text:
# Firecast

Esse repositório foi criado com o intuito de armazenar o código de projetos relacionados ao Firecast, o maior programa para jogos online de mesa do Brasil.

**Subpastas:**
- **Arts:** Imagens de uso geral para aplicações do firecast.
- **Macros:** Códigos de macros de chat.
- **Plugins:** Plugins de ficha, chat, janelas acopláveis e outros, feitos com o SDK3.
- **SDK3:** Informação e detalhes sobre o SDK3.

# Ferramentas

- [**SDK**](https://firecast.app/para-programadores/): Necessário instalar para poder compilar os projetos de plugin. No link há duas versões disponíveis, uma compatível com o RRPG antigo e mobile, o [RDK3.4b](http://www.firecast.com.br/RDK3.4b.exe), e o mais atual, [RDK 3.6c](https://firecast.app/downloads/RDK3.6c.exe).
- [**Sublime**](https://www.sublimetext.com/3): Um editor de texto, esse ou qualquer outro como [Notepad++](https://notepad-plus-plus.org/), que permita editar os arquivos do projeto.
- [**Souce Tree**](https://www.sourcetreeapp.com/): Uma ferramenta para uso do git com interface, essa ou qualquer outra, para facilitar o acesso ao Github.

# Como colaborar e adicionar seus projetos

1. Faça uma conta do [GitHub](https://github.com/join?source=login).
2. Acesse o repositório do [firecast](https://github.com/rrpgfirecast/firecast) e clique em Fork.
3. Em seu fork clique em `"Clone or Download"` e copie o URL, vai ser algo como `https://github.com/{Seu_username_do_github}/firecast.git`.
4. Abra o Source Tree, aperte Ctrl+N, cole o url copiado e clique em Clone. Espere o Source Tree baixar o repositório.
5. Quando o Source Tree terminar de baixar o repositório adicione suas alterações, como Macros, Plugins, e etc. Teste suas alterações, tenha certeza que tudo está funcionando e salve.
6. No Source Tree vá em File `Status > Working copy`. Verifique as alterações que você fez no projeto. Tudo que tiver sido adicionado vai estar marcado em verde. Tudo que foi removido vai estar em vermelho. Selecione os arquivos que desejar mandar para o repositório e clique em `Stage`.
7. Escreva uma mensagem de _commit_ explicando o que você alterou e por quê. Clique em `Commit` e depois `Push`.
8. Vá até o topo da janela do _Source Tree_ em `Repository > Create Pull Request` e clique em `Create Pull Request on Web`.
9. Na janela web que apareceu descreva o que está alterando no seu _pull request_ e clique em `Create pull request`.
10. Aguarde os moderadores do repositório avaliarem seu _pull request_ e aprovarem-no, ou caso o reprovem dizer quais são os problemas nele para que você possa corrigí-los.

# Contato

Se é novo por aqui e deseja conhecer o programa, visite o site do [**RRPG Firecast**](https://firecast.app/pt_br/).
Para ajuda com o funcionamento dos plugins, veja a documentação do [**RRPG SDK 3**](https://firecast.app/sdk3/RRPG%20SDK%203.html?Introducao.html).
Para dúvidas e ajuda, visite a [**Wiki**](http://wiki.rrpg.com.br) do RPPG Firecast, o nosso grupo do [**Facebook**](https://www.facebook.com/groups/460782814000421/), ou o nosso servidor no [**Discord**](https://discord.gg/dqbJ66Gh95).

redpajama_set_name: RedPajamaGithub


text:
How Polaris Family Dental is Responding to COVID-19: READ HERE
Now Accepting New Patients! 614-848-3384 SCHEDULE NOW
M-W: 9-6, Th: 9-6
F: 9-2, Sat: 9-3
8949 Antares Avenue
About Polaris Family Dental
Meet Dr. Winner
Sterilization and Infection Control
Sonicare System
Dr Alexis Wang
Home / About Us / Dr Alexis Wang
Dr. Alexis Wang is proud to be a part of the Polaris Family Dental team since 2018. Dr. Wang graduated from The Ohio State University in 2013 with a Bachelor of Science Degree. She then continued her studies at The Ohio State University College of Dentistry where she graduated in 2017. Dr. Wang comes from a family with a passion for all Scarlet and Gray. She spends her free time at the gym, listening to podcasts, and trying new recipes. She loves to travel and spend time with her husband, friends and her dogs, Theo & Sally.
Our practice was established in 1997 and was built from the ground up. Since then there have been a lot of changes to the surrounding area, but one thing remains the same; we provide personal and patient-focused care to every individual in our practice.
Columbus OH 43240 M-W: 9-6, Th: 9-6
Polaris Family Dental © 2023 | Privacy Policy | Terms and Conditions | HIPPAA Policy |Website by Whiteboard Marketing

redpajama_set_name: RedPajamaCommonCrawl


This week The Alchemist learns that boron most certainly is not boring, it now being a record-breaker in a chemical flat land. For medical science, we now have nanoparticles that shimmer and change color as a butterfly wing and in the world of bioanalysis, an easy to stick together microfluidic device for testing toxicity. Thermodynamics, we learn, could be crucial to the successful use of smart materials and memory alloys in making structures better equipped to resist earthquakes. In organometallic supramolecular chemistry a team in Germany has constructed a bronze Matryoshka from copper and tin cages. Finally, this week's award goes to British chemist Peter Bruce who is working on lithium-air batteries that could last up to thirty times as long as current technology without bulking up or gaining weight.
Alexander Boldyrev and colleagues at Utah State University, Logan, working with Lai-Sheng Wang and colleagues at Brown University in Providence, Rhode Island have synthesized a planar boron compound that has the highest coordination number of any flat molecule, squeezing ten spoke-like bonds to boron atoms into a wheel. This astounding feat of engineering not only breaks previous records for such compounds but offers new insights into bonding, coordination and the development of boron chemistry. Theoretical chemist Pekka Pyykkö of the University of Helsinki, Finland, is impressed. "At a deeper, quantum mechanical level, I find the electronic structure entirely logical and a pretty example on the eighteen-electron (18e) rule," he says.
Metallic nanoparticles that have some of the photonic characteristics of a butterfly's wing could be used as simple color-change components of a new approach to medical diagnostic tests, say researchers from Attophotonics Biosciences GmbH in Austria and at the University of Applied Sciences, in Wiener Neustadt. The team has demonstrated proof of principle for the detection of a model compound, interleukin-6 an important biomarker for acute sepsis. The sensitivity is high and the nanoparticles can readily be incorporated into a lab-on-a-chip device, the team reports.
Javier Atencia and colleagues at the US National Institute of Standards & Technology have taken a few sheets of plastic, a glass slide and some double-sided sticky tape to quickly and easily construct a lab-on-a-chip device for carrying out cell assays. Their "diffusion-based gradient generator" can be used to rapidly assess how changing concentrations of specific chemicals affect living cells. The microfluidic device could be used to carry out toxicity testing inexpensively on a range of chemicals.
Researchers at the Georgia Institute of Technology are investigating whether or not shape-memory alloys might be useful construction materials for seismic-resistant structures. Reginald DesRoches and colleagues have developed a computer model to study how these materials respond thermodynamically and mechanically to loading from strong motion. Their calculations could help determine the viability of using smart alloys made from combinations of copper-zinc-aluminum-nickel, copper-aluminum-nickel or nickel-titanium in cables, bars, plates and helical springs for engineering applications. For standard materials, mechanics is usually sufficient for engineers, but with smart materials thermodynamics becomes an important component of the equations.
As with the wooden Russian effigy of the maternal life cycle, the Matryoshka doll within a doll within a doll, a capsule of twelve copper atoms encases a single tin atom. But, the copper capsule is itself enveloped by 20 additional tin atoms. According to Thomas Faessler of the Technische Universitaet Muenchen, Germany, creator of the "bronze" Matryoshka, the substance itself is a rather unassuming fine, charcoal-gray powder. However, its nondescript appearance in the bulk belies the potential of these materials, with their large surface area to volume ratio, as catalysts for hydrogen transfer and other applications.
Chemist Peter Bruce of the University of St Andrew, UK, is to be honored with the inaugural UK Science Award by AkzoNobel for his fundamental research into rechargeable lithium batteries. Bruce's work might one day transform the driving range of electric vehicles and make renewable energy more viable. Lithium-air batteries have the potential to store up to 30 times the power of current lithium batteries. This could extend the distance electric vehicles might travel between charges considerably and nudge us towards greater adoption of carbon-free transport to cut pollution in cities and facilitate greater use of electricity generation based on renewable energy sources.


text:
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text:
Biden's Approval Ratings Crash Again in the US
Posted on June 15, 2022 June 15, 2022 by Admin
US President Joe Biden's public approval rating has been falling for the third week in a row, dropping to 39%. This is evidenced by the results of a public opinion poll conducted by Reuters/Ipsos.
It is specified that this figure is approaching the lowest value for the entire time of his tenure as president.
According to the poll, more than half of US residents do not approve of Biden's work. This opinion was shared by 56% of respondents.
According to the authors of the study, Biden's approval rating has remained below 50% since August 2021, which means that the Democratic Party in the November 8, 2022 midterm elections may lose control of at least one house of Congress.
Dissatisfaction with Biden within his own party is one of the reasons for his decline in popularity. Among the supporters of the Democratic Party, 74% are supporters of the incumbent president, while in August 2021 they were 85%. At the same time, only 11% of Republicans support Biden.
It is noted that the Reuters / Ipsos survey is conducted online in English throughout the United States. The latter was attended by 1,005 adults, of whom 443 supported Democrats and 364 Republicans. The margin of error is four percentage points.
The Politico newspaper reported on June 5 that the cascade of problems facing the United States has caused a moral crisis in the White House, and Biden and his aides are frustrated by their own inability to turn the tide. Among the challenges facing the United States are rising fuel prices, global inflation, a Russian special operation in Ukraine, a potentially resurgent coronavirus pandemic, and the inability of Congress to tighten gun control in the country, the publication noted.
On June 2, TV presenter Greg Gutfeld on Fox News aired that Biden would not finish until the end of his presidential term. This conclusion was made against the backdrop of rising fuel prices and other socio-economic problems that led to a sharp drop in the rating of the American leader.
The political scientist appreciated the instruction of London to Kyiv to deal with the release of mercenaries in the DPR
Skvortsova announced the absence of a shortage of imported drugs in the Russian Federation


Antiquites Francaises is a well established family business selling decorative French antiques since 1997 in Jesmond, Newcastle. They buy all of their furniture from France each year, only selecting the most beautiful pieces.
Alpha helped Antiquites Francaises to develop their website, making it modern, stylish and fully responsive. The website also has an online shop which customers can use to register interest in items on the site. This feature means Antiquites Francaises can manage stock using the website's straightforward and accessible content management system.
The client had very specific needs with this site and we met them happily. As well, our team have hosted several training sessions since completion to make sure the client is fully able to work the website.


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I believe bodywork is a vital component of self-care, reminding us of our innate ability to restore balance and harmony, beginning from within. Bodywork is a wonderful tool to not only unwind patterns of physical pain and tension, but to help us connect to our own energy and align with our divine essence. I believe in the power of receptivity and "doing nothing," and know that deep relaxation is necessary for optimal functioning. It is also of utmost importance to me that we are grounded and embodied, so we can be fully present in our lives.


text:
The Coastal Post - March, 1998
The Flood, The BJ That Got Away, Slick Willy And Slymack, The Worst Week Of Their Life
By Stephen Simac
The first week of February roared in like a tiger. The Chinese Year of the Tiger. Clawing, man-eating, creek-flooding, mud-sliding El Ni�o ripping away the curtain in front of the man.
Of course, the president had it worse than me, and we both had it better than people who died, or lost their homes and belongings. Really, neither of us have anything to whine about, unless it's to joke about it. The blessings of bad times come from laughing about them later. Much later, in our cases.
First the president, whose sex life enthralls the media but barely concerns the populace. As long as the economy is booming, we don't care about Bill's boners, maybe it's even up because he was.
And really, why should we care? Other than voyeuristic titillation, what business is it of ours? Right-wing media pundits despise our lack of morality because we don't demand the impeachment of a popularly elected president, based on gossip.
Even if it is true, it just shows how far you can go with big lips and presidential knee pads. The media has no morality, best they don't prescribe it for the rest of us.
I write for the Coastal Post because it's a brave little paper, yadayadaya, but mainly because nobody else will print me. I appreciate the paper, the good thing about it, it will piss you off now and then. Last month it got me.
Fear and loathing of editor
When I picked up the issue, I could see one of my stories got fuktup by the typesetter. Now that pisses me off. Typesetting like cleaning is a thankless and unrewarding job. That's why it's hard to keep 'em, but you gotta have 'em.
They don't get credit when they do well, but in this case, she could have signed on as co-author. She rewrote the first several paragraphs to excise some naughty words.
Nothing really obscene, oral sex isn't even real sex by executive order. Obscene is babies being bombed, kids killing kids, the rape of the planet, and the uncaring wealth and crushing poverty here.
Mainly she made bland what I wrote as spicy. Pulled out the chilies and put in dill. The real thing is on the internet.
The British are here
Anyway, I've had worse editors, like the English bloke on my junior college paper, The Phoenix. Couldn't write his way into a paper bag, but wanted to be a journalist. That meant he got made editor-typesetter. He was jealous of me and the column they let me write, a glamorous job I demanded. We didn't like each other, but he had the power of typesetting my words. After he mangled one column by sprinkling paragraphs around in no apparent order, our professor pulled him off the hot lead. I would have dipped him in it.
Unfortunately he ended up as an editor at The Alligator, the student newspaper of the University of Florida which I also wrote on. He did one rewrite I still cringe over. Nobody ever suspects the editor, when the words are fuktup.
His worse blow came when for some inexplicable reason, The Phoenix professor sent him a state award I had won to give to me. When I found this out on a visit home, I went back to Gainesville and confronted the bloke. He claimed to have lost it.
He could tell I was ready to become a violent American, and he scuttled away. I should have called immigration. Now he's an editor on the National Enquirer. Where's the justice?
We can understand that our personality can repel as well as attract. Accept enemies along with friends as the inevitable consequence of becoming ourselves. Honestly examine our soul and knows its flaws, those gleaming cracks in smoky crystal which make us uniquely human.
We seldom laugh until later, but it's like a ray of sunshine in a gray week of floods when we do.
It was weird waking up hearing bottles clinking together, a sound you shouldn't hear unless you're sleeping on a boat. I woke up on a futon raft with a foot of gleaming black water around me. I pulled the covers over my head, must be a bad dream.
Then the electrical outlets started frying. The cats were yowling in the other room. I could hear waves crashing into houses a block away on the beach.
The ocean was rolling down the street and adding to the swollen creek which now was running through the apartment. It's not always that restful being close to nature.
Women are from Venus
I should never have asked her how she really felt about me. It always makes a man cringe, when she tells you she just wants to be friends. Mainly because what they mean is, you'd be useful to lift heavy objects, or open jars. Hearing her say that made me feel like the mud the flood left behind.
Why did I even ask? But I should never have asked why not just some recreational sex. Better with a friend, eh.
She had her reasons, of course. She said she only had casual sex with men she could see having a long-term relationship with, and she couldn't imagine bringing me home to meet her family.
I pointed out that you couldn't really call that casual sex. Her family lives in Cleveland, anyways. I had a stopover there once, I didn't even leave the airport.
I wasn't interested in her whole clan, just her. Besides if she thinks her family is special, I wouldn't even dream of letting her meet my family. Any girlfriend who ever met my family suddenly just wanted to be friends. Needless to say I didn't bring any home.
Which is why my family thought I was gay. That was still better than trying to convince the women that I was adopted, or left by the elves. The Keeblers.
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\textup{2000} Mathematics Subject Classification}

\begin{document}
\title{Stochastic heat equations driven by L\'evy processes}

\author[T. Chang]{TongKeun Chang}
\address{T. Chang: Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea}
\email{chang7357@yonsei.ac.kr}

\author[M. Yang]{Minsuk Yang}
\address{M. Yang: Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea}
\email{kusnim@gmail.com}

\thanks{The first author was supported  by the National Research Foundation of
Korea(NRF-2010-0016699)}

\begin{abstract}
We study stochastic heat equations driven by a class of L\'evy
processes:
\begin{equation*}
du = \De u dt + g dX_t \quad \mbox{in} \quad     \bR^d_T, \qquad u(0,x)= 0
\quad \mbox{in} \quad  x \in \bR^d.
\end{equation*}
We prove the corresponding estimate
\[\norm{u}_{\bH_p^k(\RT)} \le c(p,T) \norm{g}_{\bB_p^{k-\frac2p}(\RT)}\]
for $2\le p<\infty$ and $k \in \bR$.

\vspace{5mm}

\noindent
2000  {\em Mathematics Subject Classification.}  Primary; 60H15, Secondary; 35R60. \\
\\
\noindent
{\it Keywords and phrases:
Stochastic heat equation, L\'evy process, Sobolev space,  Besov space.}
\end{abstract}

\maketitle

\mysection{Introduction}                          \label{Section1}
In this paper, we study the following stochastic heat equation
\begin{equation}\label{spde1}
\begin{cases}
du  = (\De u  + f )dt + g  dX_t & \RT \vspace{2mm}\\
u|_{t=0} = u_0 & \Rd,
\end{cases}
\end{equation}
where $\bR^d_T : = (0,T) \times \Rd$ for $0 < T < \infty$.
We assume that $X_t$ is a one-dimensional L\'evy process satisfying some conditions on a probability space, which is explained in Section \ref{Section2}.
We allow $f$ and $g$ to be random.
To solve the problem \eqref{spde1}, we may consider the following two problems
\begin{equation}\label{spde2}
\begin{cases}
du  = (\De u  + f )dt & \RT \vspace{2mm}\\
u|_{t=0} = u_0 & \Rd
\end{cases}
\end{equation}
and
\begin{equation}\label{spde3}
\begin{cases}
du  = \De u  dt + g  dX_t &  \bR^d_T \vspace{2mm}\\
u|_{t=0} = 0 & \Rd.
\end{cases}
\end{equation}
Since the problem (\ref{spde2}) has been well studied, we shall focus on the problem (\ref{spde3}).
For $g \in L^p(0,T; {\mathcal S}^{'} (\Rd) )$ (where ${\mathcal S}^{'} (\Rd)$
 is the space of tempered distributions), the solution of (\ref{spde3})
can be represented by
\begin{align}\label{solution}
u(t,x) = \int_0^t T_{t-s} g(s,x) dX_s.
\end{align}
Here, $T_{t-s} g(s,x) =\Gamma(t-s,\cdot) *g(s,\cdot)(x) $, where $\Gamma(t,x) = (2\pi t)^{-\f{d}{2}} e^{-\frac{|x|^2}{4t}}$ and
$*$ is the standard convolution in $\Rd$.

For the Brownian motion case, a theory was developed by N.V. Krylov \cite{Kr94}.
Since the Burkholder-Davis-Gundy inequality implies
\[\bE \int_0^T\norm{ \nabla u(s, \cdot)}_{L^p(\bR^d)}^p ds
\leq c(p) \bE \intT \intRd \Big(\int_0^t
|\nabla T_{t-s}g(s,x)|^2 ds\Big)^{p/2} dxdt,\]
he showed that for $2\le p<\infty$ there is a positive constant $c(p)$ independent of $T$ such that
\begin{align}\label{Krylov}
\bE \intT \intRd \Big(\int_0^t  |\nabla T_{t-s}g(s,x)|^2 ds\Big)^{p/2} dxdt
\leq c(p) \bE  \int_0^T\|g(s,\cdot)\|^p_{L^p(\bR^d)}ds.
\end{align}
He proved this inequality by interpolating $L^2$ estimates via
Plancherel's theorem and sophisticated BMO estimates. Using the
properties of Sobolev spaces, it was generalized for $k\in \bR$
\begin{align}\label{0521-1}
\bE \int_0^T \| u(s, \cdot)\|^p_{ H^k_p(\bR^d)} ds
\leq c(p,T) \bE \int_0^T\| g(s, \cdot)\|^p_{H_p^{k-1}(\bR^d)} ds.
\end{align}
Here, the function space $H^{k}_p(\bR^d)$ is the usual Sobolev space
(see Section \ref{Section2}).

For the general L\'evy process case, a few results are known for these types of Sobolev estimates.
In this case, instead of the Burkholder-Davis-Gundy inequality Kunita's inequality is applicable and it produces
\begin{align}\label{0521-2}
\bE \int_0^T \norm{\nabla u(s, \cdot)}_{L^p(\bR^d)}^p ds \leq
& c\bE \intT \intRd \Big(\int_0^t |\nabla T_{t-s}g(s,x)|^2 ds\Big)^{p/2} dxdt \\
\notag&\quad+ c\bE \intT \intRd \int_0^t |\nabla T_{t-s}g(s,x)|^p dsdxdt.
\end{align}
The first term on the right-hand side is the same as in \eqref{Krylov},
but the second term on the right-hand side is new.
Recently, Z. Chen and K. Kim \cite{CK}  proved that for
$2 \le p<\infty$ and $\epsilon >0$, there is a constant
$c(\epsilon,p,T) > 0$ such that
\begin{equation}\label{ep}
 \bE \intT \intRd \int_0^t  |\nabla T_{t-s}g(s,x)|^p dsdxdt \leq
c(\epsilon,p,T)  \bE \intT \| g(s,\cdot)\|^p_{  H_p^{1 -\frac2p +\ep}(\bR^d)} ds
\end{equation}
under some assumptions on L\'evy measure.

Now we state our main results.

\begin{proposition}\label{prop}
Let $0<T<\infty$ and $1<p<\infty$.
There are positive constants $c_1(p,T)$ and $c_2(p)$ such that
\begin{equation}\label{no ep1}
\bE \intT \intRd \int_0^t  |T_{t-s}g(s,x)|^p dsdxdt \leq c_1(p,T)
\bE \intT \| g(s,\cdot)\|^p_{  B_p^{ -\frac2p}(\bR^d)} ds
\end{equation}
and
\begin{equation}\label{no ep2}
\bE \intT \intRd \int_0^t |T_{t-s}g(s,x)|^p dsdxdt \leq
c_2(p)  \bE \intT \norm{g(s,\cdot)}^p_{\dot B_p^{ -\frac2p}(\bR^d)} ds.
\end{equation}
\end{proposition}

To prove Proposition \ref{prop}, we shall use the Littlewood-Paley
theory and then prove variants of Hardy's inequality. Using
\eqref{0521-1}, \eqref{0521-2}, Proposition \ref{prop} and the
mapping properties of the pseudo-differential operators
$(I-\De)^{s/2}$ and $(-\De)^{s/2}$ (see (1) and (2) of  Remark
\ref{rem}) , we can obtain our main theorem.

\begin{theorem}\label{thm}
Let $0< T < \infty$ and $2\le p<\infty$ .
If $\beta_2<\infty$ and $\beta_p < \infty$, then
there are positive constants $c_1(p,T)$ and $c_2(p)$  such that
\begin{align*}
\norm{u}_{\bH_p^k(\RT)} &\le  c_1(p,T) \norm{g}_{\bB_p^{k-\frac2p}(\RT)}, \\
\norm{u}_{\dot \bH_p^k(\RT)} &\le  c_2(p) \norm{g}_{\dot \bB_p^{k-\frac2p}(\RT)},
\end{align*}
where $\beta_p$ is defined in \eqref{beta} and stochastic Banach spaces
$\bH_p^k(\RT), \,\, \bB_p^{k }(\RT), \,\, \dot \bH_p^k(\RT)$ and $ \dot \bB_p^{k }(\RT)$
are defined in \eqref{s-f spaces}.
\end{theorem}

A direct consequence of Theorem \ref{thm} is the following corollary
which follows from the fact that ${\mathbb H}^k_p(\bR^d)$ is
continuously embedded in ${\mathbb B}^k_p (\bR^d)$ for $k\in {\bR}$
and $2 \leq p < \infty$ and the property of real interpolation; (see
\cite{BL} Theorem 6.4.4 and Theorem 6.3.1).

\begin{corollary}\label{coro}
For $ 0< T < \infty$ and $2\le p<\infty$
\begin{align*}
\norm{u}_{\bH_p^k(\RT)} &\le c(p,T) \norm{g}_{\bH_p^{k-\frac2p}(\RT)}, \\
\norm{u}_{\bB_p^k(\RT)} &\le  c(p,T) \norm{g}_{\bB_p^{k-\frac2p}(\RT)}, \\
\norm{u}_{\dot \bH_p^k(\RT)} &\le  c(p) \norm{g}_{\dot \bH_p^{k-\frac2p}(\RT)}\\
\norm{u}_{\dot \bB_p^k(\RT)} &\le  c(p) \norm{g}_{\dot
\bB_p^{k-\frac2p}(\RT)}.
\end{align*}
\end{corollary}

The organization of the paper is as follows.
In Section 2, we introduce precise definitions of function spaces
and conditions concerning L\'evy processes.
In Section 3, we prepare basic lemmas for the heat kernel.
In Section 4, we reduce Proposition \ref{prop} to Lemma \ref{lemma3}.
In Section 5, we prove our main Lemma \ref{lemma3}.
In Section 6, we prove Theorem \ref{thm}.
In Section 7, we apply our method to SPDE with fractional Laplace operator.
\mysection{Preliminaries}                           \label{Section2}
\subsection{Sobolev and Besov spaces}
Let $ \cS:=\cS(\Rd)$ denote the class of Schwartz functions on $\Rd$.
The space $\cS':=\cS'(\Rd)$ is the dual space, i.e., the space
of continuous linear functionals on $\cS $.
Given $f\in\cS $, we define the Fourier transform and the inverse Fourier transform of $f$ by
\[\cF(f)(\xi) = \widehat{f}(\xi) = \intRd e^{-2\pi i \xi \cdot x} f(x) dx,
\quad \cF^{-1}(f)(x) = \intRd e^{2\pi i x\cdot\xi} f(\xi) d\xi.\]
The definition of Fourier transform is naturally extended to a tempered
distribution $f$; (see chapter 9 in \cite{Fo}).
We define the operators
\begin{align*}
(I-\De)^{k/2}f &= \cF^{-1}((1+4\pi^2|\xi|^2)^{k/2}\widehat{f}),\\
(-\De)^{k/2}f &= \cF^{-1}((2\pi|\xi|)^k\widehat{f})
\end{align*}
for $k\in\bR$ and for $f\in\cS(\Rd)$.
Let $k\in\bR$ and $1<p<\infty$.
The (nonhomogeneous) Sobolev space $H_p^k(\Rd)$ is defined as
\[H_p^k(\Rd) = \Set{f\in\cS'~|~\norm{f}_{H_p^k }:=\norm{(I-\De)^{k/2} f}_{L^p } < \infty},\]
and the homogeneous Sobolev space $\dot{H}_p^k(\Rd)$ is defined as
\[\dot{H}_p^k(\Rd) = \Set{f\in\cS'/\cP~|~\norm{f}_{\dot{H}_p^k }:=\norm{(-\De)^{k/2} f}_{L^p } < \infty},\]
where $\cS'/\cP$ denote the set of all tempered distributions modulo polynomials.
Note that to avoid working with equivalence classes of functions we identify two distributions in $\dot{H}_p^k(\Rd)$ whose difference is a polynomial.

Before we give the definition of Besov spaces, we prepare the setup.
We fix a function $\psi\in\cS(\Rd)$ satisfying $\widehat{\psi}(\xi)=1$ for $|\xi|\le1$ and $\widehat{\psi}(\xi)=0$ for $|\xi|\ge2$ and then define
$\widehat{\phi}(\xi)=\widehat{\psi}(\xi)-\widehat{\psi}(2\xi)$.
Note also that
\begin{equation}\label{support}
\supp\widehat{\phi}(\xi) \subset \set{1/2\le|\xi|\le2}.
\end{equation}
We define for $j\in\bZ$
\begin{equation}\label{phij}
\widehat{\phi}_j(\xi) = \widehat{\phi}(2^{-j}\xi)
\end{equation}
so that for all $\xi \in \bR^d$
\begin{equation}\label{partition1}
1=\widehat{\psi}(\xi)+\sjoi \widehat{\phi}_j(\xi)
\end{equation}
and for all $\xi\neq0$
\begin{equation}\label{partition2}
1=\sjz \widehat{\phi}_j(\xi).
\end{equation}
Let $k\in\bR$ and $1 \le p\le\infty$.
The (nonhomogeneous) Besov space $B_p^k(\Rd)$ is defined as
\[B_p^k(\Rd) = \Big\{f\in\cS'~|~\norm{f}_{B_p^k}:=\norm{\psi*f}_{L^p} +
\Big(\sjoi (2^{kj} \norm{\phi_j*f}_{L^p})^p\Big)^{1/p}<\infty\Big\},\]
and the homogeneous Besov space $\dot{B}_p^k(\Rd)$ is defined as
\[\dot{B}_p^k(\Rd) = \Big\{f\in\cS'/\cP~|~\norm{f}_{\dot{B}_p^k}: =
\Big(\sjz (2^{kj} \norm{\phi_j*f}_{L^p})^p\Big)^{1/p}<\infty\Big\},\]
where $*$ denotes the standard convolution in $\Rd$.
We note that whenever $\phi\in\cS$ and $f\in\cS'$, $\phi*f$ is a well defined function.

\begin{remark}\label{rem}
\begin{itemize}
\item[(1)]
For all $k, s\in\bR$, the pseudo-differential operator $(I-\De)^{s/2}$ is
isomorphism from $H_p^k(\Rd)$ to $H_p^{k-s}(\Rd)$ and from
$B_p^k(\Rd)$ to $B_p^{k-s}(\Rd)$.
\item[(2)]
For all $k, s\in\bR$, the pseudo-differential operator $(-\De)^{s/2}$ is
isomorphism from $\dot H_p^k(\Rd)$ to $\dot H_p^{k-s}(\Rd)$ and from
$\dot B_p^k(\Rd)$ to $\dot B_p^{k-s}(\Rd)$.
\item[(3)]
In particular, if  $1<p<\infty$ and $k$ is a nonnegative integer, then
$H_p^k(\Rd)$ is the set of functions satisfying
\[\sum_{0\le|\alpha|\le k} \intRd |\p^\alpha f(x)|^p dx < \infty,\]
where $\alpha = (\alpha_1, \alpha_2, \cdots , \alpha_d) \in (\bN\cup\set{0})^d$
and $\p^\alpha f = \p_{x_1}^{\alpha_1} \p_{x_2}^{\alpha_2} \dots \p_{x_d}^{\alpha_d} f$
is a distributional derivative.
\end{itemize}
\end{remark}
\subsection{Stochastic Banach spaces}
Let $(\Om,\cF,\set{\cF_t},\bP)$ be a probability space, where $\set{\cF_t : t\ge0}$ is a filtration of $\si$-fields $\cF_t \subset \cF$
with $\cF_0$ containing all $\bP$-null subsets of $\Om$.
Assume that a one-dimensional $\set{\cF_t}$-adapted L\'evy processes $X_t$ is defined on $(\Om,\cF,P)$.
We denote the expectation of a random variable $X(\omega)$, $\om\in\Om$
by $\bE[X]$ or simply $\bE X$.
We consider $g$ as a Banach space-valued stochastic process and so $(\Omega\times(0,T),\mathcal{P},P\bigotimes\ell((0,T]))$ is a
suitable choice for their common domain, where $\cP$ is the predictable $\si$-field
generated by $\set{\cF_t : t \geq 0}$ (see, e.g., pp. 84--85 of \cite{K1}) and $\ell((0,T])$ is the Lebesgue measure on $(0,T)$.
We define the stochastic function space
\begin{equation}\label{s-f spaces}
\bH_p^k(\RT)=L^p(\Om\times(0,T),{\mathcal P},H^k_p(\Rd))
\end{equation}
with the norm
\[\norm{f}_{\bH_p^k(\RT)}=\left(\bE\intT \norm{f(s,\cdot)}_{H_p^k(\Rd)}^pds \right)^{1/p}.\]
The stochastic function spaces $\dot\bH_p^k(\RT)$, $\bB_p^k(\RT)$ and $\dot\bB_p^k(\RT)$ are defined similarly.
\subsection{L\'evy process}
A Levy process $X_t:=X(t)$ is a stochastic process satisfying
\begin{itemize}
\item[(L1)]
$X(0)=0$ a.s.,
\item[(L2)]
$ X(t)$ has stationary and independent increments,
\item[(L3)]
$X(t)$ is stochastically continuous, i.e. for all $a>0$ and for all $s\ge0$,
\[\lim_{t\to s} \bP(|X(t)-X(s)|>a)=0.\]
\end{itemize}
A process $X_t$ is c\'adl\'ag if $X_t$ has left limit and is right continuous.
Since every L\'evy process has a c\'adl\'ag modification
that is itself a L\'evy process (see Theorem 2.1.8 in \cite{App}),
we may assume all L\'evy processes $X_t$ are c\'adl\'ag.
Let $t \geq 0$ and Borel sets $A \in \cB(\Rd\setminus\set{0})$.
We denote
\[N(t,A) = \#\set{0\le s \le t : X(s)-X(s-)\in A},\]
the intensity measure $\nu(A)=\bE[N(1,A)]$, and the compensated Poisson random measure
\[\widetilde{N}(t,A) = N(t,A)-t\nu(A).\]
Note that $\nu(A)$ is the L\'evy measure of $X_t$.
By the L\'evy-Ito decomposition (see more details in \cite{App}), there exist a constant $c\in\Rd$ and a positive-definite matrix $A$ such that
\[X_t = ct + AB_t + \int_{|z|<1} z \widetilde{N}(t, dz) + \int_{|z|\ge1} z N(t, dz),\]
where $B_t$ is a $d$-dimensional Brownian motion.
If we denote $\widetilde{c} := c + \nu(\{|z| \geq 1\})$, we may write
\begin{align*}
X_t = \tilde ct + AB_t + \int_{\bR^d} z \widetilde{N}(t, dz).
\end{align*}
Note that $\widetilde{N}(t,z) $ is martingale.
Since the result for the Brownian motion is known, we assume that $\widetilde{c}=0$ and $A=0$ for the simplicity.
Finally, we denote
\begin{equation}\label{beta}
\beta_p = \int_{\bR^d} |z|^p \nu(dz).
\end{equation}
\mysection{Basic Heat Kernel Estimates}                     \label{Section3}
We give basic lemmas for the heat kernel that will be useful in the sequel.

\begin{notation}
We denote $f \lesssim g$ if $f \le cg$ for some   positive constant $c$.
\end{notation}

\begin{lemma}\label{lemma1}
There exists a constant $c>0$ such that for all $j\in\bZ$
\[\Norm{\cF^{-1}(\widehat{\phi}_j(\xi)e^{-t|\xi|^2})}_{L^1} \lesssim e^{-ct2^{2j}},\]
where $\widehat{\phi}_j$ is defined in \eqref{phij}.
The implicit constant depends only on the dimension $d$.
\end{lemma}

\begin{proof}
Let $K_j(t,x)=\cF^{-1}(\widehat{\phi}_j(\xi)e^{-t|\xi|^2})(x)$.
By a simple scaling
\begin{align*}
K_j(t,x)
&=\intRd e^{2\pi ix\cdot\xi} \widehat{\phi}(2^{-j}\xi) e^{-t|\xi|^2} d\xi \\
&=2^{jd} \intRd e^{ 2\pi i2^jx\cdot\xi} \widehat{\phi}(\xi) e^{-2^{2j}t|\xi|^2} d\xi \\
&=2^{jd} K_0(2^{2j}t,2^jx).
\end{align*}
Observe that
\[(I-\De_\xi)e^{2\pi ix\cdot\xi}=(1+4\pi^2|x|^2)e^{2\pi ix\cdot\xi}.\]
Carrying out the repeated integrations by parts gives
\[(1+ 4\pi^2|x|^2)^N K_0(t,x) = \intRd e^{2\pi ix\cdot\xi} (I-\De_\xi)^N(\widehat{\phi}(\xi) e^{-t|\xi|^2}) d\xi\]
for all $N\in\bN$.
Since $\supp\widehat{\phi}(\xi) \subset \set{1/2\le|\xi|\le2}$, we have
\[(1+ 4\pi^2|x|^2)^N|K_0(t,x)| \lesssim
\sup_\xi|(I-\De_\xi)^N(\widehat{\phi}(\xi) e^{-t|\xi|^2})|.\]
A direct computation shows that for some $c>0$
\[\sup_\xi|(I-\De_\xi)^N(\widehat{\phi}(\xi) e^{-t|\xi|^2})| \lesssim e^{-ct}.\]
We choose $N>d$ so that
\[\intRd |K_j(t,x)| dx = \intRd |K_0(2^{2j}t,x)| dx \lesssim e^{-c2^{2j}t}.\]
This completes the proof.
\end{proof}

Given $g \in H_p^k(\Rd)$, we denote
\[T_t g(x) = \Gam(t,\cdot)*g(x).\]
In fact, for $k<0$, it is the convolution of a function with a tempered distribution, that is,
\[\Gam_t*g(x) =
\begin{cases}
\intRd \Gam(t,x-y)g(y) dy & k\ge0 \vspace{2mm}\\
\ip{g,\Gam(t,x - \cdot)} & k<0,
\end{cases}\]
where $\ip{\cdot,\cdot}$ means the duality paring between $\cS'(\Rd)$ and $\cS(\Rd)$.

\begin{lemma}\label{lemma2}
There exists a constant $c>0$ such that for all $j\in\bZ$
\[\norm{T_t(\phi_j*g)(s,\cdot)}_{L^p} \lesssim e^{-c2^{2j}t} \norm{\phi_j*g(s,\cdot)}_{L^p},\]
where $\phi*g(s,x):=\phi*g(s,\cdot)(x)$.
The implicit constant depends only on the dimension $d$.
\end{lemma}

\begin{proof}
We have
\[T_t(\phi_j*g)(s,x)=\cF^{-1}\big(e^{-t|\xi|^2} \widehat{\phi}_j(\xi) \widehat{g}(s,\xi)\big)(x).\]
From the support condition \eqref{support},
\[\widehat{\phi}_j(\xi)=\Big(\widehat{\phi}_{j-1}(\xi)+\widehat{\phi}_j(\xi)+\widehat{\phi}_{j+1}(\xi)\Big) \widehat{\phi}_j(\xi).\]
Thus we have
\[T_t(\phi_j*g)(s,x) = \sum_{-1\le k \le 1} \cF^{-1}(\widehat{\phi}_{j+k}(\xi)e^{-t|\xi|^2}) * \cF^{-1}(\widehat{\phi}_j(\xi)\widehat{g}(s,\xi))(x).\]
Young's convolution inequality gives
\[\norm{T_t(\phi_j*g)(s,\cdot)}_{L^p} \le \sum_{-1\le k \le 1} \norm{\cF^{-1}(\widehat{\phi}_{j+k}(\xi)e^{-t|\xi|^2})}_{L^1} \norm{\phi_j*g(s,\cdot)}_{L^p}\]
and therefore the result follows from Lemma \ref{lemma1}.
\end{proof}
\mysection{Proof of Proposition \ref{prop}}   \label{Section4}
First we consider (\ref{no ep1}).
Using \eqref{partition1} we decompose
\begin{equation}\label{decom}
g(s,x)= \psi*g(s,x) + \sjoi\phi_j*g(s,x).
\end{equation}
Using \eqref{decom} and Minkowski's inequality, we have
\begin{align*}
&\bE\intT \int_0^t \norm{T_{t-s}g(s,\cdot)}_{L^p}^p dsdt \\
&\lesssim \bE\intT \int_0^t \norm{T_{t-s}(\psi*g)(s,\cdot)}_{L^p}^p dsdt \\
&\quad + \bE\intT \int_0^t
\Big(\sjoi\norm{T_{t-s}(\phi_j*g)(s,\cdot)}_{L^p}\Big)^p dsdt.
\end{align*}
By Young's inequality, the first term of right-hand side is  dominated by
\begin{align*}
& \bE\intT \int_0^t \| \Gam(t-s, \cdot)\|^p_{L^1}
\norm{ \psi*g (s,\cdot)}_{L^p}^p dsdt
= \bE\intT \int_0^t \norm{\psi*g(s,\cdot)}_{L^p}^p dsdt.
\end{align*}
Using lemma \ref{lemma2}, the second term of right-hand side is dominated by
\begin{align*}
 \bE\intT \int_0^t
\Big(\sjoi e^{-c2^{2j}(t-s)} \norm{\phi_j*g(s,\cdot)}_{L^p}\Big)^p dsdt.
\end{align*}
Hence, we have
\begin{align*}
&\bE\intT \int_0^t \norm{T_{t-s}g(s,\cdot)}_{L^p}^p dsdt \\
& \qquad \lesssim    \bE\intT \int_0^t \norm{\psi*g(s,\cdot)}_{L^p}^p dsdt +
\Big(\sjoi e^{-c2^{2j}(t-s)} \norm{\phi_j*g(s,\cdot)}_{L^p}\Big)^p dsdt.
\end{align*}
If we denote
\begin{equation}\label{notation}
f_j(t-s):=e^{-c2^{2j}(t-s)} \quad \text{and} \quad
g_j(s):=\norm{\phi_j*g(s,\cdot)}_{L^p},
\end{equation}
then to prove \eqref{no ep1}, it suffices to show that
\begin{equation}\label{goal2}
\intT \int_0^t \Big(\sum_{1 \leq j < \infty} f_j(t-s) g_j(s)\Big)^p ds dt
\lesssim \intT \sum_{1 \leq j < \infty} 2^{-2j} g_j(s)^p ds.
\end{equation}

Now we consider \eqref{no ep2}.
Using \eqref{partition2}, we decompose
\[g(s,x)=\sum_{j =-\infty}^\infty \phi_j*g(s,x).\]
Using similar calculation with the above estimation, we obtain
\begin{align*}
&\bE\intT \int_0^t \norm{T_{t-s}g(s,\cdot)}_{L^p}^p dsdt \\
&\le \bE\intT \int_0^t \Big(\sjz\norm{T_{t-s}(\phi_j*g)(s,\cdot)}_{L^p}\Big)^p dsdt \\
&\le \bE\intT \int_0^t \Big(\sjz e^{-c2^{2j}(t-s)}
\norm{\phi_j*g(s,\cdot)}_{L^p}\Big)^p dsdt.
\end{align*}
Hence, to prove \eqref{no ep2}, it suffices to show that
\begin{equation}\label{goal2-2}
\intT \int_0^t \Big(\sjz f_j(t-s) g_j(s)\Big)^p ds dt
\lesssim \intT \sjz 2^{-2j} g_j(s)^p ds.
\end{equation}
We shall prove the inequalities \eqref{goal2} and \eqref{goal2-2} in Section \ref{Section5}.
\section{Proof of Main Lemma} \label{Section5}
\begin{lemma}\label{lemma3}
For $0 < T< \infty$ and  $1<p<\infty$
\begin{align}\label{L-goal2}
\intT \int_0^t \Bigabs{\sjoi f_j(t-s) g_j(s)}^p ds dt
\lesssim \intT \sjoi 2^{-2j} |g_j(s)|^p ds
\end{align}
and
\begin{align}\label{L-goal3}
\intT \int_0^t \Bigabs{\sjz f_j(t-s) g_j(s)}^p ds dt
\lesssim \intT \sjz 2^{-2j} |g_j(s)|^p ds.
\end{align}
\end{lemma}

\begin{proof}
We only prove \eqref{L-goal2} since the proof of \eqref{L-goal3} is almost the same.
In order to use the decay of the function $f_j(t)$,
we separate the indices of the summation as
\begin{equation}\label{lem3-4}
\begin{split}
&2^{1-p} \intT \int_0^t \Big(\sjoi f_j(t-s) g_j(s)\Big)^p ds dt \\
&\le \intT \int_0^t \Big(\sum_{2^{2j}(t-s)\le1} f_j(t-s) g_j(s)\Big)^p ds dt
+ \intT \int_0^t \Big(\sum_{2^{2j}(t-s)>1} f_j(t-s) g_j(s)\Big)^p ds dt\\
&:= J_1 + J_2.
\end{split}
\end{equation}
If $2^{2j}(t-s) \leq 1$, then   $f_j(t-s) \leq c$ for some
positive constant $c$ depending only on $d$ . Hence, using H\"older's inequality, we have
\[J_1 \le \intT \int_0^t \Big(\sum_{2^{2j}(t-s)\le1} 2^{j/(p-1)}\Big)^{p-1} \sum_{2^{2j}(t-s)\le1} 2^{-j} g_j(s)^p ds dt.\]
Summing a geometric series, we have
\[\Big(\sum_{2^{2j}(t-s)\le1} 2^{j/(p-1)} \Big)^{p-1} \lesssim (t-s)^{-1/2}.\]
Changing the order of integration and summation, we get
\begin{equation}\label{lem3-5}
\begin{split}
J_1& \lesssim \intT \int_0^t (t-s)^{-1/2} \sum_{2^{2j}(t-s)\le1} 2^{-j} g_j(s)^p ds dt \\
&= \intT \sjoi 2^{-j} g_j(s)^p \int_s^{s+2^{-2j}} (t-s)^{-1/2} dtds \\
&\lesssim \intT \sjoi 2^{-2j} g_j(s)^p ds.
\end{split}
\end{equation}

Now, we estimate $J_2$. Let us fix   $2<r<2p$. Using H\"older's inequality, we obtain
\begin{align*}
J_2 &= \intT \int_0^t \Big(\sum_{2^{2j}(t-s)>1} 2^{rj/p} f_j(t-s) 2^{-rj/p} g_j(s)\Big)^p ds dt \\
&\lesssim \intT \int_0^t \Big(\sum_{2^{2j}(t-s)>1} 2^{rj/(p-1)} f_j(t-s)^{p/(p-1)}\Big)^{p-1} \sum_{2^{2j}(t-s)>1} 2^{-rj} g_j(s)^p ds dt.
\end{align*}
Since $f_j(t-s) \lesssim 2^{-2j}(t-s)^{-1}  $ for  $2^{2j}(t-s)>1$,
summing a geometric series, we have
\begin{align*}
\Big(\sum_{2^{2j}(t-s)>1} 2^{rj/(p-1)} f_j(t-s)^{p/(p-1)}\Big)^{p-1}
&\lesssim \Big(\sum_{2^{2j}(t-s)>1} 2^{j(r-2 p)/(p-1)}\Big)^{p-1} (t-s)^{-p}\\
&\lesssim (t-s)^{-r/2}.
\end{align*}
By changing the order of integration and summation, we get
\begin{equation}\label{lem3-6}
\begin{split}
J_2& \lesssim\intT \int_0^t (t-s)^{-r/2} \sum_{2^{2j}(t-s)>1} 2^{-rj} g_j(s)^p ds dt \\
&\le \intT \sjoi 2^{-rj} g_j(s)^p \int_{s+2^{-2j}}^\infty (t-s)^{-r/2} dtds \\
&\lesssim \intT \sjoi 2^{-2j} g_j(s)^p ds.
\end{split}
\end{equation}
From \eqref{lem3-4}, \eqref{lem3-5} and \eqref{lem3-6}, we obtain \eqref{L-goal2}.
\end{proof}
\section{Proof of Theorem \ref{thm}}          \label{Section6}
Since the proofs are similar, we only prove the first inequality.
Let $u$ be a function defined in \eqref{solution}.
Since we have
\[(I-\De)^{k/2}u(t,x)
= \int_0^t \ip{\Gam_{t-s},(I-\De)^{\frac{k}2}g(s,\cdot)} dX_s
= \int_0^t T_{t-s}( (I - \De)^{\frac{k}2} g )(s,x) dX_s,\]
it is sufficient to prove the case $k=0$, that is,
\begin{align}\label{thm0}
\norm{u}_{\bL^p(\RT)}
=\left(\bE\int_0^t \norm{u(s,\cdot)}_{L^p(\Rd)}^pds \right)^{1/p}
\lesssim \norm{g}_{\bB_p^{-\frac2p}(\RT)}.
\end{align}
Actually, if this estimate is proved, then, by (1) of  remark \ref{rem}, we obtain
\[\norm{(I-\De)^{k/2}u}_{\bL^p(\RT)} \lesssim \norm{(I-\De)^{k/2}g}_{\bB_p^{-\frac2p}(\RT)}\]
and  hence we have
\[\norm{u}_{\bH_p^k(\RT)} \lesssim \norm{g}_{\bB_p^{k-\frac2p}(\RT)}.\]
Using Kunita's inequality
(see pp. 332-335 in \cite{Ku}, corollary 4.4.24 in \cite{App}), we have
\begin{equation}\label{Kunita}
\begin{split}
\norm{u}_{\bL^p(\RT)}^p
&=\bE \intT \int_{\bR^d} \Abs{\int_0^t T_{t-s} g(s,\cdot)(x)dX_s}^pdxdt \\
&\lesssim \bE \intT \intRd \int_0^t \intRd |T_{t-s}g(s,x)|^p|z|^p \nu(dz) dsdxdt \\
&\quad+ \bE \intT \intRd \Big(\int_0^t \intRd |T_{t-s}g(s,x)|^2|z|^2 \nu(dz)ds\Big)^{p/2} dxdt\\
&= \beta_p \bE \intT \intRd \int_0^t  |T_{t-s}g(s,x)|^p dsdxdt \\
&\quad+ \beta_2^{\frac{p}2} \bE \intT \intRd \Big(\int_0^t |T_{t-s}g(s,x)|^2 ds\Big)^{p/2} dxdt.
\end{split}
\end{equation}
From \eqref{no ep1} and \eqref{Kunita}, we get \eqref{thm0}.
\mysection{SPDE with Fractional Laplace operator}  \label{Section7}
In this section, we give an application to the SPDE with fractional Laplace operator.
\begin{equation}\label{fspde1}
\begin{cases}
du  = -(-\De)^\alpha u dt + g dX_t & \RT \vspace{2mm}\\
u|_{t=0} = u_0 & \Rd,
\end{cases}
\end{equation}
where $(-\De)^\alpha u$, $0<\alpha<1$, is the fractional Laplacian of $u$ defined by
\begin{align}\label{definition2}
(-\De)^\alpha u(x) := c(d,\alpha) \int_{\bR^d} \frac{ u(x+y) - 2 u(x) + u(x-y)}{|y|^{n+2\alpha}} dy
\end{align}
with $c(d,\alpha)$ is a normalization constant. The fractional
Laplacian of $u$ also can be defined as a pseudo-differential
operator
\begin{equation}\label{definition3}
(-\De)^\alpha u(x) = \cF^{-1}((2\pi|\xi|)^{2\alpha}\widehat{u}(\xi))(x).
\end{equation}
The solution $u$ of \eqref{fspde1} is represented by
\[u(t,x) = \int_0^t P_{t-s} g(s,x) dX_s,\]
where $P_tg(s,x) = p(t, \cdot) * g(s, \cdot)(x)$ with
fundamental solution $p(t,x)$ of the fractional Laplace equation which is given by
\[p(t,x) = \cF^{-1}(e^{-t|\xi|^{2\alpha}})(x).\]
By a slight modification of the proof of Proposition \ref{prop},
one can prove the following estimate
\begin{proposition}\label{prop2}
Let $0<T<\infty$ and $2\le p<\infty$. There is a positive constant
$c$ such that
\[\bE \intT \intRd \int_0^t  |P_{t-s}g(s,x)|^p dsdxdt
\le  c\bE \int_0^T \|g(t,\cdot)\|^p_{B_p^{-\frac{2\alpha}p}(\bR^d)} dt.\]
\end{proposition}

A direct consequence is the following theorem.
\begin{theorem}\label{thm2}
For $2 \leq p < \infty$,
\begin{align*}
\| u\|_{{\mathbb H}_p^k(\bR^d_T)}
\leq c(p,T) \| g\|_{{\mathbb B}_p^{k- \frac{2\alpha}p}(\bR^d_T)}.
\end{align*}
\end{theorem}

\begin{proof}
We sketch the proof of Theorem \ref{thm2}.
From the same reasoning in the proof of Theorem \ref{thm}, we may assume that $k=0$.
Using the Kunita's inequality, we have for some $c>0$
\begin{align*}
\bE \int_0^T \norm{u(s, \cdot)}_{L^p(\bR^d)}^p ds \leq
&  c \beta_p\bE \intT \intRd \int_0^t  |P_{t-s}g(s,x)|^p dsdxdt \\
&\quad+c \beta_2^{\frac{p}2} \bE \intT \intRd \Big(\int_0^t |P_{t-s}g(s,x)|^2 ds\Big)^{p/2} dxdt.
\end{align*}
H. Kim and I. Kim\cite{KK} showed that for $2\le p<\infty$
\[\bE \intT \intRd \Big(\int_0^t |P_{t-s}g(s,x)|^2 ds\Big)^{p/2} dxdt \le c
\bE \int_0^T \| g(t, \cdot)\|^p_{H_p^{-1}(\bR^d)} dt\]
for some $c>0$ (see also \cite{CL}).
By the same proof as in Lemma \ref{lemma1} and Lemma \ref{lemma2}, one can obtain
\[\norm{P_t(\phi_j*g)(s,\cdot)}_{L^p}
\lesssim e^{-c2^{2j\alpha}t} \norm{\phi_j*g(s,\cdot)}_{L^p}.\]
Similar to the proof of Theorem \ref{thm}, we can obtain the result.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Wynonna Judd cancels performance due to 'unforeseen circumstances'
One of country music's leading ladies has cancelled her Richmond performance as her husband recovers from a motorcycle accident.
September 7, 2012; 3:32 PM • by Nathan Cushing
Wynonna and the Big Noise have cancelled their Saturday, September 8th performance due to "unforeseen circumstances." The performance was to be a key component of CenterStage's Third Birthday celebration.
On August 18th, Wynonna's husband and drummer, Michael Scott "Cactus" Moser, sustained serious injuries in a motorcycle accident in South Dakota. His injuries are not life-threatening. Moser and Wynonna married in June.
"Given the uncertainty of Wynonna's schedule as she spends time with her husband, we felt it best to postpone our gala event," said CenterStage executive director Richard M. Parison, Jr. in a statement. "The entire CenterStage family sends our thoughts and prayers to Wynonna and Cactus, and wish him the very best during his recovery."
CenterStage's birthday celebration will be rescheduled for later this year. Details should be announced in the coming weeks. CenterStage has announced refund options to ticket holders of Saturday's cancelled event:
Full refunds are available to fans that purchased their tickets through authorized retailers at the original point of purchase. Tickets purchased by credit card through Etix.com, Etix phone charge or the Richmond CenterStage Box Office will receive their refunds automatically. Fans do not need to return their tickets. Any fans who purchased with cash at the CenterStage Box Office should return their tickets to the box office for a refund.
Nathan Cushing
Nathan Cushing is a writer, journalist, and RVANews Editor.
More from Nathan
@nathancushing
September 7, 2012 3:32 PM	2
There are 2 reader comments. Read them.


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POEA accredited agencies in Niugan Malabon City – How many are legitimate? This question is the first question that enters a Filipino overseas worker's mind. With all the scam artists in the Philippines looking to fleece the unsuspecting job seeker with offers of lucrative jobs in far-away places, it is good to have an agency like the POEA to accredit and regulate recruitment agencies in Niugan, Malabon City.
All in all, just use common sense to weed out the false from the legal bearing in mind it comes to POEA "accredited agencies" – check their license right of the bat to look if they are legit.


text:
Newton Faulkner + Charlotte at O2 Institute, Birmingham, UK – 26th April 2019
So it's clear what we're in for and boy, does he deliver. When Newton Faulkner appears for his set, the audience are immediately enamoured. He clamours onto a platform underneath the scaffold and takes a seat, surrounded by his now uncovered tools and instruments. I notice no shoes and odd socks, his trademark dreads now knotted above his head and the biggest smile on his face.
TAGS: @iamcharl0tte, @NewtonFaulkner, @ZoeShoots, April 2019, band, Birmingham, Charlotte, Fans, Gig Junkies, Gig photography, Gig Photos, Gig Review, Gig reviews, Gigjunkies, Gigs, Live, Live Music, Live Music Photography, Live Music Photos, Live Music Reviews, Music, Music Photographers, Music Photography, Music Writers, Newton Faulkner, O2, O2 Institute, Photos, review, Reviews, set list, setlist, support, The Best Of...So Far, Tour, UK, Write Up, Zoe Shannon
Album Review: The Best of… So Far by Newton Faulkner
When Newton Faulkner burst onto the scene with his unique approach to a blend of pop and folk music back in the summer of 2007, he embarked on a journey that's seen him create and release some of the most beautiful music.
TAGS: @becky2293, @NewtonFaulkner, Album Review, Becky Weaver, he Very Best Of Newton Faulkner ... So Far, Newton Faulkner, The Best Of...So Far, Tour Dates
The Kooks + Clean Cut Kid @ O2 Academy, Birmingham, UK – 22nd April 2017
Fresh into their mammoth 'comeback' tour of the UK, The Kooks presented a night of nostalgia, rock 'n' roll and explosive energy at their sold out gig at Birmingham's O2 Academy on Saturday night. The pop-rock band made up of Brighton boys Luke Pritchard, Hugh Harris, Alexis Nunez, and Peter Denton, are playing 17 gigs to coincide with the release of their current single 'Be Who You Are' whilst they ready us for the release of their fifth album, treating us to a taste of what to expect.
TAGS: 'Best Of' UK Tour, @CleanCutKid, @thekooksmusic, Academy, Alexis Nunez, April 2017, band, Bands, Be Who You Are, Birmingham, Birmingham Academy, Brownbear Photographer, Carolyn Brown, Clean Cut Kid, Concert Photography, Concert Photos, Fans, FELT, Gig Junkies, Gig photography, Gig Photos, Gig Review, Gig reviews, Gigjunkies, Gigs, Hugh Harris, Live, Live Music, Live Music Photography, Live Music Photos, Live Music Reviews, Luke Pritchard, Music, Music Photographers, Music Photography, Music Writers, O2, O2 Academy, Peter Denton, Photos, pictures, review, Reviews, support, The Best Of...So Far, The Kooks, Tour, UK, Write Up


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This mini brush is designed for nail cleaning, you may use it to remove the dust on nails before nail beauty process.
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text:
McMillen and Wife Steelers Message Board
McMillen and Wife Steelers Message Board > Classic Steelers Game Broadcasts > 1979 Steelers Season Game Threads > The Seriously Great (Yet Strangely Underrated) 1979 Pittsburgh Steelers
Full Version: The Seriously Great (Yet Strangely Underrated) 1979 Pittsburgh Steelers
The Seriously Great (Yet Strangely Underrated) 1979 Pittsburgh Steelers
History has not given this team its due. Despite bringing home Pittsburgh's 4th Lombardi in 6 years in Super Bowl XIV (a feat which is still unmatched), the 1979 Steelers remain one of the most strangely underrated and underappreciated teams in NFL history.
Coming off a dominant '78 season culminating in their thrilling win over Dallas in Super Bowl XIII, they faced what at the time was the toughest schedule the league had ever seen. This was by design, by the way, courtesy of Pete Roselle, but that's a discussion for another time.
In addition to their grueling schedule, the Steelers faced three prime challenges in 1979: turnovers, injuries and age. But re-watching these games and examining the various ways they were able to overcome these obstacles, I believe a legitimate case can be made for the '79 squad as the most talented, explosive and dangerous team in Steelers history (and therefore, NFL history).
I know this conversation is usually reserved for the 1978, 1975 and 1976 Steelers teams. But hear me out.
Pittsburgh faced a slew of key injuries in 1979. But this team was LOADED with talent -- they were absolutely stacked -- and their depth was so great that they never missed a beat. In addition to fielding 10 future Hall of Fame players (plus a Hall of Fame coach, owner and President), Pittsburgh's bench was filled with high quality reserves.
Hall of Fame WR Lynn Swann was bothered by a hamstring and other nagging injuries starting in training camp. Enter Jim Smith, who like many Steeler backups, would've been full-time starter on most other NFL squads. When Hall of Fame linebacker Jack Ham had his season ended by a broken leg, no one really noticed thanks to the outstanding play of reserve LB's Robin Cole and Dirt Winston. And when Franco Harris got dinged, Sidney Thornton came off the bench to rumble for over 300 yds and 3 TDs in a 3-week span.
The Steelers also overcame their so-called age issues beautifully that year. Joe Green, for example, was supposedly well past his prime. Yet he DOMINATED the '79 AFC Championship game and was a primary reason Earl Campbell finished with just 17 yards on 15 carries.
This supremely talented team was statistically awesome as well. Pittsburgh finished #1 in total offense, #2 in total defense and was the #1 scoring team in the league. Although at times they were a turnover factory (a league-worst 52), they were so good in every other area that they overcame their mistakes to finish with the best record in the NFL.
The '79 team is also responsible for what in my opinion is one of the most dominating 4-game regular season stretches in NFL history. After a shocking 9-turnover defeat at the hands of the previously winless Bengals, the Steelers unleashed hell on the NFL (with apologies to Mike Tomlin), shellacking the 5-2 Broncos (42-7), 7-1 Cowboys (14-3), 6-3 Redskins (38-7) and finally the Marv Levy Chiefs (30-3), by a combined score of 124-20. I encourage you to watch these games and see if you agree.
During that 4-game stretch, the Steelers also put to bed endless Cowboys' whining and moaning about questionable calls, dropped balls and bad breaks costing them Super Bowl XIII. Pittsburgh dominates, outrushing Dallas 173-79 while the Steel Curtain simply throttles Roger Staubach and Tony Dorsett, with L.C. Greenwood ultimately knocking Staubach out of the game and into retirement.
This season also features what I consider to be the most pivotal regular-season win in franchise history, the 33-30 overtime thriller vs. Cleveland in which Bradshaw and Sipe combine for over 700 yds though the air. The Steelers also amass over 600 yds of total offense in this game (to my knowledge, the only time this was done in the '70s, and one of only 5 times from 1970-1999).
The standings coming into the game were Houston 10-3, Pittsburgh 9-3 and Cleveland 8-4. The Steelers were coming off a stunning turnover-filled defeat in San Diego and a Pittsburgh loss to Cleveland would've given Houston sole possession of first place in the AFC Central and left the Steelers in a tie for second with the Browns. Instead, the Steelers improve to 10-3 to remain atop the division, the Browns fall to 8-5 and are knocked from contention and the rest, as they say, is history.
There are some exciting, great-quality recordings here. Enjoy watching/reliving the season. It's filled with entertaining games and is perhaps my favorite season as a Steelers fan.
>> CLICK HERE to watch the 1979 Steelers Season <<
P.S. -- If you're looking for offensive fun, it's hard to beat the '79 game in Cleveland (51-35).
dbsfgyd1
Great post. You make a very good case on just how good this team was. Being honest, I for one have always underrated this this team, discounting just how resilient they were. I don't know of any Super Bowl winners that got by averaging over 3 turnovers a game for a season.
Thanks for the invite to review this season.
(02-14-2021, 02:21 PM)dbsfgyd1 Wrote: [ -> ] I don't know of any Super Bowl winners that got by averaging over 3 turnovers a game for a season.
IKR? One of the crazier stats about Super Bowl XIV is that the Steelers had the ignoble honor of being #1 in the NFL for turnovers committed (52), and the LA Rams were #2 (49).
I doubt that statistical freak of nature will ever be repeated.
Chucktownsteeler
It was the beginning of the for sure.
RaynorShyne
Got the entire team in my 1979 Topps Football Card Collection. :-) Was 12 years old that season and begged Mom for cards every time we went out. Not that I have a little means, I'm going to re-collect them all as graded cards. My favorite Steeler team of all time.
Per you suggestion, I've been through most of this season and have enjoyed it much. Dang, I loved that trapping run game the used. I'd forgotten how much fun it was to watching some poor slob thinking he was going to make a big play only to get clobbered. Maybe I need some professional help....
(02-22-2021, 02:51 AM)dbsfgyd1 Wrote: [ -> ] Per you suggestion, I've been through most of this season and have enjoyed it much. Dang, I loved that trapping run game the used. I'd forgotten how much fun it was to watching some poor slob thinking he was going to make a big play only to get clobbered. Maybe I need some professional help....
Be sure to check out the 2nd half of the St. Louis game. It's not a matchup that gets a lot of attention, but it's a testament to the greatness of Rocky Bleier and that incredible offensive line.
When Franco goes down with an injury with the Steelers trailing 21-7, Bleier comes in and absolutely DOMINATES the final quarter and a half of this game. The Rock finishes with 73 yds rushing on 13 carries, scores a 4-yd TD and catches two crucial passes for 44 yds... all in the second half as the Steelers cram 17 points into the 4th quarter to win 24-21 on Bahr's FG with 0:13 left.
The depth of the '79 team was RIDICULOUS.
>> 1979 Steelers 24 at Cardinals 21
(02-22-2021, 03:34 AM)mcmillenandwife Wrote: [ -> ]
Interesting game. Rookie Dornbrook coming in for injured Courson at LG hit som key blocks and more than held his own next to Peterson at LG who was filling for Kolb. I think Peterson pitched a second half shut out. As you said the Rock was solid. Hawthorne not so much. Totally miscast in this offense. Big fast and strong, but never did grasp the overall concept.
Also, Hats off to Wilkinson. He had those guys playing really well. A few up and comers on that team too, Anderson and Green.
(02-22-2021, 03:34 AM)mcmillenandwife Wrote: [ -> ] Be sure to check out the 2nd half of the St. Louis game. It's not a matchup that gets a lot of attention, but it's a testament to the greatness of Rocky Bleier and that incredible offensive line.
Rock was ALWAYS sticking his maw into a linebacker or blitzing corner. One of my all time favorites. Hell, he even made me like Robert Urich (RIP).
(02-24-2021, 10:16 AM)RaynorShyne Wrote: [ -> ] Rock was ALWAYS sticking his maw into a linebacker or blitzing corner. One of my all time favorites. Hell, he even made me like Robert Urich (RIP).
Rocky was a truly great blocker. Countless famous plays were a direct result of huge Bleier blocks, including Swann's "Levitating Leap" and final TD pass in Super Bowl X. The dude was amazing.
aqua -park
Victory over Browns in OT was huge. A loss and Pittsburgh is probably a Wild Card and I doubt they make it to Super Bowl XIV. This was a great season but the Steelers did get some luck in the post season. Since the Oilers knocked out San Diego (playing without Campbell & Pastorini no less) Pittsburgh avoided a trip out to the west coast for the AFC Championship. The Eagles, who had beaten Pittsburgh earlier in the year, lost in the playoffs. The Cowboys lost at home to the Rams on a deflected pass that went for a Ram TD at the end of the game (Cowboys looking for revenge in the Super Bowl would've been a tough game). The Steelers got to play an Oilers team with Campbell & Pastorini still hurting. Renfro's bobble was helpful. Lots of good fortune!
(04-25-2021, 01:15 PM)aqua -park Wrote: [ -> ] This was a great season but the Steelers did get some luck in the post season. Since the Oilers knocked out San Diego (playing without Campbell & Pastorini no less) Pittsburgh avoided a trip out to the west coast for the AFC Championship.
Good post. My two cents: I think you'll find EVERY team who makes it to a Super Bowl experiences a healthy share of good fortune. Conversely, teams who fail to get there experience misfortune. That's axiomatic.
My take is, if the Chargers couldn't get past the hamstrung Oilers with NO Earl Campbell and NO Dan Pastorini -- at home, no less -- there's no way they were going to get past an experienced, proven post-season MACHINE like Pittsburgh who was out for revenge. The Steelers knew how to CRANK IT UP during the playoffs; the Chargers didn't.
That Oiler team had flippin' HEART, by the way. Physical AF and my favorite "non-Steeler" team ever. Earl Campbell is still my favorite RB of all time.
(04-25-2021, 01:15 PM)aqua -park Wrote: [ -> ] Victory over Browns in OT was huge. A loss and Pittsburgh is probably a Wild Card and I doubt they make it to Super Bowl XIV.
You're 100% right about that win being huge. The single-biggest regular season win in Steelers history, IMO. And one of my all-time favorite games.
(04-25-2021, 01:15 PM)aqua -park Wrote: [ -> ] The Eagles, who had beaten Pittsburgh earlier in the year, lost in the playoffs.
No fear of the Eagles. At all. IMO, that loss was almost as much of an aberration as the San Diego game. Injuries and terrible turnovers deep in Eagle territory were just enough to tip the scales.
Pittsburgh was missing Lynn Swann, Jim Smith, Moon Mullins, L.C. Greenwood, Ron Johnson and Steve Furness. And Larry Brown was the victim of a nonsense ejection after a tussle in the 2nd quarter. Some terrible breaks in the 4th quarter: Terry threw a bad pick in the end zone, Franco fumbled at the Eagle 1-yard line, and the refs blew a call by overturning a clear Pittsburgh recovery of an onside kick late in the game.
So the Steelers had some not-so-good fortune, too. Ironically, we also got screwed on a clear recovery of an onside kick in the week 15 loss to Houston.
(04-25-2021, 01:15 PM)aqua -park Wrote: [ -> ] The Cowboys lost at home to the Rams on a deflected pass that went for a Ram TD at the end of the game (Cowboys looking for revenge in the Super Bowl would've been a tough game).
IMO (with hindsight), the toughest Super Bowl opponent we could have faced in 1979 was the Rams, despite their regular season record. This is as much because of Bud Carson as anything, because they knew our defense, then knew our offense, they even knew our audibles. Beyond that, the Rams were a hot team, winning 6 of their last 7 before meeting Pittsburgh in the Super Bowl.
As for the Cowboys, Pittsburgh had throttled Dallas earlier in the year, just like they did seemingly every year. They were still a good team, but Roger Staubach was basically done and despite their strong 7-1 start, the Cowboys stumbled badly to finish 4-5 down the stretch.
(04-25-2021, 01:15 PM)aqua -park Wrote: [ -> ] The Steelers got to play an Oilers team with Campbell & Pastorini still hurting. Renfro's bobble was helpful. Lots of good fortune!
Okay... but we spotted them 7 points and were playing without Hall of Famer Jack Ham.
Hurting or not, Pastorini played his ass off. One of the best games of his career IMO. Renfro got screwed by the officials, no question. But we got screwed the bad onside kick call a few weeks earlier, so... what comes around, goes around.
Beyond that, the Steelers outplayed Houston in every phase of the game. We gave Vernon Perry the "gimme" TD to start, but we pretty much manhandled them from there. 24 rushing yards in 22 attempts is ridiculous, especially when compared to Pittsburgh's 161 yds on 36 carries.
So while I appreciate what you're saying here and I agree with you that the Steelers had some good fortune in 1979, IMO they had "almost" as much misfortune in the form of injuries and turnovers. Yet they overcame.
And that's kinda my point.
Remington Steelers
I noticed that Cliff Stoudts' name is missing from the 1978 and 1979 roster list. Just curious if there was a reason behind that.
It was normal for me and all my Steeler friends to utter the often said, "Stoudt Stinks!!" phrase during the early 1980's. So I don't care if he stays off the list for the SB winning years. I just find it funny.
(01-10-2022, 02:18 PM)Remington Steelers Wrote: [ -> ] I noticed that Cliff Stoudts' name is missing from the 1978 and 1979 roster list. Just curious if there was a reason behind that.
Man, that's a great observation. I'm surprised! I never noticed before. I assume it's because he never played in a game until 1980? I know he was on the practice squad in '77. Pretty sure they released him and then picked him back up. I know Stoudt played during the preseason in '78 and '79 (he's on some of the preseason videos on this site), but I looked it up and he never took a single regular season snap until week 4 of 1980 vs. Chicago.
Pretty crazy. I would've thought he'd still be considered part of the roster, but he's not listed anywhere.
(01-21-2022, 02:11 PM)mcmillenandwife Wrote: [ -> ]
I believe you. But if you look at the team picture, on the 1978 and 1979 pages, he is there in both. That's how I saw it. I guess I sort of assumed that if you were in the team picture, you were on the roster whether you played or not.
Sort of how they messed up with Dwight White not being on the Super Bowl IX team picture or roster "in reverse".
(01-21-2022, 03:40 PM)Remington Steelers Wrote: [ -> ]
(01-21-2022, 02:11 PM)mcmillenandwife Wrote: [ -> ] Man, that's a great observation. I'm surprised! I never noticed before. I assume it's because he never played in a game until 1980? I know he was on the practice squad in '77. Pretty sure they released him and then picked him back up. I know Stoudt played during the preseason in '78 and '79 (he's on some of the preseason videos on this site), but I looked it up and he never took a single regular season snap until week 4 of 1980 vs. Chicago.
Yeah, I agree with you. He was in the team pictures, in my mind he was part of the team all 3 years ('77, '78' and '79). But he never took a snap before 1980. Maybe he was never activated for a game in '77-'79? But I feel like he was at least on the sideline, although I can't say for sure without going back and looking.
It's a really strange situation and now that we're talking about it, I'm kind of determined to find an answer.
EDIT: Quick follow-up. I just looked at the '79 opener vs. New England and saw Stoudt suited up and standing on the sideline holding a clipboard. So my theory is that while he was part of the team, he wasn't considered part of the official active roster because he never took a snap.


Attached is the most recent backtest report for the technical model. This was run against the 2432 symbols of the NASDAQ composite for as far back as there are active symbols — trades start in 1981.
Portfolio equity increases as one might expect. There are losses in only 4 years of the 35 year period (1981, 1984, 1988, 2008), and three of the four years the losses are single digit. By far the worst loss is 19% in 2008.
Risk adjusted return — that is, the return of monies actually invested — is 28.89%. The average length of time of a trade is 89 trading days, or a little over 4 months.
Winners are held about 100 days longer than losers, and the win percent is not quite 50%. This is not particularly troublesome. You get out of losers quickly, and you ride the winners.
The attached backtest listing is a summary of the trades, including those that are still "open".
I would welcome any input to make this better. This is the algorithm that drives my technical charts.


text:
Britain's Place in the World: Import Controls 1945-60 by George Brennan, Alan Milward
By George Brennan, Alan Milward
Britain's position on the planet examines the institution and effectiveness of import controls, fairly quotas. putting quotas again within the centre of British background, Milward and Brennan make a few radical claims for Britain's financial functionality in an international context. having a look right into a wide selection of industries from motorized vehicles to typewriters, uncooked chemical compounds to nutrition produce, they learn the meant and genuine obstruction to imported items represented through quotas, and the political and monetary ramifications past the statistics.This is the fourth e-book to be released within the Routledge Explorations in fiscal heritage sequence.
Read or Download Britain's Place in the World: Import Controls 1945-60 (Routledge Explorations in Economic History, 4) PDF
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The disadvantage, that they impede exports more than inter-firm regulation probably does, of course, remains. In the immediate post-war period, in the United Kingdom at least, most of the civil servants employed in the administration of import quotas were simply carrying out routine bureaucratic procedures. Actual negotiations with the domestic producers affected by the quota regulations were few. Usually they were confined to the setting of bilateral quota levels, global quota levels being considered only a matter of the governments own balance of payments policy.
British bilateral agreements with agricultural suppliers whose exports to Britain were subject after 1933 to import quotas sometimes included clauses stipulating that in return for their quota a fixed proportion of their coal imports should come from the United Kingdom. 12 This was thought of not only as promoting exports but also as protecting employment in a threatened industry with a very large labour force. The use of quotas, however, was much more extensive in countries that avoided currency devaluation.
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CASHLOANSVC.COM Book Archive > International Business > Britain's Place in the World: Import Controls 1945-60 by George Brennan, Alan Milward
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text:
Home News Urgent action demanded over Edinburgh Sick Kids hospital's cladding
Urgent action demanded over Edinburgh Sick Kids hospital's cladding
In a letter to the Scottish Government, the party's leader and Edinburgh Western MSP, Alex Cole-Hamilton, said any "question mark" about the Royal Hospital for Children and Young People's safety was unacceptable.
The Edinburgh facility has components in its cladding which were also used on Grenfell Tower, where 72 people died in a fire in 2017.
The new hospital – which was hit by delays due to concerns over the safety and installation of its ventilation system and saw its opening delayed by more than a year and a half – is having compliance checks to ensure the cladding does not pose a health risk, The Scotsman revealed last month.
The new Sick Kids hospital exterior, covered in the potentially dangerous cladding.
Construction of the hospital is already subject to an ongoing public inquiry, but is fully operational after opening in 2021. However, NHS Lothian is carrying out compliance checks.
It comes after the health board discovered the hospital did not fully comply with fire safety building standards when it was built.
NHS Lothian maintains the cladding is safe but a document to executive leadership states building standards were not compliant due to the distance between "cavity barriers".
These are designed to prevent fire spreading as quickly as it otherwise would and are, the document states, required every ten meters. At the Sick Kids, however, they have only been installed every 20 meters.
In his letter to the health secretary, Humza Yousaf, Mr Cole-Hamilton said the safety concerns were "very concerning" for patients and staff.
He said: "The Grenfell Tower was a modern horror story so it is deeply worrying that a similar cladding is on the walls of one of our children's hospital.
"Of all the places to have a question mark over its safety, a hospital where parents are sending their sick children to get well, is the most concerning. This is a building that is already subject to an ongoing public inquiry, and now this.
"The government has acknowledged the safety concerns and said they'll be addressed but they haven't provided any details as to what will be done, or when any work will be completed.
"Everyone working at, or relying on, the hospital needs to have complete assurance the building they are in is safe. If there's any question mark about its safety then the government and NHS Lothian need to take action today."
A Scottish Government spokesperson said there was no cladding of "exactly" the same type used on Grenfell on any hospital or high-rise health building in Scotland.
They said: "No cladding of the type used in Grenfell Tower are used in any Scottish hospital, including the Edinburgh Royal Hospital for Children and Young People (RHCYP).
"NHS Lothian is developing a program of work to ensure all cavity barriers in use at the RHCYP are fully compliant with safety regulations.
"The board has been clear that the hospital is a safe place of care."
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