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Home » featured » INTO THE FRAY: Iniquitous indictments for invented infractions
INTO THE FRAY: Iniquitous indictments for invented infractions
martin-sherman
#Democracy
#Immunity
#Indictment
#IntoFray
#Knesset
#Netanyahu
For your perusal, my latest INTO THE FRAY column
Iniquitous indictments for invented infractions
(Kindly consider "liking", sharing, tweeting – please use hash-tag #IntoFray)
Despite his remarkable success, Netanyahu has been ceaselessly assailed by his political adversaries. Indeed, it is perhaps his very success that has generated such raw rancor against him
It appears this week on the following sites (in alphabetical order):
ISRAEL NATIONAL NEWS: http://www.israelnationalnews.com/Articles/Article.aspx/24978
ISRAPUNDIT: https://www.israpundit.org/into-the-fray-iniquitous-indictments-for-invented-infractions/
JERUSALEM HERALD: https://www.jerusalem-herald.com/single-post/2020/01/06/Bibi-Iniquitous-Indictments-For-Invented-Infractions
JEWISH PRESS: https://www.jewishpress.com/indepth/columns/into-the-fray-martin-sherman/into-the-fray-iniquitous-indictments-for-invented-infractions/2020/01/07/
JNS: https://www.jns.org/opinion/iniquitous-indictments-for-invented-infractions/
JEWS DOWN UNDER: https://jewsdownunder.com/2020/01/03/into-the-fray-iniquitous-indictments-for-invented-infractions/
MEDIUM: https://medium.com/@martinsherman/into-the-fray-iniquitous-indictments-for-invented-infractions-6ca1d02e904a
RICHSWIER: https://drrichswier.com/2020/01/03/iniquitous-indictments-for-invented-infractions/
Several short excerpts:
Israel – "a land that devours its inhabitants. Numbers 13:32.
I would have preferred to have devoted this week's column to some other topic—such as the ascendance of blatant anti-Semitism around the globe, or the heightening tension between the US and Iran, or the emerging "understanding" with Hamas for "calm" in Gaza.
Dramatic request
But Netanyahu's dramatic ten minute address on Wednesday evening (January 1, 2020), in which he announced his intention to request parliamentary immunity from the charges to be brought against him for alleged breach of trust and bribery, is sufficiently significant to put other matters on the backburner—at least temporarily. (Reports on the assassination of Qasem Soleiman were just coming in as this article was being prepared for submission.)
The rationale for immunity
President Reuven Rivlin discussing parliamentary immunity
Netanyahu…invoked the words of the President, Reuven Rivlin, who has had an overtly contentious relationship with Netanyahu, and who, arguably because of that relationship, has become the epitome of moderation and respectability for the Center-Left in Israel.: "The legislators created far reaching immunity in order to protect representatives elected by the public…If the prosecutors and investigative authorities decided, because of political reasons , to neutralize a Member of Knesset, they could open an investigation against him—and things like that have happed in the past. People were subjected to criminal investigations and indictments were handed down against them—and there is grave concern that this was done by the authorities with the intention of preventing them from serving as ministers…."
Rivlin: Politically biased investigations launched in past
Significantly, Rivlin articulated very similar views, long before the prospect of any indictments against Netanyahu emerged.
Indeed, interviewed while still in his former capacity of Knesset Speaker, he expressed the identical rationale for parliamentary immunity for elected legislators: "We must remember what the logic behind [the idea] of immunity is; what the substantive reason was, which brought the founders of the Knesset…to create over-arching immunity, immunity that is unassailable, for each and every Knesset Member. [It was] the desire to protect publicly elected representatives from the ability of the authorities to bully and intimidate them".
Recriminations resonate
Attorney-General Avichai Mandelblit
I must confess that Netanyahu's shrill recriminations regarding the odds stacked against him and the motivations of the forces ranged against him resonate strongly with me.
As I have pointed out previously – see for example here— to anyone but a rabid "Bibiphobe", they appear transparently contrived, indeed, a thinly veiled attempt at a legalistic coup, creating a deep sense of unease that Israel's legal establishment is being exploited for patent political ends—i.e. that unelected elites are using their positions of influence and authority to bring about political outcomes that do not correspond with—even contradict—the election results…
"Appropriate criteria for criminal prosecution not met…"
But it seems that not only the practical conduct of the investigation and the motivations behind it are disturbingly suspect, but so are the very conceptual foundations on which it is founded.
Professors Avi Bell & Alan Dershowiitz: Severely critical of indictments against Netanyahu
This was powerfully conveyed by prominent legal expert, Professor Alan Dershowitz in a piece written almost exactly a year ago, in the far-left daily, Haaretz, entitled Voters, Not the Police or the Courts, Should Decide Netanyahu's Future.
According to Dershowitz: "The issue at the center of these investigations seems trivial against the background of the existential crises Israel is facing…The first probe, also known as Case 1000, involves gifts of cigars and champagne Netanyahu received from close friends…I strongly believe that the appropriate criteria for criminal prosecution have not been met in the cigar and champagne case against Netanyahu… The other investigations (dubbed 2000 and 4000) pose even greater dangers to democratic governance and civil liberties
Flimsy case vs impressive achievements
I disagree with Alan Dershowitz on much regarding Israel, but I identify almost completely with his analysis of the indictments filed against Netanyahu. Indeed, not only do the substance of the indictments appear "trivial" compared to the challenges Israel faces, but also seem trivial against the background of the giant strides with which Israel has progressed under Netanyahu…Of course, Netanyahu—like every mortal on the planet—is not irreplaceable or unblemished, but his record indicates that he is by far the most capable candidate to lead Israel in these challenging times.
A transformative leader
Indeed, despite any criticism of him, it is undeniable that, in many ways, he has been a truly transformative leader.
On his watch, Israel joined the prestigious group of OECD countries, and has become a major energy exporter—things almost inconceivable before his incumbency.
"A country that devours its inhabitants?"
Yet, despite his remarkable success, Netanyahu has been ceaselessly assailed by his political adversaries, ever since he was first elected in 1996. Indeed, it is perhaps his very success that has generated such raw rancor against him. Thus, despairing of removing him via the ballot box, his political rivals and adversarial civil society elites have had to turn to the law to do so—mounting what Netanyahu has accurately dubbed a legalistic coup.
It is an initiative that is likely to backfire…
As usual your talkbacks/comments/critiques welcome,
strategic Israel
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The Israel Institute for Strategic Studies is an independent policy center devoted to the promotion of joint values shared by Israel and the United States of America.
INTO THE FRAY: Terminating terrorists and assessing assassinations January 16, 2020
INTO THE FRAY: Iniquitous indictments for invented infractions January 8, 2020
INTO THE FRAY: In Israel & America – Choreographed coups with the law as a prop December 31, 2019
INTO THE FRAY: Israel's dysfunctional electorate? December 14, 2019
INTO THE FRAY: Merited mistrust? December 1, 2019
Israeli Institute for Strategic Studies ©2019 All Rights Reserved


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Simona Kirchnerova is signed with Modelwerk Hamburg, Dominique Models Brussels, Seeds Management GmbH Berlin, Uno Models Barcelona, The Hive Management London, Why Not Model Management Milan, Premium Models Paris and Elite New York. Her mother agency is Exit Model Management in Bratislava.
In 2014 Simona Kirchnerova walked the spring runway for Maison Rabih Kayrouz.
In 2015 she walked the spring couture show for Dice Kayek. She walked the fall runways for Barbara Casasola, Christopher Raeburn, Nehera, Celine and Yves Saint Laurent. She walked the fall couture show for Schiaparelli. She walked the spring runways for Wes Gordon, Edun, Jasper Conran, Barbara Casasola, Philipp Plein, Antonio Marras, Elisabetta Franchi, Marco de Vincenzo, Julien David, Pascal Millet and Hermes. She modeled for an editorial in the July edition of Cosmopolitan Germany, which was called Dichter, Denker, Designer that was photographed by Enric Galceran. She modeled for an editorial in the July edition of The Financial Times-How To Spend It Magazine, which was called Away with the Prairies that was photographed by Damian Foxe. She modeled lookbooks for Sportmax and Lefties.
In 2016 she walked the spring runways for Kristina Ti and Lucio Vanotti. She modeled for an editorial in the March edition of Marie Claire UK, which was called Road Less Travelled.


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Home - Stewart Giles
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STEWART GILES
#1 Best Selling Author
Detective Sergeant Jason Smith is an Australian working in York. He was exiled to York when he was 16 after his sister disappeared from a beach in Western Australia. Halfway through a law degree, something happened that made Smith realise he was better suited to catching criminals than defending them …
Detective Constable Harriet Taylor is a Scot working in the peaceful town of Trotterdown in Cornwall. A scandal in her home town of Edinburgh brought her to the south-west of England.
Her past still haunts her, however and the only way to erase the memories is through sleeping pills …
The new DS Jason Smith & DC Harriet Taylor Thriller
From #1 best-selling author: Stewart Giles comes a new detective series featuring the detectives from both the series' we know and love.
In Trotterdown, Cornwall a boat, The Enigma washes in to the harbour. The skipper is dead, and Harriet Taylor is brought in to investigate. But 1 dead body is not the only crime the team have to deal with.
Meanwhile, further up the coast in Whitby a boat washes up on the rocks just outside the harbour. Everyone on board is dead. Whitby Police are out of their depth, so DS Jason Smith is called in to help.
It soon becomes clear the cases are connected and the teams from York and Trotterdown settle down to work with each other.
This is a unique story with twist, turns and shocks all the way through making it an adrenaline pumping read you will not be able to put down.
The Backpacker
DC Harriet Taylor #3
WHAT IS THE SECRET AT LANDELL'S FARM?
A girl's body is found hidden in a remote spot of a Cornish Farm. The same farm that a young girl ran towards to escape her pursuer many years before. Detective Harriet Taylor has to abandon her day out to investigate.
As Littlemore and the forensics team get to work they uncover another mystery hidden among the rocks.
Who would kill a young backpacker who hurt nobody? Is there a link between this and a mystery from many years ago?
As Harriet and the team get to work they find more questions than answers. What secrets is the sleepy Cornish village hiding?
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DS Jason Smith is dragged away from paternity leave when a woman is killed with her baby in the next room. When more women with children are brutally murdered, Smith discovers that they all have something in common – they all share a secret. When he finds out that the mother of his own child is on this list of 'Unworthy' women, he finds himself in a race against time to stop the killer before his whole world comes crashing down around him.
The Perfect Murder
Two cats are found mutilated in the same town. Detective Harriet Taylor is reluctant to investigate, but then one of their owners is killed a bizarre way that same day.
Harriet and the team step in. The dead woman has four words written on her neck — four very ambiguous words.
DS Jason Smith Books
One draining murder investigation after another has left DS Jason on the brink of losing his sanity and he needs a break. A string of armed robberies at the McDonalds in the city of York are getting in the way. Smith's DI makes him a deal; if he can get to the bottom of it, he can have two weeks off. Smith figures out who the brains behind the robberies is …
When a man is found tucked up in bed with his throat sliced open two days after Christmas, DS Jason Smith and his team are baffled. It appears he has been drugged, killed and carefully wrapped up in a duvet. The only evidence Smith finds are some strands of hair belonging to a woman. One month later, another man is killed in exactly the same manner …
DC Harriet Taylor Books
Alice Green is a beekeeper in the small Cornish village of Polgarrow. She lives with her pet jackdaw in a beautiful cottage not far from the sea. One evening, Alice finds something strange under the hollyhock bush in her garden. The gruesome discovery will change everyone's lives. And then Alice's best friend Milly disappears . . .
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"Great follow up to
"I'm a great fan of the Jason Smith novels so I looked forward to a new character – Harriet Taylor doesn't disappoint. The plot was well thought out, characters are developing nicely & it's easy to imagine the next installment."
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redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Create a scrollbar (appJar) So I am creating a scroll-bar in appJar to show 10 numbers at a time (1-10, 11-20, etc.), and I am wondering what the best way is to go about this.
Should I create a function to hide the next 10 numbers until the scrollbar (which is a scale, though this might be the wrong method), and then they appear?
I am genuinely lost right now, as I've only started learning how to use appJar about 2 days ago for a project. Any ideas would be a great help. Yes, this is just the gui part, and the Python (3.6) has not been added yet, but will be.
from appJar import gui

def press(btn):

    if btn == "1":
       app.setLabel("answer", "1!")
    elif btn == "2":
       app.setLabel("answer", "2!")
    elif btn == "3":
       app.setLabel("answer", "3!")
    elif btn == "4":
       app.setLabel("answer", "4!")
    elif btn == "5":
       app.setLabel("answer", "5!")
    elif btn == "6":
       app.setLabel("answer", "6!")
    elif btn == "7":
       app.setLabel("answer", "7!")
    elif btn == "8":
       app.setLabel("answer", "8!")
    elif btn == "9":
       app.setLabel("answer", "9!")
    elif btn == "10":
        app.setLabel("answer", "10!")
        app.startScrollPane("scroller")
    elif btn == "11":
       app.setLabel("answer", "11!")
    elif btn == "12":
       app.setLabel("answer", "12!")
       app.stopScrollPane("scroller")

app=gui()
app.setFont(20)
app.addButtons(["1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12"], press)
app.addScale("scroller")
app.setScaleChangeFunction("scroller", press)
app.setButton ("1", "1")
app.setButton ("2", "2")
app.setButton ("3", "3")
app.setButton ("4", "4")
app.setButton ("5", "5")
app.setButton ("6", "6")
app.setButton ("7", "7")
app.setButton ("8", "8")
app.setButton ("9", "9")
app.setButton ("10", "10")
app.setButton ("11", "11")
app.setButton ("12", "12")
app.addLabel ("answer", "Pick a Number!")
app.go()

A: I see two alternatives here:

*

*Create a scrollPane, and utilise the scrollPane's built in scrollbars

*Link the Scale widget to a function that updates the labels of the buttons

It really depends on how you want the GUI to look...
For option 1, try the following, instead of the scale:
app.startScrollPane("p1")
for i in range(1,101):
    app.addButton(str(i), press, 0, i)
app.stopScrollPane()

For option 2, get rid of the list of setButton() calls, and replace them with a call to a new function: updateButtons().
Then, in that function, have something like:
def updateButtons(btn=None):
    m = app.getScale("scroller") * 10
    for i in range(1, 13):
        app.setButton (str(i), str(m+i))

You'll also probably want to set the range on the scale: app.setScaleRange("scroller", 0, 9)
And, register the scale to call the updateButtons() function, instead of  press
You'll also need to do the equivalent maths for the button presses, as they return their ID, not their value.

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}
\label{sec:introduction}
Particle methods, such as the smoothed particle hydrodynamics (SPH) \cite{gingold1977smoothed,liu2010smoothed,lucy1977numerical} and moving particle semi-implicit (MPS) \cite{koshizuka1995particle,koshizuka1996moving,shakibaeinia2012mps} methods, are numerical methods for solving partial differential equations that are based on points called particles distributed in a domain. In such methods, an interpolant and several approximate differential operators are defined in terms of linear combinations of weighted interactions between neighboring particles. When such methods are applied to partial differential equations, the equations are effectively discretized in space. As the discretization procedure does not require mesh generation in the domain, particle methods can be applied to moving boundary problems, such as the deformation and destruction of structures \cite{benz1995simulations,monaghan1991simulation} and flow problems associated with free surfaces \cite{monaghan1994simulating,murotani2014development}.

The accuracy of particle methods has been widely researched.
From an engineering perspective, many studies have been conducted into the convergence of such methods in practical applications, such as Amicarelli \cite{amicarelli2011sphintp,amicarelli2011sphgrad}, Fulk \cite{fulk1996analysis}, and Quinlan et al. \cite{quinlan2006truncation}.
On the other hand, few studies in the literature have presented numerical analyses of these methods from a mathematical perspective.
In the 1980s, Mas-Gallic and Ravirt \cite{mas1987particle} and Ravirt \cite{raviart1985analysis} provided error estimates for particle methods when applied to parabolic and hyperbolic partial differential equations on unbounded domains. In the 2000s, Moussa and Via \cite{moussa2000convergence} and Moussa \cite{moussa2006convergence} provided error estimates of nonlinear conservation lows on bounded domains. In their work, the time integrations of the particle positions and volumes were obtained by solving the differential equations with respect to advection fields. However, as their method is only applicable to problems described by solvable differential equations, it cannot be used with other problems, such as those involving the Navier--Stokes equations.

Sometime later, Ishijima and Kimura \cite{ishijima2010mashfreeE} developed a truncation error estimate for an approximate gradient operator in MPS. By introducing a regularity for particle distributions based on an indicator called the equivolume partition radius, they determined the conditions that depend solely on the space distributions of the particles. However, a practical limitation is that the indicator cannot be computed.

In previous works, we established truncation error estimates for an interpolant, approximate gradient operator, and approximate Laplace operator of a generalized particle method in which the particle volumes were given as Voronoi volumes \cite{imoto2016teintpV,imoto2017tedifV}. A generalized particle method is a numerical method that typically includes conventional particle methods, such as SPH and MPS. In previous studies, we derived truncation error estimates by introducing a regularity using an indicator known as the covering radius, which is used in the numerical analysis of meshfree methods based on moving least-square methods and radial basis functions \cite{levin1998errorMLS,schaback1995error,wendland2005scattered}.
Although the formulations and conditions in those works are computable, they are difficult to deploy in practical computations as the computational costs associated with particle volumes based on Voronoi decomposition are high.

The focus of the current work was to analyze particle methods under more practical conditions by extending our results to cases with commonly used particle volumes.
We also introduce another indicator of particle volumes, which we refer to as a Voronoi deviation, that represents the deviation between particle volumes and Voronoi volumes.
Then, utilizing the Voronoi deviation, we extend the regularity and introduce two hypotheses of reference weight functions.
Using the regularity and hypotheses, we derive truncation error estimates of the interpolant, approximate gradient operator, and approximate gradient operator of the generalized particle method.
Finally, we numerically analyze our estimates and compare the results to those from the theory.

The remainder of this paper is organized as follows.
The interpolant and approximate operators of the generalized particle method are introduced in Section \ref{sec:formulation}. A regularity describing the family of discrete parameters is discussed in Section \ref{sec:truncation_error_estimate}, after which we propose our primary theorem with respect to the truncation error estimates and provide some corollaries. Then, the primary theorem is proven in Section \ref{sec:Proof}, numerical results are detailed in Section \ref{sec:Numerical_results}, and some concluding remarks are outlined in Section \ref{sec:conclusions}.

In the remainder of this section, we describe some notation and define some relevant function spaces.
Let $\mathbb{R}^{+}$, $\mathbb{R}^{+}_0$, and $\mathbb{N}_0$ be the set of positive real numbers, the set of nonnegative real numbers, and the set of nonnegative integers, respectively.
Let $d$ be the dimension of a space.
Let $\dMI{d}$ be the set of all $d$-dimensional multi-indices.
If there is no ambiguity, the symbol $|\cdot|$ is used to denote the following: $|x|$ denotes the Euclidean norm for a vector $x\in\mathbb{R}^{\Dim}$; $|S|$ denotes the volume of $S$ for open set $S\subset\mathbb{R}^{\Dim}$; $|\alpha|$ denotes $|\alpha|\deq\alpha_1+\alpha_2+\dots+\alpha_d$ for $\alpha\in\dMI{d}$.
For $S\subset\mathbb{R}^{\Dim}$, let $\textrm{diam}(S)$ be $\textrm{diam}(S)\deq \sup\Set{|\xchar-\ychar|}{\xchar,\ychar\in S}$.
For $S\subset\mathbb{R}^{\Dim}$, let $\FsCz{\Ol{S}}$ be the space of real continuous functions defined in $\Ol{S}$ with the norm $\NormCz{\,\cdot\,}{\Ol{S}}$ defined as
\begin{align}
	\NormCz{\FuncCher}{\Ol{S}} &\deq \max_{\xchar\in\Ol{S}}\BrA{\FuncCher(\xchar)}.
\end{align}
For $S\subset\mathbb{R}^{\Dim}$ and $\ell\in\mathbb{N}$, let $\FsC{\ell}{\Ol{S}}$ be the space of functions in $\FsCz{\Ol{S}}$ with derivatives up to the $\ell$th order with its seminorm $\SNormC{\,\cdot\,}{\ell}{\Ol{S}}$ and norm $\NormC{\,\cdot\,}{\ell}{\Ol{S}}$ defined as
\begin{align}
	\BrA{\FuncCher}_{\FsC{\ell}{\Ol{S}}} &\deq \max_{\alpha\in\dMI{d}, |\alpha|=\ell}\BrN{D^\alpha \FuncCher}_{\FsCz{\Ol{S}}},
	\\
	\BrN{\FuncCher}_{\FsC{\ell}{\Ol{S}}} &\deq \max_{j=0,1,\dots,\ell} \SNormC{\FuncCher}{j}{\Ol{S}},
\end{align}
respectively.
Here $D^\alpha\FuncCher\deq\partial_1^{\alpha_1} \partial_2^{\alpha_2} \dots \partial_d^{\alpha_d}\FuncCher$ with multi-index $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_d)$.

\section{Approximate operators in a generalized particle method}
\label{sec:formulation}
Let $\Omega$ be a bounded domain in $\mathbb{R}^{\Dim}$.
Let $\H$ be a fixed positive number.
For $\Omega$ and $\H$, we define extended domain $\Dm_\H$ as
\begin{equation}
	\Dm_\H \deq \left\{\xchar\in\mathbb{R}^{\Dim} \;\middle|\; \exists \ychar\in\Omega \mbox{~s.t.~} |\xchar-\ychar|<\H\right\}.
\end{equation}
For $N\in\mathbb{N}$, we define a particle distribution $\PtSetCher_{\N}$ and particle volume set $\PvSetCher_{\N}$ as
\begin{equation}
	\PtSetCher_{\N} \deq \Set{\Pt{\i}\in\Dm_\H}{\i=1,2,\dots,N,\quad\Pt{\i}\neq\Pt{\j}\,(\i\neq\j)},
	\label{def:PtSet}
\end{equation}
\begin{equation}
	\PvSetCher_{\N} \deq \Set{\Pv{\i}\in\mathbb{R}^{+}}{\i=1,2,\dots,N,\quad\sum_{\i=1}^N\Pv{\i}=\Meas{\Dm_\H}},
	\label{def:PvSet}
\end{equation}
respectively.
We refer to $\Pt{\i}\in\PtSetCher_{\N}$ and $\Pv{\i}\in\PvSetCher_{\N}$ as a particle and particle volume, respectively.
An example of the particle distribution $\PtSetCher_{\N}$ in $\Dm_\H\,(\subset\mathbb{R}^2)$ is shown in Figure \ref{fig_particle_distribution}.

\begin{figure}[t]
	\includegraphics[width=55mm,bb=13 17 1837 1185]{fig_Domain.eps}
	\caption{Particle distribution $\PtSetCher_{\N}$ in $\Dm_\H\,(\subset\mathbb{R}^2)$.}
	\label{fig_particle_distribution}
\end{figure}

We define an admissible reference weight function set $\mathcal{W}$ as
\begin{equation}
	\mathcal{W} \deq \Set{\Weight\in\FsCz{\mathbb{R}^{+}_0}}{\supp(\Weight)=[0,1],\,\int_{\mathbb{R}^{\Dim}}\Weight(|\xchar|)\IntD\x=1, \mbox{absolutely continuous}},
\end{equation}
we refer to $\Weight\in\mathcal{W}$ as a reference weight function, and
we define the influence radius $\h_{\N}\in\mathbb{R}$ as satisfying $0<\h_{\N}<\H$ and $\h_{\N}\ra0\,(N\rightarrow\infty)$.
If there is no ambiguity, we denote $\h_{\N}$ as $h$.
For reference weight function $\Weight$ and influence radius $h$, we define the weight function $\WeightH\in\FsCz{\mathbb{R}^{+}_0}$ as
\begin{equation}
	\WeightH(r) \deq \frac{1}{h^{d}}\Weight\left(\frac{r}{h}\right).
	\label{def:wh}
\end{equation}
Note that the weight function $\WeightH$ satisfies
\begin{align}
	\supp(\WeightH) = [0,h], \qquad \int_{\mathbb{R}^{\Dim}}\WeightH(|\xchar|)\IntD\x=1,
\end{align}
and is absolutely continuous.

For $\FuncCher\in\FsCz{\Ol\Dm_{\H}}$, we define interpolant $\Pi_\h{}$, approximate gradient operator $\Grad_\h{}$, and approximate Laplace operator $\Delta_\h{}$ as
\begin{align}
	\Pi_\h{} \FuncCher (\xchar) &\deq \sum_{\i\in\IndexSetXH{\xchar}{h}}\Pv{\i} \FuncCher(\Pt{\i}) \WeightH(|\Pt{\i}-\xchar|),
	\label{ih_def}\\
	\Grad_\h{} \FuncCher (\xchar) &\deq d  \sum_{\i\in\IndexSetXHAst{\xchar}{h}}\Pv{\i} \frac{\FuncCher(\Pt{\i})-\FuncCher(\xchar)}{|\Pt{\i}-\xchar|}\frac{\Pt{\i}-\xchar}{|\Pt{\i}-\xchar|} \WeightH(|\Pt{\i}-\xchar|),
	\label{gh_def}\\
	\Delta_\h{} \FuncCher (\xchar) &\deq 2d\sum_{\i\in\IndexSetXHAst{\xchar}{h}}\Pv{\i} \frac{\FuncCher(\Pt{\i})-\FuncCher(\xchar)}{|\Pt{\i}-\xchar|^2} \WeightH(|\Pt{\i}-\xchar|),
	\label{lh_def}
\end{align}
respectively.
Here, for $\xchar\subset\mathbb{R}^{\Dim}$ and $r\in\mathbb{R}^{+}$, $\IndexSetXH{\xchar}{r}$ and $\IndexSetXHAst{\xchar}{r}$ are index sets of particles defined as
\begin{align}
	\IndexSetXH{\xchar}{r} &\deq \Set{\i=1,2,\dots,N}{0\leq|\xchar-\Pt{\i}|<r}, \\
	\IndexSetXHAst{\xchar}{r} &\deq \Set{\i=1,2,\dots,N}{0<|\xchar-\Pt{\i}|<r},
\end{align}
respectively.

As discussed later in Appendix \ref{sec:appendix_conventional_particle_methods}, the approximate operators \eqref{ih_def}, \eqref{gh_def}, and \eqref{lh_def} indicate a wider class of approximate operators of particle methods that those in the SPH and MPS methods.
Therefore, we refer to the approximate operators \eqref{ih_def}, \eqref{gh_def}, and \eqref{lh_def} as generalized approximate operators and to a particle method that uses them as a generalized particle method.

\section{Truncation error estimates of approximate operators}
\label{sec:truncation_error_estimate}
We first introduce a regularity of discrete parameters.
Let $\{\VoroCell{\i}\}$ be the Voronoi decomposition of $\Dm_\H$ associated with the particle distribution $\PtSetCher_{\N}$, where $\VoroCell{\i}$ is the Voronoi region defined as
\begin{align}
	\VoroCell{\i} \deq \Set{\xchar\in\Dm_\H}{|\Pt{\i}-\xchar|<|\Pt{\j}-\xchar|,~\forall \Pt{\j}\in\PtSetCher_{\N}\,(\j\neq \i) },\quad \i=1,2,\dots,N.
	\label{def_Voronoi_cell}
\end{align}
We define a particle volume decomposition $\Xi$ as a decomposition of $\Dm_\H$ associated with particle volume set $\PvSetCher_{\N}$ defined as
\begin{equation}
	\Xi\deq\Set{\PvDm{\i}\subset\Dm_\H}{\Meas{\PvDm{\i}}= \Pv{\i},\quad\bigcup_{\i=1}^N\PvDmOl{\i}=\Ol\Dm_{\H},\quad\PvDm{\i}\cap\PvDm{\j}=\emptyset\,(\i \neq \j)}.
\end{equation}
An example of the Voronoi decomposition of $\Dm_\H$ associated with the particle distribution $\PtSetCher_{\N}$ is shown in Figure \ref{fig:Voronoi_decomposition}.
\begin{figure}[t]
	\includegraphics[width=55mm,bb=0 0 653.7mm 420.9mm]{fig_Domain_Voronoi.eps}
	\caption{Example of the Voronoi decomposition of $\Dm_\H$ associated with the particle distribution $\PtSetCher_{\N}$. }
	\label{fig:Voronoi_decomposition}
\end{figure}
We define a covering radius $r_{\N}$ for particle distribution $\PtSetCher_{\N}$ as
\begin{align}
	r_{\N} \deq \max_{\i=1,2,\dots,N} \sup_{\xchar\in\VoroCell{\i}}|\Pt{\i}-\xchar|.
	\label{def:covering_radius}
\end{align}
Moreover, we define a Voronoi deviation $d_{\N}$ for the particle distribution $\PtSetCher_{\N}$ and the particle volume set $\PvSetCher_{\N}$ as
\begin{align}
	d_{\N}\deq \inf_{\Xi} \IndPvSetLocal{\Xi}
	\label{def:Voronoi_deviation}
\end{align}
with
\begin{equation}
	\IndPvSetLocal{\Xi} \deq \maxi{\i}\BrM{\sum_{\j=1}^N\dfrac{\Meas{\VoroCell{\i}\cap\PvDm{\j}}+\Meas{\PvDm{\i}\cap\VoroCell{\j}}}{\Meas{\VoroCell{\i}}}|\Pt{\i}-\Pt{\j}|}.
	\label{def:Voronoi_deviation_local}
\end{equation}
Then, we define a regularity for a family consisting of a particle distribution $\PtSetCher_{\N}$, particle volume set $\PvSetCher_{\N}$, and influence radius $h$ as follows:
\begin{definition}
	\label{def:regular}
	Family $\{(\PtSetCher_{\N}, \PvSetCher_{\N}, \h_{\N})\}_{N\rightarrow\infty}$ is said to be regular with order $m\,(m\geq 1)$ if there exists a positive constant $c_0$ such that
	\begin{align}
		\h_{\N}^{m} \geq c_0 (r_{\N}+d_{\N}),\qquad \forallN\in\mathbb{N}.
		\label{def:regular:cond}
	\end{align}
\end{definition}

\begin{remark}
	As shown in Figure \ref{fig:covering_radius}, the covering radius $r_{\N}$ becomes large in the case of a particle distribution with both dense and sparse regions.
	Therefore, the covering radius $r_{\N}$ can be considered as an indicator representing the uniformness of particle distribution $\PtSetCher_{\N}$.
\end{remark}
\begin{figure}[t]
	\includegraphics[width=120mm,bb=0 0 1551mm 559mm]{fig_covering_radius.eps}
	\caption{Two examples of covering radii $r_{\N}$ for particle distributions with same number of particles.
		The covering radius $r_{\N}$ for the uniform particle distribution \rmn{(}left\rmn{)} is smaller than that for the uneven particle distribution \rmn{(}right\rmn{)}. }
	\label{fig:covering_radius}
\end{figure}

\begin{remark}
	A Voronoi deviation $d_{\N}$ equals zero if and only if the particle volumes are given as the Voronoi volume ($\Pv{\i}=\Meas{\VoroCell{\i}}$).
	Moreover, the Voronoi deviation $d_{\N}$ becomes large if the particle volumes are given as values far from the Voronoi volumes.
	Therefore, the Voronoi deviation $d_{\N}$ can be regarded as an indicator of the deviation between the particle volume set and the Voronoi volume set.
\end{remark}

\begin{remark}
	For a given family $\{(\PtSetCher_{\N}, \PvSetCher_{\N}, \h_{\N})\}_{N\rightarrow\infty}$ and given constant $m$ $(m\geq1)$, it is possible to determine whether or not the family is regular with order $m$ as the covering radius $r_{\N}$ and Voronoi deviation $d_{\N}$ are absolutely computable, as shown in Appendix \rmn{\ref{sec:appendix_compt_procedure_indicator}}.
\end{remark}

Next, we introduce two hypotheses of reference weight function $\Weight$:
\begin{hypothesis}
	\label{hypo:ref_weight:order}
	For $n\in\mathbb{N}$, the reference weight function $\Weight$ satisfies for all $\alpha\in\dMI{d}$ with $1\leq|\alpha|\leq n$,
	\begin{align}
		\int_{\mathbb{R}^{\Dim}} \xchar^\alpha \Weight(|\xchar|)\IntD\x = 0.
		\label{w_hypo_order_eq_01}
	\end{align}
\end{hypothesis}
\begin{hypothesis}
	\label{hypo:ref_weight:smooth}
	For $k\in\mathbb{N}_0$, the reference weight function $\Weight$ satisfies
	\begin{align}
		\max\BrM{\sup_{r\in(0,1)}\BrANd{\wn{k+1}(r)}, \sup_{r\in(0,1)}\BrA{\dfrac{d}{dr}\wn{k}(r)}} < \infty,
		\label{w_hypo_smooth_01_eq}
	\end{align}
	where for $j\in\mathbb{N}_0$, $\wn{j}(r): (0,\infty)\rightarrow\mathbb{R}$ is defined as
	\begin{align}
		\wn{j}(r) \deq
		\begin{cases}
			\displaystyle \lim_{s\downarrow 0}\dfrac{\Weight(s)}{s^{j}},\qquad & r=0,
			\\
			\dfrac{\Weight(r)}{r^{j}},\qquad & r>0.
		\end{cases}
		\label{def:wn}
	\end{align}
\end{hypothesis}

\begin{remark}
	All reference functions $\Weight\in\mathcal{W}$ satisfy Hypothesis \ref{hypo:ref_weight:order} with $n=1$.
	Moreover, for all $n\in\mathbb{N}$ and $k\in\mathbb{N}$,  reference weight functions satisfying Hypothesis \ref{hypo:ref_weight:order} with $n$ and Hypothesis \ref{hypo:ref_weight:smooth} with $k$ can be constructed as shown in Appendix \ref{sec:appendix_const_ref_weight_func}.
\end{remark}

We now state a theorem that defines truncation error estimates of approximate operators in the generalized particle method with a continuous norm:
\begin{theorem}
	\label{thm:terror_cont_norm}
	Suppose that family $\{(\PtSetCher_{\N},\PvSetCher_{\N},\h_{\N})\}_{N\rightarrow\infty}$ is regular with order $m\,(m\geq1)$ and that reference weight function $\Weight$ satisfies Hypothesis \ref{hypo:ref_weight:order} with $n$.
	Then, there exists a positive constant $c$ independent of $N$ such that
	\begin{align}
		\NormCz{\FuncCher-\Pi_\h{} \FuncCher}{\Ol{\Dm}} & \leq c\, h^{\min\{m-1,n+1\}} \NormC{\FuncCher}{n+1}{\Ol\Dm_{\H}},\quad \FuncCher\in \FsC{n+1}{\Ol\Dm_{\H}}.
		\label{ineq:terror_cont_norm_intp}
	\end{align}
	In addition, if $\Weight\in\mathcal{W}$ satisfies Hypothesis \ref{hypo:ref_weight:smooth} with $k=0$, then we have
	\begin{align}
		\NormCz{\nabla\FuncCher-\Grad_\h{}\FuncCher}{\Ol{\Dm}} \leq c\, h^{\min\{m-1,n+1\}} \NormC{\FuncCher}{n+2}{\Ol\Dm_{\H}},\quad \FuncCher\in \FsC{n+2}{\Ol\Dm_{\H}},
		\label{ineq:terror_cont_norm_grad}
	\end{align}
	and if $\Weight\in\mathcal{W}$ satisfies Hypothesis \ref{hypo:ref_weight:smooth} with $k=1$, then we have
	\begin{align}
		\NormCz{\Delta\FuncCher-\Delta_\h{}\FuncCher}{\Ol{\Dm}} \leq c\, h^{\min\{m-2,n+1\}} \BrN{\FuncCher}_{\FsC{n+3}{\Ol\Dm_{\H}}},\quad \FuncCher\in \FsC{n+3}{\Ol\Dm_{\H}}.
		\label{ineq:terror_cont_norm_lap}
	\end{align}
\end{theorem}

The proof of Theorem \ref{thm:terror_cont_norm} is presented in the next section.
As shown in the corollaries in Appendix \ref{sec:appendix_conventional_particle_methods}, the approximate operators commonly used in the SPH and MPS methods are valid for Theorem \ref{thm:terror_cont_norm} under appropriate settings.

\section{Proof of truncation error estimates}
\label{sec:Proof}
The following notation will be used in the subsequent proof of Theorem \ref{thm:terror_cont_norm}.
Hereafter, let $c$ be a generic positive constant independent of $N$.
For $\alpha\in\dMI{d}$, set $\errorfuncA{\alpha}$ as
\begin{equation}
	\errorfuncA{\alpha}(\xchar) \deq \sum_{\i\in\IndexSetXH{\xchar}{h}}\Pv{\i} (\Pt{\i}-\xchar)^{\alpha} \WeightH(|\Pt{\i}-\xchar|) - \int_{\mathbb{R}^{\Dim}}\ychar^\alpha \WeightH(|\ychar|)\IntD\y,\qquad \xchar\in\Ol{\Dm}.
\end{equation}
For $\alpha\in\dMI{d}$ and $\ell\in\mathbb{N}$, set $\errorfuncB{\alpha}{\ell}$ as
\begin{equation}
	\errorfuncB{\alpha}{\ell}(\xchar) \deq \sum_{\i\in\IndexSetXHAst{\xchar}{h}} \Pv{\i} \frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell}} \WeightH(|\Pt{\i}-\xchar|) - \int_{\mathbb{R}^{\Dim}} \frac{\ychar^\alpha}{|\ychar|^\ell} \WeightH(|\ychar|)\IntD\y,\quad  \xchar\in\Ol{\Dm}.
\end{equation}
For $\ell\in\mathbb{N}$, set $\errorfuncC{\ell}$ as
\begin{equation}
	\errorfuncC{\ell}(\xchar) \deq \sum_{\i\in\IndexSetXH{\xchar}{h}} \Pv{\i} |\Pt{\i}-\xchar|^{\ell} |\WeightH(|\Pt{\i}-\xchar|)|,\qquad\xchar\in\Ol{\Dm}.
\end{equation}
We now present the following lemma.

\begin{lemma}
	\label{lem:terror_cont_norm}
	Suppose that $\Weight\in\mathcal{W}$ satisfies Hypothesis \ref{hypo:ref_weight:order} with $n$.
	Then, there exists a positive constant $c$ independent of $N$ such that
	\begin{equation}
		\NormCz{\FuncCher-\Pi_\h{} \FuncCher}{\Ol{\Dm}}
		\leq c\BrS{\sum_{1\leq|\alpha|\leqn}\NormCz{\errorfuncA{\alpha}}{\Ol{\Dm}}+ \NormCz{\errorfuncC{n+1}}{\Ol{\Dm}}}\NormC{\FuncCher}{n+1}{\Ol\Dm_{\H}},\quad\FuncCher\in\FsC{n+1}{\Ol\Dm_{\H}},
		\label{lem:terror_cont_norm:intp}
	\end{equation}
	\begin{equation}
		\NormCz{\nabla\FuncCher-\Grad_\h{}\FuncCher}{\Ol{\Dm}}
		\leqc\BrS{\sum_{2\leq |\alpha|\leqn+2} \NormCz{\errorfuncB{\alpha}{2}}{\Ol{\Dm}}+ \NormCz{\errorfuncC{n+1}}{\Ol{\Dm}}}\NormC{\FuncCher}{n+2}{\Ol\Dm_{\H}},\quad\FuncCher\in\FsC{n+2}{\Ol\Dm_{\H}},
		\label{lem:terror_cont_norm:grad}
	\end{equation}
	\begin{equation}
		\NormCz{\Delta\FuncCher-\Delta_\h{}\FuncCher}{\Ol{\Dm}}
		\leq c\BrS{\sum_{1\leq |\alpha|\leqn+3} \NormCz{\errorfuncB{\alpha}{2}}{\Ol{\Dm}}+ \NormCz{\errorfuncC{n+1}}{\Ol{\Dm}}}\NormC{\FuncCher}{n+3}{\Ol\Dm_{\H}},\quad\FuncCher\in\FsC{n+3}{\Ol\Dm_{\H}}.
		\label{lem:terror_cont_norm:lap}
	\end{equation}
\end{lemma}

\begin{proof}
	First, we prove \eqref{lem:terror_cont_norm:intp}.
	We fix $\xchar\in\Ol{\Dm}$.
	Then, let $\OBall{\xchar}{r}$ be the open ball in $\mathbb{R}^{\Dim}$ with center $\xchar$ and radius $r$, i.e.,
	\begin{equation}
		\OBall{\xchar}{r}\deq \Set{\ychar\in\mathbb{R}^{\Dim}}{|\xchar-\ychar|<r}.
	\end{equation}
	From $h<\H$, we have $\OBall{\xchar}{h}\subset\Dm_\H$.
	Then, for all $\FuncCher\in\FsC{\ell+1}{\Ol\Dm_{\H}}$ $(\ell\in\mathbb{N})$ and $\ychar\in\OBall{\xchar}{h}$, we obtain the Taylor expansion of $\FuncCher$ as
	\begin{align}
		& \FuncCher(\Pt{\i}) = \sum_{0\leq |\alpha|\leq\ell} \frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!}(\xchar-\Pt{\i})^{\alpha} + \sum_{|\alpha|=n+1}(\xchar-\Pt{\i})^{\alpha} R_{\alpha}(\Pt{\i},\xchar),
		\label{taylor01}\\
		& R_{\alpha}(\Pt{\i},\xchar) \deq \frac{|\alpha|}{\alpha !} \int_0^1 (1-t)^{|\alpha|-1}D^{\alpha}\FuncCher(t\Pt{\i}+(1-t)\xchar)dt.
		\label{regidual01}
	\end{align}
	From \eqref{ih_def} and \eqref{taylor01} with $\ell=n+1$, we have
	\begin{align}
		\Pi_\h{} \FuncCher(\xchar)
		& = \sum_{0\leq |\alpha|\leqn} \frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!} \sum_{\i\in\IndexSetXH{\xchar}{h}} \Pv{\i} (\xchar-\Pt{\i})^{\alpha} \WeightH(|\xchar-\Pt{\i}|)
		\nonumber\\
		&\quad+\sum_{|\alpha|=n+1}\sum_{\i\in\IndexSetXH{\xchar}{h}}R_{\alpha}(\Pt{\i},\xchar)\Pv{\i} (\xchar-\Pt{\i})^{\alpha} \WeightH(|\xchar-\Pt{\i}|).
	\end{align}
	Moreover, by Hypothesis \ref{hypo:ref_weight:order}, we have
	\begin{align}
		\Pi_\h{} \FuncCher(\xchar) - \FuncCher(\xchar)
		& = \sum_{0\leq |\alpha|\leqn} \frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!} \errorfuncA{\alpha}(\xchar)
		\nonumber\\
		&\quad+\sum_{|\alpha|=n+1}\sum_{\i\in\IndexSetXH{\xchar}{h}}R_{\alpha}(\Pt{\i},\xchar)\Pv{\i} (\xchar-\Pt{\i})^{\alpha} \WeightH(|\xchar-\Pt{\i}|).
		\label{prf:lem:terror_cont_norm:intp01}
	\end{align}
	Because
	\begin{align}
		|R_{\alpha}(\ychar,\zchar)| \leq \frac{1}{\alpha!}\SNormC{\FuncCher}{|\alpha|}{\Ol\Dm_{\H}}, \qquad  \ychar\in\Ol{\Dm},~\zchar\in\OBall{\ychar}{h},~\alpha\in\dMI{d},
		\label{ineq:R_a[f](y_x)}
	\end{align}
	we have
	\begin{equation}
		\BrA{\sum_{|\alpha|=n+1}\sum_{\i\in\IndexSetXH{\xchar}{h}}R_{\alpha}(\Pt{\i},\xchar)\Pv{\i} (\xchar-\Pt{\i})^{\alpha} \WeightH(|\xchar-\Pt{\i}|)}
		\leqc|\errorfuncC{n+1}(\xchar)|\SNormC{\FuncCher}{n+1}{\Ol\Dm_{\H}}.
		\label{eq:int_y^a_wh(|y|)dy}
	\end{equation}
	Moreover, we have
	\begin{align}
		\BrA{\sum_{0\leq |\alpha|\leqn} \frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!} \errorfuncA{\alpha}(\xchar)}
		&\leqc\NormC{\FuncCher}{n}{\Ol{\Dm}}\sum_{1\leq|\alpha|\leqn}\BrA{\errorfuncA{\alpha}(\xchar)}.
		\label{ineq:Ja_intp}
	\end{align}
	Therefore, from \eqref{prf:lem:terror_cont_norm:intp01}, \eqref{eq:int_y^a_wh(|y|)dy}, and \eqref{ineq:Ja_intp}, we obtain \eqref{lem:terror_cont_norm:intp}.

	Next, we prove \eqref{lem:terror_cont_norm:grad}.
	From \eqref{gh_def} and \eqref{taylor01} with $\ell=n+2$, we have
	\begin{align}
		\Grad_\h{} \FuncCher(\xchar)
		&=d\sum_{1\leq|\alpha|\leqn+1}\frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!}\sum_{\i\in\IndexSetXHAst{\xchar}{h}}\Pv{\i}\frac{(\xchar-\Pt{\i})(\xchar-\Pt{\i})^{\alpha}}{|\xchar-\Pt{\i}|^2} \WeightH(|\xchar-\Pt{\i}|)
		\nonumber\\
		&\quad+d\sum_{|\alpha|=n+2}\sum_{\i\in\IndexSetXHAst{\xchar}{h}}R_{\alpha}(\Pt{\i},\xchar)\Pv{\i}\frac{(\xchar-\Pt{\i})(\xchar-\Pt{\i})^{\alpha}}{|\xchar-\Pt{\i}|^2}\WeightH(|\xchar-\Pt{\i}|).
	\end{align}
	Because for $\beta\in\dMI{d}$ with $|\beta|=2$,
	\begin{align}
		d \int_{\mathbb{R}^{\Dim}} \frac{\ychar^{\beta}}{|\ychar|^2}\WeightH(|\ychar|)\IntD\y =
		\begin{cases}
			1,\quad & \mbox{all elements of $\beta$ are even},
			\\
			0,\quad & \mbox{otherwise},
		\end{cases}
		\label{eq:d_int_y^b_y^-2dy}
	\end{align}
	we have
	\begin{align}
		d\sum_{|\alpha|=1}\frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!}\int_{\mathbb{R}^{\Dim}} \frac{y\ychar^\alpha}{|\ychar|^2} \WeightH(|\ychar|)\IntD\y = \nabla\FuncCher(\xchar).
		\label{eq:grad_rhs1}
	\end{align}
	Hypothesis \ref{hypo:ref_weight:order} with $n$ yields
	\begin{align}
		\int_{\mathbb{R}^{\Dim}} \frac{\ychar \ychar^\alpha}{|\ychar|^2} \WeightH(|\ychar|)\IntD\y = 0\qquad\mbox{$\alpha\in\dMI{d}$ with $2\leq|\alpha|\leqn+1$}.
		\label{eq:int_y^a_y^-2_wh_dy=0}
	\end{align}
	From \eqref{eq:grad_rhs1} and \eqref{eq:int_y^a_y^-2_wh_dy=0}, we have
	\begin{align}
		\Grad_\h{}\FuncCher(\xchar)-\nabla\FuncCher(\xchar)&=-d\sum_{1\leq|\alpha|\leqn+1}\frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!}\int_{\mathbb{R}^{\Dim}} \frac{\ychar \ychar^\alpha}{|\ychar|^2} \WeightH(|\ychar|)\IntD\y
		\\
		&\quad+d\sum_{1\leq|\alpha|\leqn+1}\frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!}\sum_{\i\in\IndexSetXHAst{\xchar}{h}}\Pv{\i}\frac{(\xchar-\Pt{\i})(\xchar-\Pt{\i})^{\alpha}}{|\xchar-\Pt{\i}|^2} \WeightH(|\xchar-\Pt{\i}|)
		\\
		&\quad+d\sum_{|\alpha|=n+2}\sum_{\i\in\IndexSetXHAst{\xchar}{h}}R_{\alpha}(\Pt{\i},\xchar) \Pv{\i}\frac{(\xchar-\Pt{\i})(\xchar-\Pt{\i})^{\alpha}}{|\xchar-\Pt{\i}|^2}\WeightH(|\xchar-\Pt{\i}|).
		\label{prf:lem:terror_cont_norm:grad01}
	\end{align}
	From \eqref{ineq:R_a[f](y_x)}, we have
	\begin{equation}
		\BrA{\sum_{|\alpha|=n+2}\sum_{\i\in\IndexSetXHAst{\xchar}{h}}R_{\alpha}(\Pt{\i},\xchar)\Pv{\i}\frac{(\xchar-\Pt{\i})(\xchar-\Pt{\i})^{\alpha}}{|\xchar-\Pt{\i}|^2} \WeightH(|\xchar-\Pt{\i}|)}
		\leqc|\errorfuncC{n+1}(\xchar)|\SNormC{\FuncCher}{n+2}{\Ol\Dm_{\H}}.
		\label{ineq:R_grad}
	\end{equation}
	Moreover, we have
	\begin{equation}
		\sum_{1\leq|\alpha|\leqn+1}\BrA{\sum_{\i\in\IndexSetXHAst{\xchar}{h}}\Pv{\i}\frac{(\xchar-\Pt{\i})(\xchar-\Pt{\i})^{\alpha}}{|\xchar-\Pt{\i}|^2}\WeightH(|\xchar-\Pt{\i}|)-\int_{\mathbb{R}^{\Dim}}\frac{\ychar\y^\alpha}{|\ychar|^2}\WeightH(|\ychar|)\IntD\y}\leqc\sum_{2\leq |\alpha|\leqn+2} \BrANd{\errorfuncB{\alpha}{2}(\xchar)}.
		\label{ineq:Ja2_grad}
	\end{equation}
	Therefore, from \eqref{prf:lem:terror_cont_norm:grad01}, \eqref{ineq:R_grad}, and \eqref{ineq:Ja2_grad}, we obtain \eqref{lem:terror_cont_norm:grad}.

	Finally, we prove \eqref{lem:terror_cont_norm:lap}.
	From \eqref{lh_def} and \eqref{taylor01} with $\ell=n+3$, we have
	\begin{align}
		\Delta_\h{} \FuncCher(\xchar)
		& = 2d \sum_{1\leq |\alpha|\leqn+2} \frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!} \sum_{\i\in\IndexSetXHAst{\xchar}{h}} \Pv{\i} \frac{(\xchar-\Pt{\i})^{\alpha}}{|\xchar-\Pt{\i}|^2} \WeightH(|\xchar-\Pt{\i}|)
		\nonumber\\
		& \quad + 2d \sum_{|\alpha|=n+3}\sum_{\i\in\IndexSetXHAst{\xchar}{h}}R_{\alpha}(\Pt{\i},\xchar)\Pv{\i} \frac{(\xchar-\Pt{\i})^{\alpha}}{|\xchar-\Pt{\i}|^2} \WeightH(|\xchar-\Pt{\i}|).
	\end{align}
	From \eqref{eq:d_int_y^b_y^-2dy}, we have
	\begin{align}
		2d\sum_{|\alpha|=2}\frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!} \int_{\mathbb{R}^{\Dim}}\frac{\ychar^\alpha}{|\ychar|^2} \WeightH(|\ychar|)\IntD\y = \Delta\FuncCher(\xchar).
	\end{align}
	Hypothesis \ref{hypo:ref_weight:order} with $n$ yields
	\begin{align}
		\int_{\mathbb{R}^{\Dim}} \frac{\ychar^\alpha}{|\ychar|^2} \WeightH(|\ychar|)\IntD\y = 0,\qquad\mbox{$\alpha\in\dMI{d}$ with $|\alpha|=1$ or $ 3\leq|\alpha|\leqn+2$}.
	\end{align}
	Therefore, we have
	\begin{align}
		\Delta\FuncCher(\xchar) - \Delta_\h{}\FuncCher(\xchar)
		&=2d\sum_{1\leq|\alpha|\leqn+2} \frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!} \errorfuncB{\alpha}{2}(\xchar)
		\\
		&\quad+ 2d\sum_{|\alpha|=n+3}\sum_{\i\in\IndexSetXHAst{\xchar}{h}}R_{\alpha}(\Pt{\i},\xchar)\Pv{\i} \frac{(\xchar-\Pt{\i})^{\alpha}}{|\xchar-\Pt{\i}|^2}\WeightH(|\xchar-\Pt{\i}|).
		\label{prf:lem:estimate:lap:01}
	\end{align}
	From \eqref{ineq:R_a[f](y_x)}, we have
	\begin{equation}
		\BrA{\sum_{|\alpha|=n+3}\sum_{\i\in\IndexSetXHAst{\xchar}{h}}R_{\alpha}(\Pt{\i},\xchar)\Pv{\i} \frac{(\xchar-\Pt{\i})^{\alpha}}{|\xchar-\Pt{\i}|^2}\WeightH(|\xchar-\Pt{\i}|)}
		\leqc|\errorfuncC{n+1}(\xchar)|\SNormC{\FuncCher}{n+3}{\Ol\Dm_{\H}}.
		\label{eq:int_y^a_wh(|y|)dy_lap}
	\end{equation}
	Moreover, we have
	\begin{align}
		\BrA{\sum_{1\leq |\alpha|\leqn+2} \frac{D^{\alpha}\FuncCher(\xchar)}{\alpha!} \errorfuncB{\alpha}{2}(\xchar)}
		&\leqc\NormC{\FuncCher}{n+2}{\Ol{\Dm}}\sum_{1\leq|\alpha|\leqn+2}\BrA{\errorfuncB{\alpha}{2}(\xchar)}.
		\label{ineq:Ja2_lap}
	\end{align}
	Therefore, from \eqref{prf:lem:estimate:lap:01}, \eqref{eq:int_y^a_wh(|y|)dy_lap}, and \eqref{ineq:Ja2_lap}, we obtain \eqref{lem:terror_cont_norm:lap}.
\end{proof}

Next, we show estimates of $\errorfuncA{\alpha}$, $\errorfuncB{\alpha}{\ell}$, and $\errorfuncC{\ell}$.

\begin{lemma}
	\label{lem:Ja}
	There exists a positive constant $c$ independent of $N$ such that
	\begin{align}
		\BrNNd{\errorfuncA{\alpha}}_{\FsCz{\Ol{\Dm}}} \leq c \BrS{1+2\frac{r_{\N}}{h}}^{d}\BrS{\frac{r_{\N}+d_{\N}}{h}},\qquad \alpha\in\dMI{d}.
		\label{lem:ineq:Ja}
	\end{align}
\end{lemma}

\begin{proof}
	We arbitrarily fix $\xchar\in\Ol{\Dm}$, $\alpha\in\dMI{d}$, and particle volume decomposition $\Xi=\{\PvDm{\i}\mid\i=1,2,\dots,N\}$ and split $\errorfuncA{\alpha}$ into
	\begin{equation}
		\errorfuncA{\alpha}(\xchar) = \prffuncA{1}(\xchar) + \prffuncA{2}(\xchar) + \prffuncA{3}(\xchar)
	\end{equation}
	with
	\begin{align}
		\prffuncA{1}(\xchar) &\deq \sum_{\i\in\IndexSetXH{\xchar}{h}}\Pv{\i} (\Pt{\i}-\xchar)^{\alpha}\WeightH(|\xchar-\Pt{\i}|)-\sum_{\i=1}^{N}\sum_{\j=1}^{N} \Meas{\VoroCell{\j}\cap\PvDm{\i}} (\Pt{\i}-\xchar)^{\alpha}\WeightH(|\xchar-\Pt{\j}|),
		\\
		\prffuncA{2}(\xchar) &\deq \sum_{\i=1}^{N}\sum_{\j=1}^{N}(\Pt{\i}-\xchar)^{\alpha}\int_{\VoroCell{\j}\cap\PvDm{\i}}\{\WeightH(|\xchar-\Pt{\j}|)-\WeightH(|\xchar-\ychar|)\}\IntD\y,
		\\
		\prffuncA{3}(\xchar) &\deq\sum_{\i=1}^{N}\sum_{\j=1}^{N}(\Pt{\i}-\xchar)^{\alpha}\int_{\VoroCell{\j}\cap\PvDm{\i}}\WeightH(|\xchar-\ychar|)\IntD\y-\int_{\mathbb{R}^{\Dim}}\ychar^\alpha \WeightH(|\ychar|)\IntD\y.
	\end{align}
	Then, we estimate $\prffuncA{1}$, $\prffuncA{2}$, and $\prffuncA{3}$.

	First, we estimate $\prffuncA{1}$.
	Because
	\begin{equation}
		\sum_{\j=1}^{N} \Meas{\VoroCell{\j}\cap\PvDm{\i}}=\Pv{\i},\qquad \i=1,2,\dots,N,
		\label{sum_|VoroCell_PvDm|}
	\end{equation}
	we can rewrite $\prffuncA{1}$ as
	\begin{equation}
		\prffuncA{1}=\sum_{\i=1}^{N}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}} (\Pt{\i}-\xchar)^{\alpha}\{\WeightH(|\xchar-\Pt{\i}|)-\WeightH(|\xchar-\Pt{\j}|)\}.
	\end{equation}
	From
	\begin{equation}
		|(\ychar-\xchar)^{\alpha}| \leq \textrm{diam}(\Dm_\H)^{|\alpha|}, \qquad\ychar\in\Dm_\H,
		\label{y-x_leq_diam}
	\end{equation}
	we obtain
	\begin{align}
		\BrA{\prffuncA{1}(\xchar)} &\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}|\WeightH(|\xchar-\Pt{\i}|)-\WeightH(|\xchar-\Pt{\j}|)|.
		\label{prf:estimate:E1:01}
	\end{align}
	From
	\begin{equation}
		|\WeightH(|\xchar-\ychar|)-\WeightH(|\xchar-\zchar|)| = 0, \qquad \forall\ychar, \zchar\in\mathbb{R}^{\Dim}\setminus\OBall{\xchar}{h},
		\label{eq:|wi-wj|=0}
	\end{equation}
	we have
	\begin{multline}
		\sum_{\i=1}^{N}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}|\WeightH(|\xchar-\Pt{\i}|)-\WeightH(|\xchar-\Pt{\j}|)|\\
		\leq\sum_{\i\in\IndexSetXH{\xchar}{h}}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}|\WeightH(|\xchar-\Pt{\i}|)-\WeightH(|\xchar-\Pt{\j}|)|
		+\sum_{\i=1}^{N}\sum_{\j\in\IndexSetXH{\xchar}{h}}\Meas{\VoroCell{\j}\cap\PvDm{\i}}|\WeightH(|\xchar-\Pt{\i}|)-\WeightH(|\xchar-\Pt{\j}|)|\\
		=\sum_{\i\in\IndexSetXH{\xchar}{h}}\sum_{\j=1}^{N}(\Meas{\VoroCell{\i}\cap\PvDm{\j}}+\Meas{\VoroCell{\j}\cap\PvDm{\i}})|\WeightH(|\xchar-\Pt{\i}|)-\WeightH(|\xchar-\Pt{\j}|)|.
		\label{prf:E1_estimate:lem:01}
	\end{multline}
	Because $\WeightH$ is absolutely continuous, we have
	\begin{align}
		\BrA{\WeightH(|\xchar-\ychar|)-\WeightH(|\xchar-\zchar|)}
		&= \BrA{\{(\xchar-\ychar)-(\xchar-\zchar)\}\int_0^1\dfrac{d}{dr}\WeightH(t |\xchar-\ychar|+(1-t)|\xchar-\zchar|)dt}
		\nonumber\\
		&\leq |\ychar-\zchar| \BrA{  \int_0^1\dfrac{d}{dr}\WeightH(t |\xchar-\ychar|+(1-t)|\xchar-\zchar|)dt}
		\nonumber\\
		&\leq |\ychar-\zchar|  \int_0^h\BrA{\dfrac{d}{dr}\WeightH(r)}dr
		\nonumber\\
		&\leq \dfrac{|\ychar-\zchar|}{h^{d+1}}\int_0^1\BrA{\dfrac{d}{dr}\Weight(r)}dr,
		\label{ineq:|e_whx|}
	\end{align}
	for all $\ychar, \zchar\in\mathbb{R}^{\Dim}$.
	Moreover, we have
	\begin{align}
		\sum_{\i\in\IndexSetXH{\xchar}{r}}|\VoroCell{\i}|\leq\Meas{\OBall{\xchar}{1}}\BrS{r+r_{\N}}^{d},\qquad \forall r\in\mathbb{R}^{+}_0.
		\label{Sum_VoroCell_leq}
	\end{align}
	From \eqref{prf:E1_estimate:lem:01}, \eqref{ineq:|e_whx|}, and \eqref{Sum_VoroCell_leq}, we have
	\begin{align}
		\sum_{\i=1}^{N}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}|\WeightH(|\xchar-\Pt{\i}|)-\WeightH(|\xchar-\Pt{\j}|)|&\leq\dfrac{c}{h^{d+1}}\sum_{\i\in\IndexSetXH{\xchar}{h}}\sum_{\j=1}^{N}(\Meas{\VoroCell{\i}\cap\PvDm{\j}}+\Meas{\VoroCell{\j}\cap\PvDm{\i}})|\Pt{\i}-\Pt{\j}|
		\nonumber\\
		&\leq\dfrac{c}{h^{d+1}}\sum_{\i\in\IndexSetXH{\xchar}{h}} |\VoroCell{\i}|\sum_{\j=1}^{N} \frac{\Meas{\VoroCell{\i}\cap\PvDm{\j}}+\Meas{\VoroCell{\j}\cap\PvDm{\i}}}{|\VoroCell{\i}|}|\Pt{\i}-\Pt{\j}|
		\nonumber\\
		&\leqc\dfrac{\IndPvSetLocal{\Xi}}{h^{d+1}}\sum_{\i\in\IndexSetXH{\xchar}{h}}|\VoroCell{\i}|
		\nonumber\\
		&\leqc\BrS{1+\frac{r_{\N}}{h}}^{d} \frac{\IndPvSetLocal{\Xi}}{h}.
		\label{prf:estimate:E1:02}
	\end{align}
	Therefore, from \eqref{prf:estimate:E1:01} and \eqref{prf:estimate:E1:02}, we obtain
	\begin{align}
		\BrA{\prffuncA{1}(\xchar)}
		& \leqc\BrS{1+\frac{r_{\N}}{h}}^{d} \frac{\IndPvSetLocal{\Xi}}{h}.
		\label{prf:E1_estimate}
	\end{align}

	Next, we estimate $\prffuncA{2}$.
	Because $\supp(\WeightH) = [0,h]$ and $\VoroCell{\j}\subset\OBall{\Pt{\j}}{r_{\N}}$, we have
	\begin{equation}
		\int_{\VoroCell{\j}\cap\PvDm{\i}}|\WeightH(|\xchar-\Pt{\j}|)-\WeightH(|\xchar-\ychar|)|\IntD\y=0,\qquad\i=1,2,\dots,N,~\j\not\in\IndexSetXH{\xchar}{h+r_{\N}}.
		\label{prf:E2_estimate:01}
	\end{equation}
	From \eqref{prf:E2_estimate:01}, we have
	\renewcommand{\quad}{\quad}
	\begin{align}
		\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}|\WeightH(|\xchar-\Pt{\j}|)-\WeightH(|\xchar-\ychar|)|\IntD\y&=\sum_{\i=1}^{N}\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\int_{\VoroCell{\j}\cap\PvDm{\i}}|\WeightH(|\xchar-\Pt{\j}|)-\WeightH(|\xchar-\ychar|)|\IntD\y\nonumber\\
		&=\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\int_{\VoroCell{\j}}|\WeightH(|\xchar-\Pt{\j}|)-\WeightH(|\xchar-\ychar|)|\IntD\y.
	\end{align}
	Moreover, from \eqref{ineq:|e_whx|} and \eqref{Sum_VoroCell_leq}, we have
	\begin{align}
		\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}|\WeightH(|\xchar-\Pt{\j}|)-\WeightH(|\xchar-\ychar|)|\IntD\y
		&\leq\dfrac{c}{h^{d+1}}\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\int_{\VoroCell{\j}}|\Pt{\j}-\ychar|\IntD\y
		\nonumber\\
		&\leq\dfrac{c}{h^{d+1}}\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\int_{\VoroCell{\j}}|\Pt{\j}-\ychar|\IntD\y
		\nonumber\\
		&\leqc\dfrac{r_{\N}}{h^{d+1}}\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}|\VoroCell{\j}|
		\nonumber\\
		&\leqc\BrS{1+2\dfrac{r_{\N}}{h}}^{d}\dfrac{r_{\N}}{h}.
		\label{prf:estimate:E2:01}
	\end{align}
	Therefore, from \eqref{y-x_leq_diam} and \eqref{prf:estimate:E2:01}, we obtain
	\begin{align}
		|\prffuncA{2}(\xchar)|
		&\leq\sum_{\i=1}^{N}\sum_{\j=1}^{N}\BrA{(\Pt{\i}-\xchar)^{\alpha}}\int_{\VoroCell{\j}\cap\PvDm{\i}}\BrA{\WeightH(|\xchar-\Pt{\j}|)-\WeightH(|\xchar-\ychar|)}\IntD\y\nonumber\\
		&\leqc \sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}\BrA{\WeightH(|\xchar-\Pt{\j}|)-\WeightH(|\xchar-\ychar|)}\IntD\y\nonumber\\
		&\leqc\BrS{1+2\dfrac{r_{\N}}{h}}^{d}\dfrac{r_{\N}}{h}.
	\end{align}

	Finally, we estimate $\prffuncA{3}$.
	Because
	\begin{equation}
		\int_{\mathbb{R}^{\Dim}}\ychar^\alpha \WeightH(|\ychar|)\IntD\y=\int_{\Dm_\H}(\ychar-\xchar)^\alpha \WeightH(|\xchar-\ychar|)\IntD\y,
	\end{equation}
	we can rewrite $\prffuncA{3}$ as
	\begin{align}
		\prffuncA{3}(\xchar) &=\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}\{(\Pt{\i}-\xchar)^{\alpha}-(\ychar-\xchar)^\alpha\}\WeightH(|\xchar-\ychar|)\IntD\y.
	\end{align}
	Because $\prffuncA{3}=0$ when $|\alpha|=0$, we estimate when $|\alpha|\geq1$.
	Let $\beta_k\,(k=1,2,\dots,|\alpha|)$ be $d$-dimensional multi-indices with satisfying
	\begin{equation}
		\sum_{k=1}^{|\alpha|}\beta_k=\alpha,\qquad|\beta_k|=1~(k=1,2,\dots,|\alpha|).
	\end{equation}
	Then, we have, for all $y,z\in\mathbb{R}^{\Dim}$,
	\begin{align}
		\BrA{\ychar^{\alpha}-\zchar^{\alpha}}
		& \leq \BrA{\ychar^{\alpha}-\ychar^{\alpha-\beta_1}\zchar^{\beta_1}} + \BrA{\ychar^{\alpha-\beta_1}\zchar^{\beta_1}-\zchar^{\alpha}}
		\nonumber\\
		& \leq \BrA{\ychar-\zchar} |\ychar|^{|\alpha|-1}+ \BrA{\ychar^{\alpha-\beta_1}-\zchar^{\alpha-\beta_1}}|\zchar|
		\nonumber\\
		& \leq \BrA{\ychar-\zchar} |\ychar|^{|\alpha|-1}+\BrA{\ychar-\zchar} |\ychar|^{|\alpha|-2}|\zchar|+ \BrA{\ychar^{\alpha-\beta_1-\beta_2}-\zchar^{\alpha-\beta_1-\beta_2}}|\zchar|^2
		\nonumber\\
		&~\vdots
		\nonumber\\
		&\leq\BrA{\ychar-\zchar}\sum_{k=1}^{|\alpha|}|\ychar|^{|\alpha|-k}|\zchar|^{k-1}.
		\label{ineq:|y^alpha-z^alpha|}
	\end{align}
	From \eqref{y-x_leq_diam} and \eqref{ineq:|y^alpha-z^alpha|}, we obtain
	\begin{align}
		|\prffuncA{3}(\xchar)|
		&\leq\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}|(\Pt{\i}-\xchar)^{\alpha}-(\ychar-\xchar)^\alpha||\WeightH(|\xchar-\ychar|)|\IntD\y\nonumber\\
		&\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}|\ychar-\Pt{\i}||\WeightH(|\xchar-\ychar|)|\IntD\y.
		\label{prf:estimate:E3:01}
	\end{align}
	By $\supp(\WeightH) = [0,h]$ and $\VoroCell{\j}\subset\OBall{\Pt{\j}}{r_{\N}}$, if $\j\not\in\IndexSetXH{\xchar}{h+r_{\N}}$, then
	\begin{equation}
		\int_{\VoroCell{\j}\cap\PvDm{\i}}|\ychar-\Pt{\i}||\WeightH(|\xchar-\ychar|)|\IntD\y=0,\qquad\i=1,2,\dots,N.
		\label{prf:estimate:E3:02}
	\end{equation}
	Moreover, from $\Weight\in\mathcal{W}\subset\FsCz{\mathbb{R}^{+}_0}$, we have
	\begin{equation}
		|\WeightH(|\xchar-\ychar|)|=\dfrac{1}{h^d}\BrA{\Weight\BrS{\dfrac{|\xchar-\ychar|}{h}}}\leq\dfrac{1}{h^d}\NormCz{\Weight}{\mathbb{R}^{+}_0},\qquad \forall \ychar\in\Dm_\H.
		\label{prf:estimate:E3:03}
	\end{equation}
	From \eqref{Sum_VoroCell_leq}, \eqref{prf:estimate:E3:02}, and \eqref{prf:estimate:E3:03}, we have
	\begin{align}
		&\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}|\ychar-\Pt{\i}||\WeightH(|\xchar-\ychar|)|\IntD\y
		\nonumber\\
		&\quad=\sum_{\i=1}^{N}\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\int_{\VoroCell{\j}\cap\PvDm{\i}}|\ychar-\Pt{\i}||\WeightH(|\xchar-\ychar|)|\IntD\y
		\nonumber\\
		&\quad\leq\dfrac{c}{h^d}\sum_{\i=1}^{N}\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\int_{\VoroCell{\j}\cap\PvDm{\i}}|\ychar-\Pt{\i}|\IntD\y
		\nonumber\\
		&\quad\leq\dfrac{c}{h^d}\sum_{\i=1}^{N}\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\int_{\VoroCell{\j}\cap\PvDm{\i}}(|\ychar-\Pt{\j}|+|\Pt{\j}-\Pt{\i}|)\IntD\y\nonumber\\
		&\quad\leq\dfrac{c}{h^d}\BrS{r_{\N}\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\Meas{\VoroCell{\j}}+\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\sum_{\i=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}|\Pt{\j}-\Pt{\i}|}\nonumber\\
		&\quad\leq\dfrac{c}{h^d}\BrS{\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\Meas{\VoroCell{\j}}}\BrM{r_{\N}+\max_{\j=1,2,\dots,N}\BrS{\sum_{\i=1}^{N}\dfrac{\Meas{\VoroCell{\i}\cap\PvDm{\j}}+\Meas{\VoroCell{\j}\cap\PvDm{\i}}}{\Meas{\VoroCell{\j}}}|\Pt{\j}-\Pt{\i}|}}\nonumber\\
		&\quad\leqc\BrS{1+\dfrac{r_{\N}}{h}}^{d}\BrS{r_{\N}+\IndPvSetLocal{\Xi}}.
		\label{prf:estimate:E3:04}
	\end{align}
	Therefore, from \eqref{prf:estimate:E3:01}, \eqref{prf:estimate:E3:04}, and $h\leq\H$, we obtain
	\begin{align}
		|\prffuncA{3}(\xchar)| &\leqc\BrS{1+2\dfrac{r_{\N}}{h}}^{d}(r_{\N}+\IndPvSetLocal{\Xi})\nonumber\\
		&\leqc\BrS{1+2\dfrac{r_{\N}}{h}}^{d}\frac{r_{\N}+\IndPvSetLocal{\Xi}}{h}.
	\end{align}
	From the estimates of $\prffuncA{1}$, $\prffuncA{2}$, and $\prffuncA{3}$, we obtain
	\begin{equation}
		\BrNNd{\errorfuncA{\alpha}}_{\FsCz{\Ol{\Dm}}} \leq c \BrS{1+2\frac{r_{\N}}{h}}^{d}\frac{r_{\N}+\IndPvSetLocal{\Xi}}{h}.
	\end{equation}
	Because $\Xi$ is arbitrary, we establish \eqref{lem:ineq:Ja}.
\end{proof}

\begin{lemma}
	\label{lem:Jam}
	Suppose that reference weight function $\Weight$ satisfies Hypothesis \ref{hypo:ref_weight:smooth} with $k$.
	Then, there exists a positive constant $c$ independent of $N$ such that for all $\alpha\in\dMI{d}$ and $\ell\in\mathbb{N}$ with $1\leq\ell-k\leq|\alpha|$,
	\begin{align}
		\BrNNd{\errorfuncB{\alpha}{\ell}}_{\FsCz{\Ol{\Dm}}} \leq c \BrS{1+2\frac{r_{\N}}{h}}^{d} \frac{r_{\N}+d_{\N}}{h^{k+1}}.
		\label{lem:ineq:Jam}
	\end{align}
\end{lemma}

\begin{proof}
	We arbitrarily fix $\xchar\in\Ol{\Dm}$, $\alpha\in\dMI{d}$, particle volume decomposition $\Xi=\{\PvDm{\i}\mid\i=1,2,\dots,N\}$, and $\ell\in\mathbb{N}$ with $1\leq\ell-k\leq|\alpha|$ and
	split $\errorfuncB{\alpha}{\ell}$ into
	\begin{equation}
		\errorfuncB{\alpha}{\ell}(\xchar) = \prffuncA{4}(\xchar) + \prffuncA{5}(\xchar) + \prffuncA{6}(\xchar)
	\end{equation}
	with
	\renewcommand{\quad}{\quad}
	\begin{align}
		\prffuncA{4}(\xchar) &\deq \sum_{\i\in\IndexSetXHAst{\xchar}{h}}\Pv{\i}\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell}} \WeightH(|\xchar-\Pt{\i}|)
		-\sum_{\i\in\IndexSetXHAst{\xchar}{\infty}}\sum_{\j\in\IndexSetXHAst{\xchar}{\infty}}\Meas{\VoroCell{\j}\cap\PvDm{\i}}\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell-k}}\dfrac{\WeightH(|\xchar-\Pt{\j}|)}{|\xchar-\Pt{\j}|^{k}},
		\\
		\prffuncA{5}(\xchar)&\deq\sum_{\i\in\IndexSetXHAst{\xchar}{\infty}}\sum_{\j\in\IndexSetXHAst{\xchar}{\infty}}\Meas{\VoroCell{\j}\cap\PvDm{\i}}\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell-k}}\dfrac{\WeightH(|\xchar-\Pt{\j}|)}{|\xchar-\Pt{\j}|^{k}}-\sum_{\i\in\IndexSetXHAst{\xchar}{\infty}}\sum_{\j=1}^{N}\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell-k}}\int_{\VoroCell{\j}\cap\PvDm{\i}}\dfrac{\WeightH(|\xchar-\ychar|)}{|\xchar-\ychar|^{k}}\IntD\y,
		\\
		\prffuncA{6}(\xchar) &\deq\sum_{\i\in\IndexSetXHAst{\xchar}{\infty}}\sum_{\j=1}^{N}\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell-k}}\int_{\VoroCell{\j}\cap\PvDm{\i}}\dfrac{\WeightH(|\xchar-\ychar|)}{|\xchar-\ychar|^{k}}\IntD\y-\int_{\mathbb{R}^{\Dim}} \frac{\ychar^\alpha}{|\ychar|^\ell} \WeightH(|\ychar|)\IntD\y.
	\end{align}
	Then, we estimate $\prffuncA{4}$, $\prffuncA{5}$, and $\prffuncA{6}$.

	First, we estimate $\prffuncA{4}$ and set $\wn{k}$ as \eqref{def:wn} and $\wnh{k}$ as
	\begin{align}
		\wnh{k}(r) & \deq \frac{1}{h^{d+k}}\wn{k}\left(\frac{r}{h}\right), \qquad r\in\mathbb{R}^{+}_0.
		\label{def:wnh}
	\end{align}
	Then, from \eqref{sum_|VoroCell_PvDm|}, we can rewrite $\prffuncA{4}$ as
	\begin{equation}
		\prffuncA{4}(\xchar) =
		\sum_{\i\in\IndexSetXHAst{\xchar}{\infty}}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell-k}}\{\wnh{k}(|\xchar-\Pt{\i}|)-\wnh{k}(|\xchar-\Pt{\j}|)\}.
	\end{equation}
	Because
	\begin{align}
		\BrA{\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell-k}}}\leq |\Pt{\i}-\xchar|^{|\alpha|-\ell+k}\leq \textrm{diam}(\Dm_\H)^{|\alpha|-\ell+k}, \quad \i\in\IndexSetXHAst{\xchar}{\infty},
		\label{ineq:|psi_yz|}
	\end{align}
	we obtain
	\begin{equation}
		|\prffuncA{4}(\xchar)|\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}\BrA{\wnh{k}(|\xchar-\Pt{\i}|)-\wnh{k}(|\xchar-\Pt{\j}|)}.
	\end{equation}
	From $\supp(\wnh{k}) = [0,h]$, we have
	\begin{align}
		\wnh{k}(|\xchar-\Pt{\i}|)-\wnh{k}(|\xchar-\Pt{\j}|)= 0,\qquad \i, \j\not\in\IndexSetXHAst{\xchar}{h}.
		\label{eq:e_wh^n=0}
	\end{align}
	Thus, we obtain
	\begin{multline}
		|\prffuncA{4}(\xchar)|
		\leqc\Bigg(\sum_{\i\in\IndexSetXHAst{\xchar}{h}}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}\BrA{\wnh{k}(|\xchar-\Pt{\i}|)-\wnh{k}(|\xchar-\Pt{\j}|)}\\
		\qquad+\sum_{\i=1}^{N}\sum_{\j\in\IndexSetXHAst{\xchar}{h}}\Meas{\VoroCell{\j}\cap\PvDm{\i}}\BrA{\wnh{k}(|\xchar-\Pt{\i}|)-\wnh{k}(|\xchar-\Pt{\j}|)}\Bigg).
		\label{prf:estimate:E4:01}
	\end{multline}
	Using an argument similar to \eqref{ineq:|e_whx|}, if $\Weight$ satisfies Hypothesis \ref{hypo:ref_weight:smooth} with $k$, then for all $\ychar, \zchar\in\mathbb{R}^{\Dim}$,
	\begin{align}
		\BrANd{\wnh{k}(|\xchar-\ychar|)-\wnh{k}(|\xchar-\zchar|)}
		& \leq  \dfrac{|\ychar-\zchar|}{h^{d+k+1}}\int_0^1\BrA{ \dfrac{d}{dr}\wn{k}(r)}dr.
		\label{ineq:|e_whx^n|}
	\end{align}
	From \eqref{prf:estimate:E4:01} and \eqref{ineq:|e_whx^n|}, we obtain
	\begin{align}
		|\prffuncA{4}(\xchar)|
		&\leq\dfrac{c}{h^{d+k+1}}\sum_{\i\in\IndexSetXHAst{\xchar}{h}}\sum_{\j=1}^{N}\BrS{\Meas{\VoroCell{\i}\cap\PvDm{\j}}+\Meas{\VoroCell{\j}\cap\PvDm{\i}}}\BrA{\Pt{\i}-\Pt{\j}}
		\nonumber\\
		&\leq\dfrac{c}{h^{d+k+1}}\sum_{\i\in\IndexSetXHAst{\xchar}{h}}\Meas{\VoroCell{\i}}\sum_{\j=1}^{N}\dfrac{\Meas{\VoroCell{\i}\cap\PvDm{\j}}+\Meas{\VoroCell{\j}\cap\PvDm{\i}}}{\Meas{\VoroCell{\i}}}\BrA{\Pt{\i}-\Pt{\j}}
		\nonumber\\
		&\leqc\BrS{1+\frac{r_{\N}}{h}}^{d} \frac{\IndPvSetLocal{\Xi}}{h^{k+1}}.
	\end{align}

	Next, we estimate $\prffuncA{5}$.
	By using $\wnh{k}$, we can rewrite $\prffuncA{5}$ as
	\begin{equation}
		\prffuncA{5}(\xchar)=\sum_{\i\in\IndexSetXHAst{\xchar}{\infty}}\sum_{\j=1}^N\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell-k}}\int_{\VoroCell{\j}\cap\PvDm{\i}}\BrM{\wnh{k}(|\xchar-\Pt{\j}|)-\wnh{k}(|\xchar-\ychar|)}\IntD\y.
	\end{equation}
	From \eqref{ineq:|psi_yz|}, we obtain
	\begin{align}
		|\prffuncA{5}(\xchar)|
		&\leqc\sum_{\i=1}^N\sum_{\j=1}^N\int_{\VoroCell{\j}\cap\PvDm{\i}}\BrA{\wnh{k}(|\xchar-\Pt{\j}|)-\wnh{k}(|\xchar-\ychar|)}\IntD\y\nonumber\\
		&\leqc\sum_{\j=1}^N\int_{\VoroCell{\j}}\BrA{\wnh{k}(|\xchar-\Pt{\j}|)-\wnh{k}(|\xchar-\ychar|)}\IntD\y.
	\end{align}
	By $\supp(\wnh{k}) = [0,h]$ and $\VoroCell{\j}\subset\OBall{\Pt{\j}}{r_{\N}}$, we have
	\begin{align}
		\int_{\VoroCell{\j}}\BrA{\wnh{k}(|\xchar-\Pt{\j}|)-\wnh{k}(|\xchar-\ychar|)}\IntD\y= 0,\qquad \j\not\in\IndexSetXHAst{\xchar}{h+r_{\N}}.
		\label{prf:estimate:E5:01}
	\end{align}
	From \eqref{ineq:|e_whx^n|} and \eqref{prf:estimate:E5:01}, we obtain
	\begin{align}
		|\prffuncA{5}(\xchar)|
		&\leqc\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\int_{\VoroCell{\j}}\BrA{\wnh{k}(|\xchar-\Pt{\j}|)-\wnh{k}(|\xchar-\ychar|)}\IntD\y
		\nonumber\\
		&\leq\dfrac{c}{h^{d+k+1}}\int_{\VoroCell{\j}}\BrA{\Pt{\j}-\ychar}\IntD\y
		\nonumber\\
		&\leqc\dfrac{r_{\N}}{h^{d+k+1}}\sum_{\j\in\IndexSetXH{\xchar}{h+r_{\N}}}\Meas{\VoroCell{\j}}
		\nonumber\\
		&\leqc\BrS{1+2\frac{r_{\N}}{h}}^{d} \dfrac{r_{\N}}{h^{k+1}}.
	\end{align}

	Finally, we estimate $\prffuncA{6}$.
	Using $\wnh{k}$, we can rewrite $\prffuncA{6}$ as
	\begin{align}
		\prffuncA{6}(\xchar)&=\sum_{\i\in\IndexSetXH{\xchar}{\infty}}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}\BrM{\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell-k}}-\frac{(\ychar-\xchar)^{\alpha}}{|\ychar-\xchar|^{\ell-k}}}\wnh{k}(|\xchar-\ychar|)\IntD\y\nonumber\\
		&\quad-\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDmFunc{\i}{\xchar}}\frac{(\ychar-\xchar)^{\alpha}}{|\ychar-\xchar|^{\ell-k}}\wnh{k}(|\xchar-\ychar|)\IntD\y,
		\label{prf:estimate:E6:02}
	\end{align}
	where $\PvDmFunc{\i}{\xchar}$ is
	\begin{equation}
		\PvDmFunc{\i}{\xchar} =
		\begin{cases}
			\PvDm{\i},\qquad &\xchar=\Pt{\i},
			\\
			0,\qquad &\mbox{otherwize}.
		\end{cases}
	\end{equation}
	For $\alpha\in\dMI{d}$, let  $\beta_j\,(j=1,2,\dots,|\alpha|)$ be $d$-dimensional multi-indices satisfying
	\begin{align}
		|\beta_j| = 1 \quad \mbox{and}  \quad \sum_{j=1}^{|\alpha|}\beta_j = \alpha.
	\end{align}
	Let $\beta_j^\ast\,(j=0,1,\dots,|\alpha|)$ be $d$-dimensional multi-indices defined as
	\begin{align}
		\beta_j^\ast \deq
		\begin{cases}
			0,\qquad & j=0,
			\\
			\displaystyle \sum_{\ell=1}^{j}\beta_\ell,\qquad & j=1,2,\dots,|\alpha|.
		\end{cases}
	\end{align}
	For all $\ychar, \zchar \in\mathbb{R}^{\Dim}\setminus\{0\}$, when $|\alpha|=\ell-k$, we have
	\begin{align}
		\BrA{\dfrac{\ychar^{\alpha}}{|\ychar|^{\ell-k}}-\dfrac{\zchar^{\alpha}}{|\zchar|^{\ell-k}}}
		&\leq \sum_{j=0}^{\ell-k-1} \BrA{\dfrac{\ychar^{\beta_{|\alpha|-j}^\ast}\zchar^{\beta_{j}^\ast}}{|\ychar|^{\ell-k-j}|\zchar|^{j}}-\dfrac{\ychar^{\beta_{|\alpha|-j-1}^\ast}\zchar^{\beta_{j+1}^\ast}}{|\ychar|^{\ell-k-j-1}|\zchar|^{j+1}}}
		\nonumber\\
		&\leq \sum_{j=0}^{\ell-k-1} \BrA{\dfrac{\ychar^{\beta_{|\alpha|-j}^\ast}\zchar^{\beta_{j}^\ast}-\ychar^{\beta_{|\alpha|-j-1}^\ast}\zchar^{\beta_{j+1}^\ast}}{|\ychar|^{\ell-k-j}|\zchar|^{j}}}+\sum_{j=0}^{\ell-k-1} \BrA{\dfrac{\ychar^{\beta_{|\alpha|-j-1}^\ast}\zchar^{\beta_{j+1}^\ast}}{|\ychar|^{\ell-k-j}|\zchar|^{j}}-\dfrac{\ychar^{\beta_{|\alpha|-j-1}^\ast}\zchar^{\beta_{j+1}^\ast}}{|\ychar|^{\ell-k-j-1}|\zchar|^{j+1}}}
		\nonumber\\
		&\leq 2(\ell-k)\dfrac{|\ychar-\zchar|}{|\ychar|}.
		\label{prf:estimate:E6:03}
	\end{align}
	Moreover, from \eqref{ineq:|y^alpha-z^alpha|} and \eqref{prf:estimate:E6:03}, when $|\alpha|>\ell-k$, we have
	\begin{align}
		\BrA{\dfrac{\ychar^{\alpha}}{|\ychar|^{\ell-k}}-\dfrac{\zchar^{\alpha}}{|\zchar|^{\ell-k}}}
		&\leq\BrA{\dfrac{\ychar^{\alpha}}{|\ychar|^{\ell-k}}-\dfrac{\ychar^{\beta_{|\alpha|-\ell+k}^\ast}\zchar^{\beta_{\ell-k}^\ast}}{|\zchar|^{\ell-k}}}+\BrA{\dfrac{\ychar^{\beta_{|\alpha|-\ell+k}^\ast}\zchar^{\beta_{\ell-k}^\ast}}{|\zchar|^{\ell-k}}-\dfrac{\zchar^{\alpha}}{|\zchar|^{\ell-k}}}
		\nonumber\\
		&\leq|\ychar|^{|\alpha|-\ell+k}\BrA{\dfrac{\ychar^{\beta_{\ell-k}^\ast}}{|\ychar|^{\ell-k}}-\dfrac{\zchar^{\beta_{\ell-k}^\ast}}{|\zchar|^{\ell-k}}} +\BrA{\ychar^{\beta_{|\alpha|-\ell+k}^\ast}-\zchar^{\beta_{|\alpha|-\ell+k}^\ast}}
		\nonumber\\
		&\leq 2(\ell-k)|\ychar-\zchar||\ychar|^{|\alpha|-\ell+k-1}+|\ychar-\zchar|\sum_{j=0}^{|\alpha|-\ell+k-1}|\ychar|^{j}|\zchar|^{|\alpha|-\ell+k-1-j}.
	\end{align}
	Therefore, when $|\alpha|\geq\ell-k$, we have for all $\ychar\in\Dm_\H\setminus\{\xchar\}$ and $\i\in\IndexSetXHAst{\xchar}{\infty}$,
	\begin{align}
		\BrA{\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell-k}}-\frac{(\ychar-\xchar)^{\alpha}}{|\ychar-\xchar|^{\ell-k}}}
		&\leq c\dfrac{|\ychar-\Pt{\i}|}{|\ychar-\xchar|}.
		\label{ineq:|psi(xy)-psy(xz)|_k=}
	\end{align}
	From \eqref{ineq:|psi(xy)-psy(xz)|_k=}, we obtain
	\begin{align}
		|\prffuncA{6}(\xchar)| &\leq\sum_{\i\neq\k}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}\BrA{\frac{(\Pt{\i}-\xchar)^{\alpha}}{|\Pt{\i}-\xchar|^{\ell-k}}-\frac{(\ychar-\xchar)^{\alpha}}{|\ychar-\xchar|^{\ell-k}}}\BrA{\wnh{k}(|\xchar-\ychar|)}\IntD\y\nonumber\\
		&\quad+\BrA{\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDmFunc{\i}{\xchar}}\frac{(\ychar-\xchar)^{\alpha}}{|\ychar-\xchar|^{\ell-k}}\wnh{k}(|\xchar-\ychar|)\IntD\y}\nonumber\\
		&\leqc\sum_{\i\neq\k}\sum_{\j=1}^{N} \int_{\VoroCell{\j}\cap\PvDm{\i}}|\ychar-\Pt{\i}|\BrA{\wnh{k+1}(|\xchar-\ychar|)}\IntD\y\nonumber\\
		&\quad+\BrA{\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDmFunc{\i}{\xchar}}\frac{(\ychar-\xchar)^{\alpha}}{|\ychar-\xchar|^{\ell-k}}\wnh{k}(|\xchar-\ychar|)\IntD\y}.
		\label{prf:estimate:E6:05}
	\end{align}
	Because $|\alpha|\geq\ell-k$, we have
	\begin{align}
		\BrA{\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDmFunc{\i}{\xchar}}\frac{(\ychar-\xchar)^{\alpha}}{|\ychar-\xchar|^{\ell-k}}\wnh{k}(|\xchar-\ychar|)\IntD\y}
		&\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDmFunc{\i}{\xchar}}\BrA{\frac{(\ychar-\xchar)^{\alpha}}{|\ychar-\xchar|^{\ell-k-1}}}\BrA{\wnh{k+1}(|\xchar-\ychar|)}\IntD\y
		\nonumber\\
		&\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDmFunc{\i}{\xchar}}\BrA{\ychar-\xchar}\BrA{\wnh{k+1}(|\xchar-\ychar|)}\IntD\y.
	\end{align}
	Therefore, we have
	\begin{equation}
		|\prffuncA{6}(\xchar)|
		\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N} \int_{\VoroCell{\j}\cap\PvDm{\i}}|\ychar-\Pt{\i}|\BrA{\wnh{k+1}(|\xchar-\ychar|)}\IntD\y.
	\end{equation}
	Because for all $\ychar\in\Dm_\H$,
	\begin{equation}
		\BrA{\wnh{k+1}(|\xchar-\ychar|)}=\dfrac{1}{h^{d+k+1}}\BrA{\wn{k+1}\BrS{\dfrac{|\xchar-\ychar|}{h}}}\leq\dfrac{1}{h^{d+k+1}}\NormCzNd{\wn{k+1}}{\mathbb{R}^{+}_0},
	\end{equation}
	by the same procedure as \eqref{prf:estimate:E3:04}, we have
	\begin{equation}
		\sum_{\i=1}^{N}\sum_{\j=1}^{N} \int_{\VoroCell{\j}\cap\PvDm{\i}}|\ychar-\Pt{\i}|\BrA{\wnh{k+1}(|\xchar-\ychar|)}\IntD\y\leqc\BrS{1+\dfrac{r_{\N}}{h}}^{d}\dfrac{r_{\N}+\IndPvSetLocal{\Xi}}{h^{k+1}}.
	\end{equation}
	Therefore, we obtain
	\begin{align}
		|\prffuncA{6}(\xchar)|\leqc\BrS{1+\dfrac{r_{\N}}{h}}^{d}\dfrac{r_{\N}+\IndPvSetLocal{\Xi}}{h^{k+1}}.
	\end{align}

	From the estimates of $\prffuncA{4}$, $\prffuncA{5}$, and $\prffuncA{6}$, we obtain
	\begin{equation}
		\BrNNd{\errorfuncB{\alpha}{\ell}}_{\FsCz{\Ol{\Dm}}}
		\leqc\BrS{1+2\frac{r_{\N}}{h}}^{d}\frac{r_{\N}+\IndPvSetLocal{\Xi}}{h^{k+1}}.
	\end{equation}
	Because $\Xi$ is arbitrary, we establish \eqref{lem:ineq:Jam}.
\end{proof}

\begin{lemma}
	\label{lem:Km}
	There exists a positive constant $c$ independent of $N$ such that
	\begin{align}
		\NormCz{\errorfuncC{\ell}}{\Ol{\Dm}} \leq c\BrM{\BrS{1+2\frac{r_{\N}}{h}}^d \dfrac{r_{\N}+d_{\N}}{h}+h^{\ell}},\quad \ell\in\mathbb{N}.
		\label{eq:lem:Km}
	\end{align}
\end{lemma}

\begin{proof}
	We arbitrarily fix $\xchar\in\Ol{\Dm}$ and particle volume decomposition $\Xi=\{\PvDm{\i}\mid\i=1,2,\dots,N\}$, and split $\errorfuncC{\ell}$ into
	\begin{align}
		\errorfuncC{\ell}(\xchar) & = \prffuncA{7}(\xchar) + \prffuncA{8}(\xchar) + \prffuncA{9}(\xchar) + \prffuncA{10}(\xchar)
	\end{align}
	with
	\begin{align}
		\prffuncA{7}(\xchar) &\deq\errorfuncC{\ell}(\xchar)- \sum_{\i=1}^{N}\sum_{\j=1}^{N} \Meas{\VoroCell{\j}\cap\PvDm{\i}} (\xchar-\Pt{\i})^{\ell}|\WeightH(|\xchar-\Pt{\j}|)|,
		\\
		\prffuncA{8}(\xchar) &\deq \sum_{\i=1}^{N}\sum_{\j=1}^{N} (\xchar-\Pt{\i})^{\ell} \int_{\VoroCell{\j}\cap\PvDm{\i}}\BrM{|\WeightH(|\xchar-\Pt{\j}|)|-|\WeightH(|\xchar-\ychar|)|}\IntD\y,
		\\
		\prffuncA{9}(\xchar) &\deq \sum_{\i=1}^{N}\sum_{\j=1}^N \int_{\VoroCell{\j}\cap\PvDm{\i}}\{(\xchar-\Pt{\i})^{\ell} - (\xchar-\ychar)^{\ell}\}  |\WeightH(|\xchar-\ychar|)| \IntD\y,
		\\
		\prffuncA{10}(\xchar) &\deq \int_{\mathbb{R}^{\Dim}}(\xchar-\ychar)^{\ell}|\WeightH(|\xchar-\ychar|)|\IntD\y.
	\end{align}
	Then, we estimate $\prffuncA{7}$, $\prffuncA{8}$, $\prffuncA{9}$, and $\prffuncA{10}$.

	From \eqref{sum_|VoroCell_PvDm|}, we can rewrite $\prffuncA{7}$ as
	\begin{equation}
		\prffuncA{7}(\xchar)=\sum_{\i=1}^{N}\sum_{\j=1}^{N} \Meas{\VoroCell{\j}\cap\PvDm{\i}}(\xchar-\Pt{\i})^{\ell} \{|\WeightH(|\xchar-\Pt{\i}|)|-|\WeightH(|\xchar-\Pt{\j}|)|\}.
	\end{equation}
	For all $\ychar\in\Dm_\H$, we have
	\begin{equation}
		|(\xchar-\ychar)^{\ell}|\leq |\xchar-\ychar|^{\ell}\leq \textrm{diam}(\Dm_\H)^{\ell}.
		\label{prf:estimate:E7:01}
	\end{equation}
	From \eqref{prf:estimate:E1:02} and \eqref{prf:estimate:E7:01}, we obtain
	\begin{align}
		\BrA{\prffuncA{7}(\xchar)}
		&\leq\sum_{\i=1}^{N}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}(\xchar-\Pt{\i})^{\ell}\big||\WeightH(|\xchar-\Pt{\i}|)|-|\WeightH(|\xchar-\Pt{\j}|)|\big|
		\nonumber\\
		&\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}\big||\WeightH(|\xchar-\Pt{\i}|)|-|\WeightH(|\xchar-\Pt{\j}|)|\big|
		\nonumber\\
		&\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N}\Meas{\VoroCell{\j}\cap\PvDm{\i}}\BrA{\WeightH(|\xchar-\Pt{\i}|)-\WeightH(|\xchar-\Pt{\j}|)}
		\nonumber\\
		&\leqc\BrS{1+\frac{r_{\N}}{h}}^{d} \frac{\IndPvSetLocal{\Xi}}{h}.
	\end{align}
	From \eqref{prf:estimate:E2:01} and \eqref{prf:estimate:E7:01}, we obtain
	\begin{align}
		|\prffuncA{8}(\xchar)|
		&\leq \sum_{\i=1}^{N}\sum_{\j=1}^{N} (\xchar-\Pt{\i})^{\ell} \int_{\VoroCell{\j}\cap\PvDm{\i}}\big||\WeightH(|\xchar-\Pt{\j}|)|-|\WeightH(|\xchar-\ychar|)|\big|\IntD\y
		\nonumber\\
		&\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}\big||\WeightH(|\xchar-\Pt{\j}|)|-|\WeightH(|\xchar-\ychar|)|\big|\IntD\y
		\nonumber\\
		&\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N}\int_{\VoroCell{\j}\cap\PvDm{\i}}\BrA{\WeightH(|\xchar-\Pt{\j}|)-\WeightH(|\xchar-\ychar|)}\IntD\y
		\nonumber\\
		& \leq c \BrS{1+2\frac{r_{\N}}{h}}^{d}\frac{r_{\N}}{h}.
	\end{align}
	For all $\Pt\i\in\PtSetCher_{\N}$ and $\ychar\in\Dm_\H$, we have
	\begin{align}
		\BrA{(\xchar-\Pt{\i})^{\ell} - (\xchar-\ychar)^{\ell}}
		&=\BrA{\{(\xchar-\Pt{\i})-(\xchar-\ychar)\} \sum_{k=1}^{\ell} (\xchar-\Pt{\i})^{k-1}(\xchar-\ychar)^{\ell-k}}\nonumber\\
		&\leq\ell\,\textrm{diam}(\Dm_\H)^\ell\BrA{\ychar-\Pt{\i}}.
		\label{prf:estimate:E9:01}
	\end{align}
	From \eqref{prf:estimate:E3:04}, \eqref{prf:estimate:E9:01}, and $h<\H$, we obtain
	\begin{align}
		|\prffuncA{9}(\xchar)|
		&\leq\sum_{\i=1}^{N}\sum_{\j=1}^N \int_{\VoroCell{\j}\cap\PvDm{\i}}\BrA{(\xchar-\Pt{\i})^{\ell} - (\xchar-\ychar)^{\ell}}  |\WeightH(|\xchar-\ychar|)| \IntD\y
		\nonumber\\
		&\leqc\sum_{\i=1}^{N}\sum_{\j=1}^{N} \int_{\VoroCell{\j}\cap\PvDm{\i}} |\ychar-\Pt{\i}| |\WeightH(|\xchar-\ychar|)| dy
		\nonumber\\
		&\leqc\BrS{1+\dfrac{r_{\N}}{h}}^{d}\BrS{r_{\N}+\IndPvSetLocal{\Xi}}.
	\end{align}
	From \eqref{def:wh}, we obtain
	\begin{align}
		|\prffuncA{10}(\xchar)|
		&=\int_{\mathbb{R}^{\Dim}}|\ychar|^{\ell}|\WeightH(|\ychar|)|\IntD\y=h^{\ell}\int_{\mathbb{R}^{\Dim}}|\ychar|^{\ell}|\Weight(|\ychar|)|\IntD\y.
	\end{align}
	From the estimates of $\prffuncA{7}$, $\prffuncA{8}$, $\prffuncA{9}$, and $\prffuncA{10}$, we obtain
	\begin{equation}
		\NormCz{\errorfuncC{\ell}}{\Ol{\Dm}} \leq c\BrM{\BrS{1+2\frac{r_{\N}}{h}}^d \dfrac{r_{\N}+\IndPvSetLocal{\Xi}}{h}+h^{\ell}}.
	\end{equation}
	Because $\Xi$ is arbitrary, we establish \eqref{eq:lem:Km}.
\end{proof}

Using the lemmas defined above, we now prove Theorem \ref{thm:terror_cont_norm}.
\begin{proof}[Theorem \ref{thm:terror_cont_norm}]
	By Lemmas \ref{lem:terror_cont_norm}, \ref{lem:Ja}, and \ref{lem:Km}, we have for all $\FuncCher\in\FsC{n+1}{\Ol\Dm_{\H}}$
	\begin{equation}
		\NormCz{\FuncCher-\Pi_\h{} \FuncCher}{\Ol{\Dm}} \leq c\BrM{ \BrS{1+2\frac{r_{\N}}{h}}^{d}\frac{r_{\N}+d_{\N}}{h}+\h_{\N}^{n+1}}\NormC{\FuncCher}{n+1}{\Ol\Dm_{\H}}.
		\label{prf:ineq:terror_cont_norm_intp:01}
	\end{equation}
	Moreover, by Lemmas \ref{lem:terror_cont_norm}, \ref{lem:Jam}, and \ref{lem:Km}, when $\Weight$ satisfies Hypothesis \ref{hypo:ref_weight:smooth} with $k=0$, we have for all $\FuncCher\in\FsC{n+2}{\Ol\Dm_{\H}}$
	\begin{equation}
		\NormCz{\nabla\FuncCher-\Grad_\h{}\FuncCher}{\Ol{\Dm}}
		\leq c\BrM{ \BrS{1+2\frac{r_{\N}}{h}}^{d}\frac{r_{\N}+d_{\N}}{h}+\h_{\N}^{n+1}}\NormC{\FuncCher}{n+2}{\Ol\Dm_{\H}},
		\label{prf:ineq:terror_cont_norm_grad:01}
	\end{equation}
	and when $\Weight$ satisfies Hypothesis \ref{hypo:ref_weight:smooth} with $k=1$ for all $\FuncCher\in\FsC{n+3}{\Ol\Dm_{\H}}$,
	\begin{equation}
		\NormCz{\Delta\FuncCher-\Delta_\h{}\FuncCher}{\Ol{\Dm}}
		\leq c\BrM{ \BrS{1+2\frac{r_{\N}}{h}}^{d}\frac{r_{\N}+d_{\N}}{h^2}+\h_{\N}^{n+1}}\NormC{\FuncCher}{n+3}{\Ol\Dm_{\H}}.
		\label{prf:ineq:terror_cont_norm_lap:01}
	\end{equation}
	Because the family $\{(\PtSetCher_{\N}, \PvSetCher_{\N}, \h_{\N})\}_{N\rightarrow\infty}$ is regular, by applying \eqref{def:regular:cond} to \eqref{prf:ineq:terror_cont_norm_intp:01}, \eqref{prf:ineq:terror_cont_norm_grad:01}, and \eqref{prf:ineq:terror_cont_norm_lap:01}, we obtain \eqref{ineq:terror_cont_norm_intp}, \eqref{ineq:terror_cont_norm_grad}, and \eqref{ineq:terror_cont_norm_lap}, respectively.
	We now conclude the proof of Theorem \ref{thm:terror_cont_norm}.
\end{proof}

\section{Numerical results}
\label{sec:Numerical_results}
Set $\Omega=(0,1)^2$ and $\H=0.1$.
Then, $\Dm_\H=(-0.1,1.1)^2$.
We now compute the truncation errors of $\FuncCher:\Dm_\H\rightarrow\mathbb{R}$, which are defined as $\FuncCher(x,y)=\sin(2\pi(x+y))$.
Particle distribution $\PtSetCher_{\N}$ is set as
\begin{equation}
	\PtSetCher_{\N} = \BrM{\BrS{(i+\eta_{ij}^{(1)})\Delta x, (j+\eta_{ij}^{(2)}) \Delta x}\in\Dm_\H ;~i,j\in\mathbb{Z}},
\end{equation}
where $\Delta\xchar$ is taken by $2^{-5}, 2^{-6}, \dots, 2^{-12}$ and $\Noise{\i}{\j}{k}\,(i,j\in\mathbb{Z}, k=1,2)$ are random numbers satisfying $|\Noise{\i}{\j}{k}|< 1/4$.
Particle distribution $\PtSetCher_{\N}$ with $\Delta\xchar=2^{-5}$ is shown in Figure \ref{fig_numerical_result_particle_distribution}.
\begin{figure}[tb]
	\includegraphics[width=50mm,bb=0 0 126.5mm 122.4mm]{fig_MM_particle_distribution_with_domain.eps}
	\caption{Particle distribution $\PtSetCher_{\N}$ with $\Delta\xchar=2^{-5}$ $(N=1,521)$. The gray area represents $\Omega$. }
	\label{fig_numerical_result_particle_distribution}
\end{figure}
Particle volume set $\PvSetCher_{\N}$ is defined as
\begin{equation}
	\PvSetCher_{\N}=\BrM{\Pv{\i} = \dfrac{\Meas{\Dm_\H}}{N}\;\middle|\; \i=1,2,\dots,N}.
	\label{pvolume_exp_02}
\end{equation}
For $m=1,3,5$, the influence radius $\h_{\N}$ is set as
\begin{align}
	\h_{\N} = 2.6\times2^{5/m-5}\Delta\xchar^{1/m}.
\end{align}
Note that if $\Delta\xchar=2^{-5}$, then $h=2.6\times2^{-5}$ for all $m$.
Using the discrete parameters above,
the covering radius $r_{\N}$ satisfies $r_{\N} \leq \sqrt{2}(1+1/4)\Delta\xchar/2$.
Moreover,
the Voronoi deviation $d_{\N}$ satisfies $d_{\N}\leq64(1+\sqrt{2})\Delta\xchar/\pi$.
Therefore, the family $\{(\PtSetCher_{\N},\PvSetCher_{\N},\h_{\N})\}$ is regular with order $m$.

For the interpolant, we consider the following three cases of reference weight functions:
\begin{align*}
	({\rm \Intp1})\quad \Weight(r) &\deq\dfrac{3}{\pi}
	\begin{cases}
		1-r,& 0 \leq r < 1,
		\\
		0,\hspace{16ex}& 1 \leq r,
	\end{cases}
	\\
	({\rm \Intp2})\quad \Weight(r) &\deq \dfrac{40}{7\pi}
	\begin{cases}
		1-6r^2+6r^3, \quad & \displaystyle 0 \leq r < \frac{1}{2},
		\vspace*{0.5ex}\\
		2(1-r)^3, \quad & \displaystyle \frac{1}{2} \leq r < 1,
		\vspace*{0.5ex}\\
		0, \hspace{13.5ex} & 1 \leq r,
	\end{cases}
	\\
	({\rm \Intp3})\quad\Weight(r) &\deq\dfrac{5}{\pi}
	\begin{cases}
		(1-r)(2-3r), & 0 \leq r < 1,
		\vspace*{1ex}\\
		0,\hspace{16ex}& 1 \leq r.
	\end{cases}
\end{align*}
$({\rm \Intp1})$ is the lowest order polynomial function belonging to $\mathcal{W}$.
$({\rm \Intp2})$ is the cubic B-spline commonly used in SPH and belonging to $\mathcal{W}$.
$({\rm \Intp3})$ is the lowest order polynomial function belonging to $\mathcal{W}$ that satisfies Hypothesis \ref{hypo:ref_weight:order} with $n=3$.

For the approximate gradient operator, we consider the following three cases of reference weight functions:
\begin{align*}
	({\rm \Grad1})\quad \Weight(r) &\deq\dfrac{6}{\pi}
	\begin{cases}
		r(1-r), \quad & 0 \leq r < 1,
		\\
		0,\hspace{19ex}& 1 \leq r,
	\end{cases}
	\\
	({\rm \Grad2})\quad \Weight(r) &\deq \dfrac{40}{7\pi}
	\begin{cases}
		6r^2-9r^3, \quad & \displaystyle 0 \leq r < \frac{1}{2},
		\vspace*{0.5ex}\\
		3r(1-r)^2, \quad & \displaystyle \frac{1}{2} \leq r < 1,
		\vspace*{0.5ex}\\
		0,\hspace{17.5ex} & 1 \leq r,
	\end{cases}
	\\
	({\rm \Grad3})\quad \Weight(r) &\deq\dfrac{15}{2\pi}
	\begin{cases}
		r(1-r)(5-7r), & 0 \leq r < 1,
		\\
		0,\hspace{18ex}& 1 \leq r.
	\end{cases}
\end{align*}
$({\rm \Grad1})$ is the lowest order polynomial function belonging to $\mathcal{W}$ and Hypothesis \ref{hypo:ref_weight:smooth} with $k=0$.
$({\rm \Grad2})$ is chosen so that approximate gradient operator \eqref{gh_def} with $({\rm \Grad2})$ coincides with that in SPH with the cubic B-spline (see Appendix \ref{sec:appendix_conventional_particle_methods}).
$({\rm \Grad3})$ is the lowest order polynomial function belonging to $\mathcal{W}$, Hypothesis \ref{hypo:ref_weight:order} with $n=3$, and Hypothesis \ref{hypo:ref_weight:smooth} with $k=0$.

For the approximate Laplace operator, we consider the following three cases of reference weight functions:
\begin{align*}
	({\rm \Lap1})\quad \Weight(r) &\deq\dfrac{10}{\pi}
	\begin{cases}
		r^2(1-r), \quad & 0 \leq r < 1,
		\\
		0,\hspace{21ex}& 1 \leq r,
	\end{cases}
	\\
	({\rm \Lap2})\quad \Weight(r) &\deq \dfrac{40}{7\pi}
	\begin{cases}
		6r^2-9r^3, \quad & \displaystyle 0 \leq r < \frac{1}{2},
		\vspace*{0.5ex}\\
		3r(1-r)^2, \quad & \displaystyle \frac{1}{2} \leq r < 1,
		\vspace*{0.5ex}\\
		0,\hspace{20.5ex} & 1 \leq r,
	\end{cases}
	\\
	({\rm \Lap3})\quad \Weight(r) &\deq\dfrac{30}{\pi}
	\begin{cases}
		r^2(1-r)(3-4r), & 0 \leq r < 1,
		\\
		0,\hspace{21ex}& 1 \leq r.
	\end{cases}
\end{align*}
$({\rm \Lap1})$ is the lowest order polynomial function belonging to $\mathcal{W}$ and Hypothesis \ref{hypo:ref_weight:smooth} with $k=1$.
$({\rm \Lap2})$ is chosen so that approximate Laplace operator \eqref{lh_def} with $({\rm \Lap2})$ coincides with that in SPH with the cubic B-spline (see Appendix \ref{sec:appendix_conventional_particle_methods}).
$({\rm \Lap3})$ is the lowest order polynomial function belonging to $\mathcal{W}$, Hypothesis \ref{hypo:ref_weight:order} with $n=3$, and Hypothesis \ref{hypo:ref_weight:smooth} with $k=1$.

The above settings were used in the computation of the following relative errors
\begin{equation}
	\dfrac{\NormDiscL{\FuncCher-\Pi_\h{} \FuncCher}{\infty}{\Omega}}{\NormCzNd{\FuncCher}{\Ol{\Dm}}},\quad
	\dfrac{\NormDiscL{\nabla\FuncCher-\Grad_\h{} \FuncCher}{\infty}{\Omega}}{\NormCzNd{\nabla\FuncCher}{\Ol{\Dm}}},\quad
	\dfrac{\NormDiscL{\Delta\FuncCher-\Delta_\h{} \FuncCher}{\infty}{\Omega}}{\NormCzNd{\Delta\FuncCher}{\Ol{\Dm}}}.
\end{equation}
Here, the discrete norm $\NormDiscL{\cdot}{\infty}{\Omega}$ is defined as
\begin{align}
	\NormDiscL{\FuncCher}{\infty}{\Omega} \deq \max_{\i\in\IndexSet{\Omega}} |\FuncCher(\Pt{\i})|.
\end{align}
Figure \ref{fig:numerical_terror} shows graphs of the relative errors of (a) interpolant $\Pi_\h{}$, (b) approximate gradient operator $\Grad_\h{}$, and (c) approximate Laplace operator $\Delta_\h{}$ versus the influence radius $\h_{\N}$ with regular orders $m=1, 3, 5$.
In Figure \ref{fig:numerical_terror}, the slopes of the triangles show the theoretical convergence rates obtained via Theorem \ref{thm:terror_cont_norm}.
Table \ref{tab:numerical_theoretical_converge_rate} lists the numerical and theoretical convergence rates obtained from the cases of $\Delta x=2^{-11}$ and $2^{-12}$, where the theoretical convergence rates correspond to Theorem \ref{thm:terror_cont_norm}.
In the case of $m=1$, as the settings could not be applied to Theorem \ref{thm:terror_cont_norm}, only numerical results without convergence were obtained.
In contrast, the settings in cases $m=3$ and $5$ could be applied Theorem \ref{thm:terror_cont_norm}; thus, the numerical results with convergence were obtained.
Moreover, the approximate operators with reference weight functions satisfying Hypothesis \ref{hypo:ref_weight:order} with $n=3$ became higher convergence orders in the cases where $m=5$ as per Theorem \ref{thm:terror_cont_norm}.
\begin{figure}[h!]
	\includegraphics[width=100mm,bb=0 0 186.0mm 286.0mm]{fig_NR_terror_general_f1_a9.eps}
	\caption{Graphs of the relative errors of (a) the interpolant, (b) approximate gradient operator, and (c) approximate Laplace operator versus the influence radius with regular orders $m=1, 3, 5$.}
	\label{fig:numerical_terror}
\end{figure}
\begin{table}[htb]
	\renewcommand{\arraystretch}{1.2}
	\caption{Numerical and theoretical convergence rates of (a) the interpolant, (b) approximate gradient operator, and (c) approximate Laplace operator with regular orders $m=1, 3, 5$. The numerical convergence rates were obtained for the cases of $\Delta x=2^{-11}$ and $2^{-12}$. }
	\label{tab:numerical_theoretical_converge_rate}
	\begin{tabular}{lrrrrrr}
		\hline
		&\multicolumn{2}{c}{$m=1$}&\multicolumn{2}{c}{$m=3$}&\multicolumn{2}{c}{$m=5$}\\
		& Numer. & Theor. & Numer. & Theor. & Numer. & Theor.
		\\\hline
		\noalign{\hrule height 1pt}
		\\[-2.5ex]
		(a) Interpolant
		\\[0.5ex]
		($\rm \Intp1$) & -0.10 & N/A &2.02 & 2 & 2.00 & 2\\
		($\rm \Intp2$) & 0.05 & N/A & 2.13 & 2 & 2.01 & 2\\
		($\rm \Intp3$) &  0.00 & N/A & 4.20 & 2 & 7.41 & 4
		\\[1.5ex]
		(b) Gradient
		\\[0.5ex]
		($\rm \Grad1$) & -0.05 & N/A & 2.11 & 2 & 2.02 & 2\\
		($\rm \Grad2$) & -0.02 & N/A & 2.08 & 2 & 2.03 & 2\\
		($\rm \Grad3$) &  -0.06 & N/A & 3.56 & 2 & 7.69 & 4
		\\[1.5ex]
		(c) Laplacian
		\\[0.5ex]
		($\rm \Lap1$) & -1.14 & N/A &2.91 & 1 & 2.05 & 2\\
		($\rm \Lap2$) & -1.09 & N/A & 2.39 & 1 & 2.23 & 2\\
		($\rm \Lap3$) & -1.02 & N/A & 2.76 & 1 & 7.50 & 3\\
		\hline
	\end{tabular}
\end{table}

\section{Conclusions}
\label{sec:conclusions}
We analyzed truncation errors in a generalized particle method, which is a wider class of particle methods that includes commonly used methods such as SPH and MPS.
In our analysis, we introduced two indicators:
the first was the covering radius, which represents the maximum radius of the Voronoi region associated with the particle distribution, while the second was the Voronoi deviation, which indicates the deviation between particle volumes and Voronoi volumes.
With the covering radius and Voronoi deviation, we introduced a regularity of a family of discrete parameters, which includes the particle distribution, particle volume set, and influence radius associated with the number of particles.
Moreover, we introduced two hypotheses of reference weight functions.
With the regularity and hypotheses of reference weight functions, we established truncation error estimates for the continuous norm.
The convergence rates are dependent on the regular order and order of the reference weight functions appearing in a hypothesis.
Moreover, as it was possible to validate the conditions by calculation, we showed the numerical convergence orders were in good agreement with the theoretical ones.

In a forthcoming paper, we plan to establish error estimates of the generalized particle method for the Poisson and heat equations.

\section*{Acknowledgments}
We would like to thank Drs. Daisuke Tagami and Hayato Waki for their helpful comments over the course of this work.
This research was supported by JSPS KAKENHI Grant Number 17K17585 and the JSPS A3 Foresight Program.

redpajama_set_name: RedPajamaArXiv


text:
Q: How can i add backgroung image in the following svg? From this website
https://smooth.ie/blogs/news/svg-wavey-transitions-between-sections

i generated the following svg

<svg viewBox="0 0 500 150" preserveAspectRatio="none" style="height: 100%; width: 100%;"><path d="M213.19,0.00 C152.69,70.06 270.03,70.06 202.98,150.00 L500.00,150.00 L500.00,0.00 Z" style="stroke: none; fill: #08f;"></path></svg>

the problem is that when i try to add background image for example here on the parent class
.parent {
    background-image: url('../../../assets/images/calendar.png');
}

then the image is hidden behind the blue color of the svg.How can i 'insert' this image to be on the blue svg color image ?

A: change the opacity level so the color is lower opacity. something like this perhaps.
background {
backgroundColor:blue,
opacity: 0.5;
}

A: As name says - it's "background-image" so it always is on the background of selected element.
I suggest that you should make an  tag in parent element and style it so that .parent would have attribute position: relative and img should have position: absolute.
Also remember to set top and left/right position for .

A: it's Answered here https://stackoverflow.com/a/3798797/9017484
You can do it by making the background into a pattern:
<defs>
  <pattern id="img1" patternUnits="userSpaceOnUse" width="100" height="100">
    <image href="wall.jpg" x="0" y="0" width="100" height="100" />
  </pattern>
</defs>

your code should be something like this:

<svg viewBox="0 0 500 150" preserveAspectRatio="none" style="height: 100%; width: 100%;">
    <defs>
      <pattern id="img1" patternUnits="userSpaceOnUse" width="100" height="100">
        <image href="ImageFile.jpg" x="0" y="0" width="100" height="100" />
      </pattern>
    </defs>
<path d="M213.19,0.00 C152.69,70.06 270.03,70.06 202.98,150.00 L500.00,150.00 L500.00,0.00 Z" style="stroke: none; fill: url(#img1);"></path>
</svg>


• Travel to customers and potential customer's facilities providing them with information and support as required to secure and maintain business. • Establish, maintain, and develop business, social and civic relationships with customers in the assigned territory to maximum profitability. • Responsible for selling Wismettac products to all customers in the assigned territory. • Grow business by anticipating customer needs, determining appropriate solutions to business problems, and providing an overall high level of customer service. • Communicate plans to customers and effectively manage resources to support sales volume. • Communicate new product and service opportunities, information or feedback gathering through field activity to appropriate company staff. • Cooperate with accounting department to assure the collection of account receivable due. • Assist in establishing sales objectives for territory in cooperation with (Deputy) Area Team Managers or (Assistant) Sales Manager. • Cooperate with assigned Operation and Logistic Company staff to achieve territory sales goals. • Provide timely reporting of activities in the field to assist in business and product planning. • Submits sales activity reports and other reports as needed. • Attend quarterly inventory count. • Other duties as assigned.
Are you energized by interacting with customers face to face while educating them on current food trends? Do you have a passion for providing awesome service while driving sales? At Wismettac, we're hiring highly motivated individuals to help with customer engagements. Our sales associates will be responsible to develop business with up-in-coming local restaurants and grocery stores. Our fast-paced sales team has a positive attitude and a drive to win and succeed. If you are looking for a career, and if you are someone that loves food and sharing that knowledge with others then we need you to join our team as a Sales Associate.
Candidate's will go through a screening process. If your application is selected one of our recruiters will contact you to schedule an initial phone interview. If your application is selected to move forward you will be invited to an in-person interview at US Career Forum. We will also be holding walk-in interviews during US Career Forum as well.
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Summarize any qualification or skills you have acquired from employment or other experience related to the sales position?
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Please watch interview with Putnam County Sheriff Don Smith. Wayne County Sheriff Barry Virts sent me this link, and Sheriff Virts agrees with the sentiments and statements of Don Smith. Sheriff Virts was co-author of the Sheriff's Official Statement about Cuomo's Legislative Action.


In the article published in this issue (pp. 1-19), "A new arena in cardiac surgery: pediatric coronary artery bypass surgery (PCABS)", the author, Dr. Soichiro Kitamura describes the development of PCABS utilizing the uni-or bilateral internal thoracic arteries (ITAs), which has become the standard operation for coronary artery obstruction due to Kawasaki disease (KD) and congenital coronary diseases in infants and children.
In this cover figure, angiograms are shown before and after surgery in a one-year-old boy who developed giant coronary aneurysms in all coronary artery systems, a typical severe KD case, at the age 5 months. At the age of 16 months, the right coronary artery was totally obstructed at the outlet of the aneurysm and 95% stenosis at the origin of the left anterior descending artery. Aneurysms of the left main coronary trunk and the circumflex artery underwent spontaneous regression, which is also a common phenomenon in KD. Two arterial grafting procedures were performed using a surgical microscope and resulted in good graft patency, and the patient is now conducting normal growth and life.
Sudden death or myocardial infarction in children occurs frequently following KD, and KD was recognized as a formidable disease in the 1970s when coronary artery sequelae occurred. Attempts to rescue these children by surgery conducted by Dr. Kitamura's group have cast light on the surgical treatment of coronary complications. PCABS utilizing the ITAs was innovative because of the capability of longitudinal and circumferential growth by the ITA graft in response to the child's subsequent somatic growth.


text:
New Brighton man brandished claw hammer during street confrontation
A MAN brandished a claw hammer during a confrontation in the street.
Liverpool crown court this week heard John McCaffery, 47, went out into the road after arming himself with the weapon from his home to argue with Wayne Carter.
But John Ballam, defending, told how McCaffery was chased for nearly 200 yards down Egerton Street, New Brighton, by Mr Carter, who was wielding a stick.
He told the crown court: "Regrettably he had a lot to drink and felt threatened and that was why he carried out the offence."
McCaffery was initially charged with affray, but after the witnesses failed to turn up for his trial, the prosecution offered no evidence against him and accepted a guilty plea to possessing an offensive weapon on May 22 last year.
The Recorder of Liverpool, Judge Henry Globe QC accepted there were "provocative circumstances".
He said he was prepared to give McCaffery, of Egerton Street, "a chance" and imposed a 12-month community order with 180 hours unpaid work.


text:
Find out Which A-List Guest Stars You'll See on 'Curb Your Enthusiasm'
Aubry D'Arminio September 28, 2017 3:00 pm
John P. Johnson/courtesy of HBO
Pictured: Larry David
Larry David won't say much about the return of his kinda-sorta autobiographical vérité farce after a six-year hiatus. And you thought Game of Thrones' producers had tight lips. Like GOT's White Walkers, though, he favors the element of surprise.
"You know when you wake up with a pimple and you're like, 'Where'd that come from?'" executive producer Jeff Schaffer says. "Larry wants Season 9 to pop up with no warning. Curb is the comedy pimple that just appeared on your face."
Larry David on the Return of 'Curb Your Enthusiasm': 'I Was Missing It!'
And how did Judge Judy end up in an episode? We've got the answer here!
That said, here's some notice: Costars Jeff Garlin, Susie Essman and JB Smoove are back. So are guests Richard Lewis, Bob Einstein, Ted Danson, Mary Steenburgen and Cheryl Hines. Cameos include Bryan Cranston (as Larry's therapist), Elizabeth Perkins (as a pal's new girlfriend) and Lauren Graham (as an NBC censor).
Bet on David's alter ego angering the lot. "Curb is a comedy-horror movie, really," Schaffer says. "Something happens and you go, 'Larry, walk away! Walk away! Oh no, he's going to turn around! Don't turn around, Larry! Don't say anything! Oh no, you did. Oh no!'"
Curb Your Enthusiasm, Season Premiere, Oct. 1, 10/9c, HBO
This article also appeared in the Sept. 18–Oct. 1 issue of TV Guide Magazine.
Subscribe to TV Guide Magazine.
The Biggest Stars on TV #3: Ted Danson


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