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Gene Starwind and the Outlaw Star search for the Galactic Leyline.
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He was often described as a curmudgeon, but his repertoire was filled with stories that reflected his concern for people.
In the days since his death, Red Fisher has been lionized as a great hockey writer and, over a career that spanned even decades, he was among the giants who wrote about grown men plying a child's game.
He was the definition of an insider, someone who shared the confidences of players, owners and executives. He changed the business of hockey and raised players' salaries with a series on the game's finances in 1970. That earned him the first of three National Newspaper Awards for sports writing. He broke the story of Guy Lafleur's premature retirement from the Canadiens in 1984. He was there when the Canadiens won 17 of their 24 Stanley Cups.
But the one quality that impressed me the most was his humanity. He was often described as a curmudgeon, but his body of work is filled with stories that reflect his concern for people.
I first noticed this trait in 1972 when Red called me back from a holiday in Nova Scotia to run the Montreal Star sports department. This was in the days when newspapers had money and the Star decided to send him to Moscow in advance of the 1972 Summit Series. The idea was to gauge Russian interest in the hockey series, but the result was an insightful look into the lives of ordinary Muscovites under Soviet rule.
My one of my favourite columns described a bus from the Nassau Coliseum to the airport after a Canadiens loss to the Islanders. The bus slowed to a crawl as it approached a multi-car accident. While the players complained about the inconvenience of being delayed, Red's concern was for the victims who were being loaded into ambulances.
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If you spent any time with Red, you knew how much importance he placed on family. He looked forward to his summer adventures with his wife, Tillie, and was devastated when she became ill and refused to leave their home.
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"They're saints," Red said when I visited him during his rehab from the first break.
My relationship with Red began when I was hired by the Montreal Star in 1965. Red and Ian MacDonald served as my mentors. Two years later, I sought advice when The Gazette offered me more money and a chance to cover the Alouettes.
The call came two years later and I didn't hesitate giving my notice to The Gazette. When my first paycheque arrived, I learned I had taken a $10 haircut, but Red soon rectified that.
When The Newspaper Guild arrived a few years later, Red was given a choice — be the editor or cover the Canadiens. He opted for the latter and I became his boss in name only.
"You have to let everyone know you're the boss, but just remember who hired you," he said.
I left a year later to cover the Montreal Olympics for the Toronto Sun but, after stops in Toronto, Windsor and Vancouver, I came back to The Gazette as the sports editor in 1987. The publisher was Clark Davey, one of the top journalists in the country who had a unorthodox management style. In 1984, Davey had suggested Red had lost his fastball, a notion that was dismissed after Red broke the Lafleur story.
Davey, who had previously hired me as a consultant at the Globe and Mail and as the sports editor of the Vancouver Sun, again suggested Red's best days were behind him. I told him I had s long history with Red and I thought he was doing fine.
The guy who had lost his fastball went on to win two more National Newspaper Awards and retired on his own terms in 2012.


The author informs the recipient that a list of donors to the Emily Hammond scholarship is enclosed.
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More than 100 volunteers from U.S. civilian militia groups are en route to the southern border in an effort to enforce the country's immigration laws, as thousands of Central American migrants make their way through Mexico to seek asylum in the U.S., reports the Washington Post.
The big picture: The volunteers and so-called "border vigilantes" were motivated by President Trump's constant warnings that the migrant caravan posed a threat to the country's national security. Trump has also claimed that "criminals and unknown Middle Easterners" are among those traveling. "My phone's been ringing nonstop for the last seven days. You got other militias, and husbands and wives, people coming from Oregon, Indiana. We've even got two from Canada," Shannon McGauley, president of the Texas Minutemen, tells the Post.
President Trump doesn't want to give Puerto Rico any more federal money for its recovery from Hurricane Maria, White House officials have told congressional appropriators and leadership. This is because he claims, without evidence, that the island's government is using federal disaster relief money to pay off debt.
The big picture: Trump also told senior officials last month that he would like to claw back some of the federal money Congress has already set aside for Puerto Rico's disaster recovery, claiming mismanagement.
Trump is as jazzed as ever about hitting foreign-made cars with steep tariffs. Just about every member of his senior economic team besides Peter Navarro believes this is a terrible idea. But they haven't swayed him.
What's happening: With each passing month, his zest for car tariffs only swells.
House Republicans and Democrats will soon elect their leaders to take them through the 2020 elections. Here's a read of the field, from well-placed Republican and Democratic sources.
What's happening: Republicans go first, holding their leadership elections on Wednesday.
Legislation designed to reduce federal prison sentences for some non-violent crimes and to help prisoners prepare for freedom is inching its way toward the Senate floor. And it just got a big boost from an unlikely ally: rank-and-file police.
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Democrats pulled off a slew of significant — yet still moderate — victories in state legislative and gubernatorial races across last week after years of brutal losses as they attempt to chip away at Republicans' sweeping control of the redistricting process before the next round begins after the 2020 census.
The backdrop: In most states, voting maps are drawn and approved by state lawmakers and governors every 10 years to reflect population change. Both parties have used their time in power to craft maps to disadvantage their opponents, but Republicans took it to the extreme in the 2011 redistricting cycle, preventing Democrats from netting more state legislative seats across this year's midterm elections.
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Driving the news tonight: Our poll finds 61% of Democrats think Republicans are racist … Inside the flood of new investigations about to hit Trump … Impeachment pressure … The new way women ran — and won.
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Thai businessman Chatchaval Jiaravanon has acquired Fortune magazine for $150 million, in just the latest example of a U.S. business publication ending up in the hands of an East Asian buyer.
Be smart: The day might not be that far off when there are no major American-owned business publications at all. Even Business Insider is German.
Top Saudi officials, including deputy intelligence chief Ahmed al-Assiri, who was blamed and fired last month for allegedly ordering the murder of journalist Jamal Khashoggi, inquired about assassinating the kingdom's Iranian enemies during a meeting with private businessmen in March 2017, reports the New York Times.
The big picture: The meeting was part of an effort by a group of international businessman to pitch the Saudis on an operation to sabotage Iran's economy. During the discussion, Saudi officials reportedly asked about killing Qassem Soleimani, a general in Iran's Islamic Revolutionary Guard Corps, but were rejected by the businessmen's lawyer. Per the Times, the episode highlights the fact that more than a year before the killing of Khashoggi, officials close to Crown Prince Mohammed bin Salman had increasingly become interested in carrying out high-risk, covert operations targeting the kingdom's enemies.


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text:
\section{Introduction}

\begin{figure}
\centering
\vspace{0mm}
\centering
\includegraphics[width=6.3cm]{images/first_07.pdf}
\caption{Reachtubes of LRT-NG~\cite{gruenbacher2020lagrangian} and GoTube for a CT-RNN controlling CartPole-v1 environment. CAPD~\cite{capd} and Flow*~\cite{flowstar} failed.}
\label{fig:intro}
\end{figure}

The use of deep-learning systems powered by con\-tin\-u\-ous-depth models continues to grow, especially due to the revival of neural ordinary differential equations (Neural ODEs)~\cite{neuralODEs}. These models parametrize the derivative of the hidden states by a neural network. The resulting system of differential equations can perform strong function approximation and generative modeling. Ensuring their safety and robustness in any of these fronts is a major imperative, particularly in high-stakes decision-making applications such as medicine, automation, and finance.

A particularly appealing approach is to construct a tight overapproximation of the set of states reached over time according to the neural network's dynamics (a bounding tube) and provide deterministic
or stochastic
guarantees for the conservativeness of the tube's bounds.

Deterministic verification approaches ensure conservative bounds~\cite{flowstar,gowal2018effectiveness,mirman2018differentiable,bunel2020lagrangian,capd,gruenbacher2020lagrangian}, but often sacrifice speed and accuracy \cite{ehlers2017formal}, and thus scalability; see CAPD, Flow*, and LRT-NG in Fig.~\ref{fig:intro} and Fig.~\ref{fig:three graphs}. Stochastic methods, on the other hand, only ensure a weaker notion of conservativeness in the form of confidence intervals (stochastic bounds).  This, however, allows them to achieve much more accurate and faster verification algorithms that scale up to much larger dynamical system~\cite{probreach,gp,Gruenbacher2021verification}.

It was recently shown theoretically that stochastic verification approaches based on Lagrangian reachability (SLR) could provably guarantee confidence intervals for continuous-depth models~\cite{Gruenbacher2021verification}. The proposed theoretical framework suggests performing both stochastic global optimization and local differential optimization~\cite{stochGlobOptim, pontryagin2018mathematical}, and uses \emph{interval arithmetic} to symbolically bound the Lipschitz constant. Thus, it can construct a bounding ball of the reachable states at every time step, and over time, a tight bounding Tube. Although these theoretical results suggest an elegant way to avoid compounding errors, the SLR algorithm has not been implemented, so is this approach computationally tractable in practice?

We implemented the SLR algorithm as instructed in \cite{Gruenbacher2021verification}. We observed that even after resolving the first-occurring inefficient sampling and their vanishing gradient problems, the algorithm still blew up in time, even for low-dimensional benchmarks such as the Dubins Car. There are three fundamental algorithmic constraints of the symbolic techniques such as stochastic Lagrangian reachability that result in them being computationally intractable:
1)~the use of interval arithmetic for computing a conservative upper bound for the Lipschitz constant of the system fundamentally limits the scalability of reachability-based verification methods, 2)~the use of local gradient descent to search for local maxima in practice is more expensive than a simple local search, and 3)~the computational overhead due to the propagation of many initial states is high.

\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{images/fig_2.pdf}
\caption{GoTube in a nutshell. The center $x_0$ of ball $\mathcal{B}_0\,{=}\,B(x_0, \delta_0)$, with $\delta_0$ the initial perturbation, and samples $x$ drawn uniformly from $\mathcal{B}_0$'s surface, are numerically integrated in time to $\chi(t_j, x_0)$ and $\chi(t_j, x)$, respectively. The Lipschitz constant of $\chi(t_j, x)$ and their distance $d_j(x)$ to $\chi(t_j, x_0)$ are then used to compute Lipschitz caps around samples $x$, and the radius $\delta_j$ of bounding ball $\mathcal{B}_j$ depending on the chosen tightness factor $\mu$. The ratio between the caps' surfaces and $\mathcal{B}_0$'s surface are correlated to the desired confidence $1\,{-}\,\gamma$. }
\label{fig:gotube}
\vspace*{-4ex}
\end{figure}

In this work, we propose technical solutions for these fundamental issues and introduce a practical stochastic verification algorithm for continuous-time models. In particular, to tackle the first fundamental challenge introduced above, we develop a new theory that allows us to compute stochastic bounds for the Lipschitz constant in order to define a spherical cap around each sample, where the maximum perturbation is stochastically bounded (Lipschitz cap). As such, we are able to remove the conservative interval-based computation of the Lipschitz constant. Furthermore, we provide convergence guarantees for computing the upper bound of the confidence interval for the maximum perturbation at time $t_j$ with confidence level $1\,{-}\,\gamma$ and tube tightness $\mu$, using the estimation of the Lipschitz constant. This eliminates the dependence on the propagation horizon and considerably reduces computational complexity in the number of samples. We directly use this new Lipschitz constant computation framework instead of the costly interval arithmetic.

We supply our global optimization scheme with a simple sampling process to propagate the initial states in parallel, according to the neural network's dynamics. This compensates for local differential optimization with additional samples. Our algorithm is called GoTube, as it solves a set of global optimization problems to construct a tight and computationally tractable enclosure (Tube) of all possible evolutions of the system for a given time horizon.

GoTube takes advantage of advanced automatic differential toolboxes such as JAX to perform highly parallel and tensorized operations to further enhance the runtime of the verification suite. On a large set of experiments with continuous-depth models, GoTube substantially outperforms state-of-the-art verification tools in terms of the size of the initial ball, speed, time-horizon, task completion, and scalability. We summarize the contributions of our paper as follows:

\begin{itemize}
    \item A novel and efficient theory for computing stochastic bounds for the Lipschitz constant of the system, which helps us achieve tight reachtubes for continuous-time dynamical systems.
    \item We prove convergence guarantees for the GoTube Algorithm, thus ensuring that the algorithm terminates in finite time even using stochastic Lipschitz caps around the samples instead of deterministic local balls.
    \item We perform a diverse set of experiments on continuous-time models with increasing complexity and demonstrate that GoTube considerably outperforms state-of-the-art verification tools.

\end{itemize}

\begin{table*}[t]
\small
\centering
\caption{Related work on the reachability analysis of continuous-time systems. Determ.= Deterministic. "No" indicates a stochastic method. Table content is partially reproduced from \cite{Gruenbacher2021verification}.}
\begin{adjustbox}{width=2\columnwidth}
\begin{tabular}{l|c|c|c|c}
\toprule
\textbf{Technique} & \textbf{Determ.} & \textbf{Parallel} & \textbf{wrapping} & \textbf{Arbitrary }\\
 & &  & \textbf{effect} & \textbf{Time-horizon}\\
\midrule
LRT \cite{Cyranka2017} with Infinitesimal strain theory & yes & no & yes & no \\
CAPD \cite{capd} implements Lohner algorithm & yes & no &  yes & no  \\
Flow-star \cite{flowstar} with Taylor models & yes & no  & yes & no  \\
$\delta$-reachability \cite{deltadecidable} with approximate satisfiability & yes & no & yes & no \\
C2E2 \cite{c2e2} with discrepancy functions & yes & no  & yes & no \\
LDFM \cite{fansimul} by simulation, matrix measures & yes & yes  & no & no \\
TIRA \cite{tira} with second-order sensitivity & yes & yes & no & no \\
Isabelle/HOL \cite{isabelle} with proof-assistant & yes & no  & yes & no \\
Breach \cite{breach,donze} by simulation & yes & yes  & no & no \\
PIRK \cite{pirk} with contraction bounds & yes & yes & no & no \\
HR \cite{hr} with hybridization & yes & no & yes & no \\
ProbReach \cite{probreach2} with $\delta$-reachability,  & no & no  & yes & no \\
VSPODE \cite{reliablecomput} using p-boxes & no &no  & yes & no \\
Gaussian process (GP) \cite{gp} & no & no & no & no \\
Stochastic Lagrangian reachability SLR \cite{Gruenbacher2021verification} & no & yes & no & no \\
\textbf{GoTube (Ours)} & no & yes & no & \textbf{yes} \\
\bottomrule
\end{tabular}
\end{adjustbox}
 \label{tab:related_works}
\end{table*}

\section{Related Work}
\textbf{Global Optimization.} Efficient local optimization methods such as gradient descent cannot be used for global optimization since such problems are typically non-convex. Thus, many advanced verification algorithms tend to use global optimization schemes \cite{NEURIPS2018_be53d253,bunel2020lagrangian}. Depending on the properties of the objective function, e.g. smoothness, various types of global optimization techniques exist. For instance, interval-based branch-and-bound (BaB) algorithms \cite{neumaier_2004,hansen2003global} work well on differentiable objectives up to a certain scale, which has recently been improved \cite{de2021improved}. There are also Lipschitz-global optimization methods for satisfying Lipschitz conditions \cite{Lipschitz,kvasov2013Lipschitz}. For example, a method for computing the Lipschitz constant of deep neural networks to assist with their robustness and verification analyses was recently proposed in~\cite{NEURIPS2019_95e1533e} and~\cite{bhowmick2021lipbab}. Additionally, there are evolutionary strategies for global optimization using the covariance matrix computation~\cite{cma,cmaes}. In our approach, for global optimization, we use random sampling and compute neighborhoods (Lipschitz caps) of the samples, where we have probabilistic knowledge about the values, such that we are able to correspondingly estimate the stochastic global optimum with high confidence.~\cite{stochGlobOptim}.

\noindent\textbf{Verification of Neural Networks.}
A large body of work tried to enhance the robustness of neural networks against adversarial examples \cite{goodfellow2014explaining}. There are efforts that show how to break the many defense mechanisms proposed \cite{athalye2018obfuscated,lechner2021adversarial}, until the arrival of methods for formally verifying robustness to adversarial attacks around neighborhoods of data \cite{henzinger2021scalable}.
The majority of these complete verification algorithms for neural networks work on piece-wise linear structures of small-to-medium-size feedforward networks \cite{NEURIPS2019_246a3c55}. For instance, \cite{bunel2020branch} has recently introduced a BaB method that outperforms state-of-the-art verification methods \cite{katz2017reluplex,NEURIPS2020_f6c2a0c4}. A more scalable approach for rectified linear unit (ReLU) networks \cite{nair2010rectified} was recently proposed based on Lagrangian decomposition; this approach significantly improves the speed and tightness of the bounds \cite{de2021improved}. The proposed approach not only improves the tightness of the bounds but also performs a novel branching that matches the performance of the learning-based methods \cite{lu2020nueral} and outperforms state-of-the-art methods \cite{NEURIPS2018_d04863f1,singheth,bak2020improved,henriksen2020efficient}. While these verification approaches work well for feedforward networks with growing complexity, they are not suitable for recurrent and continuous neural network instances, which we address in this work.

\noindent\textbf{Verification of Continuous-time Systems.} Reachability analysis is a verification approach that provides safety guarantees for a given continuous dynamical system \cite{gurung2019parallel,vinod2021stochastic}. Most dynamical systems in safety-critical applications are highly nonlinear and uncertain in nature \cite{lechner2020neural}. The uncertainty can be in the system's parameters \cite{sreach,probreach,reliablecomput}, or their initial state~\cite{reliablecomput,10.1145/3126508}. This is often handled by considering balls of a certain radius around them. Nonlinearity might be inherent in the system dynamics or due to
discrete mode-jumps \cite{e71eb65e23844408b72fe95a84f88cb6}.
We provide a summary of methods developed for the reachability analysis of continuous-time ODEs in Table~\ref{tab:related_works}.

A fundamental shortcoming of the majority of the methods described in Table~\ref{tab:related_works} is their lack of scalability while providing conservative bounds.
In this paper, we show that GoTube establishes the state-of-the-art for the verification of ODE-based systems in terms of speed, time-horizon, task completion, and scalability on a large set of experiments.

\section{Setup}
In this section, we introduce our notation, preliminary concepts, and definitions required to state and prove the stochastic bounds that GoTube guarantees for time-continuous process models.

\emph{Continuous-depth models.} These are a special case of non\-li\-near ordinary differential equations (ODEs), where the model is defined by the derivative of the unknown states $x$ computed by a vector-valued function $f\,{:}\,\mathbb{R}^n\,{\rightarrow}\,\mathbb{R}^n$, which is assumed to be Lipschitz-continuous and forward-complete:
\begin{align}\label{eq:ODE}
    \partial_t x = f(x),\quad x(t_0) \in \mathcal{B}_0\,{=}\,B(x_0, \delta_0),
\end{align}
$\mathcal{B}_0$ defines the initial ball (a region of initial states, whose radius quantifies the magnitude $\delta_0$ of a perturbation of its center $x_0$).
Time dependence can be incorporated by an additional variable $x$ with $\delta_t x = 1$. Thus this definition naturally extends to time-varying ODEs. Nonlinear ODEs do not have in general closed-form solutions, and therefore one can not compute symbolically the solution $\chi(t_j, x)$ for all $x\,{\in}\,\mathcal{B}_0$. For a sequence of $k$ timesteps from time $t_0$ until time horizon $T$: $t_0\,{<}\dots{<}\,t_k = T$, we use numerical ODE solvers to compute $\chi(t_j, x)$ of the initial value problem (IVP) in Eq.~\eqref{eq:ODE} at time $t_j$ starting at different points $x(t_0)\,{=}\,x$.

We extend this computation to the entire ball by numerically integrating the center $x_0$ and a set of points $x\,{\in}\,\mathcal{V}$, uniformly sampled from the surface of the ball, and using this information to compute stochastic upper bounds for the possible evolutions of the system.
We define the bounding ball and bounding tube as follows:
\begin{definition}[Bounding Ball]
     Given an initial ball $\mathcal{B}_0\,{=}\,B(x_0, \delta_0)$, we call $\mathcal{B}_j = B(\chi(t_j, x_0), \delta_{j}(\mathcal{B}_0))$ a {\em bounding ball} at time $t_j$, if it stochastically bounds the reachable states $x$ at time $t_j$ for all initial points around $x_0$ having the maximal initial perturbation $\delta_0$.
\end{definition}
As we do not only want to bound the perturbation at one specific time, but on a time series, we define:
\begin{definition}[Bounding Tube]
     Given an initial ball $\mathcal{B}_0 = B(x_0, \delta_0)$ and bounding balls for $t_0\,{<}\dots{<}\,t_k\,{=}\,T$, we call the series of bounding balls $\mathcal{B}_1, \mathcal{B}_2, \dots, \mathcal{B}_k$ a \emph{bounding tube}.
\end{definition}
\emph{Maximum perturbation at time $t_j$}. To compute a bounding tube, we have to compute at every timestep $t_j$ the maximum perturbation $\delta_j$, which is defined as a solution of the optimization problem:
\begin{align}\label{eq:optim}
    \delta_j \ge \max_{x\in\mathcal{B}_0}\|\chi(t_j, x) - \chi(t_j, x_0)\| = \max_{x\in\mathcal{B}_0} d(t_j, x),
\end{align}
where $d_j(x)=d(t_j,x)$ denotes the {\em distance} at time $t_j$, if the initial center $x_0$ is known from the context. As stated in~\cite{Gruenbacher2021verification}, the radius at time $t_j$ can be over-approximated by solving a global optimization problem on the surface of the initial ball $\mathcal{B}_0$: as we require Lipschitz-continuity and forward-completeness of the ODE in Eq.~\eqref{eq:ODE}, the map $x \mapsto \chi(t_j,x)$ is a homeomorphism and commutes with closure and interior operators. In particular, the image of the boundary of the set $\mathcal{B}_0$ is equal to the boundary of the image $\chi(t_j,\mathcal{B}_0)$. Thus, Eq.~\eqref{eq:optim} has its optimum on the surface of the initial ball $\mathcal{B}_0^S = \textrm{surface}(\mathcal{B}_0)$, and we will only consider points on the surface.

\section{Main Results}
Our GoTube algorithm and its theory solve fundamental scalability problems of related works (see Table~\ref{tab:related_works}) by replacing interval arithmetic used to compute deterministic caps with stochastic Lipschitz caps. This enables us to verify continuous-depth models up to an arbitrary time-horizon, a capability beyond what was achievable before.

To be able to do that, we formulated Theorems on: 1)~How to choose the radius of a Lipschitz cap using stochastic bounds of local Lipschitz constants of the samples together with the expected difference quotients. 2)~Convergence guarantees using these new stochastic caps, as they are used by GoTube to compute the probability of $\delta_j$ being an upper bound of the biggest perturbation.
In addition, we implemented tensorization and substantially increased the number of random samples, thus being able to remove the dependence on the propagation-horizon of the gradient descent and increasing the computation speed to be able to deal with continuous-depth models.

\begin{algorithm}[t]
    \caption{GoTube}
    \label{algorithm:GoTube}
    \begin{algorithmic}[1]
    \REQUIRE initial ball $\mathcal{B}_0 = B(x_0, \delta_0)$, time horizon T, sequence of timesteps $t_j$ ($t_0<\dots< t_k=T$), error tolerance $\mu\,{>}\,1$, confidence level $\gamma\,{\in}\, (0,1)$, batch size $b$, distance function $d$
    \vspace*{2mm}
    \STATE $\mathcal{V}\leftarrow\{\}$ \quad(list of visited random points)
    \STATE \textbf{sample batch} $x^B \in \mathcal{B}_0^S$
    \FOR{$(j=1; j\le k; j=j+1)$}\label{line:for loop Reachsets}
        \STATE $\bar{p}\leftarrow 0$
        \WHILE{$\bar{p} < 1 - \gamma$}
            \STATE $\mathcal{V}\leftarrow \mathcal{V} \cup \{x^B\}$
            \STATE $x_j \leftarrow \chi(t_j, x_0)$\quad (integrate initial center point)
            \STATE $\bar{m}_{j,\mathcal{V}} \leftarrow \max_{x\in\mathcal{V}} d(t_j, x)$
            \STATE \textbf{compute} local Lipschitz constants $\lambda_x$ for $x\in\mathcal{V}$
            \STATE \textbf{compute} expected local difference quotient $\Delta\lambda_{x,\mathcal{V}}$ for $x\in\mathcal{V}$
            \STATE \textbf{compute} cap radii $r_x(\lambda_x, \Delta\lambda_{x,\mathcal{V}})$ (Thm.~\ref{thm:lipschitz cap}) for $x\in\mathcal{V}$\label{line:increase radii}
            \STATE $\mathcal{S}\leftarrow \bigcup_{x\in\mathcal{V}} B(x,r_x)^S$ \quad (total covered area)
            \STATE $\bar{p} \leftarrow \mathbb{P}(\mu \cdot \bar{m}_{j,\mathcal{V}} \ge m^\star)$
            \STATE \textbf{sample batch} $x^B \in \mathcal{B}_0$
        \ENDWHILE
        \STATE $\delta_j\leftarrow \mu\cdot\bar{m}_{j,\mathcal{V}}$
        \STATE $\mathcal{B}_j\leftarrow B(x_j, \delta_j)$
    \ENDFOR
    \RETURN $(\mathcal{B}_1,\dots,\mathcal{B}_k)$
    \end{algorithmic}
\end{algorithm}

We start by describing the GoTube Algorithm. This facilitates the comprehension of the different computation and theory steps. Given a continuous-depth model as in Eq.~\eqref{eq:ODE}, an initial ball $\mathcal{B}_0$ defined by a center point $x_0$ and the maximum initial perturbation $\delta_0$, a time horizon $T$ with a sequence of timesteps $t_j~(t_0\,{<}\dots{<}\,t_k=T)$, a confidence level $\gamma\,{\in}\,(0,1)$, a tightness factor $\mu\,{>}\,1$, a batch size $b$, and a distance function $d$. The output of the GoTube algorithm is a bounding tube that stochastically over-approximates at most by $\mu$ the propagated initial perturbation from the center $x_0$ with a probability higher than $1\,{-}\,\gamma$.

GoTube starts by sampling a batch (tensor) $x^B\in\mathcal{B}_0^S$. It then iterates for the $k$ steps of the time horizon $T$ the following. After initializing the probability ensured to zero, and the visited states to the empty set, it loops until it reaches the desired confidence (probability) $1\,{-}\,\gamma$, by increasingly taking additional batches. In each iteration, it integrates the center and the already available samples from their previous time step and the possibly new batches from their initial state (for simplicity, the pseudocode does not make this distinction explicit). GoTube then computes the maximum distance from the integrated samples to the integrated center, their local Lipschitz constant
according to the variational equation of Eq.~\eqref{eq:ODE}. Based on this information GoTube then computes the mean Lipschitz statistics and the cap radii accordingly. The total surface of the caps is then employed to compute and update the achieved confidence (probability). Once the desired confidence is achieved, GoTube exits the inner loop and computes the bounding ball in terms of its center and radius, which is given by tightness factor $\mu$ times the maximum distance $\bar{m}_{j,\mathcal{V}}$. After exiting the outer loop, GoTube returns the bounding tube.
\begin{definition}[Lipschitz Cap]
    Let $\mathcal{V}$ be the set of all sampled points, $x\in\mathcal{V}$ be a sample point on the surface of the initial ball, $\bar{m}_{j,\mathcal{V}} = \max_{x\in\mathcal{V}} d_j(x)$ be the sample maximum and $B(x,r_x)^S = B(x,r_x)\cap\mathcal{B}_0^S$ be a spherical cap around that point. We call the cap $B(x,r_x)^S$ a $\gamma,t_j$-Lipschitz cap, if it holds that $\mathbb{P}\left(d_j(y) \le \mu\cdot \bar{m}_{j,\mathcal{V}}\right)\ge 1-\gamma$ for all $y\in B(x,r_x)^S$.
\end{definition}
Lipschitz caps around the samples are a stochastic version of local balls around samples, commonly used to cover state space. Intuitively, the points within a cap do not have to be explored. The difference with Lipschitz caps is, that we stochastically bound the values inside that space and develop a theory to enable us to calculate a probability of having found an upper bound of the true maximum $m_j^\star = d_j(x_j^\star) = \max_{x\in\mathcal{B}_0} d_j(x)$ of the optimization problem in Eq.~\eqref{eq:optim}.
Our objective is to avoid the usage of interval arithmetic for computing the Lipschitz constant, as it impedes scaling up to continuous depth models. Instead, we define stochastic bounds on the Lipschitz constant to set the radius $r_x$ of the Lipschitz caps, such that $\mu\cdot\bar{m}_{j,\mathcal{V}}$ is a $\gamma$-stochastic upper bound for all distances $d_j(y)$ at time $t_j$ from values inside the ball $B(x, r_x)^S$.

\begin{theorem}[Radius of Stochastic Lipschitz Caps]\label{thm:lipschitz cap}
    Given a continuous-depth model $f$ from Eq.~\eqref{eq:ODE}, $\gamma\,{\in}\,(0,1)$, $\mu\,{>}\,1$, target time $t_j$,

    the set of all sampled points $\mathcal{V}$, the number of sampled points $N\,{=}\,|\mathcal{V}|$, the sample maximum $\bar{m}_{j,\mathcal{V}}\,{=}\,\max_{x\in\mathcal{V}} d_j(x)$, the IVP solutions $\chi(t_j,x)$, and the corresponding stretching factors $\lambda_x\,{=}\,\|\partial_x\chi(t_j,x)\|$ for all $x\,{\in}\,\mathcal{V}$. Let us define $\hat{\gamma} = 1-\sqrt{1-\gamma}$.
    Let $\Delta\lambda_{\mathcal{V}}$ be the $\sqrt{1-\gamma}$-quantile of a stochastic lower bound $F_{L,\hat{\gamma}}$ as defined by Lemma 1 in the Appendix:
    \begin{align}\label{eq:expected difference quotient}
        \Delta\lambda_{\mathcal{V}}(\gamma) = F_{L,\hat{\gamma}}^{-1}(\sqrt{1-\gamma}),
    \end{align}
    Let $r_x$ be defined as:
    \begin{align}\label{eq:cap radius}
        r_{x} = \frac{\left(-\lambda_x + \sqrt{\lambda_x^2 + 4\cdot\Delta\lambda_{x,\mathcal{V}}\cdot(\mu\cdot\bar{m}_{j,\mathcal{V}}-d_j(x))}\right)}{2\cdot\Delta\lambda_{x,\mathcal{V}}},
    \end{align}
    then it holds that:
    \begin{align}\label{eq:cap probability}
        \mathbb{P}\left(d_j(y) \le \mu\cdot \bar{m}_{j,\mathcal{V}}\right)\ge 1-\gamma\quad \forall y\in B(x,r_x)^S,
    \end{align}
    and thus that $B(x, r_x)^S$ is a $\gamma, t_j$-Lipschitz cap.
\end{theorem}
The full proof is provided in the Appendix. \emph{Proof sketch:} As $\Delta\lambda_{x,\mathcal{V}}$ is the $\sqrt{1-\gamma}$-quantile of $\max_{x,y}|\lambda_x-\lambda_y|/\|x-y\|$, it holds that $\mathbb{P}(\lambda_y \le \lambda_x + \Delta\lambda_{x,\mathcal{V}} \cdot \|x-y\|)\ge 1-\gamma$. Therefore Eq.~\eqref{eq:cap radius} follows by solving the following equation: $(\mu\cdot\bar{m}_{j,\mathcal{V}}-d_j(x))=\lambda_x r_x + \Delta\lambda_{x,\mathcal{V}} r_x^2$.

Using conditional probability, we are able to state that the convergence guarantee holds for the GoTube Algorithm, thus ensuring that the Algorithm terminates in finite time even using stochastic Lipschitz caps around the samples instead of deterministic local balls.

\begin{theorem}[Convergence via Lipschitz Caps]\label{thm:stochastic guarantee} Given the tightness factor $\mu > 1$, the set of all sampled points $\mathcal{V}$ and the sample maximum $\bar{m}_{j,\mathcal{V}} = \max_{x\in\mathcal{V}} d_j(x)$. Let the initial ball maximum be defined by $m^\star_j=\max_{x\in\mathcal{B}_0} d_j(x)$. Then:
    \begin{align}\label{eq:stochastic guarantee}
        \hspace*{-2ex}\forall\gamma\in(0,1),\exists N\in\mathbb{N}\textrm{ s.t. }
        \mathbb{P}(\mu\cdot\bar{m}_{j,\mathcal{V}}\ge m^\star_j) \ge 1-\gamma
    \end{align}
    where $N=|\mathcal{V}|$ is the number of sampled points.
\end{theorem}
The full proof is provided in the Appendix. \emph{Proof sketch:} Let $x^\star_j$ be a point such that $d_j(x^\star_j) = m^\star_j$. Given $\gamma\in(0,1)$ and cap radii $r_x$, we expand the convergence guarantee from deterministic local balls to stochastic Lipschitz caps. For local balls it holds that $\exists N \in\mathbb{N}\colon\mathbb{P}(\exists x\in\mathcal{V}\colon\,B(x,r_x)^S\owns x^\star_j)\ge\sqrt{1-\gamma}$. Using a set of sampled points $\mathcal{V}$ with cardinality $N$ and using $1-\sqrt{1-\gamma}$ instead of $\gamma$
in Eq.~\eqref{eq:expected difference quotient}
and Theorem~\ref{thm:lipschitz cap}, the resulting probability is larger than $\sqrt{1-\gamma}$. From the definition of a Lipschitz cap it follows that $\mathbb{P}( d_j(x^\star) \le \mu\cdot\bar{m}_{j,\mathcal{V}} |\exists x\in\mathcal{V}\colon B(x,r_x)^S\owns x^\star)\ge \sqrt{1-\gamma}$. For any sets $A, B$ it holds that $\mathbb{P}(A)\ge \mathbb{P}(A\cap B) = \mathbb{P}(A | B) \cdot \mathbb{P}(B)$, thus we multiply both probabilities and therefore Eq.~\eqref{eq:stochastic guarantee} holds.

\begin{figure*}[t]
     \centering
     \includegraphics[width=0.7\textwidth]{images/fig_3.pdf}

        \caption{Visualization of the reachtubes constructed for the Dubin's car model with various reachability methods. While the tubes computed by existing methods (LRT-NG, Flow* and CAPD) explode at $t\,{\approx}\,20s$ (this moment is shown on the right side of the figure) due to the accumulation of over-approximation errors (the infamous wrapping effect), GoTube can keep tight bounds beyond $t\,{>}\,40s$ for a 99\% confidence level (using 20000 samples, $\mu=1.1$ and runtime of one hour). Note also the chaotic nature of 100 executions. }
        \label{fig:three graphs}

\end{figure*}

\section{Experimental Evaluation}
We perform a diverse set of experiments with GoTube to evaluate its performance and identify its characteristics and limits in verifying continuous-time systems with increasing complexity. We run our evaluations on a standard workstation machine setup (12 vCPUs, 64GB memory) equipped with a single GPU for a per-run timeout of 1 hour (except for runtimes reported in Figure \ref{fig:pareto}).

\subsection{On the volume of the bounding balls with GoTube}
Our first experimental evaluation concerns the overapproximation errors of the constructed bounding tubes.
An ideal reachability tool should be able to output an as tight as possible tube that encloses the system's executions. Consequently, as our comparison metric, we will report the average volume of the bounding balls, with less volume is better.
We use the benchmarks and settings of \cite{gruenbacher2020lagrangian} (same radii, time horizons, and models) as the basis of our evaluation. In particular, we compare GoTube to the deterministic, state-of-the-art reachability tools LRT-NG, Flow*, CAPD, and LRT. We measure the volume of GoTube's balls at the confidence levels of 90\% and 99\%, using $\mu=1.1$ as the tightness factor (in the third experiment we will talk in more detail about the trade-off between tightness and runtime).

The results are shown in Table \ref{tab:volume}. For the first five benchmarks, which are classical dynamical systems, we use the small time horizons $T$ and small initial radii $\delta_0$, which the other tools could handle. GoTube, with 99\% confidence, achieves a competitive performance to the other tools, coming out on top in 3 out of 5 benchmarks - using $\mu=1.1$ as the tightness bound. Intuitively this means, we are confident that the overapproximation includes all executions with a confidence level $1-\lambda$, but this overapproximation might not be as tight as desired. GoTube is able to achieve any desired tightness by reducing $\mu$ and increasing the runtime. The specific reachtubes and the chaotic nature of hundred executions of Dubin's car are shown in Figure \ref{fig:three graphs}. As one can see, the GoTube reachtube extends to a much longer time horizon, which we fixed at 40s. All other tools blew up before 20s. For the two problems involving neural networks, GoTube produces significantly tighter reachtubes.

\begin{table}[t]
    \centering
    \caption{Comparison of GoTube (using tightness bound $\mu=1.1$) to existing reachability methods. The first five benchmarks concern classical dynamical systems, whereas the two bottom rows correspond to time-continuous RNN models (LTC= liquid time-constant networks) in a closed feedback loop with an RL environment \cite{Hasani2021liquid,vorbach2021causal}. The numbers show the volume of the constructed tube. Lower is better; best number in bold.    }
    \begin{adjustbox}{width=1\columnwidth}
    \begin{tabular}{l|llll|ll}
    \toprule \multirow{2}{*}{Benchmark} & \multirow{2}{*}{LRT-NG} &\multirow{2}{*}{Flow*}  &\multirow{2}{*}{CAPD}  & \multirow{2}{*}{LRT} & \multicolumn{2}{c}{GoTube}\\
         &    &   &  &  & (90\%) & (99\%)\\
       \midrule
       Brusselator  & 1.5e-4 & 9.8e-5  & 3.6e-4 & 6.1e-4 & 8.6e-5 & \textbf{8.6e-5} \\
       Van Der Pol &  4.2e-4 & \textbf{3.5e-4} & 1.5e-3  &  3.5e-4 & 3.5e-4 &  \textbf{3.5e-4} \\
       Robotarm & \textbf{7.9e-11} & 8.7e-10  & 1.1e-9 & Fail &  2.5e-10 & 2.5e-10 \\
       Dubins Car & 0.131  & 4.5e-2 & 0.1181 & 385 &  2.5e-2 & \textbf{2.6e-2} \\
       Cardiac Cell &  \textbf{3.7e-9} & 1.5e-8 & 4.4e-8  & 3.2e-8 &   4.2e-8 &  4.3e-8 \\

       \midrule
       CartPole-v1+LTC  & 4.49e-33  & Fail & Fail & Fail & 2.6e-37 &  \textbf{4.9e-37}  \\
       CartPole-v1+CTRNN  & 3.9e-27  & Fail & Fail & Fail &  9.9e-34 &  \textbf{1.2e-33}   \\
      \bottomrule
    \end{tabular}
    \end{adjustbox}
    \label{tab:volume}
\end{table}

\subsection{GoTube provides safety bounds up an arbitrary time horizon}
In our second experiment, we evaluate for how long GoTube and existing methods can construct a reachtube before exploding due to overapproximation errors. To do so, we extend the benchmark setup by increasing the time horizon for which the tube should be constructed, use tightness bound $\mu=1.1$ and set a 95\% confidence level, that is, probability of being conservative.
\begin{table}[ht]
\small
    \centering
    \caption{Results of the extended benchmark by longer time horizons. The numbers show the volume of the constructed tube, ``Blowup'' indicates that the method produced \texttt{Inf} or \texttt{NaN} values due to a blowup. Lower is better; the best method is shown in bold.}
    \begin{tabular}{l|cc|cc}\toprule
         Benchmark  & \multicolumn{2}{c|}{CartPole-v1+CTRNN} & \multicolumn{2}{c}{CartPole-v1+LTC} \\
         Time horizon  & 1s & 10s & 0.35s & 10s \\\midrule
         LRT & Blowup & Blowup & Blowup & Blowup \\
         CAPD & Blowup & Blowup & Blowup & Blowup \\
         Flow* & Blowup & Blowup & Blowup & Blowup\\
         LRT-NG & 3.9e-27 & Blowup & 4.5e-33 & Blowup\\
         GoTube (ours) & \textbf{8.8e-34} & \textbf{1.1e-19} & \textbf{4.9e-37} & \textbf{8.7e-21} \\\bottomrule
         \end{tabular}
    \label{tab:time}

\end{table}

The results in Table~\ref{tab:time} demonstrate that GoTube produces significantly longer reachtubes than all considered state-of-the-art approaches, without suffering from severe overapproximation errors. Particularly, Figure~\ref{fig:intro} visualizes the difference to the existing methods and overapproximation margins for two example dimensions of the CartPole-v1 environment and its CT-RNN controller.

\begin{figure*}[t]
    \centering
    \includegraphics[width=0.8\textwidth]{images/pareto.pdf}
    \caption{GoTube's runtime (x-axis) and volume size (y-axis) as a function of the tightness factor $\mu$. Volume was normalized by the volume obtained with the lowest $\mu$ (4.3e-13, 2.4e-12, and 2.1e-38 in particular).}
    \label{fig:pareto}
\end{figure*}

\subsection{GoTube can trade runtime for reachtube tightness}
In our last experiment, we introduced a new set of benchmark models entirely based on continuous-time recurrent neural networks.
The first model is an unstable linear dynamical system of the form $\dot{x} = Ax + Bu$ that is stabilized by a CT-RNN policy via actions $u$. The second model corresponds to the inverted pendulum environment, which is similar to the CartPole environment but differs in that the control actions are applied via a torque vector on the pendulum directly instead of moving a cart. The CT-RNN policies for these two environments were trained using deep RL.
Our third new benchmark model concerns the analysis of the learned dynamics of a CT-RNN trained on supervised data. In particular, by using the reachability frameworks, we aim to assess if the learned network expressed oscillatory behavior. The CT-RNN state vector consists of 16 dimensions, which is twice as much as existing CT-RNN reachability benchmarks \cite{gruenbacher2020lagrangian}.

Here, we study how GoTube can trade runtime for the volume of the constructed reachtube through its tightness factor $\mu$.
In particular, we run GoTube on our newly proposed benchmark with various values of $\mu$. We then plot GoTube's runtime (x-axis) and volume size (y-axis) as a function of $\mu$. The resulting curves show the Pareto-front of runtime-volume trade-off achievable with GoTube.

Figure \ref{fig:pareto} shows the results for a time horizon of 10s in the first two examples, and of 2s in the last example.
Our results demonstrate that GoTube can adapt to different runtime and tightness constraints and set a new benchmark for future methods to compare with.

\section{Discussions, Scope and Conclusions}

We proposed GoTube, a new stochastic verification algorithm that provides robustness guarantees (also safety guarantees if a set of states to be avoided is given)  for high-dimensional, time-continuous systems. GoTube is stable and sets the state-of-the-art in terms of its ability to scale to time horizons well beyond what has been previously possible. It also allows a larger perturbation radius for the initial ball, for which other verification methods fail. Lastly, GoTube's scalability enables it to readily handle the verification of advanced continuous-depth neural models, a setting where state-of-the-art deterministic approaches fail.

\noindent\textbf{SLR versus GoTube?} SLR combines symbolic with statistical reachability techniques. However, no implementation is available to date. For comparison purposes, we implemented SLR on our own and observed that while it does not blow up in space, it blows up in time. As a consequence, we were not able to use SLR to construct reachtubes for our high-dimensional benchmarks.

\noindent\textbf{Sample blow up in GoTube?}
As a pure Monte-Carlo technique, the number of samples $N$ to be taken depends on both the confidence coefficient $\lambda$ and the tightness coefficient $\mu$ as well as on the system's dimensionality. As a consequence, for very small values of these coefficients, the number of samples tends to blow up. The goal of symbolic techniques is exactly the one to avoid such a blowup. However, in our experiments, we observed that GoTube outperformed in all cases the symbolic techniques. This implies that the overapproximation error of symbolic techniques is more problematic than the blowup in the number of samples for a large number of dimensions.

\noindent\textbf{What about Gaussian Processes?} Gaussian Processes (GPs) are powerful stochastic models which can be used for stochastic reachability analysis~\cite{gp} and uncertainty estimation for stochastic dynamical systems~\cite{gal2016uncertainty}. The major shortcoming of GPs is that they simply cannot scale to the complex continuous-time systems that we tested here. Moreover, Gaussian Processes have a large number of hyperparameters, which can be challenging to tune across different benchmarks.

\noindent\textbf{Limitations of GoTube.} GoTube does not necessarily perform better in terms of average volume of the bounding balls for smaller tasks and shorter time horizons if not choosing a very small $\mu$, as shown in Table~\ref{tab:volume}. GoTube is not yet suitable for the verification of stochastic dynamical systems, for instance, Neural Stochastic Differential Equations (Neural SDEs) \cite{li2020scalable,xu2021infinitely}. Although GoTube is considerably more computationally efficient than existing methods, the dimensionality of the system, as well as the type of numerical ODE solver exponentially, affect their performance. We can improve on this limitation by using Hypersolvers \cite{poli2020hypersolvers}, closed-form continuous depth models, and compressed representations of neural ODEs.

\noindent\textbf{Future of GoTube.} GoTube opens many avenues for future research. The most straightforward next step is to search for better intermediate steps in Algorithm \ref{algorithm:GoTube}. For instance, better ways to compute the Lipschitz constant and to improve the sampling process. GoTube is now applicable for complex deterministic ODE systems; it would be an important line of work to find ways to marry reachability analysis with machine learning approaches to verify neural SDEs as well.
Last but not least, we believe that there is a close relationship between stochastic reachability analysis and uncertainty estimation techniques used for deep learning models \cite{abdar2021review}. Uncertainty-aware verification could be worth exploring based on what we learned with GoTube.

\section{Appendix\\ {\large Proofs of the Theorems}}
\begin{lemma}[Stochastic lower bound $F_{L,\gamma}$]\label{app_lemma:F_L}
    Consider the experiment of randomly sampling two times $m$ points of the initial ball $\mathcal{B}_0$: $(a_1,\dots,a_m)$ and $(b_1,\dots,b_m)$.
    Let $g\,:\,\mathbb{R}^n\,\rightarrow\,\mathbb{R}$ be a real-valued function and $X\,{=}\,\max_{i=1}^m|g(a_i)\,{-}\,g(b_i)|/\|a_i\,{-}\,b_i\|$ be a random variable with the unknown cumulative distribution function $F$. Let $(x_1, \dots, x_n)\sim X$ be independent, identically distributed samples with the empirical distribution function $\hat{F}_n(x)\,=\,\sum_{i=1}^n\mathbbm{1}_{x_i\le x}$. Let $G_n$ be a generalized extreme value distribution fitted to the empirical distribution function $\hat{F}_n$ and let $D_n^-$ describe the goodness of fit, being the one-sided Kolmogorov–Smirnov statistic:
    \begin{align}\label{app_eq:ks-statistic}
        D_n^- \,=\,\sup_x (G_n(x) - \hat{F}_n(x))
    \end{align}
    Given the confidence level $\gamma$ and $\alpha = \min(\gamma, 0.5)$, then let us define $\epsilon_{n,\gamma}$ and $F_{L,\gamma}$ as follows:

    \begin{align}
        \epsilon_{n,\gamma} &= \sqrt{\frac{\ln{\frac1{\alpha}}}{2n}}\label{app_eq:dkw_epsilon}\\
        F_{L,\gamma}(x) &= G_n(x) - \epsilon_{n,\gamma} - D_n^-\label{app_eq:F_L bound}
    \end{align}
    Then it holds that:
    \begin{align}\label{app_eq:prob F_L bound}
        Pr(\sup_x(F_{L,\gamma}(x)-F(x))\le 0)\ge 1-\gamma,
    \end{align}
    which intuitively means that $F_{L,\gamma}$ is a lower bound of $F$ with confidence $\gamma$.
\end{lemma}
\emph{Proof.} The Fisher-Tippett-Gnedenko theorem states that the distribution of a normalized maximum converges to the generalized extreme value distribution, if the distribution of the normalized maximum does converge. So intuitively that theorem is similar to the central limit theorem for the averages, but for the normalized maxima. Consequently, we start by fitting the empirical distribution function $\hat{F}_n$ by a generalized extreme value distribution $G_n$ and compute Eq.~\eqref{app_eq:ks-statistic}.

The Dvoretzky-Kiefer-Wolfowitz inequality~\cite{dkw1956} with a tight constant determined by~\cite{massart1990}, states that for all $\epsilon\,\ge\,\sqrt{\frac1{2n}\ln2}$, it holds that:
\begin{align}
    Pr(\sup_x(\hat{F}_n(x)-F(x))>\epsilon)\le e^{-2n\epsilon^2}
\end{align}
Solving $\gamma = e^{-2n\epsilon^2}$ for $\epsilon$ and considering Massarts lower bound for $\epsilon$, yields:
\begin{align}\label{app_eq:DKW inequality}
    \begin{split}
        &Pr(\sup_x(\hat{F}_n(x)-F(x))>\epsilon_{n,\gamma}) \le \gamma,
    \end{split}
\end{align}
with $\epsilon_{n,\gamma}$ as defined in Eq.~\eqref{app_eq:dkw_epsilon}. We use the triangular inequality for supremum and the monotony of the probability measure as follows:
\begin{align}
    &Pr\big(\sup_x\big(G_n(x)-F(x)\big)>\epsilon_{n,\gamma} + D_n^-\big) =\\
    &= Pr\Big(\sup_x\big(G_n(x)-\hat{F}_n(x)+\\
    &\qquad + \hat{F}_n(x)-F(x)\big)>\epsilon_{n,\gamma} + D_n^-\Big)\\
    &\le Pr\Big(\sup_x\big(G_n(x)-\hat{F}_n(x)\big) + \\
    &\qquad +\sup_x\big(\hat{F}_n(x)-F(x)\big)>\epsilon_{n,\gamma} + D_n^-\Big)\\
    &\overset{\eqref{app_eq:ks-statistic}}{=} Pr\big(\sup_x\big(\hat{F}_n(x)-F(x)\big)>\epsilon_{n,\gamma}\big)\\
    & \overset{\eqref{app_eq:DKW inequality}}{\le} \gamma,
\end{align}
from which it follows directly, that Eq.~\eqref{app_eq:prob F_L bound} hold.
\begin{figure}[h]
    \centering
    \includegraphics[width=0.48\textwidth]{images/Lemma1.pdf}
    \caption{Visualisation of the stochastic lower bound $F_{L,\gamma}$ of Lemma~\ref{app_lemma:F_L}.}
    \label{app_fig:lemma1}
\end{figure}
\begin{theorem}[Radius of Stochastic Lipschitz Caps]\label{app_thm:lipschitz cap}
    Given a continuous-depth model $f$ from Eq.~(1) in the main paper ($\partial_t x\,{=}\,f(x)$ with $x(t_0)\,{\in}\,\,B(x_0, \delta_0)$), $\gamma\,{\in}\,(0,1)$, $\mu\,{>}\,1$, target time $t_j$,

    the set of all sampled points $\mathcal{V}$, the number of sampled points $N\,{=}\,|\mathcal{V}|$, the sample maximum $\bar{m}_{j,\mathcal{V}}\,{=}\,\max_{x\in\mathcal{V}} d_j(x)$, the IVP solutions $\chi(t_j,x)$, and the corresponding stretching factors $\lambda_x\,{=}\,\|\partial_x\chi(t_j,x)\|$ for all $x\,{\in}\,\mathcal{V}$. Let us define $\hat{\gamma} = 1-\sqrt{1-\gamma}$.
    Let $\Delta\lambda_{\mathcal{V}}$ be the $\sqrt{1-\gamma}$-quantile of a stochastic lower bound $F_{L,\hat{\gamma}}$ as defined in Eq.~\eqref{app_eq:F_L bound} of Lemma~\ref{app_lemma:F_L}:
    \begin{align}\label{app_eq:expected difference quotient}
        \Delta\lambda_{\mathcal{V}}(\gamma) = F_{L,\hat{\gamma}}^{-1}(\sqrt{1-\gamma}),
    \end{align}
    Let $r_x$ be defined as:
    \begin{align}\label{app_eq:cap radius}
        r_{x} = \frac{\left(-\lambda_x + \sqrt{\lambda_x^2 + 4\cdot\Delta\lambda_{x,\mathcal{V}}\cdot(\mu\cdot\bar{m}_{j,\mathcal{V}}-d_j(x))}\right)}{2\cdot\Delta\lambda_{x,\mathcal{V}}},
    \end{align}
    then it holds that:
    \begin{align}\label{app_eq:cap probability}
        \mathbb{P}\left(d_j(y) \le \mu\cdot \bar{m}_{j,\mathcal{V}}\right)\ge 1-\gamma\quad \forall y\in B(x,r_x)^S,
    \end{align}
    and thus that $B(x, r_x)^S$ is a $\gamma, t_j$-Lipschitz cap.
\end{theorem}
\emph{Proof.} Let $\{x_1,\dots,x_n\}$ be $n$ independent experiments by sampling from $X\,{=}\,\max_{i=1}^m|\lambda_{a_i}\,{-}\,\lambda_{b_i}|/\|a_i\,{-}\,b_i\|$ as defined in Lemma~\ref{app_lemma:F_L}, where each variable is the maximum of $m$ executions.
From Eq.~\eqref{app_eq:prob F_L bound} it follows that:
\begin{align}\label{app_eq:cdf bound}
    Pr(F_{L,\hat{\gamma}(x)} \le F(x))\ge 1-\hat{\gamma} = \sqrt{1-\gamma}
\end{align}
Let us now derive the probability of $X$ being less or equal to $\Delta\lambda_{\mathcal{V}}$ defined by Eq.~\eqref{app_eq:expected difference quotient}.
For any sets $A, B$ it holds that $\mathbb{P}(A)\ge \mathbb{P}(A\cap B) = \mathbb{P}(A | B) \cdot \mathbb{P}(B)$, thus:
\begin{align}
    &Pr(X \le \Delta\lambda_{\mathcal{V}}) \ge\\
    &\ge Pr\Big(X\le\Delta\lambda_{\mathcal{V}} | F_{L,\hat{\gamma}(x)} \le F(x)\Big)\cdot\label{app_eq:quantile1}\\
    &\qquad\cdot Pr\Big(F_{L,\hat{\gamma}(x)} \le F(x)\Big)
\end{align}
Let us have a look on Eq.~\eqref{app_eq:quantile1}: As $Pr(X\le \Delta\lambda)=F(\Delta\lambda)$ and we are looking for the conditional probability depending on $F_{L,\hat{\gamma}(x)} \le F(x)$, we can use $F_{L,\hat{\gamma}}(\Delta\lambda)$ as a lower bound of Eq.~\eqref{app_eq:quantile1} and thus, using Eq.~\eqref{app_eq:cdf bound}:
\begin{align}
    Pr(X\le \Delta\lambda_{\mathcal{V}}) \ge F_{L,\hat{\gamma}}(\Delta\lambda_{\mathcal{V}})\cdot \sqrt{1-\gamma}
\end{align}
As Eq.~\eqref{app_eq:expected difference quotient} defines $\Delta\lambda_{\mathcal{V}}$ as the $\sqrt{1-\gamma}$-quantile of $F_{L,\gamma}$, we can further state that
\begin{align}\label{app_eq:confidence lipschitz max}
    \begin{split}
    \mathbb{P}(X \le \Delta\lambda_{\mathcal{V}})\ge 1-\gamma, \\\quad \textrm{with}\quad X=\max_{i=1}^m\left[\frac{|\lambda_{a_i} -  \lambda_{b_i}|}{\|a_i-b_i\|}\right]
    \end{split}
\end{align}
Let $Y\,=\, |\lambda_a-\lambda_b|/\|a-b\|$ be another random variable with $a, b$ being to sample points of the initial ball $\mathcal{B}_0$. From Eq.~\eqref{app_eq:confidence lipschitz max} it holds that
\begin{align}\label{app_eq:confidence lipschitz}
    Pr(Y\le \Delta\lambda_\mathcal{V})\ge Pr(X\le \Delta\lambda_\mathcal{V})\ge 1-\gamma
\end{align}
It trivially holds for $x, y \in \mathcal{V}$ that:
\begin{align}
    \lambda_y &= \lambda_x + \frac{\lambda_y - \lambda_x}{\|x-y\|}\cdot \|x-y\|\nonumber\\
    &\le \lambda_x + \frac{|\lambda_x - \lambda_y|}{\|x-y\|}\cdot \|x-y\

\end{align}
From Eq.~\eqref{app_eq:confidence lipschitz} it follows that:
\begin{align}
    &Pr\Bigg(\lambda_x + \frac{|\lambda_x - \lambda_y|}{\|x-y\|}\cdot \|x-y\| \le \\
    &\qquad\le \lambda_x + \Delta\lambda_{x,\mathcal{V}}\cdot \|x-y\|\Bigg)\ge 1-\gamma
\end{align}
and using the monotony of the probability measure:
\begin{align}
    \mathbb{P}\big(\lambda_y &\le \lambda_x + \Delta\lambda_{x,\mathcal{V}}\cdot \|x-y\|\big) \ge 1-\gamma\label{app_eq:confidence bound}
\end{align}
Using the mean value inequality for vector-valued functions it holds that:
\begin{align*}
    & |d_j(x) - d_j(y)|= | \left\lVert \chi(t_j,x) - \chi(t_j,x_0)\right\rVert -\\ &\quad- \left\lVert \chi(t_j,y) - \chi(t_j,x_0)\right\rVert|\quad \textrm{\{triangle inequality\}}\\
    &\quad \le \|\chi(t_j,x) - \chi(t_j,y)\|\quad \textrm{\{mean value theorem\}}\\
    &\Rightarrow\exists z\in [x,y] \colon  |d_j(x) - d_j(y)|\\ &\quad\le \|\partial_x \chi(t_j,z) \| \|x-y\| = \lambda_z \cdot \|x-y\|
\end{align*}
Combining this with Eq.~\eqref{app_eq:confidence bound} and thus using $\lambda_x + \Delta\lambda_{x,\mathcal{V}}\cdot \|x-y\|$ as a probabilistic upper bound for $\lambda_z$, we obtain the following results for all $y$ with $\|x-y\| \le r_x$:
\begin{align}
    &\mathbb{P}\big(|d_j(x)-d_j(y)|\le\nonumber\\
    &\quad\le (\lambda_x + \Delta\lambda_{x,\mathcal{V}}\cdot \|x-y\|) \cdot \|x-y\|\big) \ge 1-\gamma \nonumber\\
    \begin{split}
    &\mathbb{P}\big(|d_j(x)-d_j(y)|\le\\
    &\quad\le (\lambda_x + \Delta\lambda_{x,\mathcal{V}}\cdot r_x) \cdot r_x\big) \ge 1-\gamma
    \end{split}
\end{align}
As $r_x$ defined like in Eq.~\eqref{app_eq:cap radius} is the solution of the quadratic equation $\mu\cdot\bar{m}_{j,\mathcal{V}}-d_j(x)=\lambda_x r_x + \Delta\lambda_{x,\mathcal{V}} r_x^2$, it holds that:
\begin{align}
\begin{split}
    &\mathbb{P}\big(|d_j(x) - d_j(y)| \le \mu\cdot\bar{m}_{j,\mathcal{V}}-d_j(x))\big) \ge\\
    &\quad \ge 1-\gamma \quad\forall y\in B(x, r_x)^S\label{app_eq:probabilistic lipschitz}
\end{split}
\end{align}
We now distinguish between two cases for $y$: (a)~$d_j(y)\le d_j(x)$ and (b)~$d_j(y) \ge d_j(x)$. In case (a) it is trivial: $d_j(y) \le d_j(x) \le \mu \cdot \bar{m}_{j,\mathcal{V}}$. Having case (b), Eq.~\eqref{app_eq:probabilistic lipschitz} is equivalent to
\begin{align}
    \mathbb{P}\big(d_j(y) - d_j(x) &\le \mu\cdot\bar{m}_{j,\mathcal{V}}-d_j(x))\big) \ge 1-\gamma\nonumber\\ &\Longleftrightarrow\nonumber\\
    \mathbb{P}\big(d_j(y) &\le \mu\cdot\bar{m}_{j,\mathcal{V}})\big) \ge 1-\gamma,
\end{align}
thus Eq.~\eqref{app_eq:cap probability} holds and $B(x,r_x)^S$ is a Lipschitz cap.

\begin{theorem}[Convergence via Lipschitz Caps]\label{app_thm:stochastic guarantee} Given the tightness factor $\mu > 1$, the set of all sampled points $\mathcal{V}$ and the sample maximum $\bar{m}_{j,\mathcal{V}} = \max_{x\in\mathcal{V}} d_j(x)$. Let the initial ball maximum be defined by $m^\star_j=\max_{x\in\mathcal{B}_0} d_j(x)$. Then:
    \begin{align}\label{app_eq:stochastic guarantee}
        \hspace*{-2ex}\forall\gamma\in(0,1),\exists N\in\mathbb{N}\textrm{ s.t. }
        \mathbb{P}(\mu\cdot\bar{m}_{j,\mathcal{V}}\ge m^\star_j) \ge 1-\gamma
    \end{align}
    where $N=|\mathcal{V}|$ is the number of sampled points.
\end{theorem}
\emph{Proof.} Let $x^\star_j$ be a point such that $d_j(x^\star_j) = m^\star_j$. Given $\gamma\in(0,1)$ and cap radii $r_x$ as defined in Eq.~\eqref{app_eq:cap radius}, we know from the definition of a spherical cap that
\begin{align}
    &p_{r_x} = \mathbb{P}(B(x, r_x)^S\owns x_j^\star) = \frac{\area(B(x, r_x)^S)}{\area{(\mathcal{B}_0})}\\
    &\textrm{and thus it holds that:}\nonumber\\
    &\mathbb{P}(\exists y\in\mathcal{V}\colon B(y, r_y)^S\owns x_j^\star) = 1 - \prod_{x\in\mathcal{V}} \left(
        1 - p_{r_x}
        \right)
\end{align}
We derive a lower bound of $r_x$ by using the first sample $x_{j,1}$ and replacing the values in Eq.~\eqref{app_eq:cap radius} as follows:
\begin{align}
    &\mu\cdot \bar{m}_{j,\mathcal{V}} - d_j(x) \\
    &\quad\ge \mu\cdot \bar{m}_{j,\mathcal{V}} - \bar{m}_{j,\mathcal{V}} = (\mu - 1)\cdot \bar{m}_{j,\mathcal{V}} \\
    &\quad\ge (\mu - 1) \cdot d_j(x_{j,1}),
\end{align}
thus a lower bound of all Lipschitz cap radii is given by
\begin{align}
    & r_{bound} = \nonumber\\
    &\quad=\frac{-\lambda_x + \sqrt{\lambda_x^2 + 4\cdot\Delta\lambda_{x,\mathcal{V}}\cdot(\mu - 1) \cdot d_j(x_{j,1})}}{2\cdot\Delta\lambda_{x,\mathcal{V}}} \le \nonumber\\
    &\quad\le r_x \quad\forall x\in \mathcal{V}\nonumber\\
    \begin{split}
    &\Rightarrow \mathbb{P}(\exists y\in\mathcal{V}\colon B(y, r_y)^S\owns x_j^\star) \ge \\
    &\quad\ge 1 - \left(
        1 - p_{r_{bound}}
        \right)^N\label{app_eq:lower bound probability}
    \end{split}
\end{align}
As in the limit of $N\rightarrow \infty$ the probability of Eq.~\eqref{app_eq:lower bound probability} is 1, it follows that $\forall \gamma\in (0,1)~\exists N \in\mathbb{N}\colon\mathbb{P}(\exists x\in\mathcal{V}\colon\,B(x,r_x)^S\owns x^\star_j)\ge\sqrt{1-\gamma}$.

Using a set of sampled points $\mathcal{V}$ with cardinality $N$ and using $\hat{\gamma} = 1-\sqrt{1-\gamma}$ as the error rate for the upper bound $\Delta\lambda_x$ of the confidence interval in Eq.~\eqref{app_eq:expected difference quotient}. Using the result of Theorem~\ref{app_thm:lipschitz cap}, the resulting probability $\forall y\in B(x,r_x)^S$ is:
\begin{align}\label{app_eq:proof lipschitz cap}
    \mathbb{P}\left(d_j(y) \le \mu\cdot \bar{m}_{j,\mathcal{V}}\right)\ge 1-\hat{\gamma} = \sqrt{1-\gamma}
\end{align}
If there is an $x\in\mathcal{V}$ such that $B(x,r_x)^S\owns x_j^\star$, then Eq.~\eqref{app_eq:proof lipschitz cap} obviously holds also for $x_j^\star$, thus:
\begin{align*}
    \mathbb{P}( d_j(x^\star) \le \mu\cdot\bar{m}_{j,\mathcal{V}} |\exists x\in\mathcal{V}\colon B(x,r_x)^S\owns x^\star)\ge \sqrt{1-\gamma}
\end{align*}
For any sets $A, B$ it holds that $\mathbb{P}(A)\ge \mathbb{P}(A\cap B) = \mathbb{P}(A | B) \cdot \mathbb{P}(B)$, and using:
\begin{align}
    A &= (\mu\cdot\bar{m}_{j,\mathcal{V}}\ge m^\star_j)\\
    B &= (\exists x\in\mathcal{V}\colon B(x,r_x)^S\owns x_j^\star)
\end{align}
it follows that $\mathbb{P}(\mu\cdot\bar{m}_{j,\mathcal{V}}\ge m^\star_j)\ge \mathbb{P}(A|B) \cdot \mathbb{P}(B) = 1-\gamma$ and therefore Eq.~\eqref{app_eq:stochastic guarantee} holds.

redpajama_set_name: RedPajamaArXiv


Big Macs on silver platters. What class!
The idea of McDonald's, Wendy's and Burger King being served on silver platters seems a bit off, but let's not forget that these are almost certainly lukewarm McDonald's, Wendy's and Burger King meals being served on silver platters. It just doesn't sound appetizing.
If he really respected them, he'd have gotten out the gold platters!


text:
Q: В Django не работает фильтр center В шаблоне использовал фильтр center, но он не работает.

                {% for i in categories %}
                    {{ i.title|center:"2" }}
                {% endfor %}

Только обучаюсь django, так что даже не знаю где копать.

A: В документации по django дан очень хороший пример, что означает center:
"{{ value|center:"15" }}"

здесь значить, что value будет помещен в строку из 15 знаков по середине. Т.е. если value ="Django" тогда "     Django    ".
У вас же стоит значение 2 в которое по определению не может поместиться ваша строка. Попробуйте увеличить значение в кавычках. Например:
{{ i.title|center:"21" }}

Также рассмотрите возможность использовать css для верстки ваших элементов на странице.

redpajama_set_name: RedPajamaStackExchange


text:
2015-01-06 Roehl Transport and Roehl Logistics earn prestigious AGCO award
Marshfield, WI – Roehl Transport and Roehl Logistics have earned the 2014 Logistics Award from the AGCO Corporation. Roehl won the award by providing trucking and logistics support services to AGCO for consistent on time performance, claims-free deliveries, and being easy to do business with, which all led to great value for AGCO.
The award was presented in October at AGCO's "Celebrating Success" dinner, an event that honors five suppliers for their superior performance and contribution to AGCO's success in the areas of direct materials, indirect materials and logistics.
Roehl Transport has over 2,500 employees, more than 2,000 trucks and over 5,000 trailers. The company provides truckload solutions and offers truck driving jobs >www.Roehl.Jobs across the country.
Roehl Logistics is a third party logistics (3PL) provider that arranges transportation for hundreds of shippers and offers consulting and management solutions to shippers and carriers across the United States and Canada.
AGCO Corporation, based in Duluth, Georgia, is a global leader in the design, manufacture and distribution of agricultural machinery.
Steve Elliott, Vice President of Roehl Logistics, noted, "Roehl Transport and Roehl Logistics continue to collaborate to meet customers' needs. Being one of five winners, when AGCO has over 6,000 suppliers, is a top honor. The 2014 AGCO Logistics Award is great example of the two companies working together to deliver confidence and satisfaction to our customer."
About Roehl Transport
With over 2,500 employees, Roehl Transport is one of the safest and largest truckload and logistics companies in the nation. Headquartered in Wisconsin with locations across the United States, the company delivers confidence and satisfaction by providing flawless transportation services. Learn more about Roehl Transport truck driving jobs at www.Roehl.Jobs. For additional information, please contact: Mr. Lucas Heart, Roehl Transport Marketing and Communications Manager, at 715-591-7164.
About Roehl Logistics
Roehl Logistics is a third party logistics (3PL) based in Green Bay, Wisconsin. The company was founded in 2008 and arranges transportation for hundreds of shippers while also offering consulting and management solutions to shippers and carriers across the United States and Canada. Learn more at www.RoehlLogistics.com. For additional information, please contact: Mr. Steve Elliott, Roehl Logistics Vice President, at 920-321-3224.
About AGCO
AGCO (NYSE: AGCO) is a global leader in the design, manufacture and distribution of agricultural machinery. AGCO products are distributed globally through approximately 3,100 independent dealers and distributors in more than 140 countries worldwide. Founded in 1990, AGCO is headquartered in Duluth, Georgia. In 2013, AGCO had net sales of $10.8 billion. www.AGCOcorp.com.

redpajama_set_name: RedPajamaCommonCrawl


text:
Home Tags The Providence Journal
Tag: The Providence Journal
Providence Journal Executive Editor Rosenberg to retire Dec. 1
People November 13, 2020 2:36 pm
PROVIDENCE – Alan Rosenberg, executive editor of The Providence Journal, will retire effective Dec. 1, The Providence Journal reported on Friday. Rosenberg worked for the...
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News editorials are crucial part of a community's conscience
Editorial and Op-Ed May 15, 2020 2:52 am
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Economic Activity October 31, 2019 11:27 am
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New Media posts $2.6M profit in Q2, 78% drop over the...
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Projo parent reports $9.4M quarterly loss, pays $23M in dividends
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Journal publisher departing; circulation continues falling
Economic Activity April 23, 2019 5:45 pm
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ProJo parent reports $11.7M profit in Q2
Economic Activity August 2, 2018 12:15 pm
PROVIDENCE – New Media Investment Group Inc. reported an $11.7 million profit in the second quarter of 2018 Thursday, following a $21.7 million loss...


Long hours in stuffy airplane cabins might sound like a nightmare, but with a few home comforts and a little preparation it can be quite okay. To help you feel comfortable on your next trip we have selected several long flight essentials and useful flight tips. Check them out below.
It is very important to be relaxed when travelling. Treat yourself before the trip, have a nice breakfast or lunch, go to the gym, read a nice book, just give yourself some attention and love. Make sure you have all the necessary papers, your luggage is ready and you have the essentials with you. Check in early because the last thing you want before a long flight is rushing and panicking on a busy airport.
Probably one of the best tips for long flights is to pick comfortable clothes. Choose loose and soft pants and top, you are not there for a fashion show. Do not wear your jewelry, better pack it for when you arrive. It is best to wear closed toed shoes. Layering is important because it can get cold during the night. Pack your pajamas if you like to sleep on the plane.
The best airplane pillow for long flights is a small, comfortable travel pillow which is compact and easy to carry around. Travel pillows may look funny but they are going to save your neck anytime and let you relax during the long hours on the plane.
Drink a lot of water during long flights, because you need to keep yourself hydrated. Although alcohol is served on most planes, stay away from it, because it is dehydrating and can mess up with your sleep. Do not worry about using the bathroom more often; staying hydrated is much more important at this point.
Always bring toiletries in your carry-on luggage. Make sure you have wet wipes, tooth paste, tooth brush, deodorant and clean clothes.
One of the most annoying things on a flight is when the tablet or music player runs out of battery in the middle of a game or a favorite song. To avoid this situation make sure all your devices are fully charged before the flight. Pack a pair of high quality headphones or earplugs.
A pair of noise-cancelling headphones is a good investment if you fly long hours on a regular basis. Upload some back-up movies onto your tablet in case the in-flight entertainment systems fail or your computer runs out of battery. Load a lot of podcasts on your device, they take less battery life than movies and can keep your interest for a very long time.
It is very important to stay well-nourished during a long flight. Airline food is not that plentiful, so it is important to bring your own snacks to munch on. There is no need to overdo it, but a couple of pieces of fruit or a granola bar is always good to have.
Follow the basic tips for long flights we shared with you and make your flights more enjoyable. These travel aids for long flights are very basic but they make a huge difference in how you feel while exploring the world by plane. Keep curious and hungry for new adventures and make sure you are comfortable and well even on the longest flights.


text:
Q: Using Steam DRM with XNA / Monogame (C#) I want to use Steam DRM integrated in the SteamWorks-API to protect a game implemented in C# using XNA (or Monogame/FNA for Mac and Linux). Valve itself only provides its SteamWorks-API in C++, but there are wrappers like SteamWorks.Net für C#. However, as far as I know, these only support major features (like cloud save and achievements) but that do not support Steam DRM.
Is there any way to integrate Steam DRM with a C# game?
If no, what are alternatives?

A: OP:

Is there any way to integrate Steam DRM with a C# game?

Directly? No.  Indirectly? Yes.  Let me explain...
What you really want to know is how can c# code call c++ and the answer is yes.  Tell me more.  If you look at the link, the best answer is to modify the c++ code to change it to a hybrid c++/CLI DLL.  Since that's not possible with the Steam provided API, the alternative is to create a proxy.
The thing to do (since you can't modify the c++ DLL) is create a new c++/CLI DLL (let's call it Proxy) that calls the Steam C++ DLL (S) on your behalf. This gets around the problem of p-invoking a c++ DLL which is subject to name mangling as opposed to a c DLL which does not.   Proxy exposes classes that are a mix of c++ and .NET.  Your app calls what it thinks is just .NET code inside the proxy.  Proxy then calls the raw c++ code in S.
Call sequence goes like this:
App --> Proxy --> S
If you notice you will learn that the proxy is really just your own wrapper - a very simple one for the specific functions your app is interested in, so again the reliance for a 3rd party wrapper does not seem worth it considering the risk. There should not be any special and/or complex business logic in your proxy. Keep it simple.  Be sure to keep your copy of the steam code/libs up to date with Steam's latest API.
OP:

but there are wrappers like SteamWorks.Net für C#. However, as far as I know, these only support major features (like cloud save and achievements) but that do not support Steam DRM

Personally I wouldn't want to rely on a potentially unproven; subject to becoming out of sync with Steam changes; 3rd party open source project for something as important as DRM so my advice is stick with SteamWorks proper.
What's c++/CLI?
Briefly, c++/CLI is a Microsoft technology that has been around for some time now that allows you to create a hybrid c++ and .NET DLL.  Your code has access to .NET as well as the wealth of historic c/c++ code and libs.  .NET code can call public CLR types in the hybrid dll.
More

*

*name mangling - a somewhat annoying aspect of c++ compilers that turn method names like bool article::print_to (std::ostream&) into _ZN9wikipedia7article8print_toERSo for many reasons, one of which is for the need of unique naming, but have the unwanted side effect of making it difficult to determine which export is the entry to p-invoke.

A: According to Steamworks.NET

Does the Steam DRM wrapper work with Unity or XNA/Monogame?
No it doesn't. Unfortunately the Steam DRM wrapper does not play well
  with .NET applications such as Unity and your application will likely
  fail to start if it's wrapped. Even if it does succeed and work for
  you it will often fail on other computers.

So at least one major implementation has found this exact aspect untenable.
Note that there are multiple steam games which use third-party licensing software, and instead of being integrated to steam will display a product key on startup. This may help open up your options.


text:
Billy Dukes' music background includes jobs as music director and night DJ at WQDR 94.7 in Raleigh, N.C.; writer for Recoil Magazine, a monthly out of Grand Rapids, Mich.; and writer for the Kalamazoo Gazette in Kalamazoo, Mich. The Western Michigan University graduate followed a girl to Raleigh in 2006 and wisely married her one year later. In 2014, they moved to Nashville, where they now live with their two dogs, Daisy and Clementine, and their sons. In his free time, Billy enjoys playing and watching baseball, cycling and exploring nature.
Keith Urban Shares Pictures From Las Vegas Residency Rehearsal
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Dierks Bentley's Hot Country Knights Sign With UMG Nashville
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50 Best Country Songs of the 2010s
Who do you think has the No. 1 country song of the 2010s?
Carrie Underwood Will Not Return as CMA Awards Host
The singer revealed the news on social media.
10 Most Anticipated Country Albums of 2020
Including albums from Tim McGraw, Sam Hunt, Brandy Clark and more.
Carrie Underwood Shares Cover, Release Date for New Book 'Find Your Path'
The inspirational memoir will go deep into her personal journey for overall wellness.
Kenny Chesney Expands 2020 Chillaxification Tour, Adds 18 Amphitheater Concerts
You've got 40 chances to see Kenny Chesney in 2020.
Cole Swindell Had a Wardrobe Malfunction That Involved Reba McEntire
He was one of several stars asked to share a wardrobe malfunction.
Jason Aldean's Son Has One Very Bad Habit + 4 More Things We Learned From His Live Intervi…
Moms and dads can relate to the struggle.
Blake Shelton and Gwen Stefani Have a New Duet Called 'Nobody But You'
"Nobody But You" will be a part of Fully Loaded: God's Country when it's released later this month.
Miranda Lambert Is a Fighter, a Lover and an Optimist in 'Bluebird' [Listen]
The song is delightfully optimistic, with forward-moving solutions and no drama.
Chris Young's Next Album Will Fall Between 'Raised on Country' and 'Drowning'
"I would say that those are probably the left and right ends of the record," he shares.


NEWS ON THE BREAK is an online news platform built to provide reliable, verifiable and authentic source of news from Nigeria; Politics, Security, Economy, Health, and any topic that is news. The medium operated by NEBOLABS is based in Benin City, Nigeria and was founded in July, 2012. We bring the story as it breaks followed by in-depth analysis and perspectives. Our approach is a mix of professionalism, experience, trend and creativity. NEWS ON THE BREAK is aware of the competition but not intimidated because we offer the alternative option of human interest guided by truth. We believe that humanity is increasingly hungry for news and is strongly influenced by what is read or heard. There is therefore a need to keep the tap of information flowing unhindered, yet taking the responsibility to preserve fairness and equity among mankind without tinkering with the story. Our online effort is only the beginning of multifaceted media operations. NEWS ON THE BREAK will grow into other mode of publications and extend to broadcasting through content generation for our multi-media audience.


text:
Q: Using {{outlet}} as variable name in Ember I have the following Ember template;
{{#with model}}

Order #{{id}}

            Outlet
            Date
            Link

            {{outlet}}
            {{date}}
            {{#link-to 'next-route' this}}Go{{/link-to}}

{{/with}}

{{outlet}}

How to do I keep Ember from using the first {{outlet}} (which is just the property's name) to add the next view I'm transitioning to?

A: outlet is a reserved word, that being said, if you fully qualify it it will work.
In your case, with the with block you can't fully qualify it, but if you remove the with block it will work.
Personally I'd avoid using it, it's analogous to having a variable named for or if

Order #{{id}}

            Outlet
            Date
            Link

            {{model.outlet}}
            {{date}}
            {{#link-to 'next-route' this}}Go{{/link-to}}

{{outlet}}


Living (the best of) Life in My Dreams: Cute Bedtime Story - My Review of Go to Sleep, Sheep!
Cute Bedtime Story - My Review of Go to Sleep, Sheep!
I like reading children's books and finding new ones for my nieces and nephews. I thought that Go to Sleep, Sheep! looked like a fun book and that is why I chose to read and review it.
I love the pictures in this book! So cute! And, I love how one of the sheep has glasses - I have two little nieces with glasses and I know that they will like seeing them on the sheep.
This is good for babies and toddlers and their tired parents.
My favorite part about the book is the pictures. The sheep are really cute.
This is a simple book that is good for bedtime.
There isn't really anything I didn't like, I just wish there was something more to it. Something to make it special.
Overall, this is a cute book.
I give Go to Sleep, Sheep! 4 out of 5 stars.


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redpajama_set_name: RedPajamaGithub


Knowing your wildflowers can be tricky and very difficult for a novice.
However, using Newcomb's Wildflower Guide, it becomes a little easier and forces me to remember what I learned in Botany my freshman year at UConn!!
Since our property is smack dab in the middle of the woods, we are privy to all sorts of flora and fauna.
Each Spring, I look forward to the wide variety we get.
i recognize the jack-in-the-pulpit and the ferns. Are they fiddleheads? You can eat those.
That's a brilliant idea - I am no good to the names of flowers either! Some of the flower sprouts look like they're nodding, having the flower to arise a bit later ... so sweet!
i love the jack in the pulpit. had those in wisconsin!
Let's see...I recognize the Jack-in-The-Pulpit, but that is all. I am so glad to see this book, and am going looking for it! I have two wildflower guides, and neither one seems to be that good. This may be IT!
I have a couple of wildflower guides for Ontario and they are handy to have. I think I recognize everything you shared and they're common here as well.
I have one unfortunate wildflower guide where you practically have to know the name, I think it is the Latin name, before you can look the flower up. But I have a good one too, so I use it.


This 4 star hotel is located on the coastline of Mauritius . . .
The R Beach Club fronts the secured Mont Choisy bay and provides a relaxed venue with a modern decor ideal for catching up with friends or for sunbathing The concept is based on a friendly and warm welcome authentic cuisine simplicity and good value . . .
location Located in Bel Ombre on the south coast of Mauritius The property's location makes it a great choice for those wishing to combine beach and nature as many of Mauritius's well-known natural beauties are close by rooms These suites are magnifi . . .
The hotel is located at Trou aux Biches along an extensive stretch of white sandy beach It is 18 km from the capital Port Louis 5 km from Grand Baie and 50 km from the airport The hotel offers several rooms and a junior suite for visitors to choose f . . .
Apartments situated closest to the beach Plage Bleue is over an hour drive from the airport 15 minutes from Grand-Baie and 25 minutes from Port-Louis the capital city The main appeal of the apartments is their proximity to the beach two beachfront a . . .
The resort is a hideaway for couples seeking tranquility comfort and personalised service The views across the lagoon and the bay of Tamarin with the majestic Le Morne Brabant Mountain in the distance offer grandiose scenery The over-250-metre-long w . . .
Shanti Maurice - A Nira Resort is a popular choice amongst travelers in Mauritius Island whether exploring or just passing through Offering a variety of facilities and services the hotel provides all you need for a good nights sleep 24-hour room serv . . .


text:
Civic Ed
Government Ethics
News. Debate. Community. Levers for a better democracy.
One America Movement
The One America Movement's mission is to build bridges and solve problems. We bring Americans together across racial, religious, political and geographic divides to address pressing social problems—poverty, opioids, racial tension and more. Our work is guided by faith and science. We work with experts in neuroscience, social science and conflict-resolution to design our work. And we promote the core values of our faith traditions: to treat others as we wish to be treated, and to advance a vision for a stronger and more united country.
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Polarization and the politics of love
Hanauer is director of the One America Movement, a nonprofit founded by faith and community leaders to address growing divisions in American society.
There is a false notion in our national conversation right now that you have to choose between mushy calls for civility and standing up for "truth and justice." This misunderstands the nature of polarization, and it misunderstands the multi-faceted fight for actual truth and justice.
Julie Kohler of the Democracy Alliance recently made a series of standard arguments against what she calls "love politics." Kohler argued that focusing on the tone of our politics leaves us papering over the real injustices that have taken place in our history and are still taking place today. This can be true. She argued that the unmerited anger of the oppressor is not equivalent to the righteous anger of the oppressed. Absolutely true. And she argued that being nice to our opponents will not prevent the significant wrongs being perpetrated in our country by some of the people who wield power. This is unquestionably true.
Kohler is right to challenge political narratives rooted in love to be about more than sentimental politeness. If we're going to bring our country together, we need to do more than promote civility — we need to understand polarization for what it really is.
'We need to step back from our own perspective': Q&A with ... ›
Politics of Love - Wikipedia ›
It is time for love to become a radical force in politics | Aeon Essays ›
A Politics of Love - Marianne Williamson - Hardcover ›
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Hosting an easy holiday get-together: A budget friendly BBQ.
With the holiday season almost upon us, the humble barbeque will once again be the host of casual catch-ups and extended family dinners in backyards across Australia.
We've spoken before about our ideas for more healthy and adventurous eating, but there's nothing quite as quintessentially 'Aussie' (or crowd-pleasing) as a BBQ. However, as your guest list grows, so too can the costs involved. Here, we share the best ways to ensure that your next BBQ is a hit with your guests, without your wallet taking a hit!
The fantastic thing about BBQ cooking is how diverse it is; if you stay open-minded about your menu, you can make huge savings. Keep an eye on your local grocery store or butcher for upcoming deals that take advantage of alternative cuts of meat or vegetable specials. Then, work your menu around the items that offer you the greatest savings.
There's nothing wrong with asking your guests to pitch in where food and beverages are concerned—especially if you just request something simple, like a side. It can be a mountain of work to whip up coleslaw, salads and wedges for a large group of people.
If everyone brings along a plate of their own speciality, you'll be amazed at the spread that you end up with. Your guests might bring things that you never would have thought to make! Just make sure that everyone gives you notice in advance of what they're bringing along to avoid double ups.
Don't dish out the dollars on expensive, pre-packaged marinades, sauces and spice mixes. You can whip up simple and delicious BBQ additions at home for a fraction of the cost!
Combine paprika, cayenne pepper, nutmeg, cinnamon, cloves, allspice, salt and pepper for a spicy Jamaican dry rub, or make a delicious chicken marinade by mixing honey, soy sauce, lemon juice, ginger and garlic to taste. Be creative with the ingredients that you already have in your cupboard; it'll be quick and easy to add that extra touch of deeper flavour.
De-boned meat is usually the more expensive option in supermarkets. Many people don't know that you can opt for meat that's still attached to the bone! Already de-boned meat is the more expensive option.
You can either do it yourself to save money, or (for some cuts of meat) cook and serve with the bones still present. Meat cooked with bones is more tender, will remain juicier and has more flavour. They don't say that the best meat 'falls off the bone' for nothing!
We often overlook humble vegetables at most backyard BBQs. Cook your veggies over hot coals and a flame to produce special, caramelised sides.
Bulk vegetables are also fairly cheap to find, and you can do so many creative things with them! Some of our favourites include: buttery cobs of corn (a kids' favourite), charred capsicum and hearty whole mushrooms—which can be stuffed with an array of delicious fillings.
How do you keep the costs down at the gatherings you host this time of year? It can get so hectic; how do you keep the dinner under control?