letterhead, brochures, and various print and digital ads. As a business owner, during all phases of work, you feel you are working with experienced professionals who are truly vested in helping improve your business.
Pedersen Design has been both a reliable and creative partner in nearly all aspects of our marketing and communications needs. They have successfully implemented our vision in nearly every medium we use in communicating to our industrial customers. Most of all, they truly understand our business and it shows in the work we proudly display on a global basis.
I have been working with Dave Pedersen for over ten years and have found his work to always be completed on time, within budget, and in a creative and meaningful way. Pedersen Design is easy to work with and has great suggestions for developing innovative and interesting marketing materials. They just recently completed a membership brochure for our association that uses infographics in a very colorful and easy to read brochure. I would highly recommend Pedersen Design for any and all marketing needs.
Pedersen Design is wonderful business partner. The Pedersen Team is creative, accommodating and a pleasure to work with. I believe that Pedersen's designs and concepts have helped us generate more interest in our business thus helping us generate more revenue. Thank You Pedersen Design!
The Metropolitan Fire Chiefs have been looking for an alternative to printing a Fire Chief Directory every two years that basically become outdated within a month of printing. Petersen Design came up with an online version that can be continuously updated and printed from the Metropolitan Fire Chiefs website. Since we haven't printed a directory in several years, this new option was a huge hit with the members. Petersen Design also created a new website for the organization that is nicely designed and very easy to use. Finally Petersen Design gave use a time frame that I thought was unrealistic, but they completed it on time and continue to update both the website and the directory in a very timely basis. It is nice to work with such professionals that take pride in what they do.
I recall the day I walked into your company's office and said, "I need a good website". Yours was the only company I considered to do the job. I mentioned there were varying types of quality in cars from Chevrolets to Bentleys which is my "dream car" and after some communication between us you said, "You're getting a Bentley". I'm pleased with the site and friendship that has come out of dealing with Pedersen Design. Although I'll never own a Bentley I have the one in the quality of my website and our relationship.
Dave Pedersen has been part of the Bicycle Commission for several years, and from day 1, has offered his professional services to our group. He is creative, easy going, easy to work with, and a marketing genius. He has developed many campaigns for us, from Bicycle Friendly Batavia Business, to redeveloping our bike maps, and planning/implementing a plan to use funding for Safe Routes to School by developing marketing concepts and all the physical materials we needed to use for promoting bike safety and bicycling in Batavia. In addition, I have worked with Karen and Pedersen Design on other campaigns, including the ones they do for the Batavia Fine Arts Centre. Again, they are incredibly creative and so easy to work with. Pedersen Design will always be my go-to business anytime I need help with marketing or advertising.
It is my pleasure to write a recommendation for Pedersen Design. Their marketing skills, enthusiasm and ideas have a been a great boost to the Batavia Bike Commission. Dave has come to the commission with a number of new ideas such as: the Great 8, Bike Friendly Businesses and town bike maps. His enthusiasm and marketing skills have made them become some of the stronger programs and marketing pieces that the commission does and the community sees. As a result of his design skills and experience it has raised the awareness of the community in the Bike Commission as well as really helped the commission become more visible. More importantly it has also helped us be more productive and have a higher bar of expectations. I would encourage your consideration in working with Pedersen Design: Dave is devoted to his work, a joy to work with and very enthusiastic about what he does.
I met with and interviewed Pedersen Design as a possible marketing designer for our startup company, and I instantly saw Dave and his staff's ability to present new and creative design ideas that I had never considered. Pedersen Design was able to complete our new company image on time and within our budget. I would highly recommend Dave and Pedersen Design to anyone looking for creative Digital Marketing.
Pedersen Design is the best. Dave is great to work with, professional and has great ideas. The artwork they produce is top quality and installation is quick. We will continue to use Pedersen Design for all of our needs. Superb and very highly recommended!
I was looking for a local person to handle the photography needs for all of our marketing communications. The nature of our business requires that we create several different designs (packages) to serve the various applications. So finding a local, responsive, knowledgeable, detail-oriented company that could handle our growing needs was important. We found that in Pedersen Design.
The entire team at Pedersen Design continuously listen carefully to my vision and deliver a product that causes me to sit back in my leather chair and try to decide if I really live up to the image they put forward. They capture the essence of my organization in a way that motivates me to be a better company, and allows me to to absorb the depth of my accomplishments. I would recommend Pedersen Design to any company of any size – you will not be disappointed!


Up to now, I think that the image such as New Year's cards, corporate documents, school reports, etc. are printed on paper as most printers.
However, in the last few years the scope of what the term printer is defined has been expanded.
That's because "3D printers" for making "solid" ones have appeared.
How does 3D printer become three dimensional?
It is faster to actually see it in movies, but what 3D printers share in common is to make shapes into 3 dimensional solid by stacking materials of 2 dimensions or planes little by little .
Also, in recent years technology has developed and the noise etc. in the making process has been reduced to a great extent.
Also, since the price is also cheap, its hurd is falling to purchase even for the general public.
The number of places to actually introduce is increasing even at schools and small business establishments (like my place).
For that reason, "making things" themselves have become more and more familiar.
There may have been an image of "MONODZUKURI = Factory", but that is changing drastically.
An era in which various things can be easily made easily in a few places in everyday life is coming so far.
The above is a manufacturing café located in Sapporo, but this way manufacturing becomes more and more familiar.
I myself thought that it was only plastic accessories for small things that can be made with a 3D printer the other day.
However, now 3D printers that are active in various fields are appearing.
Large ones to be used at construction sites, things for medical use that can be used at hospitals, and foods and foods can also be output with 3D printers.
Now, do not you feel that the life of the future will change dramatically just by introducing it easily?
The utilization of the 3D printer itself has just begun.
For the general public it may be felt that the threshold is high, but it seems to become more and more familiar.
This is a premonitory trend! What is "Tiny Whoop" that you can enjoy drone racing at home?


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Top 4 Music Acts In Dublin Tonight!
By	Craig Marshall	/	November 7, 2016
Dublin has the honor of hosting some of the top names in music from around the world on a nightly basis! Tonight there are some big stars that fit a genre for everybody whether you're young or old. Here is a list of the Top 4 artists that are worth checking out tonight.
Oh Pep!
Olivia Hally and Pepita Emmerichs make up the folk/pop group that is Oh Pep! The two met in college back in 2009 and began writing songs together before they became Oh Pep! There music can be classified as folk/pop/country attributing to musical influences like Johnny Cash, Bob Dylan, Bright Eyes, and Arcade Fire. The duo are touring to promote their first studio album title Stadium Cake.
Location – Whelan's
Time – 20:00
Admission – €15
Jimmy Eat World most likely will strike a chord for every 90's or millennial kid. There song The Middle, was an anthem for teenagers and the band's first big commercial success. The band is a mix of alternative, pop, and punk rock that will for sure give you an energetic show. Jimmy Eat World just released their ninth studio album titled Integrity Blues just last month!
Location – Olympia Theatre
With over 60 years of experience, grammy award winning country superstar Kenny Rogers graces Dublin tonight only. While the successful country artist has had a great deal of international fame he is throwing in the towel and celebrating by touring on his Final World Tour. Don't miss this opportunity to witness the legendary Kenny Rogers.
Location – 3Arena
Admission – €45/€90
Donal Kirk & Anto Drennan
Vocalist Donal Kirk and guitarist Anto Drennan team up this Monday at JJ Smyth's to perform a series of covers. There styles cover a wide array of artists like Stevie Ray Vaughn to Crosby Stills Nash to The Beatles. Guitarist Anto Drennan is noted for his involvement with Genesis as well as other top name performers.
Location – JJ Smyth's
Time- 21:30
Admission – Free
Chromesthesia: Listening to Music in Colour
Jan 17, 2021 / By	Melanie Romero
Disco revolution: how disco liberated society in the 1970s
Jan 15, 2021 / By	Rachel McLoughlin
Modern Music Preserves Cultural Heritage for Future Generations
Jan 15, 2021 / By	Meladi Brewer
Craig Marshall

redpajama_set_name: RedPajamaCommonCrawl


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Monitors Review
Monitor TV Reviews
Predator XB271HU Vs. XB270HU
Acer has launched their Predator series, which are some of the best gaming monitors in the market. In the Acer Predator series, there are plenty of monitors; therefore, the confusion is apparent. You cannot buy a monitor without having proper knowledge about what you are investing your money in, and to save you from that, we brought you a brief comparison guide between the Predator XB271HU Vs. XB270HU.
Comparison Guide Between Predator XB271HU Vs. XB270HU
Both the monitors of the Predator series have a 27-inch display. Even though there are not many significant differences in terms of design and build of the two monitors, you can still see some improvements.
The Acer XB270HU is a rectangular structured monitor that has a sleek design. The design of this model of the series by Acer is one of the best that you can find in the market, and it is incredibly stylish, which nowadays is a major buying factor. The stand of the monitor is flexible; you can tilt, swivel and change the position. On the other hand, the Acer XB271HU has rounded edges with thin bezels. The overall structure of this Acer predator model is one of the best to create a gaming setup. It has a bipod stand that offers a grip on the overall structure. In short, this monitor gives the best look.
As we mentioned earlier, both Predator XB271HU Vs. XB270HU have a 27-inch display that can make you think about the difference in screen size that we will discuss here. Even though the screen size of both monitors is the same, the significant difference is the bezel.
The bezel in a monitor or laptop plays a significant role in giving a sleek and attractive look to the device and ensuring the amount of space available on the screen to see the images. In Predator XB270HU, you don't get a bezel-free screen. Bezel covers a lot of space on the monitor. However, in Predator XB271HU, the screen-to-bezel ratio is less, giving you better visuals and making the screen size big and impressive compared to the Acer Predator XB270HU. So in the comparison category, Acer Predator XB271HU takes the lead.
The following comparison category is the display panel. Display panels are of various types, such as VA, TN, LED, IPS, etc., and each panel has flaws and advantages in offering great visuals and colors to the users. In the display panel category, both Predator XB271HU and XB270HU have the same screen panels that are IPS. So, there's no difference as far as this category is concerned.
The IPS display of both monitors gives outstanding color reproduction. Therefore both monitors are an excellent choice for competitive gaming and higher and graphic work. In this category, there is a tie between the Predator XB271HU Vs. XB270HU.
Since the Acer Predator series is specially built for gaming purposes, we cannot leave out the comparison of the two monitors based on the refresh rate. Refresh rate is a significant aspect that makes the gaming performance smooth and exhilarating. If you see at the end, you will realize that both the models of the Acer Predator are pretty similar to each other, and the small thing that differentiates them from each other is the refresh rate. Both monitors come with 2K displays, but Predator XB271HU has a 165 Hz refresh rate, and Predator XB270HU comes with 144 Hz.
A 144 Hz refresh rate is impressive and widely used in most gaming monitors, but when you compare it with 165 Hz, you have to give an upper hand to the latter. If you are looking for a gaming monitor that comes with an excellent refresh rate, then Predator XB271HU is the best choice.
Often gaming monitors do not have good built-in speakers, due to which gamers use external audio speakers while playing games. You want to experience the outstanding quality of an inbuilt speaker in a monitor; the Predator XB271HU is the one you should go for.
For a good quality sound, manufacturers have added 2×2w speakers in this model of Acer Predator, which, unfortunately, you do not get in the Predator XB270HU. So, the clear winner in this category is also the Predator XB271HU.
Weight and Power Consumption
Weight is not a significant part as it is in a laptop because you do not carry a monitor from one place to another in your bag. But still, if we have to look at the weight, then Predator XB270HU weighs 5kgs, and Predator XB271HU weighs around 7kgs.
As far as the weight is concerned, the former is lighter in comparison with Predator XB271HU. However, in terms of power consumption, we have to give an upper hand to the Predator XB271HU because it consumes less power than the Predator XB270HU.
Acer Predator XB271HU
IPS display panel
Impressive color reproduction
Outstanding build and structure
Anti-Tearing technology
Two inputs
Less input lag
Outstanding color reproduction
Nvidia G-sync technology
Slow response time
Average stand
Bad bezel to screen ratio
Which One to Choose?
This article discussed the difference between the two models and which one is better than the other in which category. Both have an IPS display which is excellent for a 2K gaming experience and streaming purpose.
But when you see Predator XB271HU and XB270HU from a larger angle, you have to agree that Predator XB271HU is better than the other model. You can have a high refresh rate for a smooth gaming experience. The calibration of this monitor is outstanding, and with less power consumption, you can use it for as long as you want.
Brand ACER
Resolution QHD 2560×1440
Screen Size 27-Inch
Specific Use Gaming, Personal
Price Under $400
The Acer Predator series is known for outstanding monitors with a high level of specifications that make the gaming experience all the more enhanced and enjoyable. Predator XB271HU and XB270HU are the two best models of this series.
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Prinz Kaspian von Narnia bzw. Wiedersehen in Narnia bzw. Die unverhoffte Wiederkehr oder: Prinz Kaspian (engl. Originaltitel Prince Caspian) ist ein Roman des britischen Schriftstellers C. S. Lewis und erschien 1951 als Teil der Reihe Die Chroniken von Narnia.

Obwohl als zweites Buch geschrieben, ist Prinz Kaspian von Narnia nach C. S. Lewis' Lesart der vierte Roman der Reihe.

Inhalt
Das Buch erzählt die zweite Geschichte von Peter, Susan, Edmund und Lucy. Sie waren ein Jahr lang nicht mehr in Narnia, während dort jedoch 1300 Jahre vergangen sind. Gerade auf dem Weg in die Schule – sie stehen auf einem Bahnhof –, finden sie sich plötzlich in den Ruinen ihres alten Schlosses Cair Paravel (in früheren Übersetzungen "Feeneden") wieder, wo sie einen Zwerg namens Trumpkin treffen. Dieser erzählt, dass die Telmarer, ein Menschenvolk aus dem Westen, Narnia erobert und die alten Bewohner Narnias vertrieben hätten. Zur Zeit herrscht der unrechtmäßig an die Macht gekommene Miraz, dessen Neffe Kaspian, eigentlich der rechtmäßige Herrscher wäre. Nachdem ein Sohn von Miraz geboren wurde, flüchtet Kaspian mit seinem zwergenstämmigen Lehrer, da er nun befürchtet, von Miraz umgebracht zu werden.

Zusammen starten sie eine Revolution der Alt-Narnianen und beginnen anschließend einen offenen Kampf gegen die Telmarer. Da die Situation jedoch hoffnungslos aussieht, benutzt Kaspian das alte Zauberhorn von Susan, um die vier Kinder zurückzuholen. Gemeinsam mit Trumpkin machen sich die Kinder auf den gefährlichen Weg zum Lager der Narnianen, während um sie die alten Geister des Waldes wieder in Erscheinung treten.

Und auch Aslan erscheint wieder. Zuerst können die Kinder ihn nicht sehen, nach und nach aber schlüpfen sie in ihre alten Rollen und erblicken ihn damit, die Jüngste unter ihnen, Lucy, zuerst, Susan zuletzt. Mit der Hilfe der Alt-Narnianen kämpfen die Kinder, Prinz Kaspian und Aslan gegen Miraz. Im Zweikampf besiegt Peter zuletzt Miraz, Kaspian wird König. Am Ende erfahren Peter und Susan, dass sie nun nicht mehr nach Narnia kommen dürfen, da sie schon zu alt seien.

Christliche Deutungen
Im Buch Prinz Kaspian von Narnia können fünf wesentliche christliche Themen identifiziert werden:
 Der Konflikt zwischen Gut und Böse
 Gottes planvolles Lenken (Voraussicht und Versorgung)
 Glauben (der Zusammenhang zwischen Vertrauen und Gotteserkenntnis)
 Jüngerschaft (Nachfolge)
 Gottes Gegenwart (in Reden und Handeln)

Der Konflikt zwischen Gut und Böse
Der Kampf zwischen Gut und Böse spitzt sich in besonderer Weise in den verborgenen und verwinkelten Gängen und Räumen in Aslans Hügel zu. Hier wird der Bezug zum Christ-gleichen Sieg Aslans über den Tod in Der König von Narnia hergestellt.

Gottes planvolles Lenken
Auf ihrem Weg hin zu Prinz Kaspian werden menschliche Vernunft und kindliches Gottvertrauen einander gegenübergestellt. Allein letzteres führt zum Erfolg, während das Vertrauen auf die eigene Vernunft katastrophal schiefläuft, als die fünf auf ihrem Weg zu Aslans Hügel in einen Pfeilhagel von Miraz Soldaten geraten. Ein konkreter Bezug besteht hier zu .

Glauben
Im Glauben geht es nicht allein darum, die Existenz Gottes, bzw. Aslans hier in der Geschichte für wahr zu halten, sondern vor allem auf Gottes Hilfe zu vertrauen. Dieses unerschütterliche Vertrauen wird denn auch am Ende des Romans belohnt, als die Kinderfrau Kaspians endlich Aslan begegnet. Hier kann auch ein Bezug zum Glauben Abrahams genommen werden, wie er in  beschrieben wird.

Jüngerschaft
Einen besonderen Raum nimmt die Treue in der Nachfolge ein, die Aslan von Lucy ohne Rücksicht auf die Reaktion ihrer Geschwister einfordert. Und Lucy ist nach anfänglichem Scheitern gewillt, sie zu geben, jedoch zunächst ohne zu wissen, ob sie überhaupt dazu im Stande ist. Der Evangelist Johannes schreibt hierzu in : "Wer meine Gebote hat und sie hält, der ist es, der mich liebt; wer mich aber liebt, wird von meinem Vater geliebt werden und auch ich werde ihn lieben und mich ihm offenbaren." Und so ist es zunächst auch Lucy allein, die Aslan sehen kann.

Gottes Gegenwart
Die Parallelen zwischen Roman und Bibel sind in diesem Roman weniger vordergründig, als in Der König von Narnia. Hier sind sie hingegen stärker in der Person Aslans selbst, also ihrem Reden und Handeln zu ziehen. Das gilt für die Ehrfurcht, mit der die vier englischen Schulkinder und narnianischen Könige ihm als dem wahren Richter begegnen, wie auch für Aslans vergebendes, heilendes und wiederherstellendes Handeln.

Verfilmungen

 1989 – Die Chroniken von Narnia: Prinz Kaspian & Die Reise auf der Morgenröte ("Prince Caspian and the Voyage of the Dawn Treader"), Fernsehrealfilmserie,  die ersten beiden Episoden.
 Am 16. Mai 2008 (in Deutschland am 22. Juli 2008) erschien die Kinoverfilmung Die Chroniken von Narnia: Prinz Kaspian von Narnia ("The Chronicles of Narnia: Prince Caspian"); unter der Regie von Andrew Adamson spielen Ben Barnes, Georgie Henley, William Moseley, Anna Popplewell und Skandar Keynes die Hauptrollen.

Ausgaben

 Die Chroniken von Narnia Band 4: Prinz Kaspian von Narnia; ISBN 978-3-86506-084-6
 Die Chroniken von Narnia (gebundene Gesamtausgabe); ISBN 3-8000-5186-9

Literatur

Weblinks

 Zusammenfassung der Handlung

Einzelnachweise

Literarisches Werk
Literatur (Englisch)
Literatur (20. Jahrhundert)
Kinder- und Jugendliteratur
Fantasyliteratur
Narnia
Werk von C. S. Lewis
Roman, Epik

redpajama_set_name: RedPajamaWikipedia