text:
Q: Is there a reservoir sampling algorithm that makes the expected number of steps an item remains in the reservoir constant? I'm trying to sample from an infinite stream of data without having to store all past items somewhere.
I immediately stumbled upon reservoir sampling and from what I understand, it consist in having a reservoir of $k$ items and progressively replace items from the reservoir with items coming from the stream in a way the reservoir is a true sample of the distribution at the end of the execution.
So assuming $n$ steps have passed and the data distribution of items of the stream is $D$, my understanding is that:

*

*each sampled $x_i \sim D$, so if I stop at any point, I am guaranteed to have sampled $k$ items from $D$ in the reservoir

*the probability of replacing an item of the reservoir by a new item decreases with the running time $n$, in the order of $O(1/n)$
This second point is what bothers me, I would like to have items of the reservoir all remain the same amount of time in expectation.
In other words, I would like to have:
$\forall i \in \mathbb N, \mathbb E[t_i]=c$ where $t_i$ is the number of steps item $i$ was in the reservoir and $c$ a constant.
This is because I would like to have a machine learning algorithm learn from items in the reservoir and avoid over-fitting on the items that would have remained longer in the reservoir. I am not sure if this is already a property of reservoir algorithms, but intuitively I would say it is not since at the start of the algorithm many more replacements should happen, lowering the expected time of these items compared to items coming in later replacements.
So I have some questions:

*

*is it possible to design such sampling algorithm that respect equal sampling probability and equal time in the reservoir in expectation? Or are the two objectives fundamentally incompatible?

*If yes, what is an algorithm that meet requirements? Is the constant $c$ an hyperparameter of the algorithm?

*If no, what would be an algorithm fulfilling the second requirement but "close" in spirit to the first?


text:
Run & Jump
After a stroke leaves her husband disabled and fundamentally changed, a spirited Irish wife struggles to keep her family members together. All the while they are under the microscope of an American researcher documenting their recovery process.
Genre : Drama
Actors : Brendan Morris, Ciara Gallagher, Clare Barrett, Edward MacLiam, Kelby Guilfoyle, Maxine Peake, Michael Harding, Ri Galway, Ruth McCabe, Sharon Horgan, Will Forte
Director : Steph Green
Country : Germany, Ireland
Quality: ComingSoon
Keywords: Лечението Voll und ganz und mittendrin A megfigyelő Люби
The Meddler
An aging widow from New York City follows her daughter to Los Angeles in hopes of starting a new life after her husband passes away.
Set in a world with memory implants, Robin Williams plays a cutter, someone with the power of final edit over people's recorded histories. His latest assignment is one that puts him in danger.
Country: Canada, Germany, United States of America, USA
Genre: Drama, Sci-Fi, Thriller
An ambitious Hollywood hustler becomes involved with a reclusive female star whom he tries to lure out of retirement.
Country: France, Germany, West Germany
This multiple-Oscar-winning film by Roman Polanski is an exquisite, richly layered adaptation of Thomas Hardy's Tess of the d'Urbervilles. A strong-willed peasant girl (Nastassja Kinski, in a gorgeous breakthrough) is sent by her father to the estate of some local aristocrats to capitalize on a rumor that their families are from the same line. This fateful visit commences an epic narrative of sex, class, betrayal, and revenge, which Polanski unfolds with deliberation and finesse. With its earthy visual textures, achieved by two world-class cinematographers—Geoffrey Unsworth (Cabaret) and Ghislain Cloquet (Au hasard Balthazar)—Tess is a work of great pastoral beauty as well as vivid storytelling.
Country: France, UK, United Kingdom
A Month in the Country
Set in the early 1920s, the film follows Tom Birkin, who has been employed to carry out restoration work on a Medieval mural discovered in a church in the small rural community of Oxgodby, Yorkshire. The escape to the idyllic countryside is cathartic for Birkin, haunted by his experiences in World War I. Birkin soon fits into the slow-paced life of the remote village, and over the course of the summer uncovering the painting begins to lose his trauma-induced stammer and tics.
A journey inside the world of real life caped crusaders. From all over America, these self-proclaimed crime fighters, don masks, homemade costumes and elaborate utility belts in an attempt to bring justice to evildoers everywhere.
Genre: Action, Comedy, Documentary, Drama
The Apostle Peter: Redemption
Tormented by his denial of Christ, Peter spent his life attempting to atone for his failures. Now as he faces certain death at the hand of Nero, will he falter again, his weakness betray him or will he rise up triumphant in his final moment?
Two corrupt cops have a successful, seemingly perfect money making scheme- they sell drugs that they seize from dealers, kill the dealers, and blame the crimes on street gangs. Their scheme is going along smoothly until they kill an undercover DEA agent posing as a dealer, and then try to cover-up their crime.
Genre: Action, Crime, Drama, Romance, Thriller
Mekong Hotel
Shifting between fact and fiction in a hotel situated along the Mekong River, a filmmaker rehearses a movie expressing the bonds between a vampire-like mother and daughter…
Country: Thailand, UK
Chronesthesia
A romcom with a time-travel twist. Daniel Duncombe is not a committer. When he's not working at a café or running to stay in shape, he's perfectly happy to chill at home. Then he starts waking up from weird dreams to find cryptic messages scrawled on his bedroom window. Slowly it dawns on Daniel that the same mysterious force is guiding him into contact with a strange assortment of equally unattached strangers who intrude on his routine.
Genre: Comedy, Drama, Romance, Sci-Fi
Woman Thou Art Loosed: On the 7th Day
The Ames' seem to have built the perfect life until their six year old daughter is kidnapped; over the course of seven days they begin to uncover secrets about their past that could rip their marriage and lives apart.

redpajama_set_name: RedPajamaCommonCrawl


text:
Trump signs 'seriously flawed' Russia sanctions bill
By Vivian Salama - Associated Press
WASHINGTON — President Donald Trump on Wednesday signed what he called a "seriously flawed" bill imposing new sanctions on Russia, pressured by his Republican Party not to move on his own toward a warmer relationship with Moscow in light of Russian actions.
The legislation is aimed at punishing Moscow for interfering in the 2016 U.S. presidential election and for its military aggression in Ukraine and Syria, where the Kremlin has backed President Bashar Assad. The law also imposes financial sanctions against Iran and North Korea.
Trump said the law will "punish and deter bad behavior by the rogue regimes in Tehran and Pyongyang" and enhance existing sanctions on Moscow.
The president had been reluctant to proceed with the bill, even after it was revised to include some changes that American and European companies sought to ensure that business deals were not stifled by new sanctions. Trump has expressed frustration over Congress' ability to limit or override the power of the White House on national security matters, saying that it is complicating efforts to coordinate with allies — a sentiment he expressed in Wednesday's statement.
Last week, the House overwhelmingly backed the bill, 419-3, and the Senate rapidly followed with a 98-2 vote. Those margins guaranteed that Congress would be able to beat back any attempt by Trump to veto the measure.
The president said Wednesday that he signed the bill "for the sake of national unity."
"The bill remains seriously flawed — particularly because it encroaches on the executive branch's authority to negotiate," Trump said. "By limiting the executive's flexibility, this bill makes it harder for the United States to strike good deals for the American people, and will drive China, Russia, and North Korea much closer together."
Trump's talk of extending a hand of cooperation to Russian President Vladimir Putin has been met with resistance as skeptical lawmakers look to limit his leeway. The new measure targets Russia's energy sector as part of legislation that prevents Trump from easing sanctions on Moscow without congressional approval.
Those limits, backed by Republicans as well as Democrats, resulted from lawmakers' worries that Trump might ease the financial hits without first securing concessions from Putin. Republicans refused to budge even after the White House complained that the "congressional review" infringed on Trump's executive authority.
Moscow responded to a White House announcement last week that Trump intended to sign the bill, ordering a reduction in the number of U.S. diplomats in Russia.
Top members of Trump's administration voiced their unhappiness with the bill anew this week, echoing his sentiments that it poses more diplomatic hindrances than solutions.
"Neither the president nor I are very happy about that," Secretary of State Rex Tillerson said Tuesday of the sanctions bill, which he had urged lawmakers not to approve.
"We were clear that we didn't think that was going to be helpful to our efforts, but that's the decision they made," he said.
Vice President Mike Pence, traveling Tuesday in the former Soviet republic of Georgia, sought to reframe the sanctions as a "further sign of our commitment" to counter Russian aggression in the region.
"The president and our Congress are unified in our message to Russia: A better relationship, the lifting of sanctions will require Russia to reverse the actions that caused the sanctions to be imposed in the first place," Pence said. "And not before."
Lawmakers on both sides of the aisle celebrated the passage of the sanctions bill.
By Vivian Salama
Hi! A visitor to our site felt the following article might be of interest to you: Trump signs 'seriously flawed' Russia sanctions bill. Here is a link to that story: https://www.delgazette.com/news/59686/trump-signs-seriously-flawed-russia-sanctions-bill


text:
MEGGS – NEW MURAL MELBOURNE
David Hooke, AKA Meggs, is back living in Melbourne and recently painted this 25 metre mural on the outskirts of the city in Croydon. "I grew up in Croydon North so it feels good to now have a large public piece in the area I spent my childhood. This mural is inspired by the beauty…
Coverage: Kaws Museum Exhibition At NGV Melbourne
Earlier today, Kaws opened his new museum exhibition at NGV Melbourne. Take a look at all the exclusive images by Dean Sunshine below.
PHLEGM for Provocare Melbourne
UK street artist Phlegm was just in Melbourne, commissioned by the Chapel Street Precinct Association as part of the PROVOCARÉ Festival Of The Arts. John Lotton from CSPA spent two and half years planning this project, choosing the right artist, and locating the right wall for a large mural with a vision to become a landmark piece…
Heesco and Mayo for The Hive Melbourne
Melbourne artists Heesco and Mayo collaborate for the first time on a notorious wall in Richmond not far from the city centre. For years there were multiple attempts to give this wall a well-needed makeover and finally, this goal was realised in just seven days in May. The aim of the project was to bring joy…
Heesco for World's 50 Best Restaurant Awards – Melbourne
Melbourne based street artist Heesco was commissioned by Visit Victoria to paint five portraits for the World's 50 Best Restaurant Awards which are held in London every year but this year, the massive event was brought to Melbourne. In just three and a half days the Mongolian born artist completed all five artworks in Chin Chin lane in…
RONE The Alpha Project Melbourne
Our good friend RONE recently launched The Alpha Project – a cluster of four massive portrait murals painted secretly over three months in an large abandoned paper mill due for demolition. In an nearby building, another smaller portrait is visible surrounded by rubble in a scene resembling a war zone not inner suburban Melbourne. The murals were…
Adnate paints the Dalai Lama in Melbourne
Australian street artist and old mate ADNATE recently spent a few days painting the rear of a building in Fitzroy on the streets of Melbourne in Australia. The three storey mural features the Dalai Lama whom he met in person last year in India. He was very moved after the meeting and wanted honoured to paint…
Per L'Australia / Passenger by Alice Pasquini Melbourne
Italian street artist Alice Pasquini recently visited Melbourne to paint the exterior wall of the Museo Italiano in Carlton. The large ten metre mural took seven days to complete. When asked to describe the work, this is what she had to say: "The mural is dedicated to the history of Italian emigration to Australia, inspired by the…
"Grandparents" by Smug in Melbourne
Glasgow-based muralist SMUG recently finished this stunning mural in the centre of Melbourne's business district. The four storey mural on a former power station features the artist's grandparents in his signature photo realistic style and took a bit longer to complete with the wet and wild weather conditions. If you are in the area and wish…
#street art news


Envy is probably the most famous of sins as it applies to the Financial World/Blogosphere.
Envy and Luxuria are the driving forces behind the never ending desire to "keep up with the Joneses," a phrase which first entered the American Lexicon in 1913 in, of all things, a comic strip.
It is ingrained in human nature to compare ourselves with our peers.
Finding where one stands amongst their cohorts is not a bad thing in and of itself.
However the Sin of Envy first rears its ugly head when we start to begrudge those who are ranked higher and have our own happiness diminish because of it.
In the world of social media where every achievement is chronicled on Instagram, Facebook, etc. it is easy to fall into the trap of Envy.
This is exacerbated by the fact that people tend to only put the positives out on social media for the public to see and not the negatives which do occur with all of us.
The casual social media viewer therefore gets a highly distorted view of how others are apparently living and become envious as their own lives fall short.
Coupled with the fact that there are people who pretend to be rich using rentals and borrowed jewelry in order to create a facade for Instagram and you have the makings for unfounded Envy.
The financial impact of this is that we then try to emulate the people we feel are above us by purchasing items we do not need but think will help us bridge that gap.
It is interesting to note that it is not the actual object that is desired by someone who is envious.
Rather the feelings of envy are directed to the possessor of that object.
I am not envious of a private jet for example.
I am however envious of what it must feel like being the owner of a private jet.
In my mind I create my own artificial world which likely surpasses the actual reality of owning a private jet.
Envy is also the underlying force behind a phenomenon called Schadenfreude, which causes us to take pleasure in the downfall of someone who was originally above us in stature.
How many of you took some form of pleasure when, after all the generated hype of Bitcoin and its stratospheric rise suddenly disappeared and those who were Bitcoin multimillionaires suddenly fell back down to Earth?
Part of the reason is because we actually did have these feelings of Envy (whether we admit it or not) about these (initially) lucky individuals who became instantly rich.
Once they were no longer wealthy, the envy disappeared.
To counter the Cardinal Sin of Envy is the Heavenly Virtue of Kindness/Admiration.
When you can truly admire someone for what they have achieved, even if it puts them at a higher level than your own, the sin of Envy melts away.


text:
\section{Introduction}

Partial differential equations (PDEs) provide a natural description for the temporal evolution of spatially extended systems in various fields of science and engineering.
Historically and practically important examples include wave equations arising in many areas of physics, the Schr\"{o}dinger equation in quantum mechanics, the Navier-Stokes equations in fluid dynamics, and reaction-diffusion equations used to model physical, chemical, or biological systems.
In the past, models of such systems were almost always constructed from first principles or using a suitable empirical approach.
However, in recent years, a data-driven paradigm for learning the dynamics has emerged, which leverages the modern prevalence of data and computational power to create models when the underlying governing laws have eluded first-principles derivation.

Many indirect methods for learning the dynamics that do not require a PDE have been proposed.
Notable examples include equation-free modeling \cite{kevrekidis2003}, artificial neural networks \cite{hsu1997,raissi2018deep,pathak2018}, dynamic mode decomposition \cite{tu2013} and Koopman operator approaches \cite{mezic2013}, balanced truncation \cite{rowley2005}, and resolvent-based analysis \cite{mckeon2010}.
While these techniques can provide an economical approximate description of the dynamics, this is done at the cost of losing the mathematical structure that affords physical intuition or interpretability.
Symbolic regression, which was originally used to derive nonlinear ordinary differential equations describing low-dimensional systems \cite{bomgard2007, schmidt2009}, offers an enticing alternative by allowing construction of exact models and discovery of conservation laws.
The genetic algorithms used in these earlier studies are however computationally expensive, preventing application of this approach to high-dimensional systems.
Thus, the recent emergence of a sparse regression approach for model discovery \cite{xu_2008,khanmohamadi_2009,brunton2016,rudy2017,reinbold_2019} has made a significant impact.
Applied to spatially extended systems, this approach allows data-driven discovery of governing equations in the form of PDEs by evaluating a library of candidate terms containing partial derivatives at a large number of points and using a regularized regression procedure to compute the coefficients of each term and select a parsimonious model.

Sparse regression has proven computationally efficient and capable of reconstructing numerous canonical PDEs \cite{rudy2017,li_2019}, but it faces serious difficulties when used for analysis of experimental data.
One complication is that the proper choice of parsimonious model is often unclear.
In many implementations, it relies on a manual Pareto analysis to balance model accuracy and complexity \cite{brunton2016} or on an automatic but complex thresholding procedure (e.g. sequential threshold ridge regression \cite{rudy2017}) that tends to be sensitive to the choice of parameters.
More importantly, existing sparse regression methods often suffer from low accuracy even in the absence of noise and completely break down at noise levels characteristic of realistic applications.
This is because they inherently require explicit numerical evaluation of partial derivatives of the data, which is a notably ill-conditioned problem.

In this paper, we present a weak formulation of the sparse regression problem that eliminates this fundamental issue.
We also suggest a simple thresholding procedure that can always identify the correct form of the governing PDE even in the presence of extremely high noise.
Finally, we explore how this extremely flexible and robust approach can be optimized and tuned to the properties of the underlying data set to maximize accuracy.
This paper has the following structure.
Section \ref{sec:methods} describes our approach and the system used to test it.
Results are presented and interpreted in Section \ref{sec:results}, and conclusions are discussed in Section \ref{sec:conclusions}.

\section{Methods}
\label{sec:methods}

We consider the problem of using the data ${\bf u}({\bf x},t)$ to identify a parsimonious mathematical model in the form of a PDE
\begin{align}
\sum_{n=1}^N c_n \mathbf{f}_n(\mathbf{x},t,\mathbf{u}, \partial_t\mathbf{u},\partial^2_t\mathbf{u}, \nabla\mathbf{u},\nabla^2\mathbf{u}, \cdots) = 0
\label{eq:pde}
\end{align}
where each term in the sum is a function of ${\bf u}$ and its partial derivatives in space and time with constant coefficients $c_n$.
In most applications, the form of the basis functions $\mathbf{f}_n$ can be restricted based on physical considerations, such as symmetries, conservation laws, etc. \cite{bar_1999,reinbold_2019}.
Typically, $\mathbf{f}_n$ are taken to be products of powers of independent variables (${\bf x}$, $t$) and dependent variables (${\bf u}$ and its various derivatives), although the form can be arbitrary in theory.
Our goal is to determine the constants $c_n$ for the terms that should be present in the model while eliminating the dynamically insignificant and thus likely spurious terms.
Sparse regression aims to convert the PDE (\ref{eq:pde}) to a tractable (and ideally, robust) linear algebra problem.
Conventionally this is done by evaluating all of the terms in the PDE at a random collection of points $({\bf x}_k,t_k)$ using finite differences \cite{vallette_1997,bar_1999}, spectral methods \cite{xu_2008,khanmohamadi_2009}, or polynomial approximation \cite{rudy2017,reinbold_2019}.
All of these approaches are extremely sensitive to noise, especially when high-order derivatives are present.
We will instead pursue a weak formulation of the problem that can be obtained by multiplying (\ref{eq:pde}) by a weight $\mathbf{w}_j(\mathbf{x},t)$ and then integrating the result over a domain $\Omega_k$.
Repeating the process for $K$ distinct combinations of weight functions and integration domains yields the linear system
\begin{align}
Q\mathbf{c} = 0
\label{eq:lin}
\end{align}
where $\mathbf{c} = [c_1, \dots, c_N]^T$ and $Q = [\mathbf{q}_1, \dots, \mathbf{q}_N]$ is a ``library'' matrix, with each column $\mathbf{q}_n\in\mathbb{R}^K$ consisting of the integrals of the function $\mathbf{f}_n$ with all $K$ combinations of weights $\mathbf{w}_j$ and domains $\Omega_k$.

Note that there is an extra degree of freedom in \eqref{eq:lin} corresponding to the normalization of $\mathbf{c}$.
Conventionally this is dealt with by assuming that ${\bf f}_1=\partial_t{\bf u}$, setting $c_1=1$, and solving the overdetermined system that corresponds to the choice $K\gg N$ using least squares or some regularized version of it \cite{rudy2017}.
This is however not always a valid assumption: it is usually unknown {\it a priori} whether any given temporal derivative should be included in the PDE at all, whereas in this case a particular term is forced into the model.
Moreover, even if this term should be present in the model, the regression effectively assumes that the time derivative was computed without error, which reduces the practical accuracy of the procedure.

We will therefore not make the assumption that the model has the form of an evolution equation and consider the linear problem \eqref{eq:lin} in its most general form.
The normalization of $\mathbf{c}$ can be fixed by adding an extra row with arbitrary nonzero elements to $Q$, after which the resulting equation (\ref{eq:lin}) can be solved by
ordinary least squares.
A more elegant solution pursued in the present study is to instead compute $\mathbf{c}$
as the right singular vector of $Q$ corresponding to the smallest singular value.
Note that this corresponds to the solution of a constrained least squares problem for $Q^TQ\mathbf{c}=0$:
\begin{align}
\mathbf{c}=\argmin_{\|\mathbf{c}\|=1}\|Q^TQ\mathbf{c}\|.
\label{eq:linn}
\end{align}
Once a suitable solution has been obtained by further constraining the problem, the resulting parsimonious model can be rewritten in the form of an evolution equation by solving for a term such as $\partial_t\mathbf{u}$ (or $\partial_t^2\mathbf{u}$ for a wave equation).

To obtain a parsimonious model, we employ an iterative procedure to eliminate unnecessary terms from \eqref{eq:pde}.
At each step $i$, singular value decomposition is used to obtain the solution ${\bf c}^i$ given the matrix $Q^i$, and the residual $\eta^i = \|Q^i {\bf c}^i\|$ is computed.
We then find the term with the smallest $\|c^i_n\mathbf{q}_n\|/\|\mathbf{q}_n\|$
and construct $Q^{i+1}$ by eliminating the column ${\bf q}_n$ from $Q^i$.
The corresponding term is eliminated from the model if $\eta^{i+1}<\gamma\eta^i$, where $\gamma>1$ is some fixed constant (we use $\gamma=1.4$ in the present study).
The iteration terminates at step $i$ if $\eta^{i+1}>\gamma\eta^i$, yielding a parsimonious model.
We find that this method compares favorably to alternatives such as sequential threshold ridge regression \cite{rudy2017} as it robustly eliminates spurious terms without requiring extremely careful choice of parameters.
Moreover, the sparsification parameter has a simple interpretation: $\gamma-1$ is the maximum acceptable relative increase in the residual resulting from discarding a single library term.

We illustrate the advantages of our approach by applying it to the Kuramoto-Sivashinsky equation \cite{kuramoto_1978, sivashinsky_1985}
\begin{align}\label{eq:kse}
c_1\partial_t u+c_2u\partial_x u+c_3\partial_x^2u+c_4\partial_x^4u = 0
\end{align}
which has posed a significant challenge in past studies of sparse regression \cite{xu_2008,rudy2017} because it contains a fourth-order partial derivative that is difficult to evaluate numerically with adequate accuracy.
Here $c_1=\cdots=c_4=1$ are all constants, although our approach can easily be extended even to the case when these coefficients are functions of time and/or space, as discussed below.
Since this is a scalar equation in one spatial and one temporal dimension, we use scalar weight functions $w^j(x,t)$.
If we denote the terms in the model \eqref{eq:kse} by $f_1,\cdots,f_4$, then
\begin{align}
&q_1^{jk} = \int_{\Omega_k} w_j\partial_t u\,d\Omega,\quad
 q_2^{jk} = \int_{\Omega_k} w_ju\partial_x u\,d\Omega, \nonumber\\
&q_3^{jk} = \int_{\Omega_k} w_j\partial^2_x u\,d\Omega,\quad
 q_4^{jk} = \int_{\Omega_k} w_j\partial^4_x u\, d\Omega,
\end{align}
where $d\Omega=dx\,dt$.
The key feature of the weak formulation is that it can almost always be used to completely eliminate, or at least reduce the order of, the derivatives acting on the noisy data by integrating by parts.
In our particular case,
\begin{align}
&q_1^{jk} = -\int_{\Omega_k} u\partial_tw_j\, d\Omega,\quad
 q_2^{jk} = -\frac{1}{2}\int_{\Omega_k} u^2\partial_xw_j\, d\Omega,\nonumber\\
&q_3^{jk} = \int_{\Omega_k} u\partial^2_x w_j\, d\Omega,\quad
 q_4^{jk} = \int_{\Omega_k} u\partial^4_xw_j\, d\Omega
\end{align}
under the assumption that $w_j$ and its first three partial derivatives with respect to $x$ vanish on the boundary $\partial\Omega_k$.
In our implementation, we use the composite trapezoidal rule to evaluate the integrals numerically.

Note that although this particular PDE features constant coefficients, terms with variable coefficients can be treated in a similar manner.
For instance, suppose that the coefficient of the term $\partial_x^4u$ is a function of ${\bf x}$ and $t$ that can be expanded in some (finite) basis as
\begin{align}
c_4({\bf x},t) = \sum_p c'_pg_p({\bf x},t)
\end{align}
with some constants $c'_p$. Then
\begin{align}
\int_{\Omega_k} w_jc_4\partial^4_x u\, d\Omega = \sum_p c'_p q_p^{jk},
\end{align}
where
\begin{align}
q_p^{jk}=\int_{\Omega_k} u\partial^4_x(g_pw_j)\, d\Omega.
\end{align}
Sparse regression for a model including such a term would then simply require expanding the library $Q$ to include additional columns ${\bf q}_p$ with entries $q_p^{jk}$.
In this case as well, no derivatives of the noisy $u$ are used in finding the elements of $Q$.

Although in principle integration domains of any shape can be used, here we will only consider rectangular domains of a fixed size
\begin{align}
\Omega_k = \{(x,t) \ : \ |x-x_k| \leq H_x, |t-t_k| \leq H_t\}
\end{align}
where the centers $(x_k, t_k)$ of the rectangles $\Omega_k$ are chosen randomly.
Similarly, there are many possible choices for the weight functions satisfying the boundary conditions on $\partial\Omega_k$; we focus on functions of the form
\begin{align}\label{eq:wf}
w_j = (\underline{x}^2-1)^\alpha(\underline{t}^2-1)^\beta e^{\pm il\pi\underline{x}}e^{\pm im\pi\underline{t}},
\end{align}
where $\underline{x} = (x-x_k)/H_x$, $\underline{t} = (t-t_k)/H_t$ are nondimensionalized independent variables and $\alpha\ge 4$, $\beta\ge 1$, $l\ge 0$, and $m \ge 0$ are integers.
Note that there are four weight functions (corresponding to the four different choices of the signs in the exponentials) for each pair of nonzero $l$ and $m$. The integrals $q_n^{jk}$ are all of the form
\begin{align}
F_n^{lm}=\int_{-1}^1d\underline{t}\int_{-1}^1d\underline{x} f_n^{\alpha\beta}(\underline{x},\underline{t}) e^{\pm il\pi\underline{x}}e^{\pm im\pi\underline{t}},
\end{align}
where
\begin{align}
f_n^{\alpha\beta}(\underline{x},\underline{t})=f_n (u,\underline{x},\underline{t}) (\underline{x}^2-1)^\alpha(\underline{t}^2-1)^\beta,
\end{align}
so $F_n^{lm}$ are the coefficients of the two-dimensional Fourier series for $f_n^{\alpha\beta}(\underline{x},\underline{t})$.
Although $f_n(u,\underline{x},\underline{t})$ is not periodic on $\Omega_k$, the functions $f_n^{\alpha\beta}(\underline{x},\underline{t})$ are.
Moreover, $f_n^{\alpha\beta}(\underline{x},\underline{t})$ has at least $\alpha-1$ continuous derivatives in $\underline{x}$ and ${\beta-1}$ continuous derivatives in $\underline{t}$, so the Fourier coefficients decay according to $F_n^{lm}\sim l^{-\alpha}m^{-\beta}$.
The powers $\alpha$ and $\beta$ therefore control the width of the Fourier spectrum of the entries $q_n^{jk}$ in the library $Q$, while the choice of $l$ and $m$ allows us to tune the frequencies of the weights to the spectral properties of the data.
The convergence rate of Fourier series turns out to control the accuracy with which the integrals are evaluated using data that are available only on a discrete grid.
For simplicity, we will assume that the same weight functions are integrated on every domain.
It is possible to use either weight functions involving only a single pair of frequencies (e.g., $l$ and $m$) or a range of frequencies in space and/or time.

To test our sparse regression approach, we computed a solution of the Kuramoto-Sivashinsky equation, using the integrator described in Ref. \onlinecite{rudy2017} to generate data on a physical domain with dimensions $L_x = 32\pi$ and $L_t = 500$.
The numerical integration generated data with spatial resolution $\Delta x=0.0491$ using a time step $\Delta t=0.005$, which was then downsampled to a lower spatial resolution $\delta_x$ and temporal resolution $\delta_t$.
Unless noted otherwise, the results presented below are for $\delta_x = 0.1964$ and $\delta_t = 1$.
For reference, the solution has a correlation length $\ell_x \sim1.67 \approx 8.5 \delta_x$ and correlation time $\ell_t\sim 8=8\delta_t$.
To test the effects of noise, Gaussian noise with standard deviation $\sigma s_u$ was added to the data for various choices of $\sigma$, where $s_u \approx 1.3$ is the sample standard deviation of $u$ on the whole domain.

\begin{figure*}[t]
\subfigure[]{\includegraphics[width=\columnwidth]{FourierX.pdf}} \hspace{4mm}
\subfigure[]{\includegraphics[width=\columnwidth]{FourierT.pdf}}
\caption{The power spectrum over (a) space and (b) time, normalized so that the maximum is 1. The black dots show the spectrum of the original data. The symbols correspond to the spectra of the windowed data multiplied by envelopes $E^{\alpha\beta}(\underline{x}, \underline{t})$ (with different choices of $\alpha$ or $\beta$) on a ``typical'' integration domain $\Omega_k$ (i.e., averaged over 1000 uniformly distributed choices of $(x_k, t_k)$. The spatial and temporal frequencies correspond to $\kappa_l=2\pi l/F_x$ and $\omega_m=2\pi m/F_t$ for the windowed data.
}
\label{fig:Fourier}
\end{figure*}

To test the ability of the algorithm to eliminate spurious terms, in addition to the terms present in the Kuramoto-Sivashinsky equation \eqref{eq:kse}, we also included terms $\partial_x u$, $\partial_x^3 u$, $u$, $u^2$, $u^3$, and $1$ (which represents a hypothetical forcing) in our library.
The corresponding integrals were rewritten using integration by parts to remove derivatives acting on $u$, as described previously.
In the next section, we quantify the performance of our sparse regression approach using two key metrics: how well the algorithm can discriminate between the essential and spurious terms and how accurately it can determine the coefficients of the essential terms.
Since the data were generated using a known model, we know which terms are essential (those contained in the PDE \eqref{eq:kse}).
If the reference model is unavailable, ensemble regression \cite{reinbold_2019} may be used instead to help distinguish essential terms from spurious ones.

\section{Results}
\label{sec:results}

As discussed previously, the elements of the library matrix $Q$ are given by the Fourier coefficients of the different terms included in the generalized model (windowed by the envelope $E^{\alpha\beta}(\underline{x}, \underline{t}) = (\underline{x}^2-1)^\alpha(\underline{t}^2-1)^\beta$ on each domain $\Omega_k$);
hence knowledge of the Fourier spectrum of the data is crucial for an optimal choice of the size of the integration domains $\Omega_k$ and the weight functions $w_j$.
The power spectrum (or, more precisely, the absolute value of the Fourier coefficients) of the noiseless data on the entire physical domain
is shown in Figure \ref{fig:Fourier}.
In space, the spectrum is sharply peaked around a wave number $\kappa\approx 0.625$.
At high wave numbers, the spectrum decays exponentially, $P\propto e^{-\kappa/\bar{\kappa}}$ where $\bar{\kappa}\approx 0.3$.
In time, the spectrum is peaked at zero frequency $\omega$ and decays as a power law, $P\propto \omega^{-\chi}$ with $\chi\approx 2.5$.

Having characterized the data, we turn to the investigation of how the performance of our algorithm depends on the choice of various parameters.
Since the number of parameters is quite large, instead of exploring the entire parameter space, we focus on the dependence on one or two parameters at a time, with the remaining parameters staying fixed.
Specifically, the noise level $\sigma$ is fixed to 3\% and we use the following near-optimal parameters in the sparse regression.
The dimensions of the integration domain are $F_x =2H_x=14.73$ and $F_t =2H_t=75$. This choice corresponds to an equal number of grid points in both directions, $F_x/\delta^*_x=F_t/\delta^*_t=75$.
Unless noted otherwise, we use a single set of weights with $\alpha=\beta=8$, $l=1$, $m=2$, and the sparsification parameter is $\gamma = 1.4$.
We generally use every combination of 4 weight functions over 50 integration domains, so that the total number of library rows is $K=200$.
To characterize the stochastic effects, for each set of parameters, we used an ensemble of $M=100$ trials featuring different random distributions of the integration domains and realizations of noise.

First, we tested the ability of the method to reconstruct the correct form of the PDE \eqref{eq:kse} for various values of $\gamma$ with all other parameters fixed at their near-optimal values.
Our iterative regression procedure proved very robust for a fairly wide range of values of $\gamma$.
In particular, at a noise level of 30\%, it performed perfectly for $1.1\leq\gamma\leq2$, with the reconstructed model containing no missing or spurious terms in all of the trials.
For the highest noise level considered here (100\%), we found perfect performance for $1.2\leq\gamma\leq1.5$. In some fraction of the trials, spurious terms appeared at lower $\gamma$ and missing terms at higher $\gamma$, as shown in Figure \ref{fig:Spur}.
For reference, without the benefit of the weak formulation, sparse regression failed \cite{rudy2017} to correctly reconstruct the lambda-omega reaction-diffusion system, which is only second-order, for noise level as low as 1\%.

The accuracy of regression (i.e., model identification) was quantified by computing the relative error in each parameter of the Kuramoto-Sivashinsky equation
\begin{align}
\Delta c_n = \left|\frac{c_n-\bar{c}_n}{\bar{c}_n}\right|,
\end{align}
where $\bar{c}_n$ and $c_n$ are the true and estimated values of the model parameters, respectively, for $n=2,3,4$.
We normalize the estimated parameters so that $c_1 = 1$.
In all of the following figures, we plot the estimated mean value of $\Delta c_n$ with 95\% error bars, where all of the parameters in the regression procedure are held at their near-optimal values stated previously unless noted otherwise.

In particular, Fig. \ref{fig:Noise} shows the accuracy of regression as a function of $\sigma$ for two different choices of data resolution ($\delta_x$ and $\delta_t$).
It is worth noting that the average relative error is $\Delta c_n\sim 10^{-10}$ for all of the parameters for noiseless data with the higher of the two resolutions.
However, even for $1\%$ noise, $\Delta c_n\sim 2\times10^{-4}$, which is more than three orders of magnitude smaller than what had been achieved in previous studies \cite{rudy2017}.
The results are very similar for all three parameters; as this is generally the case, in subsequent figures, we only show the generally largest error $\Delta c_4$, which corresponds to the term $\partial_x^4 u$ involving the highest-order derivative.
We find two distinct regimes.
At higher noise levels, the error in evaluating the library matrix entries is due primarily to the averaged effect of noise. Applying the central limit theorem, we find that the relative error scales as
\begin{align}\label{eq:epsn}
\varepsilon_n\sim\sigma u_s\sqrt{\frac{\delta_x\delta_t}{F_xF_t}}.
\end{align}

\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{Spur.pdf}}
\caption{Fraction $p$ of identified models with spurious (circles) or missing (squares) terms at maximum noise level ($\sigma=1$) as a function of the sparsification parameter $\gamma$.
}
\label{fig:Spur}
\end{figure}

At low noise levels, the parameter accuracy is controlled by numerical error, which has two different sources.
The first source is a numerical error in the data itself, which is due to the finite accuracy of the integrator that ``solves'' the Kuramoto-Sivashinsky equation.
This source dominates for smaller $\delta_x$ and $\delta_t$.
For experimental data, this source would correspond to systematic error.
For larger $\delta_x$ and $\delta_t$, the parameter inaccuracy is mainly due to the error in computing the library matrix entries based on data that are available on a discrete grid.
Suppose we want to use numerical quadratures to evaluate an integral
\begin{align}
I=\int_0^L g(x)dx,
\end{align}
where $g(x)\in C^{m}$ (i.e., has $m$ continuous derivatives) and $g^{(i)}(0)=g^{(i)}(L)$ for all $0\leq i<m$.
Then, for the composite trapezoidal rule on a grid with spacing $h$, the relative error associated with the discretization can be estimated using exact Euler-Maclaurin formulas\cite{trefethen2014} and is found to scale as $h^{m+2}|g^{(m+2)}|$ for $m$ even (or $h^{m+1}|g^{(m+1)}|$ for $m$ odd), where a characteristic value of the derivative on the interval $[0,L]$ is used.

\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{Noise.pdf}}
\caption{Parameter errors $\Delta c_n$ as a function of noise level. The circles, triangles, and squares correspond to $n=2, 3, 4$ respectively, and the empty and filled symbols indicate results for data with double resolution ($\delta_x = 0.0982, \delta_t = 0.5$) and half resolution ($\delta_x = 0.393, \delta_t = 2$), respectively. The dashed lines show the predicted scaling.
}
\label{fig:Noise}
\end{figure}

Generalizing this result to two dimensions (and assuming $\delta_x\ll \min(\ell_x,F_x)$, $\delta_t\ll \min(\ell_t,F_t)$), we find an estimate of the relative discretization error for an element of the library matrix $Q$ that involves a temporal derivative of order $\nu_t$ and/or spatial derivative of order $\nu_x$:
\begin{align}\label{eq:mu}
\varepsilon_d\sim
\begin{cases}
h^{\mu+2},& \mu\ \mathrm{even}\\
h^{\mu+1},& \mu\ \mathrm{odd}
\end{cases}
\end{align}
where $h=\delta_t/\ell_t\approx\delta_x/\ell_x$ and
$\mu=\min(\alpha-\nu_x,\beta-\nu_t)$.
It is easy to check that, due to the conditions on $\alpha$ and $\beta$, we always have $\mu\ge 0$, as it should be for the trapezoidal rule.
The Kuramoto-Sivashinsky equation features terms that all involve derivatives, with the lowest order being one and the highest being four;
hence, for even $\alpha=\beta\ge 4$, the exponent $\mu$ ranges between $\alpha-2$ and $\alpha+1$.
Therefore the scaling
\begin{align} \label{eq:low_h}
\varepsilon_d\sim h^{\alpha-2}
\end{align}
dominates for lower $h$, while the scaling
\begin{align} \label{eq:high_h}
\varepsilon_d\sim h^{\alpha}
\end{align}
dominates for higher $h$.

\begin{figure*}[t]
\subfigure[]{\includegraphics[width=\columnwidth]{normX.pdf}} \hspace{4mm}
\subfigure[]{\includegraphics[width=\columnwidth]{normT.pdf}}
\caption{Scaling of the first four columns of the library $Q$ with the size of the integration domain in the (a) spatial and (b) temporal directions. The columns correspond to $\partial_t u$ (squares), $u\partial_x u$ (circles), $\partial_x^2 u$ (triangles), and $\partial_x^4 u$ (diamonds).
}
\label{fig:norm}
\end{figure*}

The error $\Delta c_n$ can be found using perturbation theory.
Let $\bar{Q}$ be the library matrix evaluated using a continuous noiseless solution so that $\bar{Q}\bar{\mathbf{c}}=0$ exactly (we assume that $\bar{Q}$ corresponds to the parsimonious model).
In the presence of measurement noise and/or discretization error, the error in evaluating each entry $q_n^{jk}$ of the library matrix is proportional to $\varepsilon=\max(\varepsilon_d,\varepsilon_n)$, so
\begin{align}
Q=\bar{Q}+\varepsilon\hat{Q}
\end{align}
for some matrix $\hat{Q}$ whose entries are distributed as white Gaussian noise.
Note that the entries of $\hat{Q}$ are $O(F_xF_t)$.
The entries of $\bar{Q}$ have a more complicated scaling that is determined by the Fourier spectrum of the data (i.e., exponential in space, power law in time).
Specifically, we find (cf. Fig. \ref{fig:norm})
\begin{align}\label{eq:psin}
\|\bar{\bf q}_n\|\propto
\begin{cases}
F_xF_t, & F_x\ll\ell_x,F_t\ll\ell_t\\
e^{-\lambda_nF_x}\left(\frac{\ell_t}{F_t}\right)^{\xi_n}\ell_x\ell_t,& F_x\gg\ell_x,F_t\gg\ell_t
\end{cases}
\end{align}
where $\lambda_n=O(\ell_x^{-1})$ and $\xi_n=O(1)$ are some positive constants.
To leading order in $\varepsilon$, the least squares solution to \eqref{eq:lin} is given by
\begin{align}
\mathbf{c} = \bar{\mathbf{c}}-\varepsilon \bar{Q}^+\hat{Q}\bar{\mathbf{c}},
\end{align}
where $\bar{Q}^+$ is the Moore-Penrose pseudoinverse of $\bar{Q}$.
Since the elements of $\hat{Q}$ can be considered uncorrelated, we have for $F_x\gg\ell_x$ and $F_t\gg\ell_t$
\begin{align}\label{eq:pow0}
\Delta c_n \propto \varepsilon \frac{F_xF_tK^{-1/2}}{\psi(F_x,F_t)},
\end{align}
where the numerator and denominator describe the scaling of the entries of $\hat{Q}$ and $\bar{Q}$, respectively. Following from \eqref{eq:psin},
\begin{align}\label{eq:psi}
\psi(F_x,F_t)= e^{-\lambda F_x}\left(\frac{\ell_t}{F_t}\right)^{\xi}\ell_x\ell_t
\end{align}
with some positive constants $\lambda=O(\ell_x^{-1})$ and $\xi=O(1)$.
For low $\sigma$, we have $\varepsilon=\varepsilon_d$ and therefore $\Delta c_n$ is independent of $\sigma$.
For high $\sigma$, we have $\varepsilon=\varepsilon_n$, so combining \eqref{eq:pow0} and \eqref{eq:epsn} we find
\begin{align}\label{eq:pow}
\Delta c_n \propto \sigma \sqrt{\frac{\delta_x\delta_t}{KF_xF_t}}\frac{F_xF_t}{\psi(F_x,F_t)}.
\end{align}
The predicted scaling of $\Delta c_n$ with $\sigma$ in both regimes is consistent with the results shown in Fig. \ref{fig:Noise}.
In particular, we find that the effect of changing the resolution of the data is quite minor at high $\sigma$, where $\Delta c_n\propto h$ according to \eqref{eq:pow}. At low $\sigma$, the effect is much stronger: for $\alpha=\beta=8$, we have $\Delta c_n\propto h^6$ according to \eqref{eq:mu}.
The dependence of the scaling in \eqref{eq:mu} on $\alpha$ and $\beta$ is further confirmed by Fig. \ref{fig:res}, which shows results for noiseless data.
In the $\alpha=\beta=4$ case, we observe the scaling law $\Delta c_n\propto h^2$ corresponding to \eqref{eq:low_h} in the entire range of $h$ we examined.
When $\alpha=\beta=6$, the parameter error scales according to $\Delta c_n\propto h^4$ for small $h$ and $\Delta c_n\propto h^6$ for large $h$, which correspond to the limiting cases \eqref{eq:low_h} and \eqref{eq:high_h}, respectively.
We should also note that for $h$ as large as $1/4$, the accuracy remains very good.
Thus, the method is suitable for fairly sparse data.

\begin{figure}[t]
\subfigure[]{\includegraphics[width=\columnwidth]{res.pdf}}
\caption{Parameter error $\Delta c_4$ as a function of the resolution of noiseless data for $\alpha=\beta=4$ (circles) and $\alpha=\beta=6$ (squares). The dashed lines show the predicted scaling.
}
\label{fig:res}
\end{figure}

As illustrated in Fig. \ref{fig:numDom}, we also observe the scaling for $\Delta c_n$ with $K$ predicted by \eqref{eq:pow}.
This scaling is expected to break down when the total area of the integration domains exceeds the area of the physical domain due to the loss of statistical independence between the data on different integration domains, leading to an increased linear dependence of the rows of the library matrix $Q$.
We can expect the error to asymptote to
\begin{align}\label{eq:scaleNd}
\Delta c_n\propto\varepsilon N_d^{-1/2}
\end{align}
for $K\gg N_d$, where
\begin{align}
N_d=\frac{L_xL_t}{F_xF_t}
\end{align}
is the area ratio.
For the reference set of parameters, saturation did not occur over the range of $K$ we tested.
To more easily observe the saturation effect, we set $l=m=0$, so that only one weight function is used and the number of integration domains equals $K$ (rather than $K/4$ for nonzero $l$ and $m$).
Furthermore, we reduce the size of the physical domain to $L_x=16\pi$ and $L_t = 250$, so that $N_d \approx 11$ is relatively small.
As Fig. \ref{fig:numDom} illustrates, for large $K$, the parameter accuracy indeed asymptotes to a constant.

\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{numDom.pdf}}
\caption{Parameter error $\Delta c_4$ as a function of the number of library rows $K$. Only the $l=m=0$ weight function is used and the physical domain size is reduced to $L_x=16\pi$, $L_t = 250$. The dashed lines show the predicted scaling.
}
\label{fig:numDom}
\end{figure}

The scaling described by \eqref{eq:scaleNd} can also be observed in the dependence of $\Delta c_n$ on the size of the physical domain (and hence $N_d$) with all other parameters fixed.
This dependence is quite important, since it determines how much data needs to be collected to identify the model with meaningful precision.
As Fig. \ref{fig:L} illustrates, choosing the physical domain to be just double the size of the (optimal) integration domain in both directions (which corresponds to $N_d=4$) already yields a rather acceptable accuracy when only one weight function is used.
When $l$ and $m$ are nonzero, accurate reconstruction is possible even if $N_d$ is only slightly greater than $1$.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{L.pdf}
\caption{Parameter error $\Delta c_4$ as a function of $N_d$ for $l=m=0$ and $K=500$. Squares correspond to fixing $L_t = 100$ and varying $L_x$ from $15.7$ to $98.2$. Circles correspond to fixing $L_x=19.6$ and varying $L_t$ from $80$ to $500$. The dashed line shows the predicted scaling.
}
\label{fig:L}
\end{figure}

\begin{figure*}[t]
\subfigure[]{\includegraphics[width=\columnwidth]{sizeX.pdf}}
\subfigure[]{\includegraphics[width=\columnwidth]{sizeT.pdf}}
\caption{Parameter error $\Delta c_4$ as a function of the (a) spatial and (b) temporal dimensions of integration domains when only the $l=m=0$ weight function is used (squares) and for the optimal choice of $l$ and $m$ (circles).
}
\label{fig:size}
\end{figure*}

Next, we consider how the error in the estimated coefficients depends on the choice of the integration domain size.
Figure \ref{fig:size} shows the dependence of the error $\Delta c_4$ on the size of the integration domains for two different choices of the weight functions.
In panel (a), $F_t$ is fixed to 75 and $F_x$ is taken to vary, and in panel (b), $F_x$ is fixed to 14.73 with $F_t$ varying.
In both cases, we find that there is an optimal domain size with $F_x \approx 14.73$ and $F_t \approx 75$; moreover, the optimal values remain approximately the same even if we vary the size of the other dimension or the choice of weight functions.
For small $F_x$ and/or $F_t$, the error is large because (a) the integration domain is too small to effectively average out the influence of noise and (b) the numerical quadrature error is large (both $\epsilon_n$ and $\epsilon_d$ increase as $F_x$ and/or $F_t$ decrease).
For large $F_x$ and $F_t$, we enter the regime described by \eqref{eq:pow0}, which predicts that the error should grow exponentially in $F_x$ and as a power of $F_t$.
Indeed, this is exactly what we observe in Fig. \ref{fig:size}.
Based on \eqref{eq:psin}, it appears that the optimal choice of $F_x$ and $F_t$ corresponds to the crossover between these two regimes, i.e., $F_x\propto\ell_x$ and $F_t\propto\ell_t$.
Our numerical results suggest that the optimal choice corresponds to $F_x/\ell_x\approx F_t/\ell_t\approx 8$.

\begin{figure*}[t]
\subfigure[]{\includegraphics[width=\columnwidth]{freqX.pdf}}
\subfigure[]{\includegraphics[width=\columnwidth]{freqT.pdf}}
\caption{Parameter error $\Delta c_4$ as a function of (a) the wave number $\kappa_l= 2\pi l/F_x$ and (b) the frequency $\omega_m= 2\pi m/F_t$.
}
\label{fig:freq}
\end{figure*}

Finally, let us address the optimal choice of frequencies appearing in the weight functions \eqref{eq:wf}.
Figure \ref{fig:freq} shows the effect of varying either $l$ or $m$ with all other parameters fixed at their reference values.
Specifically, we plot $\Delta c_4$ versus $\kappa_l = 2\pi l/F_x$ and $\omega_m = 2\pi m/F_t$.
(Note that when $l$ or $m$ is 0, the number of distinct weight functions is halved, so we correspondingly double the number of integration domains to keep the number of rows in the library constant.)
One could assume that the optimal values would be given by the dominant frequencies of the original data (as we discussed previously, the dominant wave number is $\kappa\approx 0.625$ and the dominant temporal frequency is $\omega=0$).
According to Fig. \ref{fig:Fourier}, windowing the data broadens the peaks but leaves both dominant frequencies roughly the same: $\kappa \approx 0.8$ ($l=2$) and $\omega = 0$ ($m=0$).
Unfortunately, it turns out that we cannot use the spectra to exactly predict the optimal frequencies, which are $\kappa \approx 0.4$ ($l=1$) and $\omega \approx 0.2$ ($m = 2$ or $3$).
However, choosing the frequencies based on the spectra still produces reasonably good accuracy (within a factor of $4$ or so of the optimal result).

These results suggest that using weight functions with a combination of different frequencies may be more robust and/or accurate.
To test this hypothesis, we considered the case in which weight functions with a range of frequencies in space or time were included, with the total number of library rows fixed at $200$.
However, this approach yielded a decrease in the accuracy, as the broader choice of weight functions did not compensate for a decrease in the number of integration domains.
This suggests that the optimal strategy is to use a large number of integration domains while keeping the frequencies of the weight functions fixed.

\section{Conclusions}
\label{sec:conclusions}

We have introduced a robust and flexible approach to data-driven discovery of models in the form of nonlinear PDEs.
The approach uses a weak formulation, coupled with a novel sparse regression procedure, to obtain a parsimonious description.
We have demonstrated its capability to identify PDEs, even with high-order derivatives, from extremely noisy data with unprecedented accuracy.
For instance, with 1\% noise, we were able to reduce the error in estimating the parameters of the 4th-order Kuramoto-Sivashinsky equation from 50\% \cite{rudy2017} to just $2 \times 10^{-4}$.
Furthermore, whereas correct identification of the functional form of the underlying PDE has been far from guaranteed at any noise level using past approaches, our algorithm was able to reconstruct the Kuramoto-Sivashinsky equation accurately in $100\%$ of cases from data with a signal-to-noise ratio of 100\%.

This impressive performance is achieved by shifting the partial derivatives from the data onto a known smooth weight function using integration by parts, thus avoiding the large errors incurred by repeated numerical differentiation.
Our method also proved to be well-adapted to sparse data, maintaining errors of less than $0.1\%$ for a grid resolution only $4$ times finer than the correlation length/time.
Such reliability and high accuracy in the presence of noisy or sparse data is indispensable for analysis of experimental data.
Notably, even in the absence of noise, our results compare very favorably with those of previous studies \cite{xu_2008,rudy2017} because the discretization error of the algorithm can be made extremely small: for the Kuramoto-Sivashinsky equation, the relative error in all parameters can easily be reduced to $10^{-10}$.
It is also important to mention that the computational cost of our algorithm is comparable to that of existing sparse regression methods.

We also derived the scaling laws that describe the accuracy of the regression as a function of the parameters used in the algorithm and the properties of the data.
These scaling laws can be used to fully exploit the flexibility of the weak formulation approach by tuning its various paramters.
In particular, the size of the input used by the regression can be controlled by choosing both the number of different integration domains and the number of different weight functions.
We have shown that the number of integration domains plays a much more important role than the number of weight functions: the best results can be obtained by using a set of weight functions with a fixed shape (frequency and envelope) and a large number of integration domains.
Furthermore, we have determined the optimal shape of the weights and the optimal size of the integration domains.
The latter turned out to be determined by the correlation length and time describing the data (with the size roughly an order of magnitude larger than these characteristic scales).
We have also shown that, although the error can be reduced further by using data on ever-larger physical domains, satisfactory results can be obtained for physical domains that are just a factor of two larger than the optimal integration domain in each dimension.

\begin{acknowledgments}
This material is based upon work supported by the National Science Foundation under Grant No. CMMI-1725587. DG gratefully acknowledges the support of the Letson Undergraduate Research Scholarship.
\end{acknowledgments}

\section{References}
\input{Chaos2019.bbl}

\end{document}

redpajama_set_name: RedPajamaArXiv


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How did we get to the middle of July already? It still hasn't rained significantly in my garden. Strange weather pattern, only a few kilometers away, they are complaining about too much rain. They should send some in my direction. Much of the garden is doing is doing amazingly well with just the municipal water and the tiny bit of rain.
The daylilies are popping out with different varieties every day. I have too many duplicate bunches of daylilies so I'm monitoring them carefully as they bloom. If I remove some of the duplicates, I'll have room for some new ones. One of the joys of daylilies is their ability to be moved at almost any time. I can dig them up and divide them while they are in bloom and, although I should remove the scapes, they will actually continue to try and bloom if I let them. This allows me to move them when I can actually see, that they are the same as the one, in that other bed.
How did I get so many duplicates? When the garden was new and I was eager to fill it; I would take the double clump of new daylilies and divide it into two, in order to save money and fill the garden. Now I'm digging out huge clumps. Have I got the energy to pot and label them all, so that I can hold a sale. Last year I made enough money to almost cover the cost of the new varieties that I "had" to have.
Despite the attacks of the red lily bug I am renewing my old love affair with the many types of true lilies. The size and colour variations are wonderful and they take up very little garden room relative to the impressive show that they put on. I've developed a routine and simple program to control, (not eliminate,) the red lily bug and I can keep their damage to a minimum with a little effort and still be able to produce some amazing flowers. Lily bulbs are best planted in the fall and are usually available when the rest of the fall bulbs are in the garden centres. The 9 ft monsters here, take a few years to develop that large but they will flower from the first year forward. Just treat them well, good soil preparation prior to planting, and each year they will be a bit bigger.
Throughout the very warm June, the Pansies in the porch planters kept on blooming while the Impatiens in the cold frame, (destined to replace the Pansies) continued to grow. They were becoming a bit anemic and the Pansies finally started to stretch so on Friday the 13th, I finally made the switch. Three days later and the Impatiens are looking great. The picture is an hour after they were planted. It's bit of a pain to keep the Impatiens watered and fed but the midsummer switch is worth the effort.
Every year about now, the routine of wandering around the vegetable garden, looking for something for dinner, explodes into a wealth of tasty delights. I ate the first beans this morning. No! Of course I didn't cook them. It's discover - pick - eat. The Zucchini are doing what Zucchini do and I have already started giving them away to the neighbours. Despite the heat bolting (see questions, below,) of, many of the cabbage family plants, the Broccoli have recovered enough to make it to tonight's dinner table.
The other big treat tonight is the stolen potatoes. The first fresh potatoes of the summer are a rare treat and not understood by those who have only eaten potatoes from a 10lb supermarket bag. Tonight we had "All Red," a variety with a reddish flesh as well as a red skin. They are stolen because you dig carefully around the plant looking for a few small potatoes that you can remove, 'steal,' from the plant without digging it up. It then carries on through the season to produce some more delectable delights. Lightly boiled - a touch of butter - life doesn't get much better!
Readers of Dallying In The Dirt get to send me questions. Just 'reply' to this email newsletter and I will attempt to provide answers. The most universal questions I will answer in the newsletter so that we may all, hopefully, learn something.
I have a too large an expanse of lawn to even attempt to water it. It is now very dry and brown. Will It survive or am I re-seeding in the fall?
Ken Answers. Kentucky Bluegrass, the dominant plant in most temperate climate lawns, has an amazing ability to go completely dormant, (as in dry and brown,) during extended periods of hot dry weather and then make a miraculous comeback when the temperature cools and sufficient moisture returns. Relax and enjoy the time when mowing ceases to be a frequent chore.
My Cabbage, Broccoli, Cauliflower and Brussel Sprouts appear to be healthy plants but the growing point, that produces the edible portion, has withered, turned black and curled up. Who's responsible?
Ken Answers. I have had a similar problem on a few plants each year and been able to ignore it. This year the majority of the crop is affected and there appears to be no causative agent. My research suggests that it is the very warm weather in June that is causing this "bolting" on the 'cole family' crops. They are usually cool weather crops. Some of mine are recovering and I have started new ones of some others, that will produce their crops in the, hopefully, cool autumn weather.


Directory Opus 11.5.7 (beta) is now available for all Opus 11 users.
Added new OnAddCommands and OnAddColumns script methods. These provide an alternative way for a script to add commands and columns. These methods have two advantages over the old OnInit way of adding command and columns. Firstly, the script's configuration is available in these methods whereas OnInit is unable to access it. Secondly, a script can modify its own commands and columns dynamically by calling the new Script.InitCommands and Script.InitColumns methods.
Note that if either of these methods is provided the equivalent method will no longer be available in the OnInit function, and also if a script is disabled via the Preferences page then commands and columns added by these methods will not be available.
Added new OnScriptConfigChange method which is called whenever the user modifies a script's configuration. A script could respond to this event by calling, for example, Script.InitColumns to update its columns based on the user's settings.
Added Dialog.defid property that allows the default button in a dialog to be changed from the first one.
The OnDoubleClick event can abort further processing when multiple files are double-clicked, by setting the DoubleClickData.cont property to False. Also added the DoubleClickData.call property, which lets a script stop the OnDoubleClick method from being called for subsequent files but allows normal processing to continue.
The DoubleClickData object now has a multiple property to indicate that multiple files were double-clicked.
Scripts launched via a drag and drop event are now able to determine the source tab of the drop if it came from within Opus.
Opus will now recognise scripts with a .txt extension (e.g. .js.txt, .vbs.txt). Also, dropping a (for example) .js.txt file on the scripts Preferences page now strips off the .txt suffix when copying it to the scripts folder. These changes are designed to make it easier to install scripts downloaded from the resource centre.
When a script error occurs and no log is currently open to display it, an error icon will now appear in the status bar that the user can click to open the log pane.
The Script configuration editor dialog, and the multiple-line text editor that can open from that, are now both resizeable.
When Prefs LAYOUT=xyz is run from a script, command.results.newlisters is now populated with any new Listers which were opened. Similarly, Prefs LAYOUT=xyz LAYOUTTHISLISTER=paths will populate command.results.newtabs with any new tabs.
It's now possible for an OnBeforeFolderChange script event to run its own Go command (the NOSCRIPT argument must be used).
If the script config editor was open and the script in question was edited, the names in the script editor could become corrupted.
The Command.SetQualifiers method was ignored.
Fixed crash with Python enumeration of Opus script collections.
Fixed desktop context menu sorting Lister layouts to the end of the list if their names contained non-ascii characters.
Lister Styles no longer start with the Toolbar checkbox set when new. Updating a Style now leaves the checkbox as it is.
Opening an empty tab (Go NEWTAB) did not update the Lister's title bar.
If a Lister style is already using an automatically generated folder tab group, and you update the style, the existing tab group is now overwritten. Previously, a new and unique tab group would be generated, alongside the old one, but this was more clutter than useful.
Fixed using Update on a Lister Style where it would only save the tabs on one side of the Lister if there was only a single tab on the other side, and the style had not previously specified a folder on the side with just one tab.
Fixed double-clicking a Style in Preferences and changing its name resulting in two copies of the Style.
Commands like Set SORTBY=+modified SORTREVERSE=ToggleSmart which mix a generic field name ("modified", which matches any column showing modified dates and/or times) with SORTREVERSE=ToggleSmart, now respect the ToggleSmart request if the column is already present as the sort column.
Fixed opening symbolic links to zip files.


byte.to Bureaucratizing Islam Morocco and the War on Terror - Filme, Spiele, Musik, Bücher und mehr kostenlos downloaden.
Ann Marie Wainscott, "Bureaucratizing Islam: Morocco and the War on Terror"
How have states in the Middle East and North Africa responded to the War on Terror? While much scholarship has focused on terrorism in the region, there is need for critical studies of Middle Eastern states counter-terrorism policies. This book addresses that need by investigating Moroccos unique approach to counter-terrorism: the bureaucratization of religion. Moroccos strategy is unique in the degree to which it relies on reforms that seek to make the countrys religious institutions into tools for rewarding loyalty and discouraging dissent from religious elites. Through these measures they have limited opposition through an enduring form of institutional control, accommodating some of the countrys most virulent critics. This book will be of great use to researchers and scholars of Middle Eastern politics, and it will also appeal to those policymakers interested in security studies and counter-terrorism policies.


I received my BA in Comparative Literature from Ludwig-Maximilians-Universität Munich in 2013 with a minor in political science. Because I wanted to broaden my MA by adding cultural studies I continued with American Studies in Leipzig.
I am passionate about translation and contemporary literature. My research interests so far have focused on the relationship between trauma and narrative and the representations of war in literature. Those foci have elicited my interest in how writers deal with dislocation, with disrupted memoirs and the feeling of (not-)belonging.


How much is the fish, for cod's sake?
re:publica 2017 - Agnes Lisik: How much is the fish, for cod's sake?
"Fishy" technology to stop overfishing: This session will address the problem of overfishing, its causes and possible solutions. In order to maintain healthy marine ecosystems, we need to be informed and keep our governments accountable over their decisions that concern the environment and our future. Technology can help us in the quest to manage fish stocks and marine protected areas in a better and more transparent way.
Have you ever wondered where your seafood comes from? Do you know if this natural renewable resource is sustainable? Are you aware that overfishing is still a big problem in Europe and globally? Have you heard of pirate fishing in the 21st century?
This session will address the problem of overfishing, its causes and possible solutions. Currently, only a minority of European fish stocks can be considered well managed, and although there is serious concern about the state of some fisheries, the potential for recovery is huge and has been quantified by scientists. In order to maintain healthy marine ecosystems, we need to be informed and keep our governments accountable over their decisions that concern the environment and our future.
Technology can help to manage fish stocks and marine protected areas in a better and more transparent way. Satellite systems monitor fishing vessels' activity, and the big data generated can be visualized in a publically accessible way. In addition, mobile applications and websites help fish identification, and blockchain use may increase seafood traceability. Technology also goes hand in hand with research and is indispensable for aiding in the exploration of the oceans. Monitoring can be conducted on the surface by drones and underwater by gliders (AUV), while we glimpse the deep-sea bottom by means of submarine robots (ROV). All these devices can be used to better know and protect our oceans.
Join this session and release the Kraken!


There must be a dozen or more lakes in Michigan named Pickerel Lake, but this one is special. This Pickerel Lake is located on Ramsdell Drive in Cannon Township in central Kent County.
There are not many areas like this any more. Pickeral Lake is an undeveloped lake of eighty acres nestled within two hundred acres of woods, trails and nature located in what is called The Pickerel Lake and Fred Meijer Nature Preserve.
If you like to fish you may find Bluegill, Large Mouth Bass and Sunfish. Or just watch the wildlife and be someplace secluded, check out this West Michigan treasure.


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Q: Unable to Manually Trigger GitHub Action I recently started working on using some GitHub actions on my projects. I am able to setup them up to run automatically but am struggling with having them run manually. I know that you need the have the workflow_dispatch in the on section. I'm not sure if it's not working because I have it automatically run too. Is someone able to tell me what I am doing wrong?
Here is one of my workflow YAML files
name: Create-Doc-Nightly

on:
  push:
    branches: [ "nightly" ]
    paths:
      - 'src/**'
      - 'pom.xml'

  workflow_dispatch:

jobs:
  doc:
    name: Create Doc
    runs-on: ubuntu-latest

    steps:
      - uses: actions/checkout@v3
        name: Step 1 - Checkout Nightly Branch
        with:
          persist-credentials: false
          fetch-depth: 0
      - name: Step 2 - Setup JDK 17
        uses: actions/setup-java@v3.4.1
        with:
          java-version: 17
          distribution: 'temurin'
      - name: Step 3 - Remove Doc
        run: |
          git remote set-url origin https://jnstockley:${{ secrets.TOKEN }}@github.com/jnstockley/BTTN.git
          git config user.email "jack@jstockley.com"
          git config --local user.name "Jack Stockley"
          git rm -r docs
          git commit -m "Removed Docs"
          git push origin nightly
      - name: Step 4 - Create Doc
        run: mvn dokka:dokka -f pom.xml
      - name: Step 5 - Move Docs
        run: |
           rm -rf docs
           mkdir -p docs
           mv target/dokka/* docs
      - name: Step 6 - Publish docs
        run: |
          git remote set-url origin https://jnstockley:${{ secrets.TOKEN }}@github.com/jnstockley/BTTN.git
          git config user.email "jack@jstockley.com"
          git config --local user.name "Jack Stockley"
          git add -f docs
          git commit -m "Updated Docs"
          git push origin nightly

Link to GitHub repo, nightly branch: https://github.com/jnstockley/BTTN/tree/nightly

A: The workflow must be on your default branch in order to use workflow_dispatch.
I believe in your case it's only on the branch nightly while it should also be on main.

A: To manually trigger a workflow, use the workflow_dispatch event. You can manually trigger a workflow run using the GitHub API, GitHub CLI, or GitHub browser interface. For more information, see Manually running a workflow
on: workflow_dispatch

Providing inputs
You can configure custom-defined input properties, default input values, and required inputs for the event directly in your workflow. When you trigger the event, you can provide the ref and any inputs. When the workflow runs, you can access the input values in the inputs context. For more information, see Contexts
This example defines inputs called logLevel, tags, and environment. You pass values for these inputs to the workflow when you run it. This workflow then prints the values to the log, using the inputs.logLevel, inputs.tags, and inputs.environment context properties.
yaml
on:
  workflow_dispatch:
    inputs:
      logLevel:
        description: 'Log level'
        required: true
        default: 'warning'
        type: choice
        options:
        - info
        - warning
        - debug
      tags:
        description: 'Test scenario tags'
        required: false
        type: boolean
      environment:
        description: 'Environment to run tests against'
        type: environment
        required: true

jobs:
  log-the-inputs:
    runs-on: ubuntu-latest
    steps:
      - run: |
          echo "Log level: $LEVEL"
          echo "Tags: $TAGS"
          echo "Environment: $ENVIRONMENT"
        env:
          LEVEL: ${{ inputs.logLevel }}
          TAGS: ${{ inputs.tags }}
          ENVIRONMENT: ${{ inputs.environment }}

If you run this workflow from a browser you must enter values for the required inputs manually before the workflow will run.

You might like the following documentation links
workflow_dispatch
github docs - events-that-trigger-workflows

redpajama_set_name: RedPajamaStackExchange


The International Exhibition of Sherlock Holmes is just one example of the global reach and world class expertise that The Conan Doyle Estate can provide. Our international network of specialists have been closely involved in consultancy and detailing for the making of the exhibition and it was with pride that it was opened in 2013 at the Oregon Museum of Science & Industry by Richard Doyle,senior director of The Conan Doyle Estate and Sir Arthur's grandnephew. The exhibition was hosted at the Powerhouse Museum in Sydney Australia in 2017 and the end of April 2018 has seen the acclaimed Exhibition putting down roots in Houston, USA.
Working with The Conan Doyle Estate, The International Exhibition of Sherlock Holmes has been able to bring Sherlock to life. With the help of our experts the exhibition allows the visitor to step into Conan Doyle's Victorian London and work side-by-side with his legendary detective.
The clues that solve a crime inside the new "The International Exhibition of Sherlock Holmes" at the Houston Museum of Natural Science are anything but elementary.
Even impassioned Sherlockians may find themselves stumped by the mystery of a gruesome murder, which makes reference to the original fictional stories created by physician and author Sir Arthur Conan Doyle but is a fresh case for the exhibition.
At the entrance, glass cases display memorabilia and props from some of pop culture's takes on the 19th-century detective series, including a curved pipe Robert Downey Jr. smoked when he portrayed Sherlock Holmes in the Warner Bros. film franchise and a pocket watch used by Jude Law, who acted as his sidekick, Dr. Watson.
There's also the pink suitcase that was a crucial clue in the first episode of BBC Television's "Sherlock" and a trench coat actor Benedict Cumberbatch wore in the series. From CBS' television show "Elementary," fans will recognise a costume of Lucy Liu's, who was cast as a female Watson.
Those three modern-day interpretations created a new generation of Sherlock Holmes fans, said Geoffrey Curley, and that following is one reason his firm curated the exhibition, which has also made stops in Seattle and Sydney.
Curley is a former theatrical designer whose Minneapolis-based company was granted access to Doyle's estate to research the show — super fans will geek out over Doyle's original manuscripts, personal correspondence and photos on view.
But it's not all observation. Before they enter a re-creation of 1890s Victorian London, where the crime has taken place, visitors get a detective's notebook to record their findings.
They'll learn crucial details in a replica of the sitting room of 221B Baker Street in London, where Holmes and Watson regularly discussed their crimes.
With dispatches and messages from Holmes himself, guests walk through Steampunk-era London to uncover clues in a train station, penny arcade and slaughterhouse. They'll use hands-on, interactive tools for evidence analysis, including blood-spatter tests, re-creation of broken objects, footprint analysis and tracing a bullet's trajectory.
Brian Clark acknowledges he didn't solve every aspect of the mystery.
Clark heads up the John Openshaw Society, the Houston chapter of an international group of Sherlockians. He took a private tour to prepare to become a volunteer docent for the exhibition. He called the historical re-creation of London "beautiful" and said it will captivate even "hard-core Sherlockians" like himself.
Even though he was a careful observer of all these details, Clark said a twist at the end of the crime-solving plot threw him off — visitors learn if they were successful when words from a newspaper clipping are revealed.
He's looking forward to returning with his 10-year-old son and 12-year-old daughter, who are also Sherlock fans, to see how they fare.


Entropy is the most important functional in probability theory, Euler characteristic is the most important functional in topology. Similarly as the twins Apollo and Artemis displayed above they are closely related. Introduction This blog mentions some intriguing analogies between entropy and combinatorial notions. One can push the analogy in an other direction and compare random variables with simplicial complexes, Shannon … ….


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Q: PSexec is not connecting to machine using supplied username and password I'm trying to connect from a Win 7 machine to a Win2k8R2 machine using the command:
psexec \MACHINE_NAME_HERE -u MACHINE_NAME_HERE\Administrator -p PASSWORD_HERE notepad
When I try this I get the error message:
Couldn't access MACHINE_NAME_HERE Access is denied.
I tried omitting the -p and manually putting in the password, but the same issue persists.
When I check the machine I'm trying to access I see that in the security event viewer that the Administrator login was successful, so I know the username password is correct, and the machine is accepting my login, but I also noticed that it tries to also log in with my domain account, which isn't in the admin list on the remote machine. If I add my domain account to the remote machine then everything works, but in this case it defeats the purpose, sometimes I need to be able to run commands on the machine when it's no on the domain.
The machine is set to use the simple model for Sharing and Security as well.

A: This is because psexec still tries to access the ADMIN$ share with your local credentials, before executing your command as another user. According to this thread, you can cache credentials before executing psexec:
cmdkey.exe /add:MACHINE_NAME_HERE /user:MACHINE_NAME_HERE\Administrator /pass:PASSWORD_HERE
psexec.exe \\MACHINE_NAME_HERE -i notepad
cmdkey.exe /delete:MACHINE_NAME_HERE

I added -i, since you specified running notepad here. Without the -i, notepad will sit idle with no interacton. Also, please remembet that, when you use -u, psexec will send the password in plaintext.


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Playing Games in Your Relationships
New Year's Resolutions – December 2015
COMMITMENT: Your Word is Your Bond – January 2016
Get mentors around you who teach you how to fish! Progress is a need like water. If I were to define happiness, it's progress, and you can only make consistent progress by having objective feedback to see things as they actually are. With that being said, you still want someone in your life who is able to be a catalyst and who .
Sometimes our egos get the best of us though and we think we are the ones who did everything. Take a look at Tiger Woods. His best stretch came with the two coaches: Hank Haney and Butch Harmon.
Here is a by-the-numbers look at Woods' swing coaches over his pro career:
BUTCH HARMON (1996-2003)
Major wins: 8 majors won in 24 starts
Majors missed: 0
PGA Tour wins: 34 titles in 127 official starts from turning pro through 2002 (26.8 percent win percentage)
HANK HANEY (2004-2010)
Majors missed: 2 (2008 British Open, 2008 PGA)
PGA Tour wins: 31 titles in 93 starts (33.3 percent)
SEAN FOLEY (2010-2014)
Major wins: 0 in 13 starts, dating to 2010 PGA
Majors missed: 4 (2011 U.S. Open, 2011 British Open, 2014 Masters, 2014 U.S. Open)
PGA Tour wins: 8 titles in 55 starts (14.5 percent)
Information provided by the Golf Channel Editorial Research Unit
When reporters asked Hank Haney if he would coach Tiger again, he said "it's time for someone else to get second guessed".
Sometimes we think we outperform the coach or our parents. Then, when the coach isn't in our life to help us see things as they are, then we go off the rails. Cus D'Amato was Mike Tyson's rock in a way. A good coach will help you when he's not even there by planting roots so deep that you know exactly what to do.
If your coach was a catalyst for you to make more money and the coach costs a fraction of that new wealth you obtained because of the new distinctions you now have, then wouldn't it make sense that the gap would continue to grow? It's not a drain on your finances, it's an investment that has HUGE dividends in both the short term and the long term.


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ODB produces document calling citizens' protests to helipad construction "heinous and illegal"
November 18, 2016 Ryukyu Shimpo
The Okinawa Defense Bureau (ODB) has made an external informational document publicizing photographs of the citizens opposed to the construction of helipads who have entered land provided to the U.S. military. The distributed document judges the citizens' behavior as "heinously illegal acts." A lawyer commented that would be possible for the ODB to indicate damages to the police and request an investigation into the matter. However, the lawyer went on to say that the ODB has gone too far by widely supplying such information at a time when there is general inquiry concerning these citizens' actions.
It seems the ODB's appreciation of human rights has been thrown into doubt, due to its description of certain individuals' behavior as "heinous and illegal," and its spreading of content capable of fostering prejudice against citizens' campaigns opposing the overlarge share of the base burden on Okinawa.
This document explains that local citizens are the ones causing the destruction of precious vegetation and having devastating effects on the environment. The ODB stated that its reasons for making these materials were to explain the present condition of these matters, and to respond to many inquiries it receives concerning the activities of those persons opposing construction related to relocation. On the other hand, an environmental expert stated that the burden on the environment is being caused in the same places that ODB personnel, riot police, and security personnel are dispatched. This expert also explained that the construction of helipads is the actual cause of large-scale deforestation and destruction of nature.
The document was no less than nine pages long and included an attachment called "Illegal Trespassers," which presented many photographs of citizens who engage in protest activities. Another item titled "Examples of Illegal and Heinously Disruptive Behavior" held nine photographs of Chairman Hiroji Yamashiro of the Okinawa Peace Movement Center entering land provided to the U.S. military.
On the page titled "Interferences to Environmental Conservation Measures," the ODB explained that windbreak fencing and transplanted Ryukyu-chiku (a variety of bamboo found in Okinawa) will be put around the helipads as an environmental conservation plan to protect against direct wind and sunlight, as well as soil runoff. Without addressing the large-scale deforestation related to construction, the document indicates that the citizens who protest in the Northern Training Area are trampling down the environs surrounding the transplanted Ryukyu-chiku. When asked by the media, the ODB responded that it is also using the document as needed when external explanation is necessary.
Previous Article:Efforts to preserve eight endangered languages discussed at summit in Yoron
Next Article:Okinawans overjoyed at successful heart transplantation of one-year-old Himari
Ramsar secretariat demands environment ministry preserve sea of Henoko
MOD insists corals ineligible for transplantation only one month prior to Henoko soil deposits
Stand off between citizens and police officers at gate of Camp Schwab in Nago
Miyakojima residents protest construction of GDSF ammunition depot
Henoko construction should be halted to protect scarce coral from further destruction


1953 Exshaw Grande Champagne Vintage Cognac 1953 (70cl). This delightful Cognac is 70% Proof and makes an ideal gift for someone born in 1953. This vintage Cognac is shipped in a silk lined wooden presentation box with four wine accessories (corkscrew, wine pourer, drip stopper & wine thermometer) [worth £17.99] at no extra cost and now includes a free gift card with your personal message and tasting and storage notes..
1953 Exshaw Grande Champagne Vintage Cognac 1953 (70cl). This delightful Cognac is 70% Proof and makes an ideal gift for someone born in 1953. This vintage Cognac is shipped in a silk lined wooden presentation box with four wine accessories (corkscrew, wine pourer, drip stopper & wine thermometer) [worth £17.99] at no extra cost and now includes a free gift card with your personal message and tasting and storage notes.. . . . .