text:
Contraception gave women 'a completely new freedom': the latest from the German bishops' website
Oh, and gay sex is a 'fundamental form of human sexuality.'
Fri Sep 18, 2015 - 4:40 pm EST
By Maike Hickson
Follow Maike
September 18, 2015 (LifeSiteNews) – The day before publishing a theology student's apologia for same-sex "marriage," the German bishops published an article lauding the contraceptive pill for giving women a "completely new freedom."
The article, an interview with the German Church historian Arnold Angenendt, appeared September 16 on the website of the German Bishops' Conference, katholisch.de. In this interview, the historian praises the invention of contraceptives: "The invention of the pill then was the decisive revolution. Since then, the women do not have to fear becoming pregnant at each sexual intercourse. That gives them a completely new freedom [...]"
Angenendt puts this claim in certain contexts, saying that, in former times, sexuality was only regarded as a means for having children. But, he says, the Second Vatican Council "made a change there and considered sexuality also as a possibility for strengthening the personal bond between the partners." This German historian – promoted now by the German Bishops – also surprisingly claims the idea that any sexual activity outside of marriage is sinful comes from a "medieval conception" whereby any spoiling of the male sperm was considered to be sinful: "Masturbation, homosexuality, or contraception were considered to be an act of murder. In the light of today's biological knowledge, this is a strong misjudgement." He also claims that modern science has proven that the homosexual orientation has to be regarded as "its own anthropologically given and fundamental form of human sexuality, just as it is in the case of heterosexuality. Thus, one cannot describe homosexuality as unnatural."
With reference to the Book of Genesis, Angenendt claims that it is not based on any scientific reality, but, rather, should be called "a myth which describes a statement of faith." And, he concludes: "Therefore, nothing speaks against our re-assessment of homosexuality, as well."
This is not the first time that this German professor proposes a re-assessment of the Church's doctrine. In 2011, Professor Angenendt published an article in the German newspaper Süddeutsche Zeitung, entitled "The Church's Fears About Sexuality," where he claims that Jesus wanted to overcome the principles that promoted celibacy, saying: "Celibacy was based upon an archaic cult of purity which Jesus wanted to overcome."
catholic, contraception, german catholic bishops
Contraception gave women 'a completely new freedom': the latest from… Contraception gave women 'a completely new freedom': the latest from the German bishops' website Opinion By Maike Hickson


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\section{Introduction}

\noindent In recent years, pioneering research have been conducted into the application of machine learning to computational physics and engineering contexts: example works include \cite{RAISSI2017683, RAISSI2018125, RAISSI2019686, Rudye1602614, Brunton3932, Lusch_2018}. As a result, new sub-fields within the computational sciences have emerged, including, but not limited to physics-informed machine learning \cite{nature_karniadakis} and scientific machine learning (Sci-ML) \cite{osti_1478744}. Within these sub-fields, the development of machine learning based methods to infer the parameters of dynamic system governing equations, and/or the discovery of new partial differential equations (PDE) has attracted significant attention and important early success. In such early work, different inference strategies have been proposed. A popular method, known as \textit{SINDy} \cite{Brunton3932} has its foundations in building a dataset of spatial derivatives that are potentially involved in the governing equation of the observed physics, in order to perform a sparse linear regression aimed at estimating the coefficients of each derivative term in some latent PDE. This seminal work was later extended in \cite{Rudye1602614}, where the PDE derivative terms were computed either through polynomial interpolations or finite differences, and in \cite{Xu_2021, XU2021110592, Both_2021} using neural networks. A major advantage of these methods is the interpretability: the spatial derivatives involved in the PDE and their coefficients are discovered explicitly. Other methods directly approximate the differential operator using a physics-informed neural network (PINN) \cite{RAISSI2019686}. While these methods are generally highly accurate, they lack interpretability \cite{JMLR:v19:18-046} or only allow for recovering the coefficients of a PDE with the functional form already known \cite{RAISSI2019686}.
\\\\
\noindent The sparse regression based methods outlined earlier have been providing promising results, but the regression system is often poorly conditioned and the spatial derivatives estimations must be highly accurate to provide satisfying results. In \cite{Xu_2021, XU2021110592, Both_2021}, measured data from a physical system are interpolated using a neural network, and then differentiated to create the spatial derivative dataset. This requires abundant data with as little noise as possible. In practical engineering and scientific applications however, acquiring enough data (which is typically experimental measurements) may be a very expensive and time consuming process. While deep and dense neural networks are able to capture complex physics described in snapshots of physical system dynamics (e.g. shocks or sharp gradients), they are also more likely to overfit noisy measurements. Unfortunately, methods for fine tuning the networks parameters (e.g. cross validation) may be inapplicable due to the lack of abundant data. Consequently, there is a trade-off: Use more data for training and potentially obtain better derivative estimations, but with a significant risk of overfitting, or use more data for testing and limit overfitting, but with a greater risk of missing complex underlying physics.
\\\\
\noindent Bayesian methods, in particular Bayesian neural network (BNNs) \cite{10.5555/525544, 10.5555/1162264, Goan_2020} are promising for avoiding this trade-off, as they a naturally less prone to overfitting, even with very noisy and sparse data. Furthermore, BNNs provide confidence intervals on their prediction, which can be used for improving the accuracy of PDE recovery techniques. In the field of PDE recovery, Bayesian machine learning methods have been used to infer parameters of governing equations with known functionnal form. For example, \cite{RAISSI2017683} used Gaussian processes, and \cite{Yang_2021, MENG2021110361} used BNNs combined with PINNs to infer PDE parameters, but do not permit the discovery of unknown PDEs. In this paper, we extend the original approach outlined in \cite{Brunton3932, Rudye1602614, Xu_2021, Both_2021} in two manners. First, we propose to use a BNN to interpolate the snapshot measurements from the physical system. Relying on BNNs offers two major advantages: It makes the model more robust to noise and sparse data (allowing for minimal tuning of the model's parameters) and provides valuable confidence intervals over the interpolation predictions. Secondly, we propose to use the BNN confidence intervals to quantify the uncertainty over the spatial derivative dataset, and introduce this uncertainty into a sparse Bayesian linear regression model to recover the PDE coefficients.

\section{Bayesian Neural Network}
\noindent In a BNN, the weights $\textbf{w}$ are assumed to be sampled from a prior probability distribution, here taken as a Gaussian with a 0 mean and unit standard deviation:
\begin{equation}
\label{prior}
    \textbf{w}\sim\mathcal{N}(0, I)
\end{equation}
The posterior over the weights is:
\begin{equation}
    p(\textbf{w}|X,y)=\frac{p(y|X,\textbf{w})p(\textbf{w})}{p(y|X)}
\end{equation}
Due to the highly non-linear nature of a neural network, the likelihood is a very complex function and sampling from the posterior is analytically intractable. We can either approximate the posterior with e.g. a Gaussian distribution and find the parameters of the Gaussian by minimizing the KL divergence (variational inference) \cite{NIPS2011_7eb3c8be, Goan_2020, jospin2020handson, Blei_2017}, or use Monte-Carlo sampling \cite{10.5555/525544, cobb2020scaling, jospin2020handson, Goan_2020}. Variational inference for BNNs is generally inefficient and often fails to provide meaningful confidence intervals \cite{Goan_2020, Blei_2017, pmlr-vR5-wang05a, turner_sahani_2011}. Here, we rely on Hamiltonian Monte-Carlo (HMC) instead.\\\\
\noindent The intuition behind HMC is to flip the posterior probability distribution up-side-down and assimilate the regions of high probability as pits. If a virtual particle is placed on the flipped distribution and starts rolling freely, it will naturally go towards the regions of lower potential energy (pits), i.e. regions with higher probability. HMC has two steps. First it simulates the motion of such particle using Hamiltonian physics and records the position over time, which provides a set of sample candidates, and then uses Metropolis Hastings to either accept or reject these samples. The Hamiltonian step generally provides suitable samples and allows for a higher acceptance rate than other MCMC methods \cite{10.5555/525544}.

\section{Inversion Framework for PDE Recovery and Discovery}

\noindent We consider $n$ measurements from a given physical phenomenon of the form inputs-outputs $\mathcal{D}=\{X_i,y_i\}_{i\in[\![1,n]\!]}$, where the input is a time-space coordinate and the output is a corresponding measured physical quantity (potentially noisy). All the data is generated from numerical solutions of PDEs using either a standard $3^\text{rd}$ order Runge-Kutta (heat equation, section \ref{heat}), or Chebyshev polynomials with Runge-Kutta time integration (Burgers equation and Korteweg-De Vries equation, section \ref{burgers} and \ref{kdv}) using the \textit{Chebfun} matlab package \cite{Driscoll2014}.\\

\noindent Unless specified otherwise, we assume 16 measurement sensors placed at random space locations that record the physical quantity of interest over time with stepping $\Delta t$. The input data has the form $X_i=(t_i,x_i)$ where $t_i$ is time and $x_i$ is the sensor Cartesian coordinate, and the output is $y_i=u(t_i,x_i)+\epsilon$, where $u$ is the PDE solution and $\epsilon$ is noise corrupting the data, we consider three cases: $\epsilon=0$ (noiseless), $\epsilon\sim\mathcal{N}(0,0.01^2)$ and $\epsilon\sim\mathcal{N}(0,0.05^2)$.\\

\noindent The general inversion framework for PDE recovery is outlined in the following subsections.

\subsection{Bayesian Neural Network Fitting}
\noindent  We first fit the measurement data using a BNN $f(t,x|\mathbf{w})$. In the following sections, the BNN indifferently takes the same architecture: 4 fully connected hidden layers of 50 hidden units each, with $\tanh$ activation functions. Unless specified otherwise, the prior over each neural network weight is a standard Gaussian distribution (zero mean and unit standard deviation). The noise standard deviation in the likelihood function is taken as $\sigma=0.01$. In the HMC sampling we take $n_s=6000$ samples with a time stepping $\Delta \tau=5\cdot10^{-4}$.
\subsection{Surrogate Derivative Dataset}
\noindent Once the BNN is properly trained, we sample a set of $n_d$ time-space coordinates randomly: $(t_j,x_j)\sim \mathcal{U}[\Omega]$ where $\Omega$ is the time-space domain and $\mathcal{U}$ the uniform distribution. The BNN is differentiated multiple times with respect to space and time and evaluated at each input $(t_j,x_j)_{j\in[\![1,n_d]\!]}$. The derivatives are then averaged over the set of weight samples. For example, for the time derivative we have:

\begin{equation}
    \mathbb{E}[f_t]_j=\frac{1}{n_s}\sum_{s=1}^{n_s}\frac{\partial f(t_j,x_j|\mathbf{w}_s)}{\partial t}\hspace{1cm}\mathbf{w}_s\sim p(\mathbf{w}|X,y)
\end{equation}

\noindent Using this approach, we build a surrogate dataset of successive derivatives in space $\tilde{X}$ potentially comprised in the underlying governing PDE. The spatial derivative orders included in $\tilde{X}$ is arbitrary and we may include non-linear terms as well (the total number of derivative candidates is defined as $n_c$). Similarly, we compute the corresponding time derivative output vector $\tilde{y}$. We also compute the variance of each spatial derivatives and sum them up to quantify the uncertainty over each surrogate derivatives (covariance matrix $\tilde{\Sigma}$).

\begin{equation}
\Tilde{X}=
    \begin{pmatrix}
        \Tilde{X}_1\\
        \vdots\\
        \Tilde{X}_d\\
    \end{pmatrix}_{[n_d\times n_c]}
\end{equation}
\begin{equation}
\Tilde{X}_j=
\Big(
    \mathbb{E}[f]_j\,, \mathbb{E}[f_x]_j\,, \dots\,, \mathbb{E}[f_{xxxx}]_j\,,\mathbb{E}[ff_{x}]_j\,,\mathbb{E}[ff_{xx}]_j\Big)_{[1\times n_c]}
\end{equation}
\begin{equation}
\label{Sigma}
\tilde{\Sigma}=
\begin{pmatrix}
    \tilde{\sigma}^2_1 & \dots & 0\\
    \vdots & \ddots & \vdots \\
    0 & \dots & \tilde{\sigma}^2_d\\
\end{pmatrix}
\end{equation}
\begin{equation}
\tilde{\sigma}^2_j=\mathbb{V}[f]_j+\mathbb{V}[f_x]_j\dots+\mathbb{V}[f_{xxxx}]_j+\mathbb{V}[ff_x]_j+\mathbb{V}[ff_{xx}]_j
\end{equation}
\begin{equation}
\tilde{y}=\Big(\mathbb{E}[f_t]_1\,, \dots\,,\mathbb{E}[f_t]_d\Big)^\top
\end{equation}

\subsection{Bayesian Linear Regression Fitting}
\noindent The surrogate dataset of derivatives is fitted with a Bayesian linear regression (BLR) in order to approximate the coefficients associated with each derivative. This is a sparse regression problem and it is expected that the set of actual derivatives involved in the ground true PDE should be much smaller than the entire set of surrogate derivatives. Hence most coefficients should be 0. Therefore, a BLR with Gaussian priors should be well suited as it will naturally skew most coefficients towards 0. Furthermore, a BLR can provide confidence intervals over the recovered PDE coefficients and since the uncertainty over each surrogate derivatives is well quantified through $\Tilde{\Sigma}$, it will automatically discard the influence of highly uncertain derivatives. The regression model can be written as follows:
\begin{equation}
\label{blr_model}
    \tilde{y}_j=\tilde{X}_jc+\epsilon_j\hspace{1cm}
    \begin{cases}
        \epsilon_j\sim\mathcal{N}(0,\eta\tilde{\sigma}_j^2)\\[2pt]
        c\sim\mathcal{N}(0,\Gamma)
    \end{cases}
\end{equation}

\noindent Note that in a state-of-the-art BLR with Gaussian noise, the variance of $\epsilon_j$ corresponds to the assumed variance of the output $\tilde{y}_j$. However, here the variance of the spatial derivatives are more informative, so we introduce the scaled variance of the inputs $\tilde{X}$ instead. Equation \ref{blr_model} leads to the posterior distribution over the PDE coefficients:

\begin{equation}
p(c|\Tilde{X}, \Tilde{y})= \frac{p(\Tilde{y}|c,\Tilde{X})p(c)}{p(\tilde{y}|\tilde{X})}=\frac{\mathcal{N}(\Tilde{y}|\Tilde{X}c,\Tilde{\Sigma})\mathcal{N}(c|0,\Gamma)}{\displaystyle \int \mathcal{N}(\Tilde{y}|\Tilde{X}c,\Tilde{\Sigma})\mathcal{N}(c|0,\Gamma) \text{d}c}
\end{equation}

\noindent $\Gamma$ is a set of hyperparameters that maximize the marginal log-likelihood $\log p(\Tilde{y}|\Tilde{X})$. The coefficient of each derivative can be taken as the mean value $\mathbb{E}[c]$ with respect to the posterior (with confidence $\mathbb{V}[c]^{1/2}$). Here, Bayesian inference is analytically tractable and the expected values of interest can be computed exactly.

\subsection{Error Metric and Computer Implementation}

\noindent For assessing the accuracy of our models, we use the $\ell^2$ norm between the ground true and the recovered vector of PDE coefficients:
\begin{equation}
    e=|\!|\mathbb{E}[c]-c_\text{true}|\!|_2
\end{equation}

\noindent For baseline comparison, a standard deep neural network (DNN) is also trained with the same architecture as the BNN, using a mean-squared error loss with a learning rate $\alpha=2\cdot 10^{-4}$ and $30000$ iterations. We also compare BLR with a standard linear regression with and without Lasso regularization and surrogate derivatives variance weighting.
\\\\
\noindent To implement our neural network models we use \texttt{PyTorch} \cite{NEURIPS2019_9015}. HMC sampling is performed using the \texttt{Hamiltorch} add-on library \cite{cobb2020scaling} and the linear regressions for recovering PDE coefficients are performed using \texttt{scikit-learn} \cite{scikit-learn}.

\section{Application}
\subsection{\label{heat} Heat Equation}
\noindent Let us consider the following 1D heat equation:
\begin{equation}
    \frac{\partial u}{\partial t}=\nu\frac{\partial^2 u}{\partial^2 x}
\end{equation}
\begin{equation}
\begin{cases}
    \,u(t=0,x)=u(t,x=0)=0\\[2pt]
    \,u(t,x=1)=\sin(t)
\end{cases}
\hspace{1cm}
\begin{cases}
    \,(t,x)\in\Omega=[0,10]\times[0,1]\\[2pt]
    \,\nu=0.1
\end{cases}
\end{equation}
The temperature is recorded every $\Delta t=0.2~s$ for each 16 sensors ($n=800$ training points), the ground true solution is shown in figure \ref{fig1}. The predictions of the BNN and DNN are shown in figure \ref{fig2}.
\\\\

\FloatBarrier

\begin{figure}[!h]
\centering
  \centering
  \includegraphics[width=0.7\textwidth]{Heat1.pdf}
  \caption{Ground true solution for the heat equation (black dots represent measured time-space points)}
  \label{fig1}
\end{figure}

\FloatBarrier

\begin{figure}[!h]
\centering
  \centering
  \includegraphics[width=1\textwidth]{Heat2.pdf}
  \caption{BNN predictive mean and standard deviation, DNN prediction, and absolute error between the BNN predictive mean, DNN and ground true - Heat equation}
  \label{fig2}
\end{figure}

\FloatBarrier

\begin{figure}
\centering
  \centering
  \includegraphics[width=1\textwidth]{Heat3.pdf}
  \caption{Predictions of the BNN (solid blue line $95\%$ confidence intervals) and DNN (solid green line) compared to the ground true (dashed red line) at $t=1~s$, $3~s$ and $6~s$. The black circles represent sensor measurement data - Heat equation}
  \label{fig3}
\end{figure}

\FloatBarrier

\noindent Table \ref{table1} shows the RMSE between the BNN predictive mean, DNN prediction and ground true over 1000 random input points and figure \ref{fig3} represents the BNN and DNN predictions at different times. In this specific case, it appears that even in the noisy cases, the standard DNN doesn't overfit too much, most definitely because heat diffusion is a very smooth process. We now generate the dataset of surrogate derivatives with $n_d=100$ derivatives and test different linear regression models: ordinary least-squares regression, Lasso regression and Bayesian regression. For the BNN surrogate derivatives, we test both with variance weighted loss and without (in the later case the covariance matrix $\tilde{\Sigma}$ in equation \ref{Sigma} is replace by a scaled identity matrix).
\\\\
\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & BNN  & DNN \\\hline
$\epsilon=0$ & 0.0125 & \textbf{0.0099}\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & 0.0130 & \textbf{0.0114}\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & \textbf{0.0208} & 0.0277\\\hline

\end{tabular}
\caption{\centering\label{table1}RMSE of the Bayesian and Standard Deep Neural Network}
\end{table}

\noindent First we consider the case where the functional form of the PDE is known (i.e. heat equation), and we want to infer the heat diffusivity of the system. Thus, the only derivative in the input derivative dataset is the second derivative in space. The recovered diffusivity coefficients for the different models is shown in table \ref{table2}, \ref{table3} and \ref{table4}. The BNN does a good job at recovering the ground true heat diffusivity and clearly outperforms the DNN model. Introducing the derivative variance in the linear regression helps achieving better accuracy (table \ref{table2} and \ref{table3}).
\\\\

\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & Least Squares & Bayesian Regression\\\hline
$\epsilon=0$ & $u_t=0.0701u_{xx}$ & $u_t=0.0667u_{xx}$ \\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=0.0524u_{xx}$ &  $u_t=0.0485u_{xx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=0.0175u_{xx}$ &  $u_t=0.0423u_{xx}$\\\hline
True & $u_t=0.1000u_{xx}$ &  $u_t=0.1000u_{xx}$\\\hline
\end{tabular}
\caption{\centering\label{table2}Recovered heat equation using DNN}
\end{table}

\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & Least Squares & Bayesian Regression\\\hline
$\epsilon=0$ & $u_t=0.0459u_{xx}$ & $u_t=0.0832u_{xx}$ \\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=0.0432u_{xx}$ &  $u_t=0.0534u_{xx}$ \\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=0.0492u_{xx}$ &  $u_t=0.0640u_{xx}$ \\\hline
True & $u_t=0.1000u_{xx}$ &  $u_t=0.1000u_{xx}$\\\hline
\end{tabular}
\caption{\centering\label{table3}Recovered heat equation using BNN (no variance weighing)}
\end{table}

\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & Least Squares & Bayesian Regression\\\hline
$\epsilon=0$ & $u_t=0.0928u_{xx}$ & $u_t=0.1000u_{xx}$ \\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=0.0925u_{xx}$ & $u_t=0.1040u_{xx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=0.0894u_{xx}$ & $u_t=0.0808u_{xx}$\\\hline
True & $u_t=0.1000u_{xx}$ &  $u_t=0.1000u_{xx}$\\\hline
\end{tabular}
\caption{\centering\label{table4}Recovered heat equation using BNN (with variance weighing)}
\end{table}

\FloatBarrier

\noindent We now consider the case where the underlying PDE is not known and we want to find the spatial derivatives involved in it. We assume spatial derivatives up to order 4, and 3 non-linear derivatives as well. The discovered PDE for the different models is shown in table \ref{table5}, \ref{table6}, \ref{table7}, and \ref{table8}. For every noise cases, the BNN combined with the variance weighted BLR (table \ref{table5}) outperforms all the other baseline models. For larger noise, it overestimates slightly the influence of the $0^\text{th}$ order and $1^\text{st}$ order spatial derivatives, but still recovers the ground true diffusivity well, and do much better than the other models (table \ref{table6}, \ref{table7} and \ref{table8}).

\begin{table}
\hspace{-2.5cm}
\footnotesize
\begin{tabular}{c|c|c}
Noise & Recovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=-0.0047u-0.0008u_x+0.0959u_{xx}+0.0000u_{xxx}+0.0001u_{xxxx}+0.0077uu_{x}-0.0012uu_{xx}$ & $\mathbf{0.0100}$ \\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.0045u+0.0201u_x+0.0898u_{xx}+0.0004u_{xxx}+0.0001u_{xxxx}-0.0010uu_{x}+0.0026uu_{xx}$ & $\mathbf{0.0232}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.0147u+0.0265u_x+0.0996u_{xx}-0.0025u_{xxx}+0.0002u_{xxxx}+0.0007uu_{x}-0.0283uu_{xx}$ & $\mathbf{0.0416}$\\\hline
True & $u_t=+0.0000u+0.0000u_x+0.1000u_{xx}+0.0000u_{xxx}+0.0000u_{xxxx}+0.0000uu_{x}+0.0000uu_{xx}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table5}Discovered PDE using BNN and BLR (with variance weighting) - Heat equation}
\end{table}

\begin{table}
\hspace{-2.5cm}
\footnotesize
\begin{tabular}{c|c|c}
Noise & Recovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=-0.0245u+0.0706u_x+0.0908u_{xx}-0.0031u_{xxx}+0.0001u_{xxxx}-0.0403uu_{x}+0.0197uu_{xx}$ &  $0.0877$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.3319u+0.2139u_x+0.0528u_{xx}-0.0009u_{xxx}+0.0001u_{xxxx}-0.0422uu_{x}+0.0161uu_{xx}$ & $0.4002$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.8684u+0.4677u_x-0.0118u_{xx}+0.0011u_{xxx}+0.0000u_{xxxx}-0.0926uu_{x}+0.0103uu_{xx}$ & $0.9970$\\\hline
True & $u_t=+0.0000u+0.0000u_x+0.1000u_{xx}+0.0000u_{xxx}+0.0000u_{xxxx}+0.0000uu_{x}+0.0000uu_{xx}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table6}Discovered PDE using DNN and BLR - Heat equation}
\end{table}

\begin{table}
\hspace{-2.5cm}
\footnotesize
\begin{tabular}{c|c|c}
Noise & Recovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=-0.4823u+0.3564u_x+0.0122u_{xx}+0.0002u_{xxx}-0.0000u_{xxxx}+0.0000uu_{x}+0.0161uu_{xx}$ &  $0.6063$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.3189u+0.2995u_x+0.0188u_{xx}+0.0001u_{xxx}-0.0000u_{xxxx}+0.0000uu_{x}+0.0180uu_{xx}$ & $0.4453$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.5798u+0.3362u_x+0.0230u_{xx}-0.0013u_{xxx}+0.0000u_{xxxx}-0.1067uu_{x}+0.0086uu_{xx}$ & $0.6831$\\\hline
True & $u_t=+0.0000u+0.0000u_x+0.1000u_{xx}+0.0000u_{xxx}+0.0000u_{xxxx}+0.0000uu_{x}+0.0000uu_{xx}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table7}Discovered PDE using BNN and Lasso regression ($\gamma=0.005$) with variance weighting - Heat equation}
\end{table}

\begin{table}
\hspace{-2.5cm}
\footnotesize
\begin{tabular}{c|c|c}
Noise & Recovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=-0.0000u+0.0546u_x+0.0941u_{xx}-0.0032u_{xxx}+0.0001u_{xxxx}-0.0160uu_{x}+0.0183uu_{xx}$ &  $0.0601$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.0220u+0.0959u_x+0.0752u_{xx}-0.0015u_{xxx}+0.0001u_{xxxx}-0.0534uu_{x}+0.0148uu_{xx}$ & $0.1156$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.7391u+0.4273u_x-0.0065u_{xx}+0.0010u_{xxx}+0.0000u_{xxxx}-0.0907uu_{x}+0.0067uu_{xx}$ & $0.8651$\\\hline
True & $u_t=+0.0000u+0.0000u_x+0.1000u_{xx}+0.0000u_{xxx}+0.0000u_{xxxx}+0.0000uu_{x}+0.0000uu_{xx}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table8}Discovered PDE using DNN and Lasso regression ($\gamma=0.005$) - Heat equation}
\end{table}

\FloatBarrier

\subsection{\label{burgers} Burgers Equation}

\noindent Let us consider Burgers equation as originally proposed in \cite{JMLR:v19:18-046}:
\begin{equation}
    \frac{\partial u}{\partial t}=\lambda u\frac{\partial u}{\partial x}+\nu\frac{\partial^2 u}{\partial^2 x}
\end{equation}
\begin{equation}
\begin{cases}
    \,u(t=0,x)=-\sin(\pi x/8)\\[2pt]
    \,u(t,x=\pm8)=0
\end{cases}
\hspace{2cm}
\begin{cases}
    \,(t,x)\in\Omega=[0,10]\times[-8,+8]\\[2pt]
    \,\lambda=-1\hspace{1cm}\nu=0.1
\end{cases}
\end{equation}
Measurements are recorded every $\Delta t=0.2~s$ for each 16 sensors ($n=800$ training points), the ground true solution is shown in figure \ref{fig4}. The predictions of the BNN and DNN are shown in figure \ref{fig5} and \ref{fig6} and table \ref{table9} shows the RMSE between the BNN predictive mean, DNN prediction and ground true over 1000 random input points.

\begin{figure}[!h]
\centering
  \centering
  \includegraphics[width=0.7\textwidth]{Burgers1.pdf}
  \caption{Ground true solution for Burgers equation (black dots represent measured time-space points)}
  \label{fig4}
\end{figure}
\FloatBarrier

\begin{figure}[!h]
\centering
  \centering
  \includegraphics[width=1\textwidth]{Burgers4.pdf}
  \caption{BNN predictive mean and standard deviation, DNN prediction, and absolute error between the BNN predictive mean, DNN and ground true - Burgers Equation}
  \label{fig5}
\end{figure}

\begin{figure}[!h]
\centering
  \centering
  \includegraphics[width=1\textwidth]{Burgers3.pdf}
  \caption{Predictions of the BNN (solid blue line $95\%$ confidence intervals) and DNN (solid green line) compared to the ground true (dashed red line) at $t=1~s$, $3~s$ and $6~s$. The black circles represent sensor measurement data - Burgers Equation}
  \label{fig6}
\end{figure}

\FloatBarrier

\noindent Table \ref{table10}, \ref{table11} and \ref{table12} show the recovered Burgers equation for the DNN, BNN with no variance weighting and BNN with variance weighting respectively, using either least-squares regression or BLR. While the BNN combined with BLR almost do as well as the DNN for low noise cases, it completely outperforms the later for $\epsilon\sim\mathcal{N}(0,0.05^2)$. Here, the BNN is able to avoid overfitting (and yields a lower RMSE than the DNN as shown in table \ref{table9}). This can also be observed in figure \ref{fig6}: Notice how the standard DNN (green line) becomes oscillatory around noisy datapoints, while the BNN captures the underlying pattern correctly. Consequently, in this specific case the BNN is able to produce better derivative estimates, ultimately yielding higher PDE recovery accuracy.
\\\\
\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & BNN  & DNN \\\hline
$\epsilon=0$ & 0.0774 & \textbf{0.0283}\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & 0.0725 & \textbf{0.0298}\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & \textbf{0.0853} & 0.0919\\\hline
\end{tabular}
\caption{\centering\label{table9}RMSE of the Bayesian and Standard Deep Neural Network - Burgers Equation}
\end{table}

\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & Least Squares & Bayesian Regression\\\hline
$\epsilon=0$ & $u_t=-1.0035uu_x+0.0962u_{xx}$ & $u_t=-0.9988uu_x+0.0998u_{xx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.9782uu_x+0.1045u_{xx}$ & $u_t=-0.9736uu_x+0.1072u_{xx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.2070uu_x-0.0097u_{xx}$ & $u_t=-0.1737uu_x-0.0117u_{xx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-1.0000uu_x+0.1000u_{xx}$ & $u_t=-1.0000uu_x+0.1000u_{xx}$\\\hline
\end{tabular}
\caption{\centering\label{table10}Recovered Burgers equation using DNN}
\end{table}

\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & Least Squares & Bayesian Regression\\\hline
$\epsilon=0$ & $u_t=-0.0505uu_x+0.0044u_{xx}$ & $u_t=-0.0234uu_x-0.0232u_{xx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.2853uu_x+0.0304u_{xx}$ & $u_t=-0.2649uu_x+0.0248u_{xx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.3684uu_x+0.0374u_{xx}$ & $u_t=-0.3385uu_x+0.0393u_{xx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-1.0000uu_x+0.1000u_{xx}$ & $u_t=-1.0000uu_x+0.1000u_{xx}$\\\hline
\end{tabular}
\caption{\centering\label{table11}Recovered Burgers equation using BNN (no variance weighing)}
\end{table}

\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & Least Squares & Bayesian Regression\\\hline
$\epsilon=0$ & $u_t=-0.9478uu_x+0.0938u_{xx}$ & $u_t=-0.9504uu_x+0.0905u_{xx}$ \\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.9912uu_x+0.0878u_{xx}$ & $u_t=-0.9911uu_x+0.0871u_{xx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.9892uu_x+0.1003u_{xx}$ & $u_t=-0.9875uu_x+0.0990u_{xx}$\\\hline
True & $u_t=-1.0000uu_x+0.1000u_{xx}$ & $u_t=-1.0000uu_x+0.1000u_{xx}$\\\hline
\end{tabular}
\caption{\centering\label{table12}Recovered Burgers equation using BNN (with variance weighing)}
\end{table}

\noindent Table \ref{table13}, \ref{table14}, \ref{table15} and \ref{table16} show the discovered PDE when assuming that the functional form is unknown. Again we can see that the BNN with BLR outperforms the standard DNN (table \ref{table14}) in the most noisy case, and does an excellent job at discovering the non-linear differential term. The BNN does struggle however to find the diffusion coefficient. A likely explanation is that overtime, the solution of Burgers equation in the present case becomes almost equivalent to a piece-wise linear function of space. Thus the diffusion term values get very small, making it harder to recover the coefficient associated to it accurately.

\begin{table}
\hspace{-1.5cm}
\begin{tabular}{c|c|c}
Noise & Discovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=-0.0528u_x+0.0745u_{xx}+0.0014u_{xxx}+0.0002u_{xxxx}-0.9451uu_{x}$ & $0.0804$ \\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.0898u_x-0.0217u_{xx}-0.0184u_{xxx}-0.0006u_{xxxx}-1.0162uu_{x}$ & $0.1533$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.0594u_x+0.0198u_{xx}-0.0297u_{xxx}+0.0026u_{xxxx}-1.0710uu_{x}$ & $\mathbf{0.1260}$\\\hline
True & $u_t=+0.0000u_x+0.1000u_{xx}+0.0000u_{xxx}+0.0000u_{xxxx}-1.0000uu_{x}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table13}Discovered PDE using BNN and BLR (with variance weighting) - Burgers Equation}
\end{table}

\begin{table}
\hspace{-1.5cm}
\begin{tabular}{c|c|c}
Noise & Discovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=-0.0498u_x+0.1086u_{xx}+0.0007u_{xxx}-0.0000u_{xxxx}-1.0439uu_{x}$ & $\mathbf{0.0670}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.0241u_x+0.1141u_{xx}-0.0019u_{xxx}+0.0000u_{xxxx}-1.0477uu_{x}$ & $\mathbf{0.0553}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=+0.0967u_x+0.0109u_{xx}+0.0009u_{xxx}+0.0000u_{xxxx}-0.3489uu_{x}$ & $0.6642$\\\hline
True & $u_t=+0.0000u_x+0.1000u_{xx}+0.0000u_{xxx}+0.0000u_{xxxx}-1.0000uu_{x}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table14}Discovered PDE using DNN and BLR - Burgers Equation}
\end{table}

\begin{table}
\hspace{-1.5cm}
\begin{tabular}{c|c|c}
Noise & Discovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=+0.0000u_x+0.0000u_{xx}-0.0000u_{xxx}+0.0000u_{xxxx}-0.0000uu_{x}$ & $1.0050$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=+0.0000u_x+0.0031u_{xx}-0.0003u_{xxx}+0.0001u_{xxxx}-0.0000uu_{x}$ & $1.0047$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=+0.0000u_x+0.0025u_{xx}-0.0000u_{xxx}+0.0000u_{xxxx}-0.0000uu_{x}$ & $1.0047$\\\hline
True & $u_t=+0.0000u_x+0.1000u_{xx}+0.0000u_{xxx}+0.0000u_{xxxx}-1.0000uu_{x}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table15}Discovered PDE using BNN and Lasso regression ($\gamma=0.005$) with variance weighting - Burgers Equation}
\end{table}

\begin{table}
\hspace{-1.5cm}
\begin{tabular}{c|c|c}
Noise & Discovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=+0.0000u_x+0.0277u_{xx}-0.0003u_{xxx}+0.0001u_{xxxx}-0.2448uu_{x}$ & $0.7587$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=+0.0000u_x+0.0238u_{xx}-0.0005u_{xxx}+0.0001u_{xxxx}-0.2002uu_{x}$ & $0.8035$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=+0.0179u_x+0.0000u_{xx}+0.0005u_{xxx}+0.0000u_{xxxx}-0.0927uu_{x}$ & $0.9130$\\\hline
True & $u_t=+0.0000u_x+0.1000u_{xx}+0.0000u_{xxx}+0.0000u_{xxxx}-1.0000uu_{x}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table16}Discovered PDE using DNN and Lasso regression ($\gamma=0.005$) - Burgers Equation}
\end{table}

\FloatBarrier

\subsection{\label{kdv} Korteweg-De Vries Equation}

\noindent Finally, we consider Korteweg-De Vries (KdV) equation as originally proposed in \cite{JMLR:v19:18-046}:
\begin{equation}
    \frac{\partial u}{\partial t}=\lambda u\frac{\partial u}{\partial x}+\beta\frac{\partial^3u}{\partial^3 x}
\end{equation}
\begin{equation}
\begin{cases}
    \,u(t=0,x)=\cos(-\pi x/20)\\[2pt]
    \,(t,x)\in\Omega=[0,40]\times[-20,+20]
\end{cases}
\hspace{2cm}
\begin{cases}
    \,\lambda=-1\\[2pt]
    \,\beta=-1
\end{cases}
\end{equation}
Measurements are recorded every $\Delta t=0.8~s$ for each 16 sensors ($n=800$ training points), the ground true solution is shown in figure \ref{fig7}. The predictions of the BNN and DNN are shown in figure \ref{fig8} and \ref{fig9}.
\\\\

\begin{figure}[!h]
\centering
  \centering
  \includegraphics[width=0.7\textwidth]{KdV1.pdf}
  \caption{Ground true solution for KdV equation (black dots represent measured time-space points)}
  \label{fig7}
\end{figure}

\begin{figure}[!h]
\centering
  \centering
  \includegraphics[width=1\textwidth]{KdV2.pdf}
  \caption{BNN predictive mean and standard deviation, DNN prediction, and absolute error between the BNN predictive mean, DNN and ground true - KdV Equation}
  \label{fig8}
\end{figure}

\begin{figure}[!h]
\centering
  \centering
  \includegraphics[width=1\textwidth]{KdV3.pdf}
  \caption{Predictions of the BNN (solid blue line $95\%$ confidence intervals) and DNN (solid green line) compared to the ground true (dashed red line) at $t=10~s$, $20~s$ and $30~s$. The black circles represent sensor measurement data - KdV Equation}
  \label{fig9}
\end{figure}

\FloatBarrier

\noindent Table \ref{table17} shows the prediction RMSE for both the BNN and DNN. The DNN predictions outperforms the BNN, but it completely fails to recover the ground true PDE, as shown in table \ref{table18}, \ref{table19} and \ref{table20}. Even in the noiseless case, the DNN does a poor job, and especially struggles at predicting the non-linear derivative coefficient (table \ref{table18}). Conversely, the BNN combined with either the BLR or the least-square regression (table \ref{table20}) is able to recover the PDE coefficients much better. This example shows that the accuracy of the neural network predictions is independent from its capability to recover the governing PDE. Indeed, the neural network doesn't have to be accurate everywhere, it only takes a few correct derivative estimations with well quantified uncertainty to find the ground true PDE.
\\\\
\noindent

\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & BNN  & DNN \\\hline
$\epsilon=0$ & \textbf{0.1840} & 0.1018\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & \textbf{0.1411} & 0.1068\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & \textbf{0.1712} & 0.1158\\\hline
\end{tabular}
\caption{\centering\label{table17}RMSE of the Bayesian and Standard Deep Neural Network - KdV Equation}
\end{table}

\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & Least Squares & Bayesian Regression\\\hline
$\epsilon=0$ & $u_t=-0.2698uu_x-0.4873u_{xxx}$ & $u_t=-0.2679uu_x-0.4841u_{xxx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.2884uu_x-0.4898u_{xxx}$ & $u_t=-0.2845uu_x-0.4851u_{xxx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.0661uu_x-0.2687u_{xxx}$ & $u_t=-0.0771uu_x-0.2748u_{xxx}$\\\hline
True & $u_t=-1.0000uu_x-1.0000u_{xxx}$ & $u_t=-1.0000uu_x-1.0000u_{xxx}$\\\hline
\end{tabular}
\caption{\centering\label{table18}Recovered KdV equation using DNN}
\end{table}

\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & Least Squares & Bayesian Regression\\\hline
$\epsilon=0$ & $u_t=-0.3639uu_x-0.5118u_{xxx}$ & $u_t=-0.3511uu_x-0.5025u_{xxx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.5932uu_x-0.6783u_{xxx}$ & $u_t=-0.5882uu_x-0.6754u_{xxx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.4482uu_x-0.5673u_{xxx}$ & $u_t=-0.4380uu_x-0.5605u_{xxx}$\\\hline
True & $u_t=-1.0000uu_x-1.0000u_{xxx}$ & $u_t=-1.0000uu_x-1.0000u_{xxx}$\\\hline
\end{tabular}
\caption{\centering\label{table19}Recovered KdV equation using BNN (no variance weighing)}
\end{table}

\begin{table}
\centering
\begin{tabular}{c|c|c}
Noise & Least Squares & Bayesian Regression\\\hline
$\epsilon=0$ & $u_t=-0.9783uu_x-0.9303u_{xxx}$ & $u_t=-0.9582uu_x-0.9219u_{xxx}$ \\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.9472uu_x-0.8581u_{xxx}$ & $u_t=-0.9343uu_x-0.8533u_{xxx}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.9179uu_x-0.8389u_{xxx}$ & $u_t=-0.9043uu_x-0.8329u_{xxx}$\\\hline
True & $u_t=-1.0000uu_x-1.0000u_{xxx}$ & $u_t=-1.0000uu_x-1.0000u_{xxx}$\\\hline
\end{tabular}
\caption{\centering\label{table20}Recovered KdV equation using BNN (with variance weighing)}
\end{table}

\noindent Table \ref{table21}, \ref{table22}, \ref{table23} and \ref{table24} show the discovered PDE when assuming that the functional form is unknown. Again, the BNN with BLR outperforms the other models. For every noise cases however, it does overestimate the influence of the $1^\text{st}$ order derivative, and underestimates the coefficients of the $3^\text{rd}$ order derivative and the non-linear term. Indeed, the KdV equation creates strong oscillatory patterns with large gradient values, so higher order derivatives may be especially hard to estimate correctly. In fact, every models generally provide better accuracy for the non-linear derivative coefficient (which is only a $1^\text{st}$ order derivative) than with the $3^\text{rd}$ order derivative term.

\begin{table}
\hspace{-1.5cm}
\begin{tabular}{c|c|c}
Noise & Discovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=+0.1914u_x+0.0683u_{xx}-0.4423u_{xxx}-0.0520u_{xxxx}-0.5921uu_{x}$ & $\mathbf{0.7221}$ \\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=+0.1034u_x+0.0377u_{xx}-0.6702u_{xxx}-0.0481u_{xxxx}-0.6907uu_{x}$ & $\mathbf{0.4678}$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=+0.0984u_x-0.0162u_{xx}-0.6399u_{xxx}-0.0959u_{xxxx}-0.6478uu_{x}$ & $\mathbf{0.5224}$\\\hline
True & $u_t=+0.0000u_x+0.0000u_{xx}-1.0000u_{xxx}+0.0000u_{xxxx}-1.0000uu_{x}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table21}Discovered PDE using BNN and BLR (with variance weighting) - KdV Equation}
\end{table}

\begin{table}
\hspace{-1.5cm}
\begin{tabular}{c|c|c}
Noise & Discovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=+0.1732u_x-0.0394u_{xx}-0.2497u_{xxx}-0.0333u_{xxxx}-0.5978uu_{x}$ & $0.8703$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=+0.1604u_x-0.0633u_{xx}-0.2813u_{xxx}-0.0497u_{xxxx}-0.5884uu_{x}$ & $0.8474$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=+0.2417u_x+0.0522u_{xx}-0.0803u_{xxx}-0.0367u_{xxxx}-0.5005uu_{x}$ & $1.0760$\\\hline
True & $u_t=+0.0000u_x+0.0000u_{xx}-1.0000u_{xxx}+0.0000u_{xxxx}-1.0000uu_{x}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table22}Discovered PDE using DNN and BLR - KdV Equation}
\end{table}

\begin{table}
\hspace{-1.5cm}
\begin{tabular}{c|c|c}
Noise & Discovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=-0.0000u_x+0.0000u_{xx}+0.0000u_{xxx}-0.0000u_{xxxx}-0.1825uu_{x}$ & $1.2916$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=-0.0000u_x-0.0000u_{xx}-0.0000u_{xxx}-0.0000u_{xxxx}-0.1820uu_{x}$ & $1.2920$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=-0.0000u_x-0.0000u_{xx}+0.0000u_{xxx}-0.0000u_{xxxx}-0.1800uu_{x}$ & $1.2932$\\\hline
True & $u_t=+0.0000u_x+0.0000u_{xx}-1.0000u_{xxx}+0.0000u_{xxxx}-1.0000uu_{x}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table23}Discovered PDE using BNN and Lasso regression ($\gamma=0.005$) with variance weighting - KdV Equation}
\end{table}

\begin{table}
\hspace{-1.5cm}
\begin{tabular}{c|c|c}
Noise & Discovered PDE & $e$ ($\ell^2$ norm)\\\hline
$\epsilon=0$ & $u_t=+0.0268u_x-0.0000u_{xx}-0.0000u_{xxx}-0.0000u_{xxxx}-0.2191uu_{x}$ & $1.2690$\\\hline
$\epsilon\sim\mathcal{N}(0,0.01^2)$ & $u_t=+0.0246u_x-0.0000u_{xx}-0.0000u_{xxx}-0.0000u_{xxxx}-0.2117uu_{x}$ & $1.2736$\\\hline
$\epsilon\sim\mathcal{N}(0,0.05^2)$ & $u_t=+0.0158u_x-0.0000u_{xx}-0.0241u_{xxx}-0.0130u_{xxxx}-0.2198uu_{x}$ & $1.2497$\\\hline
True & $u_t=+0.0000u_x+0.0000u_{xx}-1.0000u_{xxx}+0.0000u_{xxxx}-1.0000uu_{x}$ & $0.0000$\\\hline
\end{tabular}
\caption{\centering\label{table24}Discovered PDE using DNN and Lasso regression ($\gamma=0.005$) - KdV Equation}
\end{table}

\clearpage

redpajama_set_name: RedPajamaArXiv


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CEO, RMG Group
"Each Vistage group has its own 'personality,' which is a result of the meshing of the members in the group along with the guidance of the chair. Everyone should think of their group as "the best group," because it should be so for them. The "best chair" is the artist behind the scenes that helps put the whole thing together. That's Allan. [The benefits of Vistage include] being challenged to think about things from a different point of view – to realize there are many roads available to get to a place, and maybe you should consider a different one than your own first choice. [There's value in] people telling you this with real care and support, even if you're being stubborn. Six months later, you look back and realize you achieved something you never believed you could. And this happens over and over. [My Vistage group has made] the biggest impact of anything I've done for my business in improved outcomes, both for the business and for personal life satisfaction. This is where you can learn how to not be a slave to your business."
"Six months later, you look back and realize you achieved something you never believed you could. [My Vistage group has made] the biggest impact of anything I've done for my business."
Yale Zoland
CEO, Zolands
"In the years that Allan Fried has been my Vistage chair, I've improved processes and structures, enhanced my leadership skills and grown my business. Allan reads between the lines and is quick to interpret the inner struggles I'm grappling with. He brings clarity to these struggles and pushes me to interpret them as opportunities, then helps me strive for my best efforts. During our group meetings, Allan moderates thoughtfully and pushes each of us to be the best advisors to our peers that we can be. He challenges us and confronts us with sensitivity and compassion. As a result of my association, I've not only grown as a business owner but also as an individual, and I will be forever grateful to Allan – a great coach!"
"Allan reads between the lines and is quick to interpret the inner struggles I'm grappling with. He brings clarity to these struggles and pushes me to interpret them as opportunities."
Craig Weidhorn
CEO, Softwear LLC
"While I had some success prior to meeting Allan, it was my work with him that helped me to understand my value in the market. Allan's methodology differed from other mentors that I've worked with in that he put an emphasis on his clients' happiness – as opposed to just their paychecks. In my time with Allan, I developed a plan and was held accountable for the steps I took toward my goals, ultimately exceeding even my own expectations."
"Allan's methodology differed from other mentors that I've worked with in that he puts an emphasis on his clients' happiness – as opposed to just their paychecks."
Eric Nemer
Partner, Nemer Motor Group
"The first thing that comes to mind when thinking of Allan is gratitude. His impact on me can not be overstated and I am incredibly grateful for having had the chance to work with him. Allan is truly a transformative influence; his words still resonate and he has undoubtedly left me better equipped as a leader...I would even go so far as to say a better person. Although Allan is blessed with intuition and empathy, you can tell he is driven by what he does and spends an inordinate amount of energy honing his craft. Allan is a skilled guide, but be prepared to do your part...good god, don't waste this man's time if you're not willing to put the work in!"
The first thing that comes to mind when thinking of Allan is gratitude. His impact on me can not be overstated and I am incredibly grateful for having had the chance to work with him.
Gaby Sappington
Executive Director, World Music Institute
"If you're looking to create positive change and new professional opportunities for yourself, I can only recommend working with Allan as your coach. He has helped me work my way toward a new, more satisfying job that is fully aligned with my values and interests. Allan has that rare gift of consistently challenging you, while at the same time making you feel that he's your number one champion – and he truly understands where you're coming from. His insightful, probing and at times absolutely unexpected questions helped me become more aware of my blind spots as well as (old) belief systems that blocked my growth. The tools he provided activated a heightened sense of awareness that allowed me to pursue new opportunities in a positive and productive manner."
"Allan has that rare gift of consistently challenging you, while at the same time making you feel that he's your number one champion – and he truly understands where you're coming from."
Shawn Luby
Scientist-turned-Musician
"At first I was skeptical about coaching... But I needed a change. Working with Allan has transformed my life and career. I'm now an active player in the creation of my future."
David Koretz
Co-Founder & CEO, SlantShack Jerky
"It's funny how something much more powerful and insightful comes from a large group conversation rather than individual one-offs."
Make Your Move. Evolve Your Leadership. Transcend Your Edge.
Schedule your personal consultation with Coach Allan Fried.
©2018 Allan Fried Coaching
Website Created by DesignGood

redpajama_set_name: RedPajamaCommonCrawl


text:
"""
    File-resident datatype tests.

    Tests "committed" file-resident datatype objects.
"""

import numpy as np

from .common import ut, TestCase

from h5py import Datatype

class TestCreation(TestCase):

    """
        Feature: repr() works sensibly on datatype objects
    """

    def test_repr(self):
        """ repr() on datatype objects """
        self.f['foo'] = np.dtype('S10')
        dt = self.f['foo']
        self.assertIsInstance(repr(dt), str)
        self.f.close()
        self.assertIsInstance(repr(dt), str)

    def test_appropriate_low_level_id(self):
        " Binding a group to a non-TypeID identifier fails with ValueError "
        with self.assertRaises(ValueError):
            Datatype(self.f['/'].id)

redpajama_set_name: RedPajamaGithub


text:
Q: angular chart js type line setting chart-options not working Hi I'm trying to configure the graph to be without any tooltips or legends, and it look's like all the configured options are ignored. what I'm doing wrong?
it's the js object:
$scope.options = {
    segmentShowStroke : false,
     elements: { arc: { borderWidth: 0 } ,  point: {
          radius: 0
        }},
     tooltips:{
            enabled:false
        },
    legend: {
             display: false
        },
    scales: {
      yAxes: [
        {
          id: 'y-axis-1',
          type: 'linear',
          display: false,
          position: 'left'
        }

      ]
    }
  };

And it's the canvas element :
<canvas id="line" class="chart chart-line"
chart-data="data" chart-labels="labels" chart-options="options"
chart-dataset-override="datasetOverride"
chart-click="onClick" height="40px" width="1000px"
chart-colors="graph.colors" style=" width: 1000px; ;max-height: 40px !important;">
</canvas>

A: Try this solution
options : {
    scales : {
        xAxes : [ {
            gridLines : {
                display : false
            }
        } ]
    }
}

A: I managed to make it work, only after changing the options from chartjsProvider.setOptions like this
ChartJsProvider.setOptions('line',{
        segmentShowStroke : false,
         elements: { arc: { borderWidth: 0 } ,  point: {
              radius: 0
            }},
         tooltips:{
                enabled:false
            },
        legend: {
                 display: false
            },
        scales: {
             yAxes: [
                     {

                       display: false,

                     }

                   ],
                   xAxes: [
                           {

                             display: false,

                           }

                         ]
        }
    });

redpajama_set_name: RedPajamaStackExchange


I first became aware of Dale Chihuly and his Seattle exhibit through my wife. When we were planning our trip to Seattle, she did some research online and discovered that the Chihuly Garden and Glass was rated near the top, often the number one recommended attraction, for out of town visitors.
Once I learned that, it went on the official travel itinerary. It also didn't hurt that it was within walking distance of our hotel.
Located directly under the Space Needle, finding the Chihuly Garden and Glass was very easy. Considering that our hotel was just a few blocks from the Seattle Center, home of the Space Needle, all we had to do was look up to know which way to go.
One of the things most museum, tour or attraction websites will ask you to do is purchase your ticket in advance. I failed to anticipate the popularity of the Chihuly Garden and Glass. Since we were planning to visit on a Thursday, in May, about an hour after they opened – I certainly didn't expect a long line to get in.
Lesson learned! While standing at the back of the long line of people queued up to purchase their ticket inside, I used my iPhone to go to their website and get our passes. This turned out to be a very lengthy process and I only just completed my transaction after moving 2/3's of the way up in line. The gentleman behind me noticed what I was doing and asked if he should do the same. It didn't take much encouraging from me before he had his phone out.
If there's anything I would do differently next time: Skip the line, order tickets online.
After getting through the complex* process for entering the first of four areas to explore, we entered the first of eight rooms exhibiting some of the more delicate glass-work created just for this show.
One of my favorite parts of the entire tour was the Garden. I have to say it was pleasing to not know what to expect before going. It was about here that I realized that I have seen Dale Chihuly's work before. Those past exhibitions were indoor installations, but seeing his work blended in with nature really made it come alive for me.
When the website says, "Purchase your ticket in advance," – DO IT!
I love a good museum. It doesn't matter if its art, history or science. I just have a natural love for learning and when done right, I feel especially pleased as the recipient of a good immersive experience. The Chihuly Garden and Glass was certainly one of the best experiences from my trip to Seattle this year. I can't recommend it enough.
Please let me know if you have any suggestions of similar exhibits in your city.


Real-time reporting at the student, class, grade, school, district, and state levels; local and national norms; Rate of Improvement (ROI) student growth norms; ELL reports; Common Core and Lexile reporting.
If you are ordering a new subscription, select from the "First-time purchases" section below.
If you are renewing a subscription, select from the "Renewals" section below. Your customer ID and administrator's name and address are required for renewal.
Please note there is a $300 minimum on all orders and subscription types cannot be combined (i.e., you cannot purchase both reading and math).
Please try to order training 4 - 6 weeks in advance of your needs. Training dates will be determined in consultation with a Pearson consultant.
Get a closer look at the capabilities, benefits and features of aimswebPlus.
Price per student per year. Enter number of students.
* Shaywirz DyslexiaScreen training is required if the Shaywitz DyslexiaScreen add-on is purchased and if no other aimswebPlus training is purchased.
This training is required if the Shaywitz DyslexiaScreen Add-on is purchased and if no other aimswebPlus training is purchased.
aimswebPlus provides the framework, data, and guidance that administrators and teachers need to efficiently allocate and evaluate core instruction and interventions, helping them improve outcomes for all students based on accurate, continuous, and direct student assessment.
At the foundation of aimswebPlus are general outcome measures, a form of curriculum-based measurement (CBM) used for universal screening and progress monitoring.
Pearson offers a busy webinar schedule for Transitioning from AIMSweb 1.0 to aimswebPlus.


A battle is heating up again over attempts by politicians to pry open Switzerland's restrictive shopping laws.
The lower house of parliament on Wednesday approved legislation that would allow gas station stores to remain open 24 hours a day on major highways and routes heavily used by travellers.
The approval follows earlier backing of the changes by the senate, which also approved a motion in favour of Sunday shopping and for longer minimum hours during weekdays and Saturdays.
But left-wing parties and unions are already planning to challenge any such changes in a referendum.
Among the ongoing concerns is the right of employees to work reasonable hours and to be entitled to a day off on Sundays.
Other issues include extra deliveries and traffic generated by the longer operating store hours.
"The unions will not be alone in the referendum fight," Louis Schelbert, a member of the Green party from Lucerne, told the media.
A parliamentary commission is looking into a motion from the senate to allow shops across the country to open from 6am to 8 pm from Monday to Friday, and from 6 am until 7pm on Saturday.
The cantons would be able to adopt more liberal hours, under the proposal, spearheaded by Filippo Lombardi, a Christian Democrat from Ticino.
Fabio Abate, a Liberal senator from Ticino, meanwhile, wants to relax federal laws currently limiting Sunday shopping to tourist areas, main train stations and airports.
Geneva MP Christian Lüscher, sponsor of the motion for 24-hour gas station stores, said this would allow them to offer products and services meeting the needs of travellers at all times.
Ruedi Noser, a Liberal party member from Zurich, said the change would also address an anomaly in the operations of gas stations.
Current laws require them to close retail operations from 1 to 5 am from Mondays to Saturdays, as well as on Sundays.
However, staff can still sell fuel and food products from cafeterias.


Dr. Elizabeth Cudney is an Assistant Professor at Missouri University of Science and Technology. She received her B.S. in Industrial Engineering from North Carolina State University, Master of Engineering in Mechanical Engineering and Master of Business Administration from the University of Hartford, and her doctorate in Engineering Management from the University of Missouri, Rolla. In 2010, Dr. Cudney was inducted into the ASQ International Academy for Quality. She received the 2008 ASQ A.V. Feigenbaum Medal and the 2006 SME Outstanding Young Manufacturing Engineering Award. She is an ASQ Certified Quality Engineer, Manager of Quality/Operational Excellence, and Certified Six Sigma Black Belt. She is a member of the ASEM, ASME, ASQ, IIE, SAE, and the Japan Quality Engineering Society (JQES).
Stephen Gent is an Assistant Professor with the Department of Mechanical Engineering at South Dakota State University. While at SDSU, Stephen has been working to establish a research program that utilizes high-fidelity simulation models for thermo-fluids and energy systems applications. In the fall semester of 2010, Stephen developed and taught a cross-listed undergraduate and graduate level course on simulation-based modeling and design using computational fluid dynamics. Prior to being at South Dakota State University, Stephen worked with the Simulation, Modeling, and Decision Science Division at Iowa State University's Virtual Reality Application Center. His research while at these institutions included modeling complex multiphase fluid flows and systems of models to optimize engineering designs of energy systems. Stephen has been a coordinating instructor for undergraduate engineering laboratories, including fluid mechanics and heat transfer laboratories. Also, Stephen is actively involved with the ASEE and has earned several awards for excellence in teaching.


This webinar on Microsoft Outlook covers some of the various ways to search, filter, tag, and present content within Outlook. In addition to exploring some of the basic features, we spend a significant amount of time answering questions from attendees ranging from formatting to shortcuts. To go with this training sandy Rylander has provided us with a Guide to Outlook 2013 & 2016, it contains over a hundred pages information on how to get the most out of Outlook.


text:
Adâncimea este o mărime care caracterizează distanța de la un punct aflat pe nivelul oceanului planetar până la un punct aflat mai jos. Adâncimea maximă a oceanului planetar alcătuiește 11516 m, iar cea medie – 3790 m. Cea mai adâncă depresiune a scoarței terestre este situată la nivelul Mării Moarte și măsoară 396 m.

Topografie

redpajama_set_name: RedPajamaWikipedia


for enthusiasts PC users, based on the award-winning EVEREST Technology.
advanced hardware monitoring and diagnostics capabilities to check the effects of the applied settings.
and compare it to previous states or other systems.
that offers a total of 100 pages of information about your PC.


text:
EU works hand in hand with Egypt's government in fulfilling its obligations to access safe drinking water - Daily News Egypt
Business EU works hand in hand with Egypt's government in fulfilling its obligations to access safe drinking water
EU works hand in hand with Egypt's government in fulfilling its obligations to access safe drinking water
FAO to organise FAO-Egypt partnership day, technical sessions on drought insurance, says minister
Shaimaa Al-Aees May 14, 2018 Be the first to comment
Ivan Surkoš, head of EU delegation to Egypt, said that the European Union (EU) will continue to support sustainable water management in Egypt through policy dialogue and development cooperation, in close partnership with all stakeholders in the sector.
During the kick-off meeting of the first Cairo Water Week in Cairo on Sunday, Surkoš added that promoting and protecting access to the fundamental resource means also promoting and protecting water rights and security, not only for the present, but also for future generations.
"The European Union reaffirms its commitment to work with our partner countries, such as Egypt, to assist them in fulfilling their obligations regarding access to safe drinking water, sanitation, and food security," Surkoš noted, adding, "water must be available, accessible, safe, acceptable, and affordable for all without discrimination, and the right to safe drinking water and sanitation is a human right, essential for the full enjoyment of life. Therefore, we are delighted to reaffirm our support for the Cairo Water Week and its activities, and shall work, hand in hand, with Egypt's government in the implementation of its activities and the realisation of its conclusions."
The Cairo Water Week, the first EU-Egypt cooperation in the water sector, will begin on 17 October.
Egypt's minister of irrigation and water resources, Mohamed Abdel Aaty, said that by the year 2050, most countries will resort to the reuse of water as an inevitable result of water shortages.
Abdel Aaty said that developing countries, especially in Africa, suffer from poverty, a decrease in development rates, water basin and water quality deterioration, lack of drinking water supply, and electricity and energy shortages, stressing the importance of knowledge and technology to produce modern innovations.
Abdel Aaty revealed that the Food and Agriculture Organisation of the United Nations (FAO) will organise a FAO-Egypt partnership day, in addition to technical sessions on drought insurance.
He pointed out that another important event that will take place is the first EU integrated investment planning in the water sector in Egypt, where the government of Egypt, in cooperation with the EU, is actively pursuing ambitious plans for implementing strategic programmes in the water sector.
"The Cairo Water Week will host the first young water professional forum, which will include the declaration of the first African Youth Forum," the minister noted, elaborating, "the week will include important field visits to introduce participants in Egypt's national projects and technical field trips to strategic projects and plants will be organised, in addition to some social activities."
The minister explained that the Cairo Water Week aims at promoting water awareness and fostering new thinking to face the most pressing water-related challenges, keep abreast of global moves to address water challenges, and identify modern tools and techniques for managing water resources.
He stressed that the challenges in the water sector in Egypt are similar to those faced by most arid and semi-arid countries at regional and international levels. Most of them suffer from water scarcity or are affected by climate change, and their geographical situation as river estuaries come from outside their borders, and other challenges that increase the problem of water scarcity and security.
For his part, regional programme leader at delivery manager of the Regional Initiative on Water Scarcity at FAO, Pasquale Steduto, said that the FAO seeks urgent solutions for the development of water resources and irrigation.
Steduto noted, "we need to exchange experiences, knowledge, benefits, and risks. besides the need to cooperate to meet the challenges facing the water sector."
He pointed out that Egypt faces a population increase that requires increasing water resources, adding that 90% of the country's water is consumed by the irrigation sector.
Topics: EU head of EU delegation to Egypt Ivan Surkoš water
Shaimaa Al-Aees
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https://cdn1.dailynewsegypt.com/2018/05/14/eu-works-hand-in-hand-with-egypts-government-in-fulfilling-its-obligations-to-access-safe-drinking-water/
Planning minister reaffirms Egypt's efforts on water security
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Elouise Cobell, the namesake of UM's Elouise Cobell Land and Culture Institute located at the Payne Family Native American Center, was named one of 21 recipients of the 2016 Presidential Medal of Freedom, the nation's highest civilian honor that is presented to individuals who have made especially meritorious contributions to the security or national interests of the United States, to world peace, or to cultural or other significant public or private endeavors. Cobell was a Blackfeet Tribal community leader and an advocate for Native American self-determination and financial independence. She used her expertise in accounting to champion a lawsuit that resulted in a historic settlement, restoring tribal homelands to her beloved Blackfeet Nation and many other tribes, and in so doing, inspired a new generation of Native Americans to fight for the rights of others. Cobell helped found the Native American Bank, served as director of the Native American Community Development Corporation, and inspired Native American women to seek leadership roles in their communities. Although she died in 2011, her legacy of fighting for fairness and justice for Native people lives on.
Oral Surgical Associates and Keystone Dental have invited us to a lecture on "Achieving Esthetics through Restorative Solutions" on Monday, January 9 from 5:30 to 8 p.m. at the Doubletree Missoula, 100 Madison St. While geared toward implant surgeons and restorative dentists, the lecture is open and free to the public. Guest speaker is Dr. Nurit Bittner, an assistant professor of clinical dentistry for the division of prosthodontics at Columbia University's College of Dental Medicine. Please RSVP by Monday, January 2 but there may be space available after the date, so double check. Please contact Education@KeystoneDental.com or call Brad Hoag at 612.219.2201 for more information.
The first of eight shows at Radius Gallery in 2017 kicks off at the end of this month with a show for artists Hadley Ferguson, David Hiltner, and Bobbie McKibbin. You may not even realize that you're already familiar with their work, so this is a great opportunity to get up close and personal with it. The intersection of Broadway and Higgins features large murals of historic Missoula by Hadley, who was also commissioned to paint a historic homage to Montana women in the Capitol building in Helena. David, a ceramic artist, is the executive director of the Red Lodge Clay Center, which provides a place for artists to create new (ceramic-minded) work. The Smithsonian American Art Museum has acquired work by Bobbie, a pastel artist based in Stevensville. These are three artists' work you won't want to miss. The exhibit runs January 20 through February 25 at Radius Gallery, 114 E. Main. For more information about the show and the artists, visit RadiusGallery.com.
Mountain and off-road bikers, rejoice! Syringa Park, an undeveloped county park in the upper Rattlesnake, may soon be home to a "pump track," a few "jump lines," and even some "slope-style" features, according to the new master park plan adopted by the Missoula Parks and Recreation Board. The park has been designed as a bike park, which includes topographical and man-made elements specifically designed for off-road or mountain biking. Developed with neighborhood residents and the cycling community, the plan includes features for beginning to advanced cyclists of all ages, a shade shelter with picnic tables, and walking paths.
Winter is here, but spring is right around the corner, which makes this the perfect time for a remodel project. Or maybe it's the year that you're finally going to get that home theater or that beautiful deck you've been wanting. Maybe you are even ready for a brand new home. Whatever your plans are for this year and beyond, the 2017 Missoula Building Industry Association (MBIA) Home Show is the perfect place to make your plans a reality. Come meet the builders, subcontractors, suppliers, lenders, and just about anyone else in the building industry who can help you take your project from concept to completion. The Home Show is January 20-21 at the Hilton Garden Inn, 3720 N. Reserve St. Tickets ($5) are available at TicketBud.com and more information is available at BuildMissoula.com.
Step inside for the Winter's Farmers Market at the Elks Lodge, 112 N. Pattee St. You'll find local produce, meat, eggs, and other farm goods as well as locally roasted coffee, baked goods, prepared breakfast and lunch items, and handmade arts and crafts. The market is open Saturdays through April 22. For more information, search "Missoula Valley Winter Market" on Facebook.
Five, six, five, six, seven, eight. The Roxy welcomes back the Kinetoscope Dance Film Festival, a two-day cinematic celebration of the exciting relationship of dance and film, on January 28-29. The new line-up of dance films comes from around the world and features two programs.
Program A: Saturday, January 28 at 4 p.m. and Sunday, January 29 at 6 p.m.
Program B: Saturday, January 28 at 6 p.m. and Sunday, January 29 at 4 p.m.
Single admission to one program is $10, and double feature admission to both is $15.The event is sponsored by Bare Bait Dance. For more information and tickets, visit BareBaitDance.org or TheRoxyTheater.org.
Do your friends come to you with their inquiries about all things art-related? Would you like them to? If you've answered "yes" to either question, consider becoming an Art Guide at the Missoula Art Museum. The museum provides you with everything you'll need (i.e., training) to become an esteemed local aesthetic expert. For more information, contact the museum's Education Curator Renee Taafe by calling 406.728.0447 or emailing ReneeT@MissoulaArtMuseum.org.


Today's successful job seeker needs every advantage to fast-track their search, make the right impression and land the job they want. Whether pursuing an opportunity across town or around the world, GoinGlobal is that advantage. We provide the competitive edge that helps students and professionals get noticed and get hired.


text:
/**
 * @file Free_List_Allocator.hpp
 *
 * @todo Implement free-list allocator
 *
 * @author Matthew Rodusek
 * @date   June 23, 2015
 */

#ifndef VALKNUT_CORE_ALLOCATION_POLICIES_FREE_LIST_ALLOCATOR_HPP_
#define VALKNUT_CORE_ALLOCATION_POLICIES_FREE_LIST_ALLOCATOR_HPP_

namespace valknut{

	namespace allocation_policies{

		class Free_List_Allocator{
			//----------------------------------------------------------------------
			// Public Member Types
			//----------------------------------------------------------------------
		public:

			typedef true_type allows_unordered_deallocations;
			typedef true_type allows_variable_sized_allocations;

		};

	} // namespace allocation_policies

} // namespace valknut

#endif /* VALKNUT_CORE_ALLOCATION_POLICIES_FREE_LIST_ALLOCATOR_HPP_ */


I just might be a Big Hero!
I need to work on judging a book by its cover…or rather, judging a movie by its poster. Prime example…Big Hero 6. When it came out in theaters, I didn't bother giving it one thought because of the advertisements and movie poster's I had seen for it. Eventually, it made it's way to the discount theater, which my daughter and I frequent quite a bit. We had reached a point where we had seen pretty much every movie there (for children), except Big Hero 6. Honestly, I didn't think it would be that great and I thought it was more of a movie for little boys. I just didn't think that my little one would be interested, and let's be honest, I didn't think I would be either, for that matter. One Saturday afternoon, we decided on a whim, to go to the movies. I told her prior to going that the only thing left was Big Hero 6. She reluctantly agreed to go, as long as I added a clause that popcorn and drinks would be consumed during the movie. Just a side note… I don't remember getting popcorn or drinks at every single movie that I went to when I was younger,, but hey, that's another story for another day 🙂 .
So here we are at the movie, well into the movie, close to the end of the movie, and do you know what? I am crying my eyes out! Seriously, I was in complete shock! I was happy that I was able to see it at the theater instead of DVD. Funny thing, the day after the movie, we ended up renting it on Red Box…3 days in a row! Needless to say, she and I are both IN LOVE with Baymax! So much so, that my daughter threw into her birthday party suggestions (for next year) that she could even have a Big Hero Six birthday party. It was unexpected, but I love the idea. Now, come party time next year, I'm sure her mind could change, but can't a Mama dream? As I wandered through Facebook, I ran across a video for Baymax cookies.
The video seems super simple, however, I'm sure I'll manage to find a way to mess them up 50 times before I get one right. Then I figured I should just design a cupcake topper instead. It's less time in the kitchen, and Baymax is just a series of shapes. After a couple of attempts, I finally came up with something I am happy with, and more importantly, when I showed it to my daughter, she flipped and said, "Mommy, you nailed it!" So I'm guessing, it's good and actually LOOKS like Baymax…what do you think?
Previous I scream, you scream…for Ice Cream!


Retrieves an aggregated list of machine types.
The URL uses gRPC Transcoding syntax.
Project ID for this request.
Specifies a page token to use. Set pageToken to the nextPageToken returned by a previous list request to get the next page of results.
A filter expression that filters resources listed in the response. The expression must specify the field name, a comparison operator, and the value that you want to use for filtering. The value must be a string, a number, or a boolean. The comparison operator must be either =, !=, >, or <.
For example, if you are filtering Compute Engine instances, you can exclude instances named example-instance by specifying name != example-instance.
You can also filter nested fields. For example, you could specify scheduling.automaticRestart = false to include instances only if they are not scheduled for automatic restarts. You can use filtering on nested fields to filter based on resource labels.
To filter on multiple expressions, provide each separate expression within parentheses. For example, (scheduling.automaticRestart = true) (cpuPlatform = "Intel Skylake"). By default, each expression is an AND expression. However, you can include AND and OR expressions explicitly. For example, (cpuPlatform = "Intel Skylake") OR (cpuPlatform = "Intel Broadwell") AND (scheduling.automaticRestart = true).
Sorts list results by a certain order. By default, results are returned in alphanumerical order based on the resource name.
You can also sort results in descending order based on the creation timestamp using orderBy="creationTimestamp desc". This sorts results based on the creationTimestamp field in reverse chronological order (newest result first). Use this to sort resources like operations so that the newest operation is returned first.
Currently, only sorting by name or creationTimestamp desc is supported.
The request body must be empty.
[Output Only] Unique identifier for the resource; defined by the server.
A list of MachineTypesScopedList resources.
[Output Only] Name of the scope containing this set of machine types.
[Output Only] A list of machine types contained in this scope.
[Output Only] The unique identifier for the resource. This identifier is defined by the server.
[Output Only] Creation timestamp in RFC3339 text format.
[Output Only] Name of the resource.
[Output Only] An optional textual description of the resource.
[Output Only] The number of virtual CPUs that are available to the instance.
[Output Only] The amount of physical memory available to the instance, defined in MB.
[Deprecated] This property is deprecated and will never be populated with any relevant values.
[Output Only] A list of extended scratch disks assigned to the instance.
Size of the scratch disk, defined in GB.
[Output Only] Maximum persistent disks allowed.
[Output Only] Maximum total persistent disks size (GB) allowed.
[Output Only] The deprecation status associated with this machine type.
The deprecation state of this resource. This can be ACTIVE DEPRECATED, OBSOLETE, or DELETED. Operations which communicate the end of life date for an image, can use ACTIVE. Operations which create a new resource using a DEPRECATED resource will return successfully, but with a warning indicating the deprecated resource and recommending its replacement. Operations which use OBSOLETE or DELETED resources will be rejected and result in an error.
The URL of the suggested replacement for a deprecated resource. The suggested replacement resource must be the same kind of resource as the deprecated resource.
An optional RFC3339 timestamp on or after which the state of this resource is intended to change to DEPRECATED. This is only informational and the status will not change unless the client explicitly changes it.
An optional RFC3339 timestamp on or after which the state of this resource is intended to change to OBSOLETE. This is only informational and the status will not change unless the client explicitly changes it.
An optional RFC3339 timestamp on or after which the state of this resource is intended to change to DELETED. This is only informational and the status will not change unless the client explicitly changes it.
[Output Only] The name of the zone where the machine type resides, such as us-central1-a.
[Output Only] Server-defined URL for the resource.
[Output Only] Whether this machine type has a shared CPU. See Shared-core machine types for more information.
[Output Only] The type of the resource. Always compute#machineType for machine types.
[Output Only] An informational warning that appears when the machine types list is empty.
[Output Only] A warning code, if applicable. For example, Compute Engine returns NO_RESULTS_ON_PAGE if there are no results in the response.
[Output Only] A human-readable description of the warning code.
[Output Only] A key that provides more detail on the warning being returned. For example, for warnings where there are no results in a list request for a particular zone, this key might be scope and the key value might be the zone name. Other examples might be a key indicating a deprecated resource and a suggested replacement, or a warning about invalid network settings (for example, if an instance attempts to perform IP forwarding but is not enabled for IP forwarding).
[Output Only] A warning data value corresponding to the key.
[Output Only] This token allows you to get the next page of results for list requests. If the number of results is larger than maxResults, use the nextPageToken as a value for the query parameter pageToken in the next list request. Subsequent list requests will have their own nextPageToken to continue paging through the results.
[Output Only] Server-defined URL for this resource.
[Output Only] Informational warning message.
[Output Only] Type of resource. Always compute#machineTypeAggregatedList for aggregated lists of machine types.
For more information, see the Authentication Overview.
Última actualización: Marzo 25, 2019.
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Xolo has launched its first Tegra 3 processor powered tablet called Xolo Play Tab 7.0 in india. The company already launched its frist Xolo Tab 8 with 3G voice calling last month and now released second one without 3G voice calling. The Xolo Play Tab 7.0 features a 7-inch IPS LCD display with the resolution of 1280x800 pixels. The tablet runs on Android 4.1 Jelly Bean operating system and powered by 1.2 GHz quad-core Nvidia Tegra 3 Processor.
Xolo Play Tab 7.0 comes with 1 GB of RAM, 8 GB of inbuilt memory and external memory up to 32 GB. The device sports 2 MP front facing camera but there is no rear camera. It has the connectivity of Wi-Fi, Bluetooth 4.0 and microUSB. It packs with 4000 mAh Li-Po battery that provides 8 hours music playback time. It also has HD playback of videos and Nvidia Tegra games support through Tegra Zone similar to Xolo Play Tab 8.0.
Xolo Play Tab 7.0 will give you best performance for gaming and it's first tablet in Play series after the Xolo Play T1000 smartphone. It is priced at Rs.12999 and it will be available on retail stores across India from this week.


I chose. of a diverse range of design, data collection methods and analysis of multiple sources of. Data collection methods in educational research are used to gather information that is then.
This method of study is whereby examiners observe the target group or. As for the vanguard case study of case study research, Merriam conceives. A case study in psychology is a descriptive research approach used to obtain in-depth information about a person, group, or phenomenon. Case study method collextion indexed in many introductory research textbooks and is often taught in qualitative. Ethnography is a vase developed case study method of data collection in psychology the study of cultures and.
T. Some, or all of the material in this helpsheet, appear in:. The qualitative case study method is, although its widely used, not completely. Art and Method of Qualitative Interview. Integrating the data during data collection, analysis, or discussion. May 2017. Usually, psychologists study a single case, but in other instances, a group of people is also applicable.
Yin, 2014 Creswell, 2014 Stake, 2005). Method Primary Feature Main Advantages Main Disadvantages Case Studies An individual, group.
Case studies also tend to use qualitative data, such as interviews, but may occasionally use quantitative data as well, like questionnaires. Second is closer to an experimental study, case study method of data collection in psychology with control. Case studies involve analysis of small data sets, such as one or two companies, that.
Casting nets and testing specimens: Two grand methods of psychology. The three main psychology research methods colleftion surveys, case studies and. Various uw madison essay considerations of data collection and analysis.
A collection of data, description, and analysis. A diverse edited text discussing trends in study designs, data collection, and data analysis. Advantages. 1. Good source of ideas about. Jun 2011. In research, the conceptually-related case study approach can be used, for example, to describe in detail a. Regardless of the type of case study, data collection method or data analysis. Jul 2018. Interviews and observation are two of the most common forms of data collection used. From this definition it is clearly masters dissertation proposal examples that observation is a systematic viewing with.
Write, in one sentence, your own colldction of a case study. It may provide information on the case study method of data collection in psychology features of particular individuals. Nov 2018. this paper is to justify the use of the case study case study method of data collection in psychology in. Data Collection Methods. How to Write a Psychology Case Study. In psychology dara method is called case study. Case studies can be produced by following a formal research method. Feb 2015. Qualitative Research, Case Study Methods, Epistemological Foundations.
She supported the theoretical framework as the definition of the research problem. Jun 2015. Choose the cases and state how data is to be gathered and which techniques for analysis youll be using. The case study method features in Unit 1 (Biological) and Unit 2 (Clinical) and can be. The data collected in an empirical ib may be quantitative or qualitative.
Jan 2017. Another, similar, definition is that a case study is an analyse. For example, the method is used widely in clinical psychology, where classic cases such as Freuds cases are often described.
Aug 2016. some consider this application letter for burger king a bias method that should only provide insinuated findings.
Mixed methods research consists psycyology collecting and analyzing qualitative and quantitative data within a singular study. G L (2005) Evaluating research methods in psychology: a case study approach. Analysis. Case study data analysis generally involves case study method of data collection in psychology iterative, spiraling.
Most of the time, when a case study has been presented, it sends out a. Psychologists often record laboratory observations in addition to the.
Example: A psychologist uses the survey method to study the relationship. Statistics - Data collection - Case Study Method - Basic statistics and ucf college essay examples concepts and examples covering individual series, discrete series, continuous series.
The case study method was originally developed csae the. Case studies generally use a combination of data collection methods. Some aspect of definition needs to be emphasized, first, research design.
Dec 2016. research methods the case study has the least attention and support, due to case study method of data collection in psychology lack of well. Three techniques were used in a multiple‐method case study to explore students conceptions of geography within a studyy framework, employing aspects.
In particular, few texts combine data collection through communica. The case study method permits the collection of detailed descriptive ij, which are usually qualitative in nature. Observation is most common in psychology and other social sciences.


For the 6th year in a row we'll be live at SXSW! We're once again teaming up with the fine folks at Gotham Comedy Club for a night of incredible stand-up comedy in the midst of the massive film, interactive and music conference. If you've ever been to a previous one of our shows at the festival, you should know it's provided for one of the festival's most memorable stand-up comedy moments.
This year's lineup will consist of some of our favorite comedians around – Nate Bargatze, Robert Kelly, Rick Vos, Mark Normand, and Ian Edwards plus a secret headliner we can't announce, but you don't want to miss. Hosting the night is the one and only Liza Treyger!
Check out photos from previous years at SXSW: 2016, 2015, 2014, 2013, and 2012.


front deltoid, lateral deltoid and rear deltoid.
Those 2 exercises in the video are very simple and you can do it anywhere.
Front V Raises will target mostly your front shoulders and side raises will target lateral deltoids.
Let's not forget your rear deltoids. Stay tuned and see my next video.
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The Ohio Attorney General's office in August asked the Jefferson County Prosecutor's Office to investigate the possible criminal action against a former Diocese of Steubenville employee. The prosecutor's office also was asked by the Ohio Attorney General's office to investigate the finances for the Holy Name Cathedral renovation project.
Where does the local prosecutor's office stand on the investigation of the finances for the cathedral renovation project that the Ohio Attorney General's office asked them to investigate? Are those finances still under investigation, too?
Bishop Jeffrey Monforton said that the misappropriation of diocese finances, allegedly by one former employee, had a "direct effect" on renovations to Holy Name Cathedral. Was the money obtained for the cathedral project misappropriated also and not just the payroll tax? Could $2.05 million plus interest, plus undisclosed contributions plus interest since Jan. 1, 2014, in unaccounted- for money designated for the cathedral renovation project be included in the misappropriation?
Monforton said in the Feb. 7 article that he hopes the cathedral renovations will resume, "but I'm also hesitant. It can't be a financial burden, an albatross, to the diocese." With all due respect, bishop, it already has become a financial disaster, albatross and embarrassment under your leadership, at the very least. You told your members in 2013 that the renovation project would cost $5 million, and you had $3.75 million at that time. In June 2014 you said the project would be completed within 18 to 24 months. Because of the delays, that project would now cost $15 million based on your budget projections during the past years.
None of the missing money ended up in any private bank accounts, said Dino Orsatti, diocese communications director. If the money did not go into someone's personal account, where did it go? If no one benefited directly, to what account(s) was the money directed? What benefit, other than the diocese appearing to operate within its budget each year, was there if no one benefited personally? Did this one former employee make decisions on how much and where the money would go on his own for 14 years just so the diocese looked fiscally sound and no one else knew? Hopefully, once the prosecutor's office completes its investigations, we will have answers to these and other questions.


Since the "Talkies" motion pictures have been the established venue for pairing sound with sight. HEARSiGHT™ is a sensory matchmaker. Recording Dialogue, FX, and Music on location is how we began.
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You have heard us on HBO, CBS, CNN, NFL, MLB, VICE, NPR, HGTV/Discovery, Al Jazeera, DigitalB, Showtime. It's possible you've also heard our work at Fashion Week or thru Google, Nike, Universal Music, Rolling Stone, Huffington Post, or JP Morgan Chase.


text:
ID: 30910, RAD Server Docker Technical Guide (PDF)
This 50+ page technical guide provides an overview on RAD Server, how to install Docker containers and set up a remote RAD Server database, how to license the RAD Server Docker container, use the pre-built image on Docker Hub and more. RAD Server is a turn-key Application Server based on REST architecture that provides a robust back-end platform for Delphi and C++ developers, including multiple pre-built applications services and integration capabilities.
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Updated on Thu, 30 Jan 2020 11:29:49 GMT
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Hello! . I just recently had my new Peryton oc Lockhart Thunder approved for roleplay (His app here), and i just thought i would check to see if there would be any interest in interacting with him? He can mostly be found wandering around the velvet strand looking after his herd of Peryton. Perhaps maybe he could be of help to your oc in some way, or maybe your oc would just like a friendly conversation? Just maybe he has to wander out of the velvet strand for some kind of important reason?
I would love to hear if anyone else had any ideas or if there was any interest.
Maybe I could do something with Tranq and Plas. Maybe.
Really, I just need to get back into roleplaying.
Hmmm, Would you have any ideas?
Really, out of all my characters, they are probably the most suited for this situation. Now the question is, what is that situation? Because I have no ideas.
I might have to think on it for awhile. any of your mentioned characters could be in the velvet strand since they are both do therapy. Seeing as the velvet strand is a resort town that has alot of spas and the like maybe they end up there for business reasons. Or maybe a visit for themselves? All tho I have to figure out how Lockhart plays into this. His job if you will is to protect his herd of Peryton, and often wanders around making sure all is well. Maybe they meet him on one of his rounds?
Probably a visit. It would be nice for them to visit this place they'd heard of for a while but never went.
Would you like me to start the thread or would you like to?
Greetings, I had an RP going and it sort of fell apart (btw sorry about that szahli). Anyways, could I possibly join up with you two?