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Q: CI/CD: "ResourceDeploymentFailure" when deploying an Azure Data Factory with Managed private endpoint I'm facing a problem while deploying an ADF project in the prod stage off my CI/CD pipeline.
*
*I'm using Azure DevOps.
*I'm using a release pipeline, triggered when a new build is available (each validated PR on the main branch triggers a new build).
*Each release pipeline has two stages : staging and prod.
I am using a managed private endpoint connecting an ADLS account. This was causing me some problems (example), but I got it to work in the staging phase. I can now successfully deploy in staging.
I then cloned the staging stage in the release pipeline, and changed the ARM parameters in the ARM Template deployment activity. However, that activity fail in the prod stage and gives me a generic error message :
##[error]Conflict: {
"status": "Failed",
"error": {
"code": "ResourceDeploymentFailure",
"message": "The resource operation completed with terminal provisioning state 'Failed'."
}
}
I use others activities (PowerShell scripts to stop the trigger etc.) that work correctly.
When I go to my prod ADF, I can see that the pipelines, datasets and linked services are successfully deployed, but the managed private endpoint provisioning status appear as failed. I tried different things with no success, the relevant facts are:
*
*I use an Azure DevOps service connection (ARM) with Contributor role at the prod subscription level.
*My ADF is in the rg-adf-prod ressource prod, while ADLS is in the rg-dw-prod ressource group (both rg are in the prod subscription).
I know that the problem has to do with that managed private endpoint but I cannot figure what. I deleted it several times, even tried to use a subscription deployment scope in the ARM Template deployment activity, with no luck. Also, I tried to create a new storage account and put it in the same ressource group as the ADF (as I do in staging), still not luck.
Below are the YAML definition of the ARM Template deployment activity in the staging stage (successful) and prod stage (error).
Staging:
steps:
- task: AzureResourceManagerTemplateDeployment@3
displayName: 'ARM Template deployment: Resource Group scope'
inputs:
azureResourceManagerConnection: ADevopsAdfStg
subscriptionId: '<dev test sub id>'
resourceGroupName: 'rg-adf-stg'
location: 'France Central'
csmFile: '$(System.DefaultWorkingDirectory)/ADF_ARN_BUILD/AdfArmTemplates/ARMTemplateForFactory.json'
csmParametersFile: '$(System.DefaultWorkingDirectory)/ADF_ARN_BUILD/AdfArmTemplates/ARMTemplateParametersForFactory.json'
overrideParameters: '-factoryName "adfalexpocstg" -DATALAKE_properties_typeProperties_url "https://<stg adls>.dfs.core.windows.net/" -AzureDataLakeStorage1_properties_privateLinkResourceId "/subscriptions/<dev test sub id>/resourceGroups/rg-adf-stg/providers/Microsoft.Storage/storageAccounts/<stg adls>"'
timeoutInMinutes: 20
Prod:
steps:
- task: AzureResourceManagerTemplateDeployment@3
displayName: 'ARM Template deployment: Resource Group scope'
inputs:
azureResourceManagerConnection: ADevopsProd
subscriptionId: '<prod sub id>'
resourceGroupName: 'rg-adf-prod'
location: 'France Central'
csmFile: '$(System.DefaultWorkingDirectory)/ADF_ARN_BUILD/AdfArmTemplates/ARMTemplateForFactory.json'
csmParametersFile: '$(System.DefaultWorkingDirectory)/ADF_ARN_BUILD/AdfArmTemplates/ARMTemplateParametersForFactory.json'
overrideParameters: '-factoryName "adfalexpocprod" -DATALAKE_properties_typeProperties_url "https://<prod adls>.dfs.core.windows.net/" -AzureDataLakeStorage1_properties_privateLinkResourceId "/subscriptions/<prod sub id>/resourceGroups/rg-dw-prod/providers/Microsoft.Storage/storageAccounts/<prod adls>"'
timeoutInMinutes: 20
Thank you
|
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| 144
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Boys' basketball: Windham knocks off defending state champions
Mike Gilman hits a go-ahead 3-pointer with 30 seconds remaining in the Eagles' 55-53 upset of top-seeded Portland.
By Steve CraigStaff Writer
Nazari Henderson celebrates after Windham beat Portland 55-53 in a Class AA North boys' basketball semifinal at the Cross Insurance Arena on Tuesday. Staff photo by Jill Brady
There will be a new state champion in Class AA boys' basketball.
Windham's senior backcourt of Mike Gilman and Nick Curtis combined to shoot down two-time defending champion Portland, 55-53, in a North semifinal Tuesday at Cross Insurance Arena.
Gilman's 3-pointer from the left corner with 30 seconds remaining gave Windham its first lead of the game.
The go-ahead basket came after Portland standout Terion Moss missed the front end of a 1-and-1 with 39.4 seconds left.
As Windham advanced the ball, Gilman drove his defender off screens.
"It was a great call by Coach. I couldn't have picked a better play myself," Gilman said. "I had been staying in the far corner most of the game, and I knew if I went baseline I'd lose him, and my teammates set two great screens for me and I was wide open in the (opposite) corner."
Gilman drained the shot for a 54-53 lead. Portland came up empty on its next two possessions, both 3-point misses by Griffin Foley, with a Curtis free throw in between.
The fourth-seeded Eagles (15-5) will face No. 2 Edward Little (16-4) at 6:30 p.m. Friday. Windham beat Edward Little, 69-54, on Dec. 22.
Windham has never won a boys' basketball regional title.
Gilman finished with 23 points and Curtis had 21, including two 3-pointers in the fourth quarter after top-ranked Portland opened a 51-45 lead.
"They have two real good players, and both players in the fourth quarter came through," said Portland Coach Joe Russo. "Gilman hit that big 3, and Curtis kept them in the game when we were up six. So the two best players on their team came through, and we got the ball in the hands of the guys we wanted and came up short."
Moss led Portland with 22 points. Trey Ballew played well inside and scored 17 points, but the Bulldogs' supporting cast of Foley, Pedro Fonseca, Manny Yugu and 3-point specialist Simon Chadbourne (0 points) combined for just 15 points.
"Shutting down his co-workers, his guys that he relies on, and making sure Chadbourne didn't get off and not allowing him to get comfortable was a real key to our success," Windham Coach Chad Pulkkinen said in reference to Moss.
Portland took a 13-2 lead and stayed in front until the final minute, but Windham kept it close.
"Coach kept telling us all week, when you're down on the mat, get back up, keep fighting," said Curtis, who scored 12 points after picking up his fourth foul early in the third quarter.
"We talked a lot about how we knew we were just as good as them, if not better, if we played a really good game, and we felt we were the best team in this tournament," Pulkkinen said.
Portland went 76-8 with two state titles and three regional crowns during Moss's four varsity seasons, Foley was integral the past three years, and Yugu and Fonseca were key players on last year's championship team.
"That's going to be hard, knowing we're not in it, but we know what our legacy as a team (is)," Foley said. "We know we did so much for the school and the name of the program."
Steve Craig can be reached at 791-6413 or:
Twitter: SteveCCraig
Browse more in High School Sports
Boys' Hockey
Girls' Hockey
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Receive high schools sports news and scores in your inbox from preseason to the state championship games by entering your email address below.
|
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| 9,030
|
\section{Introduction}
Chiral symmetry breaking has been studied in detail
with chiral invariant {\em and confining} quark models
in the chiral limit of a
vanishing current quark mass $m_0$.
For a finite current quark mass $m_0\neq 0 $,
studies exist with an approximate confinement
\cite{Munczek:1983dx,Jain:1991pk,Munczek:1991jb,Jain:1993qh}
or with a quadratic confining potential
\cite{Bicudo:1989sh,Bicudo:1989si,Bicudo:1989sj},
but very few studies have been performed
\cite{linear4,LlanesEstrada:2004wr}
with an exactly linear confining potential.
Since the current quark masses of the six standard flavours $u, \, d, \, s, \, c , \, b, \, t$ span over five orders of magnitude from 1.5 MeV to 171 GeV, here we develop a new numerical method to study the quark mass gap and the quark vacuum energy density, with a linear exactly confining potential, from very small to very large quark masses.
Notice that even in the chiral limit of $m_0=0$, the quark has a constituent
running mass $m(p)$ function of the momentum $p$, solution of the mass
gap equation (equivalent to the Schwinger-Dyson equation) for the quark.
Recently we have shown how to measure in the excited hadron spectra
the running mass $m(p)$
\cite{Bicudo:2009cr}.
The chiral invariant and confining quark models
have also been applied to phase studies at finite temperature
$T$ and chemical potential $\mu$
\cite{Bicudo:1993yh,Battistel:2003gn,Antunes:2005hp,Glozman:2007tv,Guo:2009ma,Lo:2009ud,Kojo:2009ha,Nefediev:2009zzb}.
A finite quark mass is relevant both for the study of hadrons
which have been investigated for decades,
and for the study the QCD phase diagram which will be
explored in the future at RHIC, LHC and FAIR.
In the phase diagram, a finite current quark mass $m_0$
affects the position of the critical point between the
crossover at low chemical potential $\mu$ and the phase transition
at higher $\mu$. Moreover the current quark mass
affects the QCD vacuum energy density $\cal E$, relevant
for the dark energy of cosmology.
This all occurs in the dynamical generation of the quark
mass $m(p)$.
While the quark condensate $ \langle \bar \psi \psi \rangle$
is a frequently used order parameter for chiral symmetry breaking,
the mass gap, {\em i. e. } the quark mass at vanishing
momentum $m(0)$ is another possible order parameter for
chiral symmetry breaking. However, due to technical difficulties,
$m(0)$ has not been computed in detail previously in confining
and chiral invariant quark models.
Here we study in detail how a finite current quark
mass $m_0\neq 0 $ affects the dynamically generated quark
mass $m(p)$ .
We utilize the linear confining potential for the
quark-antiquark interaction, in the chiral quark model or Coulomb Gauge quark model,
including both confinement and chiral symmetry.
While this model, in the framework Coulomb gauge Hamiltonian
formalism is not yet full QCD, it is presently the only model able
to include both the quark-antiquark confining potential and
quark-antiquark vacuum condensation.
Importantly, since our model is well defined and solvable,
it can be used as a simpler model than QCD, and yet
qualitatively correct, to address different aspects of hadronic physics.
Is is adequate to study
the QCD phase diagram microscopically
\cite{Bicudo:1993yh,Battistel:2003gn,Antunes:2005hp,Glozman:2007tv,Guo:2009ma,Lo:2009ud,Kojo:2009ha,Nefediev:2009zzb}.
We scan the current quark masses, from the light quarks to the
heavy quarks, computing the running quark mass $m(p)$
with detail, including the infrared limit of $m(0)$, {\em i. e.} the mass gap.
In Section II we derive in detail the mass gap equation.
In Section III we review the numerical difficulties
of this non-linear integral equation,
with cancelling infrared divergences.
We solve the mass gap equation with a
new numerical method, dedicated to determine in detail the
difficult infrared region of small momentum $p \simeq 0$.
In Section IV we discuss our results and conclude.
\section{Our framework}
\subsection{Possible link to QCD.}
Our framework can be approximately derived from QCD,
in two different gauges.
In Coulomb gauge
$
\mathbf \nabla \cdot \mathbf A(\mathbf x, t)=0
$
the interaction potential,
has been derived by Lee,
\cite{TDLee},
and by Szczepaniak and Swanson
\cite{Szczepaniak:1995cw,Szczepaniak:1996gb}.
In the present study we address the quark fields only
and thus $V_I$ reduces to the quark part of the density-density term,
\def\kbar{\rlap\slash k}
\def\qbar{\rlap\slash q}
\def\pbar{\rlap\slash p}
\def\bl{{\bf l}}
\def\p{{\bf p}}
\def\a{{\bf a}}
\def\q{{\bf q}}
\def\k{{\bf k}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\A{{\bf A}}
\def\B{{\bf B}}
\def\D{{\bf D}}
\def\bsig{{\bbox{\sigma}}}
\def\h{{1\over 2}}
\noindent
\begin{eqnarray}
V_I &=+& {1\over 2} g^2\int d\x d{\bf y} \, {\cal J}^{-1} \psi^{\dag}(\x) {\rm T}^a \psi(\x)
\langle \x,a| \times
\nonumber \\
&&(\mathbf{\nabla} \cdot \D)^{-1} (-\mathbf{\nabla}^2)
(\mathbf{\nabla}\cdot \D)^{-1} | \y,b \rangle {\cal J} \psi^{\dag}(\y) {\rm T}^b \psi(\y) \nonumber \\
\end{eqnarray}
The covariant derivative in the adjoint
representation $\D = \mathbf{\nabla} - g \A$,
and ${\cal J} = \mbox{Det}[\mathbf{\nabla}\cdot \D]$
contribute to the density-density interaction, which
is expected to be confining in QCD.
Another approximate path from QCD to our model
considers the modified coordinate gauge of Balitsky
\cite{Balitsky:1985iw},
$
\mathbf A(\mathbf 0, t)=0 \ , \, \mathbf x \cdot \mathbf A(\mathbf x, t)=0
$
and in the interaction potential for the
quark sector,
\begin{eqnarray}
V_I=\int\, d^3x \left[ \psi^{\dag}( x) \;(m_0\beta -i{\vec{\alpha}
\cdot \vec{\nabla}} )\;\psi( x)\;+
{ 1\over 2} g^2 \int d^4y\, \
\right.
\nonumber \\
\overline{\psi}( x)
\gamma^\mu{\lambda^a \over 2}\psi ( x)
\langle A_\mu^a(x) A_\nu^b(y) \rangle
\;\overline{\psi}( y)
\gamma^\nu{\lambda^b \over 2}
\psi( y) + \cdots \
\label{hamilt}
\end{eqnarray}
retains the first cumulant order, of two gluons
\cite{Dosch:1987sk,Dosch:1988ha,Bicudo:1998bz}
$
g^2 \langle A_\mu^a(x) A_\nu^b(y) \rangle
$
and this also results in a simple density-density harmonic effective
confining interaction. As in QCD, this only has one scale.
Thus our framework is similar to an expansion of
the QCD interaction, truncated to the leading
density-density term.
Moreover, to address phenomenology
where the meson spectrum fits in linear Regge
trajectories,
one also needs to assume that the confining
quark-antiquark potential is a linear potential.
Notice that the short range Coulomb potential
could also be included in the interaction, but
here we ignore it since it only affects the quark
mass through ultraviolet renormalization
\cite{Bicudo:2008kc},
which is assumed to be already included in the
current quark mass.
While this is not exactly equivalent to QCD,
our framework maintains three interesting
aspects of non-pertubative QCD,
a chiral invariant quark-antiquark interaction,
\cite{Finger:1981gm,Orsay1,Orsay2,Orsay3,Orsay4,Kalinowski}
the cancellation of infrared divergences
\cite{Bicudo:1989sh,Bicudo:1989si,Bicudo:1989sj}
and a quark-antiquark linear potential
\cite{linear4,LlanesEstrada:2004wr,Szczepaniak:1995cw,linear1,linear2,linear3,Wagenbrunn:2007ie}.
\subsection{Deriving the mass gap equation}
We derive the mass gap equation,
where constituent quarks acquire
the constituent mass $m(k)$
\cite{Bicudo_scapuz}
in the true and stable vacuum,
solving the Schwinger-Dyson
equation for the quark propagator,
\be
{\cal S}^{-1}(p) =
{{\cal S}_0}^{-1}(p)
\ - \
\begin{picture}(20,15)(0,0)
\put(0,0){\line(1,0){10}}
\put(22,0){\vector(-1,0){12}}
\put(10,-7){$_{k}$}
\put(7,18){$_{p-k}$}
\put(0,0){$\cdot$}
\put(1,4.4){$\cdot$}
\put(3,7){$\cdot$}
\put(5.6,9){$\cdot$}
\put(10,10){$\cdot$}
\put(14.4,9){$\cdot$}
\put(17,7){$\cdot$}
\put(19,4.4){$\cdot$}
\put(20,0){$\cdot$}
\end{picture}
\ \ .
\ee
We utilize the truncated Schwinger-Dyson
equation at the Rainbow level,
where the dotted line represents
the same density-density interaction $V_I$
resulting identically from the truncation of
Coulomb gauge QCD or of Balitsky gauge QCD.
This leads to the same mass gap equation and quark
dispersion relation obtained assuming a quark-antiquark $^3P_0$
condensed vacuum, computing the vacuum energy density
with the Hamiltonian, and minimizing the energy density.
The relativistic invariant Dirac-Feynman \cite{Orsay4}
propagators ${\cal S}$,
can be decomposed in the quark and antiquark Bethe-Goldstone
propagators
\cite{Bicudo_scapuz},
\begin{eqnarray}
\label{quarkpropagator}
&& {\cal S}(k_0,\vec{k})
= {i \over \not k -m +i \epsilon}
\\ \nonumber
&=& {i \over k_0 -\sqrt{k^2 +m^2} +i \epsilon} \
\sum_su_su^{\dagger}_s \beta
- {i \over -k_0 -\sqrt{k^2 +m^2} +i \epsilon} \
\sum_sv_sv^{\dagger}_s \beta \ ,
\end{eqnarray}
where $m$ is the quark mass and where the quark spinor $u_s$ and antiquark spinor $v_s$ are,
\begin{eqnarray}
u_s({\bf k}) &=& \left[
\sqrt{ 1+S \over 2} + \sqrt{1-S\ \over 2} \widehat k \cdot \vec \sigma \gamma_5
\right]u_s(0)
\nonumber \\
v_s({\bf k}) &=& \left[
\sqrt{ 1+S \over 2} - \sqrt{1-S \over 2} \widehat k \cdot \vec \sigma \gamma_5
\right]v_s(0)
\ ,
\label{propagators}
\end{eqnarray}
where $ S=m /\sqrt{k^2+m^2} $ is a function of the quark mass.
Importantly, in the free propagator, the correct quark propagator in the non condensed vacuum,
the quark mass $m$ is equal to the {\em current quark mass} $m_0 $. And it is this current
quark mass $m_0$ which effects in the current quark mass we study in great detail.
However when chiral symmetry breaking occurs, $m$ is not determined from the onset.
In the physical vacuum, the {\em constituent quark mass} $m(k)$,
is a function of the momentum, a dynamical solution of the mass gap equation.
Replacing the propagator
of eq. (\ref{propagators}) in the Schwinger-Dyson equation
and projecting it with the spinors, we get the mass gap equation
and the quark dispersion relation,
\bea
\label{2 eqs}
&&0 = u_s^\dagger(k) \left\{
k \widehat k \cdot
\mbox{\boldmath \( \alpha \)}
+ m_0 \beta
-\int {d {k_0}' \over 2 \pi} {d^3 \mbf k' \over (2\pi)^3}
i \widetilde V(k-k')
{\cal S}({k_0}',\vec{k'})
\right\} v_{s''}(k) \ \
\\ \nonumber
&&E(k) = u_s^\dagger(k) \left\{k \widehat k \cdot
\mbox{\boldmath \( \alpha \)} + m_0 \beta
-\int {d {k_0}' \over 2 \pi} {d^3 \mbf k' \over (2\pi)^3}
i \widetilde V(k-k')
{\cal S}({k_0}',\vec{k'})
\right\} u_s(k),
\label{SDE}
\eea
where the usual notation for Dirac matrices is assumed.
Writing the running mass in terms of a sine and a cosine
of $\varphi(k) = \arctan{ k \over m(k)} $, the {\em chiral angle},
\bea
S(k) &=& \sin \varphi(k) = {m( k) \over \sqrt{k^2 + m(k) ^2}} \ ,
\nonumber \\
C(k) &=& \cos \varphi(k) = { k \over \sqrt{k^2 + m(k) ^2}} \ ,
\eea
the mass gap equation and the quark energy are,
\bea
0&=& + S(p) \, B(p)- C(p)\, A(p)
\label{mass gap}
\\
E(p)&=& + S(p) \, A(p)+ C(p)\, B(p)
\label{quark energy}
\eea
where the propagator functions $A$ and $B$,
respectively replacing the quark mass $m$ and
quark momentum $|\mathbf p|$ in the one-loop
dressed quark propagator of eq. (\ref{quarkpropagator}) are,
\bea
A(p)&=& m_0 + {1 \over 2} \int {d^3 \mathbf k \over (2 \pi)^3} \widetilde{V}( \mathbf p - \mathbf k)S(k) \ ,
\nonumber \\
B(p)&=& p+ {1 \over 2} \int {d^3 \mathbf k \over (2 \pi)^3} \widetilde{V}( \mathbf p - \mathbf k)(\hat p \cdot \hat k)C(k) \ .
\label{AB}
\eea
Equivalently to solve the non-linear integral mass gap equation
(\ref{mass gap}),
we can alternatively minimize the vacuum energy density per unit volume,
\be
{\cal E}= - {g \over 2} \int {d^3 \mathbf p \over (2 \pi)^3}
S(p) \left[ A(p) + m_0 \right] + C(p) \left[ B(p) +p \right]
\ee
where $g= N_f \, N_s \, N_c$ is the degeneracy factor counting the number of different but degenerate quarks.
$N_s=2$ is the number of spins and $N_c=3$ is the number of colours.
$N_f$ is the number of degenerate flavours, but since each quark has a different
current quark mass $m_0$ one should compute separately the vacuum energy difference
for each quark flavour.
\subsection{The mass gap equation for a linear confining potential}
Notice that in the case of a linear potential, divergent in the
infrared limit of large $r$, the Fourier transform needs an infrared
regulator $\mu$.
A possible form of the linear potential,
\be
V(r)= - \sigma {e^{- \mu \, r} \over \mu} \simeq - {\sigma \over \mu} + \sigma \, r - { \sigma \mu \over 2 } \, r^2 +\cdots
\label{IRdivpot}
\ee
corresponds, in the limit of small infrared regulator $\mu$, to a model of linear confinement where the quark also has
an infinite binding energy $- {\sigma \over \mu}$.
While other infrared regulations can be used for the linear potential, say $V(r)= \sigma r \, e^{- \mu \, r} $
which has no infrared divergent binding energy, the infrared divergent constant of Eq. (\ref{IRdivpot}) is exactly
cancelled in the mass gap equation by the factor in the integrand $\bigl[S(k) C(p)
- S(p) C(k) \hat k \cdot \hat p \bigr] $.
The potential in Eq. (\ref{IRdivpot}) has a simple three-dimensional Fourier transform,
\bea
V(k)
&=& \sigma { - 8 \pi \over ( k^2 + \mu^2)^2 } \ ,
\eea
and this is the momentum space potential frequently utilized to account for linear confinement.
The integrals in the angular variable $\omega$ of Eq. (\ref{AB}) are,
\bea
&& \int_{-1}^1 d \omega
{ -8 \pi \over ( k^2 + p^2 + 2 k p \omega + \mu^2)^2 }
= { - 16 \pi \over [(k-p)^2 + \mu^2] [(k+p)^2 + \mu^2] } \ ,
\nonumber \\
&& \int_{-1}^1 d \omega
{ - 8 \pi \, \omega \over ( k^2 + p^2 + 2 k p \omega + \mu^2)^2 }
\\ \nonumber
&& \ \
= { - 16 \pi \over (2 k p)^2 }
\biggl\{
-{ 2 k p (k^2 + p^2 + \mu^2) \over [(k-p)^2 + \mu^2] [(k+p)^2 + \mu^2] }
+ {1 \over 2} \log
\left[ (k+p)^2 + \mu^2 \over (k-p)^2 + \mu^2\right]
\biggr\} \ .
\eea
We find for the propagator functions $A$ and $B$,
\bea
A(p)&=& m_0 - {\sigma \over p^2} \int_0^\infty {d k \over 2 \pi}
I_A(k,p,\mu) S(k) \ ,
\nonumber \\
I_A(k,p,\mu) &=& { p k \over (p-k)^2 + \mu^2}
- { p k \over (p+k)^2 + \mu^2} \ ,
\nonumber \\
B(p)&=& p- {\sigma \over p^2} \int_0^\infty {d k \over 2 \pi}
I_B(k,p,\mu) C(k) \ ,
\nonumber \\
I_B(k,p,\mu) &=&
{ p k \over (p-k)^2 + \mu^2}
+ { p k \over (p+k)^2 + \mu^2}
+ {1 \over 2} \log {(p-k)^2 + \mu^2 \over (p+k)^2 + \mu^2}\ ,
\eea
leading to the mass gap equation in the two equivalent forms of
a non-linear integral functional equation,
\bea
\label{massgabackbacktosincos}
0 &=& p S(p) - m_0 C(p) - { \sigma \over p^2}
\int_0^\infty {d k \over 2 \pi} \, \bigl[
\\ \nonumber
&&
I_A(k,p,\mu)\, S(k) C(p)
- I_B(k,p,\mu) \, S(p) C(k) \bigr] \ ,
\eea
and of a minimum equation of the energy density,
\bea
\label{energybacktosincos}
&& {\cal E} = { -g \over 2 \pi}\int_0^\infty {dp \over 2 \pi}
\biggl[
2p^3 C(p) + 2 p^2 m_0 S(p) + \sigma \times
\\ \nonumber
&&
\int_0^\infty {d k \over 2 \pi}
I_A(k,p,\mu) \, S(k) S(p)
+ I_B(k,p,\mu) \, C(p) C(k) \biggr] \ .
\eea
Eq. (\ref{massgabackbacktosincos}) can be
rewritten as a fixed point equation for the quark mass function $m(k)$,
\bea
&& m(p) = m_0+ { \sigma \over p^3}
\int_0^\infty {d k \over 2 \pi} {I_A(k,p,\mu)\,
m (k) p -
I_B(k,p,\mu)\,
m(p) k
\over \sqrt{k^2 + m(k)^2}} \ .
\label{fixedpointeq}
\eea
Since the potential has an infinite constant independent of the mass,
in the variational equation we search for the extremum of the energy difference
\bea
\label{energydifference}
{\cal E} - {\cal E}_0&=& { -g \over 2 \pi}\int_0^\infty {dp \over 2 \pi}
\biggl\{
2p^3 [ C(p) -C_0(p) ]
+ 2 p^2 m_0 [S(p) -S_0(p)]
\\ \nonumber
&&
\ \ \ + \sigma \, \int_0^\infty {d k \over 2 \pi} I_A(k,p,\mu) \, \left[ S(k) S(p) - S_0(k) S_0(p) \right]
\\ \nonumber
&&
\ \ \ \ \ \
+ I_B(k,p,\mu) \, \left[ C(p) C(k) - C_0(k) C_0(p) \right] \biggr\} \ .
\eea
where ${\cal E}_0$ is constant, and where we use the subindex $_0$ when the mass $m(p)$ is substituted
by the constant current mass $m_0$.
These two Eqs. (\ref{fixedpointeq}) and (\ref{energydifference}) constitute the
main object of our study.
\section{Solving the mass gap equation variationally
with rational ansatze}
\subsection{Accurate numerical cancellation of infrared and ultraviolet divergences}
We use both Eq. (\ref{fixedpointeq}) and the minimization of
Eq. (\ref{energydifference}) to find the dynamical quark mass
$m(k)$, but first we must regulate correctly their divergences.
The infrared divergences are present in the term
$ p k /[(p-k)^2 + \mu^2]$,
infrared divergent in the limit of a vanishing regulator $\mu \to 0$,
which is present bot in the functions $I_A(k,p,\mu)$ and $I_B(k,p,\mu)$.
We must show that this infrared divergence is cancelled both in
the fixed point Eq. (\ref{fixedpointeq}) and in the energy density
Eq. (\ref{energydifference}).
In what concerns the fixed point Eq. (\ref{fixedpointeq}), while the denominator
diverges quadratically in $1/(k-p)^2$, the numerator $m (k) p - m(p) k$ is of
order $(k-p)$ and thus the integrand diverges like $ 1 /(p-k)$ only, and it's integral
has finite principal value.
In the energy density difference the infrared divergence also cancels,
since the numerator common to the infrared divergent terms,
\bea
&& S(p)S(k)-S_0(p)S_0(k)+C(p)C(k)-C_0(p)C_0(k)
\nonumber \\
&=& - {1 \over 2} \times \bigl[{\varphi'}^2(p) - {\varphi'_0}^2(p) \bigr] (k-p)^2
\\ \nonumber
&& - {1 \over 2} \bigl[ \varphi'(p) \varphi''(p) - \varphi'_0(p) \varphi''_0(p) \bigr] (k-p)^3 + o (p-k)^4
\eea
is then of order $(k-p)^2$.
In the ultraviolet part of the integrals, while each sub-term in the propagator
function integrands $I_A(k,p,\mu)$ and $I_B(k,p,\mu)$ is divergent, the actual
sum is ultraviolet convergent, since in
the limit of large $k$ we have,
\bea
p \, I_A(k,p,0) &= & \sum_{n=1}^\infty 4 n { p^{2 n +1} \over k ^{2 n} }
\ ,
\nonumber
\\
k \, I_B(k,p,\mu) & = & \sum_{n=1}^\infty 4 n \left({n+1 \over n + {\scriptstyle 1 \over \scriptstyle 2} }\right){ p^{2 n +1} \over k ^{2 n} }
\ ,
\label{IAIBexpansions}
\eea
and thus the integrals in $k$ have ultraviolet integrable integrands decaying like
$ p^3 /k^2$.
Also, the ultraviolet divergence in the kinetic terms of the energy density
$2p^3 C(p) + 2 p^2 m_0 S(p)$
cancels due to the difference with $2p^3 C_0(p) + 2 p^2 m_0 S_0(p)$
if the mass difference $m(k) -m_0$ vanishes sufficiently fast in the
ultraviolet.
To address correctly both the infrared and ultraviolet divergences of
the integrals in $k$ of Eqs. (\ref{fixedpointeq}) and (\ref{energydifference}),
we divide the integral in two sections, the infrared one for
$ 0 < k < 2p $ and the ultraviolet one for $ 2p < k < \infty$. In the infrared
region we compute the respective principal value, performing a symmetric sum
centred in $k=p$, maintaining a very small regulator $ \mu$ just to cancel automatically
the contribution of $k=p$. In the ultraviolet region we use the change of variable
\cite{linear1}
of Adler and Davis $ k \to x/ (1-x)$ with Jacobian $ 1 /(1-x)^2$
and with integration between $x = 2p /(1+2p)$ and 1. The change of variable
in the ultraviolet transforms, say an integral of rational functions $ 1/(1+k)^n$ into the
integral of polynomials $ (1-x)^{n-2}$ and thus it is adequate to the integral
of rational functions as we have here. Our numerical integrals in $k$
of a generic integrand $\cal I$ singular in $p$ are thus computed in the form,
\be
\int_0^\infty dk {\cal I}(k) = \mbox{P} \left[\int_0^{2 p} dk {\cal I}(k) \right]
+ \int_{2 p \over 1+ 2p} ^1 {dx \over (1- x )^2} {\cal I}\left( x \over 1 -x\right) \ ,
\label{numerical integration}
\ee
where each of the two numerical integrals can either be computed with
a rectangular, trapezoidal or Simpson sum or with the gauss quadrature method.
I one would discretize the quark mass $ m(p) $ in a series
of momenta $p_i$, then the integrals of Eq. (\ref{numerical integration})
loses accuracy. Finite differences would require many interpolations,
both in the infrared end and in the ultraviolet end, since the principal value requires requires that $k$
has many summation points smaller than $p$ and many other
larger than $p$. Moreover the correct integration of the integrand in
the ultraviolet large $k$ limit, where the integrand behaves like $ \left(p\over k\right)^2$, also
requires an integration extending beyond any value of $p$.
To solve this problem, we utilize a well defined ansatz for $m(k)$,
formally describe the parametrized as,
\be
m(p)=m(p; c_1,c_2, \cdots, c_n) \ .
\label{formalanzats}
\ee
and this allows the numerical summation for the integrals in any point
of the integration domain.
In what concerns the numerical convergence to the solution for $m(k)$,
the fixed point equation is relatively unstable, particularly in the infrared region
of $p \simeq 0$, when we are searching for the vacuum groundstate.
Notice that the mass gap equation had not only one, but an
infinite tower of solutions
\cite{Bicudo:2003cy}. The excited solutions dominate
the fixed point iteration, converging to the larger eigenvalue of the iterated matrix,
because they minimize the denominator $\sqrt{k^2 + m^2}$.
Previously in the literature, the fixed point method was provided with extra stability
with two different methods, Adler and Davis
used a cubic equation
and relaxation, to select the best solution
\cite{linear1}.
Bicudo and Nefediev quasi-linearized
the fixed point equation and selected the desired eigenvalue,
corresponding either to the stable vacuum or to excited, false vacua
\cite{Bicudo:2003cy}.
Thus they were able to find both the stable vacuum and the excited false vacua.
But these works have not yet determined in detail $m(0)$, since this demands
a very large numerical precision, and since most previous authors have focused
in computing the function $S(p)$ which in the infrared region is $S(0)=1$
regardless of the actual finite value of $m(0)$.
Importantly, the present technique of minimizing the energy density directly
tends to the right solution, which is the groundstate vacuum.
Interestingly, the variation of the ansatz parameters
$ c_1,c_2, \cdots, c_n$
of the energy density of the vacuum
$
{\cal E}={\cal E}(c_1,c_2, \cdots, c_n)
$
utilized in minimization codes with gradient
method, utilizes the fixed point equation.
Computing the partial derivatives of the energy
density we get,
\bea
{\partial {\cal E}(c_i)
\over
\partial c_i}
&=& \int d p { \delta {\cal E} \over \delta \varphi }
{\partial \varphi
\over \partial c_i }
\nonumber \\
&=& - { g \over 2 \pi} \int_0^\infty {d p \over 2 \pi }
(-2 p^2) {\cal R}(p;c_i) \, {p \over p^2 + m^2(p;c_i) } { \partial m(p;c_i) \over \partial c_i} \ ,
\label{variationenergyparameters}
\eea
where ${\cal R}(p;c_i)$ is the right hand side of the mass gap
Eq. (\ref{mass gap}),
\be
{\cal R}(p;c_i) = +S(p;c_i)B(p;c_i) - C(p;c_i)A(p;c_i) \ ,
\ee
utilized in the fixed point Eq. (\ref{fixedpointeq}).
\subsection{Choosing variational ansatze for the quark mass $m(p)$ }
To guide our choice of ansatze $m(p; c_1,c_2, \cdots, c_n)$,
we first notice that the series expansion for $I_A$ and $I_B$ in Eq. (\ref{IAIBexpansions}),
also apply when $k \leftrightarrow p$.
When replaced in the integral of
the fixed point equation (\ref{fixedpointeq}) , the series
suggest that a series expansion of $m(p)$ should only
have even terms, {\em i. e.} $m(p)$ should be a function of $p^2$.
$m(p)$ should also be a finite function since our integrals
are finite.
In what concerns the asymptotic ultraviolet tail of the integral
of the fixed point equation (\ref{fixedpointeq}), there
are two different limits we can address. In the case
of a large current quark mass $m_0$,
$ m_0 \over k^2 +{m_0}^2 $ interpolates between $1$ in the
infrared region of the integral and $ m_0 \over k $ in the
ultraviolet region of the integral. Using these
approximation, the components of the integral are analytical,
\C{
\bea
\label{analytical A B}
\int_0^\infty {d k \over 2 \pi} p \, I_A(k,p,\mu) &=& {p^3 \over 2 \, \mu} \ ,
\\
\int_0^\infty {d k \over 2 \pi} k \, I_B(k,p,\mu) &=& {p^3 \over 2 \, \mu} \ ,
\nonumber
\\
\int_0^\infty {d k \over 2 \pi} { p \over k} I_A(k,p,\mu) &=&
{ p^2 \arctan { p \over \mu }\over \pi \, \mu}
\nonumber
\\
&=& { p^2 \over 2 \mu} - {p \over \pi} + o \left( \mu \right) \ ,
\nonumber
\\
\int_0^\infty {d k \over 2 \pi} I_B(k,p,\mu) &=&
{- p \mu + (p^2 + \mu^2) \arctan { p \over \mu }\over \pi \, \mu}
\nonumber \\
&=&
{ p^2 \over 2 \mu} - 2 {p \over \pi } + o \left( \mu \right)
\ ,
\nonumber
\eea
}
and thus adding the respective components we find that in the
infared dominated approximation the integrals cancel, while
in the ultraviolet dominated approximation the integral with
$ m_0 \over k $ produces an ultraviolet behaviour of
\C{
$m(p) - m_0 \to { m_0 \sigma \over \pi \, p^2}$.
}
Thus in the case of large $m_0$ we expect that $m(p) - m_0 $
decays in the ultraviolet like $ 1/ p^2$.
In the case where $m_0 \simeq 0$, assuming
then that in the large $p$ limit the dynamical quark
mass vanishes sufficiently fast, the fixed point equation leads to,
\bea
m(p)
&\to&
{ \sigma \over p^3}
\int_0^\infty {d k \over 2 \pi} {1 \over \sqrt{k^2 + m(k)^2}}
4 { k^2 \over p^2} m (k) p
\nonumber \\
&\to&
{ 4 \sigma \over p4}
\int_0^\infty {d k \over 2 \pi} {k ^2 m (k) \over \sqrt{k^2 + m(k)^2}}
\eea
and, providing $m(p)$ decays faster than $1 / p^2$ for a
finite integral, this decays in the ultraviolet like $1 / p^4$.
The simplest possible ansatz for $m(p)-m_0$ we may have, function
of $p^2$, and encompassing both
the behaviour in $1 / p^2$ for a large current quark mass
$ m_0$ and the behaviour in $1 / p^4$ for a small current
quark mass is the rational function,
\bea
\label{ansatz4}
{\cal A}_3 (p ) &=& {1 \over c_0 + c_2 p^2 + c_4 p^4} \ .
\eea
\C{
This ansatz is a Pad\' e approximant, and to check whether our
simple ansatz is sufficient, it is convenient to check that the next
Pad\' e approximant,
}
a more flexible rational ansatz with
two more parameters,
\bea
\label{ansatz6}
{\cal A}_5 (p ) &=& {1 + n_2 p^2\over d_0 + d_2 p^2 + d_4 p^4+ d_6 p^6}
\eea
leads to the same result.
In both ansatze of Eq. (\ref{ansatz4})
and Eq. (\ref{ansatz6}) we assume that the parameters are positive.
While Eq. (\ref{ansatz4}) is a decreasing
function, Eq. (\ref{ansatz6}) may have a different behaviour at the origin,
either with an initial increase, or with a steeper decrease, an thus
it has room in it's parameter space to verify if the ansatz of
Eq. (\ref{ansatz4}) is close to the correct solution of the mass gap
equation.
We also check numerically that ansatze with steeper infrared behaviours,
including in the denominator terms like a $c_1 k$ or a $c_{-1} /k$ would
not improve the solution since the best solution would have $c_1=c_{-1}=0$.
A better ansatz than ${\cal A}_3(p)$ is ${\cal A}_5(p)$, however the improvement
of the solution is very small, almost invisible to the naked eye in graphics.
The partial redundancy between the numerator and denominator parameters
of ${\cal A}_5(p)$ already slows the convergence to the minimum of the energy
density $\cal E$. Thus an ansatz with more parameters than ${\cal A}_5(p)$
is not necessary.
The only ansatze we adopt here are the ones of the rational functions
${\cal A}_3(p)$ and ${\cal A}_5(p)$.
\subsection{One loop results for the large current mass $m_0 >> \sqrt \sigma $ limit }
We now compute the first iteration of the fixed point method starting with $m(k)=m_0$.
In the simple case case of a constant mass $m(k)=m_0$,
we can compute the integral in the fixed point
equation with a large precision. Defining the mass
difference $ {\cal D}$,
\be
{\cal D} (p) = m(p) - m_0 \ ,
\ee
we compute ${\cal D} (p) $ in the case a constant mass $m_0$ is used in the integrand,
\be
{\cal D} _0 (p) = { m_0 \sigma \over p^3}
\int_0^\infty {d k \over 2 \pi} {I_A(k,p,\mu)\,
p -
I_B(k,p,\mu)\,
k \over \sqrt{k^2 + m_0^2}} \, .
\ee
We get an accurate result for the integral $ {\cal D}_0 (p) $,
with a numerical integration decomposed according to Eq. (\ref{numerical integration}).
This provides a good quark mass solution to $m(p)=m_0+ {\cal D}(p)$
whenever the current quark mass is large, i. e. when $m_0 >> {\cal D}(p)$.
In that case the integral $ {\cal D}_0(p) $ only needs to be computed once,
since this one-loop approximation is already excellent.
Moreover we can rescale in $m_0$, and then with a single computation
we get $m(p)$ for for any constant mass
$m_0$. Denoting $\tilde k = k / m_0$ and so forth we get,
\bea
{\cal D}_0 (p) &=& { \sigma \over m_0 \tilde p^3}
\int_0^\infty {d \tilde k \over 2 \pi} {I_A (\tilde k, \tilde p, \tilde \mu)\,
\tilde p -
I_B(\tilde k, \tilde p, \tilde \mu)\,
\tilde k \over \sqrt{ \tilde k^2 + 1}}
\nonumber \\
&=& {\sigma \over m_0} {\cal F} ({p \over m_0}) \ ,
\nonumber \\
{\cal F}(p) &=&
{ 1 \over p^3}
\int_0^\infty {d k \over 2 \pi} {I_A (k, p, \mu)\,
p -
I_B( k, p, \mu)\,
k \over \sqrt{ k^2 + 1}} \ .
\label{D0}
\eea
Thus we only need to compute the dimensionless function ${\cal F}(p)$ and this
will produce the dynamical quark mass for any current quark
mass $m_0$ larger than the typical scale $\sqrt \sigma$ of our problem,
\be
m(p) \simeq m_0 + {\sigma \over m_0} {\cal F} ({p \over m_0}) \ .
\ee
The solution of the integration, obtained with the simplest numerical
rectangular sum in both integrals, but needing $10^3$ integration points at least,
is represented in Fig. \ref{massdiff1fop}.
To be able to perform an accurate integration both in the infrared
and the ultraviolet, it is necessary to have an analytical function.
We obtain an analytical function by fitting the function ${\cal F} (p )$ ,
and we get an excellent fit already with the very simple ansatz
${\cal A}_3(p)$ of Eq. (\ref{ansatz4}) for the parameter set,
\bea
c_0&=& 4.7664, \ c_2= 3.4762, \ c_4= 0.0000, \
\label{parset3}
\eea
with almost no graphically visible difference
in Fig. \ref{massdiff1fop}
from the ansatz ${\cal A}_5(p)$
with two more parameters of Eq. (\ref{ansatz6})
for the parameter set,
\bea
d_0&=& 4.7250, \ d_2=4.9818, \ d_4= 0.8392, \
d_6 = 0.0000, \ n_2= 0.2626,
\label{parset5}
\eea
and both fits confirm the $1/p^2$ decay of the mass in the ultraviolet,
for large current quark masses.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8\columnwidth]{largemomassdiffit46.eps}
\end{center}
\caption{Dimensionless mass difference function ${\cal F}(\tilde p)$ computed in
the limit when the current quark mass $m_0$ is large.
The dots show the numerical integral of Eq. (\ref{D0})
and the two almost overlapping solid lines show our fits with the
two different rational ansatze of Eq. (\ref{ansatz4}) and Eq. (\ref{ansatz6}).
\label{massdiff1fop}}
\end{figure}
Now we can include the large quark limit scaling in $m_0$,
and also apply to the quark mass difference ${\cal D}_0 (p )$
the ansatz $ {\cal A}_3 (p ) $ of Eq. (\ref{ansatz4}), and in this case the
ansatz parameters should be respectively,
\bea
{c_0} = 4.7664 \, m_0 \ ,
{c_2} = 3.4762 \, m_0^{-1} \ ,
{c_4} = 0.0000 \, m_0^{-3} \ .
\label{parsofm0}
\eea
We further compute the vacuum energy density difference for a large current quark mass $m_0$.
Although the mass difference ${\cal D}_0(p)$ in Eq. (\ref{D0}) decreases like
$ {m_0}^{-1}$ in the large current quark mass $m_0$ limit,
a dimensional analysis of ${\cal E} -{\cal E}_0$ shows
that possibly it does not vanish. Since the energy difference is finite
and independent of the infrared cutoff $\mu$, then our
only dimensionfull parameters are the string tension
$\sigma$ (scaling like a mass square) and the current quark mass $m_0$ (scaling like a mass).
Because the vacuum energy density per volume scales like a mass to the fourth power,
a large mass expansion may have terms of the
form ${m_0}^4$, ${m_0}^2 \sigma$, $\sigma^2$, $\sigma^3 {m_0}^{-2}, \cdots$
However the first term in this series clearly vanishes in the
vacuum energy difference ${\cal E} -{\cal E}_0$. Then the question
is what is the first non-vanishing term in this expansion.
\begin{table}[t!]
\begin{center}
\begin{tabular}{c| c c c c}
\hline
$m_0$ & $ 10^4 {{\cal E} - {\cal E}_0 \over g} $ & $c_0$ & $c_2$ & $c_4$ \\
\hline
$ 0.000100000 $ & $ -0.473813 $ & $ 6.24900 $ & $ 26.7910 $ & $ 17.5059 $ \\
$ 0.000316228 $ & $ -0.478133 $ & $ 6.24287 $ & $ 26.7090 $ & $ 17.3392 $ \\
$ 0.00100000 $ & $ -0.491581 $ & $ 6.22441 $ & $ 26.4168 $ & $ 16.8674 $ \\
$ 0.00316228 $ & $ -0.532245 $ & $ 6.17597 $ & $ 25.4076 $ & $ 15.8086 $ \\
$ 0.0100000 $ & $ -0.649468 $ & $ 6.01623 $ & $ 23.2517 $ & $ 12.0965 $ \\
$ 0.0316228 $ & $ -0.958565 $ & $ 5.68682 $ & $ 18.5897 $ & $ 6.38068 $ \\
$ 0.100000 $ & $ -1.62048 $ & $ 5.22374 $ & $ 12.5110 $ & $ 1.50433 $ \\
$ 0.316228 $ & $ -2.58117 $ & $ 5.04452 $ & $ 7.22212 $ & $ 0.051756 $ \\
$ 1.00000 $ & $ -3.27613 $ & $ 7.03428 $ & $ 3.06500 $ & $ 0.000000 $ \\
$ 3.16228 $ & $ -3.46213 $ & $ 17.1365 $ & $ 1.04266 $ & $ 0.000000 $ \\
$ 10.0000 $ & $ -3.47122 $ & $ 51.7243 $ & $ 0.336172 $ & $ 0.000000 $ \\
$ 31.6228 $ & $ -3.39336 $ & $ 153.536 $ & $ 0.113085 $ & $ 0.000000 $ \\
\hline
\end{tabular}
\caption{The results for the ansatz parameters of ${\cal A}_3(p)$
\C{
for $m(p)-m_0$
}
obtained with our minimization code. All results are in dimensionless units of $\sigma=0.19$ GeV$^2=1$.
\label{table 1} }
\end{center}
\vspace{0.5cm}
\end{table}
\begin{table}[t!]
\begin{center}
\begin{tabular}{c| c c c c c c}
\hline
$m_0$ & $ 10^4 {{\cal E} - {\cal E}_0 \over g} $ & $d_0$ & $d_2$ & $d_4$ & $d_6$ & $n_2$ \\
\hline
0.000100000 & -0.473908 & 6.170 & 31.9 & 22. & 14. & 0.5 \\
0.000316228 & -0.478221 & 6.163 & 32.3 & 23. & 15. & 0.6 \\
0.00100000 & -0.491647 & 6.147 & 32.2 & 24. & 16. & 0.6 \\
0.00316228 & -0.532266 & 6.09 & 33.1 & 30. & 19. & 0.9 \\
0.0100000 & -0.649397 & 5.9705 & 29.582 & 28.37 & 12.83 & 0.871 \\
0.0316228 & -0.958497 & 5.7202 & 25.358 & 32.40 & 7.4897 & 1.301 \\
0.100000 & -1.624761 & 5.369 & 11.875 & 5.05 & 0.0000 & 0.149 \\
0.316228 & -2.59104 & 5.467 & 6.720 & 2.08 & 0.0000 & 0.205 \\
1.00000 & -3.27722 & 7.5349 & 4.4398 & 0.8105 & 0.0000 & 0.2546 \\
3.16228 & -3.46240 & 16.665 & 1.2612 & 0.0105 & 0.0000 & 0.0104 \\
10.0000 & -3.47148 & 49. & 0.6 & 0.00 & 0.0000 & 0.00 \\
31.6228 & -3.76155 & 92. & 0.2 & 0.0000 & 0.0000 & 0.0000 \\
\hline
\end{tabular}
\caption{\label{table 2} The results for the ansatz parameters of ${\cal A}_5(p)$
\C{
for $m(p)-m_0$
}
obtained with our minimization code. All results are in dimensionless units of $\sigma=0.19$ GeV$^2=1$.}
\end{center}
\vspace{0.5cm}
\end{table}
To answer this question, we expand the vacuum energy density in a $\sigma \over {m_0}^2$ series,
similar to the variation in Eq. (\ref{variationenergyparameters}), but now
based in the expansion of the quark dynamical mass,
\be
m(p) = m_0 + {\sigma \over {m_0}} m_1 \left(p \over m_0 \right)
+ {\sigma^2 \over {m_0}^3} m_2 \left(p \over m_0 \right) + \cdots
\ee
and then we get
\bea
{\cal E}&=&
{ -g \over 2 \pi}\int_0^\infty {dp \over 2 \pi}
\biggl\{ 2 p^2
\biggl[ p C_0(p) + m_0 S_0(p)
+ { p \delta C_0(p) + m_0 \delta S_0(p) \over \delta m_0 } \left( m(p) -m_0 \right)
\nonumber \\ &&
\hspace{2.5cm}
+{ 1 \over 2} { p \delta^2 C(p) + m_0 \delta^2 S_0 (p) \over \delta {m_0} ^2 } \left( m(p) -m_0 \right)^2
+ \cdots \biggr]
\\ &&
+ \sigma \, \int_0^\infty {d k \over 2 \pi} I_A(k,p,\mu) \, \biggl[ S_0(p) S_0(k)
+ 2 { \delta S_0(p) \over \delta m_0 }S_0(k ) \left( m(p) -m_0 \right) + \cdots \biggr]
\nonumber \\ \nonumber &&
+ I_B(k,p,\mu) \, \biggl[ C_0(p) C_0(k)
+ 2 { \delta C_0(p) \over \delta m_0 }C_0(k ) \left( m(p) -m_0 \right) + \cdots \biggr]
\biggr\} \ .
\eea
The zeroth order term vanishes when we perform the difference of the
vacuum energy densities ${\cal E}-{\cal E}_0$. Then the first
order term in the kinetic energy density also vanishes since the kinetic
energy density is minimized by $m(p)=m_0$.
Thus the leading term is of second order in the kinetic energy density,
and of first order in the potential energy density,
and both are proportional to $ \sigma^2$.
Using the intermediate steps,
\bea
&& { p \delta^2 C_0(p) + m_0 \delta^2 S_0 (p) \over \delta {m_0} ^2 } =
- {p^2 \over \left( p^2 + {m_0}^2 \right)^{3/2}} \ ,
\\
\nonumber
&& \sigma \int_0^\infty {d k \over 2 \pi} \biggl[
{2 \delta S_0(p) \over \delta m_0 } I_A(k,p,\mu) \, S_0(k) + I_B(k,p,\mu) \, C_0(k) \biggr]
{2 \delta C_0(p) \over \delta m_0 }
\\
\nonumber
&&
\hspace{4.1cm}
= 2 {p^4 \over \left( p^2 + {m_0}^2 \right)^{3/2}} {\sigma \over m_0} D_0 \left( p \over m_0 \right) \ ,
\eea
and changing variable to the dimensionless $ \tilde p $, we get,
\bea
{\cal E}-{\cal E}_0
&=& \sigma^2 { -g \over 2 \pi} \int_0^\infty {d \tilde p \over 2 \pi}
{ \tilde p ^4 [ {\cal F} (\tilde p )]^2 \over ( \tilde p^2 + 1 )^{7 \over 2} }
+ o\left( \sigma^3 \over {m_0}^2 \right) .
\eea
Finally using the ansatz $ {\cal A}_3 (p ) $ of Eq. (\ref{ansatz4}),
with the parameter set of Eq. (\ref{parset3})
this results in
\bea
{\cal E}-{\cal E}_0
&\simeq& - - 3.47701 \times 10^{-4} \, \sigma^2 \, g \ .
\label{enerofm0}
\eea
\begin{figure}[t!]
\includegraphics[width=.5\columnwidth]{energyofmass.eps}
\hspace{0.1cm}
\includegraphics[width=.5\columnwidth]{parc0ofmass.eps}
\\
\\
\vspace{0.0cm}
\includegraphics[width=.5\columnwidth]{parc2ofmass.eps}
\hspace{0.1cm}
\includegraphics[width=.5\columnwidth]{parc4ofmass.eps}
\\
\caption{
Plots of our numerical solution with the ansatz ${\cal A}_3(p)$,
as a function of the current quark mass $m_0$:
(a) the vacuum energy density shift ${\cal E} -{\cal E}_0$,
(b) parameter $c_0$ ,
(c) parameter $c_2$ ,
(d) Parameter $c_4$.
The dots show our numerical solution, the solid line is the large $m_0$
limit obtained with Eqs. (\ref{parsofm0}) and (\ref{enerofm0}), and
the vertical dot-dashed lines represent the
current masses of the quarks $u$, $d$, $s$, $c$, $b$, $t$.
\label{fig_solutionmodel4}}
\end{figure}
\begin{figure}[t!]
\includegraphics[width=.5\columnwidth]{massdifofmass.eps}
\hspace{0.1cm}
\includegraphics[width=.5\columnwidth]{massgap.eps}
\\
\caption{
Plots of the mass gap as a function of the current quark mass $m_0$:
(a) the mass gap difference $m(0)-m_0$, measuring the amount
of generated dynamical mass,
(b) the mass gap $m(0)$ .
The dots show
our numerical solution, the solid line is the leading order
obtained when $m_0 \to \infty$, and the vertical dot-dashed lines represent the
current masses of the quarks $u$, $d$, $s$, $c$, $b$, $t$.
Notice that the dynamical mass generation has a maximum
for finite quark masses close to the strange quark mass.
All results are in dimensionless units of $\sigma=0.19$ GeV$^2=1$.
\label{fig_massgap}}
\end{figure}
\section{Results}
Utilizing our ansatze of Eq. (\ref{ansatz4}) and Eq. (\ref{ansatz6}),
we may now compute with great accuracy the
integrals of Eq. (\ref{energydifference}) which now are a function of the ansatz parameters,
and apply a standard minimizing code to determine the optimal
parameters.
We use 1000 $\times$ 1000 integration points and up
to 40 decimal digits in order to be able to find a convergence of the method
in the case of the ansatz ${\cal A}_5(p)$, due to the partial redundancy of the parameters.
Then we also minimize the energy starting from different randomly generated initial values for the parameters.
In Tables \ref{table 1} and \ref{table 2} we only show the digits which are stable,
in the sense that they do not depend on the initial values.
Notice that the energy density obtained with the two different ansatze differ only
by a few per mil and that the ansatz of Eq. (\ref{ansatz6}) already exhibits some
redundance in the parameters. This shows that the ansatz of Eq. (\ref{ansatz4}) is
already quite accurate for the parametrization of the quark mass $m(p)$.
Unlike the fixed point method, converging quite fast (with a single iteration) for large current quark masses $m_0 >> \sqrt \sigma $,
the variational method converges faster for small current quark masses. Thus both methods
are complementary. In Tables \ref{table 1} and \ref{table 2}
and in Fig. \ref{fig_solutionmodel4}
we show the results of our minimization for two ansatze and for different current quark masses spanning over
five orders of magnitude. In Figs. \ref{fig_massgap} and \ref{fig_massinsomecases} we illustrate the
mass difference at the momentum origin $m(0)-m_0$, the mass gap $m(0)$ and the running mass $m(p)$
for different current masses $m_0$.
\C{
Now that we have an excellent and simple ansatz for the running quark mass, we may compute the
regularized quark condensate and the quark dispersion relation. The quark condensate $\langle \bar \psi \psi \rangle$
is another possible order parameter, to be compared with the other order parameter
we computed, {\it i. e. } the quark mass at vanishing momentum $m(0)$.
The quark condensate is computed from the one-loop quark propagator functions in Eq. (\ref{AB}), and
it is ultraviolet divergent for finite quark masses. Thus we regularize the quark condensate, subtracting
the quark condensate for a current quark,
\be
\langle \bar \psi \psi \rangle -\langle \bar \psi \psi \rangle_0
= -{g \over \pi^2}
\int_0^\infty k^2 \, d k \left[ {m (k) \over \sqrt{k^2 + m(k)^2}} - {m_0 \over \sqrt{k^2 + {m_0}^2}} \right]
\ee
}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8\columnwidth]{Dmofpdifm0.eps}
\end{center}
\caption{The quark mass function difference ${\cal D}(p)=m(p)-m_0$
measuring the extent of dynamical mass generation, is
represented with increasing number of dashes per curve
for five different current quark masses $m_0$
with values $10^{-4}, 10^{-1}, 10^{-1/2}, 10^{0}, 10^{1/2}$,
in dimensionless units of $\sigma=0.19$ GeV$^2=1$.
\label{fig_massinsomecases}}
\end{figure}
\C{
The one quark dispersion relation $E(p)$, defined in Eq. (\ref{quark energy}),
is relevant for the boundstate equation of mesons or of baryons.
For instance, in the instantaneous Salpeter equation,
a hamiltonian $H= E_q+ E_{\bar q} + V_{q \, \bar q}$ can be defined for mesons
(actually the hamiltonian is a matrix
\cite{Bicudo_scapuz}
including negative and positive energy components).
The dispersion relation $E(p)$
is infrared divergent due to the infinite constant of quark-antiquark potential detailed
in Eq. (\ref{IRdivpot}). In momentum space, this leads to an infinite Dirac delta
in the integral present in $E(p)$. We can regularize the numerical integral, subtracting
a term to cancel the integrand when $k=p$, a term that we add back analytically,
\bea
&& E(p) =
p C(p) + m_0 S(p) + {\sigma \over p^2}\, \int_0^\infty {d k \over 2 \pi} \biggl\{
I_A(k,p,\mu) \, \left[ S(k) - S(p) \right ] S(p)
\label{regularized E}
\\
\nonumber
&& + I_B(k,p,\mu) \, \left[ C(p) -C(p) \right] C(k) + \left[ I_A(k,p,\mu) S^2(p)+ I_B(k,p,\mu) C^2(p) \right] \biggr\} \ ,
\eea
in particular the integral of $I_A \, S^2+I_B \, C^2$ is analytical thanks to Eq. (\ref{analytical A B}),
and in the limit of a vanishing infrared regulator $\mu \to 0$ this term, that we subtracted and must now
add back to the quark energy, reduces to a simple infrared divergence plus a finite term,
\be
{\sigma \over p^2}\, \int_0^\infty {d k \over 2 \pi} I_A(k,p,\mu) S^2(p)+ I_B(k,p,\mu) C^2(p)
\to
{\sigma \over 2 \mu} - { 2 \sigma \over \pi} { p \over p^2 - m(p)^2 }
\ .
\label{analytical regulator}
\ee
\C{
\begin{table}[t!]
\begin{center}
\begin{tabular}{c| c c c c c}
\hline
$m_0$ & $ \left(-\langle \bar \psi \psi \rangle +\langle \bar \psi \psi \rangle_0 \over g\right)^{1\over 4} $ & $e_0$ & $e_1$ & $e_3$ & $e_5$ \\
\hline
$ 0.000100000 $ & $ 0.255569 $ & $ 0.21104 $ & $ 1.9549 $ & $ 102.65 $ & $ 451.90 $ \\
$ 0.000316228 $ & $ 0.255847 $ & $ 0.21156 $ & $ 1.9539 $ & $ 102.23 $ & $ 448.34 $ \\
$ 0.00100000 $ & $ 0.256701 $ & $ 0.21321 $ & $ 1.9518 $ & $ 100.94 $ & $ 436.59 $ \\
$ 0.00316228 $ & $ 0.259029 $ & $ 0.21814 $ & $ 1.9488 $ & $ 97.304 $ & $ 399.80 $ \\
$ 0.0100000 $ & $ 0.267020 $ & $ 0.23387 $ & $ 1.9269 $ & $ 86.717 $ & $ 317.83 $ \\
$ 0.0316228 $ & $ 0.287014 $ & $ 0.27830 $ & $ 1.8859 $ & $ 65.176 $ & $ 173.48 $ \\
$ 0.100000 $ & $ 0.331664 $ & $ 0.39429 $ & $ 1.8758 $ & $ 35.484 $ & $ 49.785 $\\
$ 0.316228 $ & $ 0.428943 $ & $ 0.71809 $ & $ 1.8628 $ & $ 13.582 $ & $ 3.6035 $ \\
$ 1.00000 $ & $ 0.771803 $ & $ 1.6210 $ & $ 2.2704 $ & $ 2.8947 $ & $ 0.0076 $ \\
$ 3.16228 $ & $ 0.984801 $ & $ 4.7134 $ & $ 2.4773 $ & $ 0.34516 $ & $ 0.0000 $ \\
$ 10.0000 $ & $ 1.26682 $ & $ 14.999 $ & $ 2.4199 $ & $ 0.03598$ & $ 0.0000 $ \\
$ 31.6228 $ & $ 1.60590 $ & $ 48.113 $ & $ 2.3283 $ & $ 0.00371 $ & $ 0.0000 $ \\
\hline
\end{tabular}
\caption{The results for the regularized quark condensate and for the
regularized quark dispersion relation obtained with the ansatz of Eq. (\ref{quarkenergy}).
All results are in dimensionless units of $\sigma=0.19$ GeV$^2=1$.
\label{table 3} }
\end{center}
\vspace{0.5cm}
\end{table}
}
Importantly, this divergence is physically irrelevant since, in the hamiltonian of any colour singlet
hadron, the sum of the divergences of the quark and of the antiquark energies cancel
with the infrared divergence of the quark-antiquark potential detailed in Eq. (\ref{IRdivpot}).
Finally for the purpose of future computations of the hadron spectra, it is convenient
to write the dispersion relation as a sum of the analytical infrared term of Eq. (\ref{analytical regulator}),
plus the free quark dispersion relation dominating the ultraviolet,
and plus a finite and compact term $\widetilde E(p)$, an integral that we compute numerically,
\bea
E(p) & = & {\sigma \over 2 \mu} - { 2 \sigma \over \pi} { p \over p^2 - m(p)^2 } + p C(p) + m_0 S(p) + \tilde E(p) \ .
\label{compact energy}
\eea
The numerical integral $\widetilde E(p)$ decays like $1 /p^5$ in the chiral limit of small current quark
masses and decays like $1/p^3$ in the case of large current quark masses. $\widetilde E(p)$ is negative
and we can conveniently fit it with the rational function, or Pad\' e approximant with odd powers of $p$ only,
\be
\widetilde E(p) \simeq -{1 \over e_0 +e_1 p + e_3 p^3 + e_5 p^5} \ .
\label{quarkenergy}
\ee
We show the best fitting parameters $e_0, e_1, e_3, e_5$ in Table \ref{table 3}. With our
excellent fits of the dynamical quark mass $m(p)$ and of the quark dispersion relation $E(p)$
we are well equipped to address further problems,
such as the breaking of chiral symmetry or the hadronic excited spectra at finite temperature $T$.
}
\section{Conclusion}
While the chiral limit of
$m_0 << \sqrt \sigma $ was already well known in the literature, we find unanticipated effects for finite $m_0 \simeq
\sqrt \sigma$ and for heavy $m_0 >> \sqrt \sigma$ current quark masses.
We study in detail the large $m_0$ limit, performing an one-loop expansion in the dimensionless
number $\sigma / {m_0 }^2$.
We also develop a new technical approach to solve the mass gap equation, utilizing the variational
principle to increase the precision of our mass solution $m(p)$ in the infrared limit of
$p \to 0$, relevant to compute the mass gap $m(0)$, an order parameter for the chiral
phase transition.
We also show that the dynamical generated constituent quark mass $m(p)$ can be quite well fitted by
our inverse even quartic polynomial ansatz,
\C{
a Pad\' e approximant
}
with parameters $c_0, \, c_2$ and $c_4$ depending only on the
current mass $m_0$.
\begin{figure}[t!]
\includegraphics[width=.55\columnwidth]{quarkcondensate.eps}
\hspace{0.1cm}
\includegraphics[width=.45\columnwidth]{onequarkenergy.eps}
\caption{
(a) We show a log log plot of minus the regularized quark condensate $- \langle \bar \psi \psi \rangle + \langle \bar \psi \psi \rangle_0$
as a function of the current quark mass $m_0$, with vertical dot-dashed lines representing the
current masses of the quarks $u$, $d$, $s$, $c$, $b$, $t$.
(b) We represent the regularized quark dispersion relation $E(p) - {\sigma \over 2 \mu}$
with increasing number of dashes per curve
for five different current quark masses $m_0$
with respective values $10^{-4}, 10^{-1}, 10^{-1/2}, 10^{0}, 10^{1/2}$.
The quark condensate has an inflection point
for finite quark masses close to the strange quark mass, and
for large masses it rises linearly with $m_0$.
All results are in dimensionless units of $\sigma=0.19$ GeV$^2=1$.
\label{solutionmodel4}}
\end{figure}
Our surprising results are that the dynamical mass generation has finite effects persistent beyond
the chiral limit. At $m_0 \simeq \sqrt \sigma$ , in particular for masses similar to the strange quark
$s$ mass, the quark mass generation $ m(0) -m_0$ is maximum,
as shown in Fig. \ref{fig_massinsomecases},
while one would naively expect the quark mass
generation to be maximum in the chiral limit close to the up $u$ or down $d$ quark masses.
A second order parameter,
the regularized quark condensate $\langle \bar \psi \psi \rangle - \langle \bar \psi \psi \rangle_0$,
monotonously grows in absolute value with $m_0$
and shows an inflexion point for masses similar to the strange quark $s$ mass,
as depicted in Fig. \ref{solutionmodel4}.
In the limit of heavy quark masses $m_0 >> \sqrt \sigma$ of the charm $c$, bottom $b$ and top $t$ quarks,
although the mass gap difference $ m(0) -m_0$ vanishes like $ \sigma / m_0$, it occurs that the
energy density difference ${\cal E} - {\cal E}_0$ is maximum. In the heavy quark limit, the energy
density difference converges to a constant limit when $m_0 \to \infty$,
and we show in Fig. \ref{fig_solutionmodel4} that it is one order of magnitude larger than in the chiral limit.
This may be relevant for cosmology, contributing to the dark energy.
\C{
The numerical variational technique developed here, together with the detailed
solutions for the running quark mass $m(p)$ and for the quark dispersion relation $E(p)$
as a function of the current quark mass $m_0$ and of the string tension
$\sigma$, are necessary tools for the continuation of our program to study the QCD
phase diagram and the hadron spectrum at finite temperature $T$ and density $ \mu$,
when the string tension $\sigma$ becomes quite small, possibly smaller than
the light current quark masses.
}
{\bf acknowledgements}
\\
I am very grateful to Gast\~ao Krein on the variational method,
and to Marlene Nahrgang, to Pedro Sacramento and to Jan Pawlowski
for discussions on the QCD phase diagram motivating this paper.
I acknowledge the financial support
of the FCT grants CFTP,
CERN/ FP/ 109327/ 2009
and
CERN/ FP/ 109307/ 2009.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,889
|
\section*{Abstract}
Poloidal asymmetries in tokamaks are usually investigated in the context of various transport processes, usually invoking neoclassical physics. A simpler approach based on magnetohydrodynamics (MHD), focusing on the effects rather than the causes of asymmetries, yields useful insights into the generation of shear flows and radial electric field.
The crucial point to recognize is that an MHD equilibrium in which the plasma pressure is not a flux function can be maintained only by contributions from mass flows.
Coupling between the asymmetry-generated forces and toroidal geometry results in a strongly up-down asymmetric effect, where the flows exhibit a strong dependence on the location of the asymmetry with respect to the midplane. This location-dependence can be used as an effective control mechanism for the edge and thus the global confinement in tokamaks. It can also explain a number of poorly-understood observations. For instance, strong dependence of the low to high (L-H) confinement transition power threshold $P_{LH}$ on the magnetic topology can be qualitatively explained within this framework. Similarly, upper-lower midplane dependence of the poloidal flow direction after massive gas injections (MGI) naturally follows from this discussion. Similar arguments suggest that the ITER fueling ports above the midplane, to the extent they can generate a positive pressure asymmetry at the edge, are misplaced and may lead to higher input power requirements.
\maketitle
\section{Introduction}
In an idealized magnetohydrodynamic (MHD) description of the axisymmetric plasma equilibrium, rapid parallel transport removes any density and temperature gradients along the field lines and leads to a highly symmetric state in which the plasma pressure is constant in both the poloidal and toroidal directions on flux surfaces.
In modern, diverted tokamaks, however, at least the poloidal symmetry is broken near the plasma edge due to changes in magnetic topology and the presence of strongly asymmetric perpendicular transport processes. Some commonly recognized sources of symmetry breaking are intrinsic error fields and externally imposed magnetic perturbations (see, for example, \cite{evans2015, in2017}). Possibly more importantly, the presence of a divertor in modern tokamaks and the accompanying neutral recycling near the X-point, a major source of fueling\cite{fukuda2000, groth2011}, results in a poloidally localized density and pressure increase within the separatrix in the same area\cite{carreras1998}.
In addition, gas or pellet injection, either for fueling or as a disruption mitigation mechanism, are also fundamental sources of at least poloidal symmetry breaking (see Baylor\cite{baylor2007b} for ITER ports).
The potential for asymmetries to drive plasma flows have been recognized for quite sometime.
The first neoclassical studies of the role of poloidally asymmetric transport in the spin-up of a tokamak plasma can be traced back to Stringer\cite{stringer1969}. Others have expanded upon the ``Stringer mechanism'' to offer an explanation for the origin of the low to high (L-H) confinement transition\cite{wagner1982} in terms of the flows generated by inboard-outboard asymmetry of the neoclassical and turbulent particle fluxes\cite{hassam1991, hassam1993}. There have been also a number of studies examining the role of asymmetric neutral particle transport at the edge in driving toroidal flows (see, for example, Singh {\em et al.}\cite{singh2004}). However, in all these the emphasis has been on the asymmetry of the transport rather than the plasma profiles, and the theoretical treatment has involved intricate neoclassical arguments.
In his work we assume that the nonuniform distribution of particle sources at the plasma edge can lead to poloidally nonuniform density and pressure profiles. We show that the MHD equilibria consistent with these asymmetries necessarily have strong edge flows and driven radial electric fields. In turn, the flows and fields can have a profound effect on confinement through reduced turbulence\cite{shaing1989, hahm1995} and improved macroscopic stability\cite{bondeson1994, fitzpatrick1996, pustovitov2007b}. However, the focus of this work is on the generation of shear flows and radial electric field, not on their specific role in stability or confinement.
There is also a known inverse relationship between plasma dynamics and poloidal asymmetries. Centrifugal forces due to strong toroidal mass flows can introduce an in-out asymmetry in the density and pressure profiles\cite{zehrfeld1972, hameiri1983, guazzotto2004}. Similarly, Alfv\'enic poloidal flows can lead to shocks with sharp poloidal density gradients\cite{shaing1992, seol2016}. Thus, there is a close connection between poloidal density or pressure asymmetries and mass flows. A great deal of this connection can be understood within a generalized MHD equilibrium framework. An earlier work\cite{aydemir2018c} addressed limited-tokamak equilibria. This work will focus on the more realistic, diverted configurations where the poloidal field at the edge is allowed to have one or more null points.
At this point a brief review of the axisymmetric tokamak equilibrium with flows will be useful.
\subsection{Some properties equilibrium flows}
Most general form of the plasma flow consistent with an axisymmetric MHD equilibrium can be obtained simply as follows\cite{zehrfeld1972, hameiri1983}: In steady-state, Faraday's law combined with the ideal MHD Ohm's law leads to ${\bm u}\times{\bm B} =\nabla\phi,$ where $\phi$ is the electrostatic potential. Writing the axisymmetric field in the form ${\bm B}=\nabla\zeta\times\nabla\psi + F\nabla\zeta$, where $F=R^2\nabla\zeta\cdot{\bm B}$ and $\zeta$ is the usual toroidal angle, the resulting $E\times B$ velocity becomes
\begin{equation}
{\bm u}_\perp = -\frac{\Omega F}{B^2}{\bm B} + \Omega R^2\nabla\zeta. \label{eqn:uPerp}
\end{equation}
In general, the mass flux due to ${\bm u}_\perp$ will not satisfy the steady-state continuity equation: $\nabla\cdot\rho_m{\bm u}_\perp\ne 0$, where $\rho_m$ is the mass density. Thus, a parallel ``return flow'' (similar to the ``return'' Pfirsch-Schl\"uter current) is required to make the mass flux incompressible. Letting ${\bm u} = (u_\parallel/B){\bm B} + {\bm u}_\perp$ and requiring $\nabla\cdot\rho_m{\bm u} = 0$ leads to
\begin{equation}
\frac{u_\parallel}{B} - \frac{\Omega F}{B^2} = \frac{\Phi(\psi)}{\rho_m},
\end{equation}
where $\Phi$ is an arbitrary flux function. Combining with Eq.~\ref{eqn:uPerp} finally gives the general expression for the equilibrium flow velocity,
\begin{equation}
{\bm u} = \frac{\Phi(\psi)}{\rho_m}{\bm B} + \Omega(\psi) R^2\nabla\zeta. \label{eqn:equilFlow}
\end{equation}
A purely toroidal flow ($\Phi\rightarrow 0$) is possible only if there is a parallel flow of magnitude $u_\parallel = \Omega F/B$, which allows the poloidal projection of the parallel flow to cancel the poloidal component of ${\bm u}_\perp.$ However, a purely poloidal flow requires an unrealistic density profile of the form $\rho_m =f(\psi)/R^2$, where $f=-\Phi F/\Omega$ is an arbitrary flux function.
In the presence of equilibrium flows, it is convenient to write the momentum equation in the form
\begin{equation}
\rho_m\frac{\partial{\bm u}}{\partial t} = -\rho_m\nabla(u^2/2) - \rho_m{\bm w}\times{\bm u} + {\bm J}\times{\bm B} - \nabla p - \nabla\cdot\stackrel{\leftrightarrow}{\bm{\pi}},\label{eqn:momentum}
\end{equation}
where ${\bm w} \equiv \nabla\times{\bm u}$ is the vorticity, and $\stackrel{\leftrightarrow}{\bm{\pi}}$ is the viscous stress tensor. In this work, the viscous stress tensor term is assumed to have the following form that is common in fluid calculations:
\begin{equation}
\nabla\cdot\stackrel{\leftrightarrow}{\bm{\pi}} = \rho_m\nu \nabla\times\curl {\bm u} - \frac{4\rho_m\nu}{3}\nabla\nabla\cdot{\bm u} - \rho_m\gamma_p{\bm u}_p, \label{eqn:stressT2}
\end{equation}
where $\nu$ is the kinematic viscosity (momentum diffusivity), $\nu = \mu/\rho_m$, and $\mu$ is the usual scalar viscosity coefficient. The last term represents poloidal flow damping due to ``magnetic pumping''\cite{hassam1978}, where we assume $\gamma_p=0.68\nu_{ii}/\epsilon$\cite{shaing2015}.
Here the scalar pressure term has to be treated carefully. The often-assumed adiabatic equation of state, with non-vanishing poloidal flows ($\Phi\ne 0$) leads to
\begin{equation}
\frac{p}{\rho_m^\gamma} = S(\psi), \label{eqn:adiabatic}
\end{equation}
where $\gamma$ is the ratio of specific heats, and $S$, a measure of the entropy, is a flux function. However, in modern tokamaks, rapid parallel thermal transport ensures that the temperature itself is a flux function, $T=T(\psi)$, which is consistent with $p=\rho_mT$ and Eq.~\ref{eqn:adiabatic} only if $\gamma=1$ and $S\rightarrow T,$ thus effectively forcing an isothermal equation of state.
Assuming $\nabla\cdot\stackrel{\leftrightarrow}{\bm{\pi}}=0$ for the moment and using an isothermal equation of state with $p=\rho_mT(\psi),$ the parallel component of Eq.~\ref{eqn:momentum} in steady-state leads to the ``Bernoulli equation''\cite{hameiri1983}
\begin{equation}
\frac{\Phi^2B^2}{2\rho_m^2} + T\ln\rho_m - \frac{R^2\Omega^2}{2} = H(\psi), \label{eqn:bernoulli}
\end{equation}
where $H(\psi)$ is another arbitrary flux function. This equation, without the viscous stress tensor contribution, puts a constraint on the possible location of poloidal asymmetries. Taking the poloidal derivative gives
\begin{eqnarray}
\left(\frac{T(\psi)}{\rho_m} - \frac{\Phi^2B^2}{\rho_m^3}\right) \frac{\partial\rho_m}{\partial\theta} & = & \frac{\Omega^2}{2}\frac{\partial R^2}{\partial\theta} - \frac{\Phi^2}{2\rho_m^2}\frac{\partial B^2}{\partial\theta}. \label{eqn:constraint}
\end{eqnarray}
For a tokamak we generally have $B\propto 1/R,$ and the two terms on the right-hand side have opposite signs since $(\partial R^2/\partial\theta) / (\partial B^2/\partial\theta) < 0.$ Thus, the density (and pressure) can have an extremum at a given point only if both terms vanish independently, which can happen only at the midplane, $\theta=\{0,\pi\}.$
A necessary conclusion is that to support a poloidal density extremum at an arbitrary point $\theta\ne \{0,\pi\}$, the viscous stress tensor in Eq.~\ref{eqn:momentum} is needed.
Although the arguments of this section are useful for an intuitive understanding of the basic properties of equilibrium flows, they omit the effects of particle sources. Also the discussion leading up to Eq.~\ref{eqn:equilFlow} assumes the existence of flux surfaces and fails near $X$-points. Effects of pressure asymmetries associated with external sources are introduced below. Complications due to poloidal field nulls have to be dealt with numerically and are discussed in subsequent sections.
\subsection{Poloidal torque and equilibrium calculations}
Independent of any transport mechanism, a poloidal pressure asymmetry on a flux surface can generate a net torque resulting in poloidal and toroidal flows\cite{aydemir2018c}. This purely geometric effect can be understood easily: For convenience using axisymmetric flux coordinates $(\psi,\theta,\zeta)$ and starting with a localized pressure perturbation so that
\begin{equation}
p(\psi,\theta) = p_0(\psi) + \delta p(\psi,\theta), \label{eqn:p}
\end{equation}
the poloidal force ${\bm F}_t$ within a flux surface and the torque $T_\zeta$ that drives a poloidal flow can be calculated as (see Fig.~\ref{fig:fig1_aydemir}(a))
\begin{eqnarray}
{\bm F}_t & = & -\frac{1}{h_\theta^2}{\bm t}({\bm t}\cdot\nabla p),~~{\bm t} \equiv \frac{\partial {\bm \rho}}{\partial\theta},~~h_\theta \equiv |{\bm t}|, \label{eqn:Ft} \\
T_\zeta & = & ({\bm \rho}\times {\bm F}_t)_\zeta. \label{eqn:Tzeta}
\end{eqnarray}
For convenience we let $\delta p(\psi,\theta) = \delta p(\psi)f(\theta)$ and assume a wrapped Gaussian profile for $f(\theta)$ centered around $\theta=\theta_0$ to ensure periodicity. Then the net torque can be obtained by a simple surface average: $\fsAver{T_\zeta}_{s}=\oint T_\zeta dS_\psi/\oint dS_\psi.$ In circular geometry, it is given by\cite{aydemir2018c}
\begin{equation}
\fsAver{T_\zeta}_s = -\frac{r\delta p(r)}{2\pi R_0}\sum_{k=-\infty}^\infty \oint e^{-(\theta-\theta_0 + 2\pi k)^2/w^2} \sin\theta d\theta. \label{eqn:averTorque}
\end{equation}
For $w \ll 2\pi$, $\fsAver{T_\zeta}_s$ is approximately a sinusoidal function of $\theta_0$. The torque is positive if a positive pressure perturbation is in the lower half-plane $(\pi < \theta_0 < 2\pi)$, and vice versa. Note that the sign of the torque is entirely due to geometric effects in a torus and is independent of the current or toroidal field directions.
But of course the radial electric field associated with the driven flows will be a function of the magnetic field.
Equation~\ref{eqn:averTorque} implies that the sign of the pressure perturbation, in addition to its location, plays an essential role; thus, it will be helpful to consider some experimental influences that broadly help determine the sign of $\delta p$. Neutral recycling near the $X$-point, and gas puffing or pellet injection, to the extent they add particles but not energy to the plasma, can be treated as being adiabatic to a first approximation. Therefore, for these processes we should have $\delta p/p \simeq \gamma\delta n/n,$ where $\gamma$ is the adiabatic index and $\delta n$ is the change in the particle density. Thus, naturally-occurring fueling through neutral recycling in the divertor chamber, or direct fueling using main-ion gas or pellets should lead to a positive pressure perturbation in quasi-steady state.
However, disruption mitigation efforts using massive gas or (shattered) pellet injection of high-Z materials (MGI, or SPI) should generate a negative pressure perturbation since their goal
is the fast radiative collapse of the temperature (from keV to eV-range), a process that is by no means adiabatic. Thus, despite the increase in the particle density, we should expect $\delta p < 0$ for both MGI and SPI. Implications of this point will be discussed further in the section on massive impurity-injection-driven flows.
The equilibrium flows driven by the torque $T_\zeta$ of Eq.~\ref{eqn:averTorque} need to be calculated taking into account both the resulting asymmetric electromagnetic forces and various damping mechanisms. This is accomplished using the CTD code\cite{aydemir2015}, which solves the momentum equation shown in Eq~\ref{eqn:momentum}, with the stress tensor term shown in Eq.~\ref{eqn:stressT2}.
After perturbing a static, symmetric equilibrium with a perturbation of the form shown in Eq.~\ref{eqn:p}, a new equilibrium with flows is obtained in the asymptotic limit $t\rightarrow \infty,~\partial{\bm u}/\partial t \rightarrow 0.$ During this relaxation process, the plasma current and toroidal flux are held constant using appropriate boundary conditions, while $\delta p$ is maintained externally.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=2in]{fig1_aydemir.pdf}
\caption{\em \baselineskip 14pt (a) A localized positive pressure perturbation (red) near the separatrix in a double-null (DN) magnetic geometry. The wrapped Gaussian center is at $\theta_0=1.60$. (b) Resulting negative poloidal shear flow. The toroidal field and plasma current are directed out of the plane of the figure (``the standard configuration''). The figures can be magnified arbitrarily to see the details.}
\label{fig:fig1_aydemir}
\end{center}
\end{figure}
Since configurations without a field null were discussed in some detail earlier\cite{aydemir2018c}, here we will concentrate on geometries with one or more $X$-points. In an up-down asymmetric magnetic geometry, i.e., an upper or lower single null (USN, or LSN), various transport processes can generate average mass flows also\cite{strauss1995, aydemir2007a, aydemir2007b}. Therefore, the effects of poloidal pressure asymmetries are best studied in isolation in a balanced double-null (DN) geometry, and that will be the focus of the next section.
\section{Poloidal pressure asymmetries in double-null (DN) geometry}
In a DN geometry, transport-driven effects from the upper and lower nulls cancel, and without a pressure asymmetry there is no net flow or a radial electric field within our MHD model; thus, $\fsAver{u_\theta}=\fsAver{u_\zeta}=\fsAver{E_\rho}=0.$
(From hereon $\fsAver{...}$ refers to the usual flux-surface average and not the ``surface-average'' of Eq.~\ref{eqn:averTorque}.)
With a pressure asymmetry above the midplane (Fig.~\ref{fig:fig1_aydemir}(a)), however, there is a negative average net torque, which in turn drives negative (in the ion diamagnetic drift direction) poloidal flows (Fig.~\ref{fig:fig1_aydemir}(b)). The shear flows are localized mostly around the separatrix.
As suggested by the earlier discussion (and Eq.~\ref{eqn:averTorque}), location of the asymmetry has a strong influence on both the sign and amplitude of the shear flows, and the resulting radial electric field. This point is explicitly demonstrated in Fig.~\ref{fig:fig2_aydemir}, where the flux-surface-averaged field is plotted as a function of the poloidal angle $\theta_0$, center of the wrapped Gaussian used for the pressure perturbation (See Eq.~\ref{eqn:averTorque} and Fig.~\ref{fig:fig1_aydemir}). The figure shows the field extremum, which occurs near the separatrix (Fig.~\ref{fig:fig3_aydemir}(b)). If the points had been chosen more symmetrically about the midplane, we would expect the curve to exhibit more closely the simple sinusoidal symmetry $\fsAver{E_\rho}(\theta_0) =-\fsAver{E_\rho}(2\pi-\theta_0)$. It is clear, however, that the field extremum is positive for perturbations in the upper midplane ($0 < \theta_0 < \pi$) and negative for those in the lower midplane ($\pi < \theta_0 < 2\pi$), as implied by our earlier arguments.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=2in]{fig2_aydemir.pdf}
\caption{\em \baselineskip 14pt Extremum of the flux-surface-averaged radial electric field as a function of the angle $\theta_0$, center of the wrapped Gaussian pressure perturbation.}
\label{fig:fig2_aydemir}
\end{center}
\end{figure}
Radial details of the flux-surface-averaged poloidal and toroidal velocities for the perturbation of Fig.~\ref{fig:fig1_aydemir} ($\theta_0=1.6$) are plotted in Fig.~\ref{fig:fig3_aydemir}(a), showing their highly sheared nature near the separatrix, which is located at $\rho=0.81$ as measured at the midplane. In Fig.~\ref{fig:fig3_aydemir}(b), the average radial electric field is shown, which exhibits a positive peak just inside the separatrix. Here the electric field is calculated using
\begin{equation}
E_\rho = -u_\theta B_\zeta + u_\zeta B_\theta. \label{eqn:E}
\end{equation}
Thus a large negative poloidal flow is correlated with a positive radial electric field.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=1.5in]{fig3_aydemir.pdf}
\caption{\em \baselineskip 14pt (a) Flux-surface-averaged poloidal (solid red) and toroidal (dashed blue) velocities (normalized) produced by the pressure asymmetry of Fig.~\ref{fig:fig1_aydemir}(a) $(\theta_0=1.60)$. (b) Flux-surface-averaged radial electric field (normalized). The separatrix is at $\rho=0.81$.}
\label{fig:fig3_aydemir}
\end{center}
\end{figure}
In general, the shear flows and the associated electric field shown above would be expected to have a stabilizing influence on turbulence and the MHD modes localized near the separatrix. However, recall that other physics such as the ion orbit loss mechanism\cite{shaing1989} generate a {\em negative} radial electric field inside the separatrix, and in general, the L-H transition is accompanied by a deepening electric-field well\cite{hahm1995}. Thus, the positive field in Fig.~\ref{fig:fig3_aydemir}(b) would oppose and possibly neutralize this process, and for this reason this location for the poloidal pressure asymmetry would have to be considered as {\em unfavorable} from the point of view of turbulent transport and stability.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=1.5in]{fig4_aydemir.pdf}
\caption{\em \baselineskip 14pt (a) Flux-surface-averaged poloidal and toroidal velocities (normalized) produced by a positive pressure asymmetry at $\theta_0=4.2$, the lower X-point. (b) Flux-surface-averaged radial electric field for $\delta p > 0$ (dotted), and $\delta p < 0$ (solid).}
\label{fig:fig4_aydemir}
\end{center}
\end{figure}
Moving the pressure asymmetry below the midplane, on the other hand, leads to quite favorable results. The deepest $E_\rho$ well is produced for $\theta_0\simeq 5.2$ (see Fig.~\ref{fig:fig2_aydemir}), an approximate mirror-reflection about the midplane of the perturbation in Fig.~\ref{fig:fig1_aydemir}(a). However, since we anticipate naturally-occurring pressure asymmetries near the $X$-points, we next examine a case with $\theta_0=4.2$, which places the perturbation near the lower $X$-point in Fig.~\ref{fig:fig1_aydemir}(a).
As seen in Fig.~\ref{fig:fig4_aydemir}(a) both the poloidal and toroidal shear flows reverse direction now, and a negative radial-electric-field well forms just inside the separatrix, with $E_\rho$ becoming positive further inside (Fig.~\ref{fig:fig4_aydemir}(b), the dotted curve labelled ``$\delta p > 0$''). This negative potential well should help with the L-H transition; in fact, it may even be sufficient to trigger it. Note that the positive toroidal rotation driven inside (where $\fsAver{E_\rho}>0$) is consistent with co-current ``intrinsic'' rotation observations after the L-H transition\cite{rice2004b} and also with its source being at the plasma edge.
In Fig.~\ref{fig:fig4_aydemir}(b) for $\fsAver{E_\rho}$, the curve labelled ``$\delta p < 0$'' is for a {\em negative} pressure perturbation of equal magnitude at the same location. As the earlier simple torque-based arguments imply, with $\delta p < 0$ the torque reverses sign, leading to a reversal of the driven flows (not shown) and the resulting radial electric field (solid curve).
\section{Poloidal pressure asymmetries in single-null magnetic geometry}
In this section, the flows and radial electric field driven by pressure asymmetries in a lower-single-null (LSN) are investigated. Results for an upper-single-null (USN) are similar and can be obtained using symmetry arguments. As stated earlier, an up-down asymmetric magnetic geometry by itself can generate average mass flows\cite{strauss1995, aydemir2007a, aydemir2007b}; therefore, to isolate the effects of pressure asymmetry, a baseline LSN equilibrium with $\delta p=0$ that has only transport-driven flows is used for comparison.
Figure~\ref{fig:fig5_aydemir} corresponds to Fig.~\ref{fig:fig3_aydemir} of the DN configuration above and shows a positive perturbation above the midplane at $\theta_0=1.6$. Contribution to the electric field, now defined as
$\delta\fsAver{E_\rho}\equiv \fsAver{E_\rho} - \fsAver{E_\rho}_0$, where $\fsAver{E_\rho}_0$ is for the baseline equilibrium with $\delta p=0$, is again positive (Fig.~\ref{fig:fig5_aydemir}(b), driven by the negative poloidal flow near the separatrix.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=1.5in]{fig5_aydemir.pdf}
\caption{\em \baselineskip 14pt LSN geometry. (a) Flux surfaces and a positive pressure perturbation at $\theta_0=1.6$, upper half-plane. (b) Flux-surface-averaged radial electric field (magenta, dotted) and the poloidal velocity (red, solid). Changes with respect to the baseline equilibrium with no perturbation are shown. The separatrix is at $\rho=0.86.$}
\label{fig:fig5_aydemir}
\end{center}
\end{figure}
Similarly, Figure~\ref{fig:fig6_aydemir} corresponds to Fig.~\ref{fig:fig4_aydemir} of the DN configuration and shows a positive perturbation around the $X$-point. Because of the flux expansion near the null point, the perturbation in Fig.~\ref{fig:fig6_aydemir}(a) is wider in real space, although its width in $\psi$-space is the same as in Fig.~\ref{fig:fig5_aydemir}(a). Again, there is a pronounced and negative contribution to $\fsAver{E_\rho}$ by a positive poloidal flow localized around the separatrix (Fig.~\ref{fig:fig6_aydemir}(b)).
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=1.5in]{fig6_aydemir.pdf}
\caption{\em \baselineskip 14pt LSN geometry. (a) Flux surfaces and a positive pressure perturbation at $\theta_0=4.2$, the X-point. (b) Flux-surface-averaged radial electric field (magenta, dotted) and the poloidal velocity (red, solid). Changes with respect to the baseline equilibrium with no perturbation are shown.}
\label{fig:fig6_aydemir}
\end{center}
\end{figure}
\section{Role in the L-H transition}
Recall that neutral recycling around the $X$-point (in the divertor chamber) is a significant source of plasma fueling\cite{groth2011, carreras1998}, which can lead to a localized, positive pressure perturbation in this region (this point may not be valid for the superD-X divertor geometry\cite{savarkar2018}); thus, we can expect the configuration of Fig.~\ref{fig:fig6_aydemir}(a) to occur naturally in diverted tokamaks. The resulting flows and radial electric field can play a significant role in the L-H transition, as already outlined in a preliminary version of this work\cite{aydemir2018c}.
Unfortunately, despite its long history, we still lack a quantitative theory of how tokamaks spontaneously enter the H-mode; therefore, the arguments regarding the role of these poloidal asymmetries in the transition are necessarily qualitative, although they are robust and easily explained. Similar arguments were used earlier in discussions of transport-driven flows\cite{aydemir2012}. The crucial point here is that the L-H transition is accompanied by a deepening radial-electric-field well inside the separatrix (see, for example, \cite{burrell1990}). The primary driver of this field can be the ion orbit loss mechanism\cite{shaing1989}, or a combination of other physics not relevant to our discussion. We will assume that the electric field due to these processes is represented by $\fsAver{E_\rho}_{misc}$, with $\fsAver{E_\rho}_{misc} < 0$ at the edge. Then at any time the total electric field can be written as a sum of two terms:
\begin{equation}
\fsAver{E_\rho}_{tot} = \fsAver{E_\rho}_{misc} + \fsAver{E_\rho}_{\delta p}, \label{eqn:totE}
\end{equation}
where $\fsAver{E_\rho}_{\delta p}$ is the contribution by a poloidally asymmetric pressure profile. Note that if we move beyond simple MHD and invoke, for example, a two-fluid theory, we would see that the radial ion pressure gradient also contributes to the electric field of Eq.~\ref{eqn:E}; we will assume this is contained within the $\fsAver{E_\rho}_{misc}$ term. The input power threshold for the transition, $P_{LH}$, is in general a complicated function of the plasma and machine parameters\cite{ryter1996}. Without making a causal connection, we can assume the transition is associated with a critical radial electric field level, $\fsAver{E_\rho}_{crit}$. Then we have $P_{LH} = f(\fsAver{E_\rho}_{crit})$, where $f$ is a possibly machine-dependent function of the edge electric field.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=1.8in]{fig7_aydemir.pdf}
\caption{\em \baselineskip 14pt Effect of a poloidal pressure asymmetry near the $X$-point on the L-H transition power threshold for lower and upper single-null magnetic geometries. ``Standard configuration'' of the fields is assumed.
(LSN): (a) No asymmetry, $\delta p = 0.$ (b) $\delta p > 0$, as in Fig.~\ref{fig:fig6_aydemir}(a). (c) $\delta p < 0$. Note that $P_{LH}^+ < P_{LH}^0$. (USN): Because the asymmetry is in the upper half-plane, positions of the lines (b) and (c) are switched. Now $P_{LH}^+ > P_{LH}^0$.}
\label{fig:fig7_aydemir}
\end{center}
\end{figure}
Then in the input power versus edge electric field space shown in Fig.~\ref{fig:fig7_aydemir} we have the following picture, where we assume a linear relationship between $P_{in}$ and the edge electric field $\fsAver{E_\rho}_{tot}$ because we lack a quantitative theory of the L-H transition:
\begin{itemize}
\item (LSN): Without a contribution from a poloidal asymmetry (line (a), $\delta p=0$), the radial electric field starts at some negative value, $\fsAver{E_\rho}_0$ and becomes progressively more negative as the input power is increased. The L-H transition is triggered as $\fsAver{E_\rho}_{tot}$ crosses the critical value at $\fsAver{E_\rho}_{crit},$ resulting in the power threshold $P_{LH}^0.$ Here changes in $\fsAver{E_\rho}_{tot}$ would be entirely due to $\fsAver{E_\rho}_{misc}$ of Eq.~\ref{eqn:totE}, driven by the increasing input power. If we allow for a poloidal pressure asymmetry near the $X$-point as in Fig.~\ref{fig:fig6_aydemir}(a), then the initial point is lower (line (b), $\delta p > 0$) due to the negative $\fsAver{E_\rho}_{\delta p}$ contribution by the asymmetry (see Fig.~\ref{fig:fig6_aydemir}) (b)). With a more negative starting point, less power is needed to cross the $\fsAver{E_\rho}_{crit}$ level, resulting in the lower value $P_{LH}^+ < P_{LH}^0$.
A corollary of these arguments is that, if the asymmetry is negative as in line (c) (not very physical in the present context), then the initial point would be higher, since an asymmetry with $\delta p < 0$ in the lower half-plane leads to $\fsAver{E_\rho}_{\delta p} > 0$, thus producing the higher value $P_{LH}^-$.
\item (USN): A positive pressure asymmetry near the $X$-point in USN magnetic geometry, because it is located in the upper half-plane, contributes a positive electric field, $\fsAver{E_\rho}_{\delta p} > 0$, thus reversing the role of the asymmetry in the L-H transition. Now the lines (b) and (c) switch positions, and we get $P_{LH}^+ > P_{LH}^0$.
\item As explained earlier, maintaining the LSN magnetic geometry but reversing the toroidal field reverses the sign of $\fsAver{E_\rho}_{\delta p}$ without reversing the sign of the poloidal flows (see also \cite{aydemir2007b}). Thus, the reversal $B_\zeta \rightarrow -B_\zeta$ would switch the positions of the lines (b) and (c) in the (LSN) panel of Fig.~\ref{fig:fig7_aydemir}, resulting in a state qualitatively like the (USN) panel. Now a positive pressure asymmetry would increase the threshold power, giving $P_{LH}^+ > P_{LH}^0$.
\item The differences between the LSN and USN magnetic configurations in the standard configuration of the fields, and those between the standard and field-reversed states in the LSN configuration are consistent with the experimental observations where $P_{LH}$ increases approximately by a factor of two when the ion $\nabla B$-drift direction points away from the active $X$-point\cite{ryter1996}. Note that in Fig.~\ref{fig:fig6_aydemir}(a) with the standard configuration of the fields, the drift is downward towards the $X$-point.
\item It is helpful to keep in mind that a negative perturbation in the lower half-plane produces poloidal flows in the same direction as a positive perturbation in the upper half-plane. This symmetry follows from the basic physics of the flow generation mechanism discussed earlier.
\end{itemize}
In a double-null (DN), effects of the positive pressure asymmetries at the two $X$-points would cancel each other, qualitatively leading to the state with $\delta p=0$ in the LSN or USN panels of Fig.~\ref{fig:fig7_aydemir}. Then we can conclude that the expected power threshold levels for the three magnetic geometries, in the standard configuration of the fields, will have the order
\begin{equation}
P_{LSN} < P_{DN} < P_{USN}. \label{eqn:P_LH_order}
\end{equation}
Reversing the toroidal field direction will also reverse this order.
Up to now we were mainly concerned with positive pressure asymmetries produced by external fueling of the plasma using main ions. Next we examine poloidal flows due to negative pressure asymmetries produced by massive gas or pellet injections with a significant impurity content.
\section{Role in poloidal flows generated by impurity injections}
As mentioned earlier, impurities that accompany MGI or SPI for disruption mitigation necessarily leads to collapse of the plasma temperature from keV to eV range on a millisecond time scale. The result is a ``pressure hole'' near the injection site with $\delta p < 0$. The flows driven in this case are in opposite direction to those of the previous section where we had $\delta p > 0$ (analogous to the movement of an electron hole in a sea of electrons). And importantly, they can explain the poloidal flow patterns observed during the MGI shutdown events.
The experimental observations are summarized in the two panels of Fig.~\ref{fig:fig8_aydemir}, reproduced here with permission from \cite{hollmann2015} and \cite{eidietis2017}. In (I), the top panel, ``[p]oloidal flows measured by
fast bolometry during MGI shutdowns [...] in JET'' are shown. Here the injection site is in the upper low-field side (LFS). Apparently the counter-clockwise flows, ``from the outer midplane over the top of the machine and down to the center post,'' shown in the figure are commonly seen in other devices also: see for example, Fig.~4 of \cite{hollmann2015} for DIII-D and AUG results, and Fig.~2 of \cite{eidietis2017} for more recent DIII-D observations.
When the injection site is in the lower half-plane, as in panel (II) of Fig.~\ref{fig:fig8_aydemir}, which is again from DIII-D (Fig.~3 of \cite{eidietis2017}), ``the emission exhibits either no poloidal flow and spreads directly across the plasma to the LFS or else a clockwise flow under the plasma and across the X-point''\cite{eidietis2017}. Moreover, both the upper and lower injection observations are independent of the toroidal field direction and thus have nothing to do with $E\times B$ flows\cite{eidietis2017},\cite{hollmann2013}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=6in]{fig8_aydemir.pdf}
\caption{\em \baselineskip 14pt (I) Counter-clockwise poloidal flows measured by fast bolometry during a MGI from an upper half-plane injection site in JET (reproduced with permission from Fig. 4 of \cite{hollmann2015}). (II) No flow or a possible clockwise flow during a MGI from a lower half-plane injection site in DIII-D (reproduced with permission from Fig.~3 of \cite{eidietis2017}). }
\label{fig:fig8_aydemir}
\end{center}
\end{figure}
However, the poloidal flows generated both by the upper and lower half-plane MGI injections, and their insensitivity to the reversal of the toroidal field, can be explained in terms of the poloidal-pressure-asymmetry-driven flows. Although most of the physics has already been discussed, we will explicitly demonstrate the connection between MGI-generated negative pressure asymmetries and the experimental observations:
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=6in]{fig9_aydemir.pdf}
\caption{\em \baselineskip 14pt Results of MGI in upper LFS. (a) Assumed negative pressure perturbation ($\delta p < 0$). (b) The resulting counter-clockwise poloidal flows. (c) Flux surface-averaged poloidal velocity for both $\delta p < 0$ (due to MGI, solid red line) and $\delta p > 0$ (due to fueling, dashed blue line). The separatrix is at $\rho=0.86$ at the midplane. The figure can be enlarged for a more detailed view. }
\label{fig:fig9_aydemir}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=6in]{fig10_aydemir.pdf}
\caption{\em \baselineskip 14pt Results of MGI in lower LFS. a) Location of the negative pressure perturbation ($\delta p < 0$). (b) The resulting poloidal flows with a stagnation point near the center of the MGI perturbation; the inset shows a blow-up of the stagnation area.
(c) Flux surface-averaged poloidal velocity for both $\delta p < 0$ (due to MGI, solid red line) and $\delta p > 0$ (due to fueling, dashed blue line).}
\label{fig:fig10_aydemir}
\end{center}
\end{figure}
\begin{itemize}
\item These flows are driven entirely by the toroidal geometry and their direction is determined by the location of the asymmetry with respect to the midplane. Asymmetries with $\delta p > 0$ in the upper half-plane produce negative (clockwise) flows. This was shown unambiguously in the DN magnetic geometry of Fig.~\ref{fig:fig1_aydemir}, where other sources of flows cancel.
\item The driving torque is independent of the fields (see Eq.~\ref{eqn:averTorque}); the resulting poloidal flows are independent of the toroidal field direction, although the sign of the generated radial electric field $E_\rho$ does depend on $B_\zeta$.
\item The driving torque, the resulting poloidal flows and radial electric field all reverse directions when $\delta p < 0$ (see Eq.~\ref{eqn:averTorque} and Fig.~\ref{fig:fig4_aydemir} (b)).
\item In LSN magnetic geometry, the expected positive pressure asymmetry around the $X$-point makes a positive contribution to the transport-driven poloidal flows, as we saw in Fig.~\ref{fig:fig6_aydemir}. An MGI-produced negative asymmetry in the upper half-plane also adds constructively and results in the positive (counter-clockwise) flows as seen Fig.~\ref{fig:fig9_aydemir}.
Panel (a) shows the assumed negative pressure perturbation ($\delta p < 0$). Panel (b) shows the resulting counter-clockwise poloidal flows, which would normally take the radiation pattern over the top to the HFS. Recall, however, that these are equilibrium calculations in which we seek a quasi steady-state with flows in the presence of a prescribed poloidal asymmetry that is held stationary. Thus, the effect of the flows on the pressure asymmetry is not calculated here and left for a future work.
For comparison, in (c) the flux surface-averaged poloidal velocity for both $\delta p < 0$ (due to MGI, solid red line) and $\delta p > 0$ (due to fueling, dashed blue line) are shown. Note that although fueling at this location would reverse the flows (see Fig.~\ref{fig:fig5_aydemir} (b)), MGI makes a strong positive contribution and the total flow shown here is positive.
\item Results of MGI in lower LFS are quite different, as seen in Fig.~\ref{fig:fig10_aydemir}. Again the panel (a) shows the location of the negative pressure perturbation ($\delta p < 0$). The resulting poloidal flows are seen in (b). On the inner flux surfaces, the transport-driven flows continue nearly unchanged, but on the outer surfaces affected by MGI, the poloidal flow reverses, creating a stagnation point near the center of the MGI perturbation, where $u_\theta \simeq 0.$
Until the temperature collapses due to radiation and sets up $\delta p < 0$, it is possible the existing positive flows may carry injected the material up towards the midplane. But once the pressure hole is established, it is clear the radiation pattern will be stationary or move downward with the negative flows, consistent with the experimental observations in Fig.~\ref{fig:fig8_aydemir}. As stated earlier, these equilibrium calculations do not follow the time evolution of the asymmetry; this interesting exercise is left for a future work.
\end{itemize}
\section{Discussion and Summary}
The numerical results above were presented in non-dimensional units. In order to get a sense of the possible magnitude of the actual flows and electric fields generated, we will use the following parameters for the edge plasma (similar to those used in \cite{aydemir2018c}): Toroidal field $B_{\zeta 0}=3T$, deuterium density $n=10^{19}m^{-3}$, minor radius $a=1m$, and the inverse aspect ratio $\epsilon=a/R_0=1/3$. With these we get the poloidal Alfv\'en speed $v_{Ap} = \epsilon B_{\zeta 0}/\sqrt{\mu_0\rho_m}=5\times 10^6 ms^{-1}$. Thus the flow velocities, normalized with $v_{Ap}$, are of order $10^{-3}v_{Ap} = 5km s^{-1}$. The electric field is normalized with $E_0=\epsilon v_{Ap}B_{\zeta 0} = 5\times 10^6 Vm^{-1}$,
which leads to $\fsAver{E_\rho}_{min}\simeq 6\times 10^{-3}E_0= 30kVm^{-1}$ for Fig.~\ref{fig:fig2_aydemir}, a substantial radial electric field. With the same parameters we get $\fsAver{E_\rho}_{max}\simeq +10kVm^{-1}$ for Fig.~\ref{fig:fig5_aydemir} (b), clearly indicating that any density or pressure asymmetry above the low-field-side midplane (an ITER gas injection site) would be detrimental to confinement.
The quasi-steady state values for $\fsAver{u_\theta}$ and $\fsAver{E_\rho}$ are of course affected by the assumed transport coefficients. Typical values used for the poloidal damping rate $\gamma_p$ and viscosity coefficient $\mu=\rho_m\nu$ that appear in Eq.~\ref{eqn:stressT2} are as follows: $\gamma_p=10^{-4}$, $\mu=5.0\times 10^{-6}$, or in dimensional units $\gamma_p=10^{-4}(v_{Ap}/a)\simeq 5\times 10^2 s^{-1}$ and $\mu=5\times 10^{-6}(av_{Ap})=25m^2s^{-1}$, both of which are somewhat larger than physical estimates.
Since the torque $\fsAver{T_\zeta}$ (but not necessarily the full MHD results) is linear in the perturbation amplitude $\delta p$ (see Eq.~\ref{eqn:averTorque}), and since this quantity is not easily available experimentally, our numerical results should be interpreted in the light of this uncertainty. The numerical calculations typically used $\delta p/p_0\sim 10^{-4}$, where $p_0$ is the pressure on axis.
In summary, a relatively small-amplitude poloidal pressure asymmetry at the plasma edge, possibly maintained by a neutral particle source, is shown to drive substantial mass flows near the plasma boundary, entirely within an MHD equilibrium framework. The mechanism is robust and relies essentially on force-balance arguments. Whereas the transport-theory based analyses of poloidal asymmetries tend to focus on inboard vs. outboard differences, we find that the position with respect to the midplane plays a more significant role. For the ``standard configuration'' of the plasma current and toroidal field, asymmetries below the midplane produce a negative radial electric field inside the separatrix. This field can then enhance or even supplant other sources of edge radial electric field (e.g., the ion orbit-loss), and thus play a significant role in suppressing edge turbulence and MHD activity. The result would be an easier L-H transition and a lower transition power threshold, $P_{LH}.$
The generated poloidal and toroidal flows, and the associated $E_\rho$ can be easily shown to have the correct symmetries\cite{aydemir2007b} to explain the ion $\nabla B$-drift-dependence of $P_{LH}$. Briefly, the sign of the net torque and the poloidal flow due to the asymmetry depends only on its location with respect to the midplane. In a discharge in the ``standard configuration,'' if the asymmetry is near the lower $X$-point, for instance, due to neutral recycling, reversing the toroidal field will leave the poloidal (but not the toroidal) velocity intact and thus reverse $E_\rho,$ along with the ion drift direction. These changes will convert the original favorable configuration into an unfavorable one with a higher $P_{LH},$ consistent with experimental observations.
Note that one of the ITER fueling ports\cite{baylor2007b} is approximately at the location of the asymmetry in Fig.~\ref{fig:fig1_aydemir}(a). If fueling at that location can penetrate the plasma edge and produce a poloidal pressure asymmetry, it will generate negative shear flows and a positive radial electric field, which will tend to increase the L-H transition power threshold. Fortunately other ports near the $X$-point are quite favorably located from this point of view.
More generally, since the edge is the most easily accessible part of the discharge, a deliberately-induced poloidal asymmetry at an appropriate location along the plasma periphery can be used as an effective tool in generating highly beneficial edge shear flows. These flows, depending on their location with respect to the midplane, can be used to enhance or degrade edge confinement as needed.
Finally, MGI-produced pressure holes ($\delta p < 0$) near the plasma boundary drive poloidal flows in opposite direction to those that result from $\delta p>0$. These can explain the quite different flow patterns seen after MGI at the upper and lower LFS injection sites in DIII-D and other devices.
\section*{Acknowledgements}
This work was supported by MSIP, the Korean Ministry of Science, ICT and Future Planning, through the KSTAR project.
\section*{References}
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| 6,599
|
using System.Windows.Controls;
namespace TreehopperShowcase.Pages
{
/// <summary>
/// Interaction logic for SettingsPage.xaml
/// </summary>
public partial class SettingsPage : UserControl
{
public SettingsPage()
{
InitializeComponent();
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,262
|
"The Friend" Explores The Depth Of Grief
By Andy Killebrew 10 months ago
The Friend by Sigrid Nunez, winner of the 2018 National Book Award, is an elegiac, probing and unapologetic masterpiece of modern fiction. Nunez's eighth book, The Friend, is primarily a story about the attempts of a writer to deal with the sudden death of her longtime friend and writing mentor, complicated by the presence of his estranged Great Dane Apollo (the narrator remains unnamed). Aside from Apollo's immediate threat to the protagonist's tenure in her rent-stabilized New York City apartment, the narrator is also forced to care for an animal caught in the throes of mourning. As the novel moves along, the unlikely pair come to rely on each other more and more for their respective wellbeing. The two are forced to brave a world suddenly absent of their shared best friend and retreat into each other for consolation in their shared books (she reads aloud to Apollo, believing it soothes him) and walks.
The prose of The Friend is remarkable in its sparsity, poetic in its ambition. The novel only spans 200 pages, made even shorter by the ample space between paragraphs and page breaks between chapters. These absences let the writing breathe, giving the reader ample opportunity to reflect on the abundance of life within each paragraph. The Friend eschews traditional novel expectations at every turn, alternating between memoir and letter, fiction and nonfiction (one of Nunez's friends passed during the writing of The Friend, leaving the irresistible temptation to read the novel as Nunez's personal account of grief). It's hard to pin the novel into any box, and its individuality yields it an undeniable sense of truth and authority. Nunez's prowess as an author is present at every period, every word; she is an accomplished writer who is entirely in control of her narrative toolbox.
Though the majority of The Friend is focused on the relationship between dog and master during trying times, the novel also devotes significant attention to the current state of the literary world. Nunez says writing appealed to her because it was "something that she could do alone," and this perspective comes through with the book's unabashed cynicism regarding the field. She pulls no punches in attacking writers and the publishing industry at large, even going so far as to compare writers to vampires. It's interesting to see how Nunez puts these thoughts into dialogue with the canine-human relationship that occupies most of the book, and both subjects are handled with grace and wit.
While The Friend is difficult to distill into a brief review, it is a revelatory literary experience for any lover of fiction, writing or dogs. It's the sort of read that lingers with you even after the last word; it's hard to keep either Apollo or his new, grieving master out of mind. Nunez has long been a hidden gem within the literary world and her break into relevance is a gift for any reader. The Friend is an absolute must-read for its scope, power and beauty.
Andy Killebrew
killae17@wfu.edu
Previous Giving, Receiving Consent Is Important In All Situations
Next "HerCampus" Chapter Chosen To Host "New York Times" Event
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,006
|
package org.apache.flink.hdfstests;
import java.io.FileOutputStream;
import org.apache.commons.io.FileUtils;
import org.apache.flink.api.common.ExecutionConfig;
import org.apache.flink.api.common.io.FileInputFormat;
import org.apache.flink.api.common.typeinfo.TypeInformation;
import org.apache.flink.api.java.io.TextInputFormat;
import org.apache.flink.api.java.tuple.Tuple2;
import org.apache.flink.api.java.typeutils.TypeExtractor;
import org.apache.flink.core.fs.FileInputSplit;
import org.apache.flink.core.fs.Path;
import org.apache.flink.core.testutils.OneShotLatch;
import org.apache.flink.streaming.api.TimeCharacteristic;
import org.apache.flink.streaming.api.functions.source.ContinuousFileMonitoringFunction;
import org.apache.flink.streaming.api.functions.source.ContinuousFileReaderOperator;
import org.apache.flink.streaming.api.functions.source.FileProcessingMode;
import org.apache.flink.streaming.api.functions.source.SourceFunction;
import org.apache.flink.streaming.api.functions.source.TimestampedFileInputSplit;
import org.apache.flink.streaming.api.operators.StreamSource;
import org.apache.flink.streaming.api.watermark.Watermark;
import org.apache.flink.streaming.runtime.streamrecord.StreamRecord;
import org.apache.flink.streaming.runtime.tasks.OperatorStateHandles;
import org.apache.flink.streaming.util.AbstractStreamOperatorTestHarness;
import org.apache.flink.streaming.util.OneInputStreamOperatorTestHarness;
import org.apache.flink.streaming.util.OperatorSnapshotUtil;
import org.apache.flink.util.Preconditions;
import org.junit.Assert;
import org.junit.ClassRule;
import org.junit.Ignore;
import org.junit.Test;
import org.junit.rules.TemporaryFolder;
import java.io.File;
import java.io.IOException;
public class ContinuousFileProcessingFrom12MigrationTest {
private static final int LINES_PER_FILE = 10;
private static final long INTERVAL = 100;
@ClassRule
public static TemporaryFolder tempFolder = new TemporaryFolder();
/**
* Manually run this to write binary snapshot data. Remove @Ignore to run.
*/
@Ignore
@Test
public void writeReaderSnapshot() throws Exception {
File testFolder = tempFolder.newFolder();
TimestampedFileInputSplit split1 =
new TimestampedFileInputSplit(0, 3, new Path("test/test1"), 0, 100, null);
TimestampedFileInputSplit split2 =
new TimestampedFileInputSplit(10, 2, new Path("test/test2"), 101, 200, null);
TimestampedFileInputSplit split3 =
new TimestampedFileInputSplit(10, 1, new Path("test/test2"), 0, 100, null);
TimestampedFileInputSplit split4 =
new TimestampedFileInputSplit(11, 0, new Path("test/test3"), 0, 100, null);
// this always blocks to ensure that the reader doesn't to any actual processing so that
// we keep the state for the four splits
final OneShotLatch blockingLatch = new OneShotLatch();
BlockingFileInputFormat format = new BlockingFileInputFormat(blockingLatch, new Path(testFolder.getAbsolutePath()));
TypeInformation<FileInputSplit> typeInfo = TypeExtractor.getInputFormatTypes(format);
ContinuousFileReaderOperator<FileInputSplit> initReader = new ContinuousFileReaderOperator<>(
format);
initReader.setOutputType(typeInfo, new ExecutionConfig());
OneInputStreamOperatorTestHarness<TimestampedFileInputSplit, FileInputSplit> testHarness =
new OneInputStreamOperatorTestHarness<>(initReader);
testHarness.setTimeCharacteristic(TimeCharacteristic.EventTime);
testHarness.open();
// create some state in the reader
testHarness.processElement(new StreamRecord<>(split1));
testHarness.processElement(new StreamRecord<>(split2));
testHarness.processElement(new StreamRecord<>(split3));
testHarness.processElement(new StreamRecord<>(split4));
// take a snapshot of the operator's state. This will be used
// to initialize another reader and compare the results of the
// two operators.
final OperatorStateHandles snapshot;
synchronized (testHarness.getCheckpointLock()) {
snapshot = testHarness.snapshot(0L, 0L);
}
OperatorSnapshotUtil.writeStateHandle(snapshot, "src/test/resources/reader-migration-test-flink1.2-snapshot");
}
@Test
public void testReaderRestore() throws Exception {
File testFolder = tempFolder.newFolder();
final OneShotLatch latch = new OneShotLatch();
BlockingFileInputFormat format = new BlockingFileInputFormat(latch, new Path(testFolder.getAbsolutePath()));
TypeInformation<FileInputSplit> typeInfo = TypeExtractor.getInputFormatTypes(format);
ContinuousFileReaderOperator<FileInputSplit> initReader = new ContinuousFileReaderOperator<>(format);
initReader.setOutputType(typeInfo, new ExecutionConfig());
OneInputStreamOperatorTestHarness<TimestampedFileInputSplit, FileInputSplit> testHarness =
new OneInputStreamOperatorTestHarness<>(initReader);
testHarness.setTimeCharacteristic(TimeCharacteristic.EventTime);
testHarness.setup();
OperatorStateHandles operatorStateHandles = OperatorSnapshotUtil.readStateHandle(
OperatorSnapshotUtil.getResourceFilename(
"reader-migration-test-flink1.2-snapshot"));
testHarness.initializeState(operatorStateHandles);
testHarness.open();
latch.trigger();
// ... and wait for the operators to close gracefully
synchronized (testHarness.getCheckpointLock()) {
testHarness.close();
}
TimestampedFileInputSplit split1 =
new TimestampedFileInputSplit(0, 3, new Path("test/test1"), 0, 100, null);
TimestampedFileInputSplit split2 =
new TimestampedFileInputSplit(10, 2, new Path("test/test2"), 101, 200, null);
TimestampedFileInputSplit split3 =
new TimestampedFileInputSplit(10, 1, new Path("test/test2"), 0, 100, null);
TimestampedFileInputSplit split4 =
new TimestampedFileInputSplit(11, 0, new Path("test/test3"), 0, 100, null);
// compare if the results contain what they should contain and also if
// they are the same, as they should.
Assert.assertTrue(testHarness.getOutput().contains(new StreamRecord<>(split1)));
Assert.assertTrue(testHarness.getOutput().contains(new StreamRecord<>(split2)));
Assert.assertTrue(testHarness.getOutput().contains(new StreamRecord<>(split3)));
Assert.assertTrue(testHarness.getOutput().contains(new StreamRecord<>(split4)));
}
/**
* Manually run this to write binary snapshot data. Remove @Ignore to run.
*/
@Ignore
@Test
public void writeMonitoringSourceSnapshot() throws Exception {
File testFolder = tempFolder.newFolder();
long fileModTime = Long.MIN_VALUE;
for (int i = 0; i < 1; i++) {
Tuple2<File, String> file = createFileAndFillWithData(testFolder, "file", i, "This is test line.");
fileModTime = file.f0.lastModified();
}
TextInputFormat format = new TextInputFormat(new Path(testFolder.getAbsolutePath()));
final ContinuousFileMonitoringFunction<String> monitoringFunction =
new ContinuousFileMonitoringFunction<>(format, FileProcessingMode.PROCESS_CONTINUOUSLY, 1, INTERVAL);
StreamSource<TimestampedFileInputSplit, ContinuousFileMonitoringFunction<String>> src =
new StreamSource<>(monitoringFunction);
final AbstractStreamOperatorTestHarness<TimestampedFileInputSplit> testHarness =
new AbstractStreamOperatorTestHarness<>(src, 1, 1, 0);
testHarness.open();
final Throwable[] error = new Throwable[1];
final OneShotLatch latch = new OneShotLatch();
// run the source asynchronously
Thread runner = new Thread() {
@Override
public void run() {
try {
monitoringFunction.run(new DummySourceContext() {
@Override
public void collect(TimestampedFileInputSplit element) {
latch.trigger();
}
@Override
public void markAsTemporarilyIdle() {
}
});
}
catch (Throwable t) {
t.printStackTrace();
error[0] = t;
}
}
};
runner.start();
if (!latch.isTriggered()) {
latch.await();
}
final OperatorStateHandles snapshot;
synchronized (testHarness.getCheckpointLock()) {
snapshot = testHarness.snapshot(0L, 0L);
}
OperatorSnapshotUtil.writeStateHandle(
snapshot,
"src/test/resources/monitoring-function-migration-test-" + fileModTime +"-flink1.2-snapshot");
monitoringFunction.cancel();
runner.join();
testHarness.close();
}
@Test
public void testMonitoringSourceRestore() throws Exception {
File testFolder = tempFolder.newFolder();
Long expectedModTime = Long.parseLong("1493116191000");
TextInputFormat format = new TextInputFormat(new Path(testFolder.getAbsolutePath()));
final ContinuousFileMonitoringFunction<String> monitoringFunction =
new ContinuousFileMonitoringFunction<>(format, FileProcessingMode.PROCESS_CONTINUOUSLY, 1, INTERVAL);
StreamSource<TimestampedFileInputSplit, ContinuousFileMonitoringFunction<String>> src =
new StreamSource<>(monitoringFunction);
final AbstractStreamOperatorTestHarness<TimestampedFileInputSplit> testHarness =
new AbstractStreamOperatorTestHarness<>(src, 1, 1, 0);
testHarness.setup();
OperatorStateHandles operatorStateHandles = OperatorSnapshotUtil.readStateHandle(
OperatorSnapshotUtil.getResourceFilename(
"monitoring-function-migration-test-1493116191000-flink1.2-snapshot"));
testHarness.initializeState(operatorStateHandles);
testHarness.open();
Assert.assertEquals((long) expectedModTime, monitoringFunction.getGlobalModificationTime());
}
private static class BlockingFileInputFormat extends FileInputFormat<FileInputSplit> {
private static final long serialVersionUID = -6727603565381560267L;
private final OneShotLatch latch;
private FileInputSplit split;
private boolean reachedEnd;
BlockingFileInputFormat(OneShotLatch latch, Path filePath) {
super(filePath);
this.latch = latch;
this.reachedEnd = false;
}
@Override
public void open(FileInputSplit fileSplit) throws IOException {
this.split = fileSplit;
this.reachedEnd = false;
}
@Override
public boolean reachedEnd() throws IOException {
if (!latch.isTriggered()) {
try {
latch.await();
} catch (InterruptedException e) {
e.printStackTrace();
}
}
return reachedEnd;
}
@Override
public FileInputSplit nextRecord(FileInputSplit reuse) throws IOException {
this.reachedEnd = true;
return split;
}
@Override
public void close() {
}
}
private static abstract class DummySourceContext
implements SourceFunction.SourceContext<TimestampedFileInputSplit> {
private final Object lock = new Object();
@Override
public void collectWithTimestamp(TimestampedFileInputSplit element, long timestamp) {
}
@Override
public void emitWatermark(Watermark mark) {
}
@Override
public Object getCheckpointLock() {
return lock;
}
@Override
public void close() {
}
}
/**
* Create a file with pre-determined String format of the form:
* {@code fileIdx +": "+ sampleLine +" "+ lineNo}.
* */
private Tuple2<File, String> createFileAndFillWithData(
File base, String fileName, int fileIdx, String sampleLine) throws IOException {
File file = new File(base, fileName + fileIdx);
Assert.assertFalse(file.exists());
File tmp = new File(base, "." + fileName + fileIdx);
FileOutputStream stream = new FileOutputStream(tmp);
StringBuilder str = new StringBuilder();
for (int i = 0; i < LINES_PER_FILE; i++) {
String line = fileIdx +": "+ sampleLine + " " + i +"\n";
str.append(line);
stream.write(line.getBytes());
}
stream.close();
FileUtils.moveFile(tmp, file);
Assert.assertTrue("No result file present", file.exists());
return new Tuple2<>(file, str.toString());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,568
|
{"url":"https:\/\/www.accountingcapital.com\/quiz-25-debit-note-answers\/","text":"# Quiz 25 \u2013 Debit Note \u2013 (Answers)\n\nNote \u2013 Please take our quiz \u2018Quiz 25 \u2013 Debit Note \u2013 (Answers)\u2018 on this page before going through the below answers. Only the top 3% of our audience scored 10\/10 in all of them.\n\n\u2022 Name \u2013 Quiz 25 \u2013 Debit Note \u2013 (Answers)\n\u2022 Topic \u2013 Debit Note\n\n##### Q1. Debit Note may be treated as an invoice, however, it is different from that.\n\nAns. The given statement is True. A debit note is different from an invoice, though it may be treated as an invoice.\n\n##### Q2. Debit Note is issued for the value of goods _____.\n\nAns. A debit note is issued to the seller by the buyer for the value of goods that have been returned.\n\n##### Q3. Debit Note shows a _____ currency value (amount).\n\nAns.\u00a0It shows a positive amount and resembles a regular invoice with the words (debit note).\n\n##### Q4. Debit Note is usually sent for reasons such as incomplete & damaged goods received.\n\nAns. The given statement is True. Typically, debit notes are sent because of incomplete, damaged, or inaccurate goods sent.\n\n##### Q5. Debit Note reduces _____ for the buyer.\n\nAns. When the buyer sends the seller a debit note it reduces the buyer\u2019s current liability (payables).\n\n##### Q6. Debit Note is used both in the case of cash & credit sales.\n\nAns. The given statement is False. Debit notes only come into play when credit sales are made.\n\n##### Q7. Debit Note reduces _____ in books of the _____.\n\nAns. A debit note is a document that reveals details about returned goods and creates an obligation for the seller to cancel the corresponding dues. This means that the creditors (payables) are being reduced for the buyer.\n\n##### Q8. The buyer updates the \u201cPurchase Returns Book\u2019 on the basis of a debit note.\n\nAns. The given statement is True. The purchase return book is updated when a debit note is sent. (in the books of a buyer)\n\n##### Q9. The _____ is obligated to cancel the debt upon acceptance of the debit note.\n\nAns.\u00a0 Debit Note creates an obligation for the seller to cancel the related dues.\n\n##### Q10. Debit Note is a document sent by a _____ to confirm the details of goods returned.\n\nAns.\u00a0 A debit note is a document issued by a buyer to the seller to confirm the details of goods returned.","date":"2023-03-22 16:46:45","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.85270756483078, \"perplexity\": 4498.865552362346}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296943845.78\/warc\/CC-MAIN-20230322145537-20230322175537-00594.warc.gz\"}"}
| null | null |
Q: Redirecting Urls with Query String to pretty Url in NGINX I am using nginx as reverse proxy with apache for my website. I have pretty url's enabled for SEO purpose, with NGINX rewrite URL's.
for example the pretty url :
mysite.com/song/I-love-music
works for this ugly url :
mysite.com/song.php?url=I-love-music
And it works perfectly.
But I have an issue now, if someone visits this type of urls directly
mysite.com/song.php?url=I-love-music
Then this url is loading properly , same as
mysite.com/song.php?url=I-love-music
It is a big problem for SEO purpose, I need this ugly url to permanently redirected to this pretty url :
mysite.com/song/I-love-music
Please help me how do I rewrite urls in NGINX for this. I tried with apache rewrite rules in htaccess but that gives me server error..
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,262
|
\section{Introduction}
We consider an arbitrary Poisson bracket of a Poisson algebra of functionals of field variables ${\bm\chi}({\bf x})$ given by
\begin{equation}
\label{eqn:PBgene}
\{F,G\}=\int \!d^n x \, F_{\bm \chi} \cdot {\mathbb J} ({\bm\chi}) \cdot G_{\bm\chi}\,,
\end{equation}
where ${\bf x}\in {\mathbb R}^n$, ${\bm \chi}\colon{\mathbb R}^n\rightarrow{\mathbb R}^d$, and $F_{\bm \chi} \cdot {\mathbb J} \cdot G_{\bm\chi}= F_{\chi^i} \,{\mathbb J}^{ij} \, G_{\chi^j}$ with repeated indices summed. By Poisson algebra we mean a Lie algebra realization on functionals with an associative product of functionals that satisfies the Leibniz law. Also, we assume that the resulting equations of motion given by $\dot{\bm\chi}=\{{\bm\chi},H\}$, for some Hamiltonian functional $H[{\bm\chi}]$, possess a conservation law ${\cal Q}[{\bm\chi}]=0$, where ${\cal Q}$ is a functional of the field variables and their derivatives. Here we address the specific case where these conservation laws are obtained regardless of the choice of Hamiltonian $H$, so ${\cal Q}=0$ is an intrinsic property of the bracket of the Poisson algebra.
There are two ways to define such a constrained Poisson algebra. The usual way is to place a restriction on the set of field variables ${\bm\chi}$ in the Poisson algebra. However, this definition raises the question of how to appropriately compute the constrained functional derivatives $F_{\bm\chi}$. The second way is to define a Poisson algebra that does not include any constraint on the field variables and, consequently, there is no ambiguity in defining the functional derivatives -- conservation laws such as ${\cal Q}=0$ take the form of Casimir invariants.
In this article we investigate the links between these two ways of defining constrained Hamiltonian structures, and we propose a way to lift Poisson structures defined via the constrained field variables approach to ones that have the constraints as Casimir invariants. As can be expected, the difficulty resides in assuring the validity of the Jacobi identity. If we keep the same Poisson bracket but extended to the bigger algebra (the one without any constraint on the field variables), then in general, the Poisson structure is only obtained when the constraint is satisfied, i.e., the Jacobi identity is satisfied conditionally when ${\cal Q}[{\bm\chi}]=0$. It turns out that one can remedy this limitation by modifying the bracket with the inclusion of suitable projectors that leave the functional derivatives unconstrained and guarantee the Jacobi identity unconditionally. We identify such projectors acting on the functional derivatives and on the explicit dependence of the bracket on ${\bm\chi}$. We discuss the various choices of projectors and highlight a particularly relevant one obtained from Dirac's theory of constrained Hamiltonian systems.
In order to illustrate our purpose, consider the relatively simple and common example, the vorticity equation of a compressible or incompressible fluid in ${\mathbb R}^3$. The vorticity ${\bm\omega}=\nabla\times {\bf v}$, with ${\bf v}$ the velocity field, satisfies
\begin{equation}
\frac{\partial {\bm\omega}}{\partial t}=\nabla \times ({\bf v}\times {\bm\omega}).
\label{eqn:vort}
\end{equation}
In terms of a commonly used Poisson bracket (see, e.g., Ref.~\cite{zakh97}),
\begin{equation}
\{F,G\}_0=\int d^3x \, {\bm\omega}\cdot (\nabla \times F_{\bm\omega})\times(\nabla\times G_{\bm\omega})\,,
\label{eqn:brackvort}
\end{equation}
Eq.~(\ref{eqn:vort}) has the from $\dot{F}=\{F,H\}_0$ with the Hamiltonian $H=\int d^3x \,v^2/2$.
However, if one forgets about the constraint on the vector fields ${\bm\omega}$ or if one wants to lift the algebra of functionals of divergence-free ${\bm\omega}$ to the algebra of functionals of any vector field ${\bm\omega}$, then the bracket~(\ref{eqn:brackvort}) does not satisfy the Jacobi identity. This is easily seen by the following counter example:
\[
F_1=\frac1{2}\int d^3x \,{\bm \omega}\cdot\hat{\bf x}\, {y^2}\,,\quad
F_2=\frac1{2}\int d^3 x\, {\bm \omega}\cdot\hat{\bf y}\, {z^2}\,,\quad
F_3=\int d^3 x \,{\bm \omega}\cdot \hat{\bf z}\, x\,,
\]
which yields,
\begin{equation*}
\{F_1,\{F_2,F_3\}_0\}_0 + \circlearrowleft = - \int d^3 x \,{\bm \omega}\cdot\nabla(yz)\neq0\,.
\label{counter}
\end{equation*}
Evidently, the bracket~(\ref{eqn:brackvort}) satisfies the Jacobi identity only if $\nabla \cdot {\bm\omega}=0$. We refer to such Poisson brackets that only satisfy the Jacobi identity conditionally as {\em tainted brackets}. One of the questions we address in this article is how to correct a tainted bracket so that it satisfies the Jacobi identity unconditionally.
For this particular example, the correction is obtained by inserting a projection operator, following Ref.~\cite{chan11}, given by ${\cal P}_\perp=1-\nabla \Delta^{-1}\nabla\cdot$, so that it defines a new bracket
$$
\{F,G\}=\int d^3 x \, ({\cal P} {\bm\omega}) \cdot (\nabla \times F_{\bm\omega})\times(\nabla\times G_{\bm\omega}).
$$
It is rather straightforward (see Ref.~\cite{chan11}) to show that this bracket satisfies the Jacobi identity unconditionally. We notice that $\nabla \cdot {\bm\omega}$ is a Casimir invariant of the modified bracket, i.e.\ $\{\nabla\cdot {\bm\omega},G\}=0$ for any functional $G$.
As mentioned above, projectors are not only useful to lift algebras so as to satisfy the Jacobi identity, they are also involved in the way functional derivatives are computed when the field variables are constrained. As an illustration, we consider the incompressible Euler equation for the velocity field ${\bf v}({\bf x},t)$,
$$
\dot{\bf v}=-{\bf v}\cdot \nabla {\bf v} -\nabla P,
$$
where $P$ is determined by the constraint $\nabla\cdot{\bf v}=0$. This equation has a Hamiltonian structure~\cite{arno66,holm85,holm98,mars02} given by the Hamiltonian $H[{\bf v}]=\int d^3x \, v^2/2$
and the Poisson bracket
$$
\{F,G\}=\int d^3 x \,{\bf v}\cdot [F_{\bf v},G_{\bf v}]_L,
$$
where $F_{\bf v}$ are the functional derivatives of an observable $F$ with respect to the field variable ${\bf v}$ and the Lie bracket $[{\bf V},{\bf W}]_L$ is given by
$$
[{\bf V},{\bf W}]_L=({\bf W}\cdot \nabla){\bf V}-({\bf V}\cdot\nabla ){\bf W}.
$$
It should be noted that the incompressible Euler equation cannot be directly obtained from $\dot{F}=\{F,H\}$ using unconstrained functional derivatives $F_{\bf v}$ since $\nabla\cdot {\bf v}=0$ would not be conserved by the flow. One way of correcting the bracket is to use an orthogonal projector~\cite{holm85}. For divergence-free fields, this orthogonal projector is again given by ${\cal P}_\perp=1-\nabla \Delta^{-1}\nabla \cdot$ (see also Refs.~\cite{zakh97,chan11}). In other words, the constrained functional derivative $F_{\bf v}$ must be computed such that it satisfies $\nabla \cdot F_{\bf v}=0$. However, the fundamental reason for this constraint on the functional derivative is unclear, even though it yields the correct equation of motion. For a more general constraint ${\cal Q}[{\bm\chi}]=0$, is it still the orthogonal projector that has to be used for the constrained functional derivatives? In addition, this projector is in general not unique. It therefore raises natural questions such as which is the most relevant projector and how is it obtained in a systematic way?
In this article, we investigate two possible placements of a projectors: one is on the explicit dependence on the field variables, while the other is on the computation of the functional derivatives. We clarify the choice of the relevant projector by using Dirac's theory of constrained Hamiltonian systems. In order to prove the relevance of these projectors, we consider three examples taken from plasma physics. The first one is magnetohydrodynamics (MHD), both compressible and incompressible, the second one is the Vlasov-Maxwell system, and the third example involves semi-local constraints on linear Vlasov equations with two species.
The goal of this paper is twofold: The first purpose is to show that using some relevant projector, the tainted brackets can be corrected such that the new brackets satisfy the Jacobi identity unconditionally. The second purpose is to connect these corrected brackets to the ones obtained from Dirac's theory of constrained Hamiltonian systems.
\section{Formulation of the general method}
\subsection{Projected functional derivatives}
At the outset we assume that the bracket~(\ref{eqn:PBgene}) is a Poisson bracket on the algebra of functionals of ${\bm \chi}$, where ${\bm\chi}$ denotes a $d$-tuple of fields such that ${\cal Q}[ {\bm\chi}]({\bf x})=0$ and ${\cal Q}[{\bm\chi}]$ is function of ${\bm\chi}$ and its derivatives. These fields will be referred to as ${\cal Q}$-free fields. In this section, our aim is to get a corresponding Poisson bracket on the algebra of any functionals of ${\bm\chi}$, satisfying ${\cal Q}[{\bm\chi}]({\bf x})=0$ or not. The functional derivatives ${\bar F}_{\bm\chi}$ are defined in the following way:
\begin{equation}
\delta F=\int d^n x \,{\bar F}_{\bm\chi}\cdot \delta{\bm\chi},
\label{eqn:delf}
\end{equation}
for all ${\cal Q}$-free $\delta {\bm\chi}$, which here means that $\hat{\cal Q}\cdot \delta {\bm\chi}=0$ where $\hat{\cal Q}$ is the Fr\'echet derivative of ${\cal Q}$ defined by
$$
{\cal Q}[{\bm\chi}+\delta {\bm\chi}]({\bf x})-{\cal Q}[{\bm\chi}]({\bf x})=\hat{\cal Q} \delta {\bm\chi}+O(\Vert \delta {\bm\chi}\Vert^2).
$$
This means that ${\bar F}_{\bm\chi}$ is not uniquely defined: it is arbitrary up to an element of $\mbox{Rg } \hat{\cal Q}^\dagger$, since $\int d^nx\, {\bar F}_{\bm\chi}\cdot\delta{\bm\chi}=\int d^nx \,({\bar F}_{\bm\chi}+\hat{\cal Q}^\dagger {\bf g})\cdot \delta {\bm\chi}$ where ${\bf g}$ is arbitrary. We define the constrained functional derivative $\bar{F}_{\bm\chi}$ from the unconstrained one $F_{\bm\chi}$ by the following equation:
\begin{equation}
\label{eqn:1var}
\int d^n x \,\bar{F}_{\bm\chi}\cdot \delta {\bm\chi}=\int d^n x \,F_{\bm\chi}\cdot \delta \bar{\bm\chi},
\end{equation}
where now $\delta \bar{\bm\chi}$ is the constrained (${\cal Q}$-free) variation and $\delta {\bm\chi}$ the unconstrained one.
For the unconstrained variation $\delta {\bm\chi}$, we use a linear operator ${\cal P}$ acting as $\delta \bar{\bm\chi}={\cal P}^\dagger \delta {\bm\chi}$ such that $\hat{\cal Q}{\cal P}^\dagger=0$. Moreover, the range of this operator ${\cal P}^\dagger$ should be $\mbox{Ker } \hat{\cal Q}$ and, in addition, ${\cal P}^\dagger$ should act as the identity on $\mbox{ Ker } \hat{\cal Q}$. This is equivalent to requiring that ${\cal P}$ be a projector. Consequently, this leads to a condition on the possible projectors ${\cal P}$ such that ${\bar F}_{\bm \chi}={\cal P}F_{\bm\chi}$, viz.
\begin{equation}
\label{eqn:comm}
\mbox{Ker }{\cal P}= \mbox{Rg }\hat{\cal Q}^\dagger.
\end{equation}
Note that given this condition, ${\cal Q}[{\bm\chi}]({\bf x})$ is a Casimir invariant that is naturally preserved by the flow.
Still this projector is not unique. In the literature (see, e.g., Ref.~\cite{holm85}), the functional derivative is chosen such that $\hat{\cal Q} F_{\bm\chi}=0$, so that the projector satisfies $\hat{\cal Q} {\cal P}=0$. This corresponds to the orthogonal projector
\begin{equation}
\label{eqn:orth}
{\cal P}_\perp=1-\hat{\cal Q}^\dagger (\hat{\cal Q}\hat{\cal Q}^\dagger)^{-1}\hat{\cal Q},
\end{equation}
provided $\hat{\cal Q}\hat{\cal Q}^\dagger$ is invertible on $\mbox{Rg }\hat{\cal Q}$. However it is not clear if it is the best choice for the projection. Other solutions satisfy
\begin{eqnarray*}
&& {\cal P}_\perp {\cal P}={\cal P}_\perp, \\
&& {\cal P} {\cal P}_\perp ={\cal P},
\end{eqnarray*}
which are needed in order to satisfy Eq.~(\ref{eqn:comm}).
Given a particular projector ${\cal P}$ the bracket~(\ref{eqn:PBgene}) becomes
\begin{equation}
\label{eqn:PBp}
\{F,G\}_{\rm t}=\int d^n x \,({\cal P} F_{\bm\chi})\cdot {\mathbb J}({\bm\chi}) \cdot ({\cal P} G_{\bm\chi}),
\end{equation}
where now the functional derivatives are the unconstrained ones. So we have released the constraint on the functional derivatives but, in general the Poisson bracket~(\ref{eqn:PBp}) does not satisfy the Jacobi identity for functionals of arbitrary ${\bm\chi}$, ones no longer restricted to ${\cal Q}$-free fields, because ${\mathbb J}({\bm\chi})$ may give contributions that do not satisfy the Jacobi identify when ${\cal Q}[\bm\chi]\not= 0$. However, if the projector ${\cal P}$ does not depend on the field variables ${\bm\chi}$, as is the case for the examples we deal with in this article, then
a bracket that satisfies the Jacobi identity for all functionals of ${\bm\chi}$, satisfying ${\cal Q}[{\bm\chi}]=0$ or not, is given by
\begin{equation}
\label{eqn:PBproj}
\{F,G\}=\int d^n x \, ({\cal P} F_{\bm\chi}) \cdot {\mathbb J}({\cal P} {\bm\chi})\cdot ({\cal P} G_{\bm\chi}).
\end{equation}
In order to verify the Jacobi identity, we perform the change of variables ${\bm \chi}_P={\cal P}{\bm\chi}$ and ${\bm \chi}_Q={\bm\chi}-{\cal P} {\bm\chi}$ so that bracket~(\ref{eqn:PBproj}) formally becomes bracket~(\ref{eqn:PBp}) with ${\bm\chi}_P$ instead of ${\bm\chi}$. Since ${\bm \chi}_P$ is by definition ${\cal Q}$-free, the Jacobi identity is satisfied.
For the Poisson bracket~(\ref{eqn:PBproj}), we notice that ${\cal Q}[{\bm \chi}]({\bf x})$ is a Casimir invariant, and that the equations of motion for ${\bm\chi}_P$ are identical to the ones given by the Poisson bracket~(\ref{eqn:PBgene}) or (\ref{eqn:PBp}).
\subsection{Dirac brackets}
\subsubsection{Local constraints}
\label{local}
In order to identify the most appropriate projector, we use Dirac's theory of constrained Hamiltonian systems~\cite{Dira50}. We begin with the following good Poisson bracket:
\begin{equation}
\label{eqn:PBg}
\{F,G\}=\int d^nx \,F_{\bm\chi}\cdot \mathbb{J}({\bm\chi})\cdot G_{\bm\chi}
\end{equation}
and then impose the local constraint $\Phi({\bf x}):= {\cal Q}[{\bm\chi}]({\bf x})=0$, where as before ${\cal Q}[{\bm\chi}]({\bf x})$ is a function of ${\bm \chi}({\bf x})$ and its derivatives. The Dirac procedure begins with the computation of the matrix of Poisson brackets between the local constraints,
$$
C({\bf x},{\bf x}')\equiv \{\Phi({\bf x}),\Phi({\bf x}')\}=\hat{\cal Q}\mathbb{J}\hat{\cal Q}^\dagger \delta({\bf x}-{\bf x}').
$$
We set ${\cal A}:= \hat{\cal Q}\mathbb{J}\hat{\cal Q}^\dagger$ and we assume that this quantity is invertible. Then, the Dirac correction to the bracket~(\ref{eqn:PBg}) is given by
$$
-\int \int d^n x \, d^nx' \, \{F,\Phi({\bf x})\}D({\bf x},{\bf x}')\{\Phi({\bf x}'),G\},
$$
where $D({\bf x},{\bf x}')={\cal A}^{-1}({\bm \chi}({\bf x}))\delta({\bf x}-{\bf x}')$.
Since $\{F,\Phi({\bf x})\}=-\hat{\cal Q}\mathbb{J}\cdot F_{\bm\chi}$, this contribution is equal to
$$
-\int d^n x \,F_{\bm\chi}\cdot \mathbb{J}\hat{\cal Q}^\dagger {\cal A}^{-1}\hat{\cal Q}\mathbb{J}\cdot G_{\bm\chi}.
$$
Therefore, the Dirac bracket is given by
\begin{equation}
\label{eq:DB}
\{F,G\}_*=\int d^nx \,F_{\bm\chi}\cdot \mathbb{J}_*({\bm\chi})\cdot G_{\bm\chi},
\end{equation}
where
$$
\mathbb{J}_*=\mathbb{J}-\mathbb{J}\hat{\cal Q}^\dagger {\cal A}^{-1}\hat{\cal Q}\mathbb{J}.
$$
We notice that we only need to verify that ${\cal A}$ is invertible on the range of ${\cal Q}$ in order to define the Dirac bracket~(\ref{eq:DB}). It is straightforward to verify that $\mathbb{J}_*$ is antisymmetric because ${\cal A}$ is antisymmetric.
We notice that $\hat{\cal Q}\mathbb{J}_*=0$ (and therefore $\mathbb{J}_*\hat{\cal Q}^\dagger=0$). As a consequence, the constraint $\Phi$ is a Casimir invariant.
The Poisson brackets obtained by the Dirac procedure are Poisson brackets of the form~(\ref{eqn:PBp}) but untainted, i.e., they satisfy the Jacobi identity unconditionally even though they are not of the form~(\ref{eqn:PBproj}) in general. This can be seen by considering a projector ${\cal P}$ as discussed in the previous section. Under the assumption that $\mbox{Ker }{\cal P}=\mbox{Rg }\hat{\cal Q}^\dagger$, we deduce that ${\mathbb J}_*(1-{\cal P})=0$, and consequently:
$$
\mathbb{J}_*={\cal P}^\dagger \mathbb{J}_*{\cal P}.
$$
With this equality, the Poisson bracket becomes
$$
\{F,G\}_*=\int d^n x \,({\cal P} F_{\bm\chi})\cdot {\mathbb J}_*({\bm\chi}) \cdot ({\cal P} G_{\bm\chi}).
$$
The additional feature is that, a priori, the Poisson matrix $\mathbb{J}_*$ is a function of both ${\cal P} {\bm \chi}$ and $(1-{\cal P}) {\bm\chi}$. However, it is straightforward to check that $(1-{\cal P}) {\bm\chi}$ is a Casimir invariant.
Dirac's procedure shows that among the possible projectors ${\cal P}$ satisfying Eq.~(\ref{eqn:comm}), one turns out to be most convenient. The matrix ${\mathbb J}_*$ can be rewritten using the Dirac projector
\begin{equation}
\label{eqn:defProjDir}
{\cal P}_*=1-\hat{\cal Q}^\dagger {\cal A}^{-1}\hat{\cal Q} {\mathbb J},
\end{equation}
as
$$
{\mathbb J}_*={\cal P}_*^\dagger {\mathbb J} {\cal P}_*,
$$
so that the Dirac bracket becomes the same as the original one~(\ref{eqn:PBg}) with the exception that the functional derivatives are projected using the Dirac projector,
\begin{equation}
\label{eqn:compute_DB}
\{F,G\}_*=\int d^n x \,({\cal P}_* F_{\bm\chi}) \cdot {\mathbb J}({\bm \chi})\cdot ({\cal P}_* G_{\bm\chi}),
\end{equation}
where we notice that the Poisson matrix is ${\mathbb J}$ and not ${\mathbb J}_*$. The main difference between the the orthogonal projector ${\cal P}_\perp$ and the Dirac projector ${\cal P}_*$ is that ${\cal P}_\perp$ is a purely geometric object since it only depends on the constraints, and ${\cal P}_*$ is a dynamical object since it involves the Poisson matrix.
{\em Remark:}
We observe that the matrix corresponding to the Dirac bracket has the following property:
$${\mathbb J}_*={\cal P}_*^\dagger {\mathbb J} {\cal P}_*= {\mathbb J} {\cal P}_*= {\cal P}_*^\dagger {\mathbb J}, $$
i.e., the Dirac bracket can be rewritten from Eq.~(\ref{eqn:compute_DB}) using only one Dirac projector, e.g.,
\begin{eqnarray*}
\{F,G\}_*= \int d^n x \, F_{\bm\chi} \cdot {\mathbb J}({\bm \chi}) \cdot {\cal P}_* G_{\bm\chi}.
\end{eqnarray*}
As a result, the computation of the Dirac bracket is made easier.
\subsubsection{Semi-local constraints}
The calculation of Sec.~\ref{local} can be generalized to allow semi-local constraints in phase space. To this end we split the set of coordinates into two pieces, i.e., ${\bf x}=({\bf x}_1,{\bf x}_2)$ where ${\bf x}_1\in {\mathbb R}^{n-m}$ and ${\bf x}_2\in {\mathbb R}^{m}$. The semi-local constraints are given by
$$
\Phi({\bf x}_1)=\bar{\cal Q}[{\bm \chi}]({\bf x}_1)=\int d^m x_2\, {\cal Q}[{\bm \chi}]({\bf x}),
$$
where ${\cal Q}[{\bm \chi}]({\bf x})$ is a function of ${\bm \chi}({\bf x})$ and its derivatives.
The linear operator $\hat{\bar{\cal Q}}$ is defined by the linear operator associated with the function ${\cal Q}$ by
$$
{\hat{\bar {\cal Q}}}=\int d^m x_2 \,\hat{{\cal Q}}.
$$
Since ${\hat{\bar {\cal Q}}}$ acting on a function of ${\bf x}$ is only a function of ${\bf x}_1$, the linear operator $\hat{\bar{\cal Q}}^\dagger$ is defined by the equation
$$
\int d^{n-m }x_1 \,{\hat{\bar {\cal Q}}} {\bm\chi} \cdot {\bf w}({\bf x}_1)=\int d^n x \,{\bm\chi}({\bf x}) \cdot
\hat{\bar{\cal Q}}^\dagger {\bf w}.
$$
Consequently, $\hat{\bar {\cal Q}}^\dagger$ is a linear operator acting on functions of ${\bf x}_1$ as ${\hat{{\cal Q}}}^\dagger$, i.e., ${\hat{\bar {\cal Q}}}^\dagger w({\bf x}_1)={\hat{ {\cal Q}}}^\dagger w({\bf x}_1)$.
In a manner similar to that of Sec.~\ref{local}, the computation of the Dirac bracket shows that the operator
$$
{\cal A}=\hat{\bar {\cal Q}} {\mathbb J} \hat{\bar {\cal Q}}^\dagger ,
$$
must be invertible.
More explicitly, the linear operator ${\cal A}$ acts on functions of ${\bf x}_1$ as
$$
{\cal A} w({\bf x}_1)=\int d^m x_2 {\hat{\cal Q}}{\mathbb J} {\hat{\cal Q}}^\dagger w({\bf x}_1).
$$
The expression of the Dirac projector is given by
$$
{\cal P}_*=1- {\hat{\bar {\cal Q}}}^\dagger {\cal A}^{-1} {\hat{\bar {\cal Q}}} {\mathbb J},
$$
in a very similar way as the case of the local constraints. We notice that the linear operator ${\cal A}$ only needs to be invertible on $\mbox{ Rg } {\hat{\bar {\cal Q}}}$. Another important projector is the orthogonal projector given by
$$
{\cal P}_\perp= 1-{\hat{\bar {\cal Q}}}^\dagger ({\hat{\bar {\cal Q}}}{\hat{\bar {\cal Q}}}^\dagger)^{-1}{\hat{\bar {\cal Q}}}.
$$
As in the case of local constraints, these two projectors satisfy ${\mathbb J}_*={\cal P}^\dagger {\mathbb J}_* {\cal P}$, along with the two properties ${\cal P}_\perp {\cal P}_*={\cal P}_\perp$ and ${\cal P}_*{\cal P}_\perp={\cal P}_*$. In addition, the Dirac projector satisfies ${\mathbb J}_*={\cal P}_*^\dagger {\mathbb J}{\cal P}_*={\cal P}_*^\dagger {\mathbb J}={\mathbb J}{\cal P}_*$.
\section{Example 1: magnetohydrodynamics}
\subsection{Compressible magnetohydrodynamics}
\label{compMHD}
A particularly interesting example is afforded by the Hamiltonian structure of magnetohydrodynamics. The equations for the velocity field ${\bf v}({\bf x},t)$, the density $\rho({\bf x},t)$, the magnetic field ${\bf B}({\bf x},t)$, and the entropy $s({\bf x},t)$ are given by
\begin{eqnarray*}
&& \dot{\rho}=-\nabla\cdot (\rho{\bf v}),\\
&& \dot{\bf v}=-{\bf v}\cdot\nabla{\bf v}-\rho^{-1}\nabla (\rho^2 U_\rho) +\rho^{-1} (\nabla \times{\bf B})\times {\bf B},\\
&& \dot{\bf B}=\nabla\times({\bf v}\times{\bf B}),\\
&& \dot{s}=-{\bf v}\cdot\nabla s,
\end{eqnarray*}
where $U$ is the internal energy and $U_\rho$ here denotes the partial derivative of $U$ with respect to $\rho$.
The dynamical variables are $\rho({\bf x})$, ${\bf v}({\bf x})$, ${\bf B}({\bf x})$ and $s({\bf x})$ where ${\bf x}$ belongs to ${\mathbb R}^3$. The observables of the system are functionals of these vector fields, denoted generically by $F(\rho,{\bf v},{\bf B},s)$.
In these coordinates, this system has the following Hamiltonian
\begin{equation*}
\label{eqn:H}
H(\rho,{\bf v},{\bf B},s)=\int d^3x\left( \frac{1}{2}\rho{\bf v}^2 +\rho U(\rho,s)+\frac{{\bf B}^2}{2}\right).
\end{equation*}
There are two slightly different Poisson brackets that have been proposed in Refs.~\cite{morr80a,morr82e,morr82}.
A first one was given in Ref.~\cite{morr80a},
\begin{eqnarray}
\{F,G\}&=&-\int d^3x \,\left[ F_\rho \nabla\cdot G_{\bf v} +F_{\bf v}\cdot\nabla G_\rho -\rho^{-1}(\nabla\times {\bf v})\cdot \left( F_{\bf v}\times G_{\bf v}\right)\right.\nonumber \\
&&\qquad \qquad \left.+\rho^{-1}\nabla s \cdot \left( F_s G_{\bf v}-F_{\bf v} G_s\right)\right] +\{F,G\}_B, \label{eqn:PB1}
\end{eqnarray}
where the magnetic part $\{F,G\}_B$ of the Poisson bracket is chosen as $\{F,G\}_B=\{F,G\}_{B,{\rm t}}$
\begin{equation}
\{F,G\}_{B,{\rm t}}=-\int d^3x \, \rho^{-1}\left( F_{\bf v}\cdot{\bf B}\times( \nabla\times G_{\bf B})-G_{\bf v}\cdot{\bf B}\times( \nabla\times F_{\bf B})\right). \label{eq:PBmpt}
\end{equation}
It was pointed out in Ref.~\cite{morr82} that this bracket satisfies the Jacobi identity only when $\nabla\cdot {\bf B}=0$, and also that $\nabla\cdot {\bf B}$ commutes with any other functionals, i.e., $\{F,\nabla\cdot {\bf B}\}=0$ for all $F$ (it is a Casimir-like property, even though we cannot call it a Casimir invariant since the Jacobi identity is only satisfied when $\nabla\cdot {\bf B}=0$). As was the case for the vorticity equation~(\ref{eqn:vort}), the functional derivatives with respect to ${\bf B}$ must be divergence-free for coherence. However, we notice that here, since only $\nabla\times F_{\bf B}$ are involved in the expression of the magnetic part~(\ref{eqn:PB1}) of the Poisson bracket, it does not make any difference whether $F_{\bf B}$ is divergence-free or not.
In order to extend the definition of the Poisson bracket to functionals of any ${\bf B}$, ones not necessarily divergence-free, a second Poisson bracket was proposed in Refs.~\cite{morr82e,morr82}. There the magnetic part of the Poisson bracket~(\ref{eqn:PB1}) was replaced by
\begin{eqnarray*}
\{F,G\}_{B,1}&=& -\int d^3x \,\left[\left( \rho^{-1}F_{\bf v} \cdot [\nabla G_{\bf B}] - \rho^{-1}G_{\bf v} \cdot [\nabla F_{\bf B}]\right) \cdot {\bf B}\right. \\
&& \qquad \qquad \left.+ {\bf B}\cdot \left( [\nabla \left(\rho^{-1}F_{\bf v}\right)]\cdot G_{\bf B}- [\nabla \left(\rho^{-1}G_{\bf v}\right)] \cdot F_{\bf B}\right)\right]. \label{eqn:PB}
\end{eqnarray*}
Here the notation ${\bf a}\cdot [M] \cdot{\bf b}$ is a scalar explicitly given by $\sum_{ij}a_i M_{ij} b_j$ for any vectors ${\bf a}$ and ${\bf b}$ and any matrix $[M]$. It was shown that this bracket satisfies the Jacobi identity for all functionals of $(\rho,{\bf v},s,{\bf B})$ regardless of the condition $\nabla\cdot{\bf B}=0$.
The magnetic part of this Poisson bracket is rewritten as
\begin{eqnarray}
\{F,G\}_{B,1}&=&-\int d^3x\,\rho^{-1}\left[ F_{\bf v}\cdot{\bf B}\times( \nabla\times G_{\bf B})-G_{\bf v}\cdot{\bf B}\times( \nabla\times F_{\bf B})\right] \nonumber \\
&& +\int d^3x \,\rho^{-1}\nabla \cdot {\bf B} \left( F_{\bf v}\cdot G_{\bf B}-F_{\bf B}\cdot G_{\bf v}\right).\label{eq:MHDPB2}
\end{eqnarray}
The first line of the above bracket corresponds to the Poisson bracket introduced in Ref.~\cite{morr80a} [see Eq.~(\ref{eq:PBmpt})]. With the additional terms (proportional to $\nabla\cdot {\bf B}$) the Jacobi identity is unconditionally satisfied for any functionals of $(\rho,{\bf v},s,{\bf B})$. However, a property of the bracket~(\ref{eqn:PB1}) with the magnetic part~(\ref{eq:PBmpt}) has been lost, $\nabla\cdot {\bf B}$ does not Poisson-commute with any functional, so it is not a Casimir invariant.
In order to have both the Jacobi identity unconditionally satisfied and $\nabla\cdot {\bf B}$ a Casimir invariant, we apply the prescription~(\ref{eqn:PBproj}) on the magnetic part~(\ref{eq:PBmpt}). At every instance in the Poisson bracket where ${\bf B}$ is explicitly mentioned, we replace ${\bf B}$ with
$\bar{\bf B}={\bf B}-\nabla\Delta^{-1}\nabla\cdot {\bf B}$. The magnetic part becomes
$$
\{F,G\}_B=-\int d^3x\,\rho^{-1}\left( F_{\bf v}\cdot(\bar{\bf B}\times( \nabla\times G_{\bf B})-G_{\bf v}\cdot(\bar{\bf B}\times( \nabla\times F_{\bf B}))\right),
$$
and it is rewritten as
\begin{eqnarray}
&& \{F,G\}_B =-\int d^3x\,\rho^{-1}\left( F_{\bf v}\cdot({\bf B}\times( \nabla\times G_{\bf B})-G_{\bf v}\cdot({\bf B}\times( \nabla\times F_{\bf B}))\right)\nonumber \\
&& \quad +\int d^3x \,\nabla \cdot {\bf B} \, \Delta^{-1}\nabla\cdot\left(\rho^{-1}F_{\bf v}\times(\nabla\times G_{\bf B})-\rho^{-1}G_{\bf v}\times(\nabla\times F_{\bf B})\right).\label{eq:MHDPB3}
\end{eqnarray}
Here we notice that the correction term still contains terms proportional to $\nabla \cdot {\bf B}$ but is different from the one in Eq.~(\ref{eq:MHDPB2}). The main difference is that $\nabla\cdot {\bf B}$ is not a Casimir invariant for the Poisson bracket~(\ref{eq:MHDPB2}) whereas it is one for the Poisson bracket~(\ref{eq:MHDPB3}) since it only involves terms like $\nabla\times G_{\bf B}$.
\subsection{Incompressible magnetohydrodynamics}
\label{sec:MHDincomp}
For incompressible MHD we begin with the equations for compressible magnetohydrodynamics from Sec.~\ref{compMHD} and apply constraints. The Poisson bracket given by Eqs.~(\ref{eqn:PB1})-(\ref{eq:MHDPB3}) is of the form~(\ref{eqn:PBg}) with
$$
{\mathbb J}=\left( \begin{array}{cccc}
0 & -\nabla \cdot & 0 & 0\\
-\nabla & -\rho^{-1}(\nabla \times {\bf v})\times & -\rho^{-1}\bar{\bf B} \times (\nabla \times) & \rho^{-1} \nabla s \\
0 & -\nabla\times (\rho^{-1} \bar{\bf B}\times) & 0 & 0\\
0 & -\rho^{-1}\nabla s & 0 & 0
\end{array}\right).
$$
We impose the following local constraints on the field variables ${\bm\chi}=(\rho,{\bf v}, {\bf B},s)$,
$$
{\cal Q}[{\bm\chi}]({\bf x})=(\rho,\nabla\cdot {\bf v}).
$$
The reduction to incompressible MHD using Dirac's theory has already been done in Ref.~\cite{chan11} and the reduction to incompressible Euler equation in Refs.~\cite{nguy99,nguy01}. Here we propose a more compact way to present this reduction using the operators introduced in the previous sections. The expressions of the intermediate operators are
\begin{eqnarray*}
&& \hat{\cal Q}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\
0 & \nabla \cdot & 0 & 0
\end{array}\right),\\
&& \hat{\cal Q}^\dagger = \left(
\begin{array}{cc} 1 & 0\\
0 & -\nabla \\
0 & 0\\
0 & 0
\end{array}\right),\\
&& {\cal A}=\left( \begin{array}{cc} 0 & \Delta \\ -\Delta & \nabla \cdot (\rho^{-1}(\nabla\times {\bf v})\times \nabla) \end{array}\right),\\
&& {\cal A}^{-1}=\left( \begin{array}{cc} \Delta^{-1}\nabla \cdot (\rho^{-1}(\nabla\times {\bf v})\times \nabla) & -\Delta^{-1} \\ \Delta^{-1} & 0 \end{array}\right).
\end{eqnarray*}
The orthogonal projector is given by Eq.~(\ref{eqn:orth}) and its expression is
$$
{\cal P}_\perp F_{\bm \chi}=(0,\bar{F}_{\bf v}, F_{\bf B}, F_s),
$$
where $\bar{F}_{\bf v}=F_{\bf v}-\nabla \Delta^{-1}\nabla\cdot F_{\bf v}$.
The Dirac projector, computed from the Poisson bracket~(\ref{eqn:PB1}) where ${\bf B}$ has been replaced by $\bar{\bf B}={\bf B}-\nabla \Delta^{-1}\nabla \cdot {\bf B}$, is given by
$$
{\cal P}_*F_{\bm\chi}=(F_*,\bar{F}_{\bf v}, F_{\bf B}, F_s),
$$
where
$$
F_*=\Delta^{-1}\nabla\cdot\left( \rho^{-1}\left( (\nabla\times{\bf v})\times \bar{F}_{\bf v}-\bar{\bf B}\times (\nabla \times F_{\bf B})- F_s \nabla s \right) \right).
$$
We notice that the two projectors differ in the first component. Even though the two projectors ${\cal P}_\perp$ and ${\cal P}_*$ are different, both of these projectors satisfy the equation ${\mathbb J}_*={\cal P}^\dagger {\mathbb J}{\cal P}$, which is always the case for the Dirac projector but not true in general for the orthogonal projector. Actually any projector ${\cal P}F_{\bm\chi}=(F_*(\bar{F}_{\bf v},F_{\bf B},F_s), \bar{F}_{\bf v},F_{\bf B},F_s)$ satisfies ${\mathbb J}_*={\cal P}^\dagger {\mathbb J}{\cal P}$ for any function $F_*$. The first component is thus irrelevant, and consequently the orthogonal projector is the simplest projector to be used for constrained functional derivatives.
From this projector, we compute the Dirac bracket from Eq.~(\ref{eqn:compute_DB}), and it gives the same bracket as that produced in Ref.~\cite{chan11}:
\begin{eqnarray*}
\{F,G\}_*&=& \int d^3 x \,\rho^{-1}\left( (\nabla\times {\bf v})\cdot ( \bar{F}_{\bf v}\times \bar{G}_{\bf v}) -\nabla s \cdot ( F_s \bar{G}_{\bf v}-\bar{F}_{\bf v} G_s)\right.\\
&& \qquad \qquad \left. + \bar{\bf B} \cdot ( \bar{F}_{\bf v}\times( \nabla\times G_{\bf B})+ ( \nabla\times F_{\bf B})\times \bar{G}_{\bf v})\right),
\end{eqnarray*}
where $\bar{F}_{\bf v}=F_{\bf v}-\nabla\Delta^{-1}\nabla\cdot F_{\bf v}$.
\section{Example 2: Vlasov-Maxwell equations}
\subsection{Vlasov-Maxwell modified bracket as a Dirac bracket}
As a second example, we consider the Vlasov-Maxwell equations for the distribution of charged particles in phase space $f({\bf x},{\bf v},t)$ and the electromagnetic fields ${\bf E}({\bf x},t)$ and ${\bf B}({\bf x},t)$ given by
\begin{eqnarray*}
&& \dot{f}=-{\bf v}\cdot \nabla f -({\bf E}+{\bf v}\times {\bf B})\cdot \partial_{\bf v} f,\\
&& {\dot {\bf E}}=\nabla \times {\bf B}-{\bf J},\\
&& \dot{\bf B}=-\nabla \times {\bf E},
\end{eqnarray*}
where ${\bf J}=\int d^3v \,{\bf v} f$. The Hamiltonian of this system is given by
$$
H=\int d^6z \,f \frac{{\bf v}^2}{2} +\int d^3x\,\frac{{\bf E}^2+{\bf B}^2}{2},
$$
where we denote ${\bf z}=({\bf x},{\bf v})$. The Poisson bracket between two functionals of $f({\bf x},{\bf v})$, ${\bf E}({\bf x})$ and ${\bf B}({\bf x})$ is given by
\begin{eqnarray}
\{F,G\}_{\rm t}&=&\int d^6 z \, f \left( [F_f,G_f]_{\rm c}+[F_f,G_f]_{\rm B}+G_{\bf E}\cdot \partial_{\bf v}F_f-F_{\bf E}\cdot \partial_{\bf v}G_f\right)\nonumber \\
&& +\int d^3x \,\left( F_{\bf E} \cdot\nabla \times G_{\bf B}-\nabla \times F_{\bf B}\cdot G_{\bf E}\right),
\label{eq:PBMV}
\end{eqnarray}
where the two brackets $[\cdot ,\cdot]_{\rm c}$ and $[\cdot ,\cdot ]_{\rm B}$ are defined by
\begin{eqnarray}
&& [f,g]_{\rm c}=\nabla f\cdot \partial_{\bf v}g-\partial_{\bf v}f\cdot \nabla g,\label{eqn:bc}\\
&& [f,g]_{\rm B}={\bf B}\cdot(\partial_{\bf v}f\times\partial_{\bf v}g). \label{eqn:bbp}
\end{eqnarray}
The Poisson bracket~(\ref{eq:PBMV}) was given in Ref.~\cite{mars82} based on an earlier work of Ref.~\cite{morr80b} (see also Ref.~\cite{bial84}). It was pointed out in Ref.~\cite{morr82} that the Poisson bracket~(\ref{eq:PBMV}) only satisfies the Jacobi identity when $\nabla \cdot {\bf B}=0$, which is to say that it does not satisfy the Jacobi identity for arbitrary functionals of $(f,{\bf E},{\bf B})$. This problem is actually already present in the Lagrangian description (for the dynamics of charged particles) since $[\cdot,\cdot]_{\rm c}+[\cdot,\cdot]_{\rm B}$ only satisfies the Jacobi identity for functions ${\bf B}$ such that $\nabla\cdot {\bf B}=0$, whereas, individually, $[\cdot,\cdot]_{\rm c}$ and $[\cdot,\cdot]_{\rm B}$ satisfy the Jacobi identity for an arbitrary function ${\bf B}$.
In order to remedy this problem, we modify the bracket $[\cdot,\cdot]_{\rm B}$ to take the form of (\ref{eqn:PBproj}),
$$
[f,g]_{{\rm B}_P}=({\bf B}-\nabla \Delta^{-1}\nabla\cdot {\bf B})\cdot(\partial_{\bf v}f\times\partial_{\bf v}g).
$$
With this modified gyrobracket, we readily check that $[\cdot,\cdot]_{\rm c}+[\cdot,\cdot]_{{\rm B}_P}$ satisfies the Jacobi identity. Next, we consider the modified Poisson bracket~(\ref{eq:PBMV}) obtained by replacing $[\cdot,\cdot]_{\rm B}$ by $[\cdot ,\cdot ]_{{\rm B}_P}$, i.e., we consider the Poisson bracket
\begin{eqnarray}
\{F,G\}_{\rm VM}&=&\int d^6 z \,f \left( [F_f,G_f]_{\rm c}+[F_f,G_f]_{{\rm B}_P}+G_{\bf E}\cdot \partial_{\bf v}F_f-F_{\bf E}\cdot \partial_{\bf v}G_f\right)\nonumber \\
&& +\int d^3x \,\left( F_{\bf E} \cdot\nabla \times G_{\bf B}-\nabla \times F_{\bf B}\cdot G_{\bf E}\right),
\label{eq:PBMVproj}
\end{eqnarray}
which satisfies the Jacobi identity unconditionally. This follows from the change of variable
${\bf B}_P={\bf B}-\nabla \Delta^{-1}\nabla\cdot {\bf B}$ and ${\bf B}_Q=\nabla \Delta^{-1}\nabla\cdot {\bf B}$
where it should be noted that
$$
\nabla\times G_{\bf B}=\nabla \times G_{{\bf B}_P},
$$
since the operator ${\cal P}=1-\nabla\Delta^{-1}\nabla \cdot $ satisfies ${\cal P}\nabla\times =\nabla\times$.
Here it should be noticed that $\nabla\cdot {\bf B}$ is a Casimir invariant for the Poisson bracket~(\ref{eq:PBMVproj}). The untainted form of the Vlasov-Maxwell bracket~(\ref{eq:PBMVproj}) gives the Hamiltonian structure of the Vlasov-Maxwell equations in terms of physical fields without introducing the vector potential, i.e., without the restriction of $\nabla\cdot{\bf B}=0$.
In order to realize the link between brackets defined using projectors and Dirac brackets, we show below that the Poisson bracket~(\ref{eq:PBMVproj}) is a Dirac bracket of some parent bracket obtained using two constraints which, by definition, are Casimir invariants of the bracket~(\ref{eq:PBMVproj})
\begin{eqnarray*}
&& {\cal Q}[f,{\bf E},{\bf B}]({\bf x})=(\nabla\cdot {\bf E}-\rho, \nabla\cdot {\bf B}),
\end{eqnarray*}
where $\rho=\int d^3 v \,f$. As expected there is an infinite number of solutions for the parent bracket. A family of solutions is given by
\begin{equation}
\label{eq:MVparent}
\{F,G\}=\{F,G\}_{\rm VM}+\int d^3x \,\left( \nabla \cdot F_{\bf B} {\cal D} \nabla \cdot G_{\bf E}-\nabla \cdot F_{\bf E} {\cal D}^\dagger \nabla\cdot G_{\bf B}\right),
\end{equation}
where ${\cal D}$ is a linear operator independent of the field variables, so that the Jacobi identity is guaranteed by Morrison's lemma of Ref.~\cite{morr82}. This statement uses the fact that the Vlasov-Maxwell bracket has been made untainted; it would not be true if the original tainted Vlasov-Maxwell bracket~(\ref{eq:PBMV}) was considered instead of the Poisson bracket~(\ref{eq:PBMVproj}).
Now, if we apply the Dirac procedure on the extended Poisson bracket~(\ref{eq:MVparent}) with the primary constraint $\nabla \cdot\mathbf{E}-\rho$, we get the secondary constraint $\nabla \cdot\mathbf{B}$, and the reduced Dirac bracket is obtained from $\mathbb{J}_*=\mathcal{P}_*^\dagger\mathbb{J}\mathcal{P}_*$ where $\mathcal{P}_*$ is the Dirac projector~(\ref{eqn:defProjDir}). The Dirac projector can be explicitly computed. However, in order to further simplify the computation of the Dirac bracket, we use the orthogonal projector since, as in the case of incompressible MHD (see Sec.~\ref{sec:MHDincomp}), it satisfies the same relation as the Dirac projector, i.e., $\mathbb{J}_*=\mathcal{P}_*^\dagger\mathbb{J}\mathcal{P}_*=\mathcal{P}_\bot^\dagger\mathbb{J}\mathcal{P}_\bot$, where
$$
\mathcal{P}_\bot F_{\bm\chi}
=(F_\mathbf{B}-\nabla \Delta^{-1}\nabla \cdot F_{\bf B}, F_\mathbf{E}, F_f) .
$$
This implies the expected result that the Vlasov-Maxwell bracket~(\ref{eq:PBMVproj}) is the Dirac bracket of the bracket~(\ref{eq:MVparent}) with Dirac constraints $(\nabla \cdot\mathbf{E}-\rho,\nabla \cdot\mathbf{B})$.
With the extended bracket~(\ref{eq:MVparent}), the Casimirs $(\nabla \cdot\mathbf{E}-\rho,\nabla \cdot\mathbf{B})$ of the Vlasov-Maxwell system now have dynamics given by
\begin{eqnarray*}
\frac{\partial }{\partial t}\left(\nabla \cdot \mathbf{E} -\rho \right)
&=&
\Delta\mathcal{D}^\dagger\nabla \cdot\mathbf{B} ,
\\
\frac{\partial }{\partial t}\nabla \cdot \mathbf{B}
&=&
-\Delta\mathcal{D}\nabla \cdot\mathbf{E} .
\end{eqnarray*}
These equations suggest two particularly interesting choices for our still undetermined operator $\mathcal{D}$. Defining $\mathcal{D}=\Delta^{-1}$ gives to $\nabla \cdot\mathbf{E}-\rho$ and $\nabla \cdot\mathbf{B}$ the dynamics of stationary waves when $\rho=0$, whereas defining $\mathcal{D}=(-\Delta)^{-1/2}$ gives them the dynamics of propagating waves. We note that these operators always act on divergences of vector fields.
{\em Remark:} As a side note, we point out that the choice of ${\cal D}=(-\Delta)^{-1/2}$ naturally exhibits the operator $\nabla *:=\nabla (-\Delta)^{-1/2} \nabla \cdot$ which corresponds to $\nabla \times $ for the compressible part of a vector field. Indeed, the operator
$$
-\nabla * \Delta^{-1}\nabla * =\nabla \Delta^{-1}\nabla \cdot ,
$$
is the orthogonal projector onto the kernel of $\nabla \times $, just as $-\nabla \times \Delta^{-1}\nabla \times =1-\nabla \Delta^{-1}\nabla \cdot$ is the complementary projector onto the kernel of $\nabla *$. With this choice, the resulting dynamical equations associated with the Poisson bracket~(\ref{eq:MVparent}) for the solenoidal and the compressible parts of the electromagnetic fields become independent and similar:
\begin{eqnarray*}
& \square \mathbf{E}_S = - \dot{\mathbf J}_S ,
\qquad
& \square \mathbf{E}_C = - \dot{\mathbf J}_C ,
\\
& \square \mathbf{B}_S = \nabla \times \ {\mathbf J}_S ,
\qquad
& \square \mathbf{B}_C = \nabla *\ {\mathbf J}_C ,
\end{eqnarray*}
where $\square$ is the d'Alembert operator $\square=\partial ^2/\partial t^2 -\Delta$ and ${\bm \psi}_S$ is the solenoidal part of the vector field ${\bm\psi}$, i.e., ${\bm \psi}_S= - \nabla \times \Delta^{-1}\nabla \times {\bm \psi}=(1-\nabla \Delta^{-1}\nabla) \cdot{\bm \psi}$ and ${\bm \psi}_C$ is its compressible part, which is ${\bm \psi}_C= - \nabla * \Delta^{-1}\nabla * {\bm \psi}= \nabla \Delta^{-1}\nabla \cdot{\bm \psi}$. In the absence of matter, the fields ${\bm \psi}_S$ and ${\bm \psi}_C$ propagate as independent free waves.
\subsection{From Vlasov-Maxwell to Vlasov-Poisson equations}
In order to obtain Vlasov-Poisson equations from the Vlasov-Maxwell equations we impose two constraints:
$$
{\cal Q}[f,{\bf E},{\bf B}]({\bf x})=({\bf B}-{\bf B}_0({\bf x}),\nabla \times {\bf E}),
$$
where ${\bf B}_0$ is a non-uniform background magnetic field. The operators $\hat{\cal Q}$ and $\hat{\cal Q}^\dagger$ are given by
$$
\hat{\cal Q}=\left( \begin{array}{ccc} 0 & \nabla \cdot & 0\\ 0 & 0 & 1
\end{array}\right),
$$
and
$$
\hat{\cal Q}^\dagger=\left( \begin{array}{cc} 0 & 0\\ \nabla \cdot & 0\\ 0 & 1
\end{array}\right).
$$
The orthogonal projector ${\cal P}_\perp$ given by Eq.~(\ref{eqn:orth}) is given by
$$
{\cal P}_\perp=\left( \begin{array}{ccc} 1 & 0 & 0\\ 0 & \nabla \Delta^{-1} \nabla \cdot & 0\\
0 & 0 & 0
\end{array}\right).
$$
Contrary to the orthogonal projector, the expression of the Dirac projector depends on the dynamics, and in particular on the Poisson matrix ${\mathbb J}$ which is given by
$$
{\mathbb J}=\left( \begin{array}{ccc} -[f,\cdot] & -\partial_{\bf v} f & 0\\
-f\partial_{\bf v} & 0 & \nabla \times \\
0 & - \nabla \times & 0
\end{array}\right),
$$
where the small bracket $[\cdot,\cdot]$ is given by $[\cdot,\cdot]=[\cdot,\cdot]_c+[\cdot,\cdot]_{B_P}$ with these two brackets given by Eqs.~(\ref{eqn:bc})-(\ref{eqn:bbp}).
The operator ${\cal A}$ is given by
$$
{\cal A}=\left( \begin{array}{cc} 0 & (\nabla \times)^2\\ -(\nabla\times)^2 & 0
\end{array}\right).
$$
The operator ${\cal A}$ is not invertible, but one can give an expression for ${\cal A}^{-1} \hat{\cal Q}$ by
$$
{\cal A}^{-1}\hat{\cal Q}=\left( \begin{array}{ccc} 0 & 0 & \Delta^{-1}(1-\nabla \Delta^{-1}\nabla\cdot)\\
0 & -\Delta^{-1}\nabla \times & 0
\end{array}\right).
$$
As a result, the Dirac projector is computed,
$$
{\cal P}_*=\left( \begin{array}{ccc} 1 & 0 & 0\\
0 & \nabla \Delta^{-1} \nabla \cdot & 0\\
-\Delta^{-1}\nabla \times f\partial_{\bf v} & 0 & \nabla \Delta^{-1}\nabla \cdot
\end{array}\right).
$$
We notice that both projectors ${\cal P}_\perp$ and ${\cal P}_*$ satisfy the
equation ${\mathbb J}_*={\cal P}^\dagger {\mathbb J}{\cal P}$ and the Poisson matrix of the Vlasov-Poisson equations is given by
$$
{\mathbb J}_*=\left( \begin{array}{ccc}
-[f,\cdot] & -\nabla \Delta^{-1} \nabla \cdot \partial_{\bf v}f & 0\\
-\nabla \Delta^{-1} \nabla \cdot(f \partial_{\bf v}) & 0 & 0\\
0 & 0 & 0
\end{array}\right).
$$
It leads to the expression of the Poisson bracket,
\begin{eqnarray*}
\{F,G\}_*= \int d^6z \,f[F_f-\Delta^{-1}\nabla\cdot F_{\bf E},G_f-\Delta^{-1}\nabla\cdot G_{\bf E}].
\end{eqnarray*}
Like for the incompressible MHD equations, even if the Dirac and orthogonal projectors are different, both of them can be used to compute the Dirac bracket from the Poisson matrix ${\mathbb J}$, the orthogonal projector being slightly simpler and more straightforward to compute.
\section{Example 3: Quasi-neutrality as semi-local constraints}
In this section, we give an example of a set of physically relevant constraints where the orthogonal projector does not exist, and where the Dirac projector is a natural replacement for computing the constrained functional derivatives. We consider the following Vlasov equation with two species, ions and electrons, linearized about spatially homogeneous distribution functions $\alpha_i ({\bf v})$ and $\alpha_e ({\bf v})$. The equations for the phase space density {\em fluctuation} of ions $f_{\rm i}({\bf x},{\bf v})$ and electrons $f_{\rm e}({\bf x},{\bf v})$, are given by
$$
\dot{f_s}=-{\bf v}\cdot \nabla f_s +e_s \, \partial_{\bf v} \alpha_s \cdot\nabla \phi,
$$
where $\Delta \phi=-\sum_s e_s \int d^3v \,f_s$ and $e_s=\pm 1$ is the charge of the particles of each species $s={\rm i},{\rm e}$.
The field variables are ${\bm\chi}({\bf z})= (f_{\rm i}({\bf z}), f_{\rm e}({\bf z}))$, which are functions of ${\bf z}=({\bf x},{\bf v})$. The Poisson bracket, in this case, is defined by the Poisson matrix ${\mathbb J}$ given by
$$
{\mathbb J} = -\left(\begin{array}{cc} [ \alpha_{\rm i}({\bf v}), \cdot\, ] & 0 \\0 & [ \alpha_{\rm e}({\bf v}), \cdot\, ]\end{array}\right),
$$
with $[ \alpha_s({\bf v}) , G ] =- \partial_{\bf v}\alpha_s \cdot \nabla G$. According to Ref.~\cite{holm85} (see also Ref.~\cite{morr98}), the corresponding Hamiltonian can be found as the quadratic functional corresponding to the second derivative of the Hamiltonian (plus Casimirs) of the nonlinear Vlasov-Poisson system, evaluated at $f_s=\alpha_s ({\bf v})$. This is true for ion and electron densities close to equilibria that are isotropic in velocity (like a Maxwellian for instance).
We impose the set of two semi-local constraints
$$
\bar{\cal Q}[{\bm\chi}]({\bf x}) = \left( \int d^3 v\, (f_{\rm i}-f_{\rm e}), \int d^3 v \, {\bf v} \cdot \nabla (f_{\rm i}-f_{\rm e})\right).
$$
The first component of the constraint is the quasi-neutrality. The second component is a secondary constraint associated with quasi-neutrality according to Dirac's theory of constrained Hamiltonian systems~\cite{Bhans76}.
Since the constraints are linear with respect to the field variables, the operators $ \hat{\bar{\cal Q}}$ and $ \hat{\bar{\cal Q}}^\dagger $ are given by
$$
{\hat{\bar{\cal Q}}}= \int {d^3 v} \left(\begin{array}{cc} 1 & -1 \\ {\bf v}\cdot \nabla &-{\bf v}\cdot \nabla \end{array}\right),~~~~\textrm{and } ~~~~~ \hat{\bar{\cal Q}}^\dagger = \left(\begin{array}{cc} 1 & -{\bf v}\cdot \nabla \\ -1 & {\bf v}\cdot \nabla \end{array}\right).
$$
The operator $\hat{\bar{\cal Q}}^\dagger$ acts on functions of ${\bf x}$ only.
In order to compute the Dirac projector, the matrix ${\cal A}=\hat{\bar{\cal Q}}{\mathbb J}\hat{\bar{\cal Q}}^\dagger$ needs to be computed
$$
{\cal A} = \left(\begin{array}{cc} 0 & ( \bar{\alpha}_{\rm i} + \bar{\alpha}_{\rm e})\Delta \\ -( \bar{\alpha}_{\rm i} + \bar{\alpha}_{\rm e})\Delta & 2( {\bm \beta}_{\rm i} + {\bm \beta}_{\rm e})\cdot \nabla \Delta \end{array}\right),
$$
with $\bar{\alpha}_{s} =\int d^3 v\, \alpha_s$ and ${\bm \beta}_s =\int d^3 v\, {\bf v} \alpha_s$.
This operator is invertible and its inverse is
$$
{\cal A}^{-1} = \frac{1}{ \bar{\alpha}_{\rm i} + \bar{\alpha}_{\rm e}} \left(\begin{array}{cc} \displaystyle \frac{2( {\bm\beta}_{\rm i}+ {\bm\beta}_{\rm e}) }{ \bar{\alpha}_{\rm i} + \bar{\alpha}_{\rm e}}\cdot \nabla \Delta^{-1} & -\Delta^{-1} \\ \Delta^{-1} & 0 \end{array}\right).
$$
Note that the operators ${\cal A}$ and ${\cal A}^{-1}$ act on functions of ${\bf x}$ and return a function of ${\bf x}$.
The Dirac projector ${\cal P}_*$ has the form
$$
{\cal P}_*= 1 - \frac{\Delta^{-1} \nabla \cdot}{\bar{\alpha}_{\rm i} + \bar{\alpha}_{\rm e}} \int d^3 v'\, \, \left({\bf v} +{\bf v}'-2\bar{\bf v}\right) \left(\begin{array}{cc} [ \alpha_{\rm i} , \cdot ] & - [ \alpha_{\rm e} , \cdot ] \\ -[ \alpha_{\rm i} , \cdot ] & [ \alpha_{\rm e} , \cdot ] \end{array}\right),
$$
where ${\bar{\bf v}}=({\bm\beta}_{\rm i}+{\bm\beta}_{\rm e})/(\bar{\alpha}_{\rm i}+\bar{\alpha}_{\rm e})$.
Concerning the orthogonal projector, we note that $\hat{\bar{\cal Q}} \hat{\bar{\cal Q}}^\dagger$ given by
$$
{\hat{\bar{\cal Q}}} {\hat{\bar{\cal Q}}}^\dagger = 2\int d^3 v\, \left(\begin{array}{cc} 1 & -{\bf v} \cdot\nabla \\ {\bf v} \cdot \nabla & - ({\bf v}\cdot \nabla)^2 \end{array}\right),
$$
does not exist since it is unbounded when it acts on functions of ${\bf x}$. As a consequence, the orthogonal projector cannot be a solution for the computation of constrained functional derivatives. Here a convenient choice is afforded by the Dirac projector.
\section*{Acknowledgments}
We acknowledge financial support from the Agence Nationale de la Recherche (ANR GYPSI). This work was also supported by the European Community under the contract of Association between EURATOM, CEA, and the French Research Federation for fusion study. The views and opinions expressed herein do not necessarily reflect those of the European Commission. P J M was supported by U.S. Department of Energy contract DE-FG05-80ET-53088. The authors also acknowledge fruitful discussions with the \'Equipe de Dynamique Nonlin\'eaire of the Centre de Physique Th\'eorique of Marseille.
\section*{References}
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Oscar Palmer Robertson (nascido em 24 de novembro de 1938), apelidado de "The Big O", é um ex-jogador de basquete profissional que jogou pelo Cincinnati Royals e pelo Milwaukee Bucks na National Basketball Association (NBA). Robertson jogou como armador e foi 12 vezes All-Star, 11 vezes membro da Equipe All-NBA e venceu o Prêmio de MVP. Sua carreira de jogador, especialmente durante o ensino médio e a faculdade, foi atormentada pelo racismo.
Em 1962, ele se tornou o primeiro jogador na história da NBA a obter uma média de triplo-duplo em uma temporada. Na temporada de 1970-71, ele foi um jogador-chave da equipe que trouxe ao Bucks seu primeiro título da NBA.
Robertson entrou duas vezes no Hall da Fama do Naismith Memorial Basketball, tendo sido apresentado em 1980 por sua carreira individual e em 2010 como membro da equipe olímpica de basquete dos Estados Unidos em 1960 e presidente da Associação Nacional de Jogadores de Basquete. Ele também foi eleito um dos 50 Maiores Jogadores da História da NBA em 1996. A Associação de Escritores de Basquete dos Estados Unidos renomeou seu Prêmio de Jogador do Ano da Universidade como Troféu Oscar Robertson em sua homenagem em 1998, e ele foi uma das cinco pessoas escolhidas para representar a classe inaugural do Hall da Fama do Basquete Universitário em 2006. Ele foi classificado como o 36º melhor atleta americano do século XX pela ESPN.
Robertson também era parte integrante do Robertson v. National Basketball Ass'n de 1970. O processo contra a NBA, que foi feito quando Robertson era o presidente da Associação de Jogadores da NBA, levou a uma extensa reforma da free agency da liga e um aumento dos salários de todos os jogadores. Ele foi introduzido no Hall da Fama da FIBA em 2009.
Primeiros anos
Robertson nasceu na pobreza em Charlotte, Tennessee, e cresceu em um conjunto habitacional segregado em Indianápolis.
Ao contrário de muitos outros garotos que preferiam jogar beisebol, ele foi atraído pelo basquete porque era "um jogo de crianças pobres". Como sua família não tinha dinheiro para comprar uma bola de basquete, ele aprendeu a arremessar jogando bolas de tênis e trapos amarrados com elásticos em uma cesta de pêssego atrás da casa de sua família.
Carreira no ensino médio
Robertson frequentou a Crispus Attucks High School, uma escola totalmente negra. Na Crispus Attucks, Robertson foi treinado por Ray Crowe, cuja ênfase em um jogo fundamentalmente sólido teve um efeito positivo no estilo de jogo de Robertson.
Em segundo ano do ensino médio, em 1954, ele atuou em uma equipe da Attucks que perdeu nas semifinais (quartas de final estaduais) para o eventual campeão estadual Milan, cuja história seria mais tarde a base do clássico filme de 1986, Hoosiers.
Quando Robertson estava no terceiro ano, Crispus Attucks dominou seus adversários, tendo um recorde de 31-1 e vencendo o campeonato estadual de 1955, o primeiro de uma escola totalmente negra do país.
No ano seguinte, o time terminou com um recorde perfeito de 31-0 e conquistou o segundo título consecutivo no estado de Indiana, tornando-se o primeiro time em Indiana a garantir uma temporada perfeita e compilando 45 vitórias consecutivas, um recorde estadual. Após o título, o time foi desfilando pela cidade em uma tradição regular, mas eles foram levados para um parque fora do centro da cidade para continuar sua celebração, ao contrário de outros times. Robertson afirmou: "[Os policiais] pensaram que os negros iriam destruir a cidade e eles pensaram que os brancos não iriam gostar".
Robertson foi nomeado "Mr. Basketball" de Indiana em 1956. Depois de se formar naquele ano, ele se matriculou na Universidade de Cincinnati.
Carreira na faculdade
Robertson continuou a se destacar na Universidade de Cincinnati, registrando uma média incrível de 33,8 pontos por jogo, a terceira maior da história da faculdade. Em cada um de seus três anos, ele ganhou o título nacional de pontuação, foi nomeado All-American e foi escolhido Jogador do Ano na Universidade, enquanto estabeleceu 14 recordes da NCAA e 19 recordes escolares.
A atuação estelar de Robertson levou a Universidade de Cincinnati a um recorde geral de 79-9 durante suas três temporadas no time, incluindo duas aparições no Final Four do Torneio da NCAA. Quando Robertson deixou a universidade, ele era o artilheiro de todos os tempos da NCAA, até que Pete Maravich o superou em 1970. Robertson levou Cincinnati ao destaque nacional mas o maior sucesso da Universidade no basquete ocorreu imediatamente após sua partida, quando a equipe conquistou dois títulos nacionais em 1961 e 1962 e perdeu o terceiro título em 1963.
Ele continua no topo do livro de registros dos Bearcats. Os muitos recordes que ele ainda detém incluem: pontos em um jogo: 62; triplos-duplos: 10; rebotes por jogo: 15,2 e pontos totais: 2.973.
Seu melhor jogo foi quando ele registrou 45 pontos, 23 rebotes e 10 assistências contra Universidade Estadual de Indiana em 1959.
Apesar de seu sucesso na quadra, a carreira de Robertson na universidade foi azedada pelo racismo. Naqueles dias, programas universitários do sul, como os de Kentucky, Duke e Carolina do Norte, não recrutavam atletas negros, e as viagens para cidades segregadas eram especialmente difíceis, com Robertson frequentemente dormindo em dormitórios de universidades em vez de hotéis. "Nunca os perdoarei", disse ele ao The Indianapolis Star anos depois.
Décadas após seus dias de faculdade, a carreira estelar na NCAA de Robertson foi recompensada pela Associação de Escritores de Basquete dos Estados Unidos quando, em 1998, renomeou o troféu concedido ao ao Jogador do Ano da Divisão I da NCAA.
Jogos Olímpicos de 1960
Depois da universidade, Robertson e Jerry West co-capitularam a Seleção Americana de Basquetebol nos Jogos Olímpicos de Verão de 1960. A equipe, descrita como a maior assembleia de talentos amadores de basquete de todos os tempos, participou da competição para ganhar a medalha de ouro.
Robertson foi o artilheiro da equipe, já que a equipe dos EUA venceu seus nove jogos por uma margem de 42,4 pontos. Dez dos doze jogadores universitários da equipe americana jogaram mais tarde na NBA, incluindo Robertson e os futuros Halls da Fama, West, Walt Bellamy e Jerry Lucas.
Carreira profissional
Cincinnati Royals (1960–1970)
Antes da temporada de 1960-61, Robertson se qualificou para o Draft da NBA de 1960. Ele foi selecionado pelo Cincinnati Royals como uma escolha territorial. Os Royals deram a Robertson um bônus de assinatura de US$ 33.000.
Em sua estreia na NBA, Robertson registrou 21 pontos, 12 rebotes e 10 assistências na vitória de 140-123 sobre o Los Angeles Lakers. Em 27 de dezembro de 1960, ele registrou 45 pontos, 12 rebotes e 13 assistências na vitória por 129-124 sobre o Syracuse Nationals.
Em sua temporada de estreia, Robertson obteve média de 30,5 pontos, 10,1 rebotes e 9,7 assistências (liderando a liga), quase com média de um triplo-duplo durante toda a temporada. Ele foi nomeado Novato do Ano da NBA, foi eleito para o o Primeiro Time da NBA - o que aconteceria em cada uma das nove primeiras temporadas de Robertson - e fez a primeira de 12 participações consecutivas no All-Star Game. Além disso, ele foi nomeado MVP do NBA All-Star Game de 1961, após seu desempenho de 23 pontos, 14 assistências e 9 rebotes em uma vitória no Oeste. No entanto, os Royals terminaram com um recorde de 33-46 e ficou nas últimas posições da Divisão Oeste.
Em 15 de novembro de 1961, Robertson registrou 49 pontos, 22 rebotes e 7 assistências na vitória de 145-133 sobre o Cincinnati Royals. Na temporada de 1961–62, ele se tornou o primeiro jogador na história da NBA a obter um triplo-duplo de média da temporada com 30,8 pontos, 12,5 rebotes e 11,4 assistências. Robertson também estabeleceu um recorde na NBA para mais triplos duplos durante a temporada regular, com 41 triplos duplos; o recorde permaneceria por mais de meio século quando, em 2016–17, Russell Westbrook teve 42 e se juntou a Robertson como o único outro jogador a obter o triplo do dobro da média por uma temporada inteira. Ele quebrou o recorde de assistências de Bob Cousy, que havia registrado 715 assistências duas temporadas antes, registrando 899. Além disso, ele também se junta a Johnny Green e Elgin Baylor como os únicos jogadores na história da NBA com 1,95 ou menos de altura que pegou 900 rebotes em uma temporada. Os Royals ganhou uma vaga nos playoffs; no entanto, eles foram eliminados na primeira rodada pelo Detroit Pistons.
Na temporada seguinte, Robertson se estabeleceu como um dos maiores jogadores de sua geração, tendo médias de 28,3 pontos, 10,4 rebotes e 9,5 assistências, perdendo por pouco mais uma temporada de triplo-duplo. Seu melhor jogo nessa temporada foi em 23 de janeiro de 1963, quando ele registrou 43 pontos, 9 rebotes e 13 assistências em uma vitória por 138-133 sobre o Boston Celtics. Os Royals avançaram para as finais da Divisão Leste, mas sucumbiu em uma série de sete jogos contra o Boston Celtics liderado por Bill Russell.
Na temporada de 1963-64, os Royals alcançaram um recorde de 55–25, o que o colocou em segundo lugar na Divisão Leste. Sob o comando do novo treinador Jack McMahon, Robertson floresceu, ele liderou a NBA em porcentagem de lances livres, marcou 31,4 pontos por jogo e teve média de 9,9 rebotes e 11,0 assistências por jogo. Ele ganhou o prêmio de MVP da NBA e se tornou o único jogador além de Bill Russell e Wilt Chamberlain a vencê-lo de 1960 a 1968. Robertson também ganhou seu segundo prêmio de MVP do NBA All-Star Game naquele ano depois de registrar 26 pontos, 14 rebotes e 8 assistências na vitória do Leste. Na pós-temporada, os Royals derrotaram o Philadelphia 76ers, mas depois foi dominado pelos Celtics por 4-1.
Robertson teve uma média de um triplo-duplo em suas primeiras cinco temporadas na NBA com os Royals, registrando médias de 30,3 pontos, 10,4 rebotes e 10,6 assistências por jogo em 451 partidas.
Em 18 de dezembro de 1964, Robertson registrou 56 pontos, além de nove rebotes e 12 assistências na vitória por 111-107 sobre o Los Angeles Lakers.
A partir da temporada de 1964-65, as coisas começaram a azedar na franquia. Apesar de Robertson registrar médias de pelo menos 24,7 pontos, 6,0 rebotes e 8,1 assistências nas seis temporadas seguintes, os Royals foram eliminados na primeira rodadas de 1965 a 1967 e não foi para os playoffs de 1968 a 1970. Na temporada de 1969-70, pela sexta temporada consecutiva, o apoio dos fãs estava diminuindo. Para ajudar a atrair o público, o técnico de 41 anos, Bob Cousy, fez um breve retorno como jogador. Durante sete jogos, o ex-armador dos Celtics fez uma parceria com Robertson mas mesmo assim a equipe perdeu os playoffs.
Milwaukee Bucks (1970–1974)
Antes da temporada de 1970-71, os Royals surpreendeu o mundo do basquete ao trocar Robertson para o Milwaukee Bucks em troca de Flynn Robinson e Charlie Paulk. Nenhum motivo foi apresentado oficialmente, mas muitos especialistas suspeitavam que o técnico Bob Cousy estivesse com ciúmes de toda a atenção que Robertson estava recebendo. O próprio Robertson disse: "Acho que ele estava errado e nunca vou esquecê-lo". O relacionamento entre Oscar e os Royals havia azedado a ponto de Cincinnati também ter abordado o Los Angeles Lakers e o New York Knicks sobre acordos envolvendo seu jogador principal (os jogadores dos Knicks que foram discutidos nesses cenários são desconhecidos, mas Los Angeles declarou publicamente que os Royals perguntou sobre Jerry West e Wilt Chamberlain, com os Lakers dizendo que não considerariam negociar nenhuma estrela).
No entanto, o negócio provou ser altamente benéfico para Robertson. Depois de ficar preso a uma equipe com baixo desempenho nos últimos seis anos, ele agora estava emparelhado com o jovem Lew Alcindor, que anos depois se tornaria o líder em pontuação da NBA como Kareem Abdul-Jabbar. Com Alcindor e Robertson, os Bucks alcançaram o melhor recorde da liga, 66-16, incluindo uma sequência de 20 vitórias. Eles tiveram um recorde de 12-2 nos playoffs e coroou a temporada com o título da NBA, varrendo o Baltimore Bullets por 4-0 nas finais da NBA de 1971. Em seu primeiro jogo nas finais da NBA, Robertson registrou 22 pontos, 7 rebotes e 7 assistências. Pela primeira vez em sua carreira, ele ganhou um título da NBA.
De uma perspectiva histórica, no entanto, a contribuição mais importante de Robertson não foi feita em uma quadra de basquete, mas em uma quadra de justiça. Foi o ano do marco histórico de Robertson contra a NBA, uma ação movida pela Associação de Jogadores da NBA contra a liga. Como Robertson era o presidente da Associação de Jogadores, o caso teve seu nome. Nesse processo, a fusão proposta entre a NBA e a ABA foi adiada até 1976, e os Drafts e as cláusulas da free agency foram reformadas. O próprio Robertson afirmou que o principal motivo era que os clubes possuíam basicamente seus jogadores: os jogadores eram proibidos de conversar com outros clubes após o término do contrato, porque a free agency não existia até 1988. Seis anos após o processo ter sido aberto, a NBA finalmente alcançou um acordo, ocorreu a fusão ABA-NBA e o processo de Oscar Robertson incentivou a contratação de mais free agency e, por fim, levou a salários mais altos para todos os jogadores.
Nas quadras, o veterano Robertson ainda provou que era um jogador valioso. Junto com Abdul-Jabbar, eles ganharam mais dois títulos de divisão com os Bucks nas temporadas de 1971-72 e de 1972-73. Na última temporada de Robertson, ele ajudou Milwaukee a alcançar o melhor recorde da liga, 59–23, e os ajudou a chegar às finais da NBA de 1974. Lá, Robertson teve a chance de terminar sua carreira estelar com um segundo títulos. Os Bucks perderam para uma equipe do Boston Celtics com o inspirado Dave Cowens.
Como prova da importância de Robertson para os Bucks, na temporada seguinte à sua aposentadoria, a equipe caiu para o último lugar em sua divisão, com um recorde de 38-44, apesar da presença contínua de Abdul-Jabbar.
Robertson foi eleito para o Hall da Fama do Wisconsin Athletic em 1995.
Pós-Carreira
Depois de se aposentar como jogador, Robertson continuou envolvido nos esforços para melhorar as condições de vida em sua cidade natal, Indianápolis, principalmente no que se refere aos afro-americanos. Além disso, ele trabalhou como comentarista em jogos televisionados pela CBS durante a temporada de 1974-75.
Após sua aposentadoria, o Kansas City Kings (os Royals se mudaram para lá enquanto Robertson estava com o Bucks) aposentou o número 14; a aposentadoria continua a ser honrada pelos Kings em sua atual casa em Sacramento. O Bucks também aposentou o número 1 que ele usava em Milwaukee.
Em 1994, uma estátua de bronze de dois metros e meio de Robertson foi erguida do lado de fora do Shoemaker Center, a atual casa do basquete da Universidade de Cincinnati. Robertson participa de muitos dos jogos, vendo os Bearcats de uma cadeira na quadra. Em 2006, a estátua foi realocada para a entrada do Centro de Atletismo Richard E. Lindner da Universidade de Cincinnati.
A partir de 2000, Robertson atuou como diretor da Countrywide Financial Corporation, até a venda da empresa ao Bank of America em 2008.
Depois de muitos anos fora dos holofotes, Robertson foi reconhecido em 17 de novembro de 2006 por seu impacto no basquete universitário, quando foi escolhido para ser membro da classe fundadora do Hall da Fama do Basquete Universitário. Ele era uma das cinco pessoas, juntamente com John Wooden, Bill Russell, Dean Smith e Dr. James Naismith, selecionados para representar a classe inaugural.
Em julho de 2004, Robertson foi nomeado treinador interino do time de basquete masculino da Universidade de Cincinnati por aproximadamente um mês, enquanto o técnico Bob Huggins cumpriu uma suspensão por uma condenação por dirigir embriagado.
Em janeiro de 2011, Robertson ingressou em uma ação coletiva contra a NCAA, desafiando a organização pelo uso das imagens de seus ex-alunos atletas.
Em 2015, Robertson estava entre um grupo de investidores que colocou uma iniciativa de legalização da maconha em Ohio. A iniciativa buscou direitos exclusivos de cultivo para os membros do grupo, enquanto proibia todos os outros cultivos, exceto pequenas quantidades para uso pessoal. Robertson apareceu em um anúncio de televisão defendendo a aprovação da iniciativa, mas acabou sendo derrotada.
Legado
Robertson é considerado um dos maiores jogadores da história da NBA. Sua média 30,5 pontos como novato é a terceira maior de todos os novatos da história da NBA. Ele obteve médias de mais de 30 pontos por jogo em seis de suas sete primeiras temporadas. Apenas três outros jogadores da NBA tiveram mais de 30 pontos por temporada em sua carreira. Robertson foi o primeiro jogador a obter média de mais de 10 assistências, fazendo isso em um momento em que os critérios para assistências eram mais rigorosos do que hoje. Além disso, Robertson é o primeiro armador da história da NBA a obter em média mais de 10 rebotes por jogo, fazendo isso três vezes. Foi um feito que não seria repetido até Russell Westbrook conseguir alcançá-lo durante a temporada de 2016-17.
Ele terminou sua carreira com 26.710 pontos (25,7 por jogo, nono mais alto de todos os tempos), 9.887 assistências (9,5 por jogo) e 7.804 rebotes (7,5 por jogo). Ele liderou a liga em assistências seis vezes e, no momento de sua aposentadoria, ele era o líder de todos os tempos da NBA em assistências e lances livres, e era o segundo maior artilheiro de todos os tempos atrás de Wilt Chamberlain.
Em sua carreira, Robertson teve 181 triplos duplos, um recorde que só foi superado em maio 2021, por Russell Westbrook. Ele ganhou um total de seis títulos de assistência da NBA durante sua carreira. Ele alcançou uma média de 0,85 de acerto de arremessos e liderou a liga em porcentagem de lances livres duas vezes - nas temporadas de 1963-64 e 1967-68.
No Cincinnati Royals, agora realocado e nomeado Sacramento Kings, ele marcou 22.009 pontos e 7.731 assistências, e é o líder de todos os tempos nas duas estatísticas das equipes combinadas Royals / Kings.
Robertson foi consagrado no Hall da Fama do Naismith Memorial Basketball em 28 de abril de 1980. Ele recebeu o prêmio "Jogador do Século" pela Associação Nacional de Treinadores de Basquete em 2000 e foi classificado em terceiro lugar entre os 75 melhores jogadores da NAM em 2003. Além disso, em 2006, a ESPN nomeou Robertson o segundo maior armador de todos os tempos, apenas atrás de Magic Johnson.
Em 1959, o Prêmio Jogador do Ano foi criado para reconhecer o melhor jogador de basquete universitário do ano pela Associação de Escritores de Basquete dos Estados Unidos. Cinco indicados são apresentados e o indivíduo com mais votos recebe o prêmio durante o Final Four da NCAA. Em 1998, foi renomeado para o Oscar Robertson Trophy em homenagem ao jogador que ganhou os dois primeiros prêmios por causa de sua carreira marcante e de seus esforços contínuos para promover o jogo de basquete.
Estatísticas
Temporada Regular
Playoffs
Prêmios e Homenagens
NBA:
Membro do Naismith Basketball Hall of Fame: 2006
Campeão da NBA: 1971;
NBA Most Valuable Player (MVP): 1964;
NBA Rookie of the Year: 1961;
12x NBA All-Star: 1961, 1962, 1963, 1964, 1965, 1966, 1967, 1968, 1969, 1970, 1971, 1972;
3x MVP do All-Star Game: 1961, 1964, 1969;
11x All-NBA Team:
primeiro time: 1961, 1962, 1964, 1965, 1966, 1969;
segundo time: 1970 e 1971;
6x líder em assistências na temporada: 1961, 1962, 1964, 1965, 1966, 1967, 1968, 1969;
Um dos 50 grandes jogadores da história da NBA
Número 14 aposentado pelo Sacramento Kings
Número 1 aposentado pelo Milwaukee Bucks
Número 12 aposentado pela Universidade de Cincinnati
Seleção dos Estados Unidos:
Jogos Pan-Americanos:
medalha de ouro: 1959
Jogos Olímpicos:
medalha de ouro: 1960
Vida pessoal
Robertson é filho de Mazell e Bailey Robertson. Ele tem dois irmãos, Bailey Jr. e Henry. Ele se lembra de uma infância difícil, atormentada pela pobreza e pelo racismo. Quando uma biografia seria escrita sobre ele nos anos 90, Robertson brincou dizendo que sua vida havia sido "monótona" e que ele estava "casado com a mesma mulher há muito tempo".
Em 1997, Robertson doou um de seus rins para sua filha Tia, que sofreu insuficiência renal relacionada ao lúpus. Ele tem sido um porta-voz honorário da National Kidney Foundation desde então. Em 2003, ele publicou sua própria autobiografia, The Big O: Minha Vida, Meus Tempos, Meu Jogo. Robertson também é dono da empresa química Orchem, com sede em Cincinnati, Ohio.
Em relação ao basquete, Robertson afirmou que os lendários jogadores do Harlem Globetrotters, Marques Haynes e o "príncipe palhaço" Goose Tatum, eram seus ídolos. Agora, aos oitenta anos, ele está aposentado há muito tempo do basquete, embora ainda o siga na TV e assista à maioria dos jogos em casa na Universidade de Cincinnati, sua alma mater. Ele agora lista a marcenaria como seu hobby principal. Robertson acrescenta que ele ainda conseguiria uma média de triplo-duplo em uma temporada no basquete de hoje, e que ele é altamente cético quanto a qualquer outra pessoa conseguir o feito (isso foi feito mais tarde por Russell Westbrook na temporada de 2016–17).
Em 9 de junho de 2007, Oscar recebeu um Doutorado Honorário de Letras Humanas da Universidade de Cincinnati por seus esforços filantrópicos e empresariais. Ele também é membro do capítulo Beta Eta da fraternidade Kappa Alpha Psi.
Em agosto de 2018, Robertson leiloou seu anel do título de 1971, o anel do Hall da Fama e uma de suas camisas de jogo do Milwaukee Bucks. Cada item foi vendido entre US $ 50.000 e US $ 91.000.
Ver também
Hall da Fama da FIBA
Ligações externas
Biografia de Oscar Robertson
Oscar Robertson - The Art of Basketball
Estatísticas de Oscar Robertson
Basquetebolistas dos Estados Unidos
Basquetebolistas do Milwaukee Bucks
Naturais do Tennessee
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}
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TORONTO MOURNS FORMER MAYOR MEL LASTMAN
By Steve Kee
Former Toronto Mayor Mel Lastman has died. He was 88.
Known for his wit and his outspokenness, Lastman was the first mayor of 'megacity' — the name given to Toronto after the amalgamation of six municipalities in 1998.
He was known to be a brash, outspoken pitchman-turned-politician whose array of gaffes, missteps and personal scandals did little to diminish a remarkable career as mayor of Canada's largest city.
Lastman once summoned the Canadian army to help during a snowstorm, pleaded with the Spice Girls to stay together and even threatened to kill a journalist.
But today, politicians from all levels are praising Lastman. Premier Doug Ford and Mayor John Tory sent condolences through social media.
"Mel was a true leader and builder for (the City of Toronto.) He was a great mayor and he touched many lives," Ford wrote.
Tory has ordered that all municipal buildings and Toronto City Hall will fly flags at half-staff to honour the former mayor.
"He was a kind, good-hearted man with a larger-than-life personality who always wanted to do the right thing for people," Tory said in a statement.
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Tamayama (jap. , -mura) war ein Dorf in der japanischen Präfektur Iwate. Es ist bekannt als Heimat des Dichters Ishikawa Takuboku. Es werden vor allem Reisanbau, Forstwirtschaft und Viehzucht betrieben.
Derzeitige Gebietsunterteilung
Als Bezirk (Ku) der Stadt Morioka unterteilt Tamayama sich in folgende Stadtteile:
Baba ()
Hinoto ()
Imoda ()
Kawamata ()
Kawasaki ()
Kōma ()
Makibori ()
Matsunai ()
Monzenji ()
Nagai ()
Shibutami ()
Shimoda ()
Tamayama ()
Terabayashi ()
Ueda ()
Yabukawa ()
Diese Bezeichnungen entsprechen den Namen der Dörfer, die seit dem 1. April 1889 ganz oder teilweise in Tamayama eingemeindet wurden.
Geschichte
1. April 1889: Im Rahmen einer Neuordnung der städtischen Struktur werden folgende Maßnahmen durchgeführt:
Das Dorf Tamayama wird mit den Dörfern Hinoto und Kawamata sowie einem Teil des Dorfes Ueda zusammengelegt und dem Landkreis Minamiiwate untergeordnet.
Das Dorf Yabukawa wird selbständig und ebenfalls dem Landkreis Minamiiwate untergeordnet.
Das Dorf Shibutami wird mit den Dörfern Shimoda, Kawasaki, Matsunai, Imoda und Monzenji zusammengelegt und dem Landkreis Iwate untergeordnet.
Das Dorf Makibori wird mit den Dörfern Terabayashi, Baba, Nagai und Kōma zusammengelegt und dem Landkreis Kitaiwate untergeordnet.
29. März 1896: Die Landkreise Kitaiwate und Minamiiwate werden zum Landkreis Iwate zusammengelegt.
1. April 1954: Dem Dorf Tamayama werden die Dörfer Yabukawa und Shibutami angeschlossen.
1. Februar 1955: Ein Teil des neugebildeten Dorfes Tamayama wird in die Stadt Morioka eingemeindet.
1. Juni 1955: Das Dorf Makibori wird Teil des Dorfes Tamayama.
1. Februar 1961: Ein Teil der Stadt Morioka wird dem Dorf Tamyama angeschlossen.
1. Oktober 2006: Das Dorf Tamayama wird vollständig in die Stadt Morioka als Stadtbezirk Tamayama (, Tamayama-ku) eingemeindet.
Persönlichkeiten
Ishikawa Takuboku (1886–1912), Dichter, Tankaist und Literaturkritiker
Ehemalige Gemeinde in der Präfektur Iwate
Morioka
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{
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\section{Improved algorithm}
The algorithm of the previous section
is efficient when the given SLP is small, i.e., $nm = o(N)$.
However, $n$ can be as large as $O(N)$,
and hence it can be slower than the existing FFT-based $O(N \log m)$-time algorithm.
To overcome this, we use the following result:
\begin{lemma}[\cite{Goto2012SUq}]
For any SLP $\mathcal{S}$ of size $n$ describing a text $S$ of length $N$,
there exists a trie $T$ of size $O(\min\{nm, N - \alpha\})$ with $\alpha \geq 0$,
such that for any substring $Q$ of length $m$ of $S$,
there exists a directed path in $T$ that spells out $Q$.
The trie $T$ can be computed in linear time in its size.
\end{lemma}
Here $\alpha$ is a value that represents the amount of
redundancy that the SLP captures with respect to the length-$m$ substrings, which is defined by
$\alpha = \sum \{(\mathit{vOcc}(X_j) - 1) \cdot (|t_j| - (m-1)) \mid |X_j| \geq m, j = 1, \dots, n \}$,
where $\mathit{vOcc}(X_j)$ denotes the number of times a variable $X_j$ occurs
in the derivation tree, i.e.,
$\mathit{vOcc}(X_j) = |\{ v \mid \variable{v}=X_j\}|$.
By the above lemma,
computing the convolution between an SLP-compressed string
and a pattern reduces to computing
the convolution between a trie and a pattern.
In the following subsection, we will present
our efficient algorithm to compute the convolution between a trie and a pattern.
\subsection{Convolution between trie and pattern}
Here we consider the convolution between a trie $T$ and a pattern $P \in \Sigma^{+}$.
For any node $v$ of $T$ and a positive integer $k$,
let $\mathit{str}_{T}(v, k)$ be the suffix of the path from the root of $T$
to $v$ of length $\min\{k, \mathit{depth}(v)\}$.
The subproblem to solve is formalized as follows:
\begin{problem}\label{prob:convolution_trie}
Given a trie $T$ and a pattern $P$ of length $m$,
for all nodes $v$ of $T$ whose depth is at least $m$,
compute $C_{T}(v) = \sum_{j = 1}^{m}\mathit{str}_{T}(v, m)[j]P[j]$.
\end{problem}
Figure~\ref{fig:input_trie} illustrates an instance of Problem~\ref{prob:convolution_trie}.
Figure~\ref{fig:conv_52413} shows the values of the
convolution between the trie of Figure~\ref{fig:input_trie} and pattern $5 \ 2 \ 4 \ 1 \ 3$.
\begin{figure}[tb]
\centerline{\includegraphics[width=0.6\textwidth]{trie_sample.eps}}
\caption{Instance of input trie $T$.}
\label{fig:input_trie}
\end{figure}
\begin{figure}[tb]
\centerline{\includegraphics[width=0.6\textwidth]{trie_sample_52413.eps}}
\caption{The convolution between $T$ and pattern $5 \ 2 \ 4 \ 1 \ 3$. The value in each node
is the value of the convolution for the node and the pattern. The nodes with depth less than $|P| = 5$ are left blank.}
\label{fig:conv_52413}
\end{figure}
\begin{theorem}
Problem~\ref{prob:convolution_trie} can be solved in $O(r \log m)$ time,
where $r$ is the size of $T$ and $m = |P|$.
\end{theorem}
\begin{proof}
Assume that the height of $T$ is at least $m$ since otherwise no computation is needed.
We show how to compute $C_{T}(v)$ in $O(\log m)$ amortized time for each node $v \in T$.
We consider the \emph{long path decomposition}
such that $T$ is decomposed into its longest path
and a forest consisting of the nodes that are not contained in the longest path.
We recursively apply the above decomposition to all trees in the forest,
until each subtree consists only of a single path.
Figure~\ref{fig:trie_lpd} shows the long path decomposition
of the trie shown in Figure~\ref{fig:input_trie}.
\begin{figure}[tb]
\centerline{\includegraphics[width=0.6\textwidth]{trie_lpd.eps}}
\caption{The long path decomposition of the trie shown in Figure~\ref{fig:input_trie}.}
\label{fig:trie_lpd}
\end{figure}
It is easy to see that we can compute the long path decomposition in $O(r)$ time.
For each path, we compute the convolution by FFT.
Let $(w_{1}, w_{2}, \dots, w_{d})$ be one of the long paths,
where $d$ is the number of nodes on the path.
\begin{itemize}
\item When $d \geq m$:
It is enough to compute the convolution between $\mathit{str}_{T}(w_{d}, d+m-1)$ and $P$,
which takes $O((d+m-1) \log m)$ time, i.e., $O(\frac{(d+m-1)}{d} \log m) = O(\log m)$ time per node.
\item When $d < m$: The same method costs too much, i.e.,
$O(\frac{(d+m-1)}{d} \log m)$ time per node, and thus we need
a trick.
The assumption that the height of $T$ is at least $m$ implies that $w_{1}$ is not the root of $T$
since otherwise, the longest path in $T$ would be
$(w_{1}, w_{2}, \dots, w_{d})$,
and $d-1 (< m)$ would be the height of $T$, a contradiction.
Consequently, from the definition of the long path decomposition,
there must exist a path (not necessarily a long path) $(z_{1}, z_{2}, \dots, z_{d})$ such that $w_{1} \neq z_{1}$ and $\mathit{parent}(w_{1}) = \mathit{parent}(z_{1})$.
%
For any $1 \leq i \leq d$ with $\mathit{depth}(w_{i}) \geq m$, $C_{T}(w_{i})$ can be written as follows:
\begin{eqnarray*}
C_{T}(w_{i}) &=& \sum_{j = 1}^{m-d}\mathit{str}_{T}(w_{i}, m)[j]P[j] + \sum_{j=m-d+1}^{m}\mathit{str}_{T}(w_{i}, m)[j]P[j]\\
&=& \sum_{j = 1}^{m-d}\mathit{str}_{T}(z_{i}, m)[j]P[j] + \sum_{j=m-d+1}^{m}\mathit{str}_{T}(w_{i}, m)[j]P[j]\\
&=& C_{T}(z_{i}) - \sum_{j=m-d+1}^{m}\mathit{str}_{T}(z_{i}, m)[j]P[j] + \sum_{j=m-d+1}^{m}\mathit{str}_{T}(w_{i}, m)[j]P[j]\\
&=& C_{T}(z_{i}) - C'_{T}(z_{i}) + C'_{T}(w_{i}),
\end{eqnarray*}
where $C'_{T}(v) = \sum_{j=m-d+1}^{m}\mathit{str}_{T}(v, m)[j]P[j]$.
%
For all $1 \leq i \leq d$, $C'_{T}(w_{i})$ (resp. $C'_{T}(z_{i})$) can be computed in $O((d+d-1) \log d)$ time
by convolution between $\mathit{str}_{T}(w_{d}, d+d-1)$ (resp. $\mathit{str}_{T}(z_{d}, d+d-1)$) and $P[m-d+1:m]$.
Therefore, assuming that $C_{T}(z_{i})$ is already computed for all $1 \leq i \leq d$,
we can compute $C_{T}(w_{i})$ for all $1 \leq i \leq d$ in $O(\frac{(d+d-1)}{d} \log d) = O(\log m)$ time per node.
\end{itemize}
It follows from the above discussion that we can solve Problem~\ref{prob:convolution_trie} in $O(r \log m)$ time
by computing values of convolution by the longest path first and making use of the result when encountering a short path whose length is less than $m$.
\end{proof}
We obtain the main result of this paper:
\begin{theorem}
Given SLP $\mathcal{S}$ of size $n$ representing
a string $S$ of length $N$, and pattern $P$ of length $m$,
we can compute an $O(\min\{nm, N-\alpha\})$-size representation of convolution $C$ for $S$ and $P$
in $O(\min\{nm, N-\alpha\} \log m)$ time, where $\alpha \geq 0$.
Given a text position $1 \leq i \leq N - m + 1$,
our representation returns $C[i]$ in $O(\log N)$ time.
\end{theorem}
We note that a similar result to Theorem~\ref{theo:matching}
holds for our $O(\min\{nm, N-\alpha\})$-size representation of convolution,
and hence we can compute all approximate occurrences in time linear in its size.
\section{Conclusions and future work}
In this paper we showed how,
given an SLP-compressed text of size $n$ and an uncompressed pattern of length $m$,
we can compute the convolution between the text and the pattern efficiently.
We employed an $O(\min\{nm, N-\alpha\})$-size trie representation of all substrings
of length $m$ in the text,
which never exceeds the uncompressed size $N$ of the text.
By introducing a new technique to compute the convolution between a trie of size $r$
and a pattern of length $m$ in $O(r \log m)$ time,
we achieve an $O(\min\{nm, N-\alpha\} \log m)$-time solution to the problem.
A consequence of this result is that,
for any string matching problem reducible to convolution,
there exists a CPS algorithm that does not require decompression
of the entire compressed string.
However, it is not yet obvious whether
we can straightforwardly adapt an algorithm which also uses
techniques other than convolution, such as the one in~\cite{Amir2004Fas}.
Future work of interest is to clarify the above matter,
and to implement our algorithms and conduct experiments on highly compressible texts.
\section{Introduction}
String matching is a task of find all occurrences of
a pattern of length $m$ in a text of length $N$.
In various fields of computer science such as bioinformatics,
image analysis and data compression,
detecting approximate occurrences of a pattern is of great importance.
Fischer and Paterson~\cite{Fischer1974SMa} found that
various approximate string matching problems can be solved efficiently
by reduction to convolution,
and many studies have followed since.
For instance,
it was shown in~\cite{Fischer1974SMa} that the string matching problem with don't
cares can be solved in $O(N \log m \log \sigma)$ time,
where $\sigma$ is the alphabet size.
This was later improved to $O(N \log m)$ time~\cite{Cole2002Vcm,Clifford2007Sdw}.
An $O(N \sqrt{m \log m})$-time algorithm for computing
the Hamming distances between the pattern and all text substrings of length $m$
was proposed in~\cite{Abrahamson1987GSM}.
Many, if not all, large string data sets are stored in a compressed form,
and are later decompressed in order to be used and/or analyzed.
\emph{Compressed string processing (CSP)} arose from
the recent rapid increase of digital data,
as an approach to process a given compressed string
\emph{without explicitly decompressing the entire string}.
A lot of CSP algorithms have been proposed in the last two decades,
which improve on algorithms working on uncompressed strings,
both in theory~\cite{NJC97,crochemore03:_subquad_sequen_align_algor_unres_scorin_matric,hermelin09:_unified_algor_accel_edit_distan,gawrychowski11:_LZ_comp_str_fast_}
and in practice~\cite{shibata00:_speed_up_patter_match_text_compr,goto11:_fast_minin_slp_compr_strin,Goto2012SUq}.
The goal of this paper is
efficient computation of the convolution between
a compressed text and an uncompressed pattern.
In this paper, we assume that the input string
is represented by a \emph{straight-line program (SLP)},
which is a context free grammar in the Chomsky normal form
that derives a single string.
It is well known that outputs of various
grammar based compression algorithms~\cite{SEQUITUR,LarssonDCC99},
as well as those of dictionary compression algorithms~\cite{LZ78,LZW,LZ77,LZSS},
can be regarded as, or be quickly transformed to, SLPs~\cite{rytter03:_applic_lempel_ziv}.
Hence, algorithmic research working on SLPs is of great significance.
We present two efficient algorithms that compute
the convolution between an SLP-compressed text of size $n$
and an uncompressed pattern of length $m$.
The first one runs in $O(nm \log m)$ time and space,
which is based on \emph{partial decompression} of
the SLP-compressed text.
Whenever $nm = o(N)$,
this is more efficient than the existing FFT-based $O(N \log m)$-time algorithm
for computing the convolution of a string of length $N$ and a pattern of length $m$.
However, in the worst case $n$ can be as large as $O(N)$.
Our second algorithm deals with such a case.
The key is a reduction of the covolution of an SLP and a pattern,
to the convolution of a trie and a pattern.
We show how, given a trie of size $r$ and pattern of length $m$,
we can compute the convolution between all strings of length $m$ in the
trie and the pattern in $O(r \log m)$ time.
This result gives us an $O(\min\{nm, N-\alpha\} \log m)$-time algorithm
for computing the convolution between an SLP-compressed text and a pattern,
where $\alpha \geq 0$ represents a quantity of redundancy of
the SLP w.r.t. the substrings of length $m$.
Notice that our second method is at least as efficient as
the existing $O(N \log m)$ algorithm,
and can be much more efficient when a given SLP is small.
Further, our result implies that \emph{any} string matching problems
which are reducible to convolution can be efficiently solved on SLP-compressed text.
\subsection{Related work}
In~\cite{freschi10:_lz78},
an algorithm which computes the convolution between
a text and a pattern, using Lempel-Ziv 78 factorization~\cite{LZ78},
was proposed.
Given a text of length $N$ and a pattern of length $m$,
the algorithm in~\cite{freschi10:_lz78} computes the convolution
in $O(N + mL)$ time and space,
where $L$ is the number of LZ78 factors of the text.
The authors claimed that
$L = O(\frac{N}{\log N} h)$,
where $0 \leq h \leq 1$ is the entropy of the text.
However, this holds only on some strings over a constant alphabet,
and even on a constant alphabet
there exist strings with $L = O(\frac{N}{\log N})$~\cite{crochemore03:_subquad_sequen_align_algor_unres_scorin_matric}.
Moreover, when the text is drawn from integer alphabet $\Sigma = [1,N]$,
then clearly $L = \Theta(N)$.
In this case, the algorithm of~\cite{freschi10:_lz78} takes at least $O(mN)$ time
(excluding the time cost to compute the LZ78 factorization).
Since the LZ78 encoding of a text can be seen as an SLP,
and since the running time of our algorithm is independent of the alphabet size,
this paper presents a more efficient algorithm to compute the convolution
on LZ78-compressed text over an integer alphabet.
Furthermore, our algorithm is much more general and can be applied to arbitrary SLPs.
\section{Preliminaries}
\subsection{Strings}
Let $\Sigma$ be a finite {\em alphabet}.
An element of $\Sigma^*$ is called a {\em string}.
The length of a string $S$ is denoted by $|S|$.
The empty string $\varepsilon$ is a string of length 0,
namely, $|\varepsilon| = 0$.
For a string $S = XYZ$, $X$, $Y$ and $Z$ are called
a \emph{prefix}, \emph{substring}, and \emph{suffix} of $S$, respectively.
The $i$-th character of a string $S$ is denoted by $S[i]$, where $1 \leq i \leq |S|$.
For a string $S$ and two integers $1 \leq i \leq j \leq |S|$,
let $S[i:j]$ denote the substring of $S$ that begins at position $i$ and ends at
position $j$.
Our model of computation is the word RAM:
We shall assume that the computer word size is at least $\log_2 |S|$,
and hence, standard operations on
values representing lengths and positions of string $S$
can be manipulated in constant time.
Space complexities will be determined by the number of computer words (not bits).
\subsection{Convolution}
Let $V_S$ and $V_P$ be two vectors on some field whose lengths are $N$ and $m$, respectively,
with $m \leq N$.
The \emph{convolution} $C$ between $V_S$ and $V_P$ is defined by
\begin{eqnarray}
C[i] = \sum_{j=1}^{m}V_P[j] \cdot V_S[i+j-1]
\end{eqnarray}
for $1 \leq i \leq N-m+1$.
It is well-known that
the vector $C$ can be computed in $O(N \log m)$ time by FFT.
The algorithm samples $V_S$ at
every $(km+1)$-th position of $V_S$ for $0 \leq k \leq \lfloor \frac{N}{m} \rfloor$.
For each sampled position
the algorithm is able to compute
the convolution between the subvector $V_S[km+1:(k+2)m]$ of length $2m$
and $V_P$ in $O(m \log m)$ time,
and therefore the whole vector $C$ can be computed in
a total of $O(N \log m)$ time.
We can solve several types of approximate matching problems
for a text $S$ of length $N$ and a pattern $P$ of length $m$,
by suitably mapping characters $P[j]$ and $S[i+j-1]$ to numerical values.
For example, let $\phi_{a}(x) = 1$ if $x = a$ and $0$ otherwise, for any $a \in \Sigma$,
then $\sum_{a \in \Sigma} \sum_{j=1}^{m} \phi_{a}(P[j]) \cdot \phi_{a}(S[i+j-1])$ represents
the number of matching positions when the pattern is aligned at position $i$ of the text.
Consequently, the Hamming distances of the pattern and the text substrings
for all positions $1 \leq i \leq N-m+1$ can be computed in a total of $O(|\Sigma| N \log m)$ time,
by computing convolution using mappings $\phi_{a}$ for all $a \in \Sigma$ and summing them up,
which is a classic result in~\cite{Fischer1974SMa}.
For convenience, in what follows we assume strings $S$ and $P$ on integer alphabet,
and consider convolution between $S$ and $P$.
\subsection{Straight Line Programs}
\label{sec:slp}
%
\begin{figure}[tb]
\centerline{\includegraphics[width=0.5\textwidth]{slp.eps}}
\caption{
The derivation tree of
SLP $\mathcal S = \{ X_1 \rightarrow \mathtt{a}$, $X_2 \rightarrow \mathtt{b}$, $X_3 \rightarrow X_1X_2$,
$X_4 \rightarrow X_1X_3$, $X_5 \rightarrow X_3X_4$, $X_6
\rightarrow X_4X_5$, $X_7 \rightarrow X_6X_5 \}$,
representing string $S = \mathit{val}(X_7) = \mathtt{aababaababaab}$.
}
\label{fig:SLP}
\end{figure}
A {\em straight line program} ({\em SLP}) is a set of assignments
$\mathcal S = \{ X_1 \rightarrow expr_1, X_2 \rightarrow expr_2, \ldots, X_n \rightarrow expr_n\}$,
where each $X_i$ is a variable and each $expr_i$ is an expression, where
$expr_i = a$ ($a\in\Sigma$), or $expr_i = X_{\ell(i)} X_{r(i)}$~($i > \ell(i),r(i)$).
It is essentially a context free grammar in the Chomsky normal form, that derives a single string.
Let $\mathit{val}(X_i)$ represent the string derived from variable $X_i$.
To ease notation, we sometimes associate $\mathit{val}(X_i)$ with $X_i$ and
denote $|\mathit{val}(X_i)|$ as $|X_i|$,
and $\mathit{val}(X_i)([u:v])$ as $X_i([u:v])$ for any interval $[u:v]$.
An SLP $\mathcal{S}$ {\em represents} the string $S = \mathit{val}(X_n)$.
The \emph{size} of the program $\mathcal{S}$ is the number $n$ of
assignments in $\mathcal{S}$.
Note that $|S|$ can be as large as $\Theta(2^n)$. However, we assume
as in various previous work on SLP,
that the computer word size is at least $\log_2 |S|$, and hence,
values representing lengths and positions of $S$
in our algorithms can be manipulated in constant time.
The derivation tree of SLP $\mathcal{S}$ is a labeled
ordered binary tree where each internal node is labeled with a
non-terminal variable in $\{X_1,\ldots,X_n\}$, and each leaf is labeled with a terminal character in $\Sigma$.
The root node has label $X_n$.
Let $\mathcal{V}$ denote the set of internal nodes in
the derivation tree.
For any internal node $v\in\mathcal{V}$,
let $\langle v\rangle$ denote the index of its label
$\variable{v}$.
Node $v$ has a single child which is a leaf labeled with $c$
when $(\variable{v} \rightarrow c) \in \mathcal{S}$ for some $c\in\Sigma$,
or
$v$ has a left-child and right-child respectively denoted $\ell(v)$ and $r(v)$,
when
$(\variable{v}\rightarrow \variable{\ell(v)}\variable{r(v)}) \in \mathcal{S}$.
Each node $v$ of the tree derives $\mathit{val}(\variable{v})$,
a substring of $S$,
whose corresponding interval $\mathit{itv}(v)$,
with $S(\mathit{itv}(v)) = \mathit{val}(\variable{v})$,
can be defined recursively as follows.
If $v$ is the root node, then $\mathit{itv}(v) = [1:|S|]$.
Otherwise, if $(\variable{v}\rightarrow
\variable{\ell(v)}\variable{r(v)})\in\mathcal{S}$,
then,
$\mathit{itv}(\ell(v)) = [b_v:b_v+|\variable{\ell(v)}|-1]$
and
$\mathit{itv}(r(v)) = [b_v+|\variable{\ell(v)}|:e_v]$,
where $[b_v:e_v] = \mathit{itv}(v)$.
For any interval $[b:e]$ of $S (1\leq b \leq e \leq |S|)$,
let $\xi_{\mathcal{S}}(b,e)$ denote the deepest node $v$ in the derivation tree,
which derives an interval containing $[b:e]$, that is,
$\mathit{itv}(v)\supseteq [b:e]$,
and no proper descendant of $v$
satisfies this condition.
We say that node $v$ {\em stabs} interval $[b:e]$,
and $\variable{v}$ is called the variable that stabs the interval.
If $b = e$, we have that
$(\variable{v} \rightarrow c) \in \mathcal{S}$ for some $c\in\Sigma$,
and $\mathit{itv}(v) = b = e$.
If $b < e$, then
we have $(\variable{v} \rightarrow
\variable{\ell(v)}\variable{r(v)})\in\mathcal{S}$,
$b\in \mathit{itv}(\ell(v))$, and $e\in\mathit{itv}(r(v))$.
\begin{theorem}[\cite{philip11:_random_acces_gramm_compr_strin}]
\label{theo:random_access}
Given an SLP $\mathcal{S} = \{ X_i \rightarrow \mathit{expr}_i \}_{i=1}^n$,
it is possible to pre-process $\mathcal{S}$ in $O(n)$ time and
space, so that for any interval $[b:e]$ of $S$, $1 \leq b \leq e \leq N$,
its stabbing variable %
%
$\variable{\xi_\mathcal{S}(b,e)}$
can be computed in $O(\log N)$ time.
\end{theorem}
SLPs can be efficiently pre-processed to hold various information.
$|X_i|$ can be computed for all variables $X_i
(1\leq i\leq n)$ in a total of $O(n)$ time by a simple dynamic
programming algorithm.
Also, the following lemma is useful for partial decompression of
a prefix of a variable.
\begin{lemma}[\cite{gasieniec05:_real_time_traver_gramm_based_compr_files}]
\label{label:prefix_decompression}
Given an SLP $\mathcal{S} = \{ X_i \rightarrow \mathit{expr}_i \}_{i=1}^n$,
it is possible to pre-process $\mathcal{S}$ in $O(n)$ time and
space, so that for any variable $X_i$ and $1 \leq q \leq |X_i|$,
the prefix of $\mathit{val}(X_i)$ of length $q$,
i.e. $\mathit{val}(X_i)[1:q]$, can be computed in $O(q)$ time.
\end{lemma}
\subsection{Problem}
In this paper we tackle the following problem.
\begin{problem}
Given an SLP $\mathcal{S} = \{ X_i \rightarrow \mathit{expr}_i \}_{i=1}^n$
describing a text $S$
and an uncompressed pattern $P$ of length $m$,
compute a compact representation of the convolution $C$
between $S$ and $P$.
\end{problem}
By ``compact representation'' above,
we mean a representation of convolution $C$ whose size
is dependent (and polynomial) on $n$ and $m$, and not on $N = |S|$.
In the following sections,
we will present our algorithms to solve this problem.
We will also show that given a position $i$ of the uncompressed text $S$
with $1 \leq i \leq N - m + 1$,
our representation is able to return $C[i]$ quickly.
\section{Basic algorithm}
In this section, we describe our compact representation
of the convolution $C$ for a string $S$ represented as
an SLP $\mathcal{S}$ of size $n$ and a pattern $P$ of length $m$.
Our representation is based on the fact that the value of the
convolution depends only on the substrings of length $m$ of $S$.
We use compact representations of all substrings of length $m$ of $S$,
which were proposed in~\cite{goto11:_fast_minin_slp_compr_strin,Goto2012SUq}.
For any variable $X_j = X_\ell X_r$, let
$t_j = \mathit{suf}(X_\ell, m-1) \mathit{pre}(X_r,m-1)$.
Namely, $t_j$ is the substring of $\mathit{val}(X_j)$
obtained by concatenating the suffix of $\mathit{val}(X_\ell)$ of length at most $m-1$,
and the prefix of $\mathit{val}(X_r)$ of length at most $m-1$ (see also Figure~\ref{fig:2(m-1)}).
By the arguments of Section~\ref{sec:slp},
there exists a unique variable $X_j$ that stabs the interval $[i:i+m-1]$.
Hence, computing $C$ reduces
to computing the convolution between $t_j$ and pattern $P$
for all variables $X_j$.
\begin{figure}[tb]
\centerline{\includegraphics[width=0.6\textwidth]{SLP_kushi.eps}}
\caption{Substring $t_j$ of $\mathit{val}(X_j)$.}
\label{fig:2(m-1)}
\end{figure}
\begin{theorem}
Given an SLP $\mathcal{S}$ of size $n$ representing
a string $S$ of length $N$, and pattern $P$ of length $m$,
we can compute an $O(nm)$-size representation of convolution $C$ for $S$ and $P$
in $O(nm \log m)$ time.
Given a text position $1 \leq i \leq N - m + 1$,
our representation returns $C[i]$ in $O(\log N)$ time.
\end{theorem}
\begin{proof}
Let $t_j = \mathit{suf}(X_\ell, m-1) \mathit{pre}(X_r,m-1)$ for any variable $X_j = X_\ell X_r$.
Since $|t_j| \leq 2m-2$,
we can compute each $t_j$ in $O(m)$ time by Lemma~\ref{label:prefix_decompression}.
We then compute the convolution
between $t_j$ and $P$ in $O(m\log m)$ time using the FFT algorithm.
Since there are $n$ variables, it takes a total of $O(n m \log m)$ time
and the total size of our representation is $O(nm)$.
By Theorem~\ref{theo:random_access} we can compute
the stabbing variable in $O(\log N)$ time.
It is also possible to compute in $O(\log N)$ time
the text position corresponding to the node of the derivation tree of $\mathcal{S}$
representing the stabbing variable~\cite{bannaiIT12:_LZ78_grammar_compr_}.
Thus $C[i]$ can be answered in $O(\log N)$ time.
\end{proof}
By a similar argument to Section 7 in~\cite{philip11:_random_acces_gramm_compr_strin},
we obtain the following:
\begin{theorem} \label{theo:matching}
Given a compact representation of the convolution between
a string $S$ and a pattern $P$ described above,
we can output the set $occ$ of all approximate occurrences of $P$ in $S$
in $O(|occ|)$ time.
\end{theorem}
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Q: ImportError: cannot import name 'Optional' from 'torch.jit.annotations' I have installed cpuonly pytorch and torchvision in anaconda. But when i try to import torchvision i get the following error.
ImportError: cannot import name 'Optional' from 'torch.jit.annotations'(C:\Users\MSI\Anaconda3\lib\site-packages\torch\jit\annotations.py)
How can i fix this?
A: Not sure if you are installing the correct versions of the libraries. This combination seems to work:
conda create --name test5 python=3.6
conda install -c pytorch pytorch torchvision cpuonly
python
>>> import torchvision
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Q: Selecting last row, range, vba I'm fairly new to vba and have a rather simple problem. Can someone please help:
Instead of selecting the specific cell, I want my vba macro to go to the bottom of the column of interest, skip, and two cells below it do the following:
Range("W72").Select
Selection.NumberFormat = "General"
ActiveCell.FormulaR1C1 = "Null_value"
Range("X72").Select
Selection.NumberFormat = "General"
ActiveCell.FormulaR1C1 = "=R[-2]C[1]-SUM(R[-2]C[-8]:R[-2]C[-6])"
As you can see the code above refers to the specific cells W72 and X72. Currently, the last entry in these columns are in W70 and X70 but next month my dataset will get bigger so W72 and X72 aren't the right locations to do the actions above. How do I correct for this such that my vba code is automatically going to the bottom of W(n):X(n), skips one row and in W(n+2), X(n+2) performs the code above.
Also, my formula above (ActiveCell.FormulaR1C1) also is referring to specific cells, in my case Row 70 several columns to the left, but as you probably tell, this too has the same issue since the row referencing changes each month. I need to get my vba to have the formula pick up the last row of those columns, the columns are P,Q,R.
Thanks for any help you can provide.
Update: Part of my same working project, I would greatly appreciate if anyone can help with this too. Thank you:
Hi All,
I currently have an input box for a variable that changes everymonth:
r_mo = Application.InputBox(prompt:="Enter the reporting month as YYYYMM (Eg:201604). Errors in this entry will result in errors in the results.")
This prompts an input box which one has to manually enter into... However, I want to automate this process and eliminate the need for an input box. Isn't there a now function in vba that will automatically generate today's date.
From a now, or system function all I want to do is extract the year in four digits and the month in two digits.
So for example, if we're in decemeber 2016
Sub asasdas ()
"Now function"
r_mo = YYYYMM ' automatically updated from "now function"
End Sub
I appreciate any help you can give me and thank you so much all.
A: You can get the last populated row of a given column (W in my example) in VBA with the following code:
Dim ws As Worksheet : Set ws = ThisWorkbook.Worksheets("MySheetNameHere")
lastRow = ws.Cells(ws.Rows.Count, "W").End(xlUp).Row
Naturally, if you then add 2 to lastRow you have the cell you are looking for.
A: I'd do it like
Sub asdf()
Range("w1048576").End(xlUp).Offset(2, 0).Select 'gets the last row
With Selection
.NumberFormat = "General"
.FormulaR1C1 = "Null_value"
End With
ActiveCell.Offset(, 1).Select
With Selection
.NumberFormat = "General"
.FormulaR1C1 = "=R[-2]C[1]-SUM(R[-2]C[-8]:R[-2]C[-6])"
End With
End Sub
A: If you want a more detailed answer you're going to have to make a new question but for your second question try this.
Sub Now()
Dim myDate As String
myDate = Date
myDate = Format(myDate, "yyyymm")
Debug.Print myDate '201606 output for June 10th 2016
End Sub
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\section{Introduction}
The Tactile Internet (TI)~\cite{Fettweis_TI} is a fairly recent concept that involves the transmission of tactile sensations along with data, text, and multimedia.
The ability to receive tactile stimulation enhances the immersion of the user in Virtual Reality (VR), and improves control performance in teleoperation, in which a human operator (HO) controls and manipulates a remotely located robot or object~\cite{EMG_teleoperation_1}.
This involves a two-way exchange of data: commands from the operator and tactile and feedback from the remote environment, creating a closed-loop system \cite{antonakoglou2018toward}.
Advanced Human-to-Machine Interaction (HMI) in teleoperation involves the exchange of abundant sensory data (including textual, visual, and tactile information), which ensures that the HO can have an intuitive and precise interaction with the remote environment, improving the task execution accuracy and efficiency, but also putting a significant strain on the communication system.
Due to its interactive nature, teleoperation is highly sensitive to communication impairments.
The concept of motion-to-photon delay~\cite{zhao2017estimating}, often used in VR, is extremely relevant here: if we measure the closed-loop latency between the moment an action is performed by the operator and the moment that they get the related feedback, we can gauge the Quality of Experience (QoE) for the operator, as well as the final control performance.
Besides low latency, teleoperation requires a high reliability, as the communication channel losses may degrade the control precision.
Moreover, reliable low-latency systems are effectively~\emph{transparent}~\cite{transparency_work}, i.e., the operator does not notice the remote nature of the environment, and feels as if they were controlling the robot directly.
This ensures the immersion of the operator in the remote environment, improving both QoE and control performance.
On the other hand, long delays and high jitter can both deteriorate the user experience and jeopardize the stability of the
closed-loop teleoperation system, as the reactions to the events in the remote environment are delayed and sluggish~\cite{stability_td_hri}. {{For example, comfort zone for performing telesurgery is with round trip latency of around $300$ ms \cite{barba2022remote}, whereas it is around $800$ ms in VR-based haptic teleoperation \cite{valenzuela2019virtual}. }}
Communication is not the only bottleneck in the system, as closed-loop latency also includes computation and processing times, while the robot might have limited on-board computation capabilities.
As the decoding and execution of the operator's commands along with the compression and processing of the sensing data are part of the teleoperation control loop, these operations can have a significant effect on the latency.
In effect, the success of teleoperation applications strongly depends on the performance of both the communication and the computational segments. In order to fully optimize the teleoperation system to meet the closed-loop latency constraints required by the application, which can be very stringent in mission-critical industrial scenarios, we need to consider all steps of the process.
Latency in networked, closed-loop control systems is not easy to control, as the randomness of the wireless channel and the variable amount of available bandwidth and computation resources make the closed-loop latency a stochastic quantity.
While the effects of a higher latency and jitter are understood from laboratory experiments~\cite{effect_of_delay, low_latency_h2m, closing_the_force_loop}, existing schemes optimize only for the \emph{average} latency~\cite{IoT_TI, NFV_e2e_QoS,onu_2}, without providing any reliability guarantees.
Furthermore, most teleoperation system designs~\cite{MEC_tactility, MEC_TII, hybrid_caching_TI, TelSurg} do not take into account the computational delay into the latency calculation, effectively leaving out part of the control loop and potentially disregarding an important component of the closed-loop latency.
\begin{comment}
\begin{figure}[!t]
\centering
{{\includegraphics[height=4.25cm]{figure/HO_BS_Robot}}}
\caption{System model for closed-loop MEC-enabled teleoperation system. \CS{Maybe to specifically denote the processed feedback from MEC to operator in the figure - e.g., by a different kind of line?}} \label{fig:sys_mod_MEC}
\end{figure}
\end{comment}
\begin{figure}[t!]
\centering
\input{tikz_figs/sysmodel}
\caption{System model for closed-loop MEC-enabled teleoperation system.}
\label{fig:sys_mod_MEC_TI}
\end{figure}
In this work, we present an analysis of a Mobile Edge Computing (MEC)~\cite{MEC_survey, MEC_survey_IoTJ} system for teleoperation, in which the most computationally-intensive tasks are offloaded to the MEC-enabled Base Station (BS) handles as shown in Fig. \ref{fig:sys_mod_MEC_TI}. A preliminary version of this work was presented in~\cite{our_icc}, focusing only on the latency of an uplink connection, rather than a closed-loop latency, from a multi-sensor robot to the BS. This work generalizes the framework to analyze the latency of the closed-loop teleoperation system and formulate an optimization problem to minimize the closed-loop latency with high reliability under statistical constraints. The key contributions of the paper are as follows:
\begin{enumerate}
\item The system model for closed-loop MEC-enabled teleoperation system is presented, where the command and sensing data are exchanged over wireless network through a BS. A MEC server on the BS processes the command and sensing data in order to be sent to the robot and the feedback for the HO, respectively, enabling the closed-loop operation.
\item Two system design possibilities are considered and compared. In the first one, the robot located in the remote environment compresses the raw sensing data first, then transmits it to the BS, which decompresses and processes it to extract the user readable feedback that is transmitted back to the HO. In the second scenario, the robot transmits the raw sensing data to the BS without compression, where it is processed by the MEC server.
\item The closed-loop latency for both scenarios is analyzed by dividing it into three components: the duration of the compression operation on the sensing data, the transmission delays for commands and feedback, and the decompression and computation at the MEC server. All these latency components are modelled as independent random variables (RVs). Thus, the closed-loop latency, which is the sum of all these components, is also an RV. We characterize the nature of its Probability Density Function (PDF) and Cumulative Distribution Function (CDF) and obtain tractable upper and lower bounds on the closed-loop latency for both scenarios.
\item A non-convex optimization problem is formulated, aiming to minimize the the closed-loop latency in the statistical sense.
Using the obtained upper and lower bounds on the closed-loop latency, we present a computationally efficient procedure to estimate the optimal closed-loop latency and solve the problem.
\item We analyze the performance of the schemes by simulation, and find that the simulation results reveal that compression of sensing data is not always beneficial. The decision on data compression depends on the communication system parameters, as well as the computational capability of the robot. The proposed approach is also compared with the system design that is optimized in average sense, as reported in the prior works. The comparative analysis reveals that system design in average sense leads to under-provisioning and causes a significant performance degradation.
\end{enumerate}
While this work focuses on the latency analysis for teleoperation, the basic framework we propose can be used for any cascaded system with random latency components, such as Open Radio Access Network (O-RAN) \cite{o-ran} systems. The O-RAN architecture is envisioned to execute networking processes in software, making network components' behavior programmable. Telecom operators will use the standardized interfaces to control multi-vendor infrastructures. In the context of O-RAN, the proposed framework will be very useful to analyze the latency incurred across multiple software and hardware components from multiple vendors in order to maximize user QoE.
The rest of this paper is organized as follows. Section~\ref{sec:related} presents related work on the subject, while the basic model of a MEC-enabled teleoperation system is presented in Section~\ref{sec:system}. The different delay components of the closed-loop latency and their distributions are discussed in Section~\ref{sec:latency}, and the overall closed-loop latency distribution is estimated and optimized in Section~\ref{sec:analysis}. Finally, the numerical simulation results are discussed in Section~\ref{sec:results}, followed by the concluding remarks in Section~\ref{sec:conc}.
\section{Related Work}\label{sec:related}
Latency is a major issue in teleoperation systems, and several studies \cite{effect_of_delay, low_latency_h2m, closing_the_force_loop} have investigated its impact on control performance. The study in \cite{effect_of_delay} describes an experiment that uses a haptic device to generate feedback, presenting the visual three-dimensional environment to the user on a monitor and studying the effect of latency between the participant's actual action and the visible movement on the monitor. A commercial haptic teleoperation system \cite{cyber_glove} was used in \cite{low_latency_h2m}, which allowed HOs to touch and grasp the computer-generated virtual objects. This experiment demonstrated that the average latency increases significantly with the network load. A closed-loop compensatory tracking task is performed in the closed-loop control using tactile input in \cite{s_dosen_ET_fb}, where the feedback is encoded to the user using frequency and amplitude modulation schemes. Significant time delay on the order of several hundred milliseconds has been noticed in this experiment.
Most existing low-latency teleoperation schemes have tried to minimize the \emph{average} latency, which is an easier target, and neglect the required reliability targets in statistical sense. Even with fiber-wireless (FiWi) networks, existing optimization works are limited to average guarantees~\cite{onu_2}.
In this scenario, the limiting factors are the availability, skill set, distance to task location, and remaining energy of robots~\cite{onu_1}, or the association between tasks and HOs~\cite{onu_3}. The study in \cite{coll_computing} presents a task allocation strategy by combining suitable host robot selection and computation task offloading onto collaborative nodes in the FiWi infrastructure. The conventional Cloud, decentralized cloudlets, and neighboring robots as collaborative nodes are used for computation offloading.
Cross-layer techniques for low-latency teleoperation have also been considered in the literature~\cite{cross_layer_3, TI_channel_access, cross_layer_1}. TI cross-layer transmission optimization is investigated in~\cite{cross_layer_3} by considering the transmission delay, error probability and statistical queuing delay requirements, using a proactive packet dropping mechanism to limit latency. A resource allocation mechanism to maximize the uplink sum rate of traditional data while satisfying the delay requirements for tactile data is presented in \cite{cross_layer_1} using sparse code multiple access. The study in \cite{TI_channel_access} estimates the average latency from a hub to an access point for tactile body-worn devices connected using an IEEE 802.11 network.
In general, support for teleoperation applications based on Network Function Virtualization (NFV) is included in the 5G network architecture~\cite{NFV_JSAC, IoT_TI, NFV_e2e_QoS}. The study in~\cite{IoT_TI} presents a utility function based model to evaluate the performance of the NFV-based TI by considering the human perception resolution and the network cost of completing services. The utility function depends on the average round-trip delay, network link bandwidth, and node virtual resource consumption. The joint radio and NFV resources for a heterogeneous network are allocated in \cite{NFV_e2e_QoS} by guaranteeing average end-to-end delay of each tactile user, including the queuing, transmission, and computation delays. MEC offloading is another possibility for TI applications~\cite{TelSurg,MEC_TII}. A trade-off between the average service response time and power usage efficiency is investigated in \cite{MEC_tactility} for local and cooperative MEC. This can be optimized according to QoE metrics as well~\cite{MEC_TII}, or using more advanced caching techniques~\cite{hybrid_caching_TI}. Finally, a real-time network architecture for remote surgery application is presented in \cite{TelSurg}, employing cloud and MEC networks to satisfy the timing constraints, which are still expressed in terms of the average end-to-end delay.
Overall, existing experimental works have mostly considered simple links with controllable latency, while existing architectures only dealt with average latency, and often disregarded the contribution of the computational component of the closed-loop latency.
The main novelty of our work is in the consideration of these factors, estimating the complete distribution of the closed-loop latency in a public Internet setup along with optimizing it for the worst-case scenario by considering statistical guarantees to ensure a more stable performance than average latency minimization.
\begin{comment}
\begin{table} [!h]
\begin{center}
\caption{ LIST OF MAJOR VARIABLES ALONG WITH THEIR DESCRIPTIONS }
\label{tab:nature_rv}
\begin{tabular}{|p{0.07\textwidth}|p{0.35\textwidth}|p{0.07\textwidth}|p{0.35\textwidth}|}
\hline
\textbf{ } & \textbf{ } & \textbf{ } & \textbf{ } \\
\hline
$D_c$ & Volume of command data & $T_{tx}(N,\epsilon)$ & Time to transmit $N$ packet with outage $\epsilon$ \\
$D_s$ & Volume of sensing data & $T_c$ & Time elapsed in computation \\
$d_r$ & Distance between BS and robot & $T_{cp}$ & Time elapsed in compression \\
$d_h$ & Distance between BS and HO & $T_{d}$ & Time elapsed in decompression \\
$h$ & Channel gain & $X_c$ & Number of cycles required for computation \\
$P_{\text{tx}}^{ho}$ & Power transmitted by HO & $X_{cp}$ & Number of cycles required for compression \\
$P_{\text{tx}}^{bs}$ & Power transmitted by BS & $X_d$ & Number of cycles required for decompression \\
$P_{\text{tx}}^{r}$ & Power transmitted by robot & $f_{bs}$ & Frequency of MEC server at BS \\
$B$ & Bandwidth & $f_{r}$ & Frequency of processor at robot \\
$R(\epsilon)$ & Rate for outage $\epsilon$ & $\kappa_{bs}$ & Shape parameter of server at BS \\
$\gamma $ & Signal-to-noise ratio (SNR) & $\kappa_{r}$ & Shape parameter of server at robot \\
$\gamma_{th}$ & SNR threshold & $\beta_{c}$ & Scale parameter for computation operation \\
$n_p$ & Length of a packet & $\beta_{cp}$ & Scale parameter for compression operation \\
$t_p$ & Time to transmit a packet & $\beta_{cp}$ & Scale parameter for decompression operation \\
$F_{T}(\cdot)$ & CDF of random variable $T$ & $\varrho_{th}$ & Reliability threshold \\
$\tau_U$ & Upper bound on closed-loop latency & $C(Q)$ & CPU cycles required for compression ratio $Q$ \\
$\tau_L$ & Lower bound on closed-loop latency & $Q$ & Compression ratio \\
\hline
\end{tabular}
\end{center}
\end{table}
\end{comment}
\section{System Model} \label{sec:system}
The model of the considered MEC-enabled teleoperation system is shown in Fig. \ref{fig:sys_mod_MEC_TI}: the robot, which is located in the remote environment, is instructed by the HO, who receives feedback information that guides his decisions.
The instruction sent to the robot is denoted as the \emph{command signal}, whereas the the information received from the robot as the \emph{sensing data}. The command signal from the HO and the sensing data from the robot are exchanged over a wireless connection to the BS. The volume of sensing data generated is much larger than the command signal, because it can include throughput-intensive formats such as video and tactile data, along with other types of data such as audio, image, or text. The sensing data acts as feedback to the HO for further command instructions to the remote robot. The transmission of this potentially large volume of data over a wireless connection is expensive in terms of required radio resources, and may require compression before the data is transmitted.
The alternative is to process the sensing data at the robot itself and extract the user-readable feedback signal, but this may be very computational-intensive and even far beyond the capacity of the robot. On the other hand, commands from the HO are typically not compressed, and represent high-level instructions to the robot. The MEC server then translates these high-level commands to low-level commands to the robot's actuators, which can be directly executed. As the size of command signals is typically much smaller than the sensing data from the robot (commands usually come from a limited set), the compression of command data is not within the scope of our work \footnote{Note that the analytical framework presented here can be straightforwardly extended if the command data also gets compressed at the HO side before transmission.}.
The MEC server equipped at the BS acts as a decision-support system that handles the data-intensive computation task by processing the sensing data from the robot.
Such processed data is communicated to the HO.
In the system model shown in Fig. \ref{fig:sys_mod_MEC_TI}, the robot compresses the sensing data locally and transmits this compressed data to the BS. The MEC server at the BS first decompresses the data, processes it, and sends the processed data having user readable command to the HO. Based on the received feedback, the HO decides on the command signal for the robot located in the remote environment.
Thus, the command and sensing signals exchanged over wireless medium through a BS form a closed-loop teleoperation system connected over two communication links.
\begin{remark}
The focus of this work is on communication and computation aspects of the telemanupulation system. Therefore, the latency due to executing the command signal at the robot is not taken into account, because its value is negligible as compared to the latency components shown in Fig.~\ref{fig:depiction_delay}. Further, it depends upon the mechanical property of the robot and application at hand. Apart from it, the latency incurred in the HO's reaction is not considered, as this component is beyond the scope of the work.
\end{remark}
\begin{remark}
In the following, we assume that the data related to teleoperation application is allocated resources in a private slice by the BS avoiding queuing delay, as its latency-constrained nature requires network support.
\end{remark}
\begin{figure}[t!]
\centering
\input{tikz_figs/latency_components}
\caption{Depiction of different components of the closed-loop latency.}
\label{fig:depiction_delay}
\end{figure}
The different delay components of the closed-loop latency, which constitute the delay incurred in exchanging command and sensing signals, are shown in Fig. \ref{fig:depiction_delay} for the system model in Fig.~\ref{fig:sys_mod_MEC_TI}.
The feed-forward delay consists of the delay incurred in the transmission of the high-level command signal, the processing at MEC (if required), and the transmission of processed data (low-level command).
The feedback delay consists of the delay incurred in compressing the sensing data at the robot, the transmission of these compressed data, the decompression and processing at the MEC BS, and finally, in the transmission of this processed user-readable feedback signal to the H
\footnote{The propagation delay
is not taken into consideration here, as it will be several orders of magnitude lower than the other components of the closed-loop delay for the distances relevant for the considered teleoperation scenario.
}.
\begin{comment}
\begin{remark}
\CS{Suggest to remove this remark}
The sensing data contains variety of data, such as video, image, audio, tactile, and \CS{mechanical?}, which characterizes the working of the robot in the remote environment. It is important to consider all these data not only to provide the command signal, but also to assess the health condition and safety of the robot in the remote environment.
This demands for the consideration of all available sensing data rather than the recent latest useful piece of information \cite{age_of_info}. Therefore, the closed-loop latency is analyzed here. The analysis of the age of the loop from the information freshness perspective will be discussed in our next work.
\end{remark}
\end{comment}
\section{Latency Components}\label{sec:latency}
In the given context of a MEC-enabled teleoperation system, the closed-loop latency is mainly due to the delay incurred in transmission of command and sensing data, compression at the robot, and decompression and computation at the MEC server. The data transmission delay is random in nature due to uncertainty in the wireless channel, while the randomness of the computation time at the MEC server is related to the uncertainty in the amount of resources allocated for processing. Thus, the overall closed-loop latency is also an RV. In the following, we characterize the probability distributions of different delay components of the closed-loop MEC-enabled teleoperation system.
\subsection{Latency Incurred in Data Transmission}
We consider a wireless channel with Rayleigh block fading, such that the channel gain $h$ is constant over the length of the packet. Hence, the fading gain $g=|h|^2\sim \exp(1)$, and the gain over subsequent packets is independent and identically distributed.
We consider a transmitter that uses a constant power $P_\text{tx}$ located at a distance $d$ from the receiver.
As teleoperation applications are extremely sensitive to delay and require dedicated resources, we consider a slicing-enabled 5G system in which a bandwidth $B$ is reserved for the transmission~\cite{aijaz2017shaping}. The signal-to-noise ratio (SNR) $\gamma$ at the receiver is then:
\begin{align} \label{eq:snr}
\gamma(P_\text{tx},d, B) & = K_0 \frac{P_\text{tx} |h|^2}{ d^{\ell} N_0 B} = \gamma_0(P_\text{tx},d, B) \cdot g,
\end{align}
where $K_0$ is the Friss equation parameter, $\ell$ is a path loss exponent depending on the propagation scenario, $N_0$ is the noise power spectral density, and $\gamma_0(P_\text{tx}, d, B) = K_0 \frac{P_\text{tx} }{ d^{\ell} N_0 B}$ is the average SNR.
We assume that the transmitter can choose the transmission rate, guaranteeing $\epsilon$-outage of the communication link at the receiver~\cite{goldsmith2005wireless} by using Shannon's bound.
The outage probability $\epsilon$ characterizes the probability of packet loss in case of deep fading, when the transmission cannot be decoded.
We assume that the data is correctly received if the instantaneous received SNR is higher than $\gamma_\text{th}$. Thus, for a threshold SNR $\gamma_\text{th}$ with outage probability $\epsilon$, the rate $R(\epsilon)$ is
\begin{equation} \label{eq:rate}
R(\epsilon) = B\log_2(1+\gamma_\text{th}).
\end{equation}
The outage probability $\epsilon$ in a Rayleigh fading channel is given by
\begin{equation}\label{eq:epsilon}
\epsilon = \Pr \left\{ \gamma < \gamma_\text{th} \right\} = \Pr \left[ g < \frac{\gamma_\text{th}}{\gamma_{0}} \right] = 1 - e^{\left(-\frac{\gamma_\text{th}}{\gamma_{0}}\right)},
\end{equation}
where $\mbox{Pr}\{ \cdot \}$ denotes the probability of an event.
From \eqref{eq:rate} and \eqref{eq:epsilon}, $R(\epsilon)$ can be rewritten as
\begin{equation} \label{eq:rate_epsilon}
R(\epsilon) = B\log_2(1+\gamma_\text{th}) = B\log_2 \left( 1+\gamma_{0} \ln \left(\frac{1}{1-\epsilon} \right) \right).
\end{equation}
\begin{remark} \label{rem:rate_reliability}
$R(\epsilon)$ is a monotonically increasing function of $\epsilon$
\end{remark}
If the transmitter divides data into packets with a constant length $n_p$ and uses a pass-band modulation, the time $t_p$ to transmit is simply given by:
\begin{equation}
t_p = \frac{n_p}{2BR(\epsilon)}.
\end{equation}
As the erasure probability for a packet is $\epsilon$, the total time until correct reception is a geometrically distributed RV. The time elapsed in receiving acknowledgement is very small compared to the data transmission time, and hence it is ignored.
Thus, the probability mass function (PMF) of the time $\mathcal{T}$ required to transmit a packet is then given by
\begin{equation}
\mbox{Pr}(\mathcal{T}=kt_p) = \epsilon^{k-1} (1-\epsilon), \;\; k \geq 1.
\end{equation}
The mean and variance of $\mathcal{T}$ are then:
\begin{equation}
\mathbb{E}[\mathcal{T}] = \frac{1}{1-\epsilon}t_p, \;\; \mathbb{E}[(\mathcal{T}-\mathbb{E}[\mathcal{T}])^2]=\frac{\epsilon}{(1-\epsilon)^2} t_p^2.
\end{equation}
If a data block is composed of $N$ packets, the total transmission time for the block is:
\begin{equation}
T_{\text{tx}}(N, \epsilon) = \sum_{i=1}^{N} \mathcal{T}_i.
\end{equation}
We note that $T_{\text{tx}}$ is the sum of $N$ identical and independent geometrically distributed RVs. It's PMF is then a negative binomial distribution as follows:
\begin{equation}
\mbox{Pr}(T_{\text{tx}}=kt_p|N,\epsilon)=\binom{k+N-1}{N-1}\epsilon^{k-N}(1-\epsilon)^N, \;\; k \geq N.
\end{equation}
In most practical cases, $N$ will be relatively large, and we can use the Central Limit Theorem to approximate this distribution to a Gaussian RV as follows:
\begin{equation} \label{eq:t_tx_dist}
T_{\text{tx}}(N, \epsilon) \sim \mathcal{N}(\mu_{\text{tx}}, \sigma_{\text{tx}}^2),
\end{equation}
where $\mu_{\text{tx}} = N \frac{1}{1-\epsilon}t_p$ and $\sigma_{\text{tx}}^2 = N \frac{\epsilon}{(1-\epsilon)^2} t_p^2$.
This approximation substitutes the transmission time from a discrete domain with only positive values with the real domain.
However, as we assume $N\gg1$, the approximation error is negligible.
\begin{remark}
The transmission time, modeled here using the Gaussian distribution, should be non-negative, but the domain of the Gaussian distribution is $(-\infty, \infty)$.
It is known that around 99.73\% of the data points will fall within three standard deviations from the mean for a Gaussian distribution. We have noted that $\mu_{\text{tx}} - 4 \sigma_{\text{tx}} \geq 0$ for the numerical values considered in the simulation. Therefore, the left tail of the distribution has a negligible effect on the analysis, and the latency will never be negative.
\end{remark}
\begin{remark} \label{rem:gaussian_dist_nature}
The PDF of Gaussian distribution is neither a convex nor a concave function. It is symmetric and exhibits unimodal variation. Further, it is also a log-concave function.
\end{remark}
\subsection{{Latency Incurred in Data Processing }}
An MEC server performs tasks related to computation, and specifically in this case, compression and decompression also.
The time elapsed in these tasks depend upon the number of central processing unit (CPU) cycles required to process one bit of data, the clock frequency of the MEC server, and the volume of the data to be processed.
In the context of the considered closed-loop teleoperation system (see Fig.~\ref{fig:sys_mod_MEC_TI}), the processing capability of the MEC mounted at BS will be much higher than the robot's~\cite{robot_comput}. Therefore, the computation to extract the low-level command from the raw data is accomplished at the BS rather than at the robot. Here, the time elapsed in different processes are modelled, which will be helpful in analysing the closed-loop latency.
\subsubsection{Latency incurred in computation}
The time elapsed in computation $T_c$ to compute $D_0$ volume of data is given as
\begin{equation} \label{eq:comp_time_1}
T_{c} = \frac{D_0 X_c}{f_{0}},
\end{equation}
where $X_c$ is the number of CPU cycles and $f_0$ is the frequency of the MEC server.
A recent study~\cite{MEC_random_var} shows that the number of cycles allocated to compute one bit at the MEC server is stochastic in nature. This is because the CPU cycles are allocated to different ongoing tasks at the MEC server simultaneously. The number of CPU cycles required to compute one bit of data is modeled in the relevant literature~\cite{MEC_gamma_1, MEC_gamma_2} as an RV following a Gamma distribution.
Thus, the PDF of $X_c$ is given as
\begin{equation} \label{eq:no_of_cycles_1}
X_c \sim \mbox{Gamma}(\kappa_1, \beta_1) \Rightarrow f_X(x; \kappa_1, \beta_1) = \frac{1}{(\beta_1)^\kappa_1 \Gamma(\kappa_1)} x^{\kappa_1-1} \exp(-{x}/{\beta_1}),
\end{equation}
where $\kappa_1$ is the shape parameter and $\beta_1$ is the scale parameter. $\Gamma(s)= \int_{0}^{\infty} t^{s-1} e^{-t} dt$ is the Gamma function.
Note that $\mathbb{E}[X_c] = \kappa_1 \beta_1$.
Now, from \eqref{eq:comp_time_1}, \eqref{eq:no_of_cycles_1}, and the transformation of the PDF of $X_c$, the distribution of computation time at MEC server is given as
\begin{equation} \label{eq:t_mec_com}
T_c \sim \mbox{Gamma}\bigg(\kappa_1, \frac{D_0 \beta_1}{f_0}\bigg) \Rightarrow f_{T_c}(t; \kappa_1, \beta_1, D_0, f_{0}) = \bigg( \frac{f_{0}}{D_0 \beta_1}\bigg)^{\kappa_1} \frac{1}{\Gamma(\kappa_1)} t^{\kappa_1-1} \exp \bigg(\frac{-tf_{0}}{D_0 \beta_1}\bigg).
\end{equation}
The expected computation delay $\bar{T}_{c}$ is obtained as follows
\begin{equation}
\bar{T}_{c}(\kappa_1, \beta_1, D_0, f_{0}) = \mathbb{E}[T_{c}] = \frac{D_0 \kappa_1 \beta_1}{f_{0}}.
\end{equation}
\subsubsection{Latency incurred in compression}
The latency of data compression depends on the data volume and computational properties of the device's processor.
Specifically, the time elapsed $T_\text{cp}$ in compressing volume of data $D_0$ is given as \cite{jsac_comp_model}
\begin{equation} \label{eq:comp_time}
T_\text{cp} = \frac{D_0 X_{cp}}{f_\text{0}},
\end{equation}
where $X_{cp}$ is the number of CPU cycles required to compress one bit of data, and $f_\text{0}$ is the frequency (i.e., clock speed) of the processor.
Analogously to the previous case, $X_{cp}$ is stochastic in nature and follows the Gamma distribution given as follows
\begin{equation} \label{eq:comp_time_exp}
X_{cp} \sim \mbox{Gamma}(\kappa_2, \beta_2),
\end{equation}
where $\kappa_2$ and $\beta_2$ are respectively the shape and scale parameters.
Note that $\mathbb{E}[X_{cp}] = \kappa_2 \beta_2$.
\begin{comment}
Specifically,
\begin{equation} \label{eq:no_of_cycles}
f_{X_{cp}}(x) = \frac{1}{ (\beta_2)^{\kappa_2} \Gamma(\kappa_2)} x^{\kappa_2-1} \exp(-{x}/{\beta_2})
\end{equation}
where $\kappa_2$ and $\beta_2$ are respectively the shape and scale parameters.
Note that $\mathbb{E}[X_{cp}] = \kappa_2 \beta_2$.
\end{comment}
Thus, $T_\text{cp}$ is also an RV and its PDF is given as
\begin{equation} \label{eq:t_mec}
\begin{split}
T_{cp} \sim \mbox{Gamma}\bigg(\kappa_1, \frac{D_0 \beta_2}{f_0}\bigg). &
\end{split}
\end{equation}
A lossless data compression is assumed, such that the original data can be perfectly reconstructed from the compressed data without error\footnote{Huffman, run-length, Lempel-Ziv, and bzip2 are some of the most commonly used compression techniques to achieve lossless data compression \cite{data_comp_decomp_1}.} \cite{data_comp_decomp_4}. For lossless compression, the average number of CPU cycles required to compress one bit of raw data is given as~\cite{comm_lett_comp,jsac_comp_model}
\begin{equation} \label{eq:compresiion_ratio}
\mathbb{E}[X_{cp}] = \kappa_2 \beta_2 = \exp(Q \psi) - \exp(\psi) = C(Q),
\end{equation}
where $Q\geq 1$ is the compression ratio (i.e., the ratio of the sizes of raw and compressed data) and $\psi$ is a positive constant.
Using \eqref{eq:compresiion_ratio}, the PDF of compression time ${T_\text{cp}}$ is also a RV, which is given as
\begin{equation} \label{eq:t_mec_comp}
\begin{split}
T_{cp} \sim \mbox{Gamma}\bigg(\kappa_2, \frac{D_0 C(Q)}{\kappa_2 f_0}\bigg) . & \\
\end{split}
\end{equation}
The expected value of ${T_\text{cp}}$, $\bar{T}_\text{cp}$ for the compression ratio $Q$ is given as
\begin{equation}
\bar{T}_\text{cp}(\kappa_2, \beta_2, D_0, f_{0}, Q) = \mathbb{E}[T_\text{cp}] = \frac{D_0 \mathbb{E}[X_{cp}]}{f_\text{0}} = \frac{D_0 C(Q)}{f_\text{0}}.
\end{equation}
\subsubsection{Latency incurred in decompression}
Decompression refers to the process of restoring compressed data to its original form.
It is also a type of computation, and can be performed on MEC server.
The latency incurred $T_d$ in decompressing $D_0$ amount of data is given as
\begin{equation}
T_d = \frac{D_0 X_{d}}{f_\text{0}},
\end{equation}
where $X_d$ denotes the number of cycles required to decompress one bit of data, which will also follow the Gamma distribution given as
\begin{equation} \label{eq:decomp_time}
X_d \sim \mbox{Gamma}(\kappa_3, \beta_3),
\end{equation}
where $\kappa_3$ and $\beta_3$ are respectively the shape and scale parameters. Note that $\mathbb{E}[X_{d}] = \kappa_3 \beta_3$.
Recent work in \cite{data_comp_decomp_1, data_comp_decomp_2, data_comp_decomp_3} shows that if same volume of data is compressed and decompressed then the time elapsed in the decompression process is less than that elapsed in the compression process. Thus, the average number of cycles required in decompression and compression process can be related as follows
\begin{equation} \label{eq:comp_decomp_1}
\mathbb{E}[X_d] = \zeta \mathbb{E}[X_{cp}],
\end{equation}
where $0 < \zeta < 1$ is a constant.
Using \eqref{eq:comp_decomp_1}, the following can be written,
\begin{equation} \label{eq:comp_decomp}
\kappa_3 \beta_3 = \zeta \kappa_2 \beta_2 = \zeta C(Q).
\end{equation}
Thus, the decompression time $T_d$ is also an RV, $T_\text{d} \sim \mbox{Gamma}\bigg( \kappa_3, \frac{D_0 \zeta C(Q)}{f_0 \kappa_3} \bigg)$, and its PDF with compression ratio $Q$ is given as
\begin{equation} \label{eq:t_mec_decomp}
\begin{split}
T_\text{d} \sim \mbox{Gamma}\bigg( \kappa_3, \frac{D_0 \zeta C(Q)}{f_0 \kappa_3} \bigg) & . \\
\end{split}
\end{equation}
The expected value of $T_d$, $\bar{T}_{d}$, for the compression ratio $Q$, $\bar{T}_{d}$, is obtained as
\begin{equation}
\bar{T}_{d}(\kappa_3, \beta_3, D_0, f_{0}, Q) = \mathbb{E}[T_{d}] = \frac{D_0 \kappa_3 \beta_3}{f_{0}} = \frac{D_0 \zeta \kappa_2 \beta_2}{ f_{0}} = \frac{D_0 \zeta C(Q)}{f_{0}} .
\end{equation}
\begin{remark} \label{rem:gamma_dist_nature}
The PDF of Gamma distribution is neither a convex nor a concave function, but it exhibits a unimodal variation. Also, it is not a symmetric distribution \cite{gamma_pdf}. Thus, the PDFs of $T_{c}$, $T_{cp}$, and $T_{d}$ are neither convex nor concave functions, but are all unimodal functions.
In addition, Gamma distribution is a log-concave function, and hence the PDFs of $T_{c}$, $T_{cp}$, and $T_{d}$ are log-concave functions.
\end{remark}
\begin{comment}
\begin{remark}
If $\mathcal{A} \sim \mbox{Gamma}(\kappa, \beta_1)$ and $\mathcal{B} \sim \mbox{Gamma}(\kappa, \beta_2)$ are two independent RV following Gamma distribution then the distribution of $\mathcal{C}=\mathcal{A}+\mathcal{B}$ is given as
\begin{equation}
\nonumber
f_{\mathcal{C}}(c) = c^{2\kappa-1} \exp(-c/\beta_2) \int_{0}^{1} [c(1-c)]^{\kappa-1} \exp \Big( -zc \frac{\beta_1-\beta_2}{\beta_1 \beta_2} \Big) dc
\end{equation}
If $\kappa=1$ then $f_{\mathcal{C}}(c) = \frac{\beta_1 \beta_2}{\beta_1 - \beta_2} \big( \exp(-\beta_2 c) - \exp(-\beta_1 c) \big)$.
\end{remark}
\begin{remark}
If $ \mathbb{A} \sim \mbox{Gamma}(\kappa_1, \beta)$ and $ \mathbb{B} \sim \mbox{Gamma}(\kappa_2, \beta)$ are two independent RV following Gamma distribution then the distribution of $ \mathbb{C}= \mathbb{A}+ \mathbb{B}$ is given as
\begin{equation}
\nonumber
\mathbb{C} \sim \mbox{Gamma($\kappa_1+\kappa_2, \beta$)}
\end{equation}
\end{remark}
\begin{remark}
If $\mathtt{A} \sim \mbox{Gamma}(\kappa_1, \beta_1)$ and $\mathtt{B} \sim \mbox{Gamma}(\kappa_2, \beta_2)$ are two independent RV following Gamma distribution then the distribution of $\mathtt{C}= \mathtt{A}+ \mathtt{B}$ is given as
\begin{equation}
\nonumber
f_{\mathtt{C}}(c) =
\end{equation}
\end{remark}
\textcolor{blue}{Missing equation?}
\begin{remark}
If $P_1 \sim \mathcal{N}(\mu_1, \sigma_1^2)$ and $P_2 \sim \mathcal{N}(\mu_2, \sigma_2^2)$ are two independent Gaussian RV then $P_3=P_2+P_3$ will also be a Gaussian RV with $P_3 \sim \mathcal{N}(\mu_1+\mu_2, \sigma_1^2 + \sigma_2^2)$.
\end{remark}
\end{comment}
\section{Analysis of Closed-Loop Latency of MEC-enabled Teleoperation System} \label{sec:analysis}
Using the results from the previous section, the analytical framework to estimate the closed-loop latency of MEC-enabled teleoperation system shown in Fig. \ref{fig:sys_mod_MEC_TI} is developed here. We assume that the HO transmits all the command data to the BS for processing at the MEC server. On the other hand, two scenarios are analyzed regarding the processing of the raw sensing data at the robot. In the first case, the robot located in the remote environment compresses the raw sensing data first and then transmits these compressed data. The MEC server then decompresses the data to recover the original version, which is processed to extract the user readable command to be transmitted to the HO.
In the second case, the robot does not compress the sensing data, but transmits them in raw form to the BS for further processing at the MEC server to extract the user readable command for the HO.
Let $D_c$ and $D_s$ be the volume of command and sensing signal, respectively. Let the distance between the HO and the BS be $d_{ho}$, and the same between BS and the robot be $d_{r}$.
The transmission powers of the HO, the BS, and the robot are $P_{\text{tx}}^{ho}$, $P_{\text{tx}}^{bs}$, and $P_{\text{tx}}^{r}$, respectively. Assume that $B$ amount of bandwidth is dedicated for this closed-loop operation, and the HO, the BS, and the robot transmit over this bandwidth. In the given context, it may be noted that the compression process will be performed at the robot, whereas the decompression and computation processes will be performed at the BS.
Since all processes are executed over the same server, the shape parameter (see \eqref{eq:no_of_cycles_1}, \eqref{eq:comp_time_exp}, and \eqref{eq:decomp_time}) will remain the same for a given processor. On the other hand, the scale parameter will be different for different tasks (computation, compression, or decompression), as different amounts of resources in terms of CPU cycles need to be allocated. Let the shape parameter of the MEC associated with BS be $\kappa_{bs}$, and the scale parameter for computation and decompression at the BS be $\beta_c$ and $\beta_d$, respectively.
Further, let the shape parameter of the processor embedded with robot be $\kappa_r$, and the scale parameter for the compression process be $\beta_{cp}$.
Finally, let the frequency of the MEC server associated with BS and processor at the robot be $f_{\text{MEC}}$ and $f_R$, respectively.
Thus, from \eqref{eq:compresiion_ratio} and \eqref{eq:comp_decomp}, we get
\begin{equation}
\begin{split}
& \kappa_r \beta_{cp} = C(Q), \;\;\; \kappa_{bs} \beta_{d} = \zeta C(Q).
\end{split}
\end{equation}
\subsection{Case 1: Data Compression at Robot}
The closed-loop latency is composed of the latency incurred during transmitting the command data (from the HO to the robot) and the sensing data (from the robot to the HO).
{The latency incurred in transmitting command data from HO to robot is composed of the data transmission time, extracting the low-level command from raw command signal at MEC associated at BS, and transmitting the low-level command to the robot from the BS.}
Thus, referring to Fig. \ref{fig:sys_mod_MEC_TI}, the latency involved in transmitting the command signal ${T_1^c}$ when $N_c$ data packets are to be sent with outage probability $\epsilon$ is given as
\begin{equation}
T_1^c = T_{\text{tx}}^c(N_c, \epsilon) + T_c^c(\kappa_{bs}, \beta_c, D_c,f_{bs}) + T_{\text{tx}}^{pc}(N_c^p, \epsilon),
\end{equation}
where $T_{\text{tx}}^c$ is the time elapsed in transmitting the command signal, see \eqref{eq:t_tx_dist}, and $T_c^c$ is the time taken by MEC server to estimate the low-level command, see \eqref{eq:t_mec_com}.
$T_{\text{tx}}^{pc}$ denotes the time elapsed in transmitting the low-level command (consisting of $N_c^p$ packets) extracted from command signal, see \eqref{eq:t_tx_dist}.
All the constituents of $T_1^c$ are independent RVs.
The latency incurred in transmitting sensing data from the robot to HO comprises of compression at robot, transmission of compressed data, decompressing the compressed data at MEC associated with BS, extracting the low-level command from sensing data, and transmitting the low-level command to the HO from BS. Thus, referring to Fig. \ref{fig:sys_mod_MEC_TI}, the latency $T_1^f$ incurred in transmitting the compressed sensing data when $N_f$ data packets are to be transmitted with outage $\epsilon$ is given as
\begin{equation}
\begin{aligned}\label{eq:clt_11}
T_1^f = {} & T_{cp}^f(\kappa_r, \beta_{cp}, D_s, f_r, Q) + T_{\text{tx}}^f(N_f, \epsilon) + T_d^f(\kappa_{bs}, \beta_d, \frac{D_s}{Q}, f_{bs}, Q) \\
& + T_c^f(\kappa_{bs}, \beta_c, D_s, f_{bs}, Q)+ T_{\text{tx}}^{pf}(N_f^p, \epsilon)
\end{aligned}
\end{equation}
where $T_r$ denotes the time elapsed in compressing the raw sensing data with compression ratio $Q$ (see \eqref{eq:t_mec_comp}), $T_{\text{tx}}^f$ denotes the time elapsed in transmitting the compressed data (see \eqref{eq:t_tx_dist}), $T_d^f$ denotes the delay incurred in decompressing the compressed sensing data (see \eqref{eq:t_mec_decomp}), and $T_c^f$ denotes the time elapsed in processing the sensing data (see \eqref{eq:t_mec}). $T_{\text{tx}}^{pf}$ is the time elapsed in transmitting the low-level command (having number of packets $N_f^p$) extracted from sensing data to the HO (see \eqref{eq:t_tx_dist}).
All the constituents of $T_1^f$ are independent RVs.
The volume of the low-level command is much less than the original raw data, i.e., $N_c >> N_c^p$ and $N_f >> N_f^p$, and will not contribute significantly in the closed-loop latency. Using this fact, $T_1^c$ and $T_1^f$ can be written as,
\begin{equation}
\begin{aligned}\label{eq:delay_cd_fb}
T_1^c \approx {} & T_{\text{tx}}^c(N_c, \epsilon) + T_c^c(\kappa_{bs}, \beta_c, D_c,f_{bs}) ; \\
T_1^f \approx {} & T_{cp}^f(\kappa_r, \beta_{cp}, D_s, f_r, Q) + T_{\text{tx}}^f(N_f, \epsilon) + T_d^f(\kappa_{bs}, \beta_d, \frac{D_s}{Q}, f_{bs}, Q) + T_c^f(\kappa_{bs}, \beta_c, D_s, f_{bs}, Q) .
\end{aligned}
\end{equation}
The closed-loop latency $T_1$ is estimated as
\begin{equation}
\begin{aligned}\label{eq:clt_p}
T_1 = {} & T_1^c + T_1^f.
\end{aligned}
\end{equation}
Using \eqref{eq:delay_cd_fb}, \eqref{eq:clt_p} can be written as
\begin{equation}
\begin{aligned}
T_1 = {} & T_{\text{tx}}^c(N_c, \epsilon) + T_c^c(\kappa_{bs}, \beta_c, D_c,f_{bs}) + T_{cp}^f(\kappa_r, \beta_{cp}, D_s, f_r, Q) + T_{\text{tx}}^f(N_f, \epsilon) \\
{} & + T_d^f(\kappa_{bs}, \beta_d, \frac{D_s}{Q}, f_{bs}, Q) + T_c^f(\kappa_{bs}, \beta_c, D_s, f_{bs}, Q).
\end{aligned}
\end{equation}
This can be rewritten as
\begin{equation} \label{eq:clt_alpha_0_1}
\begin{aligned}
T_1 = {} & \big[T_{\text{tx}}^c(N_c, \epsilon) + T_{\text{tx}}^f(N_f, \epsilon) \big] + T_{cp}^f(\kappa_r, \beta_{cp}, D_s, f_r, Q) \\
& + \left[ T_c^c(\kappa_{bs}, \beta_c, D_c,f_{bs}) + T_d^f(\kappa_{bs}, \beta_d, \frac{D_s}{Q}, f_{bs}, Q) + T_c^f(\kappa_{bs}, \beta_c, D_s, f_{bs}, Q) \right] .
\end{aligned}
\end{equation}
The expected value of $T_1$, $\mu_{T_1}$, is given as
\begin{equation} \label{eq:mean_t_1}
\begin{aligned}
\mu_{T_1} = {} & N_c \frac{1}{1-\epsilon}t_p + \frac{ D_c \kappa_{bs} \beta_c}{f_{bs}} + \frac{D_s C(Q)}{f_r} + N_f \frac{1}{1-\epsilon}t_p + \frac{\zeta D_s C(Q)}{Q f_{bs}} + \frac{ D_s \kappa_{bs} \beta_{c}}{f_{bs}} .
\end{aligned}
\end{equation}
{ It may be noted that all the latency constituents of $T_1$ (see \eqref{eq:clt_alpha_0_1}) are independent RVs, and hence the closed-loop latency $T_1$ is also an RV. $T_1$'s constituent distributions $T_c^c, T_d^f$, and $T_c^f$ follow Gamma distribution with different scale and shape parameters, whereas $T_{\text{tx}}^c$ and $T_{\text{tx}}^f$ follow Gaussian distribution. It is very difficult to estimate the closed-form expression of the PDF of the RV $T_1$. Therefore, it is very important to characterize its properties for further analysis. }
\begin{lemma} \label{lemma_cvx}
The PDF of the sum of two independent RVs is convex iff at least one of them is convex. In the same way, the PDF of the sum of two independent RVs is concave iff at least one of them is concave.
\end{lemma}
\begin{proof}
See Appendix \ref{lemma_1}.
\end{proof}
\begin{remark} \label{rem:pdf_t_1}
The PDF of $T_1$ is neither a convex nor a concave function of $t$, because none of its constituent distributions are either convex or concave (see Lemma \ref{lemma_cvx}).
\end{remark}
\begin{theorem}
The CDF of $T_1$ is neither a convex nor a concave function of time.
\end{theorem}
\begin{proof}
See Appendix \ref{theorem_1}.
\end{proof}
\subsection{Case 2: Raw Data Offloading to MEC }
Here, no data compression happens at the robot, and the whole raw sensing data is transmitted to the BS, where MEC processes it in order to estimate the user readable command. Thus, referring to Fig. \ref{fig:sys_mod_MEC_TI}, the latency involved $T_2^c$ in transmitting the command signal from HO to the robot is given as
\begin{equation}
\nonumber
T_2^c = T_1^c.
\end{equation}
Now, referring to Fig. \ref{fig:sys_mod_MEC_TI}, the latency $T_2^f$ involved in transmitting the raw sensing data having $M_f$ packets from the robot to the HO is given as
\begin{equation}
\begin{aligned}\label{eq:clt_22}
T_2^f = {} & T_{\text{tx}}^f(M_f, \epsilon) + T_c^f(\kappa_{bs}, \beta_c, D_s, f_{bs})+ T_{\text{tx}}^{pf}(N_f^p, \epsilon)
\end{aligned}
\end{equation}
where the details of the parameters are mentioned in \eqref{eq:clt_11}.
Ignoring the latency elapsed in sending the low-level commands to robot and HO (see \eqref{eq:delay_cd_fb}), the closed-loop latency $T_2$ in this case is given as
\begin{equation}
\begin{aligned}
T_2 = {} & T_2^c + T_2^f \\
= {} & T_{\text{tx}}^c(N_c, \epsilon) + T_c^c(\kappa_{bs}, \beta_c, D_c,f_{bs}) + T_{\text{tx}}^f(N_f, \epsilon) + T_c^f(\kappa_{bs}, \beta_c, D_s, f_{bs}).
\end{aligned}
\end{equation}
This can be rewritten as
\begin{equation} \label{eq:clt_alpha_1_2}
\begin{aligned}
T_2 = {} & \big[ T_c^c(\kappa_{bs}, \beta_c, D_c,f_{bs}) + T_c^f(\kappa_{bs}, \beta_c, D_s, f_{bs}) \big] + \big[ T_{\text{tx}}^c(N_c, \epsilon) + T_{\text{tx}}^f(M_f, \epsilon) \big].
\end{aligned}
\end{equation}
The expected value of $T_2$, $\mu_{T_2}$, is given as
\begin{equation} \label{eq:mean_t_2}
\begin{aligned}
\begin{aligned}
\mu_{T_2} = {} & N_c \frac{1}{1-\epsilon}t_p + \frac{ D_c \kappa_{bs} \beta_c}{f_{bs}} + M_f \frac{1}{1-\epsilon}t_p + \frac{ D_s \kappa_{bs} \beta_{c}}{f_{bs}}.
\end{aligned}
\end{aligned}
\end{equation}
As the constituents of $T_2$ are all independent RVs, we can make some of the same inferences that we proved for $T_1$.
\begin{remark} \label{rem:pdf_t_2}
The PDF of $T_{2}$ is neither a convex nor a concave function of time, because none of its constituent distributions are either convex or concave (see Lemma \ref{lemma_cvx}).
\end{remark}
\begin{theorem}
The CDF of $T_{2}$ is neither a convex nor a concave function of time.
\end{theorem}
\begin{proof}
See Appendix \ref{theorem_2}.
\end{proof}
\subsection{Optimization of Closed-loop Teleoperation System}
It is important to optimize the closed-loop latency $\tau_i$ ($i=1$ for Case 1 and $i=2$ for Case 2) of the MEC-enabled teleoperation system for a given bandwidth $B$. This also requires to estimate the optimal compression ratio locally at the robot itself before transmitting it to the BS. For this purpose, an optimization problem is formulated as follows
\begin{equation}
\nonumber
\begin{aligned}
\textbf{(P1)}:~~ & \min_{Q, \epsilon} \quad \tau_i, \;\;\; i \in \{ 1,2 \} \\
\text{s. t.} \;\; {\textbf{(C1)}}:~ & ~ F_{T_i}(\tau_i) \geq \varrho_\text{th}, \;\;\; i \in \{ 1,2 \} \\
{\textbf{(C2)}}:~ & ~ 0 \leq \epsilon \leq \epsilon_{th} \\
{\textbf{(C3)}}:~ & ~ Q \geq Q_{th} =1
\end{aligned}
\label{eq:opt_prob_p1}
\end{equation}
Constraint {\textbf{(C1)}} ensures the closed-loop latency $\tau$ in probabilistic sense, where $F_{T}(\cdot)$ denotes the CDF of $T$.
$\varrho_\text{th}$ is an statistical parameter, which indicates the probability of the closed-loop latency be at most $\tau$. Constraint {\textbf{(C2)}} restricts the range of link outage probability. Constraint {\textbf{(C3)}} indicates the range of compression ratio.
In order to solve the optimization problem {\textbf{(P1)}}, we need to obtain the distribution of $T_1$ and $T_2$ to meet the statistical guarantee on the closed-loop latency criterion in {\textbf{(C1)}}.
However, as we noted above, these PDFs are hard to obtain, because the closed-form expression of the distribution of the sum of arbitrary Gamma RVs is not known. However, several approximation methods \cite{gamma_approx_1, gamma_approx_2, gamma_approx_3} to estimate the distribution of sum of Gamma RVs are reported in the literature. These methods offer better accuracy when the number of RVs to be summed are very high. Here, the closed-loop latency $T_1$ comprises of only four Gamma RVs (see \eqref{eq:clt_alpha_0_1}), whereas the closed-loop latency $T_2$ comprises of only two Gamma RVs (see \eqref{eq:clt_alpha_1_2}). The approximation error may then be high, which is unacceptable for the teleoperation system design as we are dealing with time-critical application scenario. Therefore, these approximation methods are not viable options in the given context. However, finding lower and upper bounds on the values of $T_1$ and $T_2$ will be helpful to solve the optimization problem \textbf{(P1)}.
\begin{theorem} \label{th:bound_case_1}
The closed-loop latency $T_1$ for a given $\varrho_{th}$ with $F_{T_1}(\tau_1) = \varrho_{th}$ is bounded as follows:
\begin{equation}
\nonumber
\tau_{1,L}(Q, \epsilon, \varrho_{th}) \leq \tau_1 \leq \tau_{1,U}(Q, \epsilon, \varrho_{th})
\end{equation}
where
\begin{equation}
\nonumber
\begin{aligned}
\tau_{1,L}(Q, \epsilon, \varrho_{th}) ={} & \max\left( F_{T_{tx}^c}^{-1}(\varrho_{th}), F_{T_{tx}^f}^{-1}(\varrho_{th}), F_{T_{cp}^f}^{-1}(\varrho_{th}), F_{T_{c}^c}^{-1}(\varrho_{th}), F_{T_{d}^f}^{-1}(\varrho_{th}), F_{T_{c}^f}^{-1}(\varrho_{th}) \right); \\
\tau_{1,U}(Q, \epsilon, \varrho_{th}) ={} & \min\bigg(F_{T_{tx}^c}^{-1}(\varrho_{th}) + F_{T_{tx}^f}^{-1}(\varrho_{th}) + F_{T_{cp}^f}^{-1}(\varrho_{th}) + F_{T_{c}^c}^{-1}(\varrho_{th}) + F_{T_{d}^f}^{-1}(\varrho_{th}) + F_{T_{c}^f}^{-1}(\varrho_{th}),\\
&\frac{\mu_{T_1}}{1-\varrho_{th}} \bigg).
\end{aligned}
\end{equation}
$F_{\mathcal{Z}}^{-1}(\cdot)$ denotes the inverse of CDF of RV $\mathcal{Z}$.
\end{theorem}
\begin{proof}
See Appendix \ref{theorem_4}.
\end{proof}
\begin{theorem} \label{th:bound_case_2}
The closed-loop latency $T_2$ for a given $\varrho_{th}$ with $F_{T_2}(\tau_2) = \varrho_{th}$ is bounded as follows:
\begin{equation}
\nonumber
\tau_{2,L}(\epsilon, \varrho_{th}) \leq \tau_2 \leq \tau_{2,U}(\epsilon, \varrho_{th})
\end{equation}
where
\begin{equation}
\nonumber
\begin{split}
\tau_{2,L}(\epsilon, \varrho_{th}) = {} & \mbox{max}\left( F_{T_{tx}^c}^{-1}(\varrho_{th}), F_{T_{tx}^f}^{-1}(\varrho_{th}), F_{T_{c}^c}^{-1}(\varrho_{th}), F_{T_{c}^f}^{-1}(\varrho_{th}) \right); \\
\tau_{2,U}(\epsilon, \varrho_{th}) = {} & \mbox{min}\left(F_{T_{tx}^c}^{-1}(\varrho_{th}) + F_{T_{tx}^f}^{-1}(\varrho_{th}) + F_{T_{c}^c}^{-1}(\varrho_{th}) + F_{T_{c}^f}^{-1}(\varrho_{th}), \frac{\mu_{T_2}}{1-\varrho_{th}} \right).
\end{split}
\end{equation}
\end{theorem}
\begin{proof}
See Appendix \ref{theorem_5}.
\end{proof}
\begin{comment}
\subsubsection{Latency-aware bandwidth optimization}
In another type of scenario of a fixed latency budget, say $T_\text{th}$, is allotted.
Then, it is important to estimate the optimal BW required to meet within the latency budget. The optimization problem for this case is formulated as follows:
\begin{equation}
\nonumber
\begin{aligned}
\textbf{(P2)}:~~ & \min_{Q, \epsilon} \quad \mbox{B}_i, \;\;\; i \in \{ 1,2 \} \\
\text{s. t.} \;\; {\textbf{(C1)}}:~ & ~ F_{T_i}(t=T_\text{th} ) \geq \varrho_{th}, \;\;\; i \in \{ 1,2 \} \\
{\textbf{(C2)}}:~ & ~ 0 \leq \epsilon \leq \epsilon_{th} \\
{\textbf{(C3)}}:~ & ~ Q \geq Q_{th} =1
\end{aligned}
\label{eq:opt_prob_p1}
\end{equation}
Constraint {\textbf{(C1)}} ensures the closed-loop latency a fixed latency budget $T_\text{th}$ in probabilistic sense, where $F_{T}(\cdot)$ denotes the CDF of $T$. Constraint {\textbf{(C2)}} restricts the range of fraction of data to be offloaded to the BS. $\varrho_{th}$ is an statistical parameter, which indicates the probability of the closed-loop latency be at most $\tau$. Constraint {\textbf{(C3)}} indicates the range of compression ratio.
\end{comment}
The closed-loop latency is the sum of independent RVs, and it doesn't have the closed-form expression. However, it is well known that the PDF of the sum of RVs is obtained by convolving the constituent PDFs. In continuous domain, the convolution leads to integration which is difficult to compute since it requires the solution of a complicated multidimensional integral. Therefore, the convolution in discrete time domain is adopted here due to simplicity. Towards this, the constituent PDFs are discretized by sampling with same sampling interval (here $1$ millisecond is considered) in order to obtain the approximate probability mass functions (PMF). Then, the constituent PMFs are convolved to obtain the PMF of the closed-loop latency. Thus, we choose to use the numerical technique to solve the optimization problem {\textbf{(P1)}}.
However, the estimation of the optimal value of the latency along with link outage $\epsilon$ and compression ratio $Q$ requires the exhaustive search, which is computationally extensive. Therefore, it requires a computationally-efficient procedure to solve {\textbf{(P1)}}, which can be achieved by reducing the search space of $\epsilon$ and $Q$.
One can observe that, as the value of $\epsilon_{\text{th}}$ increases, the time elapsed in transmitting the data decreases (see Remark \ref{rem:rate_reliability}), and hence $\epsilon = \epsilon_{\text{th}}$ will be the optimal value. Now, the problem is to find the optimal value of the compression ratio $Q$, and we will use the obtained upper and lower bounds of the closed-loop latency for this purpose.
Using the bounds on the closed-loop latency, following can be written
\begin{equation}
\tau_{i,L}^{\mbox{opt}} \leq \tau_{i}^{\mbox{opt}} \leq \tau_{i,U}^{\mbox{opt}}, \;\;\; i \in \{ 1,2 \}
\end{equation}
where $\tau_{i,L}^{\mbox{opt}} = \underset{Q, \epsilon_{\text{th}}, \varrho_{th} }{\mathrm{argmin}} \; \tau_{i,L}$ and
$\tau_{i,U}^{\mbox{opt}} = \underset{Q, \epsilon_{\text{th}}, \varrho_{th}}{\mathrm{argmin}} \; \tau_{i,U}$.
Then, the reduced search interval of compression ratio $Q_{i, \mbox{SI}}$ can be obtained as follows
\begin{equation}
Q_{i, \mbox{SI}} = \big\{Q \;| \; \tau_{i,L} \leq \tau_{i,U}^{\text{opt}} \big\}.
\end{equation}
The $Q_{i, \mbox{SI}}$ is divided into linearly spaced equidistant points (here at an interval of $0.01$), and then the PMF of the closed-loop latency is estimated for each of the values by convolving constituent PMFs. Finally, the optimal value of $Q$ is the one which offers the minimum closed-loop latency by satisfying the constraint {\textbf{(C1)}} of the optimization problem {\textbf{(P1)}}.
Thus, the knowledge of $Q_{i, \text{SI}}$ leads to reduce the search space for compression ratio, and speeds up the procedure to solve the optimization problem {\textbf{(P1)}}.
\begin{comment}
We can then use numerical techniques to solve the optimization problem {\textbf{(P1)}}. This is a challenging issue due to lack of the closed-form expression for the CDF of closed-loop latency.
One can observe that, as the value of $\epsilon_{\text{th}}$ increases, the time elapsed in transmitting the data decreases (see Remark \ref{rem:rate_reliability}), and hence $\epsilon = \epsilon_{\text{th}}$ will be the optimal value. Now, we have to find the optimal value of the compression ratio $Q$. Using the bounds on the closed-loop latency, we can write:
\begin{equation}
\tau_L^{\mbox{min}} \leq \tau_{\mbox{min}} \leq \tau_U^{\mbox{min}}.
\end{equation}
Then, the search interval of compression ratio, $Q_{\mbox{SI}}$, can be obtained using following:
\begin{equation}
Q_{\mbox{SI}} = \big\{Q \;| \; \tau_L(Q, \epsilon_{\text{th}}, \varrho_{th}) \leq \tau_U^{\text{min}} \big\},
\end{equation}
where $ \tau_U^{\text{min}} = \underset{Q}{\mathrm{argmin}} \; \tau_U(Q, \epsilon_{\text{th}}, \varrho_{th})$.
The knowledge of $Q_{\text{SI}}$ reduces the search space for compression ratio.
The roots of the equation $\tau_L(Q, \epsilon_{\mbox{th}}, \varrho_{th}) = \tau_U^{\text{min}}$ can be obtained, and then $Q_{\text{SI}}$ can be estimated.
The closed-loop latency is the sum of independent RVs, and the closed-form expression is not known.
However, it is well known that the distribution of the sum of RVs is obtained by convolving their distributions. Since this requires the solution of a complicated multidimensional integral, we use the Riemann method to estimate the PDF of the closed-loop latency $T$ numerically.
\end{comment}
\begin{comment}
\begin{equation}
\tau_L^{\mbox{min}} = \underset{Q}{\mathrm{argmin}} \; \tau_L (Q, \epsilon_{\mbox{th}}, \varrho_{th}) \Rightarrow \tau_L (Q_1, \epsilon_{\mbox{th}}, \varrho_{th}) = \tau_L^{\mbox{min}}
\end{equation}
\begin{equation}
\tau_U^{\mbox{min}} = \underset{Q}{\mathrm{argmin}} \; \tau_U(Q, \epsilon_{\mbox{th}}, \varrho_{th}) \Rightarrow \tau_U(Q_2, \epsilon_{\mbox{th}}, \varrho_{th}) = \tau_U^{\mbox{min}}
\end{equation}
\begin{equation} \label{eq:search_interval}
\tau_L(Q, \epsilon_{\mbox{th}}, \varrho_{th}) = \tau_U^{\mbox{min}} = \tau_U (Q_2, \epsilon_{\mbox{th}}, \varrho_{th})
\end{equation}
\end{comment}
\section{Simulation Results}\label{sec:results}
We illustrate the analysis presented in the previous sections through numerical evaluations. The values of the parameters considered are: $\ell = 2$, $D_c=0.15$ Mb, $D_s=0.5$ Mb, $f_{bs}=15$ GHz, $\kappa_{bs}=1.25$, $\kappa_{r}=1.5$, $\zeta=0.1$, $\Psi=3.5$, $B=10$ MHz, $T_0=0.5 \mu s$, $K_0=-27$ dB, $N_0=-110$ dB, $d_{r-bs} = d_{bs-ho} =2$ km, $P_{\text{tx}}^{ho}=P_{\text{tx}}^{r}=0.5$ W. The computational capabilities of the MEC-enabled BS are consistent with the NVIDIA Jetson TX1, a common embedded processor for edge computing applications~\cite{halawa2017nvidia}.
\begin{comment}
\begin{table} [!h]
\begin{center}
\caption{ LIST OF MAJOR VARIABLES ALONG WITH THEIR DESCRIPTIONS }
\label{tab:nature_rv}
\begin{tabular}{|p{0.07\textwidth}|p{0.35\textwidth}|p{0.07\textwidth}|p{0.35\textwidth}|}
\hline
\textbf{ } & \textbf{ } & \textbf{ } & \textbf{ } \\
\hline
$D_c$ & $0.5$ Mb & $\kappa_{r}$ & $1.5$ \\
$\kappa_{bs}$ & $1.25$ & $\Psi$ & $3.5$ \\
$\zeta$ & $0.1$ & & \\
$T_0$ & $0.5 \mu s$ & $B$ & $10$ MHz \\
$K_0$ & $-27$ dB & $N_0$ & $-110$ dBm \\
$d_{r-bs}$ & $2$ km & $d_{bs-ho}$ & $2$ km \\
$p_{tx}$ & $0.5$ W & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\end{comment}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{.4\linewidth}
\flushleft
\input{tikz_figs/case_1_lb}
\caption{Lower bound ($\tau_1^{opt} - \tau_{1,L}$ (ms)).}
\label{fig:case1_lb_val}
\end{subfigure}
\begin{subfigure}[b]{.4\linewidth}
\flushright
\input{tikz_figs/case_1_ub}
\caption{Upper bound ($\tau_{1,U} - \tau_1^{opt}$ (ms)).}
\label{fig:case1_ub_val}
\end{subfigure}
\caption{ Validation of bounds on closed-loop latency for Case 1 with $f_R=1$ GHz, $\rho_\text{th}=0.95$. }
\label{fig:bound_case_1}
\end{figure}
\begin{figure}[t!]
\centering
\input{tikz_figs/case_2_lower_bound}
\caption{Validation of bounds on closed-loop latency for Case 2 with $\rho_\text{th}=0.95$.}
\label{fig:bound_case_2}
\end{figure}
\subsection{Validation of Bounds}
The validation of the bounds on the closed-loop latency obtained for Case 1 in Theorem \ref{th:bound_case_1} is shown in Fig. \ref{fig:bound_case_1}.
Fig. \ref{fig:case1_lb_val} validates the lower bound as the difference between the optimal closed-loop latency and its lower bound, i.e., $\tau_1^{opt} - \tau_{1,L}$, is always positive. On the other hand, the difference between the upper bound on the closed-loop latency and its optimal value, i.e., $\tau_{1,U} - \tau_1^{opt}$, is always positive as shown in Fig. \ref{fig:case1_ub_val} {{\footnote{ We note that same behavior is observed for any arbitrary range of $Q$ and $\epsilon$. }}}. The bounds on the closed-loop latency obtained for Case 2 in Theorem \ref{th:bound_case_2} is shown in Fig. \ref{fig:bound_case_2}, which indicates that the optimal value of closed-loop latency lies well within the bounds. Here, the compression ratio is $1$, because sensing data is not compressed at the robot located in the remote environment. From Fig. \ref{fig:bound_case_1} and Fig. \ref{fig:bound_case_2}, it can be noted that the optimal closed-loop latency in both cases is in the proximity of its upper bound.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{.45\linewidth}
\centering
\input{./tikz_figs/case_1_fr_latency.tex}
\caption{Transmission latency.}
\label{fig:t_opt_lat}
\end{subfigure}
\begin{subfigure}[b]{.45\linewidth}
\centering
\input{./tikz_figs/case_1_fr_compr.tex}
\caption{Compression latency.}
\label{fig:t_opt_compr}
\end{subfigure}
\caption{CDF of latency incurred in data transmission and compression.}
\label{fig:cdf_t_tx}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{.45\linewidth}
\centering
\input{./tikz_figs/topt_eps.tex}
\caption{Closed-loop latency.}
\label{fig:t_opt_eps}
\end{subfigure}
\begin{subfigure}[b]{.45\linewidth}
\centering
\input{./tikz_figs/topt_q.tex}
\caption{Optimal compression ratio.}
\label{fig:q_opt_eps}
\end{subfigure}
\caption{Variation of the optimal closed-loop latency and compression ratio for different cases with $\rho_\text{th}=0.95$.}
\label{fig:t_opt_vs_reliability}
\end{figure}
\subsection{Optimal System Design}
The latency incurred in data transmission and compression is shown in Fig. \ref{fig:cdf_t_tx} through their CDF variation. The CDF of the latency incurred in transmitting the sensing data of volume $0.5$ Mb is shown in Fig. \ref{fig:cdf_t_tx}a for different levels of link outage. It may be noted that the transmission time increases significantly as the link outage level becomes stringent. This indicates that the data transmission with high level of accuracy demands for relatively higher transmission time. The CDF of the latency incurred in compressing the sensing data of volume $0.5$ Mb is shown in Fig. \ref{fig:t_opt_compr} for different compression ratio $Q$ with $f_R=5$GHz. The compression time increases significantly with increase in compression ratio. However, the higher compression ratio reduces the volume of sensing data which demands for relatively less time in transmission, and vice-versa. Thus, there is a trade-off between compression and transmission time.
Variation of the optimal closed-loop latency against outage $\epsilon$ is shown in Fig. \ref{fig:t_opt_eps} for both cases.
The optimal latency is high for stringent outage requirement and it decreases as the outage probability increases. This happens because higher outage offers higher data rate which takes lower latency in data transmission.
Although the re-transmission due to higher outage will not affect the transmission time severely due to higher data transmission rate. The effect of the computational capability of the robot has a significant impact on the latency, and the optimal value of closed-loop latency decreases as the computational capability of the robot increases.
It may be noted that the optimal latency for Case 1 is much lower than that compared to Case 2 for stringent outage requirement.
However, the optimal latency converges towards Case 2 as the outage increases even for higher computational capability of the robot.
This happens because higher outage offers higher data transmission rate, and hence transmission of raw sensing data is beneficial rather than compressing it.
The optimal compression ratio against outage probability is shown in Fig.~\ref{fig:q_opt_eps}, which depends upon the computational capability of the robot as well as the outage probability.
The optimal compression ratio decreases as outage probability increases, and the compression ratio is highest for the robot having highest computational capability.
As outage increases the optimal compression ratio converges towards $1$, i.e., no compression as in Case 2.
This observation also justifies the fact that the optimal latency converges towards Case 2 as outage increases as shown in Fig.~\ref{fig:t_opt_eps}.
\begin{remark}
Data compression is not always beneficial. The decision about whether to compress the sensing data or not depends upon the required outage constraint as well as the computational capability of the robot.
\end{remark}
\subsection{Statistical vs Expected Sense System Design}
The works reported in \cite{onu_1, onu_2, onu_3, coll_computing, IoT_TI, NFV_e2e_QoS} consider the system design in expected or average sense rather than in stochastic sense. Average sense design is far from the real-life deployment scenario because the data transmission over wireless network suffers from several impairments and one of them is jitter. But, the average sense design doesn't consider the reliability criteria on random latency or jitter.
Therefore, the reliability criteria on latency must be taken into consideration while designing the teleoperation system with low latency and high reliability requirement. Here, we will investigate the comparative analysis of the optimal closed-loop latency with different reliability criteria in stochastic sense and that in average sense for Case 1 only. However, similar shortcomings are also noted for Case 2, which is omitted due to space constraints.
The closed-loop latency in average sense is optimized using the average latency obtained in \eqref{eq:mean_t_1} from the following
optimization problem
\begin{equation}
\nonumber
\begin{aligned}
\textbf{(P2)}:~~ & \min_{Q, \epsilon} \quad \mu_{T_1}, \;\; \text{s. t.} \;\; {\textbf{(C2)}} \;\; \mbox{and} \;\; {\textbf{(C3)}} \\
\end{aligned}
\label{eq:opt_prob_p1_avg}
\end{equation}
The average sense design in {\textbf{(P2)}} doesn't take into account the statistical guarantee, as the average closed-loop latency is optimized. {\textbf{(P2)}} is found to be a convex optimization problem that can be solved using CVX. The proof of convexity of {\textbf{(P2)}} is omitted for brevity.
Variation of the optimal closed-loop latency against link outage is shown in Fig. \ref{fig:comparison_with_expected_sense}a for different stochastic reliability level, such as $\varrho_\text{th}=0.95, 0.99, 0.999$. It may be noted that the optimal latency increases significantly with increase in $\varrho_\text{th}$. The amount of increase in the optimal latency is higher for $\varrho_\text{th}=0.99$ to $\varrho_\text{th}=0.95$ as compared to $\varrho_\text{th}=0.99$. The comparative view of statistical and average sense design is depicted in Fig. \ref{fig:comparison_with_expected_sense}b for $\epsilon=10^{-4}$ and $f_R=5$GHz. It may be noted that the average sense design satisfies the statistical guarantee on the closed-loop latency around $\varrho_\text{th}=0.54$ only, which is not acceptable for low-latency and high reliability applications in real-life deployment scenario. This will lead to under-resource provisioning with potential performance degradation. Thus, the system design in average sense is not suitable, {as it leads to excessively high error rates for high-reliability applications}. This justifies the consideration of statistical guarantee on low-latency and high reliability system design, as reliability of several $9$'s will be required for several applications in coming days.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.45\linewidth}
\centering
\input{./tikz_figs/topt_avg_eps.tex}
\caption{Comparison as a function of $\epsilon$.}
\label{fig:t_opt_vs_avg_lat}
\end{subfigure}
\begin{subfigure}[b]{0.45\linewidth}
\centering
\input{./tikz_figs/topt_avg_rho.tex}
\caption{Comparison as a function of $\varrho_{\text{th}}$.}
\label{fig:t_opt_vs_avg_rho}
\end{subfigure}
\caption{Performance comparison of the proposed framework with average sense design for Case 1 with $f_R = 5$ GHz.}
\label{fig:comparison_with_expected_sense}
\end{figure}
\begin{remark}
The system design in average sense leads to under-provisioning and a potential performance degradation, which may have severe consequences in ultra-low latency and high reliability applications.
\end{remark}
\section{Concluding Remarks}\label{sec:conc}
In this paper we have introduced a framework to analyze the closed-loop latency of the teleoperation system, where the command data from the HO and sensing data from the robot are exchanged over a wireless connection through a BS with MEC capabilities.
The sensing data is compressed before transmitting it to the BS where it decompressed first to get the original raw data.
The high-level command from the HO and the sensing data from the robot are processed by the BS to extract the low-level command for the robot and the feedback signal, respectively, and then sent to the robot, and HO, respectively.
We have analyzed the closed-loop latency, which is found to be a sum of several RVs. Thus, the closed-loop latency is also an RV, which demands for statistical guarantee (termed as reliability) on the closed-loop latency.
We obtained upper and lower bounds to its distribution, and formulated an optimization problem to control the transmission rate and compression ratio.
We then investigated different trade-offs in the achievable performance in terms of latency, link outage, and transmission reliability: the decision on whether to compress mostly depends on the computational capability of the robot and link outage probability. We have also observed the shortcomings of the design approaches that only consider the expected value of the latency, disregarding worst-case outcomes.
Further work includes investigations of the age of the closed-loop of the teleoperation system from the information freshness perspective. Another interesting direction is the consideration of heterogeneous sensing data having different compression profiles.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 1,355
|
{"url":"http:\/\/www.koreascience.or.kr\/article\/ArticleFullRecord.jsp?cn=HGKGB6_2015_v14n6_77","text":"A Study on Automated Multi-Channel Combination System for the Closest Target Weight\n\nTitle & Authors\nA Study on Automated Multi-Channel Combination System for the Closest Target Weight\nAhn, Yong-Woo; Ban, Kap-Soo;\n\nAbstract\nThis paper is a study of the functions required for the system to quantify the closest target weight by combining several random weights such as chips, snacks, fruits, and vegetables. The multi-head weigher is designed for high-performance applications requiring increased production rates and tight accuracy tolerances. This combination system has 12 heads considered in the form of a rectangular array of $\\small{2{\\times}6}$ or $\\small{3{\\times}4}$. Channel combination can usually occur between 1 and n, and the frequency was the highest with two or three combinations. Experimental result of a combination system for a total target weight was measured at the range from 100g to 500g by increments of 50g, and the average success rate was about 70%. The average elapsed time was about 1.7 seconds, which means it can be used for the packaging of agricultural products with a variety of items.\nKeywords\nMulti-head;Weighing Machine;Computerized Combination System;Total Target Weight;Unit Weighing Scale;\nLanguage\nKorean\nCited by\nReferences\n1.\nKeraita, J. N. and Kim K. H., \"A Study on the Optimum Scheme for Determination of Operation Time of Line Feeders in Automatic Combination Weighers,\" J. Mech. Sci. Technol., Vol. 20, No. 10, pp. 1567-1575, 2006.\n\n2.\nHoffmann, K. (2001), \"Applying the Wheatstone Bridge Circuit,\" Retrieved 1. Oct., 2015, from http:\/\/www.hbm.com\/fileadmin\/mediapool\/hbmdoc\/technical\/s1569.pdf.\n\n3.\nHoffmann, K. (2012), \"An Introduction to Stress Analysis and Transducer Design using Strain Gauges,\" Retrieved 1. Oct., 2015, from http:\/\/www.kk-group.ru\/help\/Strain_Gauge_Measurements_Book_2012_01.pdf.\n\n4.\nLee J. H. and Lee W. R., \"A Study on the Manufacturing of a High-Efficiency Load Cell Using a Single Surface Design\" J. Korean Soc. Manuf. Technol. Eng., Vol. 19, No. 6, pp. 724-730, 2010.\n\n5.\nKim J. O., Ko, J. B. and Park, H. S., \"A Study on the Measurement of Bending Constraint Force of STS304 Thin Plate Using the Load Cell,\" Trans. Korean Soc. Mach. Tool Eng., Vol. 16, No. 6, pp. 86-93, 2007.\n\n6.\nHan D. S., Ha J. M., Han G. J., \"Creative Design of Large-Angle Pin Type Load-cell for Overload Limited of a Movable Crane,\" J. Korean Soc. Manuf. Process Eng., Vol. 9, No. 1, pp. 35-41, 2010.\n\n7.\nLee D. W., Park M. H., Lee G. G., Kin I. H., Lee S. S., \"Development of the Pin Type Load-cell Using Strain Gauge,\" J. Korean Soc. Manuf. Process Eng., Vol. 13, No. 4, pp. 75-82, 2014.\n\n8.\nLim M. R., \"A Study on Development of Combination Weigher,\" A Thesis for a Master, Yeungnam University, Republic of Korea, 2002.\n\n9.\nRhee, H. W., \"Experimental Evaluation of Fatigue Threshold for SA-508 Reactor Vessel Steel,\" J. Korean Soc. Manuf. Process Eng., Vol. 11, No. 4, pp. 160-167, 2012.","date":"2018-08-19 23:20:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 2, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.43682625889778137, \"perplexity\": 5672.150272861207}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221215404.70\/warc\/CC-MAIN-20180819224020-20180820004020-00171.warc.gz\"}"}
| null | null |
var hashmerge = require('hashmerge');
var readYaml = require('read-yaml');
var shelljs = require('shelljs');
var shelljs_global = require('shelljs/global');
var web3 = require('web3');
var express = require('express');
var compression = require('compression');
var commander = require('commander');
var wrench = require('wrench');
var python = require('python');
var syncMe = require('sync-me');
var methodmissing = require('methodmissing');
var jasmine = require('jasmine');
var Tests = require('./test.js');
var Blockchain = require('./blockchain.js');
var Deploy = require('./deploy.js');
var Release = require('./ipfs.js');
var Config = require('./config/config.js');
var Compiler = require('./compiler.js');
Embark = {
init: function() {
this.blockchainConfig = (new Config.Blockchain());
this.compiler = (new Compiler(this.blockchainConfig));
this.contractsConfig = (new Config.Contracts(this.blockchainConfig, this.compiler));
},
tests: function(contractFiles) {
return new Tests(this.contractsConfig, contractFiles);
},
startBlockchain: function(env, use_tmp) {
var chain = new Blockchain(this.blockchainConfig.config(env));
chain.startChain(use_tmp);
},
deployContracts: function(env, contractFiles, destFile) {
this.contractsConfig.init(contractFiles);
var deploy = new Deploy(env, contractFiles, this.blockchainConfig.config(env), this.contractsConfig);
deploy.deploy_contracts(env);
return deploy.generate_abi_file(destFile);
},
release: Release
}
module.exports = Embark;
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 7,798
|
Die Felsenmühle ist eine Mühle innerhalb des Gebiets der rheinland-pfälzischen Ortsgemeinde Neuleiningen. Mittlerweile fungiert sie als Ausflugslokal und Gästehaus.
Lage
Die Mühle befindet sich am Eckbach, etwa 800 Meter bachabwärts vom Eckbachweiher und gehört zum Weiler Neuleiningen-Tal.
Aufbau
Sie besteht aus dem Haupthaus im Norden, einem Wohnflügel im Osten, einer großen alten Scheune im Süden und dem Mühlenflügel im Westen; dazwischen erstreckt sich ein gepflasterter Innenhof. Das Erdgeschoss des Haupthauses liegt ein Stockwerk höher als der Hof, der über eine mittig angebrachte Doppeltreppe erreicht wird.
Geschichte
Die Felsenmühle wurde 1490 erstmals genannt. Mitte des 18. Jahrhunderts wurde die Felsenmühle durch den Müller Matthias Geißler, den Eigentümer der Obermühle, ersteigert, nachdem er zu einer List gegriffen hatte: Statt – wie seine Vorgänger – sein Brauchwasser über einen parallel zum Eckbach verlaufenden Kanal dem Betreiber der Felsenmühle zu überlassen, leitete Geißler es direkt in den Eckbach ab. Da dieser 50 m südlich der Felsenmühle vorbeifließt, war die Mühle trockengelegt, so dass ihr die Existenzgrundlage genommen war. Geißler erwarb sie, um sie anschließend selbst zu betreiben. 1749 erhielt er die Genehmigung für einen Weinausschank in der Mühle. Im 19. und 20. Jahrhundert diente sie verschiedenen Zwecken: Glasurherstellung für die damalige Steingutfabrik Jacobi, Adler & Co. in der Obermühle, dann Bierausschank, im Zweiten Weltkrieg schließlich Gefangenenlager.
Seit 1994 war sie wiederum Gastwirtschaft, in der zusätzlich eine Weinstube und ein Hotel garni betrieben wurden. In der Gaststube konnte ein riesiges unterschlächtiges Wasserrad besichtigt werden. Nachdem 2004 der Inhaber aus Altersgründen die Anlage schließen musste, stand sie leer und erlitt Einbruch- und Frostschäden. Nach Renovierung ist sie seit Sommer 2007 wieder geöffnet.
Anbindung
Die Mühle befindet sich entlang des Eckbach-Mühlenwanderwegs.
Weblinks
https://www.felsenmuehleneuleiningen.de/
Neuleiningen
Umgenutztes Bauwerk im Landkreis Bad Dürkheim
Wassermühle in Rheinland-Pfalz
Ersterwähnung 1490
|
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"redpajama_set_name": "RedPajamaWikipedia"
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| 4,694
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{"url":"https:\/\/www.greencarcongress.com\/2009\/07\/posco-sk-20090730.html","text":"## South Korea\u2019s POSCO and SK Energy to Develop Coal to Synthetic Natural Gas\n\n##### 30 July 2009\n\nSouth Korea-based POSCO, a global steel producer, and SK Energy Co., Ltd., an energy and petrochemical producer, have signed a memorandum of understanding with the South Korean government to develop coal conversion technologies, including synthetic natural gas (SNG) produced through a coal-to-gas conversion process utilizing low-quality coal.\n\nDomestic production of synthetic natural gas is expected to substantially reduce Korea\u2019s dependence on imported liquefied natural gas.\n\nPOSCO is planning to invest approximately 7.8 billion won (US$6.3 million) for research and development of coal conversion technologies, with SK Energy and the Korean government planning to contribute additional funding in the amounts of 17.2 billion won and 25 billion won (US$14 million and US\\$20 million), respectively.\n\nPOSCO plans to build a coal-to-SNG plant by 2013 with an annual production capacity of 500,000 tons.\n\nAnything new here. Wasn't that done, on a very large scale, almost 100 years ago? Cities were supplied with NG derived from coal in the early 1900's.\n\n\"100 years ago?\"\n\nIt was called 'town gas' and it was a mixture of the calorific gases: hydrogen, carbon monoxide, methane and volatile hydrocarbons, with small amounts of noncalorific gases - carbon dioxide and nitrogen - as impurities.\n\nIf you are going to make CH4 methane out of coal, you are going to end up with one heck of a lot of carbon left over. You could combine it with H2 from solar electrolysis and make more methane, sequester the carbon or find some use for it, but you have to do something with it.\n\nThe US has a coal to methane (natural gas) factory in operation in North Dakota. They produce ammonia and other by products as well. Interestingly enough they sell much of their CO2 to Canadian oil fields nearly 300 miles away.\n\nThere is no reason to require doing anything with the CO2. CO2 is not evil. Even plants, bacteria and animals produce CO2. Interestingly enough composting is a very large producer of waste CO2 as well as wasting the energy. Anaerobic digestion is used in Europe to save energy and lower CO2 release. Car owners are not required to even use the smallest highest efficiency engines for the transportation they do.\n\nIn spite of the known relationship of higher CO2 emissions per mile at high speeds, more efficient road speeds are not not required or even announced.\n\nPeople are allowing themselves to believe that power companies and coal companies are responsible for the CO2 they release rather than the consumers of the power. Many power companies can implement ZERO carbon power right now for people who wish to pay a higher price for it.\n\nThey can also implement wind power for those who want unsteady power. The already turning gigawatt turbines are required to fraudulently hide the unreliability of wind power whose representatives claim falsly that no storage is needed. Norsk Hydro and braun coal generators of Germany and Denmark hide the unreliability of the many windturbines in Denmark and Germany.\n\nPeople are allowed to take as many plane flights as they wish and live as far away from where they work as they want. They are also allowed to use the largest houses they want with the largest heating bills and air-conditioning bills.\n\nLarge and small buildings are not required to use cogeneration for heating and electrical generation to reduce CO2 use.\n\nKorea uses considerable nuclear power instead of coal and are certainly entitled to use as much coal for methane production as they would have for electrical production. Direct current transmission of electricity all the way to the home with buried cables is economically possible and efficient in a mass market with millions of units. Old computer power supplies can tap into 300 volt DC steel cables to provide 12 volts at about 100 watts for the poorest communities. Heat pumps like ECOcute can provide hot water for heat so no gas needs to be made or burned at all. Where a flame is desired, small amounts of hydrogen can be made.\n\nThe US should also devote its coal to liquid and gaseous fuel production and install fairly quick to build with local industry CANDU power plants.\n\nNuclear heat may be the cheapest heat except conservation, but cogeneration is the quicker than nuclear to install.\n\nPeople who require million years of proven safe storage for radioactive materials and drive cars as safe as they possibly can have no concept of how low a danger nuclear power presents compared to the safest driving. Most of them do not know that food, people, plants and animals have always been radioactive. ..HG..\n\nI would say that the coal fields of Montana could sell the CO2 to oil fields that are in decline, with the proper investments in pipelines.\n\nI am not saying CO2 is evil, you said that, but if you do not do something with it and just vent it to the atmosphere there will be lots of angry people around the world. I would just as soon avoid all of that if possible.\n\nThe comments to this entry are closed.","date":"2023-02-07 18:05:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.23373174667358398, \"perplexity\": 2543.0441890698153}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500628.77\/warc\/CC-MAIN-20230207170138-20230207200138-00098.warc.gz\"}"}
| null | null |
Q: How do I find moderators on this site? There are users listed under the "Users" tab. How do I know who the moderators are? Is there any specific thing to look for on a user, I mean a icon or some other sign? How do I contact a moderator? Is the a way other than using "flag" link?
A: The moderators are listed on the "moderators" tab under the "Users" page.
A: How do I know who the moderators are?
There's a moderators tab on the users page that lists them all.
Is there any specific thing to look for on a user, I mean a icon or some other sign?
Moderators have a diamond (♦) next to their names, in posts and comments:
How do I contact a moderator? Is there a way other than using "flag" link?
You generally should just use the "flag for moderator attention" button. You can also post on the site's meta (linked from the top bar), but that is obviously public and can be seen by other users on the site. There is no other built-in way to contact moderators, but they often list e-mail addresses on their profiles you can use if absolutely necessary. You can contact the site developers via the "contact us" link at the bottom of any page, but that's not the same group.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 1,123
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\section{Introduction}
One of the most fascinating features of Einstein's theory of General Relativity (GR) consists in the fact that spacetime may be curved and topologically nontrivial, describing intriguing objects like black holes and wormholes. Black hole spacetimes appear under rather natural conditions in GR, and they are expected to form in nature, for instance, by the collapse of sufficiently massive stars at the end of their life. Furthermore, there is by now compelling evidence for their existence in our Universe which has recently been reinforced by the observation of gravitational waves from binary black hole mergers~\cite{Abbott:2016blz} and the first image of the shadow of the supermassive black hole in the center of the galaxy M87~\cite{Akiyama:2019cqa}. In contrast to this, the occurrence of wormholes (\footnote{In this article, when talking about wormholes, we always refer to \emph{traversable} Lorentzian wormhole spacetimes in a metric theory of gravity.}) is much more speculative, and so far, there is no observational evidence for the existence of such structures. From the theoretical point of view, there are important constraints, such as the topological censorship theorem~\cite{jFkSdW93}. This theorem implies that asymptotically flat, globally hyperbolic wormhole spacetimes (including those relevant to this paper whose Cauchy surfaces have topology $\mathbb{R}\times S^2$, representing a throat connecting two asymptotically flat ends) require the existence of ``exotic'' matter to support the throat, that is, they require matter whose stress-energy-momentum tensor violates the (averaged) null energy condition. Intuitively, the need for exotic matter can be understood by the fact that a light bundle that traverses a wormhole throat must focus as it approaches the throat, but then must expand again as it moves away from the throat, which is opposite to the focusing effect for light due to ordinary matter~\cite{HawkingEllis-Book}. On the other hand, it has also been shown that an infinitesimally small quantity of matter violating the averaged null condition is sufficient to support the throat~\cite{mVsKnD03}. This leads to the hope that quantum effects may give rise to a semiclassical theory in which wormhole spacetimes are allowed, in a similar way to how quantum effects (Hawking radiation) induce black hole evaporation although an area decrease of the event horizon is forbidden in classical GR with matter fields satisfying the null energy condition~\cite{HawkingEllis-Book}. Nevertheless, it remains to be seen whether or not such effects are strong enough to give rise to a traversable wormhole throat of macroscopic size~\cite{eFrW96}.
Instead of invoking quantum effects, an alternative way to violate the null energy condition (which has received important motivation from cosmology, see for example~\cite{Caldwell:1999ew}) is the consideration of phantom scalar fields, i.e. scalar fields that have a negative kinetic energy (see for instance~\cite{Bro2018} and references therein). Due to this property, such fields may lead to gravitational repulsion, and hence induce interesting effects like the accelerated expansion in the Universe, universes with no particle horizon~\cite{dFmGlP19} or the ability of supporting a wormhole throat~\cite{hE73,kB73}. On the other hand, the presence of unbounded negative kinetic energy might cast doubt on the possibility that any stationary solution found in this theory could ever be stable. (\footnote{However, the presence of unbounded negative kinetic energy by itself does not imply that any stationary solution in the theory is \emph{necessarily} unstable. For example, it turns out that the Minkowski spacetime is nonlinearly stable in Einstein theory minimally coupled to a massless scalar field irrespectively of the sign of the gravitational coupling constant (see the comments and references in appendix B.5 of~\cite{mDiR08}).}) Therefore, a pressing question regarding the relevance of static wormhole solutions in such theories (or other GR theories involving exotic matter fields) is their dynamical stability under small perturbations.
The most widely studied wormhole models (including those analyzed in the present article) are based on static, spherically symmetric spacetimes in which the world sheet of the throat consists of spheres of minimal area~\cite{mMkT88}. Within the context of phantom scalar fields, many such solutions have been found; the simplest ones are obtained for a real scalar field and are due to pioneering work by Ellis~\cite{hE73} and by Bronnikov~\cite{kB73}.
Since then, these solutions have been generalized to arbitrary dimensions~\cite{jEtZ08,Torii}
and to the following supporting fields: a scalar with a self-interaction potential~\cite{Dzhunushaliev:2008bq,kBrKaZ12}, a complex phantom scalar~\cite{Dzhunushaliev:2017syc}, a family of conventional and/or phantom scalars~\cite{hSsH02,oStZ12,Carvente:2019gkd}, a phantom scalar and an electromagnetic field~\cite{jGfGoS09c}, and, very recently, a k-essence scalar \cite{kBjFdR2019}. For the linear stability analysis of many of these solutions, see Refs.~\cite{jGfGoS09a,jGfGoS09c,oStZ12,kBjFaZ11,kBrKaZ12,fCfPlP19,kBjFdR2019}. All these studies conclude that the static, spherically symmetric wormhole solutions are linearly unstable, with numerical simulations~\cite{hSsH02,jGfGoS09b,aDnKdNiN09} revealing that the throat either collapses to a black hole or expands on timescales comparable to the light-crossing time of the radius of the throat. Therefore, finding a static, spherically symmetric wormhole solution in GR with exotic matter which can be shown to be linearly stable (or unstable with a large timescale associated with all the unstable modes) remains a challenging open problem. (\footnote{See also~\cite{pKbKjK11} for the construction of static, spherically symmetric wormholes in Einstein-dilaton-Gauss-Bonnet theory, a modified gravity theory, which does not require exotic matter. However, a careful stability analysis has recently revealed that these solutions are linearly unstable as well~\cite{mCrKaZ18}.})
In this work, we focus on GR minimally coupled to a single, real phantom scalar field $\Phi$ with an arbitrary self-interaction potential $V(\Phi)$ and provide a general, gauge-invariant framework to analyze the linear stability of static, spherically symmetric wormhole solutions in these theories; the latter is tested in specific applications.
In order to clarify which are the novelties of the paper, it is necessary to sketch
the previous state of the art in this area. Linearized perturbations of wormhole
solutions of the Einstein's equations have been previously discussed, even in a gauge-invariant language.
However, most of the previous approaches are based
on fixing the radial coordinate and deriving a linearized wave equation for perturbations of the scalar field;
due to the fact that the radial coordinate has a critical point at the throat,
the effective potential appearing in this wave equation is necessarily \emph{singular} at the throat.
As explained in~\cite{jGfGoS09a} (see also~\cite{kBjFaZ11}) this yields an artificial (mirrorlike)
boundary condition at the throat which prevents perturbations from traversing the wormhole.
This artificial boundary condition effectively restricts the class of physically admissible perturbations,
and as it turns out, the unstable modes associated with the wormhole are precluded from this class,
leading to the erroneous conclusion that the wormhole is linearly stable.
To overcome these problems, a method for transforming the singular wave equation
to a regular one was introduced in~\cite{jGfGoS09a} to treat the linearized perturbations
of the Ellis-Bronnikov wormhole; this approach was subsequently generalized
and referred to as ``S-deformation method'' in~\cite{kBjFaZ11}. Both \cite{jGfGoS09a}
and \cite{kBjFaZ11} refer to $(3+1)$- dimensional spacetimes; higher dimensional
extensions were considered in \cite{Torii}, where the
reflection-symmetric Ellis-Bronnikov wormhole solution was generalized
to any spacetime dimension $d+1$ (with $d \geqslant 3$) and its linear
stability analysis was performed, using again the S-deformation method
to overcome singularity problems at the throat and eventually showing that the wormhole
under consideration is unstable in any dimension.
We are now ready to describe the novelties of the present paper.
Here we work in spacetime dimension $3+1$, in
the framework already outlined
(a phantom scalar with self-interaction minimally coupled
to gravity, the spherically symmetric wormhole solutions arising from this
setting and their linear stability analysis). Our first result
is the derivation of a coupled, $2\times 2$ linear wave system subject to a constraint,
describing the linearized dynamics of time-dependent perturbations of such solutions in terms of two gauge-invariant
linear combinations of the linearized metric coefficients and of the scalar field;
a key feature of this system is that
it is \emph{regular} at the throat, provided the scalar field does not have a critical point there.
The second result of our work is that,
provided a nontrivial time-independent solution of the coupled $2\times 2$ system is known,
it is possible to decouple the system, obtaining a single wave equation for an appropriate,
gauge-invariant linear combination of the perturbed fields,
from which all other perturbations can be reconstructed;
in most situations, such a time-independent solution can be found by
varying the parameters of a known family of static wormhole
solutions. The above two results provide a general frame for spherically
symmetric wormholes and their linear stability analysis, alternative
to the S-deformation approach of \cite{jGfGoS09a} \cite{kBjFaZ11} \cite{Torii}:
no S-deformation of the linearized perturbation equations is necessary in the approach
of this paper, since there is no singularity to be eliminated.
For the Ellis-Bronnikov wormhole, we show that the master equation obtained by our method agrees precisely with the one obtained in~\cite{jGfGoS09a} by the S-method. Furthermore, we show that our gauge-invariant method for obtaining a master equation through the decoupling of the $2\times 2$ system also works for wormhole solutions whose stability has not been addressed so far. As an explicit example, we consider a static, spherically symmetric Anti de Sitter (AdS)-type wormhole which connects two asymptotic AdS ends (this is a special case of a family of static solutions of the Einstein-scalar equations derived by Bronnikov and Fabris in~\cite{kBjF06}\cite{Bro2018}); we prove that the above AdS wormhole is linearly unstable, a fact that we presume to be a third novelty of the present work.
Finally, in this paper we provide a detailed analysis for the behavior of the solution of the master equations in both the Ellis-Bronnikov and the AdS case, based on a rigorous spectral analysis of the Schr\"odinger operator appearing therein. A negative eigenvalue of the Schr\"odinger operator gives rise to a pair of modes, one exponentially growing and the other one exponentially decaying with respect to the time variable;
a positive eigenvalue gives rise to a pair of oscillating modes, while a positive energy level
lying in the continuous spectrum gives rise to a pair of non-normalizable oscillating modes,
corresponding to generalized eigenfunctions of the Schr\"odinger operator.
If zero is an eigenvalue it gives rise to a pair of normalizable modes,
one of them constant and the other one linearly growing with time.
We show that in the Ellis-Bronnikov case, the solution can be expanded in terms of an exponentially growing, an exponentially decaying, a constant, a linearly growing mode
and a continuum of oscillators associated with non-normalizable modes.
In contrast to this, in the AdS case the spectrum is a pure point spectrum giving rise to an exponentially growing, an exponentially decaying, and to an infinite, discrete set of oscillating normalizable modes. This is due to the Dirichlet-type boundary conditions imposed at the AdS boundary, which give rise to a regular Sturm-Liouville problem.
The AdS wormhole has a de Sitter (dS) analog which, however, presents horizons; to go beyond the horizons it is necessary to consider a Kruskal-type extension of the dS wormhole spacetime, which, however, is nonstatic and thus is outside the mainstream of the paper. In any case, in the final part of the paper we discuss the above issues and also present a partial result of linear instability, concerning the static part of the wormhole spacetime (we think this is another novelty of this article, foreshadowing future developments).
The article is organized as follows. In section~\ref{Sec:SphSymFEQ} we specify our metric ansatz, make a few general comments regarding the coordinate conditions that will be relevant in this work and derive the field equations for a spherically symmetric, time-dependent configuration. In section~\ref{Sec:Static} we mainly discuss two static wormhole solutions that will serve as examples and applications for our perturbation formalism and stability analysis: the Ellis-Bronnikov solution and the previously mentioned wormhole between two AdS universes. In the same section we spend a few words on the dS analog of this wormhole, to be reconsidered in the final part of the article. In section~\ref{Sec:LinearPerturbation} we derive the relevant set of linearized equations in a gauge-fixed setting in which the scalar field is held fixed. In section~\ref{Sec:GaugeInvariant} we introduce a set of combinations of the linearized fields which are invariant with respect to infinitesimal coordinate transformations, and the linearized field equations are cast into a constrained wave system for two of these gauge-invariant fields. In section~\ref{Sec:Decoupling} we show how to decouple this wave system, provided a static solution of the linearized field equations is available, in which case a single master wave equation is obtained. This method is then applied to the examples of section~\ref{Sec:Static}, and it is shown that in each case the associated Schr\"odinger operator possesses a unique bound state with negative energy, implying that these wormholes are linearly unstable. In section~\ref{Sec:Spectral} we provide a detailed discussion on the spectral decomposition of the Schr\"odinger operator and the corresponding master equations (based on rigorous techniques from functional analysis) and contrast the Ellis-Bronnikov case with the one of the AdS wormhole. In section~\ref{Section:dS} we describe the dS wormhole, including the nonstatic extension beyond the horizons of its spacetime; we also derive a linear instability result concerning the static part of this spacetime. Conclusions, limitations and possible future applications of our method are given in section~\ref{Sec:Conclusions}. Technical details regarding the spectral theory of Schr\"odinger operators are given in the appendices.
Throughout this work, we use the signature convention $(-,+,+,+,)$ and choose units in which $c=1$, $\hbar=1$.
\section{Spherically symmetric field equations and background}
\label{Sec:SphSymFEQ}
We consider a four-dimensional spacetime $(M,{\bf g})$ in which the gravitational field ${\bf g}$ is minimally coupled to a massless phantom scalar field $\Phi$, that is, a scalar field with the reversed sign in its kinetic term that self-interacts according to a potential $V(\Phi)$. The action functional of this system is
\[S[\mathbf{g},\Phi]:=\int\bigg(\frac{R}{2\kappa}
+\frac{1}{2} \nabla^\mu\Phi\cdot \nabla_\mu \Phi -V(\Phi)\bigg)dv\,, \]
where $\kappa = 8\pi G$ is the usual coupling constant while $R$ and
$dv=\sqrt{|\textnormal{det}(g_{\mu\nu})|}\prod\limits_{\mu=0}^3 dx^{\mu}$ are the scalar curvature and the volume element associated with the metric $\mathbf{g}$. The corresponding field equations are
\begin{eqnarray}
R_{\mu\nu} &=& \kappa\left[ -\nabla_\mu\Phi \cdot \nabla_\nu\Phi + V(\Phi) g_{\mu\nu} \right]\,,
\label{Eq:Einstein}\\
0 &=& \nabla^\mu\nabla_\mu\Phi + V'(\Phi)\,,
\label{Eq:KleinGordon}
\end{eqnarray}
with $\nabla_\mu$ and $R_{\mu\nu}$ denoting the covariant
derivative and Ricci tensor, respectively, associated with ${\bf
g}$.
In this work, we focus on spherically symmetric spacetimes $(M,{\bf g})$ of the form $M = \tilde{M}\times S^2$ with metric
\begin{equation}
{\bf g} = -\alpha(t,x)^2 dt^2 + \gamma(t,x)^2 \left( dx + \beta(t,x) dt \right)^2
+ r(t,x)^2\left( d\vartheta^2 + \sin^2\vartheta\; d\varphi^2 \right)\,,
\label{Eq:SphericalMetric}
\end{equation}
which, in a general spherically symmetric coordinate system $(t,x,\vartheta,\varphi)$, is parametrized in terms of the four functions $\alpha,\beta,\gamma,r$ on the two-dimensional manifold $\tilde{M}$. Of course, the number of these functions can be reduced from four to two by an appropriate choice of the coordinates $(t,x)$ on $\tilde{M}$. There are several ``natural'' choices one can make. For example, given a smooth function $f: \tilde{M}\to \mathbb{R}$ with the property that its gradient is everywhere spacelike, one can choose an orthogonal coordinate system $(t,x)$ on $\tilde{M}$ such that $x = f$ and $\beta = 0$. (Likewise, if the gradient of $f$ is everywhere timelike one can choose $(t,x)$ such that $\beta = 0$ and $t = f$.) In particular, if the gradient of the areal radius $r$ is everywhere spacelike one can choose $f = r$ and one is left with the two functions $\alpha$ and $\gamma$ on $\tilde{M}$. Usually, however, the gradient of $r$ is not everywhere spacelike due to the presence of minimal or trapped surfaces, and the resulting coordinate system is only locally defined on $\tilde{M}$.
The field equations~(\ref{Eq:Einstein},\ref{Eq:KleinGordon}) for a spherically symmetric metric~(\ref{Eq:SphericalMetric}) in any gauge such that $\beta = 0$ can be written as
\begin{eqnarray}
\frac{\partial}{\partial t}\left( \frac{\dot{\gamma}}{\alpha} \right)
- \frac{\partial}{\partial x}\left( \frac{\alpha'}{\gamma} \right)
- \frac{\gamma}{\alpha}\frac{\dot{r}^2}{r^2}
+ \frac{\alpha}{\gamma}\frac{r'^2}{r^2}
- \frac{\alpha\gamma}{r^2}
&=& \frac{\kappa}{2}\left[ \frac{\gamma}{\alpha}\dot{\Phi}^2 - \frac{\alpha}{\gamma}\Phi'^2 \right]\,,
\label{Eq:Ev1}\\
\frac{\partial}{\partial t} \left[ \frac{\gamma}{\alpha} r\dot{r} \right]
- \frac{\partial}{\partial x} \left[ \frac{\alpha}{\gamma} r r' \right]
&=& \alpha\gamma\left( \kappa r^2 V(\Phi)-1\right)\,,
\label{Eq:Ev2}\\
\frac{\partial}{\partial t} \left[ \frac{\gamma}{\alpha} r^2\dot{\Phi} \right]
- \frac{\partial}{\partial x}\left[ \frac{\alpha}{\gamma} r^2\Phi' \right] &=& \alpha \gamma r^2 V'(\Phi)\,,
\label{Eq:Ev3}
\end{eqnarray}
with the constraints
\begin{eqnarray}
{\cal H} &:=& \frac{\alpha}{\gamma}\left[ 2\frac{r''}{r} + \frac{r'}{r}\left( \frac{r'}{r} - 2\frac{\gamma'}{\gamma} \right) \right]
- \frac{\gamma}{\alpha}\frac{\dot{r}}{r}\left( \frac{\dot{r}}{r} + 2\frac{\dot{\gamma}}{\gamma} \right)
- \frac{\alpha\gamma}{r^2}
-\frac{\kappa}{2}\left[ \frac{\gamma}{\alpha}\dot{\Phi}^2 + \frac{\alpha}{\gamma}\Phi'^2 \right]
+ \kappa\alpha \gamma V(\Phi) = 0\,,
\label{Eq:Evh}
\\
{\cal M} &:=& 2\frac{\dot{r}'}{r} - 2\frac{\dot{r}}{r}\frac{\alpha'}{\alpha}
- 2\frac{r'}{r}\frac{\dot{\gamma}}{\gamma}
-\kappa\dot{\Phi}\Phi' = 0\,.
\label{Eq:Evm}
\end{eqnarray}
Here and in the following, a dot and a prime refer to partial differentiation with respect to $t$ and $x$, respectively. In the conformally flat gauge, in which $\alpha=\gamma$, Eqs.~(\ref{Eq:Ev1},\ref{Eq:Ev2},\ref{Eq:Ev3}) yield a hyperbolic wave system for the quantities $(\alpha,r,\Phi)$ which is subject to the constraints~(\ref{Eq:Evh},\ref{Eq:Evm}). This system (or slight variants thereof) is suitable for numerical time evolutions, see for instance~\cite{jGfGoS09b}.
\section{Static wormhole solutions}
\label{Sec:Static}
In this section we deal with some examples of static wormhole solutions that have been considered previously in the literature. Most of our attention will be devoted to the Ellis-Bronnikov wormhole connecting two asymptotically flat ends~\cite{hE73,kB73} and to a reflection-symmetric wormhole connecting two AdS ends~\cite{Bro2018}; these will be the main applications of the general technique for linear stability analysis proposed in the present work (sections \ref{Sec:LinearPerturbation}-\ref{Sec:Spectral}). We will also mention a wormhole with ``dS-asymptotics'' \cite{Bro2018}; this case is essentially different from the previous two since it has horizons, a feature which is essentially outside the mainstream of the present work. We will return to this dS wormhole in section \ref{Section:dS}, where we will give a first draft of the treatment of this wormhole, including hints on its linear stability analysis; we hope to reconsider this subject in future works.
\subsection{Ellis-Bronnikov wormhole}
\label{Subsection:Ellis}
Let us assume a zero potential: $V(\Phi)=0$. In the static case, the functions $\alpha$, $\gamma$ and $r$ are $t$-independent and one
can further adjust the coordinate $x$ so that $\alpha\gamma = 1$. In this case, the field equations can be reduced to the three differential equations
$$
[ \alpha^2 r^2]'' = 2\,,\qquad
[ \alpha^2 r r']' = 1\,,\qquad
[\alpha^2 r^2\Phi']' = 0\,,
$$
which arise, respectively, from a recombination of Eqs.~(\ref{Eq:Ev1},\ref{Eq:Ev2},\ref{Eq:Evh}),
from Eq.~(\ref{Eq:Ev2}) and from Eq.~(\ref{Eq:Ev3}).
These can easily be integrated with the result
\begin{equation}
\alpha = \gamma^{-1} = e^{\gamma_1 \arctan {x \over b}}\,,\qquad
r^2 = (x^2 + b^2)\gamma^2\,,\qquad
\Phi = \Phi_1\arctan {x \over b}\,.
\label{Eq:StaticSolutions}
\end{equation}
Here, $b > 0$, $\gamma_1$ and $\Phi_1$ are integration constants, and the Hamiltonian constraint ${\cal H} = 0$ enforces the relation $\kappa\Phi_1^2 = 2(1 + \gamma_1^2)$ (while the momentum constraint ${\cal M}=0$ is obviously satisfied). This solution was obtained long time ago by Ellis~\cite{hE73} and Bronnikov~\cite{kB73} and describes a traversable wormhole whose throat is located at $x = \gamma_1 b$ (see also~\cite{jGfGoS09a} for its physical properties). The reflection-symmetric case $\gamma_1 = 0$ for which $\alpha = \gamma = 1$ results in a particularly simple form of the wormhole metric which has been posed as an exercise in general relativity in the popular article by Morris and Thorne~\cite{mMkT88}.
\subsection{A wormhole connecting two AdS universes}
\label{Subsection:AdS}
We now look for a static solution in the gauge $\alpha \gamma=1$, allowing $V(\Phi)$ to be nonzero. Let us show that a simple solution of this form can be obtained by setting as before
$r = \sqrt{x^2+b^2}$, where $b>0$. With these choices it is easy to show that the
combination (Eq.~(\ref{Eq:Ev2})$+r^2$Eq.~(\ref{Eq:Evh})) is satisfied if $\Phi = \sqrt{2/\kappa}\arctan(x/b)+\Phi_0$
with $\Phi_0$ a constant. With this expression for the scalar field, Eq.~(\ref{Eq:Ev1})
leads to
$$
\alpha=\sqrt{1-K \left(b^2+x^2\right)+M \left(b^2+x^2\right) \arctan\frac{x}{b} + b M x}\,,
$$
where $K$ and $M$ are two constants. The remaining two equations, Eqs.~(\ref{Eq:Ev2},\ref{Eq:Ev3}) (or, alternatively, Eqs.~(\ref{Eq:Ev3},\ref{Eq:Evh})), can be solved by setting
$$
V(\Phi(x))=\frac{ K(b^2+3 x^2) -M \left(b^2+3 x^2\right) \arctan\frac{x}{b} - 3 b M x}{\kappa\left(b^2+x^2\right)}\,.
$$
Choosing, without loss of generality, $\Phi_0=0$, we obtain for $V(\Phi)$
$$
V(\Phi)=\frac{K}{\kappa}\left[ 3 -2 \cos^2\left( \sqrt{\frac{\kappa}{2}}\Phi\right) \right]
- \frac{M}{\kappa}\left\{ 3 \sin \left( \sqrt{\frac{\kappa}{2}}\Phi\right) \cos \left( \sqrt{\frac{\kappa}{2}}\Phi\right)+ \sqrt{\frac{\kappa}{2}}\Phi\left[ 3 -2 \cos ^2\left( \sqrt{\frac{\kappa}{2}}\Phi\right)\right]\right\}\,.
$$
Actually, this solution is exactly the general solution given by Bronnikov and Fabris in~\cite{kBjF06} and reconsidered in the recent survey~\cite{Bro2018} (with some reparametrization of the involved constants).
From here to the end of the paper we make the choice
\begin{equation}
M = 0\,
\label{Eq:ConstantM}
\end{equation}
corresponding to a wormhole metric which is reflection-symmetric with respect to the throat;
hereafter and in most of the paper we also set
\begin{equation}
K\equiv - k^2\,, \qquad (k>0)\,.
\label{Eq:ConstantK}
\end{equation}
With the choices (\ref{Eq:ConstantM}.\ref{Eq:ConstantK}), the solution simplifies to
\begin{equation}
V(\Phi)= -\frac{k^2}{\kappa}
\left[ 3 -2 \cos ^2\left( \sqrt{\frac{\kappa}{2}}\Phi\right) \right]\,,\quad
\alpha = \gamma^{-1} = \sqrt{1 + k^2(x^2+b^2)}\,,\quad
r = \sqrt{x^2 + b^2}\,,\quad
\Phi = \sqrt{\frac{2}{\kappa}}\arctan\frac{x}{b}\,.
\label{Eq:StaticSolutionsPot1}
\end{equation}
In the limit case $b\to0$ we should replace the third equality in~(\ref{Eq:StaticSolutionsPot1}) with $r=x>0$; the corresponding metric describes an AdS universe with cosmological constant $\Lambda = -3k^2$. From now on, we intend ($b>0$, as already stated and)
\begin{equation}
x\in(-\infty , +\infty)\,;
\end{equation}
since $r(x)\sim |x|$ for $x\to \pm \infty$, we can interpret the metric in~(\ref{Eq:StaticSolutionsPot1}) as describing a wormhole connecting two separate asymptotically AdS universes with the same cosmological constant $\Lambda = -3k^2$ and minimal areal radius $b$ at the throat.
For this reason one could call the solution~(\ref{Eq:StaticSolutionsPot1}) an ``AdS-AdS wormhole'';
in the sequel this expression will always be shortened to ``AdS wormhole''.
Let us note that, for $k\to0$, the potential $V(\Phi)$ vanishes and the AdS wormhole (with $b$ fixed) becomes the reflection-symmetric Ellis-Bronnikov wormhole (as in Eq. (\ref{Eq:StaticSolutions}), with $\gamma_1=0$).
For further convenience, we introduce the change of variables
\begin{equation}
t = \frac{s}{2k \sqrt{1+B^2}}\,,\qquad x=\frac{\sqrt{1+B^2}}{k}\tan{\frac{u}{2}}\,,\qquad
B:=b k \,,\qquad
s\in (-\infty,+\infty)\,,\quad u\in (-\pi,\pi)\,,
\label{coordsu}
\end{equation}
so that in the new coordinate system the metric corresponding to the solution~(\ref{Eq:StaticSolutionsPot1}) is transformed into a metric of the form~(\ref{Eq:SphericalMetric}) with $(t,x)$ replaced by $(s,u)$ and
\begin{equation}
\alpha=\gamma = \frac{1}{2 k \cos{\frac{u}{2}}}\,,\qquad \beta = 0\,,\qquad
r= \frac{\sqrt{1+2 B^2-\cos u}}{\sqrt{2} k \cos \frac{u}{2}}\,,\qquad\Phi=\sqrt{\frac{2}{\kappa}}\arctan\left(\frac{\sqrt{1+B^2}\tan \frac{u}{2}}{B} \right)
\label{Eq:StaticSolutionsPot2}
\end{equation}
(of course, $V(\Phi)$ is still as in~(\ref{Eq:StaticSolutionsPot1})). Let us observe that the limits $x\to\pm\infty$, describing the far ends of the wormhole, are equivalent to the limits $u\to\pm \pi$.
\subsection{A dS wormhole}
\label{Subsection:dS}
As anticipated in the first paragraph of this section, we will consider later a dS-type wormhole, differing substantially from the Ellis-Bronnikov and AdS wormholes due to the presence of horizons. For the moment, we just mention that this dS wormhole is the case $M=0$, $K= k^2>0$ of the Bronnikov-Fabris solution described at the beginning of section~\ref{Subsection:AdS}; we will return to this wormhole in section \ref{Section:dS}, after acquiring experience on linearized perturbation theory through the analysis of the AdS case.
\section{Linear perturbations and the $\delta\Phi = 0$ gauge}
\label{Sec:LinearPerturbation}
In the sequel we consider, for an arbitrary potential $V(\Phi)$, a family of static solutions $(\alpha,\gamma,r,\Phi)$ of Eqs.~(\ref{Eq:Ev1}-\ref{Eq:Evm}) (without necessarily assuming the gauge condition $\alpha\gamma = 1$). This family may depend on certain parameters (like the constants $b,\gamma_1$ in subsection~\ref{Subsection:Ellis} or the parameter $B$ in subsection~\ref{Subsection:AdS}). In addition, we consider a (nonstatic) perturbation $(\delta\alpha,\delta\gamma,\delta r,\delta\Phi)$ of this static solution, which is treated by linearizing Eqs.~(\ref{Eq:Ev1}-\ref{Eq:Evm}); let us recall that Eqs.~(\ref{Eq:Ev1}-\ref{Eq:Evm}) assume $\beta = 0$ for the metric~(\ref{Eq:SphericalMetric}), so their linearization corresponds to taking $\delta \beta=0$.
For the particular case in which the potential vanishes ($V = 0$) it can be shown (see e.g. Ref.~\cite{jGfGoS09a}) that the linearized constraint equations $\delta {\cal H} = \delta {\cal M} = 0$ can be integrated. It turns out this is still the case for solutions with a nontrivial potential, yielding the conclusion that
\begin{equation}
\sigma:={\alpha r \over \gamma}\left(
\delta r'-\frac{\alpha'}{\alpha}\delta r
-r'\frac{\delta\gamma}{\gamma}- \frac{\kappa}{2} r\Phi'\delta\Phi
\right)
\label{Eq:Pert1}
\end{equation}
is a constant. This constant indeed describes a zero mode, that is,
a perturbation corresponding to an infinitesimal variation of
the parameters labeling the static solution (see section 3.1 of~\cite{jGfGoS09a}
for more details in the $V=0$ case). Since we are mainly interested in dynamical perturbations (rather than infinitesimal deformations along the static branch in the solution space), we assume from now on that
\begin{equation}
\sigma = 0\,.
\label{Eq:Sigma0}
\end{equation}
For future use, it is advantageous to introduce the quantities (\footnote{This choice of notation is somehow awkward; however the reason for it is to maintain compatibility with the notation used in Ref.~\cite{jGfGoS08}.})
\begin{equation}
\mathcal{D} := \frac{\delta\alpha}{\alpha}\,,\qquad
\mathcal{A} := \frac{\delta\gamma}{\gamma}\,,\qquad
\mathcal{C} := \frac{\delta r}{r}\,.
\label{Eq:DAC}
\end{equation}
Then Eqs.~(\ref{Eq:Pert1}-\ref{Eq:Sigma0}) become
\begin{equation}
\sigma=0\,,\qquad \sigma:={\alpha r^2 \over \gamma}\left[ \mathcal{C}' - \left(\frac{\alpha'}{\alpha} - \frac{r'}{r} \right) \mathcal{C} - \frac{r'}{r} \mathcal{A} - \frac{\kappa}{2}\Phi'\delta\Phi\right]\,;
\label{Eq:Pert1Bis}
\end{equation}
moreover, the linearization of Eqs.~(\ref{Eq:Ev1},\ref{Eq:Ev2},\ref{Eq:Ev3}) and the condition $\sigma=0$ give the following linear system of equations:
\begin{eqnarray}
\frac{\gamma}{\alpha}\ddot{\mathcal{A}} - \frac{\partial}{\partial x}\left( \frac{\alpha}{\gamma} \mathcal{D}' \right)
- \frac{\alpha'}{\gamma}(\mathcal{D} - \mathcal{A})' + 2\frac{\alpha}{\gamma}\frac{r'}{r} \mathcal{C}'
- \frac{2\alpha\gamma}{r^2}(\mathcal{A} - \mathcal{C}) + \kappa\frac{\alpha}{\gamma}\Phi'\delta\Phi' &=& 0\,,
\label{Eq:A}\\
\frac{\gamma}{\alpha}\ddot{\mathcal{C}} - \frac{\partial}{\partial x}\left( \frac{\alpha}{\gamma} \mathcal{C}' \right)
- \frac{\alpha}{\gamma}\frac{r'}{r}(\mathcal{D} - \mathcal{A} + 4\mathcal{C})' + \frac{2\alpha\gamma}{r^2}(\mathcal{A} - \mathcal{C})
-\kappa\alpha\gamma\left[2V(\Phi) \mathcal{A} + V'(\Phi)\delta\Phi \right] &=& 0\,,
\label{Eq:C}\\
\frac{\gamma}{\alpha}\ddot{\delta\Phi}
- \frac{\partial}{\partial x}\left( \frac{\alpha}{\gamma} \delta\Phi' \right)
- 2\frac{\alpha}{\gamma}\frac{r'}{r}\delta\Phi' - \frac{\alpha}{\gamma}\Phi'( \mathcal{D} - \mathcal{A} + 2\mathcal{C})'
- \alpha\gamma\left[ 2V'(\Phi) \mathcal{A} + V''(\Phi)\delta\Phi \right] &=& 0\,.
\label{Eq:Phi}
\end{eqnarray}
All the equations derived so far only assume the orthogonal gauge $\beta = 0$; at the linearized level there is still liberty which is related to the choice of a function $f$ on $\tilde{M}$, as explained in section~\ref{Sec:SphSymFEQ}. One possible choice is fixing the areal radius function $r(x)$ to its background form, such that $\delta r = 0$ and $\mathcal{C} = 0$. Equations~(\ref{Eq:Pert1Bis},\ref{Eq:C}) then allow us to express the metric fields $\mathcal{A}$ and $\mathcal{D}'$ in terms of $\delta\Phi$, and we obtain a master equation for the linearized scalar field $\delta\Phi$ (see e.g.~\cite{Bro2018}).
However, in this article, we are interested in deriving a master equation describing the dynamics of the linear perturbations of any one of the two wormholes in the previous subsections~\ref{Subsection:Ellis} and \ref{Subsection:AdS}. Since these solutions have $dr = 0$ at the wormhole throat, fixing the areal radius function $r(x)$ amounts to forcing the perturbations to vanish at the throat, which from a physical point of view is much too restrictive. At the mathematical level, enforcing the $\delta r = 0$ gauge results in a master equation for $\delta\Phi$ with a potential that is singular at the throat (see~\cite{jGfGoS09a,Bro2018} for more details). On the other hand, while $dr = 0$ at the wormhole throat, we note from Eq.~(\ref{Eq:StaticSolutions}) or Eq.~(\ref{Eq:StaticSolutionsPot1}) that $d\Phi =$ const$ \times dx/(x^2 + b^2)$ is everywhere spacelike, and hence the same will be true for sufficiently small perturbations of the static wormhole solution. As a consequence, we may choose the coordinates $(t,x)$ such that $\Phi$ is given by exactly the same expression as in Eq.~(\ref{Eq:StaticSolutions}) or Eq.~(\ref{Eq:StaticSolutionsPot1}), even for the perturbed spacetime. This implies, in particular, that $\delta\Phi = 0$. In this gauge, Eqs.~(\ref{Eq:Pert1Bis},\ref{Eq:Phi}) reduce to
\begin{eqnarray}
&& \sigma=0,\qquad \sigma:={\alpha r^2 \over \gamma}\left[ \mathcal{C}' - \left(\frac{\alpha'}{\alpha} - \frac{r'}{r} \right) \mathcal{C} - \frac{r'}{r} \mathcal{A} \right]\,,
\label{Eq:Pert3a}\\
&& \mathcal{D}' - \mathcal{A}' + 2\mathcal{C}' + 2 \gamma^2{ V'(\Phi)\over \Phi'} \mathcal{A} = 0\,;
\label{Eq:Pert3b}
\end{eqnarray}
using these equations in order to eliminate $\mathcal{C}'$ and $\mathcal{D}'$ and the static version of Eq.~(\ref{Eq:Ev3}) (from which one can eliminate the unperturbed quantity $\Phi''$), Eqs.~(\ref{Eq:A},\ref{Eq:C}) reduce to
\begin{eqnarray}
\frac{\gamma}{\alpha}\ddot{\mathcal{A}}
- \frac{\partial}{\partial x}\left[ \frac{\alpha}{\gamma}(\mathcal{A}' - 2\mathcal{C}') \right]
+ 2\frac{\alpha}{\gamma}\left( \frac{\alpha'}{\alpha} + \frac{r'}{r} \right) \mathcal{C}'
{- 2\frac{\alpha\gamma}{r^2}(\mathcal{A} - \mathcal{C})}
\nonumber\\
\qquad\qquad + {2\alpha\gamma V'(\Phi) \over \Phi'} \mathcal{A}'
+ 2\alpha\gamma\left[ \gamma^2{V'(\Phi)^2\over\Phi'^2}
+ \left( 3\frac{\alpha'}{\alpha} + 2\frac{r'}{r} \right) {V'(\Phi) \over \Phi'} + V''(\Phi) \right] \mathcal{A} &=& 0\,,
\label{Eq:Pert4}\\
\frac{\gamma}{\alpha}\ddot{\mathcal{C}}
- \frac{\partial}{\partial x}\left[ \frac{\alpha}{\gamma} \mathcal{C}' \right]
- 2\frac{\alpha}{\gamma}\frac{r'}{r} \mathcal{C}'
+2\frac{\alpha\gamma}{r^2}(\mathcal{A} - \mathcal{C})
+ 2\alpha\gamma \left[ \frac{r'}{r} {V'(\Phi)\over \Phi'} - \kappa V(\Phi) \right] \mathcal{A} &=& 0\,,
\label{Eq:Pert5}
\end{eqnarray}
which is still subject to the constraint given in Eq.~(\ref{Eq:Pert3a}).
As a simple example, let us consider the reflection-symmetric subcase of the Ellis-Bronnikov wormhole, already mentioned at the end of the subsection~\ref{Subsection:Ellis}, corresponding to the choice $V = 0$, $\alpha=\gamma=1$ and $r$ as in Eq.~(\ref{Eq:StaticSolutions}). In this case, the difference between Eqs.~(\ref{Eq:Pert4},\ref{Eq:Pert5}), along with Eq.~(\ref{Eq:Pert3a}), gives
\begin{equation}
\label{Eq:MasterEllis}
\ddot{\chi} - \chi'' - \frac{3b^2}{r^4}\chi = 0\,,\qquad \chi := \frac{\mathcal{A} - \mathcal{C}}{r}\,,
\end{equation}
which coincides with Eq.~(15) in Ref.~\cite{jGfGoS08}. We will return to this subcase at the end of section~\ref{Sec:GaugeInvariant}. The generalizations to the non-reflection-symmetric Ellis-Bronnikov wormhole and to the AdS wormhole (subsections~\ref{Subsection:Ellis}, \ref{Subsection:AdS}) will be discussed in section~\ref{Sec:Decoupling}.
\section{Gauge-invariant reinterpretation}
\label{Sec:GaugeInvariant}
In this section we analyze the behavior of the perturbed fields under an infinitesimal coordinate transformation
\begin{equation}
x^a \mapsto x^a + \delta x^a\,,\qquad (x^a) = (t,x)\,,
\label{Eq:InfCT}
\end{equation}
parametrized by a vector field $\mathbf{\delta x} = \delta x^a \partial_a = (\delta t) \partial_t + (\delta x) \partial_x$ on $\tilde{M}$,
and try to rewrite the equations from the previous section in terms of fields which
are manifestly gauge-invariant with respect to these transformations.
Under the transformation~(\ref{Eq:InfCT}) the linear perturbations of the radial part of the metric, $\tilde{g}_{ab} dx^a dx^b := -\alpha^2 dt^2 + \gamma^2(dx + \beta dt)^2$, of the areal radius $r$, and of scalar field $\Phi$ transform according to
$$
\delta \tilde{g}_{ab} \mapsto \delta \tilde{g}_{ab} + \pounds_{\mathbf{\delta x}}\tilde{g}_{ab}\,,\qquad
\delta r \mapsto \delta r + \pounds_{\mathbf{\delta x}} r\,,\qquad
\delta\Phi \mapsto \delta\Phi + \pounds_{\mathbf{\delta x}}\Phi\,,
$$
with $\pounds_{\mathbf{\delta x}}$ denoting the Lie derivative with respect to $\mathbf{\delta x}$.
Parametrizing the metric as in Eq.~(\ref{Eq:SphericalMetric})
and assuming that the background is static, this yields
\begin{eqnarray}
\delta\alpha &\mapsto& \delta\alpha + \alpha'\delta x
+ \alpha \, \delta \dot{t}\,,\\
\delta\beta &\mapsto& \delta\beta + \delta \dot{x}
- \frac{\alpha^2}{\gamma^2} \delta t'\,,\\
\delta\gamma &\mapsto& \delta\gamma + \left( \gamma\delta x \right)'\,,\\
\delta r &\mapsto& \delta r + r'\delta x\,,\\
\delta\Phi &\mapsto& \delta\Phi + \Phi'\delta x
\end{eqnarray}
(where $\delta \dot{t}$, $\delta \dot{x}$ and $\delta t'$ refer to $\frac{\partial}{\partial t}(\delta t)$, $\frac{\partial}{\partial t}(\delta x)$ and $\frac{\partial}{\partial x}(\delta t)$, respectively; similar notations are used hereafter in relation to $\delta \beta$ and $\delta \Phi$).
The following three quantities are invariant with respect to these transformations:
\begin{eqnarray}
A &:=& \frac{\delta\gamma}{\gamma} - \frac{1}{\gamma}\left( \gamma\frac{\delta\Phi}{\Phi'} \right)'\,,
\label{Eq:AGI}\\
C &:=& \frac{\delta r}{r} - \frac{r'}{r}\frac{\delta\Phi}{\Phi'}\,,
\label{Eq:CGI}\\
E &:=& \left( \frac{\delta\alpha}{\alpha} \right)' - \left( \frac{\alpha'}{\alpha}\frac{\delta\Phi}{\Phi'} \right)'
+ \frac{\gamma^2}{\alpha^2} \left( \delta\dot{\beta} - \frac{\delta\ddot{\Phi}}{\Phi'} \right)\,.
\label{Eq:EGI}
\end{eqnarray}
In the particular gauge used in the second half of the previous section,
for which $\delta\beta = \delta\Phi = 0$, it turns out that $A=\mathcal{A}$, $C=\mathcal{C}$ and $E=\mathcal{D}'$, where $\mathcal{A}$, $\mathcal{C}$ and $\mathcal{D}$ are defined by Eq.~(\ref{Eq:DAC}). Therefore, in this gauge, we may replace the quantities $\mathcal{A}$, $\mathcal{C}$ and $\mathcal{D}'$ in Eqs.~(\ref{Eq:Pert3a}--\ref{Eq:Pert5}) with the quantities $A,C,E$ of the present section.
Since the linearized field equations are gauge-invariant, the equations obtained in this way are valid in \emph{any gauge}.\\ Summing up, our gauge-invariant equations are
\begin{eqnarray}
&& \sigma = 0,\qquad
\sigma := \frac{\alpha r^2}{\gamma} \left[
C'+\left({r'\over r}-{\alpha'\over\alpha}\right)C-{r'\over r}A
\right]\,,
\label{Eq:Pert3abis}\\
&&
E - A' + 2C' + 2\gamma^2{V'(\Phi)\over \Phi'} A = 0\,,
\label{Eq:Pert3bbis}\\
&& \frac{\gamma}{\alpha}\ddot{A}
- \frac{\partial}{\partial x}\left[ \frac{\alpha}{\gamma}(A' - 2C') \right]
+ 2\frac{\alpha}{\gamma}\left( \frac{\alpha'}{\alpha} + \frac{r'}{r} \right) C'
{- 2\frac{\alpha\gamma}{r^2}(A - C)}
\nonumber\\
&& \qquad\qquad + {2\alpha\gamma V'(\Phi) \over \Phi'} A'
+ 2\alpha\gamma\left[ \gamma^2{V'(\Phi)^2\over\Phi'^2}
+ \left( 3\frac{\alpha'}{\alpha} + 2\frac{r'}{r} \right) {V'(\Phi) \over \Phi'} + V''(\Phi) \right] A = 0\,,
\label{Eq:Pert4bis}\\
&& \frac{\gamma}{\alpha}\ddot{C}
- \frac{\partial}{\partial x}\left[ \frac{\alpha}{\gamma} C' \right]
- 2\frac{\alpha}{\gamma}\frac{r'}{r} C'
+2\frac{\alpha\gamma}{r^2}(A - C)
+ 2\alpha\gamma \left[ \frac{r'}{r} {V'(\Phi)\over \Phi'} - \kappa V(\Phi) \right] A = 0\,.
\label{Eq:Pert5bis}
\end{eqnarray}
As a simple example, consider again the reflection-symmetric Ellis-Bronnikov wormhole, for which $V = 0$, $\alpha=\gamma=1$ and $r = \sqrt{x^2 + b^2}$, i.e. the same example as the one described at the end of section~\ref{Sec:LinearPerturbation} with Eqs.~(\ref{Eq:Pert3a}-\ref{Eq:Pert5}) yielding Eq.~(\ref{Eq:MasterEllis}). However, now Eq.~(\ref{Eq:MasterEllis}) can be reinterpreted in a gauge-invariant framework where $\chi=(A-C)/r$. The interest of this equation is that it involves only one unknown function $\chi(t,x)$ and reduces the linear stability analysis of this wormhole to the spectral analysis of the Schr\"odinger operator $-d^2/dx^2-3b^2/(x^2 + b^2)^2$. Since this has one negative eigenvalue (see Refs. \cite{jGfGoS08,jGfGoS09a}), one concludes that the wormhole is unstable.
In Ref.~\cite{jGfGoS08} an attempt was made to provide the present gauge-invariant formulation of the field equations in this particular subcase: while the two gauge-invariant quantities $A$ and $C$ were correctly defined, the quantity $D = \delta\alpha/\alpha$ defined in~\cite{jGfGoS08} is only invariant with respect to the restricted set of gauge transformations for which $\delta \dot{t} = 0$. However, in general, this restricted set is not sufficient to achieve both conditions $\delta\Phi = 0$ and $\delta\beta = 0$ simultaneously,
on which the derivation in Ref.~\cite{jGfGoS08} was based.
Finally, let us observe that the the gauge-invariant Eqs.~(\ref{Eq:Pert3abis},\ref{Eq:Pert4bis},\ref{Eq:Pert5bis})
(again in the present subcase $V = 0$, $\alpha=\gamma=1$ and $r = \sqrt{x^2 + b^2}$)
are related to the results presented in~\cite{fCfPlP19}, which are based on the gauge $\delta\alpha = \delta\beta = 0$. For example, Eq.~(\ref{Eq:Pert3abis}), when choosing the gauge $\delta\alpha = \delta\beta = 0$, yields Eq.~(3.9) in~\cite{fCfPlP19}. The analysis of~\cite{fCfPlP19} also yields a final equation similar to Eq.~(\ref{Eq:MasterEllis}), even though it uses a different approach related to the chosen gauge. (\footnote{In~\cite{fCfPlP19}, the variables $x$ and $b$ of the present paper are denoted with $\ell$ and $a$;
the field $\mathscr{R}$ fulfilling the master equation~(3.15) of the cited work is related
to the present gauge-invariant quantities $C$ and $E$ by the relation
$ \frac{\partial^2}{\partial t^2}\left[\mathscr{R}\left(\frac{t}{b},\frac{x}{b}\right)\right]
=\frac{r^2}{b^3}\left(r \ddot{C} - r' E\right)$. Using the linearized field equations, this can also be rewritten as $\frac{\partial^2}{\partial t^2}\left[\mathscr{R}\left(\frac{t}{b},\frac{x}{b}\right)\right] = -\frac{1}{br}(A - C)$, which explains why $\mathscr{R}$ satisfies the same master equation as $\chi$, up to a source term whose second time derivative vanishes.} )
\section{Decoupling of the pulsation equations}
\label{Sec:Decoupling}
After considering the simple example of the reflection-symmetric Ellis-Bronnikov wormhole, let us return to the case of an arbitrary potential $V(\Phi)$. In this section we try to reduce the gauge-invariant equations of section~\ref{Sec:GaugeInvariant} to one involving only one unknown function $\chi(t,x)$ (generalizing the considerations which lead to Eq.~(\ref{Eq:MasterEllis}) in the reflection-symmetric Ellis-Bronnikov subcase). To this purpose we note the following: setting
\begin{equation}
{\cal F} := {A-C\over r}\,,\qquad{\cal G} := {C\over r}
\label{Eq:FandG}
\end{equation}
and performing a lengthy calculation, we can reformulate the system of Eqs.~(\ref{Eq:Pert3abis},\ref{Eq:Pert4bis},\ref{Eq:Pert5bis}) as the hyperbolic system of wave equations
\begin{equation}
\left[ \frac{\partial^2}{\partial t^2}
- \left( \frac{\alpha}{\gamma}\frac{\partial}{\partial x} \right)^2
+
\left( \begin{array}{cc} Y_0 & Y_0 \\ 0 & 0 \end{array} \right)\frac{\alpha}{\gamma}\frac{\partial}{\partial x}
+ \frac{\alpha^2}{\gamma^2} \left( \begin{array}{cc} W_{11} & W_{12} \\ W_{21} & W_{22} \end{array} \right)
\right] \left( \begin{array}{cc} {\cal F} \\ {\cal G} \end{array} \right) = 0\,,
\label{Eq:WaveSystem}
\end{equation}
subject to the constraint
\begin{equation}
{\cal G}' = \left( \frac{\alpha'}{\alpha} - \frac{r'}{r} \right){\cal G} + \frac{r'}{r}{\cal F}\, .
\label{Eq:WaveConstraint}
\end{equation}
Here, the functions $ Y_0$ and $W_{ij}$ are given by the following functions of the background quantities:
\begin{eqnarray}
Y_0 &:=&2\alpha\gamma {V'(\Phi)\over{\Phi'}}\,,
\label{Eq:Y0}\\
W_{11} &:=& \frac{r'}{r}\left( 4\frac{\alpha'}{\alpha} + 3\frac{r'}{r} \right) - 3\frac{\gamma^2}{r^2}
+ Z_{11}\,,\\
W_{12} &:=& 4\frac{\alpha'^2}{\alpha^2} +Z_{12}\,,\\
W_{21} &:=& -4\frac{r'^2}{r^2} + 2\frac{\gamma^2}{r^2} + Z_{21}\,,\\
W_{22} &:=& \frac{r'}{r}\left( -4\frac{\alpha'}{\alpha} + 3\frac{r'}{r} \right) - \frac{\gamma^2}{r^2} + Z_{22}\,,
\end{eqnarray}
where
\begin{eqnarray}
Z_{11} &:=& Z_{12} + \kappa\gamma^2V(\Phi)\,,\\
Z_{12} &:=& 2\gamma^2\left[ \gamma^2{V'(\Phi)^2\over \Phi'^2}
+ \left( 3{\alpha'\over \alpha}+2{r'\over r} \right){V'(\Phi) \over \Phi'} + V''(\Phi) \right]\, ,\\
Z_{21} &:=& 2\gamma^2\left[ -\kappa V(\Phi) + {r'\over r}{V'(\Phi)\over \Phi'} \right]\,,\\
Z_{22} &:=& Z_{21} + \kappa\gamma^2 V(\Phi)\,.
\end{eqnarray}
In deriving the wave system~(\ref{Eq:WaveSystem}) we have used the background equations
\begin{eqnarray}
\frac{r'}{r}\left( 2\frac{\alpha'}{\alpha} + \frac{r'}{r} \right) - \frac{\gamma^2}{r^2}
+ \frac{\kappa}{2}\Phi'^2 + \kappa\gamma^2 V(\Phi) &=& 0\,,
\label{Eq:Background1}\\
\frac{\alpha''}{\alpha}- \frac{\alpha'}{\alpha}\left( \frac{\gamma'}{\gamma} - 2\frac{r'}{r} \right)
+ \kappa \gamma^2 V(\Phi) & = & 0\,,
\label{Eq:Background2}\\
\frac{r''}{r} - \frac{r'}{r}\left( \frac{\gamma'}{\gamma} - \frac{\alpha'}{\alpha}- \frac{r'}{r} \right) - \frac{\gamma^2}{r^2} + \kappa \gamma^2 V(\Phi) & = & 0\,.
\label{Eq:Background3}
\end{eqnarray}
Observe that in the reflection-symmetric Ellis-Bronnikov subcase $V = 0$, $\alpha=\gamma=1$ and $r = \sqrt{x^2 + b^2}$ it follows that $Y_ 0 = W_{12} = 0$, such that the equation for ${\cal F}$ in the system~(\ref{Eq:WaveSystem}) decouples trivially from the remaining ones.
In the following, we describe a general trick which allows one to decouple the constrained wave system~(\ref{Eq:WaveSystem},\ref{Eq:WaveConstraint}). Let us suppose that we know a static solution $({\cal F}_0(x),{\cal G}_0(x))$ of Eqs.~(\ref{Eq:WaveSystem},\ref{Eq:WaveConstraint}) such that ${\cal G}_0(x)\neq 0$ for all $x$. In this case, based on Leibnitz's product rule,
it is not difficult to verify that the field
$$
\tilde\chi := {\cal F} - \frac{{\cal F}_0}{{\cal G}_0}{\cal G}
$$
satisfies the decoupled wave equation
\begin{equation}
\left[ \frac{\partial^2}{\partial t^2}
- \left( \frac{\alpha}{\gamma}\frac{\partial}{\partial x} \right)^2
+ Y_0\frac{\alpha}{\gamma} \frac{\partial}{\partial x}
+ \frac{\alpha^2}{\gamma^2}\tilde{\cal {V}} \right] \tilde\chi = 0\,,
\label{Eq:Master1}
\end{equation}
with the potential
\begin{equation}
\tilde{\mathcal{ V}} = W_{11} - \frac{{\cal F}_0}{{\cal G}_0} W_{21} - 2\frac{r'}{r}\left( \frac{{\cal F}_0}{{\cal G}_0} \right)'
+\frac{\gamma}{\alpha}\frac{r'}{r} Y_0\left( \frac{{\cal F}_0}{{\cal G}_0}+1 \right)\,.
\end{equation}
Let us observe that it is possible to eliminate the first spatial derivative in Eq.~(\ref{Eq:Master1}). Indeed, let us define
\begin{equation}
\label{Eq:ChiTrans}
\chi:={\tilde\chi\over a}\,,\qquad
a(x) = a_0 e^{\int_{x_0}^{x}\frac{Y_0(y)\gamma(y)}{2\alpha(y)}dy }\,,
\end{equation}
where $a_0$ and $x_0$ are two constants. (\footnote{Note that $a$ satisfies
$$
a' = \frac{Y_0 \gamma}{2 \alpha} a\,,\qquad
a'' = \left[ \left(\frac{Y_0 \gamma}{2 \alpha}\right)' + \left(\frac{Y_0 \gamma}{2 \alpha}\right)^2 \right] a\,.
$$
}) Then, it is found that $\chi$ satisfies the wave equation
\begin{equation}
\label{Eq:Master2}
\left[ \frac{\partial^2}{\partial t^2}
- \left( \frac{\alpha}{\gamma}\frac{\partial}{\partial x} \right)^2
+ \frac{\alpha^2}{\gamma^2}{\cal {V}} \right] \chi = 0\,,
\end{equation}
with the potential (\footnote{Here $Y_0'$ can be computed by taking a derivative of Eq.~(\ref{Eq:Y0}) and eliminating $\Phi''$ via the static version of Eq.~(\ref{Eq:Ev3}). })
\begin{eqnarray}
{\cal{V}} &:=& \tilde{\mathcal{ V}}
+ \frac{1}{4}\frac{\gamma^2}{\alpha^2} Y_0^2 - \frac{1}{2}\frac{\gamma}{\alpha} Y_0'\nonumber\\
&=& \frac{r'}{r}\left( 4\frac{\alpha'}{\alpha} + 3\frac{r'}{r} \right) -
\frac{3\gamma^2}{r^2}
+ \gamma^2\left[ 2\gamma^2{V'(\Phi)^2\over \Phi'^2}
+ 4\left( \frac{\alpha'}{\alpha} + \frac{r'}{r} \right) {V'(\Phi) \over \Phi'}
+ V''(\Phi) + \kappa V(\Phi) \right]
\nonumber\\
&+& \left[ 4\frac{r'^2}{r^2} - \frac{2\gamma^2}{r^2} + 2\kappa\gamma^2 V(\Phi) \right]
\frac{{\cal F}_0}{{\cal G}_0}
- 2\frac{r'}{r}\left( \frac{{\cal F}_0}{{\cal G}_0} \right)'\,.
\label{Eq:Potential}
\end{eqnarray}
We refer to Eq.~(\ref{Eq:Master2}) as the \emph{master equation}; this reduces the linear stability analysis to the spectral analysis of the linear, Schr\"odinger-type operator $-\left( \frac{\alpha}{\gamma}\frac{d}{d x} \right)^2 + \frac{\alpha^2}{\gamma^2}\,{\cal {V}}$.
Once the master equation has been solved for the field $\chi(t,x)$, it is possible to reconstruct the gauge-invariant quantities ${\cal F}$ and ${\cal G}$ by integrating the constraint equation~(\ref{Eq:WaveConstraint}). Using the definition of $\chi$ and the fact that $({\cal F}_0,{\cal G}_0)$ satisfy the constraint, one obtains
\begin{eqnarray}
{\cal G}(t,x) &=& {\cal G}_0(x)\int\limits_{x_0}^x \frac{r'(y)}{r(y)}\frac{a(y)}{{\cal G}_0(y)} \chi(t,y) dy\,,\\
{\cal F}(t,x) &=& a(x)\chi(t,x) + \frac{{\cal F}_0(x)}{{\cal G}_0(x)} {\cal G}(t,x)\,,
\end{eqnarray}
and from this one can also reconstruct the gauge-invariant fields $A$ and $C$. Finally, the gauge-invariant field $E$ is obtained from Eq.~(\ref{Eq:Pert3bbis}).
Let us repeat that the above approach requires the knowledge of a static solution $({\cal F}_0(x),{\cal G}_0(x))$ of Eqs.~(\ref{Eq:WaveSystem},\ref{Eq:WaveConstraint}). A general strategy to obtain such a static solution is to make an infinitesimal variation along a static solution $(\alpha,\gamma,r,\Phi)$ of the Einstein-scalar equations with respect to its parameters. Since this linearization of the static solution automatically satisfies the linearized system~(\ref{Eq:Ev1}--\ref{Eq:Evm}), the corresponding gauge-invariant fields $A$ and $C$ (defined by Eqs.~(\ref{Eq:AGI},\ref{Eq:CGI})) fulfill the system~(\ref{Eq:Pert4bis},\ref{Eq:Pert5bis}), provided that we have the vanishing condition~(\ref{Eq:Pert3abis}) for $\sigma$; obviously, under the same condition, the fields $\cal F$ and $\cal G$ associated with $A$ and $C$ represent a static solution of the wave system~(\ref{Eq:WaveSystem},\ref{Eq:WaveConstraint}). In the following we will apply this general strategy to the cases of the Ellis-Bronnikov and AdS wormholes.
\subsection{Perturbed Ellis-Bronnikov wormhole}
\label{SubSec:PertEllis}
The Ellis-Bronnikov solution given in Eq.~(\ref{Eq:StaticSolutions})
can be linearized with respect to the parameters $b$ and $\gamma_1$, which yields
\begin{eqnarray}
\frac{\delta\gamma}{\gamma} &=&
- \left(\arctan\frac{x}{b} \right)\delta\gamma_1
+ \frac{\gamma_1 x}{x^2 + b^2}\delta b\,,\\
\frac{\delta r}{r} &=& -\left(\arctan \frac{x}{b} \right)\delta\gamma_1
+ \frac{b + \gamma_1 x}{x^2 + b^2}\delta b\,,\\
\frac{\delta\Phi}{\Phi'} &=& \frac{\gamma_1}{1 + \gamma_1^2}(x^2 + b^2)
\left(\arctan \frac{x}{b} \right)\frac{\delta\gamma_1}{b} - x\frac{\delta b}{b}\,.
\end{eqnarray}
Introduced into Eqs.~(\ref{Eq:AGI},\ref{Eq:CGI}) this gives rise to the gauge-invariant quantities
\begin{eqnarray}
A &=& -\frac{1}{1 + \gamma_1^2}\left[ \gamma_1
+ \left( 1 + 2\gamma_1\frac{x}{b} \right) \arctan \frac{x}{b} \right]
\delta\gamma_1 + \frac{\delta b}{b}\,,
\label{Eq:AEllis}\\
C &=& -\frac{1 + \gamma_1\frac{x}{b}}{1 + \gamma_1^2}\left(\arctan \frac{x}{b} \right) \delta\gamma_1
+ \frac{\delta b}{b}\, .
\label{Eq:CEllis}
\end{eqnarray}
As explained before, the fields $A$ and $C$ automatically satisfy the system of equations~(\ref{Eq:Pert3abis},\ref{Eq:Pert4bis},\ref{Eq:Pert5bis}). In this case the definition of $\sigma$ in Eq.~(\ref{Eq:Pert3abis}) gives $\sigma=-\delta b\gamma_1-b \delta \gamma_1=\delta(b\gamma_1)$, so that the condition $\sigma=0$ therein holds if
\begin{equation}
\delta b=-{b \delta\gamma_1\over \gamma_1}\,.
\label{Eq:SigmaEllis}
\end{equation}
Inserting Eqs.~(\ref{Eq:AEllis},\ref{Eq:CEllis},\ref{Eq:SigmaEllis}) into the definition~(\ref{Eq:FandG}) of $\cal F$ and $\cal G$ (and omitting the proportionality factor $\delta\gamma_1$), one obtains the following time-independent solution of the constrained
wave system~(\ref{Eq:WaveSystem},\ref{Eq:WaveConstraint}):
\begin{equation}
\left( \begin{array}{cc} {\cal F}_0 \\ {\cal G}_0 \end{array} \right)
:= \frac{1}{r}\left( \begin{array}{c}
\frac{\gamma_1^2}{1 + \gamma_1^2}\left[ 1 + \frac{x}{b} \arctan\frac{x}{b} \right] \\
F(x)
\end{array} \right)\,,\qquad
F(x) := 1 + \frac{\gamma_1}{1 + \gamma_1^2}\left( 1 + \gamma_1\frac{x}{b} \right)
\arctan \frac{x}{b}\,.
\label{Eq:StaticSolutionWaveSystem}
\end{equation}
Note that the function $F: \mathbb{R}\to \mathbb{R}$ is smooth and strictly positive. (\footnote{ For $\gamma_1 = 0$, $F = 1$ and the statement is trivial. When $\gamma_1\neq 0$ one has $F(x)\to +\infty$ for $x\to \pm\infty$, thus $F$ has a global minimum at some $x = x_0$, where $ 0 = (1 + \gamma_1^2) b F'(x_0) = \gamma_1\left[ \gamma_1\arctan(x_0/b) + (b + \gamma_1 x_0) b/(x_0^2 + b^2) \right]$. Eliminating the $\arctan$ term one obtains from this $(1 + \gamma_1^2) F(x_0) = (x_0 - b\gamma_1)^2/(x_0^2 + b^2)$. However, this minimum value must be strictly positive since otherwise $x_0 = b\gamma_1$ which would imply that $(1 + \gamma_1^2) b F'(x_0) = \gamma_1\left(\gamma_1\arctan\gamma_1 + 1 \right)$ which cannot be zero since $\gamma_1\neq 0$.})
Based on these observations, we can apply the general method for decoupling the
wave system~(\ref{Eq:WaveSystem},\ref{Eq:WaveConstraint}), choosing the static solution
$({\cal{F}}_0(x),{\cal{G}}_0(x))$ as in Eq.~(\ref{Eq:StaticSolutionWaveSystem}).
Note that in this case we have $Y_0 = 0$ and can choose $a = 1$ as $V = 0$,
which implies that $\mathcal{ V}=\tilde{\mathcal{ V}}$ and $\chi = \tilde{\chi}$.
One can verify that the function $\frac{\alpha^2}{\gamma^2} \mathcal{ V}$ appearing
in the master equation~(\ref{Eq:Master2}) agrees up to a rescaling with the potential
defined in Eq.~(32) of Ref.~\cite{jGfGoS09a}, denoted therein with $W$; more precisely,
\begin{equation}
\label{Eq:W}
\left(\frac{\alpha^2}{\gamma^2} \mathcal{V}\right)(x) =~{1 \over b^2} W\!\left(\frac{x}{b}\right)\,.
\end{equation}
For future mention let us point out some features of this function,
following from the analysis of $W$ in \cite{jGfGoS09a}.
First of all $\frac{\alpha^2}{\gamma^2} \mathcal{V} : \mathbb{R} \to \mathbb{R}$ is a $C^\infty$ bounded function; moreover, if $\gamma_1 \neq 0$ one has
$\left(\frac{\alpha^2}{\gamma^2} \mathcal{ V}\right)(x) \sim 2 e^{\pm 2 \pi \gamma_1}/x^2$
for $x \mapsto \pm \infty$. In the reflection-symmetric case $\gamma_1=0$, where $\alpha=\gamma=1$, one obtains
\begin{equation}
\mathcal{V}(x) = - \frac{3 b^2}{(x^2 + b^2)^2} = - \frac{3 b^2}{r^4(x)}\,,
\label{Eq:Vsim}
\end{equation}
and the master equation~(\ref{Eq:Master2}) is found to coincide with
Eq.~(\ref{Eq:MasterEllis}).
Let us now sketch some spectral features of the Schr\"odinger operator $-\left({\alpha \over\gamma}{d\over dx}\right)^2 + {\alpha^2\over\gamma^2}\, \mathcal{V}$ appearing in the master equation~(\ref{Eq:Master2}) (which can be regarded as a selfadjoint operator in $L^2(\mathbb{R},{\gamma \over\alpha} dx)$); these features allow one to infer the linear instability of the Ellis-Bronnikov wormhole both for $\gamma_1\neq 0$ and for $\gamma_1=0$. As shown in~\cite{jGfGoS09a}, the \textquotedblleft zero energy'' equation $\left[-\left({\alpha\over\gamma}{d\over dx}\right)^2 + {\alpha^2\over\gamma^2}\, \mathcal{V} \right]\chi_0 = 0$ has a solution
\begin{equation}
\chi_0(x) = \frac{x - b\gamma_1}{r(x) F(x)}\,,
\label{Eq:EBZeroMode}
\end{equation}
which has precisely one zero in the interval $(-\infty,+\infty)$. According to the Sturm oscillation theorem (see for instance~\cite{Weid-Book}, \cite{bS05} and references therein) it follows that for each $\gamma_1$ including $\gamma_1 = 0$, the Schr\"odinger operator in the master equation possesses a single bound state with negative energy. Note that for $\gamma_1 \neq 0$ the function $\chi_0$ decays as $1/|x|$ for large $|x|$, so that it describes a bound state with zero energy, while for $\gamma_1 = 0$ it reduces to $\chi_0(x) = x/\sqrt{1 + x^2}$ which is not normalizable but still has a single zero. The existence of a single bound state with negative energy implies that the master equation~(\ref{Eq:Master2}) for each Ellis-Bronnikov wormhole possesses a unique mode diverging exponentially in time, a fact of course sufficient to infer the instability of the wormhole. In the next section we will give more details on the spectral properties of the Schr\"odinger operator and on the solution of the master equation~(\ref{Eq:Master2}) within a rigorous functional setting, also allowing comparison with the corresponding problem for the perturbed AdS wormhole.
\subsection{Perturbed AdS wormhole}
\label{SubSec:Pertads}
Next, we analyze the AdS wormhole in the coordinate system $(s,u)$, as described by Eq.~(\ref{Eq:StaticSolutionsPot2}) for arbitrary parameters $k,B > 0$, and apply the general framework presented in this section with $(s,u)$ in place of $(t,x)$. Although the static solution formally depends on two parameters $B$ and $k$, it is important to note that $k$ also appears in the potential function $V(\Phi)$ (see Eq.~(\ref{Eq:StaticSolutionsPot1})). However, since we regard the potential to be fixed in our perturbation analysis, we will exclude the possibility of varying $k$. In contrast to $k$, the parameter $B$ is free, and variation of the solution~(\ref{Eq:StaticSolutionsPot2}) with respect to it gives
\begin{eqnarray}
\label{Eq:Variations1}
\delta \alpha &=&\delta \gamma=0\,,\\
\label{Eq:Variations2}
{\frac{\delta r}{r}}&=&
\frac{2B\delta B}{1+2B^2-\cos u}\,,\\
\label{Eq:Variations3}
{\frac{\delta \Phi}{\Phi'}}&=&-\sin u\frac{ \delta B}{B(1+B^2)}\,.
\end{eqnarray}
Equations~(\ref{Eq:Variations1},\ref{Eq:Variations2},\ref{Eq:Variations3}), introduced into Eqs.~(\ref{Eq:AGI},\ref{Eq:CGI}), yields the following expressions for the gauge-invariant quantities $A$ and $C$:
\begin{equation}
A = \frac{1+\cos u}{2 B(1+ B^2)} \delta B\,,\qquad
C = \frac{{\delta B}}{B}\,.
\label{Eq:ABAdS}
\end{equation}
From here and from the definition of $\sigma$ in Eq.~(\ref{Eq:Pert3abis}) we see that $\sigma=0$, as required, for every choice of the perturbation $\delta B$. Inserting Eq.~(\ref{Eq:ABAdS}) into the definition~(\ref{Eq:FandG}) of $\cal F$ and $\cal G$ (and omitting the proportionality factor $\delta B$) one obtains, also in this case, a static solution of the system~(\ref{Eq:WaveSystem},\ref{Eq:WaveConstraint}):
\begin{equation}
\left( \begin{array}{cc} {\cal F}_0 \\ {\cal G}_0 \end{array} \right)
:= \frac{\sqrt{2}k}{B} \cos\left( \frac{u}{2} \right)\times
\left( \begin{array}{c}
-\frac{\sqrt{1+2B^2-{\cos u}}}{2(1+B^2)} \\
\frac{1}{\sqrt{1+2B^2-{\cos u}}}
\end{array} \right)\,.
\label{Eq:SolStat2}
\end{equation}
Note that ${\cal G}_0$ is a strictly positive function of $u\in(-\pi,\pi)$, and that ${\cal F}_0/{\cal G}_0 = -(1 + 2B^2 - \cos u)/(2(1 + B^2))$.
Having found a nontrivial solution, we can now obtain the master equation governing the spherical symmetric linearized perturbations of the AdS wormhole, following the general method explained before. We observe that $Y_0 = -2\tan\frac{u}{2}$ and that we can choose the constants $a_0$ and $x_0$ in Eq.~(\ref{Eq:ChiTrans}) such that $a = 1/\alpha^2$; therefore Eq.~(\ref{Eq:ChiTrans}) reads
$$
\chi := \left({\cal F} - \frac{{\cal F}_0}{{\cal G}_0}{\cal G}\right)\alpha^2
$$
and the master equation~(\ref{Eq:Master2}) becomes (recalling that $\alpha/\gamma=1$ for the AdS wormhole in coordinates $(s,u)$)
\begin{equation}
\left[ \frac{\partial^2}{\partial s^2}
- \frac{\partial^2}{\partial u^2} +
{\cal {V}} \right] \chi = 0\,,
\label{Eq:MasterAdS}
\end{equation}
with the potential
\begin{equation}
{\mathcal{V}}(u) \equiv {\mathcal{V}}_B(u) = -\frac{B^2 \left(2+B^2+\cos u\right) }{\left(1+2 B^2-\cos u\right)^2}\,.
\label{Eq:PotentialB}
\end{equation}
For the following, we assume Dirichlet boundary conditions at the two asymptotic AdS ends, that is,
\begin{equation}
\chi(s,\pm \pi) = 0\,.
\label{Eq:BounCond}
\end{equation}
Since
$$
\chi = \frac{1}{\sqrt{2}\sqrt{1+2 B^2-\cos u}}\delta\gamma
- \frac{1 + \cos u}{4(1 + B^2)(1 + 2B^2-\cos u)} \delta r
$$
for $\delta \Phi=0$, a sufficient condition for Eq.~(\ref{Eq:BounCond}) to hold is that,
in the gauge $\delta \Phi=0$, the perturbed functions $\delta r$ and $\delta \gamma$
vanish at the far ends $u=\pm\pi$ of the wormhole. For general considerations on boundary conditions for field theories on AdS spaces, see~\cite{sAcIdS78}\cite{cK13}\cite{cDhFaM18}.
Now (following the same scheme of the previous subsection) let us sketch some spectral features of the Schr\"odinger operator $-d^2/u^2+\mathcal{V}(u)$ with Dirichlet boundary conditions at $u=\pm \pi$ (to be regarded as a selfadjoint operator in $L^2((-\pi,\pi),du)$); these facts will allow us to infer the linear instability of the AdS wormhole. The zero-energy Schr\"odinger equation $[-d^2/du^2 + \mathcal{V} ]\chi_0 = 0$ admits for each fixed $B > 0$ the general solution
\begin{equation}
\chi_0(u) = C_1 {\sin {u \over 2} \over \sqrt{1+2B^2 -\cos u}}
+ C_2 {-2 u \sin{u \over 2} + 4 B^2 \cos{u \over 2}\over \sqrt{1+2B^2 -\cos u}}\,,\qquad
-\pi < u < \pi\,,
\label{Eq:AdSZeroMode}
\end{equation}
with constants $C_1,C_2$. The Dirichlet boundary conditions $\chi_0(\pm\pi)=0$ are satisfied only in the trivial case $C_1=C_2=0$, which shows that none of these solutions is an eigenfunction of our Schr\"odinger operator. For $C_1 = -2\pi C_2\neq 0$ the zero-energy solution satisfies the left boundary condition, i.e. $\chi_0(-\pi) = 0$, and since this solution has precisely one zero in the interval $(-\pi,\pi)$, (\footnote{Let us justify this statement on the number of zeroes
of $\chi_0$ for the special choice $C_1 = -2\pi C_2\neq 0$. In this case we can write $\chi_0(u) =\left(-2 C_2 \cos \frac{u}{2}\right) w(u)/\sqrt{1+2 B^2-\cos u}$ where
$w : (-\pi,\pi)\to \mathbb{R}$, $u \mapsto w(u) := { (u+\pi ) \tan \frac{u}{2}-2 B^2}$.
The zeroes of $\chi_0$ in $(-\pi,\pi)$ coincide with the zeroes of the function
$w$. To find the zeroes of $w$, it is useful to note that this function has derivative $w'(u) = \left({1\over 2} \sec ^2\frac{u}{2}\right)\left(u+\sin u+\pi\right)>0$ for all $u \in (-\pi,\pi)$; from $w'>0$ it follows that $w$ is a strictly monotonic bijection of $(-\pi,\pi)$ to $(-2 B^2-2, + \infty)$, and thus possesses a unique zero.
})
it follows from the Sturm oscillation theorem (see Theorem~3.4 in~\cite{bS05}) that our Schr\"odinger operator (with Dirichlet boundary conditions) has a single bound state with negative energy $E < 0$. This state gives rise to an exponentially growing (in time) mode solution of the master equation~(\ref{Eq:MasterAdS}) which is proportional to $e^{\sqrt{-E} t}$; this establishes the linear instability of the AdS wormhole. In the next section we analyze the spectral properties of the Schr\"odinger operator and their implications for the solutions of the master equation in a rigorous functional setting.
\section{Spectral representation of the solutions of the master equations}
\label{Sec:Spectral}
In this section we provide a rigorous analysis regarding the spectral properties of the Schr\"odinger-type operators involved in the master equations discussed so far. The next subsection and the related Appendix~\ref{appebro} concern the reflection-symmetric Ellis-Bronnikov case; the subsequent subsection sketches a similar analysis for the nonsymmetric case. The third subsection and the related Appendix~\ref{appeads} treat the corresponding problem for the AdS case (not previously considered in the literature, to the best of our knowledge); in the same subsection we establish bounds on the eigenvalues of the Schr\"odinger operator. Brief comments regarding the timescale associated with the instability are made in the final subsection.
\subsection{Spectral decomposition of the master equation and instability of the Ellis-Bronnikov wormhole in the reflection-symmetric case}
\label{Sec:SpectralA}
Let us consider the reflection-symmetric Ellis-Bronnikov wormhole and the corresponding master equation~(\ref{Eq:MasterEllis}), containing the potential $\mathcal{V}(x) = - 3 b^2/(x^2 + b^2)^2$; this equation can be written as
\begin{equation}
\ddot{\chi}(t) + \Hop \chi(t) = 0 \quad (t\in \mathbb{R}),
\label{masterbro}
\end{equation}
where $\chi(t)$ stands for the function $\mathbb{R} \ni x \mapsto \chi(t,x)$, and $\Hop$ indicates the operator $- d^2/d x^2 + \mathcal{V}$.
If we want a rigorous functional setting for Eq.~(\ref{masterbro}), we are led to consider
the Hilbert space
(\footnote{Throughout the paper, the expression ``Hilbert space'' is an abbreviation
for ``complex, separable Hilbert space''.})
\begin{equation}
\label{hilbro}
\Hilb := L^2(\mathbb{R}, d x)
\end{equation}
made of the functions $f: \mathbb{R} \to \mathbb{C}$, $x \mapsto f(x)$ which
are square integrable for the Lebesgue measure $d x$; we will write $\langle ~|~\rangle$
and $\|~\|$ for the natural inner product and norm of this space, defined by
$\langle f | \ell \rangle := \int_{\mathbb{R}} d x \bar{f}(x) \ell(x)$
and $\| f \|^2 = \langle f | f \rangle$ for $f, \ell \in \Hilb$.
$\Hop$ can be regarded
as a selfadjoint operator in $\Hilb$, if we give for it the precise definition
\begin{equation}
\label{hbro}
\Hop := -\frac{d^2}{d x^2} + \mathcal{V} : \mathfrak{D} \subset \Hilb
\to \Hilb\,, \quad
\mathfrak{D} := \{ f \in \Hilb~|~f_{x x} \in \Hilb \}
\end{equation}
intending all $x$-derivatives in the distributional sense
({\footnote{The conditions $f \in \Hilb$ and $f_{x x} \in \Hilb$
imply $f_x \in \Hilb$, due to the Gagliardo-Nirenberg interpolation inequality
(see e.g. \cite{Adams-Book});
$\mathfrak{D}$ is just the usual Sobolev space $W^{2, 2}(\mathbb{R}) \equiv H^2(\mathbb{R})$.
Let us also remark that, for $f \in \Hilb$, one has automatically $\mathcal{V} f \in \Hilb$
due to the boundedness of $\mathcal{V}$.}}).
Due to general facts on Schr\"odinger operators~\cite{Ber}, and to a specific analysis performed in~\cite{jGfGoS09a} for the potential $\mathcal{V}(x) = - 3 b^2/(x^2 + b^2)^2$, we can state
that the spectrum of $\Hop$ is the union of:
\begin{itemize}
\setlength\itemsep{-0.1cm}
\item[(i)] the point spectrum, which consists of a unique, simple eigenvalue $\mu_1 < 0$; \\
\item[(ii)] the continuous spectrum $[0,+\infty)$.
\end{itemize}
One can construct a generalized orthonormal basis
of the Hilbert space $\Hilb$, in the sense explained by Appendix \ref{appebro} and by \cite{Ber}, using:
\begin{itemize}
\setlength\itemsep{-0.1cm}
\item[(i)] a normalized eigenfunction $e_1$ for the eigenvalue
$\mu_1$ ($e_1 \in \mathfrak{D}$, $\Hop e_1 = \mu_1 e_1$, $\| e_1 \|=1$;
$e_1$ is proved to be $C^\infty$);
\item[(ii)] two suitably chosen ``improper eigenfunctions'' $e_{i \lambda}$ ($i=1,2$) for each
$\lambda \in (0,+\infty)$ (i.e., for each nonzero point $\lambda$ of the continuous spectrum); these are two linearly independent $C^\infty$ functions on $\mathbb{R}$ which fulfill
$- d^2 e_{i \lambda}/ d x^2 + \mathcal{V} e_{i \lambda} = \lambda e_{i \lambda}$ but do not belong to
$\Hilb$.
\end{itemize}
Then, one can search for the solution
$\mathbb{R} \ni t \to \chi(t)$ of
Eq.~(\ref{masterbro}) with appropriate smoothness properties and with
the initial conditions
\begin{equation} \chi(0) = q\,, \quad \dot{\chi}(0) = p\,,
\label{cond}
\end{equation}
where $q: x \mapsto q(x)$ and $p : x \mapsto p(x)$ are sufficiently regular functions.
For all technical details, we
refer again to Appendix \ref{appebro}; here we introduce
the selfadjoint operator $| \Hop |^{1/2}: \mathfrak{D}^{1/2}
\subset \Hilb \to \Hilb$ and indicate how to regard
the domains $\mathfrak{D}^{1/2}$ and $\mathfrak{D}$ as Hilbert spaces with their
own inner products. One can show that, for any $q \in \mathfrak{D}$ and $p \in \mathfrak{D}^{1/2}$, Eqs.~(\ref{masterbro},\ref{cond}) have a unique solution $t \mapsto \chi(t)$ in
$C(\mathbb{R}, \mathfrak{D}) \cap C^1(\mathbb{R}, \mathfrak{D}^{1/2}) \cap C^2(\mathbb{R}, \Hilb)$,
which is as follows for all $t \in \mathbb{R}$:
\begin{equation}
\chi(t) = \left[ \langle e_1 | q \rangle \cosh( |\mu_1|^{1/2} t) +
\langle e_1 | p \rangle \frac{\sinh( |\mu_1|^{1/2} t)}{|\mu_1|^{1/2}} \right] e_1 +
\sum_{i=1}^{2} \int_{0}^{+\infty} \hspace{-0.4cm} d \lambda
\left[ \langle e_{i \lambda} | q \rangle \cos( \lambda^{1/2} t) +
\langle e_{i \lambda} | p \rangle \frac{\sin( \lambda^{1/2} t)}{\lambda^{1/2}} \right] e_{i \lambda}\, .
\label{solbro}
\end{equation}
As explained in Appendix~\ref{appebro}, the symbols $\langle \cdot | \cdot \rangle$ in the above formula indicate usual inner products in $\Hilb$, or suitably defined generalizations;
the integrals over $\lambda$ are understood in a weak sense. Of course, we are interested
in the case where $\chi(t)$ is \textsl{real valued} for each $t$, which occurs if
and only if the data $q, p$ are real-valued functions.
The coefficient of $e_1$ in Eq.~(\ref{solbro}) diverges exponentially both for
$t \to -\infty$ and for $t \to + \infty$ (except for very special choices of
$\langle e_1 | q \rangle$ and $\langle e_1 | p \rangle$
(\footnote{\label{footnote13}For $\langle e_1 | q \rangle = \langle e_1 | p \rangle = 0$,
the coefficient of $e_1$ in~(\ref{solbro}) vanishes. For
$\langle e_1 | q \rangle = \xi \langle e_1 | p \rangle/|\mu_1|^{1/2} \neq 0$,
with $\xi = \pm 1$, the coefficient of $e_1$ diverges for $t \to \xi (+\infty)$
and vanishes for $t \to \xi (-\infty)$.})); this suffices to infer the (linear) instability
of the reflection-symmetric Ellis-Bronnikov wormhole
\cite{jGfGoS08} \cite{jGfGoS09a} \cite{fCfPlP19}. In addition, let us remark that the integrals
over $\lambda$ in Eq.~(\ref{solbro}) are superpositions of ``non-normalizable'' oscillatory modes, living outside the space $\Hilb = L^2(\mathbb{R}, d x)$ like the improper eigenfunctions $e_{i \lambda}$.
\subsection{Spectral decomposition of the master equation and instability of the Ellis-Bronnikov wormhole in the nonsymmetric case}
\label{Sec:SpectralB}
Let us now pass to the non-reflection-symmetric Ellis-Bronnikov wormhole
(as in Eq.~(\ref{Eq:StaticSolutions}) with $\gamma_1 \neq 0$). In this case the master equation for $\chi(t,x)$ has the form~(\ref{Eq:Master2}), involving the operator
\begin{equation}
- \left(\frac{\alpha}{\gamma} \frac{\partial}{\partial x}\right)^2 + \frac{\alpha^2}{\gamma^2} \mathcal{V}\,, \quad
\frac{\alpha(x)}{\gamma(x)} = e^{2 \gamma_1 \arctan \frac{x}{b}}~~(x \in \mathbb{R})\,.
\end{equation}
As noted in \cite{jGfGoS09a}, the spectral analysis of this case can be simplified by
introducing the new coordinate
\begin{equation}
\label{Eq:ro}
\rho = \rho(x) := \int_{0}^x \frac{\gamma(y)}{\alpha(y)} \,dy\,;
\end{equation}
note that the mapping $x \mapsto \rho(x)$ is a diffeomorphism of $\mathbb{R}$
to itself, and $\rho(x) \sim e^{\mp \pi \gamma_1} x$ for $x \to \pm \infty$.
By construction
$\frac{\alpha}{\gamma} \frac{\partial}{\partial x}$ $= \frac{\partial}{\partial \rho}$;
so, by writing $\chi(t,\rho)$ as an abbreviation for $\chi(t,x(\rho))$ we can
rephrase the master equation~(\ref{Eq:Master2}) as
\begin{equation}
\label{Eq:Master2reph}
\left[ \frac{\partial^2}{\partial t^2}
- \left( \frac{\partial}{\partial \rho} \right)^2
+ \mathcal{U}(\rho) \right] \chi(t, \rho) = 0\,,
\quad \mathcal{U}(\rho) := \left(\frac{\alpha^2}{\gamma^2} \mathcal{V}\right)(x(\rho))
\quad (t,\rho \in \mathbb{R}).
\end{equation}
The function $\mathcal{U} : \mathbb{R} \to \mathbb{R}$ is $C^\infty$;
due to the $x \to \pm \infty$ asymptotics of
$\rho(x)$ (see after Eq.~(\ref{Eq:ro})) and $\left(\frac{\alpha^2}{\gamma^2} \mathcal{V}\right)(x)$
(see after Eq.~(\ref{Eq:W})), we have $\mathcal{U}(\rho) \sim 2/\rho^2$ for $\rho \to \pm \infty$.
A precise functional setting for Eq.~(\ref{Eq:Master2reph}) can be obtained by introducing the
Hilbert space and the selfadjoint operator
\begin{equation}
\label{hbrons}
\Hilb := L^2(\mathbb{R}, d \rho)\,; \qquad
\Hop := -\frac{d^2}{d \rho^2} + \mathcal{U} : \mathfrak{D} \subset \Hilb
\to \Hilb\,, \quad
\mathfrak{D} := \{ f \in \Hilb~|~f_{\rho \rho} \in \Hilb \}
\end{equation}
(the $\rho$-derivatives are meant distributionally);
$\langle~|~\rangle$ and $\|~\|$ indicate in the sequel
the natural inner product and norm of $\Hilb$. (\footnote{\label{footnote:Hilbert} Note that, since $d\rho={\gamma\over \alpha}dx$, working with the operator and the Hilbert space defined in Eq.~(\ref{hbrons}) is equivalent to working directly with the operator $-\left({\alpha\over\gamma} {d\over d x}\right)^2+{\alpha^2\over\gamma^2}\mathcal{ V}$ in the Hilbert space $L^2(\mathbb{R},{\gamma\over\alpha} dx)$, which is the formulation considered at the end of subsection \ref{SubSec:PertEllis}.})
After giving these prescriptions we write Eq.~(\ref{Eq:Master2reph}) in the form~(\ref{masterbro}),
where $\chi(t)$ stands for the function $\rho \mapsto \chi(t,\rho)$; obviously enough,
the treatment of this equation is reduced to a spectral analysis of the
Schr\"odinger operator $\Hop$ in Eq.~(\ref{hbrons}), which is rather
similar to the discussion of the operator~(\ref{hbro}) for the reflection-symmetric wormhole.
The main difference with respect to the symmetric case is that
the operator $\Hop$ in Eq.~(\ref{hbrons}) has a point spectrum
consisting of \textsl{two} simple eigenvalues $\mu_1 < 0$ and $\mu_2 := 0$, see the comments below Eq.~(\ref{Eq:EBZeroMode}); the continuous spectrum is $(0,+\infty)$. Due to these facts there is a generalized orthonormal basis
made of normalized eigenfunctions $e_1, e_2$ for the eigenvalues $\mu_1 < 0$
and $\mu_2 = 0$ ($e_1, e_2 \in \mathfrak{D}(\Hop)$, $\Hop e_1 = \mu_1 e_1$,
$\Hop e_2 = 0$, $\| e_1 \| = \| e_2 \| = 1$),
plus two improper eigenfunctions $e_{i \lambda}$ ($i=1,2$)
for each $\lambda$ in the continuous spectrum.
As in the symmetric case, one can define Hilbert space
structures for the domains $\mathfrak{D}$, $\mathfrak{D}^{1/2}$ of the operators
$\Hop$, $|\Hop|^{1/2}$. For $q \in \mathfrak{D}$ and $p \in \mathfrak{D}^{1/2}$,
the master equation ~(\ref{masterbro}) with initial conditions~(\ref{cond}) is proved
again to possess a unique solution $t \mapsto \chi(t)$ in
$C(\mathbb{R}, \mathfrak{D}) \cap C^1(\mathbb{R}, \mathfrak{D}^{1/2}) \cap C^2(\mathbb{R}, \Hilb)$;
this has a representation similar to~(\ref{solbro}) with an additional
term associated with the eigenvalue zero, namely:
\begin{equation}
\chi(t) = \mbox{r.h.s. of Eq.~(\ref{solbro})} +
\Big[ \langle e_2 | q \rangle + \langle e_2 | p \rangle t \Big] e_2\,.
\label{solbrons}
\end{equation}
So, besides the exponentially divergent term proportional
to $e_1$, the expression of $\chi(t)$ contains a term diverging
linearly for $t \to \pm \infty$ (if $\langle e_2 | p \rangle \neq 0$);
in any case the wormhole is
linearly unstable. Let us note that, as in Eq.~(\ref{solbro}),
the present expression for $\chi(t)$ contains an integral
over $\lambda$ of non-normalizable oscillatory modes,
proportional to the improper eigenfunctions $e_{i \lambda}$ which
live outside $\Hilb$.
\subsection{Spectral decomposition of the master equation and instability of the AdS wormhole}
In the AdS case we introduce the Hilbert space
\begin{equation}
\label{hilbu}
\Hilb := L^2((-\pi,\pi), du)
\end{equation}
formed by the functions $f : (-\pi,\pi) \to \mathbb{C}$, $u \mapsto f(u)$ which
are square integrable with respect to the Lebesgue measure $d u$; from now on we
denote by $\Braket{~|~}$ and $\|~\|$ the natural inner product and norm of $\Hilb$,
so that $\langle f | \ell \rangle := \int_{-\pi,\pi} d u \bar{f}(u) \ell(u)$
and $\| f \|^2 = \langle f | f \rangle$ for $f, \ell \in \Hilb$.
In addition,
let us consider the potential
${\mathcal{ V} }$ appearing in Eq.~(\ref{Eq:PotentialB}) (a $C^\infty$ function on
$[-\pi, \pi]$). A rigorous setting for the master equation~(\ref{Eq:MasterAdS}) with boundary conditions~(\ref{Eq:BounCond}) can be set up using the space~(\ref{hilbu}) and the selfadjoint operator
\begin{equation}
\label{hB}
\Hop := -{d^2 \over du^2}+ \mathcal{ V} : \mathfrak{D} \subset \Hilb \to \Hilb\,,
\quad
\mathfrak{D} := \{ f \in \Hilb~|~f_{u u} \in \Hilb~, f(\pm \pi) = 0 \}\,.
\end{equation}
Here and in the sequel, the $u$-derivatives like $f_{u u }$ are understood distributionally;
a function $f\in \Hilb$ with $f_{uu}\in \Hilb$ is in fact in $C^1([-\pi,\pi])$,
so it can be evaluated at $u=\pm\pi$
(\footnote{The conditions $f \in \Hilb$, $f_{u u} \in \Hilb$
imply $f_u \in \Hilb$, due to the already mentioned Gagliardo-Nirenberg interpolation inequality
\cite{Adams-Book}.
The space $\{ f \in \Hilb~|~f_{u u} \in \Hilb \}$ coincides with
the standard Sobolev space $W^{2, 2}(-\pi,\pi) \equiv H^{2}(-\pi,\pi)$, which is contained in $C^1([-\pi,\pi])$
by the Sobolev embedding theorem (see again \cite{Adams-Book}). Let us also remark that,
due to the boundedness of the function $\mathcal{V}$, for each $f \in \Hilb$
one has automatically $\mathcal{V} f \in \Hilb$.}).
As an operator in the Hilbert space $\Hilb$, $\Hop$ has the following properties:
\begin{itemize}
\setlength\itemsep{-0.1cm}
\item[(i)] it is selfadjoint;
\item[(ii)] it is bounded from below;
\item[(iii)] it has a purely discrete spectrum.
\end{itemize}
\noindent
As known in general for Hilbert space operators
satisfying properties (i-iii), it is possible to represent the eigenvalues of $\Hop$ as an increasing sequence $\mu_1 <\mu_2 < \cdots$. In addition, $\Hop$ has the following properties:
\begin{itemize}
\setlength\itemsep{-0.1cm}
\item[(iv)] any of its eigenfunctions is in the space $C^{\infty}([-\pi,\pi])$;
\item[(v)] each one of its eigenvalues is simple.
\end{itemize}
\noindent
For future mention, let us recall that the operator $\Hop^0:=-d^2/du^2$ with domain $\mathfrak{D}$ as above also has
the properties (i-v); in this case the eigenvalues are $\mu^{0}_n :=n^2/4$, with normalized
eigenfunctions $f^0_n(u):=(1/\sqrt{\pi})\sin[(n/2)(u+\pi)]$ ($n=1,2,\ldots$)
(\footnote{Let us give more complete information on the above issues (i-v).
For some general facts about Hilbert space operators with properties (i-iii)
(including the possibility to arrange their eigenvalues in an increasing sequence),
see e.g. \cite{Schmudgen-Book} (especially, pages 37, 178 and 265-67).
To go on, let us recall the following regularity result: if $f$ is a distribution on an open interval $\Omega \subset \mathbb{R}$ (with derivatives $f^{(i)}$, $i=0,1,\ldots$) and $f$ fulfills a homogeneous linear ODE
$f^{(k)} + \sum_{i=0}^{k-1} a_i f^{(i)}=0$ of any order
$k \in \{1,2,\ldots\}$ with $C^\infty$ coefficients
$a_i: \Omega \to \mathbb{C}$, then $f$ is a $C^\infty$ function on $\Omega$:
this follows from Theorem IX in \cite{Schwartz-Book}, page 130.
The properties (i-v) of $\Hop^0$ and the expressions given above
for its eigenvalues and eigenfunctions are checked ``by hand'', keeping
in mind that the eigenfunctions are smooth
due to the previously mentioned regularity result.
Now consider any function $\mathcal{V}\in C^{\infty}([-\pi,\pi],\mathbb{R})$;
then, due to the boundedness of this function,
the multiplication operator by $\cal V$ is a bounded selfadjoint operator on $\Hilb$.
As well known the properties (i), or (i-ii), or (i-iii) of an operator in an abstract Hilbert space
are preserved by the addition
of a bounded selfadjoint perturbation (see again \cite{Schmudgen-Book}); therefore the operator
$\Hop := \Hop^0+\mathcal{V}=-d^2/du^2+\mathcal{V}$ with domain $\mathfrak{D}$
fulfills (i-iii). The operator $\Hop$ also has the
properties (iv-v). For the proof of (iv) one can use again the cited
regularity result for distributional, homogeneous linear ODEs;
a derivation of (v) can be found e.g. in \cite{Poschel-Book}, page 30.
All the previous statements apply, in particular, with ${\cal V}$ as in Eq.~(\ref{Eq:PotentialB}).
}).
In the remainder of this section the notations $\mathcal{V}$, $\Hilb$, $\mathfrak{D}$, $\Hop$, $(\mu_n)_{n=1,2,\ldots}$ will always indicate, respectively, the potential $\mathcal{V}$ in Eq.~(\ref{Eq:PotentialB}), the Hilbert space in Eq.~(\ref{hilbu}), the domain and the operator in Eq.~(\ref{hB}), and the eigenvalues of this operator in increasing order. Sometimes it will be useful to emphasize that the potential $\mathcal{V}$ depends on the parameter $B \in (0,+\infty)$, thus originating in a similar dependence for the corresponding operator and its eigenvalues: $\mathcal{V} \equiv \mathcal{V}_B$, $\Hop \equiv {\Hop}_B$, $\mu_n \equiv \mu_n(B)$ ($n=1,2,\ldots$).
As discussed in section~\ref{SubSec:Pertads}, the analysis of the zero-energy solutions in Eq.~(\ref{Eq:AdSZeroMode}) implies that the ground-state energy is negative, while all other eigenvalues are positive, such that
\begin{equation}
\label{signs}
\mu_1 < 0 < \mu_2 < \mu_3 < \cdots\,,
\end{equation}
the negative eigenvalue $\mu_1$ being associated a mode of the master equation~(\ref{Eq:MasterAdS}) growing exponentially in time, whereas in contrast to this, the eigenvalues $\mu_n$ for $n\geqslant 2$ are associated with oscillatory modes.
In what follows, we provide estimates for the eigenvalues of $\mu_n(B)$. We start with an upper bound for the ground-state energy $\mu_1 \equiv \mu_1(B)$. According to the Rayleigh-Ritz variational characterization
(see e.g. \cite{Schmudgen-Book}, pages 265-266) one has
\begin{equation}
\mu_1(B)=\inf_{f \in \mathfrak{D} \setminus \{0\}} {\Braket{f|\Hop_B f} \over||f||^2} \, .
\label{Eq:InfSpectrum}
\end{equation}
Choosing in $\mathfrak{D}$ the function
\begin{equation}
f(u) := \cos{u \over 2},
\end{equation}
we get
\begin{equation}
{\Braket{f|\Hop_B f} \over||f||^2}=\frac{1}{4}-B^2+\frac{\sqrt{1 + B^2} \left(4 B^2 -3\right)}{4B}
=: \varepsilon(B)\,,
\label{Eq:epsilonB}
\end{equation}
which, together with Eq.~(\ref{Eq:InfSpectrum}), yields the estimate
\begin{equation*}
\mu_1(B)\leqslant \epsilon(B)
\end{equation*}
for each $B>0$. It can be checked that $B\mapsto\varepsilon(B)$ is a negative, monotonously increasing function on $(0,+\infty)$ with the properties
\begin{equation}
\lim\limits_{B\to 0^+} \varepsilon(B) = -\infty\,,\qquad
\lim\limits_{B\to +\infty} \varepsilon(B) = 0^-\,.
\end{equation}
Therefore, we obtain the upper bound for the ground-state energy
\begin{equation}
\mu_1(B)\leqslant \varepsilon(B) < 0
\label{Eq:mu1Bound}
\end{equation}
which provides an independent proof for the fact that it is negative, and hence also for the linear instability of the AdS wormhole.
Next, we provide two-sided bounds on the eigenvalues $\mu_n \equiv \mu_n(B)$ for arbitrary $n$. In order to achieve this, we check that for any fixed $B > 0$, one has
\begin{equation}
\min_{u \in [-\pi,\pi]} {\mathcal{V}}_B(u) = {\mathcal{V}}_B(0) = - {1 \over 4} - {3 \over 4 B^2}\,,
\qquad \max_{u \in [-\pi,\pi]} {\mathcal{V}}_B(u)
= {\mathcal{V}}_B(\pm \pi) = - {1 \over 4} + {1 \over 4 (1 + B^2)}\,.
\label{Eq:LimPot}
\end{equation}
In the Hilbert space $\Hilb$, let us consider the operators
$\Hop =-{d^2\over du^2}+{\cal V}$, $\Hop^- :=-{d^2\over du^2} - {1 \over 4} - {3 \over 4 B^2}$
and $\Hop^+ :=-{d^2\over du^2} - {1 \over 4} + {1 \over 4 (1 + B^2)}$,
all of them with the same domain $\mathfrak{D}$ as defined in Eq.~(\ref{hB})
(and all of them satisfying the properties (i-v) after the cited equation). Due to Eq.~(\ref{Eq:LimPot}) we have
$\Braket{f|\Hop^- f}\leqslant \Braket{f|\Hop f}\leqslant\Braket{f|\Hop^+ f}$
for all $f\in \mathfrak{D}$, and this implies (see e.g. \cite{Schmudgen-Book}, pages 230 and 267)
$\mu^{-}_n \leqslant \mu_n \leqslant \mu^{+}_n$ for $n=1,2,\ldots$,
where $\mu^{\mp}_1 < \mu^{\mp}_2 < \cdots$ are the eigenvalues of $\Hop^{\mp}$.
On the other hand, the eigenvalues of $\Hop^{\mp}$ are obtained by shifting those
of $\Hop^0=-d^2/du^2$, i.e., $\mu^{-}_n ={n^2 \over 4} - {1 \over 4} - {3 \over 4 B^2}$
and $\mu^{+}_n ={n^2 \over 4} - {1 \over 4} + {1 \over 4 (1 + B^2)}$. In conclusion, the eigenvalues of $\Hop$ satisfy the two-sided bounds
\begin{equation}
\label{Eq:boundsmun}
{n^2 - 1 \over 4} - {3 \over 4 B^2} \leqslant \mu_n(B) \leqslant {n^2 - 1\over 4}
+ {1 \over 4 (1 + B^2)} \qquad (n=1,2,\ldots)\,.
\end{equation}
Combining this result with Eq.~(\ref{Eq:mu1Bound}) one obtains the following two-sided bound for the ground-state energy:
\begin{equation}
\label{Eq:boundsmu1}
-\frac{3}{4B^2} \leqslant \mu_1(B) \leqslant \varepsilon(B) = -\frac{1}{2B^2} + {\cal O}\left( \frac{1}{B^4} \right)\,.
\end{equation}
After these remarks concerning the eigenvalues of the Schr\"odinger operator $\Hop$, we discuss the spectral decomposition of the master equation. To this purpose, we choose
for each $n$ a normalized eigenfunction $e_n$ for the (simple) eigenvalue $\mu_n$:
\begin{equation}
\label{basisads}
e_n \in \mathfrak{D}~, \quad \Hop e_n = \mu_n e_n~, \quad \| e_n \| = 1 \qquad (n=1,2,\ldots) \, .
\end{equation}
Then $(e_n)_{n=1,2,\ldots}$ is an orthonormal basis of $\Hilb$ (in the ordinary sense),
due to the spectral theorem for selfadjoint operators with a purely discrete spectrum. In comparison with the previous analysis for the Ellis-Bronnikov wormhole, we do not have the technical complications associated with the continuous spectrum and to the related ``improper'' eigenfunctions.
Next, we write the master equation~(\ref{Eq:MasterAdS}) in a form similar to~(\ref{masterbro}) and add initial conditions as in~(\ref{cond}); in this way we obtain the system
\begin{equation}
\ddot{\chi}(s) + \Hop \chi(s) = 0 ~~ (s \in \mathbb{R})\,,
\quad \chi(0)=q\,,\quad \dot{\chi}(0) = p\,,
\label{masterads}
\end{equation}
where $\chi(s)$ refers to the function $u \mapsto \chi(s,u)$, the dots stand
for $s$-derivatives and $q: u \mapsto q(u)$, $p: u \mapsto p(u)$ are functions with
appropriate regularity.
A technically precise framework for the discussion of the system~(\ref{masterads})
is provided by Appendix \ref{appeads} where we introduce
(similarly to the previous treatment for the Ellis-Bronnikov wormhole)
the selfadjoint operator $| \Hop |^{1/2}: \mathfrak{D}^{1/2}
\subset \Hilb \to \Hilb$ and indicate how to regard
the domains $\mathfrak{D}^{1/2}$ and $\mathfrak{D}$ as Hilbert spaces with appropriate
inner products. It turns out that, for any $q \in \mathfrak{D}$ and $p \in \mathfrak{D}^{1/2}$,
the system~(\ref{masterads}) has a unique solution $s \mapsto \chi(s)$ in
$C(\mathbb{R}, \mathfrak{D}) \cap C^1(\mathbb{R}, \mathfrak{D}^{1/2}) \cap C^2(\mathbb{R}, \Hilb)$;
using an orthonormal basis $(e_n)_{n=1,2,\ldots}$ as in Eq.~(\ref{basisads}), the solution
can be written as follows
for all $s \in \mathbb{R}$:
\begin{equation}
\chi(s) = \left[ \langle e_1 | q \rangle \cosh( |\mu_1|^{1/2} s) +
\langle e_1 | p \rangle \frac{\sinh( |\mu_1|^{1/2} s)}{|\mu_1|^{1/2}} \right] e_1 +
\sum_{n=2}^{+\infty}
\left[ \langle e_{n} | q \rangle \cos( \mu^{1/2}_n s) +
\langle e_{n} | p \rangle \frac{\sin( \mu^{1/2}_n s)}{\mu^{1/2}_n} \right] e_{n}\, .
\label{solads}
\end{equation}
The above function $\chi(s)$ is real valued for each $s$ if
and only if the data $q, p$ are real-valued functions.
The coefficient of $e_1$ in Eq.~(\ref{solads}) diverges exponentially both for
$s \to -\infty$, and for $s \to + \infty$ (except for very special choices of
$\langle e_1 | q \rangle$ and $\langle e_1 | p \rangle$
(\footnote{See the footnote \ref{footnote13} in the discussion after
Eq.~(\ref{solbro}), which is readily adapted to the present framework.}));
so, the AdS wormhole is linearly unstable. For each $n \geqslant 2$, the $n$-th term
in Eq.~(\ref{solads}) represents a ``normalizable'' oscillatory mode, living
like $e_n$ inside the Hilbert space $\Hilb$ (indeed, inside the subspace $\mathfrak{D}\subset \Hilb$). This is a relevant difference with respect to the ``non-normalizable''
oscillatory modes that we have found for the perturbed Ellis-Bronnikov wormhole,
associated with the continuous spectrum and living outside the Hilbert space of the system
(see the comments after Eqs.~(\ref{solbro}) and~(\ref{solbrons})).
\subsection{Instability times}
\label{Sec:SpectralD}
In the Ellis-Bronnikov case it has been shown~\cite{jGfGoS09a} that the timescale $\tau_\text{unstable}$ (measured with respect to proper time at the throat of the unperturbed solution) associated with the unstable mode is of the order of the throat's areal radius $r_\text{throat}$ divided by the speed of light. The estimates provided in Eq.~(\ref{Eq:boundsmu1}) allow us to estimate the corresponding timescale for the AdS wormhole, and yield
\begin{equation}
\frac{1}{\sqrt{3}} \leq \frac{\tau_\text{unstable}}{r_\text{throat}}
\leq \frac{1}{2B\sqrt{-\varepsilon(B)}}\,,
\end{equation}
with the function $\varepsilon(B)$ defined in Eq.~(\ref{Eq:epsilonB}). Since $2B\sqrt{-\varepsilon(B)} \to \sqrt{2}$ for large $B$ and since for $B\to 0$ the AdS wormhole reduces to the reflection-symmetric Ellis-Bronnikov wormhole (\footnote{See the comment on the limit $k\to 0$ after Eq. (\ref{Eq:StaticSolutionsPot1}), keeping in mind that $B=b k$.}), it follows also in this case that $\tau_\text{unstable}$ is of the order of the throat's areal radius (divided by the speed of light in physical units).
\section{A dS wormhole with horizons and its linearized perturbations}
\label{Section:dS}
Let us return to the Bronnikov-Fabris wormhole solution mentioned at the beginning of section \ref{Subsection:AdS}, depending on the parameters $M$ and $K$. Keeping the assumption (\ref{Eq:ConstantM}) that $M=0$ we can as well consider, as an alternative to (\ref{Eq:ConstantK}), the choice
\begin{equation}
K\equiv k^2\,, \qquad (k>0)\,.
\end{equation}
In this way we obtain
\begin{equation}
V(\Phi)= \frac{k^2}{\kappa}
\left[ 3 -2 \cos ^2\left( \sqrt{\frac{\kappa}{2}}\Phi\right) \right]\,,\quad
\alpha = \gamma^{-1} = \sqrt{1 - k^2(x^2+b^2)}\,,\quad
r = \sqrt{x^2 + b^2}\,,\quad
\Phi = \sqrt{\frac{2}{\kappa}}\arctan\frac{x}{b}\,.
\label{Eq:StaticSolutionsPotdS}
\end{equation}
For $b\to0$ the third equality in~(\ref{Eq:StaticSolutionsPotdS}) should be read as $r=x>0$, and the corresponding metric represents a dS universe with cosmological constant $\Lambda = 3k^2$. From now on we intend
\begin{equation}
b\in\left(0,{1\over k}\right)\,;
\label{b}
\end{equation}
the limitation $b<{1\over k}$ ensures that the expressions for $\alpha$ and $\gamma$ in Eq. (\ref{Eq:StaticSolutionsPotdS}), if taken literally, make sense near the throat $x=0$ or, more substantially, that $\partial_t$ is actually timelike and $\partial_x$ is actually spacelike near $x=0$.
We also set
\begin{equation}
B:=b\, k\in(0,1)\,,\qquad\ell:={\sqrt{1-B^2}\over k}\,;
\label{Eq:B_ell}
\end{equation}
the metric corresponding to the above coefficients reads
\begin{align}
{\bf g}& = - \left[ 1 - k^2(x^2+b^2) \right] dt^2
+ \frac{d x^2}{ 1 - k^2(x^2+b^2)}
+ \left( x^2+b^2\right) (d \vartheta^2 + \sin^2 \vartheta\, d \varphi^2)
\nonumber\\
&=
-(1-B^2) \left( 1 - \frac{x^2}{\ell^2} \right) dt^2
+ \frac{d x^2}{(1-B^2)(1 - \frac{x^2}{\ell^2})}
+ \left( x^2 + b^2 \right) (d \vartheta^2 + \sin^2 \vartheta\, d \varphi^2)\,.
\label{Eq:dSWH1}
\end{align}
By analogy with the terminology of section
\ref{Subsection:AdS}, we refer to this as a ``dS wormhole''; let us note that the
expressions for $\Phi$, $V(\Phi)$, $r$ in Eq.~\eqref{Eq:StaticSolutionsPotdS} and
the expression \eqref{Eq:dSWH1} for ${\bf g}$ can be obtained formally from
the analogous expressions of the AdS case (see Eq.~\eqref{Eq:StaticSolutionsPot1}) by making the replacement $k \mapsto i k$.
Let us consider the regions
\begin{equation*}
I:=\{(t,x)\,|\,t\in \mathbb{R}\,,\, x \in(-\ell, \ell)\}\,,
\end{equation*}
\begin{equation}
E^-:=\{(t,x)\,|\,t\in \mathbb{R}\,,\, x \in(-\infty, -\ell)\}\,,\qquad E^+:=\{(t,x)\,|\,t\in \mathbb{R}\,,\, x \in(\ell, +\infty) \}\,;
\label{region_I_E}
\end{equation}
then the expressions for $\alpha$ and $\gamma$ in Eq.(\ref{Eq:StaticSolutionsPotdS}) are well defined in a literal sense over $I$; more substantially, the metric \eqref{Eq:dSWH1} is well defined over $I \times S^2$ and the vector fields $\partial_t$ and $\partial_x$ are, respectively, timelike and spacelike on this domain. However, Eq. (\ref{Eq:dSWH1}) also gives a Lorentzian metric on each one of the regions $E^-$ and $E^+$; here $\partial_t$ is spacelike and $\partial_x$ is timelike, so the metric is nonstatic. In the sequel we often refer to $I$ as the internal region and to $E^\pm$ as the exterior regions in $(t,x)$ space. At $x=\pm \ell$ the metric seems to be ill defined but, as explained hereafter, these are just apparent singularities related to the coordinate system: the hypersurfaces $x=\pm\ell$ are indeed cosmological horizons and the metric is nonsingular across them.
Let us note that the $b\to 0$ limit of the previous statement (with $x>0$) corresponds
to well-known features of the dS universe, having a horizon at $x=\ell={1\over k}$. \par
In the next paragraph we consider an alternative coordinatization for the
internal region $I$ introducing the analogs of the AdS wormhole coordinates $(s,u)$ (see Eq. (\ref{coordsu})); in the subsequent paragraphs we consider alternative parametrizations yielding a Kruskal-type extension of the metric (\ref{Eq:dSWH1}) which is regular across $x=\pm\ell$. The extended universe constructed in this way can also be interpreted as a regular black hole with an expanding cosmology beyond the horizons, and is hence referred to as a ``black universe'' in \cite{Bro2018}.
\subsection{Another coordinate system for the internal region $I$}
Let us put
\begin{equation}
t= {\ell\over 2\, (1-B^2)} \,s\,,\qquad x=\ell \,\mbox{tanh} {u \over 2} \,,\qquad
(s, u)\in\mathbb{R}^2 \,;
\label{coordsu_dS}
\end{equation}
the map $(s,u)\mapsto(t,x)$ is one to one between $\mathbb{R}^2$ and the inner region $I$ (with inverse
$
s={2(1-B^2)\over \ell}\,t$, $ u=2\,\mbox{arctanh} {x\over\ell}= \log ({\ell+x\over\ell - x})
$).
We can regard $(s,u)$ as an alternative coordinate system for $I$; this does not eliminate the apparent singularities at $x = \pm \ell$ but sends them to infinity since the limits $x\to\pm \ell$ correspond to the limits $u\to\pm\infty$. In the new coordinates the metric \eqref{Eq:dSWH1} becomes
\begin{equation}
{\bf g} = \frac{1}{4 k^2 \cosh^2\frac{u}{2}}
\left[ -ds^2 + du^2
+2\bigg(\cosh u -(1-2 B^2)\bigg)
(d \vartheta^2 + \sin^2 \vartheta\, d \varphi^2) \right],
\label{gsu}
\end{equation}
with radial null geodesics given by the straight lines $s = \pm u + const$.
To conclude this paragraph, let us remark that the transformation \eqref{coordsu_dS} and the expression \eqref{gsu} for the metric can be obtained from their AdS analogs
(see Eqs.~(\ref{coordsu},\ref{Eq:StaticSolutionsPot2})) by
making the formal replacements $k \mapsto i k, B \mapsto i B, s \mapsto i s, u \mapsto i u$.
\subsection{A first spacetime extension}
We start our construction from the internal region $I$, that we
describe in terms of the coordinates $(s,u)$. Let us set
\begin{eqnarray}
s = \log \left(-{U\over V}\right)\,,\qquad u=-\log(-U V)\,,\qquad
U\in (0, +\infty)\,,\quad V \in(-\infty, 0)\,;
\label{coor_su}
\end{eqnarray}
the transformation $(U,V)\mapsto(s,u)$ is one to one between the sets
$(0, +\infty)\times(-\infty, 0)$
and $\mathbb{R}^2$.
By compositions with \eqref{coordsu_dS} we obtain the transformation
\begin{equation}
{t}={\ell\over 2(1-B^2)}\log\left(-{ U\over V}\right)~, \quad
x=\ell \, {1+UV\over 1-UV}~,
\label{xuv}
\end{equation}
which is a diffeomorphism between $(0, +\infty)\times(-\infty, 0)$
and the inner region
$I$.
The first cosmological horizon ${x}=-\ell$ corresponds
to $U\to +\infty$ or $V\to -\infty$, while the second cosmological horizon ${x}=\ell$ coincides with $UV=0$.
Now the metric \eqref{gsu} reads
\begin{equation}
\label{Eq:g_UV}
{\bf g} = \frac{1}{k^2 (1 - U V)^2}
\left[ -4 d U d V + \bigg( B^2 (1 - U V)^2 + (1 - B^2) (1 + U V)^2 \bigg)
(d \vartheta^2 + \sin^2 \vartheta\, d \varphi^2) \right]\,.
\end{equation}
It is evident that this metric is regular on the cone $U V=0$ and can be extended beyond the corresponding horizon to the region
\begin{equation}
{\mathscr{R}} := \{ (U,V)\in\mathbb{R}^2~|~\, U V < 1 \}\,,
\label{Region_R}
\end{equation}
which is bounded by the two branches of the hyperbola $U V = 1$, corresponding to the spacelike infinity ${x} = +\infty$. The two branches of the hyperbola $U V = -1$ correspond to the throat $x = 0$. To go on, let us extend the transformation \eqref{xuv} setting
\begin{equation}
{t}={\ell\over 2(1-B^2)}\log\left|{ U\over V}\right|~, \quad
x=\ell \, {1+UV\over 1-UV}
\label{xuvest}
\end{equation}
whenever this makes sense.
The map
$(U,V) \mapsto x$ is smooth throughout the region ${\mathscr{R}}$, while $(U,V) \mapsto t$ is well defined
and smooth on the subregion $\{ (U,V) \in {\mathscr{R}}~|~U V \neq 0 \}$. The correspondence
$(U,V) \mapsto (t,x)$ gives diffeomorphisms
between the following pairs of regions: $( 0 , +\infty)\times( -\infty , 0)$ and $I$ (as already shown);
$( -\infty,0)\times( 0,+\infty )$ and
$I$;
$\{ (U,V)\in(0,+\infty)^2\,|\, U V < 1 \}$ and the exterior region $E^+$;
$\{(U,V)\in(-\infty,0)^2\,|\, U V < 1 \}$ and the exterior region $E^+$.
Under each one of these four diffeomorphisms, the metric of
Eq.~\eqref{Eq:dSWH1} takes the form \eqref{Eq:g_UV}. To conclude we note that, writing
$\Phi$ as in Eq.~\eqref{Eq:StaticSolutionsPotdS} and $x$ as in Eq.~\eqref{xuvest} we obtain a smooth extension of the scalar field $\Phi$ to the whole region ${\mathscr{R}}$.
\subsection{Extending spacetime further}
\label{Subsubsection:extension_ds}
We now consider a ``compactification'' of the extended region ${\mathscr{R}}$ \eqref{Region_R} based on the reparametrization
\begin{equation}
U=\tan \mathscr{U}\,,\qquad V=\tan \mathscr{V}\,.
\label{uhat_vhat}
\end{equation}
We know that the cone $UV=0$ and the limits $U\to +\infty$, $V\to -\infty$ and $U\to -\infty$, $V\to +\infty$ correspond to the horizons $x=\pm\ell$ in \eqref{xuvest}; according to Eq.~\eqref{uhat_vhat} the cone and the indicated limits are associated with finite values of $\mathscr{U}$ and $\mathscr{V}$, so the effect of the above transformation is to bring both the horizons to finite distances. One could use $\mathscr{U}$ and $\mathscr{V}$ as an alternative set of coordinates and reexpress the metric~\eqref{Eq:g_UV} and so on; but the situation can be described in a simpler way by making a further transformation (essentially, a rotation of ${\pi\over 4}$ and a translation of the axes)
\begin{equation}
\mathscr{U} ={T\over 2} - {X\over 2} + {\pi\over 4}\,,\qquad
\mathscr{V} = {T\over 2} + {X\over 2} - {\pi\over 4} \,.
\label{T_X}
\end{equation}
The composition of Eqs.~(\ref{uhat_vhat},\ref{T_X}), whenever they make sense, gives
\begin{equation}
U = \tan \left({T\over 2} - {X\over 2} + {\pi\over 4}\right)\,,\qquad
V = \tan \left({T\over 2} + {X\over 2} - {\pi\over 4}\right)\,;
\label{U_V_T_X}
\end{equation}
the application $(T,X)\mapsto(U,V)$ is a bijection between the regions ${\cal R}$ and $\mathscr{R}$, where $\mathscr{R}$ is defined by Eq.~\eqref{Region_R} and
$\mathcal{R}:= \{ (T,X)\in\mathbb{R}^2 | -{\pi\over 2}< T <{\pi\over 2},\, -{\pi\over 2} < X-T,X + T<{3\over 2}\pi\}$ .
In the coordinates $(T,X, \vartheta,\varphi)$ the metric~\eqref{Eq:g_UV} assumes the form
\begin{equation}
{\bf g} = \frac{1}{k^2\cos^2 T}\bigg[ -dT^2 + dX^2
+ \bigg( {B^2}\cos^2 T +( 1-B^2) \sin^2 X \bigg) (d \vartheta^2 + \sin^2 \vartheta\, d \varphi^2) \bigg],
\label{g_T_X}
\end{equation}
which clearly admits a further extension to the region $\mathcal{ S} \times S^2$,
where we have defined
\begin{equation}
\mathcal{ S} := \left\{ (T,X)\in\mathbb{R}^2\,|\, -\frac{\pi}{2}<T<\frac{\pi}{2} \right\}\,.
\label{S}
\end{equation}
Equations~(\ref{g_T_X},\ref{S}) provide the final form of our dS wormhole spacetime; the strip $\mathcal{ S} $ is represented in Fig.~\ref{Fig:dSWH}, which also accounts for some facts illustrated hereafter. Note that the metric (\ref{g_T_X}) is invariant under the spatial translation, the spatial reflection and the time reflection
\begin{equation}
\mathfrak{T} : (T,X) \mapsto (T, X + \pi)\,,\qquad\mathfrak{S} : (T, X) \mapsto (T, \pi - X)\,,\qquad \mathfrak{R} : (T,X) \mapsto (-T,X)\,.
\label{transf}
\end{equation}
Let us also remark that, in the limit case $B\to 0$, the expression (\ref{g_T_X}) reduces to the familiar representation of the dS metric as a conformal factor times the line element of the static Einstein universe.
For any $B>0$, the connection between the spacetime (\ref{g_T_X},\ref{S}) and the original setting (\ref{Eq:dSWH1},\ref{region_I_E})
is understood by expressing the original variables $(t,x)$ in terms of the new variables $(T,X)$. To this purpose we note that the composition of the transformations (\ref{xuvest},\ref{U_V_T_X}), whenever they make sense, gives
\begin{equation}
t = {\ell\over 2(1-B^2)}\log\left| {\sin T + \cos X \over \sin T - \cos X} \right|\,,\qquad
x = \ell {\sin X\over \cos T}\,.
\label{x_T_X}
\end{equation}
The map of $(T,X)\mapsto x$ is everywhere smooth on $\mathcal{ S} $, while the map of $(T,X)\mapsto t$ has singularities at the points of $\mathcal{S}$ where the argument of the logarithm vanishes or diverges; this occurs at points where $\sin T =\mp \cos X$, which are just the points where $x=\mp \ell$. Moreover, we note that
$(t,x) \circ \mathfrak{T} = (-t ,-x)$,
$(t,x) \circ \mathfrak{S} = (-t, x)$ and
$(t,x) \circ \mathfrak{R} = (-t,x)$; the behavior of $\mathbf{g},t,x$ under $\mathfrak{T}$ implies the invariance of each one of these three objects under the translation $\mathfrak{T}^2:(T,X)\mapsto(T,X+ 2\pi)$.
To go on, let us now introduce the diamond $\mathcal{I}$ and the triangles $\mathcal{ E}^\mp$ defined by
\begin{align}
\nonumber
\mathcal{I} &:=\left\{(T,X)\in\mathbb{R}^2\,|\, - {\pi\over2} < T-X,T+X < {\pi\over2}\right \}\,,\\
\label{I_E}
\mathcal{ E}^- &:=\left\{(T,X)\in\mathbb{R}^2\,|\, T < {\pi\over2},\, T-X > {\pi\over 2},\,T+X > -{\pi\over 2}\right \}\,,
\\
\mathcal{ E}^+ &:=\left\{(T,X)\in\mathbb{R}^2\,|\, T<{\pi\over2},\,T-X > -{\pi\over 2},\, T+ X > {\pi\over 2}\right \}
\nonumber
\end{align}
(see again Fig.~\ref{Fig:dSWH}); then the map $(T,X)\to(t,x)$, described by Eq.~\eqref{x_T_X}, gives isometric diffeomorphisms between $\mathcal{I}$ and $I$, between $\mathcal{ E}^-$ and $E^-$, and between $\mathcal{ E}^+$ and $E^+$, where $I$ and $E^\mp$ are, respectively, the internal region and the two exterior regions~(\ref{region_I_E}) with the metric \eqref{Eq:dSWH1}.
Moreover, we have that $x=\pm\ell$ along the sides of $\mathcal{ I}$, $x=-\ell$ and $x=-\infty$ along the sides of $\mathcal{ E}^-$ and $x=\ell$ and $x=+\infty$ along the sides of $\mathcal{ E}^+$ (see once more Fig.~\ref{Fig:dSWH}). It is easy to construct infinitely many replicas of the previous statement using the previous information of the behavior of $\mathbf{g},t,x$ under the transformations~(\ref{transf}). For example, using the fact that $\mathbf{g},t,x$ are invariant under all the iterates $\mathfrak{T}^{2 h}: (T,X)\mapsto(T,X+ 2h\pi)$ ($h\in\mathbb{Z}$), one can readily show that for each $h\in\mathbb{Z}$, the map~\eqref{x_T_X} gives isometric diffeomorphisms between
$\mathfrak{T}^{2 h}(\mathcal{I})$ and $I$, between $\mathfrak{T}^{2 h}(\mathcal{ E}^-)$ and $E^-$, and between $\mathfrak{T}^{2 h}(\mathcal{E}^+)$ and $E^+$. Moreover, by applying the time reflection $\mathfrak{R}$ to each one of the translated triangles $\mathfrak{T}^{2 h}(\mathcal{E}^\mp)$ one gets other regions isometrically diffeomorphic to $E^\mp$.
Finally, let us recall that we have already noted that the points $(T,X)$ where Eq.~\eqref{x_T_X} gives singularities for $t$ are just the points at which the same equation gives $x=\pm\ell$; so from the viewpoint of the extended manifold $\mathcal{S}\times S^2$, the apparent singularities at $x=\pm\ell$ of the original metric \eqref{Eq:dSWH1} are just due to the singularities of $t$ as a coordinate on $\mathcal{S}$.
Up to now, we have not considered the scalar field $\Phi$. The prescription
\begin{equation}
\Phi = \sqrt{\frac{2}{\kappa}}\arctan\frac{x}{b}\,,\qquad \mbox{ with $x$ as in Eq. \eqref{x_T_X}}
\label{phi}
\end{equation}
gives a smooth function everywhere on $\mathcal{ S}$, with the properties
$\Phi\circ\mathfrak{T}=-\Phi$, $\Phi\circ\mathfrak{T}^{2}=\Phi$ and so on.
The triple $\mathcal{S}\times S^2,\mathbf g,\Phi$ in Eqs.~(\ref{S},\ref{g_T_X},\ref{phi})
is a solution to the Einstein-scalar equations (with field self-potential
$V(\Phi)$ as in \eqref{Eq:StaticSolutionsPotdS}) .
Of course, the extended spacetime $\mathcal{ S}\times S^2$ has the topology of
$\mathbb{R}^2\times S^2$. For any fixed $p=1,2,3,...$ we can take the quotient
of the strip $\mathcal{ S}$ with respect to the iterated translation $\mathfrak{T}^{p}$;
the quotient $\mathcal{ S}/\mathfrak{T}^{p}$ has the topology of $\mathbb{R}\times S^1$
and the metric \eqref{g_T_X} can be projected on $(\mathcal{ S}/\mathfrak{T}^{p})\times S^2$,
thus getting a new spacetime with the topology $\mathbb{R}\times S^1\times S^2$. The function $\Phi$ of Eq. \eqref{phi} is projectable on this quotient
spacetime for $p$ even, since in this case $\Phi\circ\mathfrak{T}^p=\Phi$;
on the contrary, $\Phi$ is not projectable for $p$ odd because
$\Phi\circ\mathfrak{T}^p=-\Phi$. Finally, let us mention that
all spacetimes $\mathcal{ S} \times S^2$ and
$(\mathcal{ S}/\mathfrak{T}^{p}) \times S^2$ ($p=1,2,3,\ldots$) are time
orientable: in fact, $\partial/\partial T$ is a smooth timelike
vector field, defined everywhere on $\mathcal{ S} \times S^2$
and projectable on $(\mathcal{ S}/\mathfrak{T}^{p}) \times S^2$
both for $p$ even and for $p$ odd. One could also consider the quotients $(\mathcal{ S}/(\mathfrak{T}^{p}\circ \mathfrak{R}))$ with $p=1,2,3,\ldots$ involving the time reflection, which yield smooth spacetimes which are, however, not time orientable.
\begin{figure}[htbp]
\begin{centering}
\includegraphics[scale=0.6]{PenroseDiagramdSWHExt.eps}
\par\end{centering}
\caption{
Penrose diagram showing the strip $\mathcal{S}$ in the final extended spacetime of our dS wormhole (Eqs.~(\ref{S},\ref{g_T_X})). The dashed lines are lines with constant $x$, determined according to Eq.~\eqref{x_T_X}. Also indicated are the red diamond region $\mathcal{I}$ and the green triangular regions $\mathcal{ E}^{\pm}$ of Eq.~\eqref{I_E} which correspond to the original regions~\eqref{region_I_E} in the $(t,x)$ coordinate space; the same can be said of the images of $\mathcal{I}$ and $\mathcal{ E}^\pm$ under any translation $\mathfrak{T}^{2 h} : (T,X)\mapsto(T,X+2h\pi)$ ($h\in \mathbb{Z}$). Applying the time reflection $\mathfrak{R}: (T,X)\mapsto(-T,X)$ to the triangles $\mathcal{E}^\mp$ and to the translated triangles mentioned before, one obtains other regions which are isometric to $E^\mp$.}
\label{Fig:dSWH}
\end{figure}
\subsection{Linear instability of the dS wormhole in the inner region}
\label{Subsubsection:intstab_ds}
If one confines the attention to the spacetime $(I\times S^2, {\bf g}, \Phi)$,
where $I$ is the inner region \eqref{region_I_E} in $(t,x)$ space and $\Phi, {\bf g}$ are as in Eqs.~(\ref{Eq:StaticSolutionsPotdS},\ref{Eq:dSWH1}), the analysis of linearized perturbations for the Einstein-scalar equations is rather simple in the framework of this paper.
First of all, one replaces the coordinates $(t,x)$ with the coordinates $(s,u) \in \mathbb{R}^2$ defined by Eq.~\eqref{coordsu_dS}. After this, one should in principle apply the general scheme of sections \ref{Sec:LinearPerturbation}-\ref{Sec:Decoupling} (in the coordinates $(s,u)$) to the linearized perturbations of this solution, ultimately yielding a master equation. As a matter of fact, it is not even necessary to carry on this construction and it suffices to use the following trick: since the dS wormhole under analysis is connected to the AdS wormhole of sections~\ref{Subsection:AdS},\ref{SubSec:Pertads} through the formal replacement rules $(k,B,s,u) \mapsto (i k, i B, i s, i u)$ (see the comments after Eqs.~(\ref{Eq:dSWH1},\ref{gsu})), the master equation for the perturbed dS wormhole can be obtained, making formally the same replacements in Eqs.~(\ref{Eq:MasterAdS},\ref{Eq:PotentialB}) of the AdS case. In conclusion, the master equation governing linear perturbations of the dS wormhole, in an unknown function $\chi(s,u)$, reads
\begin{equation}
\left[ \frac{\partial^2}{\partial s^2}
- \frac{\partial^2}{\partial u^2} +
{\cal {V}} \right] \chi = 0\,,
\end{equation}
and involves the potential
\begin{equation}
{\mathcal{V}}(u) \equiv {\mathcal{V}}_{B}(u) :=
- \frac{B^2 \left(2-B^2+\cosh u\right) }{\left(-1+2 B^2 + \cosh u\right)^2}
\end{equation}
($s,u \in \mathbb{R}$); note that ${\mathcal{V}}(u)$ is everywhere negative and
vanishes like $-1/\cosh u$ for $u \to \pm \infty$.
The corresponding Schr\"odinger operator
$$
H := - \frac{d^2}{d u^2} + \mathcal{V}
$$
is selfadjoint in $L^2(\mathbb{R}, du)$, and can be
analyzed by standard methods, including Sturm oscillation theory
(\footnote{
Application of Sturm theory relies on
the zero energy Schr\"odinger equation $[-d^2/d u^2 + \mathcal{V} ]\chi_0 = 0$.
The general solution of this equation is obtained
from the analogous solution \eqref{Eq:AdSZeroMode} for the AdS case
with the formal replacements $(u,B) \mapsto (i u, i B)$ and reads
$$
\chi_0(u) = C_1 {\sinh {u \over 2} \over \sqrt{-1+2 B^2 + \cosh u}}
+ C_2 {2 u \sinh{u \over 2} - 4 B^2 \cosh{u \over 2}\over \sqrt{-1+2 B^2 + \cosh u}}\,
$$
($C_1, C_2 \in \mathbb{C}$). One has
$\chi_0 \in L^2(\mathbb{R}, d u)$ if and
only if $C_1=C_2=0$, thus zero is not an eigenvalue of $H$. If
$C_1 \in \mathbb{R} \setminus \{0 \}$ and $C_2=0$ it is evident that $\chi_0$ has a unique
zero in $\mathbb{R}$ (namely, $u=0$). If $C_2 \in \mathbb{R} \setminus \{0 \}$ and $C_1 \in \mathbb{R}$,
one can show that $\chi_0$ possesses two
zeroes in $\mathbb{R}$ (via an analysis rather similar
to that given for the function $\chi_0$ of the AdS case \eqref{Eq:AdSZeroMode};
see, in particular, the footnote which accompanies this equation).
Summing up, the \emph{minimal} number of zeroes of the real, non identically vanishing
solutions $\chi_0$ of the zero energy equation is \emph{one}.
The Sturm oscillation theorem (see Theorem 14.8 of \cite{Weid-Book}) states that such a minimal number
of zeroes is the number of negative eigenvalues of $H$.
So, $H$ has a unique negative eigenvalue; in addition, due
to general facts on Schr\"odinger operators (and to
the previous remark that $0$ is not an eigenvalue), $H$ has
continuous spectrum $[0,+\infty)$.}).
In this way, the spectrum
of $H$ is found to consist of a unique negative eigenvalue and of the continuous
spectrum $[0,+\infty)$. The situation is similar to that of the reflection-symmmetric
Ellis-Bronnikov wormhole: the system is linearly unstable, and the general
solution of the master equation has the form given by Eq.~\eqref{solbro} (with the
variables $(t,x) \in \mathbb{R}^2$ appearing therein replaced by the present variables $(s,u) \in \mathbb{R}^2$).
\subsection{Linear instability of the extended dS wormhole?}
For a full understanding of the subject under discussion,
linearized perturbations of the Einstein-scalar equations
should be treated on the extended spacetime $\mathcal{S}\times S^2$
of subsection \ref{Subsubsection:extension_ds}
(or on the quotients $(\mathcal{S}/ \mathfrak{T}^p)\times S^2$),
possibly in a gauge-invariant fashion.
The discussion of this problem would bring us outside the scope of
the present paper, since the extended spacetime $\mathcal{S}\times S^2$ is not static.
One can reasonably expect that the instability result of the previous subsection
about the inner region will
eventually produce a precise statement of linear instability for the extended dS wormhole.
However, we prefer to postpone these matters to future works; let us also mention
that the notion of linear instability is not so obvious if one perturbs
a nonstatic spacetime, and requires in our opinion a general discussion
before reconsidering the specific case of the extended dS wormhole.
\section{Conclusions}
\label{Sec:Conclusions}
In this work we have analyzed the linear stability of a class of static, spherically symmetric wormhole solutions in GR minimally coupled with a self-interacting phantom scalar field. To this purpose, we have provided a gauge-invariant perturbation formalism that describes the dynamics of linearized, spherical but time-dependent perturbations of the metric and of the scalar field, resulting in a coupled $2\times 2$ linear wave system subject to a constraint (see Eqs.~(\ref{Eq:WaveSystem},\ref{Eq:WaveConstraint})). Provided that a nontrivial, time-independent solution is known (as is usually the case when a family of static solutions is known) we have shown that this system can be decoupled to yield a master wave equation which is manifestly gauge-invariant and regular at the throat. This construction relies on a basic requirement (of course satisfied by the examples that we treat): the derivative $\Phi'$ of the (background) scalar field should vanish nowhere. The relevance of this condition in our approach is indicated by the almost ubiquitous presence of the reciprocal $1/\Phi'$ in the equations of sections~\ref{Sec:LinearPerturbation}-\ref{Sec:Decoupling}.
Based on our formalism we have rederived the regular master equation first obtained in~\cite{jGfGoS09a}, describing linear spherical perturbations of the Ellis-Bronnikov wormhole in a fully gauge-invariant setting and without intermediate steps involving singularities at the throat. (For an alternative approach which treats the reflection-symmetric case in a fixed gauge, see~\cite{fCfPlP19}.) Furthermore, we have analyzed the linear stability of an AdS wormhole introduced in~\cite{kBjF06}, for which the scalar field is subject to a nontrivial self-interaction term, and we have shown that this solution is linearly unstable as well. In both examples, the instability is characterized by a unique mode growing exponentially in time, associated with a bound state of negative energy of the Schr\"odinger operator arising in the master equation. As discussed in section~\ref{Sec:SpectralD} the associated instability times are rather short (of the order of a light-crossing time corresponding to the areal radius of the throat.)
Based on spectral analysis, we have also provided a detailed and rigorous discussion for the mode decomposition of the solutions to the master wave equations in both the aforementioned examples, which revealed that besides the modes growing exponentially in time, there might also be linearly growing modes, while all the remaining modes are oscillatory. In particular, the AdS wormhole has infinitely many normalizable, oscillatory modes in addition to the pair of exponentially growing and decaying modes associated with the unique bound state of negative energy of the Schr\"odinger operator.
In the last section, which is admittedly outside the mainstream of the present paper, we have also sketched the discussion of a dS wormhole with horizons, whose spacetime has a natural nonstatic extension; in this case we have provided a linear instability result, which, however, refers only to the static spacetime region within the horizons.
Let us conclude with some remarks on the possible future developments of the present work. We have already mentioned that the linear stability theory for nonstatic wormhole solutions, and its application to the (extended) dS wormhole, deserves further work in our opinion. Sticking to the case of static wormholes and of their linearized perturbations, we think that the forthcoming issues (i-ii) are worthy of future investigation:
(i) A basic requirement of our approach, recalled above, is the condition that $\Phi$ have no critical points. Removing this requirement would be interesting since, recently, a large class of new wormhole solutions of the Einstein-scalar equations has been found~\cite{Carvente:2019gkd}, generalizing previous work~\cite{Dzhunushaliev:2017syc}, in which the scalar field $\Phi$ has an extremum at the throat. Since $r$ has a global minimum at the throat and $r'$ converges to zero as fast as or faster than $\Phi'$, it turns out that the gauge-invariant quantity $C$ defined in Eq.~(\ref{Eq:CGI}) is still well defined; unfortunately, it is unclear if a decoupled equation for $C$ can be obtained that is regular at the throat. In connection with this problem, one could try to recover the S-deformation method of~\cite{jGfGoS09a,kBjFaZ11} (see the discussion in the Introduction; the formulation of this method in~\cite{kBjFaZ11} indeed considers the gauge-invariant quantity $C$). However, when the potential $V(\Phi)$ is nonzero, this method seems to require the numerical integration of a Riccati-type equation to find the regularized potential, and further one still needs to justify \emph{a posteriori} the validity of the transformed equation at the throat. An alternative possibility consists in applying a variation of the approach discussed in this article, in which $\Phi'$ is absent from all denominators, thanks to the use of new gauge-invariant quantities in place of the functions $A,C,E$ of Eqs.~(\ref{Eq:AGI}-\ref{Eq:EGI}); at present, it is not clear to us whether this will be possible.
(ii) Let us propose the following question: is there a \emph{deep} geometrical reason for which our present approach succeeds, in certain cases, in decoupling the perturbation equations~(\ref{Eq:WaveSystem}-\ref{Eq:WaveConstraint}) and reducing them to a single, scalar master equation? Typically, the possibility of reducing to a simpler form a PDE or of a system of PDEs is due to the presence of a Lie group of symmetries; an interpretation of this kind could perhaps be given for our decoupling method. As already recalled, our approach uses a static solution of Eqs.~(\ref{Eq:WaveSystem}-\ref{Eq:WaveConstraint}), arising from variations with respect to the parameters of a \emph{family} of static wormhole solutions. The availability of such parametric families could perhaps be interpreted in terms of a Lie group of symmetries, acting on the static solutions of the Einstein-scalar system; if so, it would be interesting to understand the interplay of these symmetries with the linearized perturbation equations.
\acknowledgments \noindent F.C. and L.P. were supported by: INdAM,
Gruppo Nazionale per la Fisica Matematica; Universit\`{a} degli
Studi di Milano. LP was also supported by: INFN; MIUR, PRIN 2010
Research Project \textquotedblleft Geometric and analytic theory of
Hamiltonian systems in finite and infinite dimensions." O.S. was
partially supported by a CIC grant to Universidad Michoacana. We thank an anonymous referee for pointing out to us Ref.~\cite{Torii} and for the suggestion to extend the stability
analysis to the dS case.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,656
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\section{Introduction}
Major trends in the manufacturing sector are the growing individualization of products and volatility of product mixes. If taken to extremes, this scenario also counts for products being produced only one time (lot-size-1) or only on demand.
In order to reach this goal, the concept of \emph{Flexible Manufacturing Systems (FMS)}, which can change their software during runtime \cite{YILMAZ1987209}, and \emph{Reconfigurable Manufacturing Systems (RMS)}, which can adapt their software as well as their hardware \cite{KOREN1999527}, play a vital role. Moreover, standalone systems from different manufacturers are interconnected to accomplish a common production goal and the production processes can be orchestrated automatically in so-called \emph{Plug-and-Produce} scenarios \cite{ARAI20001}.
Due to frequent changes of the products being manufactured, the rapid adjustment of a factory is a major challenge to implement application scenarios of flexible production systems (often called Industry 4.0 or Cyber-Physical Production Systems) successfully. Although the high flexibility of future flexible production scenarios promises a faster market adaptation and responsiveness, it raises at the same time dependability-related concerns due to unknown configurations at runtime. Thus, apart from functional aspect (i.e.~the check if a factory is able to manufacture a specific product), safety aspects as well as product quality assurance aspects must be addressed.
Safety standards, such as ISO 13849 \cite{iso13849} or IEC 62061 \cite{iec62061} in context of industrial production systems, provide guidelines to keep the residual risks in machine operation within tolerable limits. For every production system, a comprehensive risk assessment is performed, which includes risk reduction measures if required (e.g.~by introducing specific risk protective measures such as fences). The resulting safety documentation describes the assessment principles and the resulting measures that are implemented to minimize hazards. This documentation lays the foundation for the safe operation of a machine and it proves the compliance with the Machinery Directive 2006/42/EC of the European Commission \cite{directive}.
In flexible production scenarios, risk assessment must be conducted after each reconfiguration of the production system. Since this is a prerequisite for operating the factory in the new configuration, a manual approach can no longer effectively fulfill the objectives for assuring the safety in highly flexible manufacturing scenarios. Hence, the acquisition of safety-related information from each individual production step and the analysis of possible emergent hazards must be conducted in an automated way to quickly assess a new configuration of a manufacturing plant.
To evaluate the quality of a product considering the production process, a \emph{Process Failure Mode and Effects Analysis} (process-FMEA) is typically performed. During production, every process step can negatively influence the quality of the product depending on the negative outcome of the process step. This is captured in a process-FMEA with the concept of failure modes of a process step as well as the respective severity. It also defines measures to detect and deal with unwanted effects on product quality. Such an analysis is important to document the applied quality measures and to find out where drawbacks in the production process are and how they can be addressed.
Since the factory's configuration as well as its products constantly change in adaptable and flexible factory scenarios, a process-FMEA must be performed dynamically during the production of each product based on the configuration used. This is necessary to ensure that the products requirements w.r.t.~quality will be met by the provided production process.
In this paper, we present an approach for the model-based assessment of flexible and reconfigurable manufacturing systems based on a meta-model. This integrated approach SQUADfps (machine \emph{S}afety and product \emph{QUA}lity for \emph{f}lexible \emph{p}ro\emph{D}uction \emph{s}ystems) captures all information needed to conduct both risk assessment and process-FMEA dynamically during the runtime of the manufacturing system in an automated way. In this way, the approach enables flexible manufacturing scenarios with frequent changes of the production system up to a lot-size of one. In order to provide a better understanding for our proposed approach, we assume that the considered production systems are already installed as intended and focus only on the reconfigurability in terms of equipment and process changes.
The rest of the paper is organized as follows:
In Chapter \ref{relatedwork}, an overview of related work is given. Chapter \ref{modelbasedsafetyassessment} introduces a meta-model (Section \ref{metamodel}) that enables an automated hazard and risk assessment for the factory (Section \ref{modelbasedriskassessment}) and a process-FMEA for a product to be produced in a factory to ensure that the its quality is on track (Section \ref{dynamicprocessFMEA}).
Chapter \ref{casestudy} presents a case study to show how the model can be applied.
Chapter \ref{conclusion} summarizes the main results and provides an outlook for further research topics.
\section{Related Work}
\label{relatedwork}
The usage of model-based approach to carry out safety analyses or safety assessment aims to achieve compositional, reusable safety assessment and to improve traceability of information provided from the system design phase \cite{lisagor2011model}. The collective term \textit{model-based safety assessment} includes a wide range of techniques proposed in the academia \cite{joshi2005model,lisagor2011model} that have already been applied extensively nowadays in varying domains such as the automotive industry \cite{PAPADOPOULOS200577}, IT security \cite{houmb2002towards,1137696}, aviation sector \cite{bernard2007experiments}, train protection system \cite{wang2009study} and industrial automation e.g. collaborative robots application \cite{8247648}. In most of these applications, the safety requirements of the designed systems are assessed based on the functional system models created. Different tools and modeling techniques have also been developed since then to facilitate and maintain model-based safety engineering and safety analysis \cite{cancila2009sophia,grigoleit2016qsafe,prosvirnova2013altarica}. However, most of the mentioned publications deal with model-based applications during design and development phases instead of \textit{runtime} applications, which is one of the most important aspects for highly flexible manufacturing scenarios. To facilitate flexible and reconfigurable manufacturing systems in a practical way, as safety analyses nowadays are an inherently manual tasks in which only very few steps are automated, it is necessary to support these manual processes with automation as far as possible. Frequent system changes needed for lot-size-1 scenarios require runtime safety assessment to be done in an economically feasible manner\cite{koo2018challenges}. In this paper, we propose a method to carry out safety assessment automatically at runtime using a proposed meta-model, which can facilitate human during the decision-making to approve new system configuration.
\textit{Failure mode and effects analysis} (FMEA) has its origins in military applications \cite{MIL-STD-1629A} and was used in the same decade to analyze the influence of failures in production processes \cite{FordFMEA}. Since it is an effective but costly analysis technique, automating it has a long history in functional safety \cite{cichocki00G,cichocki01G,david.2008,papadopoulos04,walker.2009.a} and also for analyzing machinery.
In \cite{IntegratedFMEA}, it is mentioned that process-FMEA is part of an integrated approach for safe products, but that classic process-FMEA does not consider the manufacturability of a product influenced by quality problems. In \cite{CPP}, the authors use FMEA among other techniques to assess the manufacturability and estimating the cost of a conceptual design in early product design phases. They introduce an approach to estimate costs of failures during manufacturing using an extended FMEA approach introduced in \cite{FMERA}. This is a manual task that is used to prioritize different manufacturing options. Their work can be used in combination with the approach presented here to include costs of potential failures during manufacturing. In \cite{PPR}, product process resource-based approach is presented that uses an ontology to model the manufacturing capabilities and the required process steps to produce a product. Similar to the approach presented here, the authors use a standardized language set in an ontology to (semi-)automate the process of mapping production steps for a product to machinery. Nevertheless, they do neither aim for quality aspects of the output nor for rejected items in the mapping process.
\section{Model-based Safety and Quality Assessment of Flexible, Adaptable Production Systems}
\label{modelbasedsafetyassessment}
\subsection{Meta-model}
\label{metamodel}
Figure \ref{fig.pfmea} shows the proposed meta-model of SQUADfps for a flexible and reconfigurable production system. The meta-model is divided into four categories, considering both machine safety and product quality aspects:
\begin{itemize}
\item{\textbf{Process Definition}: In the product category, the elements address the order and steps related to \emph{what} has to be done to produce a product. This category also addresses the required safety approval process before the production is allowed to commence.}
\item{\textbf{Abstract Services}: The model elements of the category abstract services collect common specification of services and service parameters across all factories. These elements enable the specification of a product independently from a concrete production equipment. Besides, this category provides abstract service to carry out safety assessment for any concrete production equipment.}
\item{\textbf{Production Equipment}: The elements of this category model a concrete factory or production system along with its machinery equipment, describing what it can do, what quality measures are available and what safety functions are implemented.}
\item{\textbf{Process Implementation}: In the process implementation category, the elements address the concrete process used to produce a product. Here, the process steps address concrete ordered actions that are executed to produce a product. These steps provide a solution on \emph{how} a product is produced. Besides, concrete hazards associated to the process are identified.}
\end{itemize}
\begin{figure*}[ht!]
\centerline{
\includegraphics[scale=0.5]{img/metamodel2.png}
}
\caption{SQUADfps meta-model for process-FMEA and safety assessment supporting automation}
\label{fig.pfmea}
\end{figure*}
These categories allow users to map different activities, use cases and roles in the domain of dynamic reconfigurable production scenarios to automatically generate a process-FMEA (quality of the product) and a risk assessment result (safety of the production system) for the production system under consideration.
In the \emph{process definition} category, the product owner specifies \emph{which} steps are needed and in which order they need to be executed to produce a product (recipe). The product owner addresses abstract services that can satisfy steps of its recipe. Those abstract services provide a global library of all available services. Each service declaration can have constraints and parameters that can be set for a recipe step (service property). For example, the abstract service \emph{drill} requires the rotation speed of the drill and the size of the drill hole as parameters. When the abstract service is instantiated in a recipe step, these parameters need to be set.
Different failure modes can be stored (failure mode declarations) during the abstract service declaration. Independent from the concrete equipment or machinery (equipment), failure modes are known and defined in general. For example, the service \emph{drill} has the potential failure mode \emph{skew drill hole} for all concrete machinery implementing this service.
For each addressed service declaration in a recipe step, the failure mode declarations are known to the product owner that defined the recipe. He now can specify how severe the different failure modes (using recipe step failure modes) are for his product. Thus, the first step for the quality assessment using a process-FMEA can be performed without knowledge of the concrete equipment that later produces the product. This can be done for the combination of recipe steps and failure modes rated with a severity value.
Independent from this specification scenario of a recipe, the owner of a factory can model the equipment (\emph{production equipment}) with equipment services and safety functions. Equipment services address the abstract service declarations available in the global library. Equipment property constraints are used to specify the possible operating parameters and limitations of service property declarations, while equipment failure modes address the service failure modes of the abstract service.
During declaration of \emph{production equipment}, the factory owner can specify the available machinery and the equipment service along with its parameter limitations, which can be provided for a specific recipe. With this, the factory owner gets a list of possible abstract failure modes and can specify how often the abstract failure mode occurs for the concrete service (equipment failure mode). This can be known either by previously collected data or data provided by the manufacturer of the machinery. In this case, the factory owner can provide information about the occurrence value of concrete failure modes while using the equipment during the production.
In order to consider the safety of the production, the process will require safety approval before operation (process approval) during \emph{process definition}. Process safety requirements specify the minimal safety requirements to be achieved. For instance, the product owner can specify that only safety functions with a certain minimal safety guarantee are allowed due to safety criticality of the product or enforced safety guidelines. This process approval addresses the relevant abstract service (safety assessment), which checks whether all expected hazards are covered by the available safety functions considering the risk level.
Beside modeling the failure model for process-FMEA, the factory owner can also model the safety functions provided by an equipment. For instance, an equipment protective measure such as light curtain that is installed can protect the personnel during interaction with the equipment, which provides safety guarantee in term of \emph{performance level} to describe the reliability of the safety device. A safety function covers certain hazard types, which can be described through predefined semantics. A light curtain can protect personnel against mechanical hazards (crushing, shearing etc.), as long as the hazard source lies within the allowed working area and occurs during certain interaction tasks (safety constraint).
During \emph{process implementation}, the factory owner will get a list of possible hazards in relation to the interaction tasks, in which the frequency of the task can be defined. The risk parameter frequency describes the interaction of personnel with the production system. For a lot-size-1 scenario, the frequency can still be defined as high if the responsible personnel needs to carry out manual tasks for a foreseeable high amount of time. In combination with the concrete risk parameters (severity and possibility for avoidance) provided by the equipment, an identified hazard with its evaluated risk level (hazard property) can be checked against the safety function to ensure the production safety. Further examples for hazard properties include runtime location of hazard source, moving speed of its equipment, relevant interaction tasks etc.
\subsection{Model-based Risk Assessment}
\label{modelbasedriskassessment}
As mentioned before, a production process might include some human interaction tasks in different life cycle phases, such as setup of equipment, interactions during the production or maintenance activities. These interactions are specific to the concrete process and independent from the recipe, which describes the product to be manufactured. Each interaction task can include one or more hazards for the personnel involved, which have a certain level of severity. Each hazard also possesses a possibility of avoidance, which determine how possible a person can avoid the hazard during its occurrence. According to the risk graph in the standard ISO 13849 \cite{iso13849}, the risk level of a particular hazard can be evaluated using the severity of associated hazard (S), the frequency of tasks (F) and the possibility of avoidance (P). This leads to a risk level described in term of \emph{required performance level} (PLr).
Safety functions are typically installed to protect humans against a certain hazard and have a \emph{performance level} (PL) value, which describes the overall reliability of the safety device considering the components used. Having this information provided by the machine vendors, the required performance level gained from risk assessment (PLr) can be evaluated against the provided safety function performance level (PL). In a conservative manner, the production process can only be approved manually by the factory owner when all the identified risks are covered successfully by the available safety functions considering PL value.
\subsection{Dynamic process-FMEA}
\label{dynamicprocessFMEA}
Since equipment is not only able to execute production steps in a recipe, but is also able to execute quality measures, an equipment service can therefore cover certain failure modes. These measures can be originating from the same service, from a different service of the same equipment or from a service of a different equipment. For example, a robot arm that can be used to perform pick and place can also supervise its own actions using a camera. In this case, the failure mode \emph{misplacement} of the service \emph{pick and place} can be covered by the service \emph{camera supervision} from the same equipment. Using this methodology, the factory owner can specify which machinery can be used to increase the quality of the production. Since quality measures decrease the occurrence of certain failure modes, each covered failure mode stores a decreased occurrence value.
Using the severity of a failure mode from the product specification (recipe failure mode) multiplied by the occurrence value of the equipment failure mode or with the decreased occurrence value of a quality measure, a process-FMEA can be conducted for a product produced by a certain process on a concrete set of equipment.
This model-based approach ensures a structured and systematic analysis for all known failure modes that are captured within the model. This is valuable, as systematic and complete analysis is a requirement e. g. required by safety or quality standards. If experience from production about failures that actually were observed but not yet captured in the model is included, over time the analyis should become complete with regard to present knowledge. During the first applications in real production there might be a need to at least verify completeness by a manual inspection. A manual inspection and possibly extension of a pre-generated pFMEA is much less effort than starting from scratch, so even at the introductory phase there is already a reduction in effort to be expected.
\section{Case Study}
\label{casestudy}
In this small case study, we want to demonstrate how to use the meta-model as described in Section \ref{modelbasedsafetyassessment}. The product that we investigate here is a small roll that consist of a roll body, an axle and two metal discs as depicted in Figure \ref{fig.rolle} and Figure \ref{fig.scheibe}. The entire material is delivered on a tray, see Figure \ref{fig.tray}, and is set together by a robot arm that also greases the contact area of the parts. After that, a visual inspection detects insufficient products. To rate failure modes, we use an risk priority number (RPN) based approach for the parameters \emph{severity (sev)}, \emph{occurrence (occ)} and \emph{detection (det)} with a range from one to five whereas one represents the lowest severity, a negligible occurrence rate and a sure detection and five represents a high severity, a high occurrence rate and an nearly impossible detection. For the assessment of machine safety for the setup production system, performance level (PL) is used in accordance with ISO 13849-1\cite{iso13849}. Risk level of the identified hazards can be described through \emph{required} performance level (PLr), with the risk parameters \emph{severity}, \emph{frequency} and \emph{possibility of avoidance}, whereas \emph{PL a} represents the lowest tolerable risk and \emph{PL e} represents the highest risk.
\begin{figure}[ht]
\centerline{
\includegraphics[scale=0.86]{img/rolle.pdf}
}
\caption{Roll with axle}
\label{fig.rolle}
\end{figure}
\begin{figure}[ht]
\centerline{
\includegraphics[scale=0.86]{img/scheibe.pdf}
}
\caption{Disc to be mounted and greased}
\label{fig.scheibe}
\end{figure}
\begin{figure}[ht]
\centerline{
\includegraphics[scale=0.8]{img/trayy.pdf}
}
\caption{Tray with products}
\label{fig.tray}
\end{figure}
\subsection*{Dynamic process-FMEA}
The recipe steps for production are depicted on the left side in table \ref{fig.example_neu} for recipe $R=r_1,\dots,r_6$. For the first process $P=p_1,\dots,p_{6a}$ , the tray is delivered using an abstract service \emph{convey} which is implemented by the equipment \emph{belt conveyor}. The failure modes of this service are \emph{misplacement} and \emph{shock} rated by the design team with a severity value of four and five respectively. The production equipment produces failures with a occurrence of two and one. A visual inspection can safely detect both failure modes (detection value \emph{Det} is 1).
\begin{table*}[ht]
\caption{Example product recipe and two processes using abstract services.}
\centerline{
\includegraphics[scale=0.6]{img/example_neu_2_bw.png}
}
\label{fig.example_neu}
\end{table*}
The next step is to mount the axle inside the roll. This step is fulfilled by the service \emph{pick and place} which is implemented by a robot arm. This service can fail in two ways, the object can be misplaced but can also be crimped by the clutch. Crimping is not very severe to the axle since it is made from solid metal. This cannot be detected by a visual inspection (detection value \emph{Det} is 5). Both discs need to be greased and there can be too much and too little grease. Having too little grease is quite severe, and the worker can detect it. Having too much grease is just a minor failure. Since the roll itself is made from plastic material, crimping is severe since the roll can be damaged. This failure mode can hardly be detected by the worker, since he is not doing a stress test (detection value \emph{Det} is 5).
The elements of properties and constraints are not depicted in the table for the reason of space limitations. Service properties of \emph{pick and place} would include, for example, start- and endpoint, trajectory and weight, whereas an equipment implementing this service provides limitations of those parameters and recipe steps requesting the service would provide the required information to fulfill the step.
With the failure mode information provided by the service definition, the design team can specify \emph{what} failure mode is severe (requirement) and the vendor can specify \emph{how} often the failure mode appears on its machinery and \emph{how} the effect of the failure mode can be prevented in later products (implementation). The process $P$ generally is capable to implement the recipe $R$ since the equipment fulfills the required service of each recipe step and the relative order of the process steps matches the order of the recipe steps with an additional step at the end of the process: $p_{6a}$.
Also depicted in table \ref{fig.example_neu} is an additional process $P'=p'_1,\dots,p'_{6a}$ that also fulfills recipe $R$ but with different equipment. A different robot arm is used, that has a lower probability of crimping. Additionally, the visual inspection is implemented by a more precise laser scanner that better detects crimping. With these two adoptions in place, the highest risk priority number is lowered from 100 to 20.
This approach in its basic implementation is of a qualitative nature. It therefore enables comparing different production alternatives and facilitates the selection of a appropriate schedule selection for the production of a concrete product.
In a specific domain the quality criteria might be specified in a quantitative manner, failure probabilities replacing occurence values and actual costs at risk replacing severity values. If this is possible for a certain domain or use case then for products and production quantitative goals can be specified and the selection or ranking of different schedules with regard to fulfilling quality requirements of a product can be done. A manual selection will probably still be necessary to balance e. g. quality goals with other goals not captured within this model.
\subsection*{Model-based Risk Assessment}
Using the same production process described above, an example for the conduct of safety assessment using an abstract service, as depicted in figure \ref{fig.pfmea}, can also be shown. In this production process, the operator is required to load the product parts (roll body with axle) in a frequent manner onto the belt conveyor. Besides, the robot's handling tool needs to be adjusted and maintained occasionally to ensure its high precision. Hence, the task frequency of these two interaction tasks can be described as F2 (high frequent) and F1 (low frequent) respectively. As the frequency is defined in relation to the overall process duration required, its definition is hence independent from the product lot sizes.
The initial production system with \emph{Belt conveyor} and \emph{Robot Arm} in table \ref{fig.case_study_1} introduces three different hazards for the described interaction tasks. During the loading of production parts, the movement of robot arm can cause shearing points with high severity (S2), which can hardly be avoided due to its high movement speed (P2). On the other hand, the moving belt conveyor introduces possible squeezing points to the operator with the risk parameters S1 and P1 thanks to its relatively well-considered inherent design. During the maintenance of robot arm, the operator might still be bruised by the arm movement (S2) although the movement speed is monitored by its safety control function (P1). Here, only a light curtain is provided as a safety function with the performance level PL d.
As shown in table \ref{fig.case_study_1}, the current setup does not fulfill all the safety requirements. Hazard $h_1$ with a high level of risk PL e does not receive a suitable safety function that fulfill the required performance level. In addition, there is no available safety function that can counter the hazard $h_3$. Based on this result, the responsible individual can decide whether to reduce the risks, to eliminate the risks or to provide extra protective devices to the system. This involves creative decision-making process and is not being considered in the proposed meta-model. The generated risk assessment result can then provide instant updated information after every system modification to assist the decision-making process.
It is assumed that the financial situation allows the factory owner to acquire new equipment. In order to improve the safety of the production system, a different robot arm (\emph{Robot Arm 2}) that provides the same services is now used, as depicted in table \ref{fig.case_study_2}. This robot arm is equipped with an integrated sensor skin (\emph{safety sensitive cover}) that can detect human approaches and turn off the robot once the operator violates the safety distance. This sensor skin provides a safety assurance of PL e. With this new robot arm, all previously unfulfilled safety requirements are now satisfied by the provided safety functions. The abstract service now confirms the results and awaits safety engineer to make the final approval.
\begin{table*}[ht]
\caption{Exemplary risk assessment for the provided product recipe using abstract services (safety requirements are not completely fulfilled).}
\centerline{
\includegraphics[scale=0.4]{img/case_1.png}
}
\label{fig.case_study_1}
\end{table*}
\begin{table*}[ht]
\caption{Risk assessment after implementing counter measures using a different robot arm (safety requirements are now fulfilled).}
\centerline{
\includegraphics[scale=0.4]{img/case_2.png}
}
\label{fig.case_study_2}
\end{table*}
This example shows how the usage of an abstract service allows the definition of an abstract production recipe without addressing concrete production equipment. The product design team uses abstract service declarations and properties to formulate production requirements. It can be decided (semi-)automatically if the production equipment can manufacture a product defined by a recipe. By providing information about the severity of certain failure modes, those requirements are extended by quality requirements. In a second step, a factory can map its production equipment to this abstract language and evaluate if it can produce the recipe. By providing information about the occurrence of failure modes of the existing production equipment, it can be evaluated using RPNs if the required quality can be met or if additional quality measures need to be implemented to increase the quality. By having a budget for a recipe, the vendor of a product can evaluate the economic efficiency of its possible production scenarios and decide to produce a product or to decline an offer. By comparing the RPNs of prospective processes and their economic deficiencies, an optimal process can be selected.
The same applies to the risk assessment procedure. By using abstract service definitions, the integrated production equipment can be checked automatically during runtime to guarantee the safety of operators while interacting with the production system. The known interaction tasks are firstly associated with information regarding possible involved hazards, whereas the severity and possibility of avoidance are then described concretely by the integrated production equipment. The frequency of interaction tasks can also be predefined in order to evaluate the required risk level using performance level PLr along with the other risk parameters. Considering the available safety functions along with its constraints at runtime, the production system can be assessed against the identified hazards, emphasizing hence the critical points that require further safety considerations and safety measures. This ensures a higher efficiency, quality and completeness of the risk assessment result, which is usually done manually nowadays.
\section{Conclusion \& Future Work}
\label{conclusion}
In this publication, we presented an integrated model-based approach SQUADfps that enables both the automated conduct of risk assessment and the dynamic creation of a process-FMEA for a flexible, adaptable or reconfigurable production system. Our proposed meta-model provides the foundation to enable flexible production scenarios in which individual and customer-specific productions can be manufactured up to lot-size-1. The proposed model-based risk assessment can ensure the safe operation of a new, previously unknown configuration of the manufacturing system by conducting the required risk assessment in an automated way based on the information available in the meta-model. Moreover, the evaluation on whether a specific product can be manufactured while meeting the customer's quality requirements by a specific configuration of the plant (as well as a cost-efficient use of quality assurance mechanisms within the manufacturing process) can be conducted by generating a process-FMEA in an automated manner. By applying the proposed model-based approach, all information required to perform these assessments can be provided automatically during runtime. The currently manual and time-consuming tasks to conduct assessments can be automated. This assists the decision-making process of human and thus, enables the fast reconfiguration of production systems in flexible production scenarios. In the future, this integrated model-based approach will be applied to further use cases to improve the completeness and significance of the generated results.
\section*{Acknowledgement}
\label{acknowledgement}
The work leading to this paper was funded by the German Federal Ministry of Education and Research under grant number 01IS16043Q and 01IS16043O (CrESt).
\bibliographystyle{splncs04}
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{"url":"https:\/\/mahara.org\/interaction\/forum\/topic.php?id=5793&post24806","text":"# Forums | Mahara Community\n\n## Developers \/ Custom Jquery form field in CPD plugin\n\nPosts: 6\n\n11 October 2013, 8:39 PM\n\nHello everyone, this is my first post so please be gentle with me!\n\nI'm trying to adapt the location input field in the CPD plugin (great plugin by the way!) to use Jquery (or javascript). I want to hook it up to a database with a list of locations, and when the user starts to type (and reaches around 3 or 4 \u00a0characters), the database will return a sub-dropdown menu with the choices available.\n\nI've tried replacing the entire form in a template which seems to work ok and the jquery side of things work just fine, but then I can't figure out how to get the form data back into the lib.php in order to continue with the rest of the process, as it's expecting a pieform.\n\nI've also tried creating a new pieform element which also seems to work, but then I can't figure out how to incorporate the Jquery!\n\nI'm very new to my job\/ mahara\/using templates having just finished uni so I'm kind of in at the deep end with the whole templating business. If anyone has any pointers I'd be so grateful, perhaps at I'm looking at the problem wrong, I just don't know.\n\nPosts: 108\n##### Re: Custom Jquery form field in CPD plugin\n\n13 October 2013, 10:39 PM\n\nHi Sarah\n\nGlad to hear you are finding the CPD plugin useful.\n\nHowever, probably can't help much with the jQuery (I guess you are using the jQueryUI Autocomplete widget) \/ Pieforms integration. During development of the CPD plugin, we used Pieforms, drawing heavily from the design of the existing Plans plugin.\n\nOne note about jQuery. Mahara also uses the MochiKit JavaScript library. So, to use jQuery alongside this, you may have to use the jQuery or $j function rather than$ (used by MochiKit). See the code comments in \/lib\/web.php for details.\n\nGood luck with adding new features to the CPD plugin. If they are useful, I'll be very happy to integrate them into the plugin code.\n\nPosts: 99\n##### Re: Custom Jquery form field in CPD plugin\n\n14 October 2013, 7:23 PM\n\nHi Sarah and Geoff,\n\nWe're not using the plugin and I haven't had a look at the code. But I have some notes on how to loading the JavaScript-Code. I assume you put the code into a separate file.\n\nIf the smarty-Template is created with a clear $smarty = smarty();statement, you can put the path to the javascript file in an array right into the constructor. Look at artefact\/blog\/view\/index.php for an examle. I think this doesn't work if it is created with $smarty=smarty_core()\n\nYou can also include the javascript-file in the template, look at artefact\/blog\/theme\/raw\/image_popup.tpl for an example. Seems like WWWROOT is generally accessible in the smarty-templates.\n\nI'm not sure what you mean by \"replacing the entire form in a template\", did you write your own html-code for a form from scratch instead of using pieforms and hand that to the template? Then you'd have to write your own logic for evaluating the return-data. It seems to me that changing the given pie-form is easier and less work, just replace the entry for the input-field - or just add the javascript-logic that adds the autocomplete-functionality to the input-field if that's all that it takes. If you don't replace the input-element or the new element has the same name, the submit()-function referenced in the pie-form can handle the return-data just as before.\n\n@Geoff: A question about the jQueryUI: Is that already included in Mahara or would you have to put the jQueryUI-framework into the plugin-code? I disregarded that last idea when I wanted an autocomplete-field because it felt like overkill loading the complete jQueryUI just for one little feature in one plugin. Not sure though if writing the autocomplete-plugin from scratch made more sence in the end.\n\nAnd yes, it took me some time to figure out the jQuery()-call instead of \\$(). Good hint\n\nPosts: 6\n##### Re: Custom Jquery form field in CPD plugin\n\n14 October 2013, 10:10 PM\n\nHi Geoff andTobias.\n\nThanks for your replies. I probably will use the Jquery auto complete widget, although I haven't got that far yet!\n\nAs I was getting so confused I just cut the form out of the source code from the browser and put that into a variable within the lib.php which gets passed in instead of the the pieform. All I need to do now is add in the callbacks and it should work. I have included Jquery within the function\/form but I will try taking it out and using jQuery() instead. At the moment I'm not so bothered about things being optimised as much as I am just getting things to work!\n\nI can't say I'm a fan of pieforms. I've only ever written code from scratch and I prefer having the control to do what I want without trying to figure out a whole different way of doing things. That said I can see the benefit of using template engines and did breifly use Freemarker as part of a MongoDB course I did. At any rate, this has certainly been an experience!\n\nPosts: 99\n##### Re: Custom Jquery form field in CPD plugin\n\n15 October 2013, 3:18 AM\n\nHi Sarah,\n\nI can understand your initial aversion for pieforms. It took me some time to get used to them myself. But copying the html-code, modifying a bit here and there and then plugging that back into the php-code isn't the same as programming from scratch.\n\nAnother thing to consider is that maybe someone else might have to read your code later on. In that case it will be way easier for him to understand your code if you use the same pieforms as elsewhere in mahara.\n\nI'm still not sure what to think about pieforms. I found some of the links to documentation on the sourceforge-page broken and I still don't know why the standard-renderer doesn't support container-elements. But it does solve the weirdness of having to deal with the arguments of the form in the targeted page - by calling a submit-function and redirecting.\n\nSo you might want to give the pieforms a try\n\nPosts: 6\n##### Re: Custom Jquery form field in CPD plugin\n\n15 October 2013, 4:39 AM\n\nI didn't mean to imply that I was coding from scratch here... everything I did at university was from scratch (I don't think we'd have got away with refactoring someone else's code there!), the point is I just haven't got my head around templates yet.\n\nI totally understand your comment about someone else having to read the code; deciphering what the heck is going on in mahara in general has highlighted that well enough,\u00a0but at this point in time I only have two weeks to get what I'm trying to do up and running and I've already done most of this in my own time.\n\nI'm sure I will go back to pieforms at some point as I hate not doing things properly, but for now I'm more interested in it working by any means necessary.\n\nThanks for your help, it will most definitely come in useful\n\nPosts: 896\n##### Re: Custom Jquery form field in CPD plugin\n\n15 October 2013, 2:04 PM\n\nYeah, the Mahara Developer docs are woefully incomplete and out of date at this point. I've been meaning to reorganize and revise them, but it's hard to find the time. So, I've started by just re-writing a few pages for now, with the pages I've rewritten stored here for the time being: https:\/\/wiki.mahara.org\/index.php\/User:Aaronw\n\nI'm also not a big fan of Pieforms. It is very useful to have a forms library, in order to standardize display and validation and save the devs a lot of reptition. But Pieforms is pretty PHP 4 in a lot of ways. However, it's very entrenched in the code base at present, so it would be hard to find a graceful way to switch to something else.\n\nI think my main annoyances about Pieforms are:\n\n\u2022 The way it automagically does everything when you call the pieform() function, which just looks like an innocent constructor or factory method but actually functions as more of a \"controller\" method that can throw the page onto an entirely separate execution track, including calling exit() so nothing after it runs\n\u2022 The way validation and page submission are done, by default, via callback methods following a particular naming pattern\n\u2022 The way pieform elements aren't based on classes or inheritance, but on files and functions following a naming pattern\n\u2022 The way the pieform() method takes a huge structured array to define the form and its behavior\n\nAll of that makes it really hard for the unitiated to tell what's going on, and what the API is, especially since the developer docs are pretty dusty at this point.\n\nFrom what you want to do, it sounds like all you really need is to include the text field and select menu as pieform elements, then add some JQuery javascript that will locate them by their ID and attach additional functionality to them. So long as the JQuery updates the underlying <input> and <select> elements' values, those will get passed back when the form is submitted and you should be able to handle it in a normal Pieforms submission callback function.\n\nAs for how to get that JQuery on there, Tobias' suggestion of passing it to smarty() is probably the best. You could also create a custom pieform element type and give it a pieform_element_{name}_views_js() function, but that might be overkill if you only need the JS used on one particular form. If you do want to go that route, note that Plugins can define their own Pieform elements (in 1.7 the only example of this is the artefact\/file\/form\/elements\/filebrowser.php element... which in 1.8 we've actually relocated to lib\/form since it's used by a lot of other pieces of code now. But the support for Plugins to have their own custom Pieform elements is still there!)\n\nCheers,\n\nAaron\n\n7 results","date":"2018-03-24 06:22:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.26777076721191406, \"perplexity\": 1297.8828471552024}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257649931.17\/warc\/CC-MAIN-20180324054204-20180324074204-00118.warc.gz\"}"}
| null | null |
From the warped mind of the guy who brought you LEGO Breaking Bad here is a game that will not be shown today at E3: Bluthfighter—The Arrested Development Fighting Game.
The video is by Brian K. Anderson. Enjoy.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,501
|
\section{Introduction}
\label{intro}
Low-power (FRI: \citealt{fanaroff74}) radio galaxies are commonly
found in the centres of rich galaxy groups and clusters, where they
are thought to play an important role in regulating the central gas
properties and galaxy evolution via a (currently poorly understood)
feedback process \citep[e.g.][ and references
therein]{mcnamara07,fabian12}. Among the many uncertainties about the way in
which this feedback process operates, one long-standing problem is the
unknown nature of the dominant particle or field component within the
radio lobes, which are important as the lobes are the means of energy
transfer to the surrounding gas via their expansion. The radio
synchrotron emission from the lobes provides only a combined
constraint on electron density and magnetic field strength, and so it
has been common to assume equipartition of energy in field and
radiating particles \citep[e.g.][]{burbidge56}, which corresponds
roughly to the minimum total energy the source requires in order to
produce the observed radio emission. But while the lobes of powerful FRII
radio galaxies appear to be close to equipartition
\citep[e.g.][]{croston05a,kataoka05}, it has been known for some time
that the energy content of FRI radio galaxies must be distributed
differently to that of FRIIs, as the radiating particles and magnetic
field, if at equipartition, cannot in the vast majority of cases
provide sufficient pressure to balance the measured external pressures
surrounding FRI lobes \citep[e.g.][]{morganti88,worrall00}.
The external pressure acting on the jets and lobes can now be
constrained tightly on scales of a few to several hundred kpc for many
low-power radio galaxies, using X-ray observations of the surrounding
group or cluster gas with {\it Chandra} and {\it XMM-Newton}
\citep[e.g.][]{hardcastle02b,croston03b,croston08a}. If it is assumed that the jets and lobes
are close to pressure equilibrium with the surrounding medium (likely to be
true on kpc -- hundred kpc scales for low-power sources), then the
external pressure profile must correspond closely to the run of
internal pressure along the jet as it evolves into a lobe or plume.
The internal pressure cannot be measured directly from the radio
observations of the source; however, the internal pressure in some
combination of radiating particles (electrons and positrons) and
magnetic field can be measured by modelling the radio emission. This
type of comparison has now been carried out for many low-power radio
galaxies, including large samples of cavity sources in galaxy clusters
(including so-called ``ghost'' cavities in which any radio emission is
weak or absent), and, as mentioned above, typically shows that the
radiating particles and magnetic field cannot dominate the internal
energy of the source {\it if they are at equipartition}
\citep{croston03b,croston08a,dunn04,dunn05,dunn06b,birzan08}.
Given that the lobes of low-power radio galaxies cannot be dominated
by an equipartition electron-positron plasma, other models for the
energetically dominant component of the radio lobe contents must be
considered. The two most obvious explanations are that the dominant
internal pressure is provided by a departure from equipartition or by
a significant population of non-radiating particles. There is evidence
from X-ray inverse Compton observations that powerful FRII radio
galaxies may deviate from equipartition by a small amount in the
direction of electron dominance \citep[e.g.][]{isobe02,croston05a}; however,
electron dominance by large factors would be expected to produce
detectable levels of X-ray inverse-Compton emission in at least some
FRI radio galaxies, which are inconsistent with observations
\citep{hardcastle98e,croston03b}. Recently, detailed models of magnetically dominated
jets and lobes have been developed \citep[e.g.][]{li06,nakamura06};
however, they are difficult to reconcile with observations, e.g. of
radio jet polarization properties and geometry (see later discussion).
Proton-dominated models have been discussed by a number of authors
\citep[e.g.][]{deyoung06,birzan08,mcnamara07}, but it is energetically difficult
to supply the proton population required by transport from the inner
jet \citep[e.g.][]{deyoung06}.
There are several reasons to favour instead a model in which
entrainment of material as the jet expands leads to an
energetically dominant proton population on scales of tens to hundreds
of kpc. Entrainment of the ISM and ICM is thought to be the means by
which FRI jets decelerate from relativistic to transonic speeds on kpc
scales \citep[e.g.][]{bicknell94}. There is growing observational evidence
that entrainment is occurring \citep[e.g.][]{hardcastle03b,hardcastle07a}, as well as
strong support for its importance from detailed kinematic modelling of
FRI jets \citep[e.g.][]{laing02b,laing06a}. A model in which entrainment
accounts for the apparent ``missing'' pressure in FRI radio galaxies
also has the advantage of explaining the observed difference in the
energetics of the FRI and FRII populations (the former being massively
underpressured if at equipartition, while the latter appear close to
equipartition both from IC observations and pressure comparisons)
without the need to invoke differences in the intrinsic particle
content of the inner jets, which might require different jet production
mechanisms: since FRII jets do not decelerate or interact with their
environments significantly, they would not in general be expected to
entrain significant amounts of material. Finally, we have previously
found a relationship between FRI source structure and particle/energy
content, suggesting that sources likely to be undergoing strong
entrainment have a larger contribution from non-radiating material
than those likely to be weakly entraining \citep{croston08a}. This provides
further support for an entrainment-dominated model.
In this paper we investigate in detail the observational constraints
on models for the particle and energy content of low-power radio
galaxies, by considering how the non-radiating and radiating
components of the jets in the well-studied radio galaxy 3C\,31
must evolve with distance in order to maintain pressure balance and
produce the observed radio emission. We use new deep X-ray data and
high-resolution radio data to place tight constraints on the external
pressure and internal pressure from radiating particles and field
within the radio jets and plumes of 3C\,31 . We consider in detail the constraints this
result provides for what particle population or magnetic field
structure dominates the source energetics, and
also carry out a pressure comparison for the cluster-centre source
Hydra\,A as a preliminary test of the generality of our results.
Throughout the paper we use a cosmology in which $H_0 = 70$ km
s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm m} = 0.3$ and $\Omega_\Lambda =
0.7$. At the redshifts of 3C\,31 ($z = 0.0169$) and Hydra\,A ($z = 0.0549$),
this gives luminosity distances of $D_{L} = 73.3$ Mpc and $D_{L} = 244.9$
Mpc, respectively, and angular scales of $0.3438$ kpc/arcsec (3C\,31)
and $1.067$ kpc/arcsec (Hydra\,A). Spectral indices $\alpha$ are defined in
the sense $S_{\nu} \propto \nu^{-\alpha}$. Reported errors are
1$\sigma$ for one interesting parameter, except where otherwise noted.
\section{Observational constraints}
\label{obs}
\subsection{External pressure of the hot-gas environment}
\subsubsection{3C\,31}
We used new {\it XMM-Newton} data to obtain a radial profile of the
external pressure surrounding the radio jets and plumes in 3C\,31. We
observed 3C\,31 on 2008 July 1st for $\sim 50$ ks (ObsID
0551720101). The data were processed in the standard way using {\it
XMM-Newton} SAS version 11.0.0, and the latest calibration files from
the {\it XMM-Newton} website. The pn data were filtered to include
only single and double events (PATTERN $\leq 4$), and FLAG==0, and the
MOS data were filted according to the standard flag and pattern masks
(PATTERN $\leq 12$ and \#XMMEA\_EM, excluding bad columns and
rows). Unfortunately the observation was badly affected by background
flares, and so after filtering for good time intervals, the remaining
clean exposure durations were 24, 29, and 24 ks for the MOS1, MOS2 and
pn cameras, respectively.
Surface brightness profiles in the energy range 0.3 -- 5.0 keV were
extracted from the {\it XMM-Newton} data using the closed-filter
double-background method described by \citet{croston08a}. The {\it
Chandra} surface brightness profile of \cite{hardcastle02b} was also used to
help constrain the inner profile shape. The combined {\it XMM-Newton}
(MOS1, MOS2 and pn) profile and {\it Chandra} profile were jointly
fitted with a projected double beta model \citep{croston08a}, convolved
with the appropriate point-spread function (PSF) for each telescope,
using the Markov-Chain Monte Carlo (MCMC) method described by
\citet{ineson13}. The resulting model was used to obtain a gas density
profile for the environment.
A corresponding temperature profile was obtained by extracting spectra
from six annular regions, and using the background fitting method
described by \citet{croston08a}, which correctly accounts for both
particle and X-ray background, to obtain (projected) temperature
measurements. For each region, the spectra from the three {\it
XMM-Newton} cameras were fitted jointly with an {\it apec} model
(using the energy range 0.3 -- 7.0 keV, but excluding the region
between 1.4 -- 1.6 keV, which is affected by an instrumental
line). The normalizations for the three cameras were allowed to vary,
but the temperatures were tied together. A free abundance fit led to
unphysically large values for the abundance, and so we fixed the
abundance to the best-fitting abundance from a global spectral fit ($Z
= 0.3$). The results of spectral fitting are given in
Table~\ref{spectra}. For the inner regions of the group, we used the
{\it Chandra} temperature profile of \citet{hardcastle02b} in order to
obtain more accurate pressure constraints. We used the deprojected
temperature values, although the effect of deprojection on the
temperature profile is small. In the outer regions of the group the
temperature varies only by $\sim 20$ per cent, so that any uncertainty
from not correcting for projection is small, and less than the
statistical uncertainty on the outer temperature.
In order to obtain a gas pressure profile with high resolution, we
fitted the measured temperature profile with the analytic model of
Vikhlinin et al. (2006), and obtained a finely binned look-up table
for $\Lambda (T)$, the conversion factor between volume emission
measure and gas density (obtained from {\sc xspec}). The resulting
table was used together with the analytic temperature model to obtain
a gas pressure profile, which is shown in Fig~\ref{press1}.
\subsubsection{Hydra\,A}
Although the majority of this paper focuses on 3C\,31, using our new
X-ray data, we also carried out a pressure comparison for Hydra\,A as
a preliminary test of whether our findings are likely to apply widely
to FRI radio galaxies. For Hydra A, we did not reanalyse the archival
{\it Chandra} and {\it XMM-Newton} observations, but made use of
previously published gas density and temperature profiles. For the gas
density profile we used the double beta model of \citet{wise07},
normalised to the density profile published by \citet{david01}. We
interpolated over the (projected) temperature profile of \citet{david01}
to obtain a cluster pressure profile over the radial ranges of
interest, which is also shown in Fig.~\ref{press1}.
\begin{table}
\caption{Results of spectral fitting for the environment of
3C\,31. Spectral fits were obtained for annular regions between the
radii listed, using an {\sc apec} model, were in the energy range
$0.3 - 7$ keV, assuming $N_{H} = 5.4 \times 10^{20}$ cm$^{-2}$. The abundance was fixed to the best-fitting value from a global spectral fit, as free abundance fits led to unphysical values (likely due to the additional free parameters of the background model).}
\label{spectra}
\begin{tabular}{lrrr}
\hline
Region&$kT$&$Z$&$\chi^{2}$ (d.o.f.)\\
\hline
60 -- 80 arcsec&$1.58\pm0.1$&0.1&64 (65)\\
80 -- 120 arcsec&$1.62^{+0.6}_{-0.7}$&0.3&157 (167)\\
120 -- 200 arcsec&$1.60^{+0.5}_{-0.6}$&0.3&466 (395)\\
200 -- 300 arcsec&$1.54\pm0.5$&0.3&722 (608)\\
300 - 450 arcsec&$1.36^{+0.2}_{-0.1}$&0.3&1046 (835)\\
450 -- 600 arcsec&$1.09^{+0.12}_{-0.01}$&0.3&1191 (974)\\
\hline
\end{tabular}
\end{table}
\subsection{Internal pressure from radiating particles and magnetic field}
\label{radio}
The radio emission from the sources does not place a constraint on the
total internal pressure of the lobes, as the radiating plasma could be
far from equipartition; however, it does place constraints on the
internal pressure of the radiating particles and magnetic field. We
used high-resolution radio data to obtain profiles of synchrotron
emissivity along the jets.
For 3C\,31 we used the combined 1.4-GHz map of \citet{laing08a}, which
has a resolution of 5 arcsec, and a 330-MHz map made in the standard
way from VLA archival data in the B and C configurations (Program
AL597), with a resolution of 21.3 arcsec $\times$ 18.2 arcsec, to
obtain the most reliable low-frequency measurements for outer
regions. These data enable us to measure the source geometry and radio
surface brightness accurately in the inner regions, while adequately
sampling the source structure out to the hundred kpc scale regions of
interest. For Hydra A we used the 330-MHz map of \citet{laing04} for
the outer lobes with a resolution of 15 arcsec, and to image the inner
structure with sufficient resolution for our geometric measurements we
made a map using A and B configuration archival VLA data at 1.4 GHz
\citep[e.g.][]{taylor90} with resolution of 1.4 arcsec. For 3C\,31,
roughly 20 regions per jet were used to measure the radio flux density
and jet geometry. For Hydra\,A around 20 regions were used to study
the northern jet. In the case of 3C\,31, where significant jet bending
occurs on the scales of interest, we measured the distances along the
projected jet paths as the best way of estimating the distance
travelled by material at a particular position along the jet; however,
for simplicity we assumed initially the source is in the plane of the
sky, and does not change position angle relative to the plane. The
external pressure acting at a particular position is therefore assumed
to be the pressure at the distance corresponding to the projected
radial distance from the AGN nucleus and group centre. The effects of
projection and jet bending on our results are discussed in
Section~\ref{geometry}.
To investigate the energetics of the radiating particles and magnetic
field, we initially assumed a single electron energy distribution
consisting of a power law with spectral index of 0.55, minimum
electron energy of $\gamma=10$ and maximum energy of
$\gamma=10^{4}$. This correctly describes the radio spectra of the two
sources in the inner regions (in the GHz radio regime), but is a
somewhat flatter spectral index than is measured in the outer parts of
the source. Any systematic effects of spectral steepening at GHz
frequencies on our pressure results for the outer parts of the sources
will be small, as the total electron energy content is dominated by
the low-energy electron population. Allowing the spectral index to
vary based on the observed spectral index at GHz frequencies would
introduce large systematic uncertainty in the low-energy electron
density. We therefore used a single electron distribution for all
regions, normalized to the measured radio flux density for that region
from the appropriate radio map (in the case of Hydra A flux densities
at 5 GHz were used in order to have sufficient spatial resolution out
to a distance of 40 kpc, with the 330-MHz map used beyond that
distance). In future work we will make use of new low-frequency data
from the Low-Frequency Array (LOFAR) to improve our spectral model.
In order to investigate the variations in internal conditions along
the source, a power law was fitted to the emissivity distribution so
as to provide a smooth model for the variation with distance. Although
there is some small systematic deviation of the observed emissivity
about the model, the measured profile is never more than $\sim 40$
percent different from the model (and typically within 10
percent).
Using the smoothed emissivity profiles for 3C\,31, we first determined
the internal pressure as a function of position along the jet, under
the assumptions of equipartition of energy between particles and
magnetic field, and no non-radiating particles ($\kappa=0$, where
$\kappa = U_{NR}/U_{R}$, i.e. the ratio of energy density in
non-radiating particles to that in synchrotron-emitting
particles). The internal, equipartition pressure profiles for are
shown in Fig.~\ref{press1}, together with the external pressure
profiles determined from the X-ray observations. For Hydra\,A, we
simply calculated an internal pressure profile for the existing radio
bins, under the same assumptions. This profile is shown in
Fig.~\ref{press1}, illustrating a strong qualitative similarity to the
behaviour of the 3C\,31 jets.
\section{Implications for lobe contents}
\label{imp}
\begin{figure*}
\centerline{\hbox{
\includegraphics[height=.35\textheight]{pcomp.pdf}
\hskip 1.0cm
\includegraphics[height=.35\textheight]{hydra_press.pdf}}}
\caption{External and internal (equipartition) pressure profiles for
the two sources 3C\,31 (l) and Hydra A (r). The external pressures
derived from X-ray measurements are shown as shaded regions, which
indicate the 1$\sigma$ errors, and the internal, equipartition
pressures with the assumption of no protons are given by the solid
black (3C31\,north, Hydra\,A north) and dashed red (3C\,31 south)
lines. The statistical uncertainties on the internal pressures are
negligible compared to model assumptions and so are not plotted.}
\label{press1}
\end{figure*}
Fig.~\ref{press} shows the ratio of external pressure to internal,
equipartition pressure (with no protons) for 3C\,31, determined from the external and internal pressure
profiles described in the previous section. As seen in previous work
\citep[e.g.][]{worrall00}, the internal equipartition pressure is
significantly below the external pressure at all radii. It also is
readily apparent that the apparent pressure ``deficit'' increases with
distance, apart from in the inner $\sim 10$ kpc. This
figure illustrates clearly that on scales $> 10$ kpc
the contribution of the radiating material to the total internal
pressure of the radio source, in the equipartition case, must decrease
substantially as the jet evolves out into the group or cluster
environment. Alternatively, if equipartition between radiating
particles and magnetic field does not hold, then there must be a
systematic departure from this condition that increases with distance
from the nucleus and group/cluster centre. Such an effect was first
observed in ROSAT environmental studies
\citep[e.g.][]{hardcastle98e,worrall00}, and is also seen in our
combined {\it Chandra} and {\it XMM-Newton} analysis of NGC\,6251
\citep{evans05} and 3C\,465 \citep{hardcastle05c}; however, the higher
quality of the X-ray and radio pressure constraints in the new work we
present here places the result on a much firmer footing.
We have considered in detail the possible effects of projection on
this conclusion (see Section~\ref{geometry}). Neither 3C\,31 or Hydra\,A is
thoughout to be highly projected, and for plausible jet orientations
the plots in Fig.~\ref{press1} and ~\ref{press} do not alter
significantly as the two effects of projection act in the same
direction: the internal pressure decreases with $\theta_{los}$ since
the jet volume at a given projected distance increases, and the
external pressure acting on the jet at this projected distance
decreases because it is further out in the X-ray atmosphere whose
pressure is dropping off.
Although such detailed pressure profile comparisons have not been
carried out previously, it is interesting to note that a similar
behaviour can be seen at a statistical level in the sample of cluster
cavities studied by \citet{dunn05}, where the so-called ``ghost''
cavities are typically at much larger distances from the cluster
centre than the active lobes, which are systematically closer to
pressure balance assuming $\kappa=0$.
The pressure constraints shown in Fig.~\ref{press} can be used to test
a range of models that have been proposed for the particle or field
content dominating the energy budget of low-power radio lobes. In the
following section we consider four models for the dominant energy
content of the lobes:
\begin{itemize}
\item {\bf Model I -- lepton dominance:} the jets and lobes are out of equipartition, but the contribution from protons remains negligible and it is the radiating electrons and positrons that dominate the internal pressure
\item {\bf Model II -- magnetic field dominance:} the jets and lobes are out of equipartition, but the contribution from protons remains negligible and the magnetic field dominates the internal pressire.
\item {\bf Model III -- relativistic proton or ion dominance:} the
jets and lobes are in equipartition, with relativistic protons (and/or ions) dominating the internal pressure (i.e. $\kappa >> 0$)
\item {\bf Model IV -- thermal gas dominance (entrainment):} the jets
and lobes are in equipartition, with thermal material, likely
entrained from the surrounding intragroup medium, dominating the
internal pressure (i.e. $\kappa >> 0$)
\end{itemize}
It is clear that more complex models are possible -- in particular, it
is plausible that non-radiating particles are present, but the jets and lobes are
not at equipartition in all locations along the jet. Such models
are harder to test, and so we begin by considering the four simpler
models listed above.
\subsection{Departures from equipartition (Models I and II)}
\label{sec:depeq}
\begin{figure}
\includegraphics[height=.3\textheight]{psynch.pdf}
\caption{The fraction of required internal pressure that can be
provided by the synchrotron-emitting components of the jets if at
equipartition, as a function of distance from group/cluster
centre for 3C\,31, showing that this component can provide a decreasing
fraction of the jet pressure on scales of tens to hundreds of
kpc. Line styles are as for Fig.~\ref{press1}.}
\label{press}
\end{figure}
As previously stated, it is clear from Figs~\ref{press1} and
\ref{press} that in order for a departure from equipartition to be the
explanation for the apparent ``missing'' pressure in FRI lobes, the
jets must evolve further and further from the equipartition condition
as the source expands (apart from in the very inner parts -- we will
consider the implications of the differing behaviour in the inner
$\sim 10$ kpc of 3C\,31 in Section~\ref{inner})
The energy densities and magnetic field strengths required in order
that the total energy density in the synchrotron-emitting plasma
should match the measured external pressure were determined by modelling
the electron energy distribution using the parameters discussed in
Section~\ref{radio}. Fig.~\ref{protons} shows the evolution of the
required energy ratio between magnetic fields and leptons required to
maintain pressure balance with the surrounding hot gas for Models I
and II.
In Model I (lepton domination) the particle energy dominates by a
factor $\sim 100$ in the inner regions, then, after an initial
decrease, increases to $\sim 500$ at hundred-kpc distances. For the
large electron densities required in this model, the predicted level
of X-ray inverse-Compton radiation from the radio jets and lobes would
be significant, and can be ruled out in a number of individual cases
\citep[e.g.][]{croston03b,hardcastle10b}. In particular,
\citet{hardcastle10b} have examined in detail the constraints on
inverse Compton emission from Hydra A, and conclude that relativistic
electrons (and positrons) can contribute at most $\sim 6$ per cent of the internal
pressure of the radio lobes. We can therefore conclusively rule out
this explanation. For 3C\,31, we considered the outermost region of our
profile, and calculated the predicted level of X-ray inverse Compton
emission at 1 keV using the {\sc synch} code of \citet{hardcastle98c}
under the assumptions of Model I. We find that the observed residual
level of X-ray flux in this region after background subtraction is a
factor $\sim 2000$ times lower than the prediction of this model,
consistent with results for other FRI sources.
In Model II (magnetic field domination), the energy ratio
$U_{B}/U_{E}$ evolves similarly to Model I, with the factor by which
the magnetic field dominates increasing from around 30 to $\sim 100$
by hundred kpc scale distances. Fig.~\ref{bdom} shows the magnetic
field strengths required as a function of distance to achieve pressure
balance in this model. The magnetic field strengths required are high
($\sim 10 - 40 \mu$G), decreasing by a factor of a few from the inner
parts to hundred kpc scale distances. This model requires the
generation of magnetic field energy density along the source. The
dashed and dotted lines in Fig.~\ref{bdom} show the expected evolution
of magnetic field strength due to adiabatic expansion for the case of
a predominantly radial/toroidal and predominantly longitudinal field
structure, respectively \citep[e.g.][]{baum97}. A constant velocity
profile was assumed, which is conservative, as a decreasing velocity
would steepen the losses for the perpendicular components of
$B$. Hence a passively evolving magnetic field component is
inconsistent with the observations. The results shown in
Fig.~\ref{bdom} are not consistent with previously proposed models for
cylindrical jets with helical $B$ fields \citep[e.g.][]{nakamura06},
but such models are also inconsistent with FRI jet geometries and
polarization structures \citep[e.g.][]{laing81,laing08a}. The
requirement for a slow decrease in $B$ along the jets (despite lateral
expansion of the jet) could be consistent with a model in which
turbulence increasingly amplifies the magnetic field on large scales;
however, this would need to take place with no appreciable particle
acceleration for consistency with the radio constraints, and turbulent
amplification of magnetic fields beyond equipartition values is
challenging \citep{deyoung80}. Our results show that energy would have
to be being transferred from the particle population to the magnetic
field to a greater and greater extent at larger distances. This model
cannot be ruled out directly, but from the constraints on the model
given above we conclude that magnetic domination of the jets and lobes
is highly unlikely.
\begin{figure}
\begin{center}
\includegraphics[height=.3\textheight]{protonsetc.pdf}
\caption{The evolution of the energetically dominant component of
the 3C\,31 jets with distance, showing the ratio of lepton to
magnetic field energy density for Model I, the inverse ratio for
Model II, and the proton/ion content $\kappa$ for Models III and
IV. Line styles are as for Fig.~\ref{press1}.}
\label{protons}
\end{center}
\end{figure}
\begin{figure}
\includegraphics[height=.3\textheight]{badiab.pdf}
\caption{The magnetic field strength required as a function of
distance in the case where magnetic field energy dominates the
internal pressure (shown for the northern jet of 3C\,31). The dotted and dashed lines show the
expected evolution of magnetic field strength due to adiabatic
expansion for the case of a predominantly tangential and
predominantly longitudinal field structure, respectively. Line styles are as for Fig.~\ref{press1}.}
\label{bdom}
\end{figure}
\subsection{Contributions from non-radiating particles (Models III and IV)}
\label{sec:protons}
The question of whether or not the inner jets of radio galaxies
consist of an electron-positron or electron-proton plasma is a
long-standing one, which has not yet been resolved satisfactorily,
despite substantial efforts over the past couple of decades
\citep[e.g.][]{ghisellini92,celotti93,wardle98,homan09}. On kpc scales, there is an
obvious additional source of non-radiating particles in the form of
material entrained into the jets from the surroundings: there is
substantial evidence for entrainment in FRI jets, and the standard
picture of FRI dynamics relies on entrainment to decelerate the jets
from relativistic to transonic speeds on scales of a few kpc
\citep[e.g.][]{bicknell94,laing02b}. In this section we consider models in
which relativistic protons (and/or ions (Model III) or thermal gas entrained from the surroundings (Model IV) dominate
the internal pressure.
The contribution from heavy particles (protons/ions) required to achieve pressure balance can be
determined straightforwardly under the assumption of equipartition of
energy between all particles (radiating and non-radiating) and
magnetic field. Details of this calculation for the cases of
relativistic and thermal gas are provided in
Appendix~A. In Fig.~\ref{protons} we plot the required ratio
of energy density in non-radiating particles to radiating particles for these two models.
Fig.~\ref{pdom} shows the run of energy density in relativistic
protons (or ions) required to balance the external pressure for Model III,
assuming equipartition of particles (both radiating and non-radiating) with
magnetic field. If the electron population suffers significant
radiative losses (which do not affect the proton/ion energy density), it
might be expected that the relative energy density in protons (and/or ions) would
increase with distance, as required by the external pressure
data. However, if the energy is carried by relativistic protons
injected in the jets' inner regions, then their energy density would
be expected to evolve adiabatically with distance, in the absence of
significant radiative losses or particle acceleration. Fig.~\ref{pdom}
shows that the simplest version of Model III in which protons are
injected only in the inner jet is not viable, because the proton
energy density in this model decreases much less steeply with distance
than expected as a result of adiabatic losses (calculated from 10 kpc
outwards). We can therefore rule out a model in which protons injected
in the inner regions evolve passively along the jet. For relativistic
protons and/or ions to dominate the jets and lobes over their entire length,
significant particle acceleration is required on scales of tens to
hundreds of kpc (which must not significantly affect the lepton population).
\begin{figure}
\includegraphics[height=.3\textheight]{pdom.pdf}
\caption{The energy density in relativistic protons and/or ions required to balance the
external pressure (filled squares), shown for the northern jet of 3C\,31. The dashed and dotted lines indicate
the expected evolution of energy density with distance along the
source assuming adiabatic losses. Note that the flattening of the adiabatic model between 20 and 70 kpc is caused by the jet's cylindrical geometry in that region (see also Fig.~\ref{modelres})}
\label{pdom}
\end{figure}
A model in which entrainment of surrounding material leads to an
increasing thermal gas content as the jets evolve (whether or not they
initially contain relativistic protons), such as Model IV, is more
consistent with the data as it provides a simple explanation for the
decreasing energetic importance of the radiating particles as the jet
evolves. Fig.~\ref{protons} shows how the ratio of energy density in
non-radiating particles to radiating particles must evolve along the jet in this model. This
evolution of energy density could occur either by increasing
entrainment (via an increasingly large boundary layer), or by
increased heating/acceleration of entrained thermal gas. The required entrainment rate for Model IV can be
obtained by consideration of mass, momentum and energy flux conservation along
the jet. In the following section we develop a toy model to
investigate this scenario.
\subsection{An entrainment model on for 3C\,31}
\label{sec:entrain}
We model the region of jet between 12 kpc and 140 kpc, which is where
the X-ray constraints are tight while the uncertainties on jet
geometry are acceptable (beyond this distance further jet bends and
flaring making it difficult to constrain the geometry). The inner
boundary is chosen to be beyond the initial deceleration region
according to the model of \citet{laing02b}, so that relativistic
effects can be neglected. We assume Model IV, above, i.e. the
following assumptions hold: (1) the jet internal pressure, $P_{int}$,
balances the external pressure ($P_{ext}$, as measured from the X-ray
observations) at each radius; (2) the internal pressure has
contributions from magnetic field ($P_{B}$), synchrotron radiating
leptons ($P_{E}$), and thermal gas entrained from the environment
($P_{th}$); and (3) the magnetic field strength and energy density are
assumed to be in equipartition with the total particle energy density
(from synchrotron-radiating and non-radiating particles). We later discuss
the effects of relaxing the final assumption.
\begin{figure*}
\includegraphics[height=0.8\textheight]{entrainplots_final.pdf}
\vskip -1.0cm
\caption{Jet properties vs. distance for our entrainment model,
assuming an initial temperature for the thermal component of 100
keV. Top row:
cross-sectional area (l) and velocity (r), middle row: gas density
(l) and temperature (r) for the thermal component, bottom row:
mass entrained per unit length (l) and ratio of kinetic to jet
internal energy flux (r), all shown for the northern (black solid)
and southern (red dashed) jets of
3C\,31.}
\label{modelres}
\end{figure*}
\begin{figure*}
\includegraphics[height=0.3\textheight]{energydiv.pdf}
\caption{The evolution of energy flux with distance along the jet,
for models with matched jet power, showing kinetic energy (black),
magnetic field energy (green), internal energy of thermal
particles (red) and of relativistic leptons (blue), with left and
right panels indicating the northern and southern jets, respectively.}
\label{energydiv}
\end{figure*}
By making use of the (non-relativistic) equations for conservation of
momentum and energy flux along the jet, the density and temperature of
the `missing' thermal component of the jet can be obtained, as
described in detail in Appendix~B. We require initial
conditions of density and velocity at the inner boundary. We take the
jet velocity of 3C\,31 at 12 kpc from the model of \citet{laing02b} as
our inner boundary condition, and assume a range of initial gas
temperatures. The choice of temperature for the thermal component at
12 kpc sets a boundary condition on the gas density (via the pressure
constraints), and hence determines the jet power. As discussed later,
we can therefore use the jet power as a consistency check on the
most appropriate choice of initial temperature.
Fig.~\ref{modelres} shows some illustrative results, with initial
conditions chosen to obtain jet powers matched for the two jets, and
in broad agreement with the model of \citet{laing02b} (this requires
initial temperatures at 12 kpc of 100 keV and 230 keV for the northern
and southern jets, respectively. In this model the behaviour of the
two jets is broadly similar, but with some differences driven by
variation in how the jet geometry evolves. The northern jet can be divided into
several regions on scales of tens to hundreds of kpc in which its
geometry differs. As shown in the top left panel of the figure, the
cross-sectional area initially increases steeply with distance, the
jet then becomes cylindrical between around 20 kpc to 60 kpc; and then
the jet radius increases again to 100 kpc scales and beyond. These
geometrical features show an interesting correspondence with bends in
the jet (occuring at both of those transition points), and with the
external pressure gradient, as the pressure profile flattens at around
20 kpc (plausibly moving from a galaxy-scale halo to a group-scale
atmosphere) and then steepens again between 50 and 100 kpc. The
density profile that results from an assumption of constant
temperature along the jet shows features that correspond to this
geometry, with an inner region of increasing density, followed by a
region of constant density and then a decreasing density in the outer
region as the jet/plume widens. Finally the bottom left panel shows
that in this model the entrainment must be fairly localized, with
large amounts of material ingested at the two transition points of
$\sim 20$ and $60$ kpc (note that these are distances along the jet
centre-line, rather than radial distances in the group atmosphere). At
other times the entrainment rate is low. The southern jet expands more
smoothly, and somewhat faster, consequently requiring entrainment to
be spread out over larger distances. At large distances the
cross-sectional area expands significantly more steeply than for the
northern jet, which leads to higher entrainment, deceleration, and
thermalization of kinetic energy.
The conservation-law analysis of \citet{laing02b} leads to an
entrainment rate at 12 kpc of $\sim 10^{20}$ kg s$^{-1}$
kpc$^{-1}$. The models shown in Fig~\ref{modelres} are consistent with
this level of entrainment; however, it is also possible that the
entrainment in our model results from fairly localised disruption of
the jet at its bends, which may be unconnected to the steadier
entrainment implied by the model of \citet{laing02b}, in which case
consistency with their measurement of entrainment rate is not required.
In our model the energy flux is primarily in the form of kinetic
energy in the inner parts of the jet, but is increasingly converted
into internal energy of the thermal (and presumably relativistic)
particles, as shown in Fig~\ref{energydiv}. The temperatures required
by our model, for realistic jet powers, are much higher than the
temperature of the surrounding gas, indicated that the entrained
material must be heated fairly rapidly by tapping the jet's kinetic
energy. We are assuming that all of the thermal material at a
particular distance in the jet has a single temperature, which is
simplistic; however, at any given position the majority of material
will have been in the jet for some time with recently entrained gas
comprising only a small fraction. We note that temperatures of $>100$
keV for entrained gas are consistent with the limits on the presence
of thermal material in cluster cavities obtained from limits on the
thermal X-ray emission due to this gas
\citep[e.g.][]{blanton03,sanders07}. The `thermal' component, although
very hot, remains (predominantly) sub-relativistic in this model,
although a non-thermal, relativistic tail cannot be ruled out.
Hence we conclude that our simple entrainment model is qualitatively
consistent with providing the dominant energetic contribution to the
jets and plumes of 3C\,31 on scales from 10 to 100 kpc. Most
interestingly, if entrainment does drive the source energetics, then much of the mass ingestion appears to be localised, and coincide
with regions where the jets bend and/or spread. In particular, the two
regions of the northern jet where entrainment takes place in our model
coincide with the flattening and
steepening of the external pressure profile, it is clear that the gas
distribution in the group environment determines the energetic
evolution of the radio-galaxy plasma on these scales.
Our assumption of equipartition of energy density between particles
and magnetic field may not be correct. We argue in Section~\ref{sec:depeq}
that non-equipartition models with no protons are unlikely to be
correct, but a model where thermal and relativistic particles together
dominate the energetics, with a lower magnetic field energy density,
cannot be ruled out. However, such a model would not strongly differ
in the qualitative picture for the evolution of thermal content of the
jet -- the radio synchrotron constraints mean that if the magnetic
field strength contributes a lower fraction of the internal energy
flux then the electron contribution must increase. Significant
entrainment would still be required at the locations seen in
Fig~\ref{modelres}, but the quantities of mass entrained and the
required temperature profile could be somewhat different.
\subsection{Geometrical uncertainties}
\label{geometry}
Uncertainty in the geometry of the radio jets, and in particular how
the jet orientation changes relative to the plane of the sky at the
observed jet bends, is potentially a major limitation of our
analysis. As discussed in Section~\ref{imp}, our main observational
result -- that the synchrotron-emitting components of the jet
contribute a decreasing fraction of the jet pressure, if at
equipartition -- is not affected by uncertainties in projection. In a
geometry with high inclination, the resulting larger synchrotron
emitting volume and lower external pressure acting on a particular
region due to larger radial distance in the cluster counteract each
other, which means that the overall result is largely unaffected. The
evolution of the non-radiating particle energy fraction in Models III
and IV (or of $U_{E}$ and $U_{B}$ in Models I and II) are therefore
qualitatively similar in any plausible geometry, even though the
numerical values will change somewhat. We do not attempt here to
derive precise constraints on the jet energy content at a particular
radius, but rather to develop a robust qualitative understanding of
how the components of the jet plasma evolve. Therefore, while we
acknowledge that the geometry is poorly constrained, our general
conclusions are robust.
\subsection{Uncertainties due to assumption electron energy distribution}
A further uncertainty comes from our lack of knowledge of the
low-energy electron distribution as a function of distance along the
jets. This will soon be remedied by ongoing work with LOFAR (Heesen et
al., in prep); however, at present we can only extrapolate to the
lowest frequencies from the radio spectrum at 330 MHz. As discussed in
Section~\ref{radio} we assumed a low-frequency spectral index of
$\alpha=0.55$ \citep[e.g.][]{laing13} and a value of $\gamma_{min} =
10$. Evidence for $\gamma_{min} \gg 1$ comes from the broad-band
spectra of hotspots \citep[e.g.][]{meisenheimer97,carilli99}; however, the
situation in FRI jets remains unknown. For the electron distribution
assumptions to significantly alter our results we would require an
evolution in the low-frequency properties of the jet with distance
from the nucleus. If $\gamma_{min}$ is determined by the particle
acceleration process that occurs in the inner jet, then it is
plausible that it could evolve to lower energies (e.g. via adiabatic
losses) at the plasma is advected downstream. Alternatively, the
low-frequency spectral index could evolve to become steeper at larger
distances, but there is no indication that this is the case in the
existing 330-MHz data (e.g. the spectral index between 330 MHz and 608
MHz is $\sim 0.58$ for the outermost region we consider in the
northern jet).
We investigated the electron energy distribution that would be
required to achieve pressure balance in the outermost region of the
3C\,31 northern plume, assuming equipartition (the non-equipartition
cases having been considered previously). Simply reducing
$\gamma_{min}$ to 1, while extrapolating from the observed spectral
index of 0.55, is inadequate to achieve pressure balance. It would
necessary for the radio spectrum to steepen significantly below 330
MHz, to $\alpha > 0.9$, {\it and} to have a low-energy cut-off of
$\gamma_{min} = 1$ in order for the synchrotron emitting components to
provide all of the pressure within the lobes at this distance. As the
radiation from such a component is currently unobservable with
existing radio data, this scenario is effectively indistinguishable
from Models III and IV, above; however, it is difficult to reconcile
with particle acceleration models, and would require a second
relativistic particle population that has previously been
undetected. Such a dominant lepton population with $\gamma < 1500$,
emitting below 330-MHz, cannot currently be ruled out by existing radio
or X-ray inverse Compton constraints. We also cannot at this stage
rule out more complex models in which the spectral index (and
$\gamma_{min}$) vary while the contribution from thermal gas also changes
with distance, but we look forward to being able to test such models
in the near future with LOFAR data.
\subsection{Evolution in the inner jet region}
\label{inner}
We have focused mainly on the region of the jet beyond 10 kpc, where
it is thought to be subrelativistic and evolving into the group gas
environment. As shown in Figs.~\ref{press1} and \ref{press}, the
evolution of the jet plasma appears to be different in the region
inside 10 kpc. We have made no attempt to correct for the
effects of relativistic beaming in calculating our radio emissivity
profile as our focus is on the outer regions, but the effect of
``de-beaming'' the synchrotron emissivity [assuming the velocity model of
\citet{laing02b}] is a small decrease in the pressure of the
synchrotron components of the northern jet, and an increase in their
contribution for the southern jet. Hence this does not qualitatively
alter the behaviour of the northern jet, though it brings the southern
jet to have a roughly constant ratio of $P_{ext} / P_{synch}$ in the
inner region.
If Model IV above is the correct explanation for the evolution of the
jet plasma on scales beyond 10 kpc, then other effects must be more
important in the inner region. One possibility is that the jet is
initially significantly electron (or relativistic electron and proton)
dominated (e.g. due to substantial particle acceleration in the inner
jet) before evolving towards equipartition between particles and
magnetic field, with entrainment taking over as an important mechanism
affecting the overall energetics from around 10 kpc. Such a model is
somewhat speculative, however, with the microphysics of energy
transfer between jet components poorly understood and difficult to
test.
\section{Conclusions}
We have shown that X-ray and radio measurements of external pressure
and internal pressure from radiating material as a function of
distance along the source can be used to distinguish between models
for the contents of radio lobes. Considering in detail the cases of
3C\,31 and Hydra A, we have shown that:
\begin{itemize}
\item The fractional contribution to the total energy budget from
synchrotron-emitting components (relativistic leptons and magnetic
field), if at equipartition, must decrease with distance from the
central AGN.
\item A model in which the energetics are dominated by
relativistic leptons can be ruled out by inverse-Compton limits.
\item Magnetic domination requires the magnetic field strength to
remain close to constant along the jet, which is implausible given
the jet geometry, due to the need to convert an increasing fraction
of the jet energy into magnetic field as the jet evolves, without
producing significant particle acceleration.
\item A model in which relativistic protons/ions injected in the inner jet
dominate the jet energetics and evolve adiabatically along the jet
is ruled out.
\item Finally we have demonstrated that a simple entrainment model is
consistent with the external pressure constraints and the evolution
of radio emissivity, with regions of entrainment corresponding to
locations of jet bending/disruption and changes in the external
pressure profile. Such a model requires a high temperature for the
entrained component, and an increasing temperature with distance,
consistent with a rapidly decreasing kinetic energy flux of the jet
being converted to particle and magnetic field internal energies.
\end{itemize}
The results presented here are based on consideration of a single
object, for which the highest quality radio and X-ray data on the
scales of interest are available. Our detailed pressure comparison for
Hydra\,A, as well as indications from less well constrained
comparisons for other objects \citep{evans05,hardcastle98e,worrall00}
and circumstantial evidence from observations of cluster cavities,
mean that it is plausible that our conclusion that an
entrainment-dominated model is favoured in 3C\,31 can be generalised
to low-power radio galaxies in general. In future work we will apply
these analysis methods to other systems with high-quality X-ray and
radio data, as well as incorporating new low-frequency radio
measurements to minimise uncertainties from extrapolation of the
electron energy distribution.
\section*{Acknowledgments}
JHC acknowledges support from the South-East Physics Network (SEPNet)
and from the Science and Technology Facilities Council (STFC) under
grant ST/J001600/1. We would like to thank Robert Laing for providing the
1.4-GHz map of 3C\,31. We would also like to thank the referee, Geoff
Bicknell, for a helpful report, which has enabled us to improve the paper.
\bibliographystyle{mn2e}
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Hyten Releases Commander's Strategic Intent Document
May 10, 2016 May 9, 2016 Doug Messier News Leave a comment
Gen. John E. Hyten
PETERSON AIR FORCE BASE, Colo. (USAF PR) — General John Hyten, commander, Air Force Space Command (AFSPC), revealed his updated Commander's Strategic Intent May 6. The strategic intent document serves as the overarching document guiding the command.
Click this link to access the full document: AFSPC Commander's 2016 Strategic Intent.pdf
"The global expanse of our Nation's international engagements increasingly demands that our Air Force provide Global Vigilance, Global Reach, and Global Power today and in the anticipated environment 20 years from now," General Hyten said. "More than ever, AFSPC is called upon to deliver agile, integrated, and resilient effects in, from, and through space and cyberspace that are critical to fulfilling these strategic demands."
The intent document outlines three primary priorities for the command; win today's fight, prepare for tomorrow's fight, and take care of our Airmen and our families.
Win today's fight
"Every U.S. military operation across the planet, from humanitarian operations to full spectrum combat depends upon integrated space and cyberspace effects to accomplish national objectives," General Hyten said. "Space and cyberspace are perhaps the most inherently joint of all operational domains as all Services rely equally upon the effects delivered, in, from and through these domains."
According to the National Space Policy of the United States, "Space systems allow people and governments around the world to see with clarity, communicate with certainty, navigate with accuracy, and operate with assurance."
General Hyten's Strategic Intent emphasizes the importance of delivering integrated multi-domain combat effects, saying, "The effects that AFSPC provides the Joint Force and the Nation are not services. They are combat and combat support effects that open doors and neutralize threats."
"Our nation expects AFSPC to provide the space and cyberspace contributions necessary to achieve agile information superiority. Agile information superiority is the aggregation of our multi-domain effects to control operational domains at the time and place of our choosing to push and pull trusted information to and from warfighters within a contested, degraded and operationally-limited environment," said General Hyten.
"When we deliver actionable information on the battlefield faster than our adversaries, the Joint Force can out-think, out-decide and out act the enemy."
Prepare for tomorrow's fight
As outlined in the intent, potential adversaries around the world are moving quickly, continuously adapting to counter our capabilities and reduce the asymmetric advantage our Armed Forces provide. Their vigor in pursuing advanced capabilities and their strategic goals continue to transform the dynamics of our operating environments.
Recognizing this trend, President Obama's National Space Policy of the United States of America makes clear the way ahead, stating "the United States will employ a variety of measures to help assure the use of space for all responsible parties, and, consistent with the inherent right of self-defense, deter others from interference and attack, defend our space systems and contribute to the defense of allied space systems, and, if deterrence fails, defeat efforts to attack them."
In implementing the policy, General Hyten stresses, "No one wants a conflict that extends into space or cyberspace, but we must be prepared for when and if it does."
"To preserve the space and cyberspace global commons we must partner to influence norms of behavior that preserve and improve the usefulness of the space and cyberspace domains. We must also work with the joint community to dissuade and deter conflict within these domains," General Hyten said. "And finally, as a force provider, we must organize, train, and equip forces to project power to neutralize threats and defeat actors threatening the U.S. homeland, our national interests, and our Allies and partners should deterrence fail."
The Commander's Strategic Intent highlights the importance of multi-domain integration: the Air Force as a Service is moving away from stove-piped, cross-domain solutions towards fully-integrated, multi-domain operations. Space and Cyberspace assets will act in concert with assets from all domains to deliver combat effects.
"To preserve our domains and provide our contribution to agile information superiority, the command must organize, train, equip and operate for a fight that may extend into our operational environments. We must take an enterprise view that raises us above employing our individual systems and platforms alone and unsupported." General Hyten said. "Secondly, we must embrace "resilience capacity" as the measure that informs how we experiment, prototype, design, train, integrate, and fight as an enterprise. An enterprise view and resilience capacity are the two critical concepts that inform how we fight through contested, degraded, and operationally limited environments to provide effects on the battlefield and respond to adversary actions on tactical timelines."
Take care of our Airmen and our families
General Hyten also charges commanders with protection and care of the Air Force's greatest resource – Airmen and their families – saying, "Trust enables leaders to empower Airmen to innovate, act quickly and decisively, manage and take calculated risk, learn from mistakes and rapidly adapt to achieve our shared mission, vision, and intent."
General Hyten urges the command to maintain the passion, innovation, integrity, and courage of its predecessors, calling Airmen to rededicate themselves to the profession of arms as they face new and dynamic challenges in both space and cyberspace, stressing that Airmen, not machines, deliver effects to execute the Air Force core missions of air and space superiority.
In addition, General Hyten recognizes the stressors placed on Air Force families such as war, deployments, and budget uncertainty, and charges commanders to develop and maintain a Wingman culture that does a better job taking care of family members.
"To win today's fight, prepare for tomorrow's fight, and take care of our Airmen and our Families, AFSPC must increase our ability to operate effectively in contested, degraded, and operationally limited environments, and reconnect to our profession of arms," General Hyten said. "To do so, we must increase the resilience of our enterprise and our people in everything that we do. We must view ourselves first as warfighters and Airmen, and continue to move fast."
John Hyten, U.S. Air Force, U.S. Air Force Space Command
Chinese Space Program Increases International Cooperation
Dragon Returns to Earth on Wednesday
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Biografia
È nato ad Irvine, negli Stati Uniti, figlio dell'attore e artista marziale britannico Gary Daniels e della filippina Marilyn Lontok.
Il suo nome deriva da quello dell'omonimo personaggio dei manga giapponesi, interpretato tra l'altro dal padre nel film Fist of the North Star.
Caratteristiche tecniche
Daniels è un giocatore polivalente: può essere schierato come ala sinistra o destra, trequartista, mezza punta o all'occorrenza come terzino, il suo ruolo originale.
Carriera
Club
Proveniente dalle giovanili del Kaya, viene promosso in prima squadra nel 2013.
Nazionale
Compie il suo debutto con la Nazionale filippina il 1º marzo 2014, in occasione del pareggio per 0-0 contro la Malesia. Con l'arrivo del nuovo commissario tecnico Thomas Dooley diventa un titolare degli Azkals.
Note
Collegamenti esterni
Calciatori della Nazionale filippina
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| 1,691
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Becky Ann Baker (født 17. februar 1953) er en amerikansk skuespillerinde, der måske er bedst kendt for sin skildring af Jean Weir i den Emmy Award-vindende komedieserie Freaks and Geeks. Siden 1990 har hun været gift med skuespilleren Dylan Baker.
Eksterne henvisninger
Skuespillere fra Kentucky
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Klemens Alois Baader (München, 8 april 1762 – aldaar, 23 maart 1838) was een Duits katholiek theoloog.
Biografie
Klemens Alois Baader was de zoon van Joseph Franz von Paula Baader, lijfarts van de keurvorst van Beieren. Hij volgde een gymnasium in München en studeerde aansluitend theologie aan de Universiteit van Ingolstadt. In 1785 behaalde hij daar zijn doctoraat in de filosofie. Vervolgens werkte hij aan de consistories van de bisschoppen te Augsburg en Salzburg. Baader werd op 25 augustus 1787 kanunnik in Freising. Op 30 mei 1797 werd hij lid van de Academie van Wetenschappen in München en op 10 juli 1799 van die in Erfurt. Klemens Baader werd op 7 januari 1803 aangesteld als commissaris voor het onderwijs in München; op 25 oktober van datzelfde jaar werd hij tot hoofdcommissaris bevorderd. Hij werd als kreisschulrat met de rang van landesdirektionsrat in Ulm aangesteld. In 1811 werd Baader kreisschulrat in Salzburg, maar verloor zijn functie toen die stad in 1816 Oostenrijks werd. Na een aanstelling te Burghausen keerde hij op 22 maart 1817 weer terug naar zijn geboorteplaats. Daar ging hij met pensioen en verbleef daar tot zijn dood op 23 maart 1838.
Baader publiceerde verschillende boeken waaruit zijn belangstelling voor het verlichtingsdenken en de politiek in zijn tijd blijkt. Een project waarin hij alle Beierse geleerden uit de 18e eeuw wilde behandelen kwam in eerste instantie niet verder dan het eerste deel, dat in 1804 verscheen. Na zijn pensionering zette hij het project in gewijzigde vorm voort, aangezien de grenzen van Beieren inmiddels aanzienlijk gewijzigd waren. Ook beperkte hij zich tot de overleden auteurs en behandelde niet meer de nog levende, zoals in het eerste deel wel het geval was geweest.
Publicaties (selectie)
Fragmente a. d. Tagebuche eines Menschen und Christen (1791)
Reisen durch verschiedene Gegenden Deutschlands in Briefen (twee banden; 1795 tot 1797)
Eduards Briefe über die französische Revolution (1796)
Gedanken und Vorschläge eines bairischen Patrioten in drei Briefen über Geistlichkeit und Landschulen (1801)
Aussichten, Wünsche und Beruhigung fürs Vaterland (1801)
Nothwendigkeit der individuellen Säcularisation etc. (1802)
Das gelehrte Baiern oder Lexikon aller Schriftsteller, welche Baiern im 18. Jahrhundert erzeugte, A–K (1804; geen verdere delen verschenen)
Kurze Geschichte der Kriegsvorfälle zu Ulm im Spätherbst 1805 (1806)
Lexikon verstorbener bairischer Schriftsteller des 18. und 19. Jahrhunderts (twee banden; 1824 tot 1825)
Blumen aus verschiedenen Gärten, Aphorismen etc. (1822 tot 1824)
Freundschaftliche Briefe (1823)
Literatuur
Ernst Kelchner: Baader, Klemens Alois. In: Allgemeine Deutsche Biographie (ADB). Band 1. Duncker & Humblot, Leipzig 1875, p. 712 f.
Externe links
Werke Baaders in de Bayrischen Landesbibliothek
Duits theoloog
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,014
|
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:layout_width="match_parent"
android:descendantFocusability="blocksDescendants"
android:layout_height="match_parent" >
<RelativeLayout
android:id="@+id/rl_title"
android:layout_width="match_parent"
android:layout_height="40dp"
android:layout_alignParentTop="true"
android:background="#ee3a43" >
<TextView
android:id="@+id/tv_agentpersonal"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_centerInParent="true"
android:paddingLeft="5dp"
android:text="我的二手房"
android:textColor="#ffffff"
android:textSize="14sp" />
<LinearLayout
android:id="@+id/ll_back_putongershoulist"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_alignParentLeft="true"
android:layout_centerVertical="true"
android:orientation="horizontal" >
<ImageView
android:padding="10dp"
android:id="@+id/imageView1"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_alignParentLeft="true"
android:layout_alignTop="@+id/textView1"
android:src="@drawable/white_left" />
</LinearLayout>
<TextView
android:id="@+id/tv_putongershoulist_submit"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_alignParentRight="true"
android:layout_centerVertical="true"
android:layout_marginRight="15dp"
android:text="发布"
android:textColor="#ffffff"
android:textSize="14sp" />
</RelativeLayout>
<ListView
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:divider="@null"
android:scrollbars="none"
android:layout_below="@+id/rl_title"
android:id="@+id/lv_submit_ershou_list1"
></ListView>
</RelativeLayout>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,658
|
Q: Using jQuery, how do I disable the click effect on the current tab? I have a menu with an animation going on, but I want to disable the click while the animation is happening.
<div></div>
<div></div>
<div></div>
$("div").click(function() {
$(this).animate({height: "200px"}, 2000);
return false;
});
However, I want to disable all the buttons while the event is happening, AND disable the div that was clicked.
I was thinking of adding a class to the div that's clicked and putting the click only on the divs without that class:
$("div").not("clicked").click(function() {
$(this).animate({height: "200px"}, 2000).addClass("clicked");
return false;
});
But this doesn't appear to work (I think it does logically)?
Any help appreciated.
Cheers,
Steve
A: $("div").click(function() {
if (!$(this).parent().children().is(':animated')) {
$(this).animate({height: "200px"}, 2000);
}
return false;
});
A: You could do something like this...
$(function() {
$("div").click(function() {
//check to see if any of the divs are animating
if ($("div").is(":animated")) {
alert("busy");
return;
}
//whatever your animation is
var div = $(this);
div.slideUp(3000, function(){ div.slideDown(1000); });
});
});
This will animate the click so long as any div is not currently animating. I'm most likely put all those divs in a container and not just simply refer to div, since it could affect a lot more on the page than just your menu,
A: How do I leave the clicked <div> disabled after the animation's occurred? I've added a class to the div that's been clicked, but doing the following doesn't appear to work:
<div></div>
<div class="active"></div>
<div></div>
$("div").not('active').click(function() {
if (!$(this).parent().children().is(':animated')) {
$(this).animate({height: "200px"}, 2000);
}
return false;
});
I suspect I'm missing something with regard to the way .not works
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,203
|
Q: How can i specify a range in a http request? I make a http request like this:
Outputstream os;
os.write(("GET /Lifted-"+file+"p.dat HTTP/1.0\r\n\r\n").getBytes());
But in the request I would like to specify that in that file i want to start in some offset (2000 for example) and end in offset 10000. How can i make that range?
Thanks.
A: You can add the range header in your outputstream before you flush your stream. Try something as follows:
os.write(("GET /Lifted-"+file+"p.dat HTTP/1.0\r\n").getBytes());
os.write("Range : bytes=2000-10000\r\n".getBytes());
os.write("\r\n".getBytes());
os.flush();
Maybe this could help you.
Note: Check the ietf document for Range header: IETF Range Header
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,768
|
Q: NullPointerException when moving ImageView in Javafx, scenebuilder I'm aware that similar questions have been asked on Stackoverflow before, but none of the answers I've found seem to fix my problem. Also I'd like to add that I'm entirely new to javafx and started coding java literally a week ago altogether, so I apologize for any silly errors in my code!
When trying to move an ImageView in my javafx, moving a player and crates around in the game "sokoban", I get a nullpointerexception stating that my ImageView, marked with an fx:id in scene builder and then called on in my controller class, is a null object. Here's my code:
Application:
import javafx.application.Application;
import javafx.fxml.FXMLLoader;
import javafx.scene.Scene;
import javafx.stage.Stage;
public class App extends Application{
@Override
public void start(final Stage primaryStage) throws Exception {
primaryStage.setTitle("Sokoban");
primaryStage.setScene(new Scene(FXMLLoader.load(App.class.getResource("StartScreen.fxml"))));
primaryStage.show();
}
public static void main(final String[] args) {
App.launch(args);
}
}
Controller:
import java.io.IOException;
import java.util.Arrays;
import java.util.List;
import javafx.event.ActionEvent;
import javafx.event.EventHandler;
import javafx.fxml.FXML;
import javafx.fxml.FXMLLoader;
import javafx.scene.Scene;
import javafx.scene.Node;
import javafx.scene.control.Button;
import javafx.scene.image.Image;
import javafx.scene.image.ImageView;
import javafx.scene.input.KeyCode;
import javafx.scene.input.KeyEvent;
import javafx.scene.layout.AnchorPane;
import javafx.scene.transform.Translate;
import javafx.stage.Stage;
public class SokobanController {
GameState currentGame;
int stepSize;
@FXML Button Easy;
@FXML Button Medium;
@FXML Button Hard;
@FXML AnchorPane easyGameScene;
@FXML AnchorPane mediumGameScene;
@FXML AnchorPane hardGameScene;
@FXML ImageView playerEasy;
@FXML ImageView crateEasy0;
@FXML ImageView crateEasy1;
@FXML ImageView crateEasy2;
@FXML ImageView crateEasy3;
List<ImageView> crateImagesEasy =
Arrays.asList(crateEasy0,crateEasy1,crateEasy2,crateEasy3);
@FXML void handleDifficultySelection(ActionEvent event) throws IOException {
String difficulty = ((Button) event.getSource()).getId();
System.out.println(difficulty);
if (difficulty.equals("Easy")) {
stepSize = 75;
}
else if (difficulty.equals("Medium")) {
}
else if (difficulty.equals("Hard")) {
}
currentGame = new GameState(difficulty);
// playerEasy = new ImageView(new Image("@player.png"));
Scene gameScene = new Scene(FXMLLoader.load(getClass().getResource("GameScreen.fxml")));
gameScene.setOnKeyPressed(new EventHandler<KeyEvent>() {
@Override
public void handle(KeyEvent event) {
handleOnKeyPressed(event);
}
});
Stage window = (Stage)((Node)event.getSource()).getScene().getWindow();
window.setScene(gameScene);
window.show();
}
@FXML void handleOnkey(ActionEvent event) {
System.out.println("hallo");
}
void handleOnKeyPressed(KeyEvent event) {
System.out.println("hallo");
if (event.getCode().equals(KeyCode.LEFT)) {
// Translate translate = new Translate();
// translate.setX(-stepSize);
if (!currentGame.checkCollideWall(-1, 0)) {
int i = 0;
for (Crate crate : currentGame.crates) {
if (currentGame.checkCollideCrate(crate, -1, 0)
&& !currentGame.checkCollideWall(-2, 0)) {
// crateImagesEasy.get(currentGame.crates.indexOf(crate))
// .getTransforms().addAll(translate);
crateImagesEasy.get(currentGame.crates.indexOf(crate)).relocate(-stepSize,0);
// playerEasy.getTransforms().addAll(translate);
i = 1;
playerEasy.relocate(-stepSize, 0);
}
}
if (i == 0) {
// playerEasy.getTransforms().addAll(translate);
playerEasy.relocate(-stepSize, 0);
}
}
}
else if (event.getCode().equals(KeyCode.RIGHT)) {
Translate translate = new Translate();
translate.setX(stepSize);
if (!currentGame.checkCollideWall(1, 0)) {
int i = 0;
for (Crate crate : currentGame.crates) {
if (currentGame.checkCollideCrate(crate, 1, 0)
&& !currentGame.checkCollideWall(2, 0)) {
crateImagesEasy.get(currentGame.crates.indexOf(crate))
.getTransforms().addAll(translate);
playerEasy.getTransforms().addAll(translate);
i = 1;
}
}
if (i == 0) {
playerEasy.getTransforms().addAll(translate);
}
}
}
else if (event.getCode().equals(KeyCode.UP)) {
Translate translate = new Translate();
translate.setY(-stepSize);
if (!currentGame.checkCollideWall(0, -1)) {
int i = 0;
for (Crate crate : currentGame.crates) {
if (currentGame.checkCollideCrate(crate, 0, -1)
&& !currentGame.checkCollideWall(0, -2)) {
crateImagesEasy.get(currentGame.crates.indexOf(crate))
.getTransforms().addAll(translate);
playerEasy.getTransforms().addAll(translate);
i = 1;
}
}
if (i == 0) {
playerEasy.getTransforms().addAll(translate);
}
}
}
else if (event.getCode().equals(KeyCode.DOWN)) {
Translate translate = new Translate();
translate.setY(stepSize);
if (!currentGame.checkCollideWall(0, 1)) {
int i = 0;
for (Crate crate : currentGame.crates) {
if (currentGame.checkCollideCrate(crate, 0, 1)
&& !currentGame.checkCollideWall(0, 2)) {
crateImagesEasy.get(currentGame.crates.indexOf(crate))
.getTransforms().addAll(translate);
playerEasy.getTransforms().addAll(translate);
i = 1;
}
}
if (i == 0) {
playerEasy.getTransforms().addAll(translate);
}
}
}
}
}
FXML-file:
<?import javafx.scene.image.Image?>
<?import javafx.scene.image.ImageView?>
<?import javafx.scene.layout.AnchorPane?>
<?import javafx.scene.layout.HBox?>
<?import javafx.scene.layout.StackPane?>
<?import javafx.scene.layout.VBox?>
<?import javafx.scene.shape.Circle?>
<?import javafx.scene.shape.Rectangle?>
<?import javafx.scene.text.Font?>
<?import javafx.scene.text.Text?>
<AnchorPane prefHeight="630.0" prefWidth="600.0" xmlns="http://javafx.com/javafx/11.0.1" xmlns:fx="http://javafx.com/fxml/1" fx:controller="app.SokobanController">
<children>
<VBox prefHeight="630.0" prefWidth="600.0">
<children>
<StackPane prefHeight="600.0" prefWidth="600.0">
<children>
<AnchorPane fx:id="easyGameScene" prefHeight="600.0" prefWidth="600.0" style="-fx-background-color: f3e1bb;">
<children>
<ImageView fitHeight="75.0" fitWidth="75.0" pickOnBounds="true" translateX="150.0">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="150.0" layoutY="150.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="150.0" layoutY="225.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutY="225.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutY="300.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutY="375.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="75.0" layoutY="225.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="150.0" layoutY="75.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="225.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="300.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="300.0" layoutY="75.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="300.0" layoutY="150.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="375.0" layoutY="150.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="450.0" layoutY="150.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="525.0" layoutY="150.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="525.0" layoutY="225.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="525.0" layoutY="300.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="450.0" layoutY="300.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="375.0" layoutY="300.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="375.0" layoutY="375.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="375.0" layoutY="450.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="375.0" layoutY="525.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="300.0" layoutY="525.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="225.0" layoutY="525.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="225.0" layoutY="450.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="225.0" layoutY="375.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="150.0" layoutY="375.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fitHeight="75.0" fitWidth="75.0" layoutX="75.0" layoutY="375.0" pickOnBounds="true">
<image>
<Image url="@wall.PNG" />
</image>
</ImageView>
<ImageView fx:id="crateEasy0" fitHeight="75.0" fitWidth="75.0" layoutX="225.0" layoutY="225.0" pickOnBounds="true">
<image>
<Image url="@crate.png" />
</image>
</ImageView>
<ImageView fx:id="crateEasy2" fitHeight="75.0" fitWidth="75.0" layoutX="225.0" layoutY="300.0" pickOnBounds="true">
<image>
<Image url="@crate.png" />
</image>
</ImageView>
<ImageView fx:id="crateEasy3" fitHeight="75.0" fitWidth="75.0" layoutX="300.0" layoutY="375.0" pickOnBounds="true">
<image>
<Image url="@crate.png" />
</image>
</ImageView>
<ImageView fx:id="crateEasy1" fitHeight="75.0" fitWidth="75.0" layoutX="375.0" layoutY="225.0" pickOnBounds="true">
<image>
<Image url="@crate.png" />
</image>
</ImageView>
<ImageView fx:id="playerEasy" fitHeight="75.0" fitWidth="75.0" layoutX="300.0" layoutY="300.0" pickOnBounds="true">
<image>
<Image url="@player.png" />
</image>
</ImageView>
<Circle fill="#eb6020" layoutX="113.0" layoutY="338.0" radius="15.0" stroke="BLACK" strokeType="INSIDE" />
<Circle fill="#eb6020" layoutX="263.0" layoutY="113.0" radius="15.0" stroke="BLACK" strokeType="INSIDE" />
<Rectangle arcHeight="5.0" arcWidth="5.0" fill="#f5f5f500" height="75.0" layoutX="75.0" layoutY="300.0" stroke="#eb6020" strokeType="INSIDE" strokeWidth="5.0" width="75.0" />
<Circle fill="#eb6020" layoutX="488.0" layoutY="263.0" radius="15.0" stroke="BLACK" strokeType="INSIDE" />
<Circle fill="#eb6020" layoutX="338.0" layoutY="488.0" radius="15.0" stroke="BLACK" strokeType="INSIDE" />
<Rectangle arcHeight="5.0" arcWidth="5.0" fill="#f5f5f500" height="75.0" layoutX="225.0" layoutY="75.0" stroke="#eb6020" strokeType="INSIDE" strokeWidth="5.0" width="75.0" />
<Rectangle arcHeight="5.0" arcWidth="5.0" fill="#f5f5f500" height="75.0" layoutX="450.0" layoutY="225.0" stroke="#eb6020" strokeType="INSIDE" strokeWidth="5.0" width="75.0" />
<Rectangle arcHeight="5.0" arcWidth="5.0" fill="#f5f5f500" height="75.0" layoutX="300.0" layoutY="451.0" stroke="#eb6020" strokeType="INSIDE" strokeWidth="5.0" width="75.0" />
</children>
</AnchorPane>
</children>
</StackPane>
<HBox alignment="CENTER" prefHeight="30.0" prefWidth="600.0" style="-fx-background-color: #000000;">
<children>
<Text fill="#d79420" strokeType="OUTSIDE" strokeWidth="0.0" text="Controls: move with arrow-keys and press "R" to restart the game">
<font>
<Font name="System Bold" size="15.0" />
</font>
</Text>
</children>
</HBox>
</children>
</VBox>
</children>
</AnchorPane>
My application starts with an initial difficulty selection scene and then switches over to the actual game scene, where I've encountered the problem.
This is the error I get:
Exception in thread "JavaFX Application Thread" java.lang.NullPointerException
at ovinger/app.SokobanController.handleOnKeyPressed(SokobanController.java:97)
at ovinger/app.SokobanController$1.handle(SokobanController.java:64)
at ovinger/app.SokobanController$1.handle(SokobanController.java:1)
at javafx.base/com.sun.javafx.event.CompositeEventHandler.dispatchBubblingEvent(CompositeEventHandler.java:86)
at javafx.base/com.sun.javafx.event.EventHandlerManager.dispatchBubblingEvent(EventHandlerManager.java:238)
at javafx.base/com.sun.javafx.event.EventHandlerManager.dispatchBubblingEvent(EventHandlerManager.java:191)
at javafx.base/com.sun.javafx.event.CompositeEventDispatcher.dispatchBubblingEvent(CompositeEventDispatcher.java:59)
at javafx.base/com.sun.javafx.event.BasicEventDispatcher.dispatchEvent(BasicEventDispatcher.java:58)
at javafx.base/com.sun.javafx.event.EventDispatchChainImpl.dispatchEvent(EventDispatchChainImpl.java:114)
at javafx.base/com.sun.javafx.event.BasicEventDispatcher.dispatchEvent(BasicEventDispatcher.java:56)
at javafx.base/com.sun.javafx.event.EventDispatchChainImpl.dispatchEvent(EventDispatchChainImpl.java:114)
at javafx.base/com.sun.javafx.event.EventUtil.fireEventImpl(EventUtil.java:74)
at javafx.base/com.sun.javafx.event.EventUtil.fireEvent(EventUtil.java:54)
at javafx.base/javafx.event.Event.fireEvent(Event.java:198)
at javafx.graphics/javafx.scene.Scene$KeyHandler.process(Scene.java:4058)
at javafx.graphics/javafx.scene.Scene$KeyHandler.access$1500(Scene.java:4004)
at javafx.graphics/javafx.scene.Scene.processKeyEvent(Scene.java:2121)
at javafx.graphics/javafx.scene.Scene$ScenePeerListener.keyEvent(Scene.java:2595)
at javafx.graphics/com.sun.javafx.tk.quantum.GlassViewEventHandler$KeyEventNotification.run(GlassViewEventHandler.java:217)
at javafx.graphics/com.sun.javafx.tk.quantum.GlassViewEventHandler$KeyEventNotification.run(GlassViewEventHandler.java:149)
at java.base/java.security.AccessController.doPrivileged(AccessController.java:391)
at javafx.graphics/com.sun.javafx.tk.quantum.GlassViewEventHandler.lambda$handleKeyEvent$1(GlassViewEventHandler.java:248)
at javafx.graphics/com.sun.javafx.tk.quantum.QuantumToolkit.runWithoutRenderLock(QuantumToolkit.java:390)
at javafx.graphics/com.sun.javafx.tk.quantum.GlassViewEventHandler.handleKeyEvent(GlassViewEventHandler.java:247)
at javafx.graphics/com.sun.glass.ui.View.handleKeyEvent(View.java:547)
at javafx.graphics/com.sun.glass.ui.View.notifyKey(View.java:971)
at javafx.graphics/com.sun.glass.ui.win.WinApplication._runLoop(Native Method)
at javafx.graphics/com.sun.glass.ui.win.WinApplication.lambda$runLoop$3(WinApplication.java:174)
at java.base/java.lang.Thread.run(Thread.java:830)
(Line 97 is the first point in which I try to call a method on the ImageView object playerEasy)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,448
|
{"url":"http:\/\/buyalli.site\/how-to-find-the-domain-of-coordinates","text":"# Find Domain and Range of Relations Given by Graphs.\n\nThe formula for the domain of a circle is x a - r < x < ar. The formula for the range of a circle is y b - r < y < br. The domain gives the maximum and minimum values for the X coordinates of the circle, while the range gives the maximum and minimum values for the Y coordinates of the circle. Find the domain and range of the relation given by its graph shown below and state whether the relation is a function or not. Solution: a Domain: Points A-3,-2 and B1,-2 have the smallest and the largest x-coordinates respectively, hence the domain:-3 \u2264 x \u2264 1. The domain is the set of values that can be taken by the independent or input variable -- generally the. You have four ordered pairs in your set, and all of the -coordinates are equal to 1. Therefore the domain is. The range is the set of all values of the dependent variable resulting from all values of the independent variable.\n\nMar 29, 2019\u00a0\u00b7 How to Find the Domain and Range of a Function. Every function contains two types of variables: independent variables and dependent variables, whose values literally \"depend\" on the independent variables. For example, in the function y =. First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is.\n\nFree practice questions for SAT Math - How to find domain and range of the inverse of a relation. Includes full solutions and score reporting. Finding the Domain of a Function - Cool Math has free online cool math lessons, cool math games and fun math activities. Really clear math lessons pre-algebra, algebra, precalculus, cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. Free functions domain calculator - find functions domain step-by-step.\n\nAs mentioned in the comment from @coffeemath; by domain I will assume that you mean the values for $\\theta$ to make each petal. So the interval in this case is simply $\\cfrac2\\pi3$ for each petal as the full domain $2\\pi$ must be divided equally across each petal. The cosecant and secant functions are closely tied to sine and cosine, because they\u2019re the respective reciprocals. In reference to the coordinate plane, cosecant is r\/y, and secant is r\/x. The value of r is the length of the hypotenuse of a right triangle \u2014 which is always positive and always greater than x and []. Nov 13, 2010\u00a0\u00b7 Is the domain the x coordinate? Answer. Wiki User November 13, 2010 4:03AM. Yes, and the range is the y coordinate. Related Questions. Asked in Algebra. Local extrema if any exist over the entire domain of the function while global extrema exist for a function defined on a bounded domain. Concepts of differential calculus are used to find these.\n\nApr 30, 2018\u00a0\u00b7 Therefore The Domain is x \u2208- \u221e,\u221e and the range is y\u2208 [-1,\u221e for the given parabola. Example-2. Identify the orientation and vertex of the parabola. Find the Domain & Range of given parabola thereby. Step1: Identify the orientation of the given graph. The sine and cosine functions are unique in the world of trig functions, because their ratios always have a value. No matter what angle you input, you get a resulting output. The value you get may be 0, but that\u2019s a number, too. In reference to the coordinate plane, sine is y\/r, and cosine is [].","date":"2020-06-03 16:08:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6786524057388306, \"perplexity\": 406.7500434453758}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347435238.60\/warc\/CC-MAIN-20200603144014-20200603174014-00260.warc.gz\"}"}
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\section{Introduction}
\label{sec:introduction}
The standard assumption in mainstream machine learning is that the observed data are IID
(independent and identically distributed);
we will refer to it as the \emph{IID assumption}.
Deviations from the IID assumption are known as dataset shift,
and different kinds of dataset shift have become a popular topic of research
(see, e.g., \citet{Candela/etal:2009}).
Testing the IID assumption has been a popular topic in statistics
(see, e.g., \citet{Lehmann:2006}, Chapter 7),
but the mainstream work in statistics concentrates on the batch setting
with each observation being a real number.
In the context of deciding whether a prediction algorithm needs to be retrained,
it is more important to process data online, so that at each point in time we have an idea
of the degree to which the IID assumption has been discredited.
It is also important that the observations are not just real numbers;
in the context of machine learning the most important case is where each observation
is a pair $(x,y)$ consisting of a sample $x$ (such as an image) and its label $y$.
The existing work on detecting dataset shift in machine learning
(see, e.g., \citet{Harel/etal:2014} and its literature review)
does not have these shortcomings but does not test the IID assumption directly.
At this time the only existing method of testing the IID assumption online
is based on the method of conformal prediction \citep{Vovk/etal:2005book};
see, e.g., \citet{Vovk:2021} for a recent review.
As we explain in Section~\ref{sec:martingales},
conformal prediction allows us to construct \emph{exchangeability martingales},
which can be used as tools for testing the IID assumption.
In Section~\ref{sec:Ville} we discuss an informal scheme
that uses exchangeability martingales for deciding when a prediction algorithm needs to be retrained
and illustrate it on the well-known Wine Quality dataset.
In this paper we discuss three basic procedures for raising an alarm when the IID assumption appears to become violated.
The simplest one is the \emph{Ville procedure},
which raises an alarm when a given exchangeability martingale exceeds a given threshold.
As we explain at the beginning of Section~\ref{sec:KS},
the Ville procedure works well at the beginning of the testing process
but then becomes less efficient.
To remedy this drawback,
we introduce conformal versions of the popular CUSUM and Shiryaev--Roberts procedures
and illustrate their performance on the Wine Quality dataset.
Finally, in Section~\ref{sec:proposal} we state our proposed schemes
for deciding when a prediction algorithm should be retrained.
The two main schemes, which we call the variable and fixed schedules,
are combinations of the three basic procedures.
A short Section~\ref{sec:conclusion} concludes.
In this paper we repeatedly refer to the validity vs efficiency of our procedures.
Validity refers to their behaviour when the IID assumption is satisfied (the \emph{ideal setting});
typically, it limits the probability or frequency of false alarms.
Our procedures, being based on conformal prediction,
satisfy various properties of validity automatically.
Efficiency refers to their behaviour when the IID assumption is violated,
such as raising an alarm soon after the change point,
and achieving it is often an art.
\section{Exchangeability martingales}
\label{sec:martingales}
In this section we will define conformal test martingales,
which will be our key tool.
Let $z_1,z_2,\dots$ be an infinite sequence of observations (typically pairs $z_n=(x_n,y_n)$),
elements of a measurable space $\mathbf{Z}$, our \emph{observation space}.
An \emph{inductive conformity measure} is a measurable function $A$ mapping any observation $z\in\mathbf{Z}$
to a real number $\alpha=A(z)$, the \emph{conformity score} of $z$;
the conformity score may also depend on some prior data (in a measurable manner).
Given such an $A$, the \emph{conformal p-value} computed from observations $(z_1,\dots,z_n)\in\mathbf{Z}^*$ is
\begin{equation}\label{eq:p}
p_n
:=
\frac
{
\left|
\left\{
i \mid \alpha_i<\alpha_n
\right\}
\right|
+
\theta_n
\left|
\left\{
i \mid \alpha_i=\alpha_n
\right\}
\right|
}
{n},
\end{equation}
where $i$ ranges over $\{1,\dots,n\}$,
$\alpha_1=F(z_1),\dots,\alpha_n=F(z_n)$ are the conformity scores for $z_1,\dots,z_n$,
and $\theta_n$ is a random number distributed uniformly on the interval $[0,1]$.
To state the property of validity of conformal p-values in a strong form,
we need to relax the IID assumption.
For any natural number $N$, a probability measure $P$ on $\mathbf{Z}^N$ is \emph{exchangeable}
if it is invariant with respect to permutations in the following sense:
for any measurable set $E\subseteq\mathbf{Z}^N$ and any permutation $\pi$ of the set $\{1,\dots,N\}$,
\[
P
\left(
(z_1,\dots,z_n) \in E
\right)
=
P
\left(
(z_{\pi(1)},\dots,z_{\pi(n)}) \in E
\right).
\]
The following property of validity is proved in, e.g., \citet{Vovk/etal:2005book} (Theorem~8.2).
\begin{proposition}\label{prop:validity}
Suppose $N$ is a natural number,
observations $z_1,\dots,z_N$ are exchangeable (i.e., generated from an exchangeable probability measure),
$\theta_1,\dots,\theta_N$ are IID, distributed uniformly on $[0,1]$,
and independent of the observations.
Then the p-values $p_1,\dots,p_N$ defined by \eqref{eq:p} are IID and distributed uniformly on $[0,1]$.
\end{proposition}
A probability measure $P$ on $\mathbf{Z}^{\infty}$ over the infinite sequences $(z_1,z_2,\dots)$
is \emph{exchangeable} if, for any natural number $N$,
its restriction to the first $N$ observations $(z_1,\dots,z_N)$ is exchangeable.
According to de Finetti's representation theorem
(see, e.g., \citet{Schervish:1995}, Theorem 1.49),
assuming the observation space $\mathbf{Z}$ is a Borel space,
every exchangeable probability measure is a mixture of \emph{IID measures}
(i.e., probability measures on the infinite sequences $\mathbf{Z}^{\infty}$ of observations
under which the observations are IID).
This makes the assumptions of IID and exchangeability almost indistinguishable for $\mathbf{Z}^{\infty}$;
however, the difference is essential for finite sequences of observations.
We will test the IID assumption by testing exchangeability.
The idea is to gamble against the uniform distribution of the conformal p-values
$(p_1,p_2,\dots)\in[0,1]^{\infty}$.
A \emph{betting martingale} is a measurable function $F:[0,1]^*\to[0,\infty]$
such that $F(\Box)=1$ ($\Box$ being the empty sequence)
and, for each sequence $(u_1,\dots,u_{n-1})\in[0,1]^{n-1}$, $n\ge1$,
\begin{equation*}
\int_0^1
F(u_1,\dots,u_{n-1},u)
\,\mathrm{d} u
=
F(u_1,\dots,u_{n-1}).
\end{equation*}
\emph{Conformal test martingales} are defined by
\begin{equation*}
S_n
=
F(p_1,\dots,p_n),
\quad
n=0,1,\dots,
\end{equation*}
where $p_1,p_2,\dots$ are the p-values computed by~\eqref{eq:p},
with $(\theta_1,\theta_2,\dots)$ distributed uniformly in $[0,1]^{\infty}$
and independent of the observations.
Conformal test martingales $S_n$ are bona fide martingales in the sense of satisfying
\begin{equation}\label{eq:martingale}
\Expect(S_n \mid S_1,\dots,S_{n-1})
=
S_{n-1}
\end{equation}
for all $n\ge1$,
provided the observations are exchangeable,
or, as we will say, they are \emph{exchangeability martingales};
we only consider exchangeability martingales that are nonnegative and have 1 as their initial value.
We interpret $S_n$ as the amount of evidence found against our null hypothesis, the IID assumption,
after the first $n$ observations.
In betting terms, it is the capital of a tester who starts from 1 and gambles against the null hypothesis.
\begin{remark}
Even in the case of a finite horizon, where we have $N$ observations $z_1,\dots,z_N$,
we will have \eqref{eq:martingale} for $n=1,\dots,N$.
The case of finite horizon is important for us since permuting a dataset
ensures its exchangeability but not the IID assumption.
And because of the exchangeability,
all conformal test martingales constructed in this paper
will always lose capital in the ideal setting.
\end{remark}
By Ville's inequality (\citet{Ville:1939}, p.~100, \citet{Shiryaev:2019}, Theorem 7.3.1),
for any constant $c>1$,
\[
\Prob(\exists n: S_n\ge c)
\le
1/c.
\]
This is a property of validity for exchangeability martingales.
If, for example, we raise an alarm when $S_n$ exceeds the threshold of 100,
the probability of ever raising a false alarm will not exceed~$1\%$.
\section{Ville procedure in action}
\label{sec:Ville}
In this section we discuss a possible informal scheme for deciding when to retrain a predictor.
As an example, we consider the Wine Quality dataset \citep{Cortez/etal:2009},
available at the UCI Machine Learning repository \citep{UCI:2017}.
The dataset consists of two parts, 4898 white wines and 1599 red wines.
We randomly choose a subset of 1599 white wines and refer to it as \emph{test set 0},
and the remaining white wines (randomly permuted) will be our \emph{training set}.
All 1599 red wines form our \emph{test set 1};
therefore we have two test sets of equal sizes.
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{wine_K=10_3_accuracies.pdf}
\end{center}
\caption{The accuracies of various prediction algorithms on the Wine Quality dataset,
as described in text}
\label{fig:wine-accuracies}
\end{figure}
We will be interested in two scenarios.
In scenario 0 we train our prediction algorithm on the full training set
and test the resulting model on test set 0.
We can expect the quality of prediction to be good,
since the training and test set are coming from the same distribution.
If we normalize the data by using \texttt{StandardScaler} in \texttt{scikit-learn} \citep{scikit-learn:2011},
we can achieve the test MAD (mean absolute deviation) of about 0.45.
The best values achieved by the algorithms implemented in \texttt{scikit-learn}
for the default values of the parameters are given in Figure~\ref{fig:wine-accuracies},
where RF stands for Random Forest, 1-NN for 1-Nearest Neighbour,
MLP for Multilayer Perceptron, and SVR for Support Vector Regression.
The relevant boxplots are those marked with 0 in parentheses;
the boxplots are over 1000 simulations and shown to give an idea of the dependence
on the seed used for the random number generator
(the seed affects the split into the training and test sets
and may be used internally by the prediction algorithm, e.g., by Random Forest).
The algorithms are ordered by their performance in scenario 0.
In scenario 1 we test the same trained model on test set 1;
since its distribution is different (from the very start of the test set),
the resulting test MAD will be significantly worse,
as indicated in Figure~\ref{fig:wine-accuracies} by the boxplots marked with 1 in parentheses.
To detect a possible change point in the test set (which does not exist in test set 0 and is the very start in test set 1),
the training set of 3299 white wines is randomly split into three \emph{folds} of nearly equal sizes,
1100, 1100, and 1099.
We use each fold in turn as the \emph{calibration set} and the remaining folds as the \emph{training set proper}.
For each fold $k\in\{1,2,3\}$ we train a prediction algorithm on the training set proper
and run an exchangeability martingale (based, in some way, on the resulting model)
on the $1100+1599=2699$ observations $z'_1,\dots,z'_{1100},z''_1,\dots,z''_{1599}$,
where $z'_1,\dots,z'_{1100}$ is the calibration set and $z''_1,\dots,z''_{1599}$ is the test set
(for one of the folds, 1100 should be replaced by 1099, but we will ignore this in our discussion).
This way we obtain three paths, plots of the values of the exchangeability martingales vs time.
We still have two scenarios:
scenario 0 uses test set 0, and scenario 1 uses test set 1;
thus we have 6 paths overall.
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{wine_RFdiff_Ville_full.pdf}
\end{center}
\caption{Paths of the three conformal test martingales
(based on Random Forest, conformity measure $y-\hat y$, and Simple Jumper)
applied to the Wine Quality dataset as described in text}
\label{fig:wine-RFdiff-Ville-full}
\end{figure}
Results for a specific conformity measure (based on Random Forest) and betting martingale (to be defined shortly)
are shown in Figure~\ref{fig:wine-RFdiff-Ville-full}.
For each fold we have a conformal test martingale,
and paths of these three martingales are shown using different colours,
as indicated in the legend.
We have two paths for each of the martingales:
the one over the calibration set and test set 0 (with the part over test set 0 shown in lighter colours),
and the other over the calibration set and test set 1.
The behaviour of the three martingales in scenario 1 is similar,
all achieve a high value, of the order of magnitude about $10^{70}$,
and start rising sharply soon after the change point (shown as the thin vertical line);
the presence of the change point becomes obvious shortly after it happens.
The conformity measure used in Figure~\ref{fig:wine-RFdiff-Ville-full} is
\begin{equation}\label{eq:diff}
\alpha_i
:=
y_i - \hat y_i,
\end{equation}
where $\hat y_i$ is the prediction for the label $y_i$ of the sample $x_i$
produced by the model found from the training set proper.
\begin{algorithm}[bt]
\caption{Simple Jumper ($(p_1,p_2,\dots)\mapsto(S_1,S_2,\dots)$)}
\label{alg:SJ}
\begin{algorithmic}[1]
\State $C_{-1}:=C_0:=C_1:=1/3$
\State $C:=1$
\For{$n=1,2,\dots$:}
\For{$\epsilon\in\{-1,0,1\}$:}
$C_{\epsilon} = (1-J)C_{\epsilon} + (J/3)C$
\EndFor
\For{$\epsilon\in\{-1,0,1\}$:}
$C_{\epsilon} = C_{\epsilon} f_{\epsilon}(p_n)$
\EndFor
\State $S_n := C := C_{-1}+C_0+C_1$
\EndFor
\end{algorithmic}
\end{algorithm}
To transform the p-values $p_1,p_2,\dots$ computed by \eqref{eq:p}
into a conformal test martingale
we use the betting martingale
\[
F(p_1,\dots,p_n)
:=
\int
\left(
\prod_{i=1}^n
f_{\epsilon_i}(p_i)
\right)
\mu(\mathrm{d}(\epsilon_0,\epsilon_1,\dots)),
\]
where
\begin{equation}\label{eq:f}
f_{\epsilon}(p)
:=
1 + \epsilon(p-0.5)
\end{equation}
and $\mu$ is the following Markov chain with state space $\{-1,0,1\}$:
the initial state is $\epsilon_0=-1,0,1$ with equal probabilities,
and the transition function prescribes maintaining the same state with probability $1-J$
and, with probability $J$, choosing a random state from the state space $\{-1,0,1\}$.
The intuition is that at each step $i$ we are using one of the \emph{betting functions} \eqref{eq:f};
$f_{-1}$ corresponds to betting on small values of $p_i$,
$f_{1}$ corresponds to betting on large values of $p_i$,
and $f_0$ corresponds to not betting.
We always set $J:=0.01$.
Therefore, we start from the uniform allocation of the initial capital to the states
and usually continue betting in the same way as on the previous step.
We will refer to this betting martingale as the \emph{Simple Jumper}
(it is a simplification of the Sleepy Jumper described in \citet{Vovk/etal:2005book}, Section~7.1).
The pseudocode for the Simple Jumper applied to the p-values \eqref{eq:p} is given as Algorithm~\ref{alg:SJ}.
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{wine_RFabsdiff_Ville_full.pdf}
\end{center}
\caption{Analogue of Figure~\ref{fig:wine-RFdiff-Ville-full}
with the conformity measure $y - \hat y$ replaced by $\left|y - \hat y\right|$}
\label{fig:wine-RFabsdiff-Ville-full}
\end{figure}
Replacing the conformity measure \eqref{eq:diff} by its absolute value,
\begin{equation}\label{eq:absdiff}
\alpha_i
:=
\left|y_i - \hat y_i\right|,
\end{equation}
leads to a slower growth, as illustrated in Figure~\ref{fig:wine-RFabsdiff-Ville-full},
and in this paper we concentrate on the signed version \eqref{eq:diff}.
In the case of fold~1 (the red line),
we can see a pronounced phenomenon of ``decay'' setting in around observation 2000;
as it were, the new distribution (corresponding to red wines) becomes a new normal,
and the growth of the martingale stops.
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{wine_RFpit_Ville_full.pdf}
\end{center}
\caption{Analogue of Figure~\ref{fig:wine-RFdiff-Ville-full}
for the PIT conformity measure $F_i(y_i)$ (with $F_i$ produced by Random Forest)}
\label{fig:wine-RFpit-Ville-full}
\end{figure}
In the case of Random Forest, there is an interesting alternative to the conformity measure \eqref{eq:diff}.
With the default values of the parameters in \texttt{scikit-learn},
its prediction $\hat y_i$ is computed by averaging the predictions
produced by 100 decision trees.
The \emph{PIT conformity measure} (where PIT stands for ``probability integral transform'')
is $\alpha_i:=F_i(y_i)$, where $F_i$ is the empirical distribution function
determined by the predictions $\hat y_i^j$ produced by the decision trees $j=1,\dots,100$.
In other words,
\[
\alpha_i
:=
\left|\left\{
j\mid \hat y_i^j \le y_i
\right\}\right|
/ 100.
\]
The results are shown in Figure~\ref{fig:wine-RFpit-Ville-full};
the growth of the conformal test martingales in scenario 1
becomes even weaker than for the conformity measure~\eqref{eq:absdiff}.
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{wine_NNdiff_Ville_full.pdf}
\end{center}
\caption{Analogue of Figure~\ref{fig:wine-RFdiff-Ville-full}
with $\hat y$ produced by 1-Nearest Neighbour}
\label{fig:wine-NNdiff-Ville-full}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{wine_MLPdiff_Ville_full.pdf}
\end{center}
\caption{Analogue of Figure~\ref{fig:wine-RFdiff-Ville-full}
with $\hat y$ produced by Multilayer Perceptron}
\label{fig:wine-MLPdiff-Ville-full}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{wine_SVRdiff_Ville_full.pdf}
\end{center}
\caption{Analogue of Figure~\ref{fig:wine-RFdiff-Ville-full}
with $\hat y$ produced by Support Vector Regression}
\label{fig:wine-SVRdiff-Ville-full}
\end{figure}
The conformity measure \eqref{eq:diff} can be applied to any regressor.
Figures~\ref{fig:wine-NNdiff-Ville-full}, \ref{fig:wine-MLPdiff-Ville-full}, and~\ref{fig:wine-SVRdiff-Ville-full}
are the analogues of Figure~\ref{fig:wine-RFdiff-Ville-full}
for 1-Nearest Neighbour, Multilayer Perceptron, and Support Vector Regression.
Notice that our martingales detect lack of exchangeability best
in situations where it matters most:
according to Figure~\ref{fig:wine-accuracies}, the accuracy of Multilayer Perceptron
suffers most of the dataset shift,
and according to Figure~\ref{fig:wine-MLPdiff-Ville-full},
the conformal test martingales based on this algorithm achieve the fastest growth.
In some cases (as for fold~1 in Figure~\ref{fig:wine-SVRdiff-Ville-full})
the phenomenon of decay is even more pronounced than in Figure~\ref{fig:wine-RFabsdiff-Ville-full}.
The procedure described in this section may be used for deciding when to retrain:
e.g., we may decide to retrain when one of the three martingales
exceeds the threshold 100.
In this case,
the probability of ever raising a false alarm never exceeds~3\%.
\subsection*{Detecting covariate shift}
The conformity measures that we have used so far in this section
were functions of the true labels and predictions.
If the underlying algorithm is robust to moderate covariate shift,
the resulting testing procedures will not detect deviations from exchangeability
under such covariate shift.
It makes sense since no retraining is required in this case.
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{wine_IND_Ville_full.pdf}
\end{center}
\caption{Paths of the three conformal test martingales
based on the nearest-distance conformity measure
applied to the Wine Quality dataset}
\label{fig:wine-IND-Ville-full}
\end{figure}
Figure~\ref{fig:wine-IND-Ville-full} shows the results
for the conformity score $\alpha$ of an observation $(x,y)$
(in the calibration or test set)
computed as the distance from $x$ to the nearest sample in the training set proper.
This conformity measure completely ignores the labels
but still achieves spectacular values for conformal test martingales based on it.
\section{CUSUM and Shiryaev--Roberts procedures for change detection}
\label{sec:KS}
A well-known disadvantage of the Ville procedure
is that it becomes less and less efficient as time passes and the value of the martingale goes down,
which inevitably happens in the absence of change points:
cf.\ scenario 0 (lighter colours) in Figures~\ref{fig:wine-RFdiff-Ville-full}--\ref{fig:wine-IND-Ville-full}.
It may take a long time to recover the lost capital and so to detect the change point.
We can say that, whereas the Ville procedure may be suitable during the first stages of the testing process
(while our capital is still not negligible),
it is less suitable for later stages.
Let $S$ be an exchangeability martingale that never takes value 0
(such are all martingales considered in the previous section).
The \emph{CUSUM procedure} \citep{Page:1954} raises an alarm at the time
\begin{equation}\label{eq:CUSUM}
\tau
:=
\min
\left\{
n \mid \gamma_n \ge c
\right\},
\text{ where }
\gamma_n
:=
\max_{i=0,\dots,n-1} \frac{S_n}{S_i}
\end{equation}
and $c>1$ is the parameter of the procedure.
The Shiryaev--Roberts procedure \citep{Shiryaev:1963,Roberts:1966}
modifies this by replacing the maximum with a sum:
\begin{equation}\label{eq:SR}
\sigma
:=
\min
\left\{
n \mid \psi_n \ge c
\right\},
\text{ where }
\psi_n
:=
\sum_{i=0}^{n-1} \frac{S_n}{S_i}.
\end{equation}
In both \eqref{eq:CUSUM} and \eqref{eq:SR}, $\min\emptyset:=\infty$.
The maxima $\gamma_n$ in \eqref{eq:CUSUM} (with $\gamma_0:=0$) are called the \emph{CUSUM statistics},
and the sums $\psi_n$ (with $\psi_0:=0$) in \eqref{eq:SR} are called the \emph{Shiryaev--Roberts statistics}.
It is known that, under the null hypothesis (the IID assumption in this context),
$\Expect(\sigma)\ge c$
(see, e.g., \citet{Vovk:2021}, Proposition 4.2);
moreover, $\Expect(\sigma)$ equals $c$
if we ignore the possibility of the Shiryaev--Roberts statistics overshooting the threshold $c$.
Since $\tau\ge\sigma$, we also have $\Expect(\tau)\ge c$.
It is also shown in \citet{Vovk:2021} (Proposition 4.4) that,
when the Shiryaev--Roberts procedure is applied repeatedly,
the relative frequency of false alarms will not exceed $1/c$ in the long run
(this statement is informative only when $\Expect(\sigma)=\infty$,
since otherwise it follows from $\Expect(\sigma)\ge c$ and Kolmogorov's strong law of large numbers).
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{wine_MLPdiff_SR_1200.pdf}
\end{center}
\caption{The Shiryaev--Roberts statistic over the first 1200 observations of the combined calibration and test sets
for Multilayer Perceptron and the conformity measure $y-\hat y$}
\label{fig:wine-MLPdiff-SR-1200}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{wine_MLPdiff_CUSUM_1200.pdf}
\end{center}
\caption{The analogue of Figure~\ref{fig:wine-MLPdiff-SR-1200}
for CUSUM (in the form $\gamma_n\vee1$) in place of Shiryaev--Roberts}
\label{fig:wine-MLPdiff-CUSUM-1200}
\end{figure}
Figures~\ref{fig:wine-MLPdiff-SR-1200} and~\ref{fig:wine-MLPdiff-CUSUM-1200}
show the evolution of the Shiryaev--Roberts and CUSUM statistics
over the calibration set and the first 100 observations of the test set (in scenarios 1 and 0)
for the conformity measure $y-\hat y$, $\hat y$ being computed by Multilayer Perceptron.
Both statistics will raise an alarm in scenario 1 for $c$ up to $10^9$.
Let us now compare the performance of different prediction algorithms and conformity measures
for change-point detection more systematically,
still concentrating on the Wine Quality dataset.
The betting martingale is, as before, the Simple Jumper (with $J:=0.01$).
Our results will be summarized in Table~\ref{tab:alarms}.
\begin{table}
\begin{center}
\begin{tabular}{cccc}
c.~measure & Ville & CUSUM & SR \\
\hline
$y-\hat y$, RF & 70 $[56,87]$ & 68 $[57,85]$ & 64 $[52,80]$ \\
$y-\hat y$, 1NN & 92 $[69,138]$ & 93 $[70,137]$ & 86 $[65,126]$ \\
$y-\hat y$, MLP & 55 $[44,72]$ & 55 $[45,72]$ & 51 $[42,67]$ \\
$y-\hat y$, SVR & 156 $[100,294]$ & 156 $[102,297]$ & 142 $[94,270]$ \\
$\left|y-\hat y\right|$, RF & 222 $[131,538]$ & 221 $[130,501]$ & 203 $[119,450]$ \\
$\left|y-\hat y\right|$, 1NN & 192 $[114,416]$ & 188 $[115,412]$ & 172 $[106,342]$ \\
$\left|y-\hat y\right|$, MLP & 88 $[62,143]$ & 88 $[62,142]$ & 81 $[58,130]$ \\
$\left|y-\hat y\right|$, SVR & 720 $[300,\infty]$ & 721 $[303,\infty]$ & 562 $[272,\infty]$ \\
$F(y)$, RF & 199 $[133,375]$ & 196 $[135,373]$ & 177 $[123,333]$ \\
ND & 29 $[26,32]$ & 29 $[27,31]$ & 27 $[25,29]$ \\
FND & 33 $[29,37]$ & 33 $[30,36]$ & 31 $[28,33]$
\end{tabular}
\end{center}
\caption{The median delay and the interquartile intervals for the delay
for different conformity measures and different procedures for change-point detection,
as described in text}
\label{tab:alarms}
\end{table}
We randomly choose a subset of 1000 white wines as training set,
a disjoint subset of 1000 white wines as calibration set,
and a subset of 1000 red wines,
all three subsets randomly ordered.
We train various prediction algorithms,
labelled with the same abbreviations as in Figure~\ref{fig:wine-accuracies},
on the training set,
use the conformity measure given in the column ``c.\ measure'' to obtain a conformal test martingale,
as described earlier,
and run the Ville, CUSUM, and Shiryaev--Roberts (SR) procedures
with the thresholds $c=10^2,10^4,10^6$, respectively,
on the calibration set continued by the test set.
The alarm is raised (i.e., the threshold is exceeded) on the test set, in the vast majority of cases,
and we define the \emph{delay} as the ordinal number of the observation in the test set
at which the alarm happens.
Table~\ref{tab:alarms} reports the median delay accompanied by the interquartile intervals
for the delays (i.e., the intervals whose end-points are the lower and upper quartiles)
for ten different conformity measures (already described earlier) and 1000 simulations;
ND stands for the nearest distance conformity measure
discussed at the end of Section~\ref{sec:Ville}.
One striking feature is how much the conformity measures $\left|y-\hat y\right|$ lose
as compared with $y-\hat y$.
The nearest distance conformity measure (detecting covariate shift) is quickest in raising alarms.
\section{Our proposed procedures}
\label{sec:proposal}
In this section we propose two procedures for deciding when to retrain,
which we call the fixed training schedule and the variable training schedule.
They are based on exchangeability martingales as described in the previous sections.
Both schedules use the Ville procedure at the beginning,
but then apply different strategies for deciding when to retrain.
For simplicity, instead of free parameters we will often use specific numbers.
Fix a prediction algorithm (such as Random Forest) and a conformity measure (such as $y-\hat y$).
Training the algorithm on a dataset then gives both a predictor (trained model)
and an exchangeability martingale (conformal test martingale).
\subsection*{Variable training schedule}
Train the prediction algorithm on the full training set,
and start running the resulting predictor on the stream of test observations.
Decide on the target lifespan of the predictor, say $C$.
\begin{enumerate}
\item
As the first step of the testing component,
split the training set into 3 approximately equal folds,
1, 2, and~3.
\item
For each $k\in\{1,2,3\}$:
\begin{itemize}
\item
Train the prediction algorithm on the folds different from $k$;
the resulting predictor gives rise to a conformal test martingale $S^k$,
as described earlier.
\item
Start running the conformal test martingale $S^k$ on fold $k$ (randomly permuted)
and then on the stream of test observations.
Run the Shiryaev--Roberts statistic $\phi^k_n:=\sum_{i=0}^{n-1}S^k_n/S^k_i$
on top of $S^k$.
\end{itemize}
\item
When two out of the three martingales $S^k$ raise an alarm at level 100, $S^k_n\ge100$, retrain
(i.e., retrain when $\left|\{k\mid S^k_n\ge100\}\right|\ge2$).
\item
When two out of the three Shiryaev--Roberts statistics $\phi^k$ raise an alarm at level $C$,
$\psi^k_n\ge C$, retrain.
\end{enumerate}
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{SR_plot.pdf}
\end{center}
\caption{Behaviour of the Shiryaev--Roberts statistic in the ideal setting}
\label{fig:SR-plot}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{SR_plots.pdf}
\end{center}
\caption{Behaviour of the maximum process of the Shiryaev--Roberts statistic
for three seeds of the random number generator
in the ideal setting}
\label{fig:SR-plots}
\end{figure}
The variable training schedule is based on the fact that $\Expect(\sigma)\ge C$
(where $\sigma$ is defined in \eqref{eq:SR})
and $\Expect(\sigma)\approx C$ when overshoots do not play a big role.
Suppose the target lifespan is $C=10^6$.
Figure~\ref{fig:SR-plot} shows the behaviour of the SR statistic $\psi_n$
(see \eqref{eq:SR}) and its \emph{maximum process}
\begin{equation}\label{eq:records}
\psi^*_n
:=
\max_{i\in\{0,\dots,n\}}
\psi_i
\end{equation}
in the \emph{ideal setting}, where the p-values are independent and uniformly distributed on $[0,1]$
(as they are under the IID assumption).
The black line is the compensator $n$ of the submartingale $\psi_n$, in the sense of $\psi_n-n$ being a martingale.
The typical behaviour of $\psi_n$ is illustrated by the light blue line in Figure~\ref{fig:SR-plot},
which is very different from its expectation, the black line.
Figure~\ref{fig:SR-plots} shows three paths of the maximum process $\psi^*_n$.
\begin{figure}
\begin{center}
\includegraphics[width=\picturewidth]{SR_K=10_5_hist.pdf}
\end{center}
\caption{The histogram for the time of alarm for the Shiryaev--Roberts statistic
for the target lifespan $10^6$ and $10^5$ simulations in the ideal setting}
\label{fig:SR-hist}
\end{figure}
Figure~\ref{fig:SR-hist} gives the histogram for the times of the alarm,
$\min\{n\mid\psi_n\ge10^6\}$, over $10^5$ simulations in the ideal setting.
In numbers, rounded to the nearest $10^4$:
the mean time to the alarm is $1.125\times10^6$, which exceeds $10^6$ because of overshoots,
with the large standard deviation of $1.123\times10^6$;
the median is $0.781\times10^6$, and the interquartile interval is $[0.320\times10^6,1.561\times10^6]$.
These figure and numbers illustrate the property of validity of the variable training schedule:
the expected lifespan of the trained predictor is indeed around $C=10^6$.
However, the variability of the lifespan, even in the ideal setting, may be a problem in some applications,
if retraining is a complicated process that needs to be planned in advance.
\subsection*{Fixed training schedule}
In the fixed training schedule we decide in advance when we would like to retrain our predictor,
and change our plans only when we have significant evidence that the distribution of the data has changed
(presumably, this will be a rare event).
Since we do not have any theoretical property of the form $\Expect(\tau)\approx C$
(the inequality $\Expect(\tau)\ge C$ still holds, of course, but it is very conservative),
our property of validity for the fixed training schedule will be computational.
\begin{table}
\begin{center}
\begin{tabular}{cccc}
$f$ & alarms & $99.9\%$ confidence interval \\
\hline
$3.5\times10^5$ & 969 & $[0.87\%,1.08\%]$ \\
$3.6\times10^5$ & 939 & $[0.84\%,1.04\%]$ \\
$3.7\times10^5$ & 905 & $[0.81\%,1.01\%]$ \\
$3.8\times10^5$ & 866 & $[0.77\%,0.97\%]$ \\
$4\times10^5$ & 820 & $[0.73\%,0.92\%]$ \\
$5\times10^5$ & 635 & $[0.56\%,0.72\%]$
\end{tabular}
\end{center}
\caption{The confidence intervals, based on $10^5$ simulations,
for the probability of false alarm in the fixed training schedule
for lifespan $C=10^6$ and various choices of the threshold~$f$}
\label{tab:endgame}
\end{table}
The fixed training schedule also starts from the target lifespan of the predictor, $C$.
\begin{enumerate}
\item
Split the training set into 3 approximately equal folds, 1, 2, and~3.
\item
For each $k\in\{1,2,3\}$:
\begin{itemize}
\item
Train the prediction algorithm on the folds different from $k$
getting a conformal test martingale~$S^k$.
\item
Start running the conformal test martingale $S^k$ on fold $k$ (randomly permuted)
and then on the stream of test observations.
Run the CUSUM statistic $\gamma^k_n:=\max_{i<n}S^k_n/S^k_i$
on top of each $S^k$.
\end{itemize}
\item\label{it:Ville}
When two out of the three martingales $S^k$ raise an alarm at level 100, $S^k_n\ge 100$, retrain.
\item\label{it:CUSUM}
When two out of the three CUSUM statistics $\gamma^k$ raise an alarm at level $f=f(C)$,
$\gamma^k_n\ge f$, retrain.
\end{enumerate}
The value $f=f(C)$ should be chosen in such a way that the probability of one CUSUM statistic reaching level $f$
should not exceed $1\%$ in the ideal setting.
Let us consider, for concreteness, $C=10^6$.
One possibility is to set $f$ to the 99th percentile,
over a large number $K$ of simulations,
of the empirical distribution function
of the maximum attained by the CUSUM statistic over the random path of the Simple Jumper
between 0 and $10^6$ in the ideal setting.
Setting $K:=10^5$, we obtain, for seed 0 of the NumPy random number generator,
$3.4798\times10^5$ as the 99th percentile.
To ensure the validity of $f$,
we have computed the exact confidence intervals \citep{Clopper/Pearson:1934}
for several round values for $f$ at confidence level $99.9\%$
(to allow for multiple hypothesis testing, as we are looking at several candidates for $f=f(C)$).
For each of those $f$, we computed the number of the paths of $\gamma_n$
that trigger an alarm (which is a false alarm, since we are in the ideal setting)
at level $f$ over $n=1,\dots,10^6$;
these numbers are given in the column ``alarms'' in Table~\ref{tab:endgame}.
The confidence intervals are computed using the R package \texttt{binom} \citep{binom}.
For example, according to Table~\ref{tab:endgame}, we can set $f:=4\times10^5$,
since the corresponding confidence interval is a subset of $[0,1\%]$.
(It is sometimes argued that the Clopper--Pearson confidence intervals are too conservative
and less conservative approximate intervals are desirable,
but in our current context there is no need to sacrifice exact validity
since the number of simulations is under our control.)
The overall probability of the fixed training schedule raising a false alarm is at most $3\%$.
This follows from the Ville component \ref{it:Ville} raising a false alarm with probability at most $1.5\%$
and the CUSUM component \ref{it:CUSUM} raising a false alarm with probability at most $1.5\%$.
\subsection*{Practical aspects}
For both schedules, the predictions provided to the users of our prediction algorithm
should be computed from the full training set, of course.
The predictions computed from two out of the three folds should only be used
for monitoring the validity of the IID assumption.
Not all observations on which the trained predictor is run are necessarily included in the test stream.
In general, we have a training set, a test stream, and an exploitation stream.
We should also be careful about including observations
in the test stream in order not to violate exchangeability for irrelevant reasons.
For example, new test observations can be added in randomly shuffled batches of reasonable sizes.
\section{Conclusion}
\label{sec:conclusion}
In this paper we have discussed using conformal prediction for testing exchangeability
(this is the only known way of constructing non-trivial exchangeability martingales)
and then for deciding when a prediction algorithm should be retrained.
We have not discussed the process of retraining,
which is an interesting direction of research.
A natural question is:
which part of the available data should be used for retraining?
One possible approach is to use an exchangeability martingale
(trained on recent data) backwards:
starting from the recent data,
move into the past until the martingale detects loss of exchangeability.
This paper only scratched the surface of various specific kinds of dataset shift,
such as concept shift and covariate shift.
Those kinds will require adapting the methods proposed in this paper
and developing new ones.
\subsection*{Acknowledgments}
This research was partially supported by Amazon and Stena Line.
Thanks to Emily Hector for useful comments.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,960
|
\section{Introduction}
Despite several decades' efforts, an important part of the dynamics of QCD
remains far from analytical control and in several cases numerical techniques have proved too difficult to implement.
In particular, recent experiments at RHIC seem to probe dynamical properties of the Quark Gluon Plasma (QGP)
phase which are not within the reach of lattice techniques without extra assumptions.
On the other hand large-$N_c$ techniques have promised early-on an alternative approach to the strongly coupled physics of QCD
based on an effective string theory description of glue.
This route took an interesting twist in 1997 with the advent of the Maldacena conjecture \cite{malda}, with the unexpected
result that the string theory must live in more than four dimensions. In particular there is one extra dimension, known as the holographic
dimension, that plays the role of (renormalization group) energy scale of the strongly coupled gauge theory.
Since \cite{malda}
there has been a flurry of attempts to devise such correspondences for gauge theories with less supersymmetry with the obvious final goal: QCD.
Several interesting string duals with a QCD-like low lying spectrum and confining IR physics were proposed \cite{D4}.
In the simplest $D4$ example flavor can be added via the addition of $D_8$ branes \cite{sas} and its finite temperature phase structure
has similarities with QCD \cite{weiz}.
Although such theories reproduced the qualitative features of IR QCD dynamics,
they contain Kaluza-Klein modes, not expected in QCD,
with KK masses of the same order as the dynamical scale
of the gauge theory. Above this scale the theories deviate from QCD.
Despite the hostile environment of non-critical theory,
several attempts have been made to
understand holographic physics in lower dimensions
in order to avoid the KK contamination,
based on two-derivative gravitational actions, \cite{ks}.
A different and more phenomenological approach was in the meantime developed and is now known as AdS/QCD.
The original idea described in \cite{ps} was successfully applied to the meson
sector in \cite{adsqcd1}, and its thermodynamics was analyzed in \cite{herzog}.
The bulk gravitational background consists of a slice of AdS$_5$, and a constant dilaton.
There is a UV and an IR cutoff. The confining IR physics is imposed by boundary
conditions at the IR boundary.
This approach, although crude, has
been partly successful in studying meson
physics, despite the fact that the dynamics driving chiral symmetry breaking must be
imposed by hand via IR boundary conditions.
Its shortcomings however include a glueball spectrum that does not fit well the lattice data,
the fact that magnetic quarks are confined instead of screened, and asymptotic Regge trajectories for
glueballs and mesons are quadratic instead of linear. A phenomenological fix of the last problem was suggested by
introducing a soft IR wall, \cite{soft}. Although this fixes the asymptotic spectrum, it does not allow a proper
treatment of thermodynamics. In particular, neither dilaton nor metric equations of motion are solved.
Therefore the ``on-shell" action is not really on-shell. The entropy computed from the BH horizon does
not match the entropy calculated using standard thermodynamics from the
free energy computed from the action, etc.
Phenomenological metrics for the deconfined phase were also suggested, \cite{adr,keijo}
capturing some aspects of the expected thermodynamics.
In \cite{ihqcd} an improved model for QCD was proposed. It united inputs from both gauge theory and string theory
while keeping the simplicity of a two derivative action. It could describe both the region of asymptotic freedom as well as
the strong IR dynamics of QCD. It is a 5d theory like AdS/QCD.
In this letter we present
the finite temperature dynamics in the pure gauge sector derived from the setup of \cite{ihqcd}.
We find that this setup describes very
well the basic features of large-$N_c$ Yang Mills at finite temperature.
It exhibits a first order deconfining phase transition.
The equation of state and
speed of sound of the high
temperature phase are remarkably similar to the corresponding lattice results. Moreover,
using the zero temperature potential and without adding any extra
parameter, we obtain a value for the
the critical temperature in
very good agreement with the one computed from the lattice.
A detailed derivation of the results will appear elsewhere, \cite{GKNL2}.
\section{Improved Holographic QCD at T=0}
The holographic model introduced in \cite{ihqcd} is five-dimensional.
The basic fields that are non-trivial
in the vacuum solution, and describe the pure gauge dynamics,
are the 5d metric $g_{\mu\nu}$,
a scalar $\Phi$ (the dilaton) that controls the 't Hooft coupling $\lambda_t$
of QCD, and an axion $a$, that is dual to to the QCD $\theta$ angle.
Moreover, as the kinetic term of the axion is suppressed by 1/$N_c^2$, it does not play any role neither in the
geometry, nor in the evolution of the 't Hooft coupling. It has however a non-trivial profile in the vacuum, implying an IR running
of the effective $\theta$-angle, \cite{ihqcd}.
Quarks can be added to the pure gauge theory by adding space-filling $D_4-\bar D_4$ brane pairs in the background gauge theory solution.
The $D_4-\bar D_4$ tachyon condensation then induces chiral symmetry breaking, \cite{ckp,ihqcd}.
The action for the 5D Einstein-dilaton theory reads,
\begin{equation}
S_5=M_p^3 N_c^2\left(-\int d^5x\sqrt{g}
\left[R-{4\over 3}{(\partial\lambda)^2\over \lambda^2}+V(\lambda) \right]+2\int_{\partial M}d^4x \sqrt{h}~K\right)
\label{action}\end{equation}
where $M_p$ is the Planck mass \footnote{The physical Planck mass that governs the interactions is $M_p N^{2\over 3}$. We will however call
$M_p$ the Planck mass for simplicity.} and we use the conventions of \cite{book}. The
second term in the action is the Gibbons-Hawking with $K$ being the extrinsic curvature on the boundary.
The only nontrivial input in the two-derivative action of the
graviton and the dilaton is the dilaton potential $V(\lambda)$, where
$\lambda=e^{\Phi}$. $\lambda$ is proportional to the 't Hooft coupling of
the gauge theory, $\lambda=\kappa \lambda_t$. The constant of
proportionality $\kappa$ cannot be calculated at present from
first principles but as we discuss below all of the physical
observables turn out to be independent of $\kappa$. The potential is
directly related to the gauge theory $\beta$-function once a
holographic definition of energy is chosen. Although the shape of
$V(\lambda)$ is not fixed without knowledge of the exact gauge theory
$\beta$-function, its UV and IR asymptotics can be determined.
In the UV, the input comes from perturbative QCD. We demand asymptotic freedom
with logarithmic running.
This implies in particular that the asymptotic UV geometry is that of $AdS_5$ with logarithmic corrections.
It requires a (weak-coupling) expansion of $V(\lambda)$ of the form
$V(\lambda) = 12/\ell^2 (1 + v_1 \lambda + v_2 \lambda^2 +\cdots)
$.
Here
$\ell$ is the AdS radius and $v_i$ are dimensionless parameters of the potential directly related to the perturbative $\beta$-function coefficients of QCD,
\cite{ihqcd}. In conformal coordinates, close to the $AdS_5$ boundary at
$r=0$, the metric and dilaton behave as \footnote{We will use a ``zero'' subscript to indicate quantities evaluated at zero temperature.}:
\begin{eqnarray}\label{sol0UV}
ds^2_0 &=& \frac{\ell^2}{r^2} \left(1+\frac89\frac{1}{\log r\Lambda}+\cdots\right)\left(dr^2+dx_4^2\right),\\
\lambda_0 &=& -\frac{1}{\log r\Lambda}+ \cdots\nonumber
\end{eqnarray}
where the ellipsis represent higher order corrections that arise from second and higher-order terms
in the $\beta$-function. The mass scale $\Lambda$ is an initial condition for the dilaton equation and corresponds to
$\Lambda_{QCD}$.
Demanding confinement of the color charges restricts the large-$\lambda$ asymptotics
of $V(\lambda)$. In \cite{ihqcd} we focused on potentials such that, as $\lambda\to \infty$,
$ V(\lambda) \sim \lambda^{\frac43}(\log{\lambda})^{(\alpha-1)/\alpha}$
where $\alpha$ is a positive parameter. The IR asymptotics of the solution in the Einstein frame are:
\begin{equation}\label{sol0IR}
ds^2_0 \to e^{- C \left(\frac{r}{\ell}\right)^{\alpha}}\!\!\left(dr^2+dx_4^2\right),
\quad\lambda_0 \to e^{3C/2 \left(\frac{r}{\ell}\right)^{\alpha}}\!\!\left(\frac{r}{\ell}\right)^{\frac34(\alpha-1)}
\end{equation}
where the constant $C$ is a positive constant related to $\Lambda$ in (\ref{sol0UV}). Confinement
requires $\alpha \geq 1$.
The parameter $\alpha$ characterizes the large excitation
asymptotics of the glueball spectrum, $m_n^2\sim n^{2(\alpha-1)/\alpha}$. For linear confinement, we choose $\alpha=2$.
The parameters of the holographic model a priori are: the Planck mass $M_p$, which governs the scale of interactions between the glueballs in the theory,
$\kappa$ that relates $\lambda$ and the 't Hooft coupling, the parameters $v_i$ that specify the shape of the potential, the scale $\Lambda$
that plays the role of $\Lambda_{QCD}$ and the AdS scale $\ell$.
The latter is not a physical parameter but only a choice of
scale: only $\Lambda\ell$ enters into the computation of physical
observables. A specific choice for $V(\lambda)$ was made in
\cite{ihqcd} with the appropriate asymptotic properties, that only
depended on a single parameter which can be taken as $v_1$, hence
fixing all $v_i$ for $i>1$. Furthermore, one can show that all
of the physical observables both at zero T and finite T are left
invariant under a rescaling of $\lambda$. More concretely, given a
potential $V(\lambda)$ and a dilaton profile that follows from this
potential with an integration constant $\Lambda$, there exists another
profile with a different integration constant $\Lambda_{\eta}$ which
follows from a rescaled potential $V_{\eta}(\lambda) = V(\eta\lambda)$ and
the two solutions yield the same glueball spectra and the same
thermodynamic observables. This symmetry allows one to scale away
the parameter $\kappa$. Finally, $v_1$ and $\Lambda$ are fixed by
matching to the lattice data for the first two $0^{++}$ glueball
masses. Once $\Lambda$ is fixed, all other interesting scales,
like the fundamental string scale $\ell_s$ and the effective QCD
string tension $\sigma$ are also fixed.
This determines all the parameters of the theory except the Planck mass $M_p\ell$.
We shall show below that $M_p$ can be indirectly inferred from the large temperature behavior.
\section{The deconfinement transition}
At finite temperature
there exist two distinct types of solutions to the action (\ref{action}) with AdS asymptotics, (\ref{sol0UV}):
\begin{enumerate}
\item[i.] The thermal graviton gas, obtained by compactifying the Euclidean time in the zero temperature solution with $\tau\sim \tau+1/T$ :
\begin{equation}\label{TG}
ds^2 = b^2_0(r)\left(dr^2 + d\tau^2+ dx^2_3\right), \,\, \lambda=\lambda_0(r).
\end{equation}
This solution exists for all $T\geq 0$ and corresponds to a confined phase,
if the gauge theory at zero T confines.
\item[ii.] The black hole (BH) solutions (in Euclidean time) of the form:
\begin{equation}\label{BH}
ds^2 = b^2(r)\left(\frac{dr^2}{f(r)} + f(r) d\tau^2+ dx^2_3\right), \,\, \lambda=\lambda(r).
\end{equation}
The function $f(r)$ approaches unity close to the boundary at $r=0$. There exists a singularity in the interior
at $r=\infty$ that is now hidden by a regular horizon at $r=r_h$ where $f$ vanishes. Such solutions correspond to a deconfined phase.
\end{enumerate}
As we discuss below, in confining theories
the BH solutions exist only above a certain minimum temperature, $T>T_{min}$.
The thermal gas solution has two parameters: T and $\Lambda$.
The black hole solution should also have a similar set of parameters:
the equations of motion are
second order for $\lambda$ and $f$, and first order for $b$ \cite{GKNL2}.
Thus, {\em a priori} there are 5 integration constants to be specified. A combination of two integration constants of $b$ and $\lambda$
determines $\Lambda$. (The other combination can be removed by reparametrization invariance in $r$). The condition $f\to 1$ on the
boundary removes one integration constant and demanding regularity at the horizon, $r=r_h$, in the form $f\to f_h(r_h-r)$, removes
another. The remaining integration constant can be taken as $f_h$,
related to the temperature by $4\pi T= f_h$.
From Einstein's equations one can show \cite{GKNL2}:
\begin{equation}\label{T}
4\pi\,T= b^{-3}(r_h)\left(\int_{0}^{r_h} {du\over b(u)^3}\right)^{-1}.
\end{equation}
In the large $N_c$ limit, the saddle point of the action is dominated by one of the two types of solutions.
In order to determine the one with minimum free energy, we
need to compare the actions evaluated on solutions i. and ii. with equal
temperature.
We introduce
a cutoff boundary at $r/\ell=\epsilon$ in order to regulate the infinite volume.
The difference of the two scale factors is given near the boundary as
\cite{GKNL2}:
\begin{equation}\label{bb0}
b(\epsilon) - b_0(\epsilon) = \mathcal{C}(T) \epsilon^3+\cdots
\end{equation}
By the standard rules of AdS/CFT we can relate $\mathcal{C}(T)$ to the difference of VEVs of the gluon condensate:
$\mathcal{C}(T) \propto\langle{\rm Tr} F^2\rangle_T - \langle{\rm Tr} F^2\rangle_0 $.
The free energy difference is given by \cite{GKNL2}:
\begin{eqnarray}
{\cal F} &=& M_p^3N_c^2V_3\left(15 \mathcal{C}(T)\ell^{-1} -\pi T b^3(r_h)\right)\nonumber\\
{} & = & 15 \mathcal{C}(T)\,M_p^3N_c^2V_3\ell^{-1} -{T S \over 4},\label{free2}
\end{eqnarray}
where, in the last equality, we used the fact that the entropy is given by the area of the horizon.
It is clear that the existence of a non-trivial deconfinement phase transition is driven by a non-zero value for the thermal gluon condensate $\mathcal{C}(T)$.
For a general potential we can prove the following statements, that only
require the validity of the laws of black hole thermodynamics:
\begin{itemize} \item[i.] {\em There exists a
phase transition at finite T, if and only
if the zero-T theory confines.} \item[ii.] {\em This transition is
of the {\bf first order} for {\bf all} of the confining geometries,
with a single exception described in iii:}
\item[iii.] {\em In the limit
confining geometry $b_0(r)\to \exp(-C r)$ (as $r\to \infty$), the phase
transition is of the {\bf second order} and happens at $T =
3C/4\pi$.}
\item[iv.] {\em All of the non-confining geometries at zero T are always in the
black hole phase at finite T. They exhibit a second order phase
transition at $T=0^+$.}
\end{itemize}
We now sketch a heuristic argument, limited to
asymptotics of the type (\ref{sol0IR}). A general, coordinate
independent proof will appear in \cite{GKNL2}.
The existence
of a minimum black hole temperature $T_{min}$ in confining theories follows from the small and large $r_h$ behavior of the geometries.
On one hand, the black-hole
approaches an AdS-Schwarzschild geometry near the boundary,
which obeys $T=1/\pi r_h$.
On the other hand, as the horizon approaches the deep interior {\it i.e.~}
$r_h\to \infty$, the mass of the black-hole vanishes and the black
hole solution approaches the zero-$T$ geometry in this limit. In
passing, we note that this implies vanishing of ${\cal F}$ in
this limit. Using the large $r_h$ limit in (\ref{T}), we find the
following asymptotics for $T$: \begin{equation}\label{Tbigrh} T\to
\frac{3C\alpha}{4\pi} r_h^{\alpha-1}, \; r_h\to\infty ; \quad T\to
\frac{1}{\pi r_h}, \; r_h\to 0. \end{equation}
The large $r_h$ behavior in eq. (\ref{Tbigrh}) is valid under
the assumption that the zero-$T$ solution, with IR asymptotics
(\ref{sol0IR}), can be continuously deformed
into a black hole with arbitrarily small mass and arbitrarily
large value of $r_h$. This assumption indeed holds, as we will show elsewhere \cite{GKNL2}
for a more general class
of confining backgrounds.
Eq. (\ref{Tbigrh}) shows that for
$\alpha\geq 1$, that there exists a minimum temperature $T_{min}>0$
above which the black-hole solutions exist.
Here, for
simplicity, we assume
a single extremum of the function $T(r_h)$.
We
illustrate the function $T(r_h)$ schematically in figure
\ref{illus}.
The simple convex shapes in (a) are due to our assumption of a single minimum. In general
the function $T(r_h)$ may exhibit multiple extrema. Our demonstration here can be generalized to these
cases \cite{GKNL2}.
In the
confining geometries $\alpha>1$, for a given $T>T_{min}$, there exist a big and a
small black hole solution, given by
$r_h<r_{min}$ and $r_h>r_{min}$ respectively, see fig.\ref{illus}. The big BH has
positive specific heat hence it is thermodynamically stable, whereas the small BH is unstable.
In the borderline confining geometry $\alpha=1$, there is a single BH solution.
\begin{figure}
\begin{center}
\includegraphics[height=6cm,width=9cm]{Trh.eps} \hspace{1.3cm}(a)\\
\vspace{0.8cm}
\includegraphics[height=6cm,width=9cm]{Frh.eps} \hspace{1.3cm}(b)
\end{center}
\caption[]{Schematic behavior of temperature (a) and free energy density (b)
as a function of $r_h$, for the infinite-$r$ geometries of the
type (\ref{sol0IR}), for different values of $\alpha$.}
\label{illus}
\end{figure}
Existence of a critical temperature $T_c\geq T_{min}$ for $\alpha
\geq 1$ follows from the physical requirement of positive entropy.
From the first law of thermodynamics, it follows that $d{\cal
F}/dr_h = -S\, dT/dr_h$. Then, as $S>0$ for any physical system,
extrema of ${\cal F}(r_h)$ should coincide with the extrema of
$T(r_h)$. Using also the fact that ${\cal F}(r_h)\to -\infty$ for
$r_h\to 0$ and ${\cal F}(r_h)\to 0$ near $r_h\to \infty$, we
arrive at conclusion (ii) described above: {\em There is a first
order transition for all of the confining geometries}.
An interesting case is the borderline confining geometry, where
$T_c$ coincides with $T_{min}$ and located at $r_h=\infty$. The
entropy vanishes there because the geometry shrinks to zero size.
The free energy also vanishes because this point coincides with
$T_c$. Therefore the latent heat also vanishes and one has a {\em
second order transition}. Although this geometry is not
interesting for the gauge theory, it is of some interest for GR.
We recall \cite{ihqcd}, that it corresponds to an asymptotically
AdS geometry that becomes a linear dilaton background in the deep
interior. We have shown that such a geometry exhibits a second
order Hawking-Page transition into a black-hole solution. By
similar arguments, point iv of the proposition above can also be
demonstrated without difficulty.
Finally, the small $r_h$ asymptotics also allows us to fix the
value of the Planck mass in (\ref{action}). Small $r_h$
corresponds to high $T$. This geometry corresponds to an ideal gas
of gluons with a free energy density ${\cal F}\to (\pi^2/45) N_c^2
V_3 T^4.$ On the other hand, as the geometry becomes AdS, eq.
(\ref{free2}) implies\footnote{It can be shown that the first term
in (\ref{free2}) is subleading in the high T limit.} that: ${\cal
F}\to \pi^4 (M_p\ell)^3 N_c^2 T^4 V_3.$ Hence we conclude that,
\begin{equation}\label{planck}
M_p\ell = \left(45\pi^2\right)^{-\frac13}.
\end{equation}
Using the value of $\ell$ in \cite{ihqcd}, we obtain $M_p\approx
2.32\,$ GeV.
\section{Numerical Results}
In \cite{ihqcd} an explicit form of the scalar potential with the correct asymptotics was proposed.
The resulting background, that corresponds to the choice $\alpha=2$ in (\ref{sol0IR}), exhibits asymptotic freedom, linear confinement, and
a glueball spectrum in very good quantitative agreement with the lattice
data.
Here we present a numerical computation of the relevant thermodynamic quantities
in the same theory.
The potential chosen in \cite{ihqcd} was fixed such that the UV expansion reproduces
the Yang-Mills beta-function up to two loops and has the large-$\lambda$ asymptotics
$V(\lambda) \sim \lambda^{4/3}(\log\lambda)^{1/2}$. It depends on two parameters:
the first is the overall normalization (that fixes the
$AdS$ length $\ell$ and the energy units); the second is $b_0$, that is
equivalent to the coefficient of linear term in the small $\lambda$ expansion, i.e. $v_1$.
These parameters were fit
to reproduce the lattice results for the two lowest scalar glueball masses.
Our general analysis shows that this theory has black hole solutions
above a temperature $T_{min}$ and exhibits a first order phase transition
at some $T_c>T_{min}$
To analyze the behavior of the theory at finite temperature, we have solved numerically
Einstein's equations for the metric and dilaton. The integration constants were fixed
as explained earlier.
We find a minimum temperature for the existence of black hole solutions, $T_{min}=210$ MeV.
Next, we compute the free energy difference between the black hole and thermal
gas solutions, as a function of temperature. As shown in eq. (\ref{free2}), there
are two competing contributions, which must be dealt with separately:
\begin{enumerate}
\item The term $\pi T b^3(r_h)$ can be obtained directly by evaluating the numerical solution
at the horizon.
\item The term $15 \mathcal{C}(T)\ell^{-1}$ must be
extracted by fitting the coefficient of the cubic term in the black hole
scale factor close to the boundary, $b(r) - b_0(r) \sim \mathcal{C}(T) r^3$.
This is a large source of error in our numerics, since it is a tiny quantity
arising as a difference of $O(1)$ quantities.
\end{enumerate}
The resulting free energy as a function of the temperature is shown in figure \ref{FT}, which
clearly shows the existence of a minimum temperature, and a first order
phase transition at $T=T_c$, where ${\cal F}(T_c) = 0$. For $T<T_c$,
the thermal gas dominates, and the system is in the confined phase. For $T>T_c$, the
(large) black hole dominates, corresponding to a deconfined phase. The
small black hole branch is thermodynamically disfavored at all temperatures.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{FT3.eps}
\end{center}
\caption[]{Black hole free energy}
\label{FT}\end{figure}
The value we obtain for the critical temperature, $T_c = \bm{235\pm 15}$ MeV, is close to
the value obtained for large-N Yang-Mills \cite{teperlucini},
which with our normalization of the lightest glueball would be $260\pm 11$ MeV \footnote{The physical units are obtained by fixing $m_{0++}=$1475\ MeV as
in \cite{ihqcd}. The value $260\pm11$ MeV is obtained combining the
results in \cite{teperlucini} and \cite{teperlucini2}}.
It should be emphasized that, we did not have to adjust any new parameter with
respect to the zero-temperature theory in order to obtain this result.
From the free energy we can determine all other quantities by thermodynamic identities.
However, for numerical precision it is preferable to derive the entropy directly as
the black hole area, rather than as a derivative of
the free energy. The latter suffers from the uncertainty in the determination of ${\cal C}(T)$. Also, due to the linear
dependence of all thermodynamic quantities on $V_3$, it is
convenient to use densities. The pressure, and the energy and entropy densities of the
deconfined phase are given by:
\begin{equation}\label{densities}
p = -{\cal F}/V_3, \quad s = 4 \pi M^3_p N_c^2 b^3_T(r_h), \quad \epsilon = p + T s .
\end{equation}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{esp_crop_corrected.eps} \hspace{0.5cm} (a) \\
\includegraphics[scale=0.7]{e-3p_crop.eps} \hspace{0.5cm} (b)
\end{center}
\caption[]{(a) Dimensionless thermodynamic functions and
(b) interaction measure. The dashed curves correspond to the lattice
data of \cite{karsch}}
\label{eps}\end{figure}
Next, we present some of the thermodynamic quantities that are compared with
the lattice results. It is useful to compare dimensionless quantities,
so that the $\ell$-dependence drops out.
{\bf Latent Heat}
The latent heat per unit volume is defined as the jump
in the energy at the phase transition, $L_h = T_c\Delta s(T_c)$, and it is expected
to scale as $N_c^2$ in the large $N_c$ limit \cite{teperlucini}.
From eq. (\ref{densities}) we note that this expectation is reproduced
in our theory. Quantitatively, we find $L_h^{1/4} /T_c \simeq 0.65 \sqrt{N_c}$.
This is to be compared with the value $0.77$ reported in \cite{teperlucini}.
{\bf Equation of state and the interaction measure.}
A useful indication about the thermodynamics of a system is given by the relations
between the quantities $\epsilon/T^4$, $3 (p/T^4)$, $3/4(s/T^3)$ (the normalizations
are chosen so that they all equal the same constant in the case of a free relativistic gas).
In figure \ref{eps} (a) we compare our results for these quantities with the corresponding
lattice results, reported in \cite{karsch}\footnote{These results are for $N_c=3$; we
are unaware of similar plots obtained in the large $N_c$ limit.}.
In the low temperature phase, the thermodynamic functions vanish to the leading order in $N_c^2$
and the jump in $\epsilon$ and $s$ at $T_c$ reflects the first order phase transition.
The {\em interaction measure}, $(\epsilon-3p)/T^4$
(proportional to the trace anomaly), is plotted
in figure \ref{eps} (b), together with the lattice result from \cite{karsch}.
From eq. (\ref{free2}), $\epsilon - 3p \propto \mathcal{C}(T)$, consistent
with our interpretation of ${\cal C} (T)$ as the gluon condensate.
{\bf Speed of sound.}
This quantity is defined as $c_s^2 = (\partial p / \partial \epsilon)_{S} = s/c_v$. It is
expected to be small at the phase transition, and to reach the conformal value $c_s^2 = 1/3$ at high temperatures.
In figure \ref{cs} we compare our results with the lattice data, finding
good agreement.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{cs_crop.eps}
\end{center}
\caption[]{Comparison between the speed of sound in our model and the lattice result of \cite{karsch} (dashed curves)}
\label{cs}\end{figure}
{\bf Shear viscosity.} In agreement with the general results of
\cite{Buchel}, the ratio between shear viscosity and entropy density
is $\eta/s = (4\pi)^{-1}$.
\section{Discussion}
The model presented here describes
well the basic features of large-$N_c$ Yang Mills at finite temperature:
it exhibits a first order deconfining phase transition, and the
temperature dependence
of the pressure, entropy, energy density, interaction measure and
speed of sound in the high
temperature phase behave similarly to the corresponding
lattice results.
Without adding any extra parameter, one obtains a value for
the critical temperature 10 \% off the lattice value.
On the other hand the model can be improved in many ways. The
latent heat $L_h/T_c^4$ is 40\% off the lattice value. Also,
our comparison
shows that (see e.g. fig 3a) approach to the free field limit at high T is slower than the lattice data.
This may be traced back to the relative smallness of the latent heat in our potential.
Although the UV and the IR asymptotics of the dilaton potential
are fixed by general requirements from the field theory, the intermediate region is free to modify.
The reason is that the low-level glueball spectrum and the thermodynamics near the phase transition are not controlled by the same
regions of the potential.
With a suitable deformation one hopes to obtain better agreement with the lattice data. In particular,
it is possible to obtain a fit to quantities in figs. 3 and 4, well within the errors of the lattice data
in a temperature range $T_c<T<5 T_c$ \cite{GKNL2}.
Retrofitting the potential is an interesting challenge that we plan to address in \cite{GKNL2}.
\section*{Acknowledgments}
We thank K. Rajagopal and M. Teper for useful discussions.
This work was partially supported by European Union Excellence Grant,
MEXT-CT-2003-509661. U.G. and F.N. are supported by
European Commission Marie Curie Fellowships, contracts MEIF-CT-2006-039962
and MEIF-CT-2006-039369. L.M. is supported by INFN fellowship and has partially been supported by ICTP.
\vspace{1.5cm}
{\bf Note added}
While this paper was
being written, the work \cite{gubser} appeared, discussing related
issues in a similar setup.
\newpage
\addcontentsline{toc}{section}{References}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,333
|
The Pacific Electric Sub-Station No. 14 is a former traction substation in Santa Ana, California. It was built by the Pacific Electric Railway to provide electricity to run the railway's streetcars in central Orange County, California. The building was added to the National Register of Historic Places in 1983.
Substation function
Electric trolley and interurban cars required 600 volts direct current (DC) to operate a car's DC traction motors. The function of a "substation" was to convert very high voltage alternating current (AC) from a power station, often miles away, for the necessary conversion to a lower voltage DC. High voltage AC entered the substation, was dropped to a lower voltage by a transformer, and then fed to a device called a Rotary Converter for the conversion to 600 volts DC. Substations were required on every trolley and interurban line in the United States and often still are for today's subway and light rail lines. Later the very large and cumbersome rotary converters, as much as in diameter rotating and vibrating and requiring a human round-the-clock operator, were replaced by small package solid state converters with no operator.
History
Pacific Electric Railway's Santa Ana substation #14 was built in 1907 and still stands. It is a single-story, rectangular building made of brick with minimal classical ornamentation in its design. There are hollow pipes at each end for electric wires to enter and exit. Very high voltage alternating current (AC) from the Watts Steam Generating Station ( away) entered at one end of the building and was converted (by rectification by a six foot high massive rotating machine called a rotary converter) to 600 volts direct current (DC) necessary to power the interurban trains, and the DC wires then exited the building at the opposite end. A transformer was located inside the building to drop the high voltage (delivered from Watts) to a lower voltage AC for the rotary converter. This substation powered the Watts–Santa Ana Line, the Santa Ana–Orange Line, and the Santa Ana–Huntington Beach Line and was in service from 1907 until the cessation of passenger service in 1950. It is the last former Pacific Electric substation building remaining in Orange County.
Substation #14 was listed in the National Register of Historic Places in 1983 due to its association with the Pacific Electric Railways extensive operation in Orange County as well as for its architecture. The building was partially restored in 2020, replacing the roof and converting interior spaces.
See also
Pacific Electric Railway Company Substation No. 8
Ivy Substation
National Register of Historic Places listings in Orange County, California
References
Electrical substations
Sub-Station No. 14
Buildings and structures in Santa Ana, California
History of Santa Ana, California
National Register of Historic Places in Orange County, California
Railway buildings and structures on the National Register of Historic Places in California
Railway buildings and structures on the National Register of Historic Places
Industrial buildings and structures on the National Register of Historic Places in California
Transportation buildings and structures in Orange County, California
1907 establishments in California
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,902
|
package com.wenyu.ylive.test;
import android.content.Context;
import android.content.Intent;
import android.support.v7.app.AppCompatActivity;
import android.os.Bundle;
import android.text.TextUtils;
import android.util.Log;
import android.view.View;
import android.widget.Button;
import android.widget.EditText;
import android.widget.TextView;
import com.wenyu.xmpp.XmppClient;
import com.wenyu.ylive.R;
import org.jivesoftware.smack.XMPPException;
import rx.Subscriber;
import rx.android.schedulers.AndroidSchedulers;
import rx.schedulers.Schedulers;
public class XMPPActivity extends AppCompatActivity {
private EditText mEditText;
private Button mSend;
private TextView mContent;
private XmppClient mXmppClient;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_xmpp);
mEditText = (EditText) findViewById(R.id.input);
mSend = (Button) findViewById(R.id.send);
mContent = (TextView) findViewById(R.id.content);
mXmppClient = XmppClient.getXmppClient(getApplicationContext(), "ylive", "19940525");
try {
mXmppClient.enterRoom("spark_name@conference.192.168.1.101",
new XmppClient.Callback() {
@Override
public void onMessageReceived(boolean isFromMe, String message) {
mContent.setText(message);
}
})
.subscribeOn(Schedulers.io())
.observeOn(AndroidSchedulers.mainThread())
.subscribe(new Subscriber<Void>() {
@Override
public void onCompleted() {
Log.d("chan_debug", "xmpp complete");
}
@Override
public void onError(Throwable e) {
Log.d("chan_debug", "xmpp error");
}
@Override
public void onNext(Void aVoid) {
Log.d("chan_debug", "xmpp next");
}
});
} catch (XMPPException e) {
e.printStackTrace();
}
mSend.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View v) {
String content = mEditText.getText().toString();
if (!TextUtils.isEmpty(content)) {
mXmppClient.sendDanma(content)
.subscribeOn(Schedulers.io())
.observeOn(AndroidSchedulers.mainThread())
.subscribe(new Subscriber<Void>() {
@Override
public void onCompleted() {
Log.d("chan_debug", "send complete");
}
@Override
public void onError(Throwable e) {
Log.d("chan_debug", "send error");
}
@Override
public void onNext(Void aVoid) {
Log.d("chan_debug", "send next");
}
});
}
}
});
}
public static Intent newIntent(Context context) {
return new Intent(context, XMPPActivity.class);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,587
|
I have been kinda MIA last week. I was working on getting some cards and projects done for a craft fair. The craft fair was to be Saturday but it was so cold and rainy that we couldn't set up. I was so disappointed as it was the kick off to my official Craft Fair tour. I have been doing craft fairs with my friend Jennifer for 13 years. I am so blessed to be able to do these shows with such a wonderful friend and sideline. Check out my projects on here and if you have any questions please let me know.
I will be happy to help you with the dimensions if you need them, I didn't post them this time as there are so many projects. Just let me know if you need any help and will be happy to walk you through.
Labels: 3D Projects, Halloween, Stampin' Up!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 799
|
Q: React-Native: Unexplainable space(a white line) between two Views I have two Views stacked one on top the other.
Screenshot in emulator:
What I see on my phone:
As can be seen from the screens, the emulator version is fine but i have a white line between the two Views on my phone. Code below:
import React, { Component } from 'react';
import {
Text,
View,
StyleSheet,
} from 'react-native';
class FlightSearch extends Component {
render() {
return (
<View style={styles.pageRoot}>
<View style={styles.navSection}></View>
<View style={styles.searchSection}>
<View style={styles.locationSection}></View>
<View style={styles.searchParametersSection}></View>
</View>
</View>
);
}
}
const styles = StyleSheet.create({
pageRoot: {
flex: 1,
flexDirection: 'column',
},
navSection: {
backgroundColor: '#368ec7',
flex: 25,
alignSelf: 'stretch'
},
searchSection: {
flex: 75,
alignSelf: 'stretch',
},
locationSection: {
flex: 30,
backgroundColor: '#276fa3',
padding: 10,
paddingLeft: 20,
paddingRight: 20,
borderBottomWidth: 1,
borderBottomColor: '#205e8c'
},
searchParametersSection : {
flex: 70,
backgroundColor: 'rgba(41,123,184,1)',
borderTopWidth: 1,
borderTopColor: 'rgba(69, 140, 194, 0.7)',
flexDirection: 'column'
}
});
export default FlightSearch;
A: I had the same problem within a scrollview, where few images lie horizontally without any space between them. While working on iOS I had no problems however when I switched to Android, those white lines popped out, and they were disappearing at some scroll positions. The hack that I used was adding a marginRight: -1 (horizontal images).
User won't notice the difference but this way you can resolve the issue.
return (
<ScrollView ref='sr' style={styles.container} horizontal={true}>
<Image source={im1} style={{height: h, width: 400, resizeMode: 'stretch', marginRight: -1}} />
<Image source={im2} style={{height: h, width: 400, resizeMode: 'stretch', marginRight: -1}} />
<Image source={im3} style={{height: h, width: 400, resizeMode: 'stretch', marginRight: -1}} />
<Image source={im4} style={{height: h, width: 400, resizeMode: 'stretch', marginRight: -1}} />
<Image source={im5} style={{height: h, width: 400, resizeMode: 'stretch', marginRight: -1}} />
<Image source={im6} style={{height: h, width: 400, resizeMode: 'stretch', marginRight: -1}} />
</ScrollView>
)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,750
|
using namespace boost::python;
using namespace Gaffer;
using namespace GafferBindings;
namespace
{
void setValue( StringPlug *plug, const std::string& value )
{
// we use a GIL release here to prevent a lock in the case where this triggers a graph
// evaluation which decides to go back into python on another thread:
IECorePython::ScopedGILRelease r;
plug->setValue( value );
}
std::string getValue( const StringPlug *plug, const IECore::MurmurHash *precomputedHash )
{
// Must release GIL in case computation spawns threads which need
// to reenter Python.
IECorePython::ScopedGILRelease r;
return plug->getValue( precomputedHash );
}
std::string substitutionsRepr( unsigned substitutions )
{
static const IECore::StringAlgo::Substitutions values[] = {
IECore::StringAlgo::FrameSubstitutions,
IECore::StringAlgo::VariableSubstitutions,
IECore::StringAlgo::EscapeSubstitutions,
IECore::StringAlgo::TildeSubstitutions,
IECore::StringAlgo::NoSubstitutions
};
static const char *names[] = { "FrameSubstitutions", "VariableSubstitutions", "EscapeSubstitutions", "TildeSubstitutions", nullptr };
if( substitutions == IECore::StringAlgo::AllSubstitutions )
{
return "IECore.StringAlgo.Substitutions.AllSubstitutions";
}
else if( substitutions == IECore::StringAlgo::NoSubstitutions )
{
return "IECore.StringAlgo.Substitutions.NoSubstitutions";
}
std::string result;
for( int i = 0; names[i]; ++i )
{
if( substitutions & values[i] )
{
if( result.size() )
{
result += " | ";
}
result += "IECore.StringAlgo.Substitutions." + std::string( names[i] );
}
}
return result;
}
std::string serialisationRepr( const Gaffer::StringPlug *plug, Serialisation *serialisation )
{
std::string extraArguments;
if( plug->substitutions() != IECore::StringAlgo::AllSubstitutions )
{
extraArguments = "substitutions = " + substitutionsRepr( plug->substitutions() );
if( serialisation )
{
serialisation->addModule( "IECore" );
}
}
return ValuePlugSerialiser::repr( plug, extraArguments, serialisation );
}
std::string repr( const Gaffer::StringPlug *plug )
{
return serialisationRepr( plug, nullptr );
}
class StringPlugSerialiser : public ValuePlugSerialiser
{
public :
std::string constructor( const Gaffer::GraphComponent *graphComponent, Serialisation &serialisation ) const override
{
return serialisationRepr( static_cast<const StringPlug *>( graphComponent ), &serialisation );
}
};
} // namespace
void GafferModule::bindStringPlug()
{
PlugClass<StringPlug>()
.def(
boost::python::init<const std::string &, Gaffer::Plug::Direction, const std::string &, unsigned, unsigned>(
(
boost::python::arg_( "name" )=Gaffer::GraphComponent::defaultName<StringPlug>(),
boost::python::arg_( "direction" )=Gaffer::Plug::In,
boost::python::arg_( "defaultValue" )="",
boost::python::arg_( "flags" )=Gaffer::Plug::Default,
boost::python::arg_( "substitutions" )=IECore::StringAlgo::AllSubstitutions
)
)
)
.def( "__repr__", &repr )
.def( "substitutions", &StringPlug::substitutions )
.def( "defaultValue", &StringPlug::defaultValue, return_value_policy<boost::python::copy_const_reference>() )
.def( "setValue", &setValue )
.def( "getValue", &getValue, ( boost::python::arg( "_precomputedHash" ) = object() ) )
;
Serialisation::registerSerialiser( StringPlug::staticTypeId(), new StringPlugSerialiser );
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,732
|
There are at least three known scripts which were used by Hungarians during their history. Their scripts are called Rovas Scripts. The comprehensive view of the Rovas alphabets was introduced first at the end of the twentieth century by the archaeologist-historian Gábor Vékony. He gave transcriptions of Carpathian Basin Rovas and Steppean Rovas inscriptions by using Rovas alphabets cognate to each other. The close relation of Carpathian Basin Rovas to Szekely-Hungarian Rovas has been shown by Gyula Németh in 1932. Moreover, according to the archaeologist-historian István Erdélyi the Steppean Rovas is related to Carpathian Basin Rovas and Szekely-Hungarian Rovas.
Each of these scripts are described in details in their appropriate articles. In this section, only some general, comprehensive features and comparisons should be presented.
In the middle of the 20th century, the scholars believed that this is a medieval writing system derived from the Old Turkic script used to write the medieval version of the Hungarian language. However, in the last decades of the 20th century, several new Rovas relics were explored by the archaeologists. Based on these, the Hungarian scholars were able to decipher some inscriptions, including the Rovas scripts of the Nagyszentmiklós Golden Treasure and the newly explored Szarvas relic. Based on these, the view of the Rovas scripts significantly changed.
The archaeologist-historian István Erdélyi (doctor of the Hungarian Academy of Sciences) showed that the Steppean Rovas is related to CBR and SHR.
The close relation of CBR to SHR has been shown by linguist Gyula Németh in 1932.
András Róna-Tas Turkologist (member of the Hungarian Academy of Sciences): According to him, the Bodrog-Alsóbű Szekely-Hungarian Rovas relic is an important chain between the East-European scripts and the Szekely script.
The comprehensive view of the Rovas alphabets was introduced first at the end of the twentieth century by the archaeologist-historian Gábor Vékony (1944-2004) PhD, Assoc. Prof. of the Eötvös Loránd University, Budapest. He gave transcriptions of CBR and KR inscriptions by using Rovas alphabets cognate to each other.
Its first relic is from the last third of the 7th century.
Its first relic is from the 7th century.
Its first relic is from around 900.
Used continuously from the 8th or 9th century.
Vékony, Gábor (1987): Spätvölkerwanderungszeitliche Kerbinschriften im Karpatenbecken. In: Acta Archeologica Hungarica. Vol. 39, 211-256.
This page was last modified on 18 August 2016, at 09:35.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,666
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{"url":"http:\/\/math.stackexchange.com\/questions\/132695\/fallacy-in-fermats-little-theorem","text":"# Fallacy in Fermat's Little Theorem?\n\nFermat's Little Theorem: If $p$ is prime, then for every $1 \u2264 a < p$,\n\n$a^{p-1} \u2261 1$ $(mod$ $p)$\n\nLet $p$ be 9 (a composite number), and let $a$ be 2.\nLet $S$ be the nonzero integers modulo $9$\n\n$S = (1, 2, 3, 4, 5, 6, 7, 8)(mod$ $9)$\n\n$2^8S$\n$=2^8(1, 2, 3, 4, 5, 6, 7, 8)$\n$= (1*2,2*2,3*2,4*2,5*2,6*2,7*2,8*2)$\n$= (2, 4, 6,8,10,12,14,16)$\n$= (1,2,3,4,5,6,7,8)(mod$ $9$)\n\n\u2234 $8! = 2^8*8!$\n\nDivide both sides by $8!$ and you arrive at: $2^8 \u2261 1$ $(mod$ $9)$\n\nDoesn't Fermat's Little Theorem only apply when $p$ is a prime number? The above calculations showed when $p = 9$ (composite), $a = 2$ still fulfills the equation.\n\n-\nFermat theorem doesn't imply that $a^{n-1}\\equiv 1 (mod n)$ fails for all $n$ composite. \u2013\u00a0azarel Apr 16 '12 at 21:36\nWhen you divide by the product of the numbers in $S$, you are dividing by something which is congruent to $0$ modulo $9$. That is no more permissible in the integers modulo $9$ than it is in the integers. \u2013\u00a0Andr\u00e9 Nicolas Apr 16 '12 at 21:36\nYour $S$ is a set. How do you divide by a set?? \u2013\u00a0Tara B Apr 16 '12 at 21:37\n@Farhad mod $9$ the congruence $8!\\equiv 2^8\\:8!\\:$ is simply $0\\equiv 2^8\\cdot 0$ since $9\\ |\\ 3\\cdot 6\\ |\\ 8!\\:$ so $\\rm\\:8!\\equiv 0.$ So you are inferring $\\rm\\ 0\\cdot n\\equiv 0\\:\\Rightarrow\\: 1\\equiv n,\\:$ which is false. \u2013\u00a0Bill Dubuque Apr 16 '12 at 21:56\n@Farhad: And still the same error: Fermat's Little Theorem does not say \"if $1\\leq a\\lt n$ then $a^{n-1}\\equiv 1\\pmod{n}$ if and only if $n$ is prime.\" It says \"if $n$ is prime, then $a^{n-1}\\equiv 1\\pmod{n}$.\" And you still cannot \"divide both sides\" by $8!$\", because $8!\\equiv 0\\pmod{9}$. \u2013\u00a0Arturo Magidin Apr 16 '12 at 21:56\n\nIt is perfectly possible for $a^{n-1}$ to be congruent to $1$ modulo $n$ for some $a$, even when $n$ is composite. It is even possible for $a^{n-1}$ to be congruent to $1$ modulo $n$ for all $a$ relatively prime to $n$. This happens when $n$ is a Carmichael number (the smallest Carmichael number is $561$; there are infinitely many others).\n\nBut let's go back to your calculation. You used an argument much like the one used in one of the standard proofs of Fermat's Theorem. You let $S$ be $\\{1,2,3,\\dots,8\\}$. You then observed that as $x$ travels over $S$, the number $2x$ is congruent modulo $9$ to $1,2, \\dots,8$ in some order.\n\nSo if $P_S$ is the product of the numbers in $S$, we fairly quickly conclude that $$2^8P_S \\equiv P_S \\pmod 9.\\tag{\\ast}$$ From this you concluded that $2^8 \\equiv 1\\pmod 9$. Even though $(\\ast)$ is correct, the conclusion that $2^8\\equiv 1\\pmod 9$ does not follow, and is in fact not true, since $256\\equiv 4\\pmod 9$.\n\nWhat went wrong? Note that $P_S\\equiv 0 \\pmod 9$. Division by $0$ is no more permissible in the integers modulo $9$ than it is in the integers. The correct cancellation law is that if $ax\\equiv ay \\pmod n$ and $a$ and $n$ are relatively prime, we can conclude that $x\\equiv y \\pmod n$. Somewhat more generally, if $ax\\equiv ay \\pmod {n}$ and $\\gcd(a,n)=d$, then $x\\equiv y \\pmod{n\/d}$.\n\nHowever, your idea can be made to work nicely. Instead of using the numbers $1$ to $8$, use only $1,2,4,5,7,8$, the $6$ numbers in our list relatively prime to $9$. Let $Q$ be the product of these numbers. Using exactly your argument, we find that $2^6Q\\equiv Q\\pmod{9}$. Since $Q$ is relatively prime to $9$, we can cancel, and conclude that $2^6\\equiv 1\\pmod{9}$.\n\nThis idea is one way to prove Euler's Theorem. But we must only use the numbers from $1$ to $n-1$ which are relatively prime to $n$, so that the cancellation we want to make is justified. That's where the $\\varphi(n)$ in Euler's Theorem comes from.\n\n-\nThank You. Do you mind expanding on the correct cancellation law please? I do not understand it fully. \u2013\u00a0user26649 Apr 16 '12 at 22:12\nThe simple one is that if $a$ and $n$ are relatively prime, we can cancel. The easy way to prove this is to note that in this case $a$ is invertible modulo $n$, so there is a $b$ such that $ab\\equiv 1\\pmod n$. Then multiply both sides of the congruence by $b$. \u2013\u00a0Andr\u00e9 Nicolas Apr 16 '12 at 22:15\n\ni) No see Fermat pseudoprime.\n\nii) $2^8=256$ and $2+5+6=13$ so that $2^8= 1+3 \\pmod{9}$ (the other one is right)\n\n$2^{340} = 1\\pmod{341}$ and $341=11\\cdot 31$\n\n-\nThank You. Sorry for the second question- stupid calculation mistake. The article however says the smallest pseudoprime for base 2 is 341. Is it wrong? \u2013\u00a0user26649 Apr 16 '12 at 21:55\n@FarhadYusufali: the article should be right even if I can't prove you that 341 is the smallest pseudoprime... The other problem is that you divide by 8! which is not valid as pointed by Andr\u00e9 since $8! = 0 \\pmod{9}$ (corrected...) \u2013\u00a0Raymond Manzoni Apr 16 '12 at 22:34","date":"2015-11-30 13:31:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9044625163078308, \"perplexity\": 137.73476495972443}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-48\/segments\/1448398462665.97\/warc\/CC-MAIN-20151124205422-00050-ip-10-71-132-137.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.neetprep.com\/question\/54271-solid-sphere-density----times-lighter-water-suspended-awater-tank-string-tied-its-base-shown-fig-mass-thesphere-m-tension-string-given-amgbmgcmgdmg\/126-Physics--Mechanical-Properties-Fluids\/685-Mechanical-Properties-Fluids","text":"# NEET Physics Mechanical Properties of Fluids Questions Solved\n\nA solid sphere of density\u00a0$\\mathrm{\\eta }$ ( > 1) times lighter than water is suspended in a water tank by a string tied to its base as shown in fig. If the mass of the sphere is m then the tension in the string is given by\n\n(a)\u00a0$\\left(\\frac{\\mathrm{\\eta }-1}{\\mathrm{\\eta }}\\right)\\mathrm{mg}$\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0(b)\u00a0$\\mathrm{\\eta mg}$\n\n(c)\u00a0$\\frac{mg}{\\mathrm{\\eta }}$\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (d)\u00a0$\\left(\\mathrm{\\eta }-1\\right)\\mathrm{mg}$\n\nExplanation is a part of a Paid Course. To view Explanation Please buy the course.\n\nDifficulty Level:","date":"2019-10-22 23:48:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 5, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.34998035430908203, \"perplexity\": 1155.4751345924647}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987826436.88\/warc\/CC-MAIN-20191022232751-20191023020251-00053.warc.gz\"}"}
| null | null |
Buffoonery
Claim: Queen Elizabeth is related to the warlord of the desert, Mohamed…….
Buffoonery UK
By KGSaccess_time3 years agochat_bubble_outline3
A lineage I wouldn't be proud of whatsoever…
Is the Queen related to Prophet Muhammad? Historians believe Elizabeth II is a descendant of the founder of Islam after tracing her family tree back 43 generations
Queen Elizabeth's lineage can be traced back 43 generations to the founder of Islam, according to historians
Claim first surfaced in 1986 after Burke's Peerage, a British authority of royal pedigrees, discovered the link
Although disputed by some historians, genealogical records of early medieval Spain also support the claim
By Katie French For Mailonline
PUBLISHED: 22:14 BST, 6 April 2018 | UPDATED: 02:13 BST, 7 April 2018
Historians believe the Queen is a descendant to the founder of Islam – after tracing her family tree back 43 generations.
The claim makes the British monarch a distant ancestor of the Prophet Muhammad.
The findings were first published in 1986 by Burke's Peerage, a British authority on royal pedigrees.
But the claim has recently resurfaced after a Moroccan newspaper said it had traced the queen's lineage back to the Prophet.
Written by KGS
Previous Post Previous Post Visegrád Group poses more of a threat to the EU than Brexit……
Next Post Next Post UK: Afghani fraudster entitled to funding and questioning over Grenfell apt's inferno…….
By A Swede Speaks Apr 07, 2018
This claim is highly dubious as it seems like Zaida's only confirmed child died childless.
https://genealogics.org/getperson.php?personID=I00065043&tree=LEO
By Jay Harper Apr 09, 2018
Well, Rule Britannia then. Time to assert the queen's authority into the backward parts of the Empire.
By Rana Danyal Oct 29, 2018
I think its a false news just created to create disturbance and to misguide people.
Contact the TT @ tundratabloids@gmail.com
The conflation of Islamofauxbia with that of traditional Jew-hatred has got to be one of the most insidious propaganda efforts by sharia supremacists in the modern age…
Melting pot societies are the only way to secure the individual…
Multiculturalism is a gross failure. Assimilation, where celebrating one's own heritage but as a full member of the dominant culture, wins.
There Is No Such Thing As White Cultural Heritage. The West's Legacy Is Open To All….
There's No Common Cultural Legacy For The Alt-Right
Still, is there something to it? Is there a common heritage that will cover El Greco and Hume and Dostoyevsky? Is there one that can include the Jacobites and the Jacobins? There is, but it is not racial, and white supremacists reject it because it rejects them. The unifying heritage of Europe is religious and philosophical. It is Jerusalem and Athens, in one famous formulation. Christian religion and Greek philosophy, filtered through Roman law and culture, are the foundation of European culture. The tensions, agreements, developments and settlements between these have shaped the Western world, and these roots of Western civilization are not congenial to white supremacy.
Christianity is universal in its message and Jewish in its origins. For centuries after its founding, Christianity's center was the Mediterranean world, including Asia Minor and North Africa. Christianity has never been defined by race, and locally-grown racist heresies are only sustainable among those ignorant of Christianity's teachings, origins and history.
Greek philosophy is likewise ill-suited to serve as a basis for white identity. It is either too universal (addressing the human condition in general) or too local—none of us live as citizens of an ancient Greek polis. Later philosophical developments in Europe, such as the philosophies of the Enlightenment, likewise tend to be too universal for white supremacists seeking a tribal identity. As for the scientific revolution that developed within Western culture (albeit with much borrowed from outside Europe), math doesn't care what color someone is.
"More here"
Daniel Greenfield explains Islam 101:
"Every devout Muslim is an "Islamist". Islam is not a personal religion. It is a religion of the public space. A "moderate" Muslim would have to reject Islam as a religion of the public space, as theocracy, and that secularism would be a rejection of Islam.
Nothing in Islam exists apart from anything else. While liberals view culture and religion as a buffet that they can pick and choose from, it is a single integrated system. If you accept one part, you must accept the whole. Once you accept any aspect of Islam, you must accept its legal system and once you accept that, you must accept its governance and once you accept that, you lose your rights.''
Trending Israeli News…
IDF wrapping up Gaza border crisis probes - exclusive 14 criminal probes, 2 convictions later, IDF likely moving on By YONAH JEREMY BOB JANUARY 6, 2021 08:48
Ex-Israeli envoy reveals large number of Ashkenazi Jews lived in Cairo "Cairo was like a railway terminal, Jews coming and going," ambassador Rosen said. By BENJAMIN WEINTHAL JANUARY 6, 2021 08:33
Israel's abortion rate continues 32-year decline Experts credit increased access to contraception, sex-ed By TARA KAVALER/THE MEDIA LINE JANUARY 5, 2021 18:54
Must watch video! Diana West interviewed on her book "The Red Thread"…….
https://www.youtube.com/watch?time_continue=601&v=xZhDlf7sOaE
Diana West's book offers compelling evidence of the ideological corruption of the US intelligence community in such a stunning way. The very institutions that were created to safeguard the Republic from Marxist, Communist influences, have been not only subverted, but run by the proponents of that demonstrably evil ideology…
GW/CC debunked by premier climatologist Prof.Richard Lindzen
https://www.youtube.com/watch?v=X2q9BT2LIUA
Climate change hoax, how to destroy it…
https://www.youtube.com/watch?v=bpxAIYrtGLw
Trending European News…
Malmö Man Charged in Connection with Brussels 2016 Terror Attack
Farage: Boris Set to Sell the UK 'Down the River' on Immigration
Mr. Bean' Star Rowan Atkinson Compares Cancel Culture to a 'Medieval Mob Looking for Someone to Burn'
(Than you President Trump) Israel-Europe ties improving, warming up to Abraham Accords - exclusive The Foreign Ministry is also in dialogue with the EU and individual member states to stop illegal construction in Area C of the West Bank and to coordinate any further building with Israel.
The Svensmark "The Cloud Mystery", what really drives Global Warming…
https://www.youtube.com/watch?v=yZ7IEUBUe4s
Trending Middle East News…
US Targets Iran's Steel Industry With New Sanctions Treasury Department measures come in last days of Trump presidency
(They loved the deal that allowed them to cheat) EU 'Highly Concerned' About Iran Nuclear Enrichment Spokesman says alliance seeks to preserve nuclear agreement
Iran Tests Drones in Military Exercise Iran and regional forces it backs have increasingly relied in recent years on drones in Yemen, Syria, Iraq and Strait of Hormuz at mouth of Persian Gulf
Iran temporarily frees Jewish prisoner for her crime of visiting Israel By BENJAMIN WEINTHAL
SECULARISM AND RELIGION: THE ONSLAUGHT AGAINST THE WEST'S MORAL CODES
War is being waged against Western culture from within which is in essence a war against Christianity and its moral origins in the Hebrew Bible. By attacking these Biblical foundations in the name of reason and human rights, the culture warriors of secularism are sawing off the branch on which they sit. The only way to defend Western civilisation is to reaffirm and restore its Biblical foundations. My argument is a development of ideas I first explored in my 2012 book The World Turned Upside Down: The Global Battle over God, Truth and Power.
We are living in an era which extols reason, science and human rights. These are said to be essential for progress, a civilised society and the betterment of humanity. Religion is said to be their antithesis, the source instead of superstitious mumbo-jumbo, oppression and backward-thinking.
Some of this hostility is being driven by the perceived threat from Islamic terrorism and the Islamisation of Western culture. However, this animus against religion has far deeper roots and can be traced back to what is considered the birthplace of Western reason, the 18th-century Enlightenment.
Actually, it goes back specifically to the French Enlightenment. In England and Scotland, the Enlightenment developed reason and political liberty within the framework of Biblical belief. In France, by contrast, anti-clericalism morphed into fundamental hostility to Christianity and to religion itself.
"Ecrasez l'infame," said Voltaire (crush infamy) — the infamy to which he referred being not just the Church but Christianity, which he wanted to replace with the religion of reason, virtue and liberty, "drawn from the bosom of nature".
Perfecting society
But this Enlightenment did not remove religion so much as pervert it. It took millenarian fantasies, the idea that the perfection of the world was at hand, and it secularised them. Instead of God producing heaven on earth, it would be mankind which would bring that about. Reason would create the perfect society and "progress" was the process by which utopia would be attained.
More here"
Middle East expert Mordechai Kedar: The Muslim Mind
https://www.youtube.com/watch?v=7sL_8BvqHo0
What mainstream Islam really teaches, what they believe…
https://www.youtube.com/watch?v=j4OCTslNeT8
ADL, Protecting American Muslims, Fooling American Jews…
https://www.youtube.com/watch?v=jVqqPx-Cq8k
Lord Baron Pearson on Brexit…
https://www.youtube.com/watch?v=dh-xT6951kc
Mohammed's Koran: Why Muslims Kill For Islam by Peter McLoughlin (Author), Tommy Robinson (Author)
Jenin: Massacring Truth (This documentary was made long before the term #FakeNews got started…
© 2021Tundra Tabloids.com. All Rights Reserved.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,438
|
Luke Andrew Doran (born 14 August 1991 in Baulkham Hills, Sydney, New South Wales) is an Australian cricketer who plays for the Sydney Sixers and the New South Wales Blues. He was a member of Australia's 2010 Under-19 Cricket World Cup squad which won the 2010 Under-19 Cricket World Cup in New Zealand. He has played in three List A matches for New South Wales, and 15 Big Bash games for the Sydney Sixers. Doran did not play in the 2015–16 Big Bash League season due to a side strain, although he was in the Sixers semi-final squad.
Doran previously represented the Fairfield-Liverpool Lions, and now also represents Lindisfarne in the Cricket Tasmania Premier League; in the 2015 final, Doran was dismissed for 87 and infamously threw his bat at the match official.
He is the older brother of cricketer Jake Doran and Micheal Doran who attends and is currently at Luke/s old school The Hills Sports High School.
See also
List of New South Wales representative cricketers
References
External links
Living people
1991 births
New South Wales cricketers
Australian cricketers
Sydney Sixers cricketers
Cricketers from Sydney
Sydney Thunder cricketers
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 1,835
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Q: SQL Anywhere - SELECT COUNT(PK) WHERE [Conditions] performance I have a query similar to the one below and am trying to find ways of decreasing execution time. I've tried adding normal indexes, but from what I've read using LIKE % string % means the database can't use them. I've also tried adding text indexes and using CONTAINS(AColumn, String) but that also didn't seem to speed things up (I also couldn't find a way to completely mimic the match anywhere behavior with contains). Is there a better way to find a count of rows based on string matching for this scenario? Being able to search AColumn from TableA and BColumn from TableB is a must.
SELECT COUNT(TableA.PK)
FROM TableA LEFT JOIN
TableB
WHERE (AColumn LIKE '%String%' OR BColumn LIKE '%STRING%')
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 316
|
require 'rubygems'
require 'pp'
require 'jira'
require 'uri'
class Service::Jira < Service::Base
title "Jira"
string :project_url, :placeholder => "https://domain.atlassian.net/browse/projectkey",
:label => 'URL to your Jira project: <br />' \
'This should be your URL after you select your project ' \
'under the "Projects" tab.'
string :username, :placeholder => 'username',
:label => "These values are encrypted to ensure your security. <br /><br />" \
'Your Jira username:'
password :password, :placeholder => 'password',
:label => 'Your Jira password:'
page "Project", [ :project_url ]
page "Login Information", [ :username, :password ] # TODO - See comment above, :sync_issues ]
# Create an issue on Jira
def receive_issue_impact_change(config, payload)
url_components = parse_url(config[:project_url])
client = jira_client(config, url_components[:context_path])
project = client.Project.find(url_components[:project_key])
users_text = if payload[:impacted_devices_count] == 1
'This issue is affecting at least 1 user who has crashed '
else
"This issue is affecting at least #{ payload[:impacted_devices_count] } users who have crashed "
end
crashes_text = if payload[:crashes_count] == 1
"at least 1 time.\n\n"
else
"at least #{ payload[:crashes_count] } times.\n\n"
end
issue_description = "Crashlytics detected a new issue.\n" + \
"#{ payload[:title] } in #{ payload[:method] }\n\n" + \
users_text + \
crashes_text + \
"More information: #{ payload[:url] }"
post_body = { 'fields' => {
'project' => {'id' => project.id},
'summary' => payload[:title] + ' [Crashlytics]',
'description' => issue_description,
'issuetype' => {'id' => '1'} } }
# The Jira client raises an HTTPError if the response is not of the type Net::HTTPSuccess
issue = client.Issue.build
if issue.save(post_body)
{ :jira_story_id => issue.id, :jira_story_key => issue.key }
else
raise "Jira Issue Create Failed: #{issue.respond_to?(:errors) ? issue.errors : {}}"
end
rescue JIRA::HTTPError => e
raise "Jira Issue Create Failed. Message: #{ e.message }, Status: #{ e.code }, Body: #{ e.response.body }"
end
def receive_verification(config, payload)
url_components = parse_url(config[:project_url])
client = jira_client(config, url_components[:context_path])
resp = client.Project.find(url_components[:project_key])
verification_response = [true, 'Successfully verified Jira settings']
if config[:sync_issues]
begin
register_webhook(client, payload)
rescue JIRA::HTTPError => e
log "HTTP Error: webhook request(status: #{ e.code }, message: #{ e.message })"
verification_response = [true, 'Successfully verified Jira settings but Jira\'s webhook could not be registered. You need to use an Admin account to set it up.']
end
end
verification_response
rescue JIRA::HTTPError => e
log "HTTP Error: status code: #{ e.code }, body: #{ e.response.body }"
[false, 'Oops! Please check your settings again.']
rescue => e
log "Rescued a verification error in jira: (url=#{config[:project_url]}) #{e}"
[false, 'Oops! Is your project url correct?']
end
def jira_client(config, context_path)
url = config[:project_url]
ssl_enabled = (URI(url).scheme == 'https')
ssl_verify_mode = ssl_enabled ? OpenSSL::SSL::VERIFY_PEER : OpenSSL::SSL::VERIFY_NONE
JIRA::Client.new(
:username => config[:username],
:password => config[:password],
:site => config[:project_url],
:context_path => context_path,
:auth_type => :basic,
:use_ssl => ssl_enabled,
:ssl_verify_mode => ssl_verify_mode
)
end
def parse_url(url)
matches = url.match(/(https?:\/\/.+?)(\/.+)?\/(projects|browse)\/([\w\-]+)/)
if matches.nil?
raise "Unexpected URL format"
end
{ :url_prefix => matches[1], :context_path => matches[2] || '', :project_key => matches[4] }
end
private
def callback_webhook_url(payload)
"https://www.crashlytics.com/api/v3/projects/#{ payload[:app][:id] }/service_hooks/jira/responses"
end
def register_webhook(client, payload)
new_webhook = callback_webhook_url(payload)
# Unregister webhooks that look identical to avoid duplicates
response = client.get('/rest/webhooks/1.0/webhook')
current_hooks = JSON.parse(response.body)
current_hooks.each do |hook|
if hook['url'] == new_webhook
client.delete(hook['self'])
end
end
webhook_params = {
'name' => 'Crashlytics Issue sync',
'url' => new_webhook,
'events' => ['jira:issue_updated'],
'excludeIssueDetails' => false }
client.post('/rest/webhooks/1.0/webhook', webhook_params.to_json)
end
JIRA_FIELDS = [
'assignee',
'created',
'creator',
'description',
'issuetype',
'priority',
'project',
'reporter',
'resolution',
'resolutiondate',
'status',
'summary',
'updated'
]
def format_jira_issue(jira_issue)
response = {
'id' => jira_issue.id,
'key' => jira_issue.key
}
JIRA_FIELDS.each do |field|
response[field] = jira_issue.fields[field]
if response[field].respond_to? :delete
response[field].delete 'self'
end
end
if jira_issue.comments.present? && jira_issue.comments.size
response['comments'] = []
jira_issue.comments.each do |comment|
comment.attrs.delete 'self'
response['comments'] << comment.attrs
end
end
response
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,258
|
Q: How can I access whatsapp/media directory in android 11 (Scoped storage)? I have an app that access status folder in whatsapp/media directory.
But on android 11 my app is not working due to android 11 scoped storage.
Is there a way to access statuses folder in whatsapp/media directory ?
A: In Android 11 the path has changed to /Android/media/com.whatsapp/WhatsApp/media
A: I think it might be work.
String mString= Build.VERSION.SDK_INT<=30 ? "WhatsApp/Media/.Statuses":"Android/media/com.whatsapp/WhatsApp/Media/.Statuses";
A: Google Play changed their policy in All file access permission.
Here I mentioned below the github link where he used file access for Whatsapp status Downloader but the problem is if you publish it on playstore means it won't work they will reject your app so you can find the solution for that.
https://github.com/GauthamAsir/WhatsApp_Status_Saver
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,020
|
require 'active_resource'
require 'yandex_uslugi_wrapper/hash'
module YandexUslugiWrapper
# Класс является базовым для Яндекс Услуг
# Здесь задаются api ключ, основной url для запроса, referer
# Также реализуется наследование headers и методы генерирующие url
class YandexUsluga < ActiveResource::Base
API_SITE = URI::parse("http://api.uslugi.yandex.ru")
API_PREFIX = "/1.0/"
self.site = API_SITE
self.prefix = API_PREFIX
self.format = :xml
# Задание Api key и Referer'a. Необходимы для работы с Яндекс Услугами
@@api_key = ""
@@referer = ""
class << self
# Если headers не заданы в подклассе, то назначаем ему
# headers суперкласса
def headers
if defined?(@headers)
@headers
elsif superclass != Object && superclass.headers
superclass.headers
else
@headers ||= {}
end
end
def api_key
@@api_key
end
def api_key=(key)
@@api_key = key
use_api_key
end
def referer
@@referer
end
def referer=(referer_site)
@@referer = referer_site
use_referer_site
end
# Метод генерирует url для элемента, например /banks/{id}
# Удаляем из стандратного метода .#{format.extension}, чтобы в конце запроса не было .xml
def element_path(id, prefix_options = {}, query_options = nil)
prefix_options, query_options = split_options(prefix_options) if query_options.nil?
"#{prefix(prefix_options)}#{collection_name}/#{id}#{query_string(query_options)}"
end
# Метод генерирует url для коллекции, например /banks
# Также удаляем .#{format.extension}
def collection_path(prefix_options = {}, query_options = nil)
prefix_options, query_options = split_options(prefix_options) if query_options.nil?
"#{prefix(prefix_options)}#{collection_name}#{query_string(query_options)}"
end
end
private
def self.use_api_key
self.headers['Authorization'] = @@api_key
end
def self.use_referer_site
self.headers['Referer'] = @@referer
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,580
|
Ján Bátik ( à Liptovský Mikuláš en Tchécoslovaquie) est un céiste slovaque pratiquant le slalom.
Biographie
Palmarès
Championnats du monde de canoë-kayak slalom
2007 à Foz do Iguaçu,
Médaille de bronze en relais 3xC2
2009 à La Seu d'Urgell,
Médaille d'or en relais 3xC2
Championnats d'Europe de canoë-kayak slalom
2008 à Cracovie
Médaille de bronze en relais 3xC2
2010 à Čunovo,
Médaille d'or en relais 3xC2
Liens externes
Céiste slovaque
Céiste de slalom
Naissance en janvier 1986
Naissance à Liptovský Mikuláš
Champion d'Europe de slalom (canoë-kayak)
Naissance en Tchécoslovaquie
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,223
|
Carlsbad is a prosperous community with a lot of momentum. Our economy is diversified, robust, and growing. With so many companies succeeding, our workforce is well-paid with a high disposable income. This has created incredible growth opportunities for real estate developers, retailers, restaurants, hotels, and other businesses. Now is the right time to invest in Carlsbad and those who do will have a first mover's advantage.
Businesses are not the only ones thriving here. With a high quality of life, gorgeous outdoor spaces, strong educational system, and a safe community atmosphere, our region of 70,000 is becoming a go-to destination for millennials, families, and retirees. Come visit, explore and discover why Carlsbad, NM should become your next home.
Live in Carlsbad, New Mexico.
Work in Carlsbad, New Mexico.
Play in Carlsbad, New Mexico.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,485
|
{"url":"https:\/\/lavelle.chem.ucla.edu\/forum\/viewtopic.php?f=135&t=26581&p=80661","text":"Gibbs Free Energy [ENDORSED]\n\n$\\Delta G^{\\circ}= \\Delta H^{\\circ} - T \\Delta S^{\\circ}$\n\n$\\Delta G^{\\circ}= -RT\\ln K$\n\n$\\Delta G^{\\circ}= \\sum \\Delta G_{f}^{\\circ}(products) - \\sum \\Delta G_{f}^{\\circ}(reactants)$\n\nDaniisaacson2F\nPosts: 30\nJoined: Sat Jul 22, 2017 3:00 am\n\nGibbs Free Energy\n\nIf deltaS is the disorder (for use of a better word) or a reaction, how is that different than delta G?\n\nLauren Seidl 1D\nPosts: 51\nJoined: Fri Sep 29, 2017 7:06 am\n\nRe: Gibbs Free Energy\n\nDelta G combines the values of delta H and delta S to fully determine whether a reaction is favorable or not.\n\nmayasinha1B\nPosts: 70\nJoined: Fri Sep 29, 2017 7:04 am\n\nRe: Gibbs Free Energy\n\nGibbs free energy is a term that defines whether a reaction will be spontaneous or not. It combines the entropy and enthalpy (delta H) to determine the favorability of a reaction.\n\nOliviaShearin2E\nPosts: 37\nJoined: Fri Sep 29, 2017 7:05 am\n\nRe: Gibbs Free Energy\n\nCan you further clarify your question?\n\nJessica_Singh_1J\nPosts: 50\nJoined: Fri Sep 29, 2017 7:03 am\n\nRe: Gibbs Free Energy\n\nDeltaG is a measure of the change in free energy of a reaction, whereas deltaS is a measure of the change in entropy. The equation deltaG = deltaH - TdeltaS uses the changes in enthalpy (deltaH) and entropy (deltaS) at constant temperature to calculate change in free energy (deltaG). A negative deltaG indicates a spontaneous reaction; a positive deltaG indicates the opposite.\n\nLeanne Wong 1H\nPosts: 50\nJoined: Tue Oct 10, 2017 7:13 am\nBeen\u00a0upvoted: 1 time\n\nRe: Gibbs Free Energy\u00a0\u00a0[ENDORSED]\n\nGibbs free energy determines the favorability of a reaction whereas S shows the change in entropy\/\"disorder\".\n\nSohini Halder 1G\nPosts: 58\nJoined: Thu Jul 13, 2017 3:00 am\nBeen\u00a0upvoted: 1 time\n\nRe: Gibbs Free Energy\n\nGibbs free energy also shows how much energy is \"Free\" to do nonexpansion work, whereas entropy relates to the possible number of states a system could be in.","date":"2019-08-19 17:15:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 3, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.49435675144195557, \"perplexity\": 4305.155216036607}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027314852.37\/warc\/CC-MAIN-20190819160107-20190819182107-00500.warc.gz\"}"}
| null | null |
\section{Introduction and Preliminaries}
One of the fundamental trends in
semigroup theory has been the study of how idempotents shape the structure of the semigroup. Howie's book \cite{Ho95} can be seen as an excellent survey of the results obtained from the 40s to the 90s on this general problem. Evidently, there is the analogous question for the group of units, namely, to what extent the group of units shapes the structure of the semigroup. (And a similar question can be asked about the normalizer of the semigroup; more on this below.) However, unlike the idempotents case, the group of units approach quickly leads to problems that could not be tackled with the tools available 30 years ago, let alone 70 years ago. Fortunately now the situation is totally different, since the enormous progress made in the last decades in the theory of permutation groups provides the necessary tools to develop semigroup theory from this different point of view.
A particular instance of the general problem of investigating how the group of units shapes the whole semigroup might be described as follows:
{\em classify the pairs $(a,G)$, where $a$ is a map on $X$ and $G$ is a group of permutations of $X$,
such that the semigroup $\genset{a,G}$, generated by $a$ and $G$, has a given property}. (Observe that whenever $S$ is a semigroup with group of units $G$ we have $S=\cup_{a\in S}\genset{a,G}$.)
A very important class of groups that falls under this general scheme is that of synchronizing groups, groups of permutations on a set that together with any non-invertible map on the same set generate a constant (see \cite{steinberg}, \cite{cameron}, \cite{neumann}). These groups are very interesting from a group theoretic point of view and are linked to the \v{C}ern\'y conjecture, a longstanding open problem in automata theory.
Another instance of the general problem described above is the following: {\em classify the permutation groups on a set that together with any map on that set generate a regular semigroup}. (An element $a$ in a semigroup $S$ is said to be regular if there exists $b\in S$ such that $a=aba$. The semigroup is said to be regular if all its elements are regular.) This question has been answered in \cite{ArMiSc} as follows. (From now on $\mathcal{S}_{n}$ will denote the symmetric group on the set $[n]:=\{1,\ldots,n\}$; by $\mathcal{T}_{n}$ we will denote the full transformation monoid on $[n]$. We use the notation $D(2*n)$ for the dihedral group of order $2n$, called either $D_{n}$ or $D_{2n}$ in the literature; other notation for finite groups is standard.)
\begin{thm}\label{th2}
If $n\geq 1$ and $G$ is a subgroup of $\mathcal{S}_{n}$, then the following are equivalent:
\begin{enumerate}
\item[\rm (i)] The semigroup $\genset{G,a}$ is regular for all $a\in \mathcal{T}_{n}$.
\item[\rm (ii)] One of the following is valid for $G$ and $n$:
\begin{enumerate}
\item[\rm (a)] $n=5$ and $G\cong C_5,\ D(2*5),$ or $\mbox{\rm AGL}(1,5)$;
\item[\rm (b)] $n=6$ and $G\cong \mbox{\rm PSL}(2,5)$ or $\mbox{\rm PGL}(2,5)$;
\item[\rm (c)] $n=7$ and $G\cong\mbox{\rm AGL}(1,7)$;
\item[\rm (d)] $n=8$ and $G\cong\mbox{\rm PGL}(2,7)$;
\item[\rm (e)] $n=9$ and $G\cong\mbox{\rm PSL}(2,8)$ or $\mbox{\rm P}\Gamma {\rm L}(2,8)$;
\item[\rm (f)] $G=\mathcal{A}_{n}$ or $\mathcal{S}_{n}$.
\end{enumerate}
\end{enumerate}
\end{thm}
The critical observation that led to the proof of Theorem \ref{th2} is that for $n\geq 12$, if $G\ (\leq \mathcal{S}_{n})$ satisfies the property that any rank $\lfloor\frac{n}{2}\rfloor$ map $a\in \mathcal{T}_{n}$ is regular in the semigroup $\genset{G,a}$, then $G$ contains the alternating group. Therefore Theorem \ref{th2} can be seen as an (almost) immediate corollary of the following result.
\begin{thm}\label{th3}
If $n\geq 12$ and $G$ is a subgroup of $\mathcal{S}_{n}$, then the following are equivalent:
\begin{enumerate}
\item[\rm (i)] The element $a$ is regular in $\genset{G,a}$, for all $a\in \mathcal{T}_{n}$ such that $\operatorname{rank}(a)=\lfloor\frac{n}{2}\rfloor$.
\item[\rm (ii)] $G=\mathcal{A}_{n}$ or $\mathcal{S}_{n}$.
\end{enumerate}
\end{thm}
Below a sharper version of this result is going to be stated.
The aim of this paper is to carry out a deeper group theoretic analysis in order to prove a lemma on the groups $G$ that satisfy the following property: for every rank $k$ map $a$ (with fixed $k$ such that $1\leq k\leq \lfloor\frac{n+1}{2}\rfloor$), $a$ is regular in $\genset{G,a}$; and then extract many important consequences from this result.
In order to understand these groups we need some observations. Suppose $G\leq \mathcal{S}_{n}$ and all rank $k$ maps $a\in \mathcal{T}_{n}$ are regular in $\genset{G,a}$. Then there exists $b\in \genset{G,a}$ such that $a=aba$ and hence $ab$ is an idempotent in $\genset{G,a}$ having the same rank as $a$, where $b=a$ or $b\in G$ or $b=g_{1}ag_{2}a\ldots ag_{m}$, for $m\geq 2$. Suppose $b\in G$. Then there exists $g\in G$ (namely $g=b$) such that $ag$ is idempotent and hence $[n]ab$ is a section for the partition of $[n]$ induced by the kernel of $a$. Suppose $b=g_{1}ag_{2}a\ldots ag_{m}$ with $m\geq 2$. Since $ab$ is idempotent and $ab=ag_{1}ag_{2}a\ldots ag_{m}$ it follows that there exists $g\in G$ (namely $g=g_{1}$) such that $a$ and $aga$ have the same rank. In both cases there exists $g\in G$ such that $[n]ag$ is a section for the partition of $[n]$ induced by the kernel of $a$. As this property must hold for all rank $k$ maps $a\in \mathcal{T}_{n}$, it follows that $G$ must satisfy the following $k$-\textit{universal transversal property}: for every $k$-set $I\subseteq [n]$ and every partition $P$ of $[n]$ into $k$ blocks, there exists $g\in G$ such that $Ig$ is a section for $P$. The next six results (almost) provide the classification of the groups possessing the $k$-universal transversal property.
The first theorem handles the permutation groups of small degree.
\begin{thm}\label{th4a}
For $n<11$ and $2\leq k\leq \lfloor\frac{n+1}{2}\rfloor$, a group $G\leq \mathcal{S}_{n}$ with the $k$-universal transversal property is
$k$-homogeneous, with the following exceptions:
\begin{enumerate}
\item $n=5$, $G\cong C_{5}$ or $D(2*5)$ and $k=2$;
\item $n=6$, $G\cong \mbox{\rm PSL}(2,5)$ and $k=3$;
\item $n=7$, $G\cong C_{7}$ or $G\cong D(2*7)$, and $k=2$; or
$G\cong\mbox{\rm AGL}(1,7)$ and $k=3$;
\item $n=8$, $G\cong \mbox{\rm PGL}(2,7)$ and $k=4$;
\item $n=9$, $G\cong 3^{{2}}:4$ or $G\cong 3^{{2}}:D(2*4)$ and $k=2$;
\item $n=10$, $G\cong \mathcal{A}_{5} $ or $G\cong \mathcal{S}_{5}$ and $k=2$; or
$G\cong\mbox{\rm PSL}(2,9)$ or $G\cong \mathcal{S}_{6}$ and $k=3$.
\end{enumerate}
\end{thm}
The next results deal with the groups of degree larger than 10. We start by the case of groups possessing the $k$-universal transversal property, for large values of $k$.
\begin{thm}\label{th4b}
Let $n\geq 11$, $G\leq \mathcal{S}_{n}$.
If $6\leq k\leq \lfloor \frac{n+1}{2}\rfloor$, then the following are equivalent:
\begin{enumerate}
\item $G$ has the $k$-universal transversal property;
\item $\mathcal{A}_{n}\leq G$.
\end{enumerate}
\end{thm}
The four next results deal with groups possessing the $k$-universal transversal property, when $k\in \{2,\ldots,5\}$.
\begin{thm}\label{th4c}
Let $n\geq 11$, $G\leq \mathcal{S}_{n}$ and let $2\leq k\leq \lfloor \frac{n+1}{2}\rfloor$.
The following are equivalent:
\begin{enumerate}
\item $G$ has the $5$-universal transversal property;
\item $G$ is 5-homogeneous, or $n=33$ and $G=\mbox{\rm P}\Gamma {\rm L}(2,32)$.
\end{enumerate}
\end{thm}
Unlike the previous cases, the classification of groups possessing the $4$-universal transversal property was not possible. So far we have the following results, and we believe the remaining cases require very delicate considerations.
\begin{thm}\label{th4d}
Let $n\geq 11$, $G\leq \mathcal{S}_{n}$ and let $2\leq k\leq \lfloor \frac{n+1}{2}\rfloor$. If $G$ is $4$-homogeneous, or $n=12$ and $G=\mbox{\rm M}_{11}$, then $G$ has the $4$-universal transversal property.
If there are more groups possessing the $4$-universal transversal property, then they must be groups $G$ such that $\mbox{\rm PSL}(2,q)\leq G\leq \mbox{\rm P}\Gamma {\rm L}(2,q)$, with either $q$ prime or $q=2^p$ for $p$ prime. Note, however, that the groups $\mbox{\rm PSL}(2,q)$ for $q\equiv1$ (mod~$4$) cannot possess the $4$-universal transversal property since they fail to satisfy the necessary condition of $3$-homogeneity.
\end{thm}
Similarly to the previous case, a full classification of the groups possessing the $3$-universal transversal property was not possible.
\begin{thm}\label{th4e}
Let $n\geq 11$, $G\leq \mathcal{S}_{n}$ and let $2\leq k\leq \lfloor \frac{n+1}{2}\rfloor$. $G$ has the $3$-universal transversal property if $G$ is $3$-homogenous, or one of the following groups
\begin{enumerate}
\item $\mbox{\rm PSL}(2,q)\leq G\leq \mbox{\rm P}\Sigma {\rm L}(2,q)$, where $q\equiv 1 (\mbox{mod }4)$;
\item $\mbox{\rm Sp}(2d,2)$ with $d\geq 3$, in either of its $2$-transitive representations;
\item $2^{{2d}}:\mbox{\rm Sp}(2d,2)$;
\item \mbox{\rm HS};
\item ${\mbox{\rm Co}}_{3}$;
\item $2^{{6}}:{\mbox{\rm G}}_{2}(2)$ and its subgroup of index $2$;
\item $\mbox{\rm AGL}(1,p)$ where, for all $c\in {\mbox{\rm GF}}(p)\setminus\{0,1\}$, $|\langle -1,c,c-1\rangle|=p-1$.
\end{enumerate}
If there are more groups possessing the $3$-universal transversal property, then they must be Suzuki groups $\mbox{\rm Sz}(q)$, possibly with field automorphisms adjoined, and/or subgroups of index $2$ in $\mbox{\rm AGL}(1,p)$ for $p\equiv 11$ $(mod\ 12)$.
\end{thm}
Finally, possessing the $2$-universal transversal property is just another way of saying primitive.
\begin{thm}\label{th4f}
A permutation group has the $2$-universal transversal property if and only the group is primitive.
\end{thm}
These theorems immediately imply an analogue of the Livingstone--Wagner \cite{lw} result on $k$-homogenous groups (for $2\leq k\leq \lfloor\frac{n+1}{2}\rfloor$).
\begin{cor}\label{cor1}
Let $n\geq 5$, let $2\leq k\leq \lfloor \frac{n+1}{2}\rfloor$, and let $G\leq \mathcal{S}_{n}$ be a group having the $k$-universal transversal property. Then $G$ has the $(k-1)$-universal transversal property.
\end{cor}
Now consider partitions of type $(\overbrace{1,\ldots,1}^{k-1},n-k+1)$ (that is, $k-1$ classes of size 1 and one class of size $n-k+1$). Any group $G$ satisfying the $k$-universal transversal property must also satisfy the following property: for every $k$-partition $P$ of $[n]$ of type
$({1,\ldots,1},n-k+1)$ and for every $k$-set $I$ there exists
$g\in G$ such that $Ig$ is a section for $P$. In particular this implies that the union of all the singleton blocks is contained in $Ig$; as the union of the singleton blocks can be any $(k-1)$-subset of $[n]$, it follows that any group $G$ possessing the $k$-universal transversal property must be $(k-1,k)$-homogeneous, that is, for every $(k-1)$-set $I$ and for every $k$-set $J$ there exists $g\in G$ such that $Ig\subseteq J$. In this new setting we can state the sharper version of Theorem \ref{th3} announced above.
\begin{thm}\label{th5}
If $n\geq 12$ and $G$ is a subgroup of $\mathcal{S}_{n}$, then the following are equivalent:
\begin{enumerate}
\item[\rm (i)] $G$ is $(\lfloor \frac{n}{2}\rfloor-1,\lfloor \frac{n}{2}\rfloor)$-homogeneous.
\item[\rm (ii)] The map $a$ is regular in $\genset{G,a}$ for all $a\in \mathcal{T}_{n}$ such that $\operatorname{rank} (a)=\lfloor\frac{n}{2}\rfloor$ and $a$ has kernel type $(1,\ldots,1,\lceil\frac{n}{2}\rceil+1)$.
\item[\rm (iii)] The map $a$ is regular in $\genset{G,a}$ for all $a\in \mathcal{T}_{n}$ such that $\operatorname{rank}(a)=\lfloor\frac{n}{2}\rfloor$.
\item[\rm (iv)] $G=\mathcal{A}_{n}$ or $\mathcal{S}_{n}$.
\end{enumerate}
\end{thm}
Our second main theorem on groups provides the following classification of the $(k-1,k)$-homogeneous groups.
\begin{thm}\label{th6}
If $n\geq 1$ and $2\leq k\leq \lfloor \frac{n+1}{2} \rfloor$, then the following are equivalent:
\begin{enumerate}
\item[\rm (i)] $G$ is a $(k-1,k)$-homogeneous subgroup of $\mathcal{S}_{n}$;
\item[\rm (ii)] $G$ is $(k-1)$-homogeneous or $G$ is one of the following groups
\begin{enumerate}
\item[\rm (a)] $n=5$ and $G\cong C_5,\ D(2*5),$ $k=3$;
\item[\rm (b)] $n=7$ and $G\cong\mbox{\rm AGL}(1,7)$, with $k=4$;
\item[\rm (c)] $n=9$ and $G\cong\mbox{\rm ASL}(2,3)$ or $\mbox{\rm AGL}(2,3)$, with $k=5$.
\end{enumerate}
\end{enumerate}
\end{thm}
Once again an analogue of the Livingstone--Wagner \cite{lw} result is immediate.
\begin{cor}\label{cor2}
Let $n\geq 1$, let $3\leq k\leq \lfloor \frac{n+1}{2}\rfloor$, and let $G\leq \mathcal{S}_{n}$ be a $(k-1,k)$-homogeneous group. Then $G$ is a $(k-2,k-1)$-homogeneous group, except when $n=9$ and $G\cong\mbox{\rm ASL}(2,3)$ or $\mbox{\rm AGL}(2,3)$, with $k=5$.
\end{cor}
By \cite[Theorem 2.3]{lmm} we know that every rank $k$ map $a\in \mathcal{T}_{n}$ is regular in $\genset{G,a}$ if and only if $G$ has the $k$-universal transversal property, that is, in the orbit of every $k$-set there exists a transversal (or section) for every $k$-partition. Therefore we can state our main results about semigroups.
A quasi-permutation is a transformation in which all kernel classes but one are singletons.
\begin{thm}\label{th6c}
Let $G\leq \mathcal{S}_{n}$ and let $1< k\leq \lfloor \frac{n+1}{2}\rfloor$. Then the following are equivalent
\begin{enumerate}
\item for every quasi-permutation $a$, such that $\mbox{rank}(a)=k$, the semigroup $\langle G,a\rangle$ is regular;
\item $G$ is $(k-1)$-homogeneous or $G$ is one of the following groups
\begin{enumerate}
\item[\rm (a)] $n=5$ and $G\cong C_5,\ D(2*5)$, with $k=3$;
\item[\rm (b)] $n=7$ and $G\cong\mbox{\rm AGL}(1,7)$, with $k=4$;
\end{enumerate}
\end{enumerate}
\end{thm}
We are now ready to state the result that dramatically generalizes Theorem \ref{th2} (and also \cite[Theorem 2.3]{lmm} taking advantage of the fact that if a group has the $k$-universal transversal property, then it also has the $(k-1)$-universal transversal property, by Corollary \ref{cor1}).
\begin{thm}\label{Kut1}
Let $n\geq 5$, $G\leq \mathcal{S}_{n}$ and let $1< k\leq \lfloor \frac{n+1}{2}\rfloor$. Then the following are equivalent:
\begin{enumerate}
\item all rank $k$ transformations $a\in \mathcal{T}_{n}$ are regular in $\langle a, G\rangle$;
\item for all rank $k$ transformations $a\in \mathcal{T}_{n}$, the semigroup $\langle a, G\rangle$ is regular;
\item $G$ has the $k$-universal transversal property (and hence is one of the groups listed in the classification).
\end{enumerate}
\end{thm}
One last word about the normalizer. It is well known that not every semigroup has a group of units, and hence the approach proposed in this paper might seem limited. Therefore two observations should be made here. The first is that it is commonly believed that the majority of finite semigroups have only one idempotent (which is a zero), but that did not prevent experts in semigroup theory to investigate how idempotents shape the structure of a semigroup; and the second observation is that by \cite[Theorem 2.3, $(iii)\Leftrightarrow(iv)\Leftrightarrow (v)$]{lmm}, Theorem \ref{th6c} and Theorem \ref{Kut1} admit versions in terms of conjugates. As a sample result we have the following immediate (from \cite[Theorem 2.3]{lmm}) version of the previous theorem.
\begin{thm}\label{KutNorm}
Let $n\geq 5$, $G\leq \mathcal{S}_{n}$ and let $1< k\leq \lfloor \frac{n+1}{2}\rfloor$. Then the following are equivalent:
\begin{enumerate}
\item all rank $k$ transformations $a\in \mathcal{T}_{n}$ are regular in $$\langle g^{-1}ag\mid g\in G\rangle;$$
\item for all rank $k$ transformations $a\in \mathcal{T}_{n}$ the semigroup $\langle g^{-1}ag\mid g\in G\rangle$ is regular;
\item $G$ has the $k$-universal transversal property.
\end{enumerate}
\end{thm}
This observation is important because the transformation semigroup $S\leq \mathcal{T}_{n}$ might contain no group of units, but every transformation semigroup $S$ has a normalizer and hence the results of this paper can be used to extract information about the structure of $S$ from its normalizer. For example, if $S=\langle t_{1},\ldots ,t_{m}\rangle$ is a semigroup generated by $m$ rank 3 maps (for example in $\mathcal{T}_{176}$) and it turns out that the normalizer $N(S):=\{g\in\mathcal{S}_{n} \mid g^{-1}Sg = S\}$ contains the Higman--Sims group, then we know that the semigroup $S$ is regular.
An even more striking consequence of Corollary \ref{cor1} and of the fact that possessing the $k$-universal transversal property is closed upwards (that is, if $G\leq H\leq \mathcal{S}_{n}$ and $G$ has the $k$-universal transversal property, then $H$ also has it), is the following result.
\begin{thm}
Let $n\geq 5$, $G\leq \mathcal{S}_{n}$ and let $1< k< \lfloor \frac{n+1}{2}\rfloor$. Then the following are equivalent:
\begin{enumerate}
\item[(i)] $G$ has the $k$-universal transversal property;
\item[(ii)] $G$ has the $l$-universal transversal property for all $l$ such that $1\leq l\leq k$;
\item[(iii)] $H$ has the $k$-universal transversal property for all $H$ such that $G\leq H\leq \mathcal{S}_{n}$;
\item[(iv)] $H$ has the $l$-universal transversal property for all $H$ such that $G\leq H\leq \mathcal{S}_{n}$ and for all $l$ such that $1\leq l\leq k$;
\end{enumerate}
As a consequence, the following are equivalent.
\begin{enumerate}
\item all rank $k$ transformations $a\in \mathcal{T}_{n}$ are regular in $\langle a, G\rangle$;
\item all rank $k$ transformations $a\in \mathcal{T}_{n}$ are regular in $\langle g^{-1}ag\mid g\in G\rangle$;
\item for all rank $k$ transformations $a\in \mathcal{T}_{n}$ and for all groups $H$ such that $G\leq H\leq \mathcal{S}_{n}$ we have that
$\langle h^{-1}ah\mid h\in H\rangle$ is a regular semigroup.
\item for all rank $k$ transformations $a\in \mathcal{T}_{n}$ and for all groups $H$ such that $G\leq H\leq \mathcal{S}_{n}$ we have that
$\langle a, H\rangle$ is a regular semigroup.
\end{enumerate}
\end{thm}
Finally we summarize what this paper brings to groups and to semigroups:
\begin{enumerate}
\item We have generalized the notion of $(k-1)$-homogeneity in permutation groups; we have extended it first to the obvious notion of $(k-1,k)$-homogeneous groups and then extended this to the notion of groups having the $k$-universal transversal property (for $2\leq k\leq \lfloor \frac{n}{2}\rfloor$).
\item The $(k-1,k)$-homogenous groups were fully classified, and the groups having the $k$-universal transversal property have been classified, with the exception of a class of groups (for $k=3$) and another class (for $k=4$). These two classes left undecided are surely very interesting problems for group theorists and combinatorialists.
\item As a corollary of the classification it follows that $(k-1,k)$-homogenous groups are $(k-2,k-1)$-homogenous with two exceptions; and groups having the $k$-universal transversal property have the $(k-1)$-universal transversal property. And this fact is extremely important for the impact of these results on the theory of semigroups.
\item Regarding semigroups, we take deep results out of the classification of finite simple groups and show that it is possible to follow the promising path of investigating how the group of units (or other groups associated to the semigroup such as the normalizer) shape the structure of the semigroup. This mimics what has been done in semigroup theory for the last 70 years with the set of idempotents.
\item The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.
\end{enumerate}
\section{The classification of $(k-1,k)$-homogeneous groups}
A permutation group $G$ of degree~$n$ is \emph{$k$-homogeneous} if it acts
transitively on the set of $k$-subsets of its domain. Since $k$-homogeneity
is clearly equivalent to $(n-k)$-homogeneity, it is usually assumed that
$k\le(n-1)/2$. With this assumption, Livingstone and Wagner, in an elegant
paper~\cite{lw}, proved that a $k$-homogeneous group is $(k-1)$-homogeneous,
and is $k$-transitive if $k\ge5$.
Kantor~\cite{kantor:4homog,kantor:2homog} determined all
$k$-homogeneous groups which are not $k$-transitive for $2\le k\le 4$.
The $2$-transitive groups have been determined as a consequence of the
Classification of Finite Simple Groups; lists of them can be found in
\cite{cam} and \cite{dixon}.
For $k\le l$, the permutation group $G$ is $(k,l)$-homogeneous if, given
subsets $K,L$ of the domain with $|K|=k$ and $|L|=l$, there is an element
of $G$ which maps $K$ to a subset of $L$. Note that
\begin{itemize}
\item $(k,k)$-homogeneity is equivalent to $k$-homogeneity, and for fixed $k$
the concept of $(k,l)$-homogeneity becomes formally weaker as $l$ increases;
\item $(k,l)$-homogeneity is equivalent to the ``dual'' concept of $(l,k)$-homogeneity, requiring that for given $K$ and $L$ as before, there is an
element of $G$ mapping $L$ to a superset of $K$;
\item $(k,l)$-homogeneity is equivalent to $(n-l,n-k)$-homogeneity.
\end{itemize}
In this section we are concerned with $(k,k+1)$-homogeneity. Because of the
third property above, we lose no generality in assuming, as Livingstone and
Wagner did, that $k\le(n-1)/2$; indeed this condition will be used in our
proofs several times. We had hoped to find arguments as elegant as those of
Livingstone and Wagner, but have not succeeded. We prove the following
theorem:
\begin{thm}\label{2.1}
Let $G$ be a $(k,k+1)$-homogeneous permutation group of degree $n\ge2k+1$.
Then either $G$ is $k$-homogeneous, or $G$ is one of the following groups:
\begin{enumerate}
\item $n=5$, $k=2$, $G\cong C_5$ or $D(2*5)$;
\item $n=7$, $k=3$, $G=\mbox{\rm AGL}(1,7)$;
\item $n=9$, $k=4$, $G=\mbox{\rm ASL}(2,3)$ or $\mbox{\rm AGL}(2,3)$.
\end{enumerate}
\label{kkp1}
\end{thm}
\subsection{General observations}
Let $G$ be $(k,k+1)$-homogeneous of degree $n\ge2k+1$. We begin with a few
general observations.
\begin{prop}
The number of $G$-orbits on $k$-sets is at most $k$.
\end{prop}
\begin{proof}
Since a fixed $(k+1)$-set contains a representative of every orbit on
$k$-sets, there are at most
$k+1$ orbits on $k$-sets. This bound can be reduced to $k$. For suppose there
are $k+1$ orbits; then each $(k+1)$-set contains exactly one $k$-set from each
orbit. Let $V_k$ be the $\mathbb{Q}$-vector space of functions from $k$-sets
to $\mathbb{Q}$, and let $T:V_k\to V_{k+1}$ be defined by
\[(Tf)(L)=\sum_{x\in L}f(L\setminus\{x\})\]
for $f\in V_k$ and $|L|=k+1$. Since $n\ge 2k+1$, it is known that $T$ is
injective (Kantor~\cite{kantor:inc}). However, if $f$ is the characteristic
function of any $G$-orbit, then $Tf$ is the all-$1$ function.
\end{proof}
This gives a lower bound for $|G|$, namely $|G|\ge{n\choose k}/k$. We refer
to this as the \emph{order bound}.
The right-hand side of this bound is a monotonic function of $k$
for $k<n/2$; so, whenever we rule out a group $G$ on the basis of this
inequality for a certain value of $k$, then it cannot occur for any larger
value of $k$ either.
The \emph{Ramsey number} $R(k,l,r)$, for positive integers $k,l,r$ with
$k\le l$ and $r\ge1$, is the smallest number $n$ such that, if the $k$-element
subsets of an $n$-set are coloured with $r$ colours, there exists a
$l$-element set all of whose $k$-element subsets have the same colour.
\begin{prop}
If $G$ is $(k,k+1)$-homogeneous but not $k$-homogeneous of degree $n$, then
$n<R(k,k+1,2)$.
\end{prop}
\begin{proof}
Colour the $k$-sets in one $G$-orbit
red and the others blue. Each $(k+1)$-set contains $k$-sets of each colour.
\end{proof}
It happens that the Ramsey numbers $R(2,3,2)=6$ and $R(3,4,2)=13$ are two
of the very few which are known exactly. The first is the well-known ``party
problem''; the second was computed by McKay and Radzizowski~\cite{mr} in 1991
(see \cite{rad} for a survey). The number $R(4,5,2)$ is not known, and the
known upper bounds are too large for our purpose.
In the case $k=2$, we have $n\le5$, and it is easy to see that the
cyclic and dihedral groups are examples and hence we have (1) of Theorem \ref{2.1}. So we may assume that $k\ge3$.
\medskip
Our general results allowed us to decide which groups are $(k,k+1)$-homogeneous, except for a number of groups of small degrees. To decide those cases we used \textsf{GAP}~\cite{GAP} and include here a word about those computations. For $n\le20$,
the simplest method is to compute the orbits of a given group on $k$-sets
and $(k+1)$-sets, and for each orbit representative on $(k+1)$-sets, test
whether it contains representatives of all the $k$-set orbits.
For larger $n$, the memory requirements of this method are too heavy, so
we proceed a little differently. First, as we will prove below, any candidate
group must be $2$-transitive; so we reject groups which either fail to be
$2$-transitive or are $k$-homogeneous. We also reject groups which fail the
order bound $|G|\ge{n\choose k}/k$. Then, if $G$ is $t$-transitive, we loop
over all pairs $(K',L')$, where $K'$ and $L'$ are subsets of $\{t+1,\ldots,n\}$
of cardinality $k-t$ and $k+1-t$ respectively, and check whether there is
an element of $G$ mapping $\{1,\ldots,t\}\cup K'$ to a subset of
$\{1,\ldots,t\}\cup L'$. If this fails for any pair $(K',L')$, we can terminate
the computation and report that $G$ is not $(k,k+1)$-homogeneous.
\subsection{Transitivity}
From now on, $G$ will be a $(k,k+1)$-homogeneous but not $k$-homogeneous
group of degree $n\ge2k+1$, which is not one of the exceptions listed in
the statement of Theorem~\ref{kkp1}.
\begin{prop}
$G$ is transitive.
\end{prop}
\begin{proof}
Let $O$ be an orbit of $G$. There exists a $k$-set containing at least
one point of $O$. Hence every $(k+1)$-set contains a $k$-set containing at
least one point of $O$, and thus intersects $O$. So $|O|\ge n-k > n/2$. Since
$O$ was arbitrary, there is only one orbit.
\end{proof}
\subsection{Primitivity}
\begin{prop}
$G$ is primitive.
\end{prop}
\begin{proof}
Suppose that $G$ is imprimitive, with $r$ blocks of size $s$.
If $k\le s$, then there is a $k$-set contained in a block. But, since
$k+1\ge4$, there is a $(k+1)$-set and all instances of an $k$-set or an $(k+1)$ containing at least two points of each of
two blocks; such a set cannot contain a $k$-set of the type just described.
So $k>s$, and $r=n/s>n/k>2$, so $r\ge3$. There is a $(k+1)$-set which
contains either $\lfloor(k+1)/r\rfloor$ or $\lceil(k+1)/r\rceil$ points from
each block. On the other hand, there is a $k$-set containing all the points of
a block. So $(k+1)/r\ge\lceil(k+1)/r\rceil-1\ge s-1$, whence
$k+1\ge (r-1)s\ge2n/3$, a contradiction.
\end{proof}
Since lists of primitive groups are conveniently available in computer
algebra systems such as GAP, we have checked all primitive
groups of degree at most $20$, and find no counterexamples for the statement of Theorem \ref{2.1}. In view of our
remarks about Ramsey numbers earlier, we may from now on assume that $k\ge4$.
\subsection{$2$-homogeneity}
\begin{prop}
$G$ is $2$-homogeneous.
\end{prop}
\begin{proof} Assume that $G$ is not $2$-homogeneous; let it have $r$ orbits on
$2$-element subsets. Each is the edge set of one of the symmetrised orbital
graphs for $G$; each of these graphs is vertex-primitive and edge-transitive.
First we show:
\begin{itemize}
\item each symmetrised orbital graph has valency at least $k$;
\item there are at most two such graphs (that is, $r\le2$).
\end{itemize}
For the first point, suppose that there is a graph whose valency $v-1$ is
smaller than $k$, so that $v\le k$.
Then some $k$-set contains a vertex and all its neighbours in this graph,
and hence every $(k+1)$-set does so. The number of $(k+1)$-sets is
$n\choose k+1$, whereas the number of ways of choosing the closed
neighbourhood of a vertex in the graph, and then adjoining $k+1-v$ more
points to make a $(k+1)$-set is $n{n-v\choose k+1-v}$. Since the second
method overcounts, we have ${n\choose k+1}\le n{n-v\choose k+1-v}$. A
short calculation yields $n\le k+2$, a contradiction.
For the second, suppose that $r\ge3$, and let $\Gamma_1$, $\Gamma_2$,
$\Gamma_3$ be three of the orbital graphs. By the first point, we can find
a $(k+1)$-set consisting of a vertex $x$ and $k$ of its neighbours in the
graph $\Gamma_1$. This set must contain a $k$-set consisting of a point
$y$ and $k-1$ of its neighbours in $\Gamma_2$, and a $k$-set consisting of
a point $z$ and $k-1$ of its neighbours in $\Gamma_3$. Now it is clear that
$x,y,z$ are distinct; but then the pair $\{y,z\}$ must be an edge in both
$\Gamma_2$ and $\Gamma_3$, a contradiction.
Now we conclude the proof. Suppose that $r=2$ and let $\Gamma$ be one of the
two complementary orbital graphs. Suppose first that the valency of $\Gamma$
is at least $k+1$. Then we can choose a $(k+2)$-set consisting of a vertex
$x$ and a set $X$ of $k+1$ of its neighbours. Now one point of $X$, say $y$,
must be joined to at least $k-1$ further points of $X$, say those in a
subset $Y$. Now the induced subgraph on the $k+1$ points $\{x,y\}\cup Y$
has minimum valency at least $2$, and so cannot contain a vertex which is
nonadjacent to all but one point of this set, a contradiction.
So $\Gamma$ has valency $k$, as does its complement, and $n=2k+1$.
There are $n\choose k+1$ choices of a $k+1$-set; each contains a vertex
joined to all or all but one of the remaining vertices. But each vertex
lies in just one $k+1$-set in which it is joined to all other vertices,
and to $k^2$ in which it is joined to all but one (choose one neighbour to
omit, and one non-neighbour to include). So ${n\choose k}\le nk^2$.
This inequality fails for $k\ge5$; if $k=4$ then $n=9$, which has already
been disposed of by computation.
\end{proof}
\subsection{$2$-transitivity}
\begin{prop}
$G$ is $2$-transitive.
\end{prop}
\begin{proof}
According to Kantor's classification, a 2-homogenous, but non 2-transitive group $G$ is contained in a
one-dimensional affine group, and has order at most $q(q-1)/2\cdot\log_pq$,
where $n=q$ is a power of $p$ and is congruent to $3$ (mod~$4$); so $p$ is
congruent to $3$ (mod~$4$) and $\log_pq$ is odd.
If $k=3$, then we have $q(q-1)/2\cdot\log_pq\ge q(q-1)(q-2)/18$, so
$q-2\le 9\log_pq$. It is easy to check that this inequality is satisfied only
for $q=7,11,27$. The first two cases are covered by computation.
Suppose that $q=27$, so that $|G|\le27\cdot13\cdot3$. If $k=4$, then our
inequality $|G|\ge{27\choose 4}/4$ is violated. As remarked earlier, this
settles larger values of $k$ also.
\end{proof}
\subsection{Completion of the proof}
We have a list of $2$-transitive groups. It is now a case of going through
the list.
The condition of $(k,k+1)$-homogeneity is closed upwards; so we can usually
assume that the groups we are considering are maximal subgroups of the
symmetric or alternating group. The only exception is when we are testing
the $(k,k+1)$-homogeneity of a group which has a $k$-homogeneous overgroup.
Since $k\ge4$ and we may assume that $n\ge20$, the only cases which need to
be considered are $\mbox{\rm PSL}(2,23)\le \mbox{\rm M}_{24}$ (with $n=24$, $k=4$ or~$5$) and
$\mbox{\rm PGL}(2,32)\le\mbox{\rm P}\Gamma {\rm L}(2,32)$ (with $n=33$, $k=4$).
Computation shows that neither group is $(k,k+1)$-homogeneous.
According to Burnside, the $2$-transitive groups are of two
types: \emph{affine groups}, whose minimal normal sugroup is elementary
abelian; and \emph{almost simple groups}, whose minimal normal subgroup is
simple. For the affine groups, the maximal groups are $\mbox{\rm AGL}(d,p)$ for
$p$ prime.
\paragraph{Case $G=\mbox{\rm AGL}(d,p)$, with $p$ prime}
\subparagraph{Subcase $d=1$} We have $p(p-1)\ge{p\choose 4}/4$, so $p\le 11$;
these cases are excluded by computation.
So we may assume that $d\ge2$.
Below, $\mathbf{0}$ and $\mathbf{1}$ denote the all-zero and all-one vectors.
\subparagraph{Subcase $k\le d$} There is an affine independent $(k+1)$-set.
But since $k\ge4$, there is a $k$-set containing three or four affine
dependent points ($(0,\mathbf{0})$, $(1,\mathbf{0})$ and $(2,\mathbf{0})$ if
$p>2$, and $(0,0,\mathbf{0})$, $(1,0,\mathbf{0})$, $(0,1,\mathbf{0})$ and
$(1,1,\mathbf{0})$ if $p=2$) which cannot be contained in such a $(k+1)$-set.
\subparagraph{Subcase $d+1\le k\le p^{d-1}$, excluding $k=d+1$, $p=2$, $d$ odd}
If $p$ is odd, or if $p=2$ and $d$ is odd,
there exist $d+2$ points such that every hyperplane omits at least two,
namely $\mathbf{0}$, $\mathbf{1}$, and the points with a single coordinate $1$
and all others zero. (This construction needs to be modified if $p$ is odd
and divides $d-1$: then replace the all-$1$ vector by $(2,\mathbf{1})$.)
A $(k+1)$-set containing it can contain no $k$ points contained in a
hyperplane, a contradiction.
If $p=2$ and $d$ is odd, then $d\ge5$, so we can add one more point and
find a set of size $d+3$ with the claimed property: any non-zero vector with
an even number of $1$s will do.
\subparagraph{Subcase $k=d+1$, $d$ odd, $p=2$} Since $d+2\le2^{d-2}$, we
can take a $(k+1)$-set contained in a $(d-2)$-flat and an affine independent
$k$-set.
\subparagraph{Subcase $p^{d-1}-1\le k\le p^d-d(p-1)-2$} There is a set of
$1+d(p-1)$ points meeting every hyperplane, namely those with at most one
non-zero coordinate. Its complement contains a $(k+1)$-set omitting a point
of every hyperplane. But there is a $k$-set containing a hyperplane.
\subparagraph{Subcase $k\ge p^d-d(p-1)-1$} Since $2k+1\le p^d$, we have
$p^d\le2d(p-1)+1$, which is satisfied only for $p=3$, $d=2$, giving the
known examples.
\medskip
For groups with simple socle, there are more cases.
\paragraph{Case: $G=\mbox{\rm P}\Gamma {\rm L}(2,q)$, $q=p^e$}
\subparagraph{Subcase $k\ge5$}
We have $|G|=(q+1)q(q-1)e$. If $k\ge5$, then $|G|\ge{q+1\choose 5}/5$, so
$(q-2)(q-3)\le600e$, so $q\le27$ or $q=32$, handled by computation.
For $k\ge6$ the inequality gives $q\le17$, which is covered by computation.
\subparagraph{Subcase $k=4$}
The orbits of $\mbox{\rm PGL}(2,q)$ on the $4$-tuples of distinct points are parametrised
by cross-ratio, of which there are $q-2$ distinct values. A typical $4$-set
has six distinct cross-ratios; depending on the congruence of $q$, there may
be a set with only two cross-ratios, and one with only three. So the group
$\mbox{\rm PGL}(2,q)$ has at least $(q+5)/6$ orbits on $4$-sets. Adding field
automorphisms at worst divides the number of orbits by $e$. So
$q+5\le24e$. If $e=1$ then $q\le19$, covered by our computation. For $e>1$,
the remaining values to be checked are $q=25$, $27$, $32$, $64$ and $128$;
again computation shows there are no examples.
\paragraph{Case: $G$ is a unitary, Suzuki or Ree group}
These groups are smaller than $\mbox{\rm PSL}(2,q)$ of the same degree; all are ruled
out by the order test except for $\mbox{\rm P}\Gamma {\rm U}(3,q)$ with
$q=3,4$. Now $\mbox{\rm P}\Gamma {\rm U}(3,3)$ (with degree $28$) is a
subgroup of $\mbox{\rm Sp}(6,2)$, considered below. $\mbox{\rm P}\Gamma {\rm U}(3,4)$
is handled by computation.
\paragraph{Case $G=\mbox{\rm P}\Gamma {\rm L}(d,q)$, $d\ge3$}
Here $n=(q^d-1)/(q-1)$. We follow similar arguments to the affine case.
\subparagraph{Subcase: $k\le d$} There exist $d+1$ points, no three
collinear. On the other hand, there is a $k$-set containing three
collinear points.
\subparagraph{Subcase: $d<k<(q^{d-1})/(q-1)$} There is a $k$-set containing a basis
for the vector space, and a $(k+1)$-set contained in a hyperplane.
\subparagraph{Subcase: $(q^{d-1}-1)/(q-1)\le k<n-(q+1)$} Since a line contains $q+1$
points and meets every hyperplane, there is a $(k+1)$-set containing no
hyperplane; but there is a $k$-set which contains a hyperplane.
\subparagraph{Subcase: $k\ge n-(q+1)$} In this case, $k>n/2$, contrary to assumption.
\paragraph{Case: $G=\mbox{\rm Sp}(2d,2)$, with $n=2^{2d-1}\pm2^{d-1}$}
We start with a brief description of these groups. Let $V$ be a vector space
of dimension $2d$ over the field of two elements, and $B$ a fixed nondegenerate
alternating bilinear form on $V$. Let $\mathcal{Q}$ be the set of all quadratic
forms on $V$ which polarize to $B$. These fall into two orbits under the
action of the symplectic group $\mbox{\rm Sp}(d,2)$, of sizes
$2^{2d-1}\pm2^{d-1}$,
corresponding to the two types of quadratic form, distinguished by the
dimension of their maximal totally singular subspaces. The two types are
designated $+$ and $-$, and the corresponding dimensions are $d$ and $d-1$
respectively. Let $\mathcal{Q}_\epsilon$ be the set of forms of type $\epsilon$.
The symplectic group is $2$-transitive on each orbit. We may assume that
$d\ge3$, since otherwise the degrees are smaller than $20$.
It is readily checked from the order bound that the values of $k$ which need
to be considered satisfy $k+1\le|W|$, where $W$ is a maximal totally singular
subspace of the relevant quadratic form, except for $d=3$ and type $-$ (acting
on $28$ points). This exceptional case can be handled by computation.
There is a ternary relation on $\mathcal{Q}_\epsilon$ preserved by the group.
If $Q_1,Q_2,Q_3$ are three quadratic forms of the same type, then
$Q_1+Q_2+Q_3$ is a quadratic form, which may be of the same or opposite type.
Let $\mathcal{T}$ be the set of all triples for which the sum is of the same
type. It is easily checked that, for a fixed form $Q$, and a maximal totally
singular subspace $W$ for $Q$, the set $\{Q_w:w\in W\}$, where
$Q_w(x)=Q(x)+B(x,w)$, is a set of $|W|$ forms, all triples of which belong
to $\mathcal{T}$. Since $k+1\le|W|$, and there exists a triple not belonging
to $\mathcal{T}$, we see that $G$ cannot be $(k,k+1)$-homogeneous.
\paragraph{Case: $G$ is sporadic}
The sporadic $2$-transitive groups of degree greater than $20$ are $\mbox{\rm M}_{22}$
and its automorphism group ($n=22$), $\mbox{\rm M}_{23}$ ($n=23$), $\mbox{\rm M}_{24}$ ($n=24$),
the Higman--Sims group ($n=176$) and the Conway group ${\mbox{\rm Co}}_3$ ($n=276$).
Computation handles all of these except the Conway group, which is a bit
on the large side. However, the order test shows that we only need consider
$k\le6$; the case $k=4$ yields to computation, and the other cases cannot arise
since inspection of the combinatorial object preserved by the group (a
so-called ``regular two-graph'' see \cite{taylor}) shows that there are seven substructures on
five points, and so at least seven orbits on $5$-sets and on $6$-sets.
\section{The analogue of the Livingstone--Wagner result}
Livingstone and Wagner \cite{lw} proved that if a group $G\leq S_{n}$ is $k$-homogeneous (for $2\leq k\leq \lfloor \frac{n}{2}\rfloor$), it is also $(k-1)$-homogeneous.
\begin{cor}\label{cor2b}
Let $n\geq 1$, let $3\leq k\leq \lfloor \frac{n+1}{2}\rfloor$, and let $G\leq \mathcal{S}_{n}$ be a $(k-1,k)$-homogeneous group. Then $G$ is a $(k-2,k-1)$-homogeneous group, except when $n=9$ and $G\cong\mbox{\rm ASL}(2,3)$ or $\mbox{\rm AGL}(2,3)$, with $k=5$.
\end{cor}
\begin{proof}
We know that $G$ either is $(k-1)$-homogeneous or is one of the five exceptions listed in Theorem \ref{2.1}. If the group is $(k-1)$-homogenous, then it certainly is $(k-2,k-1)$-homogeneous.
Regarding the three exceptions of degree 5 and 7, GAP shows they satisfy the corollary. Regarding the two groups of degree 9, GAP shows that they are not $(3,4)$-homogeneous as for both groups the orbit of $\{1,2,3\}$ does not contain a subset of $\{1,2,4,5\}$.
\end{proof}
Recall that if $G$ is a permutation group of degree~$n$, for $k\leq n$, we say that
$G$ possesses the $k$-universal transversal property if the orbit of any $k$-set contains a section for every $k$ partition of $[n]$. It is clear that the class of groups the $k$-universal transversal property (for some $2\leq k\leq \lfloor n/2\rfloor$) is contained in the class of $(k-1,k)$-homogeneous groups (consider a $k$-partition with $k-1$ singleton blocks); therefore the groups possessing the $k$-universal transversal property are $(k-1)$-homogenous, with the exceptions listed in Theorem \ref{2.1}, and hence they have the $(k-1)$-universal transversal property, with the possible exception of the five exceptional groups listed in Theorem \ref{2.1}. Inspection of these groups leads to the following result.
\begin{cor}\label{cor1b}
Let $n\geq 5$, let $2\leq k\leq \lfloor \frac{n+1}{2}\rfloor$, and let $G\leq \mathcal{S}_{n}$ be a group having the $k$-universal transversal property. Then $G$ has the $(k-1)$-universal transversal property.
\end{cor}
If $k>\lfloor\frac{n+1}{2}\rfloor$, then no analogue of the Livingstone--Wagner Theorem can hold. But the situation is actually very simple.
\begin{thm}\label{bigk}
Let $G$ be a subgroup of $\mathcal{S}_{n}$, and let $k$ be an integer satisfying
$\lfloor\frac{n+1}{2}\rfloor < k < n$. Then the following are equivalent:
\begin{enumerate}
\item $G$ has the $k$-universal transversal property;
\item $G$ is $(k-1,k)$-homogeneous;
\item $G$ is $k$-homogeneous.
\end{enumerate}
In particular, if these condiditions hold with $k<n-5$, then $G$ is $\mathcal{S}_{n}$
or $\mathcal{A}_{n}$.
\end{thm}
\begin{proof}
(1) implies (2): The proof of this given earlier does not depend on the
value of $k$.
(2) implies (3): Let $G$ be $(k-1,k)$-homogeneous. Then $G$ is
$(n-k,n-k+1)$-homogeneous. Because of the inequality on $k$, the exceptional
groups in Theorem~\ref{th6} do not occur; so $G$ is $(n-k)$-homogeneous, and
hence $k$-homogeneous.
(3) implies (1): Clear.
\end{proof}
\section{The classification of the groups possessing the $k$-universal transversal property}\label{kutp}
Let $G$ be a permutation group of degree~$n$. For $k\leq n$, we say that
$G$ possesses the $k$-universal transversal property if the orbit of any $k$-set contains a section for every $k$ partition of $[n]$. A group is said to have the universal transversal property if it has the $k$-universal transversal property for all $k\leq n$. In \cite{ArMiSc} the following theorem is proved.
\begin{thm} \label{superlemma}
A subgroup $G$ of $\mathcal{S}_{n}$ has the universal
transversal property if and only if one of the following is valid:
\begin{enumerate}
\item[\rm (i)] $n=5$ and $G\cong C_5,\ \dihed{5},$ or $\mbox{\rm AGL}(1,5)$;
\item[\rm (ii)] $n=6$ and $G\cong \mbox{\rm PSL}(2,5)$ or $\mbox{\rm PGL}(2,5)$;
\item[\rm (iii)] $n=7$ and $G\cong\mbox{\rm AGL}(1,7)$;
\item[\rm (iv)] $n=8$ and $G\cong\mbox{\rm PGL}(2,7)$;
\item[\rm (v)] $n=9$ and $G\cong\mbox{\rm PSL}(2,8)$ or $\mbox{\rm P}\Gamma {\rm L}(2,8)$;
\item[\rm (vi)] $G=\mathcal{A}_{n}$ or $\mathcal{S}_{n}$.
\end{enumerate}
\end{thm}
The goal of the following two sections is to prove that with some exceptions, the groups possessing the $k$-universal transversal property are $k$-homogeneous (for $2\leq k\leq \lfloor \frac{n}{2}\rfloor$).
We abbreviate ``$k$-universal transversal property'' to
\textit{$k$-ut property}.
Our main results are Theorems \ref{th4a}--\ref{th4f} stated in the introduction, and that we now state in a single theorem.
\begin{thm}\label{RegularAndKut}
Let $n\geq 11$, $G\leq \mathcal{S}_{n}$ and let $2\leq k\leq \lfloor \frac{n}{2}\rfloor$.
\begin{enumerate}
\item If $6\leq k\leq \lfloor \frac{n}{2}\rfloor$, then the following are equivalent:
\begin{enumerate}
\item $G$ has the $k$-ut property;
\item $\mathcal{A}_{n}\leq G$.
\end{enumerate}
\vspace{0.3cm}
\item The following are equivalent:
\begin{enumerate}
\item $G$ has the $5$-ut property;
\item $G$ is 5-homogeneous, or $n=33$ and $G=\mbox{\rm P}\Gamma {\rm L}(2,32)$.
\end{enumerate}
\vspace{0.3cm}
\item
\begin{enumerate}
\item if $G$ is $4$-homogenous, or $n=12$ and $G=\mbox{\rm M}_{11}$, then $G$ has the $4$-ut property.
\item Apart from the possible exception of some $G$ with $\mbox{\rm PSL}(2,q)\leq G\leq \mbox{\rm P}\Gamma {\rm L}(2,q)$, the groups listed above are the only ones having the $4$-ut property.
\end{enumerate}
\vspace{0.3cm}
\item
\begin{enumerate}
\item if $G$ is $3$-homogenous, or one of the following groups
\begin{enumerate}
\item $\mbox{\rm PSL}(2,q)\leq G\leq \mbox{\rm P}\Sigma {\rm L}(2,q)$, where $q\equiv 1 (\mbox{mod }4)$;
\item $\mbox{\rm Sp}(2d,2)$ with $d\geq 3$, in either of its $2$-transitive representations;
\item $2^{{2d}}:\mbox{\rm Sp}(2d,2)$;
\item Higman--Sims;
\item ${\mbox{\rm Co}}_{3}$;
\item $2^{{6}}:{\mbox{\rm G}}_{2}(2)$ and its subgroup of index $2$;
\item $\mbox{\rm AGL}(1,p)$ where, for all $c\in \mbox{\rm GF}(p)^{{*}}$, $|\langle -1,c,c-1\rangle|=p-1$;
\end{enumerate}
then $G$ has the $3$-ut property.
\item Apart from the possible exception of the Suzuki groups $\mbox{\rm Sz}(q)$, the groups listed above are the only ones having the 3-ut property.
\end{enumerate}
\vspace{0.3cm}
\item The following are equivalent.
\begin{enumerate}
\item $G$ has the $2$-ut property;
\item $G$ is primitive.
\end{enumerate}
\end{enumerate}
For $n<11$, a group $G\leq \mathcal{S}_{n}$ with the $k$-universal transversal property is
$k$-homogeneous, with the following exceptions:
\begin{enumerate}
\item[(i)] $n=5$, $C_{5}$ or $D(2*5)$ and $k=2$;
\item[(ii)] $n=6$, $\mbox{\rm PSL}(2,5)$ and $k=3$;
\item[(iii)] $n=7$, $C_{7}$ or $D(2*7)$, and $k=2$; or
$\mbox{\rm AGL}(1,7)$ and $k=3$;
\item[(iv)] $n=8$, $\mbox{\rm PGL}(2,7)$ and $k=4$;
\item[(v)] $n=9$, $3^{{2}}:4$ or $3^{{2}}:D(2*4)$ and $k=2$;
\item[(vi)] $n=10$, $\mathcal{A}_{5} $ or $\mathcal{S}_{5}$ and $k=2$; or
$\mbox{\rm PSL}(2,9)$ or $\mathcal{S}_{6}$ and $k=3$.
\end{enumerate}
\end{thm}
\begin{proof}
In Sections \ref{kutp} and \ref{exceptional} all the claims for $n\geq 11$ are proved. Regarding $n<11$, all the claims can be easily checked with GAP.
For $n=5,6$ all the possible groups appear in the statement of the theorem.
For $n=7$ the group $7:3$ is $2$-homogenous, but does not have the $3$-ut property as the orbit of $\{1,2,7\}$ has no transversal for $\{ \{ 1 \}, \{2, 7 \}, \{ 3, 6, 4, 5 \} \}$. The $2$-homogeneous group $L(3,2)$ also does not have the $3$-ut property as the orbit of $\{1,2,4\}$ does not contain a transversal for $\{ \{ 1 \}, \{ 2, 4 \}, \{ 3, 7, 6, 5 \} \}$.
For $n=8$, the $3$-homogenous groups $\mbox{\rm AGL}(1,8)$, $\mbox{\rm A}\Gamma {\rm L}(1,8)$, $\mbox{\rm ASL}(3,2)$ do not have the $4$-ut property. There is no section for the partition $\{ \{ 1 \}, \{ 2 \}, \{ 3, 4 \}, \{ 5, 6, 7, 8 \} \}$ in the orbits of the set $\{1,2,3,4\}$.
The $3$-homogenous group $\mbox{\rm PSL}(2,7)$ does not have the $4$-ut property as the orbit of $\{1,2,3,5\}$ has no transversal for the partition $\{\{ 1, 5, 7, 8 \}, \{ 2, 6 \}, \{ 3 \}, \{ 4 \} \}$.
For $n=9$, the $2$-homogenous groups $\mbox{\rm M}_{9}$, $\mbox{\rm AGL}(1,9)$, $\mbox{\rm A}\Gamma {\rm L}(1,9)$, $\mbox{\rm ASL}(2,3)$, $\mbox{\rm AGL}(2,3)$ do not have the $3$-ut property. Their orbits on the set $\{1,2,3\}$ contain no section for the partition $\{ \{ 1 \}, \{ 2,3 \}, \{ 4 , 5, 6, 7, 8,9 \} \}$.
For $n=10$ the $3$-homogeneous groups $\mbox{\rm PGL}(2,9)$, $\mbox{\rm M}_{10}$, $\mbox{\rm P}\Gamma {\rm L}(2,9)$ do not have the $4$-ut property as the partition $\{ \{ 1 \}, \{ 2 \}, \{ 3, 10 \}, \{ 4, 9, 8, 6, 7, 5 \} \}$ has no transversal in the orbit of $\{1,2,3,10\}$.
\end{proof}
\begin{prop}\label{critical}
\begin{enumerate}
\item A $k$-homogeneous group has the $k$-ut property.
\item A group with the $k$-ut property is $(k-1,k)$-homogeneous, and hence
is $(k-1)$-homogeneous or one of the five exceptions to Theorem~\ref{th6}.
\item If $k>n/2$, then a group has the $k$-ut property if and only if it is $k$-homogeneous
or one of the exceptions to Theorem~\ref{th6}.
\end{enumerate}
\end{prop}
The first part is trivial; the second is contained in the preamble to
Theorem~\ref{th5} and the result of Theorem~\ref{th6}; the third is contained
in the preamble to Theorem~\ref{kkp1}.
Our aim is to determine, as completely as possible, the groups with the
$k$-ut property which are not $k$-homogeneous. For $k=2$, there are many
such groups and no hope of a determination:
\begin{prop}
A subgroup of $\mathcal{S}_{n}$ has the $2$-ut property if and only if it is primitive.
\label{primlem}
\end{prop}
\begin{proof}
By Higman's Theorem~\cite{higman}, $G$ is primitive if and only if all the
non-diagonal orbital graphs are connected. (These are just the graphs whose
edge sets are the orbits on $2$-sets.) But a graph is connected if and
only if, for every 2-partition of the vertices, there is an edge which is
a section for the partition.
\end{proof}
However, for $k>2$, we are in a stronger position, due to the following pair
of results, one negative, one positive:
\begin{prop}
If $G$ is a group of automorphisms of a Steiner system $S(k-1,l,n)$
with $k-1<l<n$, then $G$ does not have the $k$-ut property.
\end{prop}
(A Steiner system is a collection of blocks or subsets of size $l$ of the
$n$-set so that each $(k-1)$-set is contained in a unique block.)
\begin{proof}
Take the partition with $k-2$ singleton parts, one part of size
$l-k+2$ consisting of the remaining points in some block containing these
points, and one part consisting of everything else. A $k$-set which is
contained in a block cannot be a section for this partition.
\end{proof}
This shows that the following $(k-1)$-homogeneous groups do not have the $k$-ut property:
\begin{enumerate}
\item Subgroups of $\mbox{\rm AGL}(d,p)$, if $d>1$ and $p>2$, with $k=3$ (these
groups preserve the geometry of affine points and lines).
\item $3$-transitive subgroups of $\mbox{\rm AGL}(d,2)$ for $d>2$, with $k=4$ (these
preserve geometry of affine points and planes).
\item Subgroups of $\mbox{\rm P}\Gamma {\rm L}(d,q)$ with $d>2$, for $k=3$ (these preserve the
geometry of projective points and lines).
\item The unitary and Ree groups, with $k=3$ (these preserve unitals).
\item Subgroups of $\mbox{\rm P}\Gamma {\rm L}(2,q)$ containing $\mbox{\rm PSL}(2,q)$, where $q$ is a proper
power of an odd prime or $q=2^e$ where $e$ is not prime, with $k=4$ (these
preserve circle geometries). Note that we can exclude subgroups of
$\mbox{\rm P}\Sigma {\rm L}(2,q)$ containing $\mbox{\rm PSL}(2,q)$ for $q\equiv1$ (mod~$4$), since these
groups are not $3$-homogeneous.
\item The Mathieu groups $\mbox{\rm M}_{11}$, $\mbox{\rm M}_{12}$, $\mbox{\rm M}_{22}$ (and $\mbox{\rm Aut\,}(\mbox{\rm M}_{22})$),
$\mbox{\rm M}_{23}$ and $\mbox{\rm M}_{24}$ in their usual representations, with $k=5,6,4,5,6$,
respectively, as these preserve famous Steiner systems.
\end{enumerate}
A permutation group $G$ is \emph{$k$-primitive} if it is $(k-1)$-transitive
and the pointwise stabiliser of $k-1$ points acts primitively on the
remaining points. It is \emph{generously $k$-transitive} if the setwise
stabiliser of $k+1$ points induces the symmetric group on these points, and
is \emph{almost generously $k$-transitive} if the setwise stabiliser of
$k+1$ points induces the symmetric or alternating group on them
(Neumann~\cite{pmn:generous}).
\begin{lem}
If a permutation group $G$ is $k$-primitive and almost generously
$k$-transitive, then every orbit of $G$ on $(k+1)$-sets contains a section
for any $(k+1)$-partition of which $k-1$ of the parts are singletons.
\end{lem}
\begin{proof}
In an almost generously $k$-transitive group, each orbit on $(k+1)$-sets
corresponds in a natural way to an orbit of the $(k-1)$-point stabiliser on
pairs of points outside the given $k-1$ points. Now the proof concludes as
in Proposition~\ref{primlem}.
\end{proof}
\begin{prop}\label{two-graph-prop}
Each of the following $2$-transitive groups $G$ has the $3$-ut property:
\begin{enumerate}
\item $\mbox{\rm PSL}(2,q)\le G\le\mbox{\rm P}\Sigma {\rm L}(2,q)$, where $q\equiv1$ ($\mathrm{mod}\;4$);
\item $\mbox{\rm Sp}(2d,2)$ with $d\ge3$, in either of its $2$-transitive representations;
\item $2^{2d}\colon\mbox{\rm Sp}(2d,2)$;
\item ${\mbox{\rm Co}}_3$.
\end{enumerate}
\end{prop}
\begin{proof}
Each of these groups is $2$-primitive and generously $2$-transitive, so by
the previous lemma it is enough to consider partitions $\{X,Y,Z\}$ in which no part is
a singleton. We assume that $X$ is the smallest part.
Also, each of these groups has just two orbits on $3$-sets, and each orbit $T$
is a \emph{regular two-graph} (Taylor~\cite{taylor}), that is,
\begin{itemize}
\item any two points lie in exactly $\lambda$ members of $T$, for some
$\lambda$;
\item any four points contain an even number of members of $T$.
\end{itemize}
Now suppose that the orbit $T$ contains no section for the partition
$\{X,Y,Z\}$. For any $x_1,x_2\in X$,
$y\in Y$ and $z\in Z$, we have $x_1yz, x_2yz\notin T$, and so both or neither
of $x_1x_2y$ and $x_1x_2z$ belong to $T$. Suppose that $x_1x_2y\in T$, for
some $y\in Y$. Then $x_1x_2z\in T$ for all $z\in Z$, and $x_1x_2y\in T$ for all
$y\in Y$. Hence we have $\lambda \ge |Y|+|Z| \ge 2n/3$. In the contrary case,
neither of these triples belong to $T$, and $\lambda \le |X|-2 \le n/3 - 2$.
However, for these groups, it is easily checked that these inequalities fail
in all cases:
\begin{itemize}
\item for $\mbox{\rm PSL}(2,q)$, $n=q+1$, $\lambda=(q-1)/2$;
\item for $\mbox{\rm Sp}(2d,2)$, $n = 2^{2d-1} \pm 2^{d-1}$, $\lambda=2^{2d-2}$ or
$\lambda=2^{2d-2}\pm 2^{d-1}-2$;
\item for $2^{2d} \colon \mbox{\rm Sp}(2d,2)$, $n = 2^{2d}$, $\lambda = 2^{2d-1}$ or
$\lambda=2^{2d-1}-2$;
\item for ${\mbox{\rm Co}}_3$, $n=276$, $\lambda=112$ or $\lambda=162$.
\end{itemize}
\end{proof}
These results, together with computation for $n\le 11$, resolve
the question of the $k$-ut property for all groups $G$ which
are $(k-1)$-homogeneous but not $k$-homogeneous, but for the following exceptional cases:
\begin{enumerate}
\item $k=3$:
\begin{enumerate}
\item $\mbox{\rm AGL}(1,p)$, $p$ prime, or its subgroup of index $2$ if $p \equiv 3$
(mod~$4$);
\item $2^6 \colon {\mbox{\rm G}}_2(2)$ and its subgroup of index $2$;
\item $\mbox{\rm Sz}(q)$;
\item The Higman--Sims group.
\end{enumerate}
\item $k=4$:
\begin{enumerate}
\item $\mbox{\rm PSL}(2,q) \le G \le\mbox{\rm P}\Gamma {\rm L}(2,q)$, with either $q$ prime (except
$\mbox{\rm PSL}(2,q)$ for $q\equiv1$ (mod~$4$), which is not $3$-homogeneous), or $q=2^p$ for $p$ prime;
\item $\mbox{\rm M}_{11}$, degree $12$.
\end{enumerate}
\item $k=5$:
\begin{enumerate}
\item $\mbox{\rm P}\Gamma {\rm L}(2,32)$, degree $33$.
\end{enumerate}
\end{enumerate}
\section{The Exceptional Cases}\label{exceptional}
In this section we are going to look at the exceptional cases listed at the end of the previous section.
\subsection{The group $\mbox{\rm M}_{11}$ \ut{4}}
The group $\mbox{\rm M}_{{11}}$ of degree 12 can easily be handled with GAP. This group has two orbits on $4$-sets (in GAP are the orbits of $\{1,2,3,4\}$ and of $\{1,2,3,7\}$). GAP checks in less than one minute that the orbit of each one of these sets contains a section for all possible $4$-partitions of $\{1,\ldots,12\}$.
\subsection{Some general results}\label{general}
In this subsection we are going to prove a number of auxiliary results. We start by associating a graph to a $t$-homogeneous group $G\leq \mathcal{S}_{n}$ as follows. Let $B\subseteq [n]$ and $c\in [n]$ such that $|B|=t-1$. Then we define the following graph on the points in $[n]\setminus B$:
\[
G(B,c)=\{ \{x,y\} \mid \{x,y\}\cup B \in (\{1,\ldots ,t,c\})G \}.
\]
In the particular case of $B=\{b\}$, we will write $G(b,c)$ rather than $G(\{b\},c)$.
Observe that for every $g\in G$ we have $G(B,c)\cong G(Bg,c)$. Therefore, as $t$ is larger than the order of $B$, it follows that $G(B,c)$ is connected if and only if $G(B',c)$ is connected, for all $B'$ such that $|B'|=|B|=t-1$.
\begin{prop}\label{connected}
If a $t$-homogeneous group $G\leq \mathcal{S}_{n}$ \ut{(t+1)}, then $G(B,c)$ is connected, for all $B\subset [n]$ (with $|B|=t-1$) and all $c\in [n]$.
\end{prop}
\begin{proof}
If $G(B,c)$ is not connected, then there exists
a connected component $D$ contained in the graph. Now consider the partition $P=(\{b_{1}\},\dots,\{b_{t-1}\},D,R)$, where $R$ contains the remaining elements, that is, $R=[n]\setminus (B\cup D)$, and $B=\{b_{1},\ldots,b_{t-1}\}$.
Any section for the partition $P$ must contain $B$. Therefore any set containing $B$ and in the orbit of $\{1,\ldots ,t,c\}$ must be of the form $\{x,y\}\cup B$ and hence $x$ and $y$ are connected in $G(B,c)$. Thus, either $x\in D$ and hence $y\in D$ (because $D$ is a connected component of $G(B,c)$) so that $\{x,y\}\cup B$ is not a section for $P$; or $x\not\in D$ and hence $y\not \in D$ thus implying $x,y\in R$. Again $\{x,y\}\cup B$ is not a section for $P$. The result follows.
\end{proof}
This proposition immediately implies that, for example, the 2-homogeneous group $G=\mbox{\rm AGL}(1,17)$ \notut{3}.
In fact, according to GAP, the graph $G(\{17\},4)$ has the following two connected components
$\{ \{1, 2, 3, 4, 6, 9, 10, 15 \}, \{ 5, 7, 8, 11, 12, 13, 14, 16 \}\}$. And it can be checked, in fact, that the orbit of $\{1,2,4\}$ under $G$ has no section for the partition $$P=( \{1, 2, 3, 4, 6, 9, 10, 15 \}, \{ 5, 7, 8, 11, 12, 13, 14, 16 \},\{17\}).$$ (More on these groups below.)
For the particular case of the $3$-ut property, another important graph is the following: for a set $C\subseteq [n]$ and $c\in [n]$, we have
\[
\Gamma(C,c)=\bigcup_{{b\in C}}\left(G(b,c)\cap \left[([n]\setminus C)\times([n]\setminus C)\right] \right).
\]
Let $G$ be a group admitting a bad 3-partition, that is, $P=(A,C,A')$ such that no set in the orbit of $\{1,2,c\}$ is a section for $P$. This means that the distance from $A$ to $A'$ in the graph $\Gamma(C,c)$ must be infinite. In fact, if it is not infinite, it must be one as every vertex in this graph either is on $A$ or in $A'$. That means that there exist $a\in A$ and $a'\in A'$ that are connected in $\Gamma(C,c)$. But, by definition, $\Gamma$ is a union of subgraphs of $G(b,c)$ and hence it follows that for some $b\in C$ we have that $\{a,a'\}$ is an edge in $G(b,c)$. Thus $b\in C$, $a\in A$, $a'\in A'$ and hence $\{a,b,a'\}$ is a section for $P$ that belongs to the orbit of $\{1,2,c\}$, by the definition of $G(b,c)$. It is proved that bad partitions $P=(A,C,A')$ induce graphs $\Gamma(C,c)$ in which the distance from $A$ to $A'$ is infinite.
This observation leads to the following procedure that (if it ends) allows to check that a group $G$ \ut{3}. We already know that if $G(n,c)$ is disconnected, then the group \notut{3}. The question is whether there exists a group $G$ with connected graph $G(n,c)$, but that \notut{k}. Therefore we assume that $G(n,c)$ is connected and start with three sets $(A,C,A')$, all contained in $[n]$, such that (without loss of generality because we only consider 2-homogeneous transitive groups) $\{1\}= A$ and $\{n\}= C$, where $n$ is the degree of the group $G$. We are going to try to build a bad partition and hence include in $A\cup A'$ and $C$ all the elements that must necessarily be in each one of this sets provided that we want $\Gamma(C,c)$ to be disconnected on $A\cup A'$. So we proceed as follows (denote the distance from $a$ to $b$ in graph $Gr$ by $D_{{Gr}}(a,b)$): for a fixed $d\in [n]$ that will be the distance in $G(n,c)$ from $A$ to $A'$,
\begin{enumerate}
\item put in $A'$ an element $y\in [n]$ such that $D_{G(n,c)}(1,y)=d$; add to $A\cup A'$ the set $\{x\in [n] \mid \{1,y\}\in G(x,c)\}$. (Observe that these $x$ must be in $A\cup A'$ because if one of them is in $C$, then $\{1,y,x\}$ would be a section for $(A,C,A')$ and hence any oversets of $A,A',C$ yielding a partition of $[n]$ would have a section in the orbit of $\{1,2,c\}$.)
\item add to $C$ the set $\{x\in [n]\mid 1-\ldots-x-\ldots -y\}$, where $a-b$ means that $\{a,b\}$ is an edge in $G(n,c)$. (Observe that if one of these $x$ is not in $C$, then $D_{G(n,c)}(A,B)<d$, contrary to our assumption.)
\item check if $A\cup A'$ is connected under $\Gamma(C,c)$; if it is connected, then all the overpartions of $(A,C,A')$ are good; if it is not connected, then
\item add to $A\cup A'$ the set $\{x\in [n]\mid A\cup A' \mbox{ is connected in } \Gamma(C\cup \{x\},n)\}$ (as such $x$ cannot go to $C$).
\item $A\cup C\supseteq \{x\mid D_{G(n,c)}(A,x)<d\}$ and $A'\cup C\supseteq \{x\mid D_{G(n,c)}(A',x)<d\}$;
\item add to $A$ the set $(A\cup C)\cap (A\cup A')$.
\item add to $A'$ the set $(A'\cup C)\cap (A\cup A')$.
\item add to $C$ the set $(A\cup C)\cap (C\cup A')$.
\item go to (3).
\end{enumerate}
\subsection{The two exceptional groups of degree 64}\label{64}
The group $H=2^{{6}} :{\mbox{\rm G}}_{2}(2)$ and its subgroup $G$, of index 2, have $G(64,c)$ connected. Therefore the guess is that they have the $3$-ut property. To test that conjecture we are going to apply to $G$ the procedure outlined at the end of the previous subsection, as if $G$ \ut{3}, then the overgroup $H$ also has. On the set $[64]$ the group $G$ has three orbits on $3$-sets, namely (in GAP) $\{1,2,3\}$, $\{1,2,5\}$ and $\{1,2,29\}$.
The points $y$ such that $D_{G(64,3)}(1,y)=2$ are
\begingroup
\everymath{\scriptstyle}
\tiny
$$\{2, 3, 4, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 27, 28, 37, 38, 39, 40, 41, 42, 43, 44, 49, 50, 51, 52, 61, 62, 63\}.$$
\endgroup
To all of them the procedure ends yielding the result that at a certain point $A$ and $A'$ are connected under $\Gamma(C,3)$. And there are no $y$ such that $D_{G(64,3)}(1,y)>2$.
The points $y$ such that $D_{G(64,5)}(1,y)=2$ are
\begingroup
\everymath{\scriptstyle}
\tiny
$$\begin{array}{c}\{5, 6, 7, 8, 9, 10, 11, 12, 15, 17, 18, 19, 20, 23, 25, 29, 30, 31, 32, 33, 34, \\ 35, 36, 40, 42, 45, 46, 47, 48, 50, 53, 54, 55, 56, 57, 58, 59, 60 \}.\end{array}$$
\endgroup
To all of them the procedure ends yielding the result that at a certain point $A$ and $A'$ are connected under $\Gamma(C,5)$. And there are no $y$ such that $D_{G(64,5)}(1,y)>2$.
Regarding $\{1,2,29\}$ (the $3$-set with the smallest orbit), the points $y$ such that $D_{G(64,29)}(1,y)=2$ are
\begingroup
\everymath{\scriptstyle}
\tiny
$$\{2, 3, 4, 13, 14, 16, 21, 22, 24, 26, 27, 28, 37, 38, 39, 41, 43, 44, 49, 51, 52, 61, 62, 63 \}.$$
\endgroup
To all of them the procedure ends yielding the result that at a certain point $A$ and $A'$ are connected under $\Gamma(C,29)$. In this case there are also some $y$ such that $D_{G(64,29)}(1,y)=3$:
\begingroup
\everymath{\scriptstyle}
\tiny
$$\{ 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 19, 20, 29, 30, 31, 32, 33, 34, 35, 36, 45, 46, 47, 48, 53, 54, 55, 56, 57, 58, 59, 60 \}.$$
\endgroup
To all of them the procedure ends yielding the result that at a certain point $A$ and $A'$ are connected under $\Gamma(C,29)$. And there are no $y$ such that $D_{G(64,29)}(1,y)>3$.
It is checked that $G$ \ut{3} and hence the same holds for $H$.
\subsection{The group $\mbox{P}\Gamma\mbox{L}(2,32)$}\label{pgaml}
The graph $G(\{1,2,3\},c)$ is connected and hence the guess is that this group \ut{5}.
It is too big to be tested directly and the algorithm used in subsection \ref{64} does not work here. Therefore we used the following algorithm (in GAP) to prove that indeed this group \ut{5}.
$G$ has three orbits on 5-sets: $\{1,\ldots,5\}$, $\{1,\ldots,4,6\}$ and $\{1,\ldots,4,10\}$ are representatives. Suppose we want to prove that $\{1,\ldots,4,10\}G$ contains a section for all partitions. To do that we are going to try to build a bad partition (one that has no section in $\{1,\ldots,4,10\}G$).
\begin{enumerate}
\item We start with the subpartition $p:=\{\{1\},\{2\},\{3\},\{4\},\{6\}\}$ and add the number $5$ (the smallest not in $\{1,2,3,4,6\}$) to the blocks of $p$ in the five possible ways. We get $5$ subpartions and remove all the partitions that have a section in $\{1,\ldots,4,10\}G$. According to GAP all the $5$ are left.
\item Repeat the previous step with $7$ being included in each one of the previous five subpartitions, and then removing the ones that have a section in $\{1,\ldots,4,10\}G$. According to GAP all the $25$ are left.
\item Repeating with $8$, we end up with $115$ subpartitions.
\item With $9$, we get $337$ subpartitions.
\item Repeating with $10,\ldots,19$, we get, respectively, the following number of subpartitions $$719,1052,1065,1193,912,229,211,137,50,2.$$
And it does not matter where we put $20$, we end always with a partition admitting a section in $\{1,\ldots,4,10\}G$. This proves that $\{1,\ldots,4,10\}G$ contains a section for all the partitions in which $1,2,3,4,6$ are all in different blocks.
\item Then we start with a subpartition $\{\{1\},\{2\},\{3\},\{4\},\{5\}\}$ and the orbit $\{1,\ldots,4,10\}G$, and follow the previous algorithm. Again we get that $\{1,\ldots,4,10\}G$ contains a section for all 5-partitions (in which $1,2,3,4,5$ are in different blocks). This proves that $\{1,\ldots,4,10\}G$ contains a section for all the 5-partitions since there is only another orbit of partitions: those in which $1,2,3,4,10$ are in different blocks and for those $\{1,\ldots,4,10\}G$ trivially contains a section.
\item Finally we repeat the same algorithm with $\{1,\ldots,5\}G$ and $\{1,\ldots,4,6\}G$. The worse case is when $P:=\{\{1\},\{2\},\{3\},\{4\},\{6\}\}$ and we have the orbit of $\{1,\ldots,4,10\}$.
\end{enumerate}
\subsection{The Higman--Sims group }
Let $G$ be the Higman--Sims group, a group of order $2^{{9}}\cdot 3^{{2}}\cdot 5^{{3}}\cdot 7\cdot 11$ and degree $176$.
Then $G$ has three orbits $O_1$, $O_2$, $O_3$ on ordered triples, with cardinalities $176\cdot175\cdot t$, for $t=12,72,90$; representatives of the orbits are $(1,2,16)$, $(1,2,3)$ and $(1,2,6)$ respectively.
According to Taylor~\cite{taylor}, $O_2$ is a regular two-graph with $\lambda=72$, and exactly the same argument as in Proposition~\ref{two-graph-prop} shows that every $3$-partition has a section belonging to the orbit $O_2$. So we only have to deal with the orbits $O_1$ and $O_3$.
Moreover, every $3$-partition is equivalent under $G$ to one with $1,2,3$ in different parts; so we started with the subpartition $\{\{1\},\{2\},\{3\}\}$ and applied the same algorithm used in the preceding subsection, concluding that this group has the $3$-ut property.
\subsection{The groups $\mbox{\rm AGL}(1,p)$ for $p$ prime}\label{numbertheory}
We proved above that if the graph $G(B,c)$ is not connected for some $B$ and $c$, then $G$ \notut{k}, for $k=|B|+2$. Unfortunately, the groups $\mbox{\rm AGL}(1,p)$ have disconnected graphs for some $p$, and connected graphs for
other $p$. Therefore we need sharper results. The aim of this section is to prove them.
We start by providing a characterization of connectedness in this setting.
We denote by $\mbox{\rm GF}(p)$ the field with $p$ elements and by $\mbox{\rm GF}(p)^{{*}}$ its non-zero elements.
\begin{prop}
Let $G=\mbox{\rm AGL}(1,p)$, with $p$ prime. If $G(0,c)$ is not connected then $H:=\langle -1,c,c-1\rangle\subset \mbox{\rm GF}(p)^{{*}}$. In such a case, the orbit of $\{0,1,c\}$ under $\mbox{\rm AGL}(1,p)$ has no section for the partition $\{\{0\},H,\mbox{rest}\}$.
\end{prop}
\begin{proof}
Observe that $\{b,x,y\}$ is in the orbit of $\{0,1,c\}$ if and only if $$\{x,y\}\in \{\{\alpha+b,c\alpha+b\},\{b-\alpha, (c-1)\alpha +b\},\{(1-c)\alpha +b, b-c\alpha\}\mid \alpha \in \mbox{\rm GF}(p)^{{*}}\}.$$
In particular, for $b=0$, we have that
$$\{x,y\}\in \{\{\alpha,c\alpha\},\{-\alpha, (c-1)\alpha\},\{(1-c)\alpha , -c\alpha\}\mid \alpha \in \mbox{\rm GF}(p)^{{*}}\}.$$
Now if $G(0,c)$ is not connected, then there exists
a set $A\subset \mbox{\rm GF}(p)^{{*}}$ such that for every $\{x,y\}\in G(0,c)$ we have
\[
x\in A \Leftrightarrow y \in A.
\]
In particular, for $\{x,y\}= \{\alpha,c\alpha\}$ we have $\alpha \in A \Leftrightarrow c\alpha \in A$, that is, $A=Ac$. In the same way we get the conditions $A=-A(c-1)$ and $Ac=A(c-1)$.
Collecting these three conditions we get that:
\[
A=Ac=A(c-1)=-A.
\]
Now, for $c_{i}\in \{-1,c,c-1\}$, we have $A=Ac_{1}=(Ac_{2})c_{1}=\ldots =Ac_{n}\cdots c_{1}$ and hence $A=A\langle -1,c,c-1\rangle$.
Clearly, saying that {\em there exists a proper subset $A\subset \mbox{\rm GF}(p)^{{*}}$ such that $A=-A=Ac=A(c-1)$} is equivalent to saying that {\em the group $\langle -1,c,c-1\rangle$ is strictly contained in $\mbox{\rm GF}(p)^{{*}}$}. The first claim follows.
Regarding the second claim, any section for the partition must have the form $\{0,c_{1},r\}$, with $c_{1}\in H$, and hence we must have $\{c_{1},r\}\in G(0,c)$. As we saw above, this means that
$$\{c_{1},r\}\in \{\{\alpha,c\alpha\},\{-\alpha, (c-1)\alpha\},\{(1-c)\alpha , -c\alpha\}\mid \alpha \in \mbox{\rm GF}(p)^{{*}}\}.$$
Checking all the possibilities always leads to the conclusion that $r\in H$.
\end{proof}
The next result is our main result regarding the groups $\mbox{\rm AGL}(1,p)$.
\begin{thm}
Let $p$ be a prime and let $c\in \mbox{\rm GF}(p)\setminus \{0,1\}$. Then the following are equivalent:
\begin{enumerate}
\item the orbit of $\{0,1,c\}$ under $\mbox{\rm AGL}(1,p)$ contains a section for all the $3$-partitions of $\{0,\ldots ,p\}$;
\item $|\langle -1,c,c-1\rangle|=p-1$.
\end{enumerate}
\end{thm}
\begin{proof}
We already proved that if $|\langle -1,c,c-1\rangle|<p-1$, then there exists
a 3-partition $P$ such that no set in the orbit of $\{0,1,c\}$ (under $\mbox{\rm AGL}(1,p)$) is a section for $P$.
Conversely, suppose that $|\langle -1,c,c-1\rangle|=p-1$ and suppose that there exists
a bad partition $P=(A,C,A')$. This implies that $\Gamma(C,c)$ is not connected, that is, for all $\{x,y\}\in \Gamma(C,c)$, either $x,y\in A$ or $x,y\in A'$. As $\Gamma(C,c)$ is a union of graphs, this means that for all $b\in C$, if $\{x,y\}\in G(b,c)$, then $$x\in A \Leftrightarrow y \in A.$$
Now, repeating the arguments in the previous result we observe that this last condition is equivalent to saying that
\[
\begin{array}{rccl}
\left(\forall b\in C,\alpha \in \mbox{\rm GF}(p)^{{*}}\right) &\alpha \in A-b &\Leftrightarrow & c\alpha\in A-b;\\
\left(\forall b\in C,\alpha \in \mbox{\rm GF}(p)^{{*}}\right) &-\alpha \in A-b &\Leftrightarrow & (c-1)\alpha \in A-b;\\
\left(\forall b\in C,\alpha \in \mbox{\rm GF}(p)^{{*}}\right) &(1-c)\alpha \in A-b &\Leftrightarrow & -c\alpha \in A-b.
\end{array}
\]
The first equivalence implies that $(A-b)=(A-b)c$, that is, $B=Bc$, for $B=A-b$; the second implies $B=B(c-1)$, and the last implies $B=-Bc^{{-1}}(1-c)^{{-1}}$. All these three together imply $B=Bc=B(c-1)=-B$. We already proved that this implies $|B|\geq |\langle -1,c,c-1\rangle|=p-1$, and clearly $|B|=|A|$. A contradiction since in $P=(A,C,A')$ the set $A$ cannot have $p-1$ elements.
\end{proof}
If $p\equiv 1$ (mod $3$) and $p>7$, then we can take $c$ to be a primitive $6^{\mbox{th}}$ root of the unity; then $c^{2}=c-1$, so $\langle c,c-1,-1\rangle$ is a subgroup of order $6$. Thus $\mbox{\rm AGL}(1,p)$ does not have the $3$-ut property if $p\equiv 1$ (mod $3$) and $p>7$.
Also, if $p\equiv 1$ (mod $4$) and $p>5$, then there are consecutive quadratic residues in $\{1,\ldots, p-1\}$; if $c$ is the larger of such a pair, then $\langle c,c-1,-1\rangle$ is contained in the subgroup of squares and again $\mbox{\rm AGL}(1,p)$ does not have the $3$-ut property. (If no two consecutive residues exist, then as $1$ and $p-1$ are residues, we see that residues and non-residues must alternate, apart from one pair of non-consecutive non-residues. But consecutive integer squares in $\{1,\ldots,p-1\}$ are an odd distance apart, and so there must be consecutive non-residues between them. So there are only two such squares, namely $1$ and $4$ and so $p=5$.)
Thus, only for primes $p\equiv 11$ (mod $12$) is the question undecided.
We have not so far considered the subgroup of index $2$ in $\mbox{\rm AGL}(1,p)$, which is $2$-homogeneous for $p\equiv 3$ (mod $4$). But if $p\equiv 7$ (mod $12$), then $\mbox{\rm AGL}(1,p)$ does not have the $3$-ut property, and neither does its subgroup. So these are also undecided only for $p\equiv 11$ (mod $12$).
\subsection{The groups $\mbox{\rm PSL}(2,q)$}
Regarding the groups $\mbox{\rm PSL}(2,q) \le G \le\mbox{\rm P}\Gamma {\rm L}(2,q)$, with either $q$ prime (with the exception of
$\mbox{\rm PSL}(2,q)$ for $q\equiv1$ (mod~$4$), which are not $3$-homogeneous), or $q=2^p$ for $p$ prime, we have the following:
\begin{enumerate}
\item Suppose $p$ is such that for some $c\in \mbox{\rm GF}(p)^{{*}}$ we have that $\langle -1,c,c-1\rangle$ is a proper subgroup of $\mbox{\rm GF}(p)^{{*}}$. Then there exists
a 3-partition $P=(A,B,C)$ such that the orbit of $\{0,1,c\}$ under $\mbox{\rm AGL}(1,p)$ has no section for $P$. Therefore the partition $(\{\infty\},A,B,C)$ has no section in the orbit of $\{\infty
,0,1,c\}$ under $\mbox{\rm PGL}(2,p)$. This follows from the fact that any set in this orbit containing $\infty$ is of the form $\{\infty\}\cup D$ where $D$ is a 3-set in the orbit of $\{0,1,c\}$ under $\mbox{\rm AGL}(1,p)$.
\item By the previous observation, it follows that, for the same $p$, $G$ \notut{4} for all $ G\leq \mbox{\rm PGL}(2,p)$.
\item for the case $\mbox{\rm PSL}(2,q)$, where $q=2^{{p}}$, we have $\mbox{\rm PSL}(2,q)=\mbox{SL}(2,q)$.
\item Below we have all the edges of $G(\{0,1\},c)$.
\end{enumerate}
\[
\begin{array}{l}
\left\{ {\frac {c{\alpha}^{2}}{c{\alpha}^{2}-c+1}},2\,{\frac {{\alpha}^{2}}{2\,{\alpha}^{2}-1}} \right\}
\left\{ 2\,{\frac {{\alpha}^{2}}{2\,{\alpha}^{2}+1}},2\,{\frac {c{\alpha}^{2}}{2\,c{\alpha}^{2}-c+2}} \right\}
\left\{ {\frac {c{\alpha}^{2}}{c{\alpha}^{2}-1+c}},2\,{\frac {c{\alpha}^{2}}{2\,c{\alpha}^{2}-2+c}} \right\} \\
\left\{ {\frac {{\alpha}^{2} \left( {c}^{2}+2-3\,c \right) }{-3\,c{\alpha}^{2}+2\,{\alpha}^{2}+{c}^{2}{\alpha}^{2}-1}},{\frac {c{\alpha}^{2} \left( c-1 \right) }{1-c{\alpha}^{2}+{c}^{2}{\alpha}^{2}}}\right\}
\left\{ {\frac {{\alpha}^{2} \left( {c}^{2}+2-3\,c \right) }{-3\,c{\alpha}^{2}+2\,{\alpha}^{2}+{c}^{2}{\alpha}^{2}+1}},{\frac {c{\alpha}^{2} \left( c-2 \right) }{2-2\,c{\alpha}^{2}+{c}^{2}{\alpha}^{2}}}\right\} \\
\left\{ {\frac {c{\alpha}^{2} \left( c-1 \right) }{-c{\alpha}^{2}-1+{c}^{2}{\alpha}^{2}}},{\frac {c{\alpha}^{2} \left( c-2 \right) }{{c}^{2}{\alpha}^{2}-2\,c{\alpha}^{2}-2}} \right\}
\left\{ {\frac {{\beta}^{2} \left( c-1 \right) }{c{\beta}^{2}-{\beta}^{2}-2+c}},{\frac {{\beta}^{2} \left( c-1 \right) }{c{\beta}^{2}-{\beta}^{2}-c}} \right\} \\
\left\{ {\frac {{\beta}^{2} \left( c-1 \right) }{c{\beta}^{2}-c-{\beta}^{2}+2}},{\frac {{\beta}^{2}}{{\beta}^{2}-2}} \right\}
\left\{ {\frac {{\beta}^{2} \left( c-1 \right) }{c{\beta}^{2}+c-{\beta}^{2}}},{\frac {{\beta}^{2}}{{\beta}^{2}+2}} \right\}
\left\{ {\frac {{\beta}^{2} \left( c-2 \right) }{c{\beta}^{2}+4\,c-2\,{\beta}^{2}-4}},{\frac {{\beta}^{2}}{{\beta}^{2}+2}} \right\} \\
\left\{ {\frac {{\beta}^{2} \left( c-2 \right) }{c{\beta}^{2}-2\,{\beta}^{2}+4-4\,c}},{\frac {{\beta}^{2} \left( c-2 \right) }{c{\beta}^{2}-2\,{\beta}^{2}-2\,c}} \right\}
\left\{ {\frac {{\beta}^{2}}{{\beta}^{2}-2}},{\frac {{\beta}^{2} \left( c-2 \right) }{c{\beta}^{2}+2\,c-2\,{\beta}^{2}}} \right\}
\end{array}
\]
But it does not seem clear how an approach similar to the one used in $\mbox{\rm AGL}(1,p)$ can be carried out here...
\subsection{The state of the art} Regarding our list of exceptional cases the situation is the following:
\begin{enumerate}
\item $k=3$:
\begin{enumerate}
\item regarding $\mbox{\rm AGL}(1,p)$, with $p$ prime, or its subgroup of index $2$ if $p \equiv 3$
(mod~$4$), we have:
\begin{enumerate}
\item $\mbox{\rm AGL}(1,p)$, or (if $p\equiv 3$ (mod $4$)) its subgroup of index $2$, does not have the $3$-ut property, unless possibly when $p\equiv 11$ (mod $12$);
\item in general $\mbox{\rm AGL}(1,p)$ \notut{3} if and only if there exists
$c\in \mbox{\rm GF}(p)\setminus \{0,1\}$ such that $|\langle c,c-1,-1\rangle|<p-1$.
\end{enumerate}
\item The group $2^6 \colon {\mbox{\rm G}}_2(2)$ \ut{3}; the same happens to its subgroup of index $2$.
\item The groups $\mbox{\rm Sz}(q)$ appear to have connected $G(b,c)$ and hence, probably, each one of them \ut{3}.
\item The Higman--Sims group \ut{3}.
\end{enumerate}
\item $k=4$:
\begin{enumerate}
\item For the groups $\mbox{\rm PSL}(2,q) \le G \le\mbox{\rm P}\Gamma {\rm L}(2,q)$, with either $q$ prime, or $q=2^p$ for $p$ prime, the situation is this:
\begin{enumerate}
\item if $q$ is prime and there exists
$c\in GF(p)^{{*}}$ such that $|\langle c,c-1,-1\rangle|<q$, then $G\leq \mbox{\rm PGL}(2,q)$ \notut{4}.
\item for $q\equiv1$ (mod~$4$) the group
$\mbox{\rm PSL}(2,q)$ is not $3$-homogeneous.
\item what happens in the other groups is undecided.
\end{enumerate}
\item $\mbox{\rm M}_{11}$, degree $12$, \ut{4}.
\end{enumerate}
\item $k=5$:
\begin{enumerate}
\item $\mbox{\rm P}\Gamma {\rm L}(2,32)$, degree $33$, \ut{5}.
\end{enumerate}
\end{enumerate}
\section{Regular semigroups}
Arguably, three of the most important classes of semigroups are groups, inverse semigroups and regular semigroups, defined as follows: for a semigroup $S$ we have that
\begin{itemize}
\item $S$ is a group if for all $a\in S$ there exists a unique $b\in S$ such that $a=aba$;
\item $S$ is inverse if for all $a\in S$ there exists a unique $b\in S$ such that $a=aba$ and $b=bab$;
\item $S$ is regular if for all $a\in S$ there exists $b\in S$ such that $a=aba$.
\end{itemize}
Recall from the introduction that to a large extent semigroup structure theory is (almost) all about trying to show how the idempotents shape the structure of the semigroup. Therefore it is no surprise that groups and inverse semigroups can be characterized by their idempotents:
\begin{itemize}
\item a semigroup is inverse if and only it is regular and the idempotents commute (see \cite[Theorem 5.5.1]{Ho95});
\item a semigroup is a group if and only if it is regular and contains exactly one idempotent (see \cite[Ex. 3.11]{Ho95}).
\end{itemize}
Inverse semigroups, apart from being the class of (non-group) semigroups with the largest number of books dedicated to them, were introduced by geometers and they keep being very important to them \cite{Pa99}.
The full transformation semigroup $T(X)$ is regular, and all regular semigroups embed in some $T(X)$; every group embeds in some $T(X)$ as a group of permutations; and every inverse semigroup embeds in some $T(X)$ as an inverse semigroup of quasi-permutations, that is, transformations in which all but one of the kernel classes are singletons. (This follows from the Vagner--Preston representation \cite[Theorem 5.1.7]{Ho95} that maps every inverse semigroup into an isomorphic semigroup of partial bijections on a set; and every partial bijection $f$ on $X$ can be extended to a quasi-permutation $\bar{f}$ on $X\cup \{\infty\}$, defining $x\bar{f}=\infty$, for all $x$ not in the domain of $f$, and $x\bar{f}=xf$ elsewhere, yielding a semigroup of full quasi-permutations isomorphic to the original one.)
In the introduction we provided the classification of the groups $G\leq \mathcal{S}_{n}$ such that the semigroup generated by $G$ and any map $a\in \mathcal{T}_{n}$ is regular. Our aim now is to dramatically improve that result by extending it to quasi-permutations and transformations of a given rank. The main observation is the following straightforward lemma.
\begin{lem}\label{auxl}
Let $a\in \mathcal{T}_{n}$ and let $G\leq \mathcal{S}_{n}$. Then $a$ is regular in $\langle a,G\rangle$ if and only if there exists $g\in G$ such that $\mbox{rank}(aga)=\mbox{rank}(a)$.
\end{lem}
\begin{proof}
Suppose that $a$ is regular in $\langle a,G\rangle$. Then there exists
$b\in \langle a,G\rangle$ such that $a=aba$. As $\mbox{rank}(uv)\leq \mbox{min}\{\mbox{rank}(u),\mbox{rank}(v)\}$ it follows that $\mbox{rank}(b)\geq \mbox{rank}(a)$. Now, either $b\in G$ and the result follows, or $b=g_{1}ag_{2}\ldots g_{m}ag_{m+1}$ and $\mbox{rank}(b)\leq \mbox{rank}(a)$, that is, $\mbox{rank}(a)=\mbox{rank}(b)$. Therefore, for every $g_{i}\in \{g_{2},\ldots,g_{m}\}$ we have $\mbox{rank}(ag_{i}a)=\mbox{rank}(a)$. It is proved that there exists
$g\in G$ such that $\mbox{rank}(aga)=\mbox{rank}(a)$.
Conversely, if $\mbox{rank}(aga)=\mbox{rank}(a)$, then $[n]ag$ is a transversal of $\mbox{Ker}(a)$ and hence $ga$ permutes $[n]a$. Therefore, for some natural $m$, $(ga)^{{m}}$ acts on $[n]a$ as the identity and hence $a(ga)^{{m}}=a$. The lemma follows.
\end{proof}
In \cite{lmm} a stronger version of the previous result is proved.
\begin{thm}(\cite[Theorem 2.3 and Corollary 2.4]{lmm})\label{mcalister}
Let $G\leq \mathcal{S}_{n}$ and let $a\in \mathcal{T}_{n}$. Then the following are equivalent:
\begin{enumerate}
\item there exists $g\in G$ such that $\mbox{rank}(aga)=\mbox{rank}(a)$;
\item $a$ is regular in $\langle G,a\rangle$;
\item every $b\in \langle G,a\rangle$, such that $\mbox{rank}(b)=\mbox{rank}(a)$, is regular in $\langle G,a\rangle$.
\end{enumerate}
\end{thm}
With the new tools developed in the previous sections we can now prove our first main theorem regarding regularity of semigroups generated by a group and a quasi-permutation. Recall that $a\in \mathcal{T}_{n}$ is a quasi-permutation if all, but one, of the $\mbox{Ker}(a)$-classes have one element.
\begin{thm}\label{quasi-permutation}
Let $G\leq \mathcal{S}_{n}$ and let $1< k< n$. Then the following are equivalent
\begin{enumerate}
\item every quasi-permutation $a$, such that $\mbox{rank}(a)=k$, is regular in $\langle G,a\rangle$;
\item $G$ is $(k-1)$-homogeneous or $G$ is one of the following groups
\begin{enumerate}
\item[\rm (a)] $n=5$ and $G\cong C_5,\ \dihed{5},$ with $k=2$;
\item[\rm (b)] $n=7$ and $G\cong\mbox{\rm AGL}(1,7)$, with $k=3$;
\item[\rm (c)] $n=9$ and $G\cong\mbox{\rm ASL}(2,3)$ or $\mbox{\rm AGL}(2,3)$, with $k=4$.
\end{enumerate}
\end{enumerate}
\end{thm}
\begin{proof}
Every quasi-permutation $a$ of rank $k$ has a kernel of the form $$(\{a_{1}\},\ldots,\{a_{k-1}\},\{a_{k},\ldots,a_{n}\})$$ and has image $\{b_{1},\ldots,b_{k}\}$. By the previous lemma we know that $a$ is going to be regular in $\langle a,G\rangle$ if and only if there exists
$g\in G$ such that $\mbox{rank}(a)=\mbox{rank}(aga)$. But this is equivalent to saying that there exists
$g\in G$ such that $([n]\setminus \{b_{1},\ldots,b_{k}\})g\subseteq \{a_{k},\ldots,a_{n}\}$. As these sets are arbitrary, it follows that $G$ satisfies the property that each quasi-permutation $a$ is regular in $\langle a,G\rangle$ if and only $G$ is $(n-k,n-k+1)$-homogenous. By Theorem \ref{2.1} this last condition is equivalent to $(2)$. It is proved that $(1)$ and $(2)$ are equivalent.
\end{proof}
Now we can state and prove our second main result about quasi-permutations.
\begin{thm}\label{quasi-permutation2}
Let $G\leq \mathcal{S}_{n}$ and let $1< k\leq \lfloor \frac{n}{2}\rfloor$. Then the following are equivalent
\begin{enumerate}
\item for every quasi-permutation $a$, such that $\mbox{rank}(a)=k$, the semigroup $\langle G,a\rangle$ is regular;
\item $G$ is $(k-1)$-homogeneous or $G$ is one of the following groups
\begin{enumerate}
\item[\rm (a)] $n=5$ and $G\cong C_5,\ \dihed{5}$, with $k=2$;
\item[\rm (b)] $n=7$ and $G\cong\mbox{\rm AGL}(1,7)$, with $k=3$;
\end{enumerate}
\end{enumerate}
\end{thm}
\begin{proof}
Clearly $(1)$ implies that every quasi-permutation $a$ is regular in $\langle a,G\rangle$ and hence, by the previous result, it follows that $G$ must be one of the groups listed in the statement of the theorem, or $n=9$ and $G\cong\mbox{\rm ASL}(2,3)$ or $\mbox{\rm AGL}(2,3)$. However, for $a:=\left(\begin{array}{cccc}\{1\}&\{2\}&\{3\}&\{4,\ldots,9\}\\ 1&4&5&2\end{array}\right)$, the semigroup $\langle a,G\rangle$ is not regular (when $G$ is $\mbox{\rm ASL}(2,3)$ or $\mbox{\rm AGL}(2,3)$).
It is proved that $(1)$ implies $(2)$.
Conversely, let $a$ be a rank $k$ quasi-permutation and let $G\leq \mathcal{S}_{n}$ be a $(k-1)$-homogenous group. By the previous theorem ($(2)$ implies $(1)$) we know that $a$ is regular in $\langle a,G\rangle$ and hence, by Theorem \ref{mcalister}, every $b\in \langle a,G\rangle$ such that $\mbox{rank}(b)=k$ is regular in $ \langle a,G\rangle$. Now suppose that $b\in \langle a,G\rangle$ and $\mbox{rank}(b)=l<k$. Then $G$ is $l$-homogenous (because $k\leq \lfloor \frac{n}{2}\rfloor$) and hence there exists
$g\in G$ such that $bgb$ has rank $l$. Thus, once again by Lemma \ref{auxl}, $b$ is regular in $ \langle b,G\rangle$. As $ \langle b,G\rangle\subseteq \langle a,G\rangle$, it follows that $b$ is regular in $ \langle a,G\rangle$.
That the groups listed in (2) (a) and (b) satisfy the condition (1) follows from Theorem \ref{th2}.
\end{proof}
Now we turn to the case of transformations of a given rank.
Our main theorem is the following.
\begin{thm}\label{Kut}
Let $n\geq 5$, $G\leq \mathcal{S}_{n}$ and let $1< k\leq \lfloor \frac{n+1}{2}\rfloor$. Then
the following are equivalent:
\begin{enumerate}
\item for all rank $k$ transformations $a\in \mathcal{T}_{n}$, we have that $a$ is regular in $\langle a, G\rangle$;
\item for all rank $k$ transformations $a\in \mathcal{T}_{n}$, the semigroup $\langle a, G\rangle$ is regular;
\item $G$ has the $k$-ut property (and hence is one of the groups listed in Theorem \ref{RegularAndKut}).
\end{enumerate}
\end{thm}
\begin{proof}
It follows from Lemma \ref{auxl} that (1) and (3) are equivalent, and (2) implies (1) trivially. In addition, (1) (together with Theorem \ref{mcalister}) implies that $b$ is regular in $\langle a,G\rangle$, for all $b\in \langle a,G\rangle$, with $\mbox{rank}(b)=\mbox{rank}(a)$. It also follows from Proposition \ref{critical} (2) that
if $G$ has the $k$-ut property, then $G$ is $(k-1)$-homogenous, or it is one of the exceptions in Theorem \ref{th6}. If $G$ is $(k-1)$-homogenous, then
for every $c\in \langle a,G\rangle$, with $\mbox{rank}(c)<\mbox{rank}(a)$, we have that $G$ is $\mbox{rank}(c)$-homogenous and hence $c$ is regular in $\langle c,G\rangle\subseteq \langle a,G\rangle$. Thus $c$ is regular in $\langle a,G\rangle$.
If $G$ is one of the three exceptions of degree $5$ and $7$, then (by Theorem \ref{th6c}) $\langle a,G\rangle$ is regular, for all rank$(k)$ maps $a$. Thus it is proved that (1) and (3) imply (2). \end{proof}
When $n=9$ and $G= \mbox{\rm ASL}(2,3)$ or $G=\mbox{\rm AGL}(2,3)$, (by Theorem \ref{quasi-permutation2}) there exist
rank-$4$ maps $a\in \mathcal{T}_{n}$ such that $\langle a,G\rangle$ is not regular. Also these groups do not have the $4$-ut property.
\section{Problems}
We start by proposing a problem to experts in number theory. If this problem can be solved, the results on $\mbox{\rm AGL}(1,p)$, in Section \ref{numbertheory}, will be dramatically sharpened.
\begin{problem}
Classify the prime numbers $p$ congruent to $11$ (mod $12$) such that for some $c\in \mbox{\rm GF}(p)^{{*}}$ we have $|\langle -1,c,c-1\rangle|<p-1$.
\end{problem}
\begin{problem}
Do the Suzuki groups $\mbox{\rm Sz}(q)$ have the $3$-ut property?
Classify the groups $G$ that have the $4$-ut property, when $\mbox{\rm PSL}(2,q) \le G \le\mbox{\rm P}\Gamma {\rm L}(2,q)$, with either $q$ prime (except
$\mbox{\rm PSL}(2,q)$ for $q\equiv1$ (mod~$4$), which is not $3$-homogeneous), or $q=2^p$ for $p$ prime.
\end{problem}
\begin{problem}
Prove results analogous to Theorem \ref{quasi-permutation2} and to Theorem \ref{Kut}, but for the transformations of rank $k>\lfloor \frac{n+1}{2}\rfloor$.
\end{problem}
The difficulty here (when rank $k>\lfloor \frac{n+1}{2}\rfloor$) is that a $k$-homogenous group is not necessarily $(k-1)$-homogenous. Therefore a rank $k$ map $a\in \mathcal{T}_{n}$ might be regular in $\langle a, G\rangle$, but we are not sure that there exists
$g\in G$ such that $\mbox{rank}(bgb)=\mbox{rank}(b)$, for $b\in \langle a, G\rangle$ such that $\mbox{rank}(b)<\mbox{rank}(a)$.
\begin{problem}\label{top}
In what concerns this paper, the most general problem that has to be handled is the classification of pairs $(a,G)$, where $a\in \mathcal{T}_{n}$ and $G\leq \mathcal{S}_{n}$, such that $\genset{a,G}$ is a regular semigroup.
\end{problem}
When investigating $(k-1)$-homogenous groups without the $k$-ut property, it was common that some of the orbits on the $k$-sets have transversals for all the partitions. Therefore the following definition is natural.
A group $G\leq \mathcal{S}_{n}$ is said to have the weak $k$-ut property if there exists
a $k$-set $S\subseteq [n]$ such that the orbit of $S$ under $G$ contains a section for all $k$-partitions. And such a set is called a $G$-universal transversal set.
\begin{problem}\label{five}
Classify the groups with the weak $k$-ut property; in addition, for each one of them, classify their $G$-universal transversal sets.
\end{problem}
In this paper we considered groups such that the orbit of every $k$-set contains a section for every $k$-partition. And this is of course a very strong requirement. In order to attack Problem \ref{top}, it seems the next step (in addition to Problem \ref{five}) is to consider groups such that the orbit of every $k$-set contains sections for some (not all) partitions.
\begin{problem}
Let $\pi$ be a partition of $n$. A map $a\in \mathcal{T}_{n}$ has kernel type $\pi$ if the partition of $n$ induced by the cardinalities of the kernel blocks is equal to $\pi$. Classify the groups $G\leq \mathcal{S}_{n}$ such that for all maps $a\in \mathcal{T}_{n}$ of a given kernel type $\pi$, the semigroup $\genset{G,a}$ is regular.
\end{problem}
In McAlister's celebrated paper \cite{mcalister} it is proved that if $e^2=e\in \mathcal{T}_{n}$ is a rank $n-1$ idempotent, then $\genset{G,e}$ is regular for all groups $G\leq \mathcal{S}_{n}$. In addition, assuming that $\{\alpha,\beta\}$ is the non-singleton kernel class of $e$ and $\alpha e=\beta$, if $\alpha$ and $\beta$ are not in the same orbit under $G$, then $\genset{e,G}$ is an orthodox semigroup (that is, the idempotents form a subsemigroup); and $\genset{G,e}$ is inverse if and only if $\alpha$ and $\beta$ are not in the same orbit under $G$ and the stabilizer of $\alpha$ is contained in the stabilizer of $\beta$.
\begin{problem}
Classify the groups $G\leq \mathcal{S}_{n}$ that together with any idempotent [rank $k$ idempotent] generate a regular [orthodox, inverse] semigroup.
Classify the pairs $(G,a)$, with $a\in \mathcal{T}_{n}$ and $G\leq \mathcal{S}_{n}$, such that $\genset{G,a}$ is inverse [orthodox]. (Recall that by \cite{schein} every element $a\in \mathcal{T}_{n}$ is contained in an inverse subsemigroup of $\mathcal{T}_{n}$; in addition it is a longstanding open problem to describe the maximal inverse subsemigroups of $\mathcal{T}_{n}$.)
\end{problem}
A group $G\leq \mathcal{S}_{n}$ has the $(n-1)$-universal transversal property if and only if it is transitive. And $\genset{G,a}$ contains all the rank $n-1$ maps of $\mathcal{T}_{n}$ if and only if $G$ is $2$-homogeneous. In this last case $\genset{G,a}$ is regular for all $a\in \mathcal{T}_{n}$ because $\genset{G,a}=\{b\in \mathcal{T}_{n}\mid |[n]b|\leq n-1\}\cup G$, and this semigroup is well known to be regular.
\begin{problem}
Classify the groups $G\leq \mathcal{S}_{n}$ such that $G$ together with any rank $n-k$ map, where $k\leq 5$, generate a regular semigroup. We already know that such $G$ must be $k$-homogeneous and so are classified.
\end{problem}
The majority of the previous problems (and theorems) admit an obvious analogous with regular replaced everywhere by {\em idempotent generated}.
\begin{problem}
Classify all the pairs $(a,G)$, where $a\in \mathcal{T}_{n}$ and $G\leq \mathcal{S}_{n}$, such that $\genset{a,G}\setminus G$ is idempotent generated (that is, $\genset{a,G}\setminus G$ is generated by its own idempotents).
Solve particular instances of this general problem analogous to the list of problems above.
\end{problem}
The theorems and problems in this paper admit linear versions that are interesting for experts in groups and semigroups, but also to experts in linear algebra and matrix theory. However, for the linear case, not even an analogue of Theorem \ref{th2} exists. All we know is that any singular matrix with any group containing the special linear group generate a regular semigroup \cite{ArSi1,ArSi2} (see also the related papers \cite{Gr,Pa,Ra}).
\begin{problem}
Prove (or disprove) that if $G\leq GL(n,q)$ such that for all singular matrix $a$ there exists
$g\in G$ with $\operatorname{rank}(a)=\operatorname{rank}(aga)$, then $G$ contains the special linear group.
\end{problem}
It is clear that such a group must satisfy the following property. If $V$ is a vector space (over a finite field) with $\dim(V)=n$, and $U,T\leq V$ are two non-null subspaces such that $\dim(U)+\dim(T)=n$, then there exists
$g\in G$ such that $V=Ug\oplus T$.
For $n=2$ and for $n=3$, this condition is equivalent to irreducibility of $G$. But we conjecture that, for sufficiently large $n$, it implies that $G$ contains the special linear group.
\begin{problem}\label{11}
Classify the groups $G\leq GL(n,q)$ such that for all rank $k$ (for a given $k$) singular matrix $a$ we have that $a$ is regular in $\genset{G,a}$ [the semigroup $\genset{G,a}$ is regular].
\end{problem}
To handle this problem it is useful to keep in mind the following results. Kantor~\cite{kantor:inc} proved that if a subgroup of $\mbox{\rm P}\Gamma {\rm L}(d,q)$ acts transitively on $k$-dimensional subspaces, then it acts transitively on $l$-dimensional subspaces for all $l\le k$ such that $k+l\le n$; in~\cite{kantor:line}, he showed that subgroups transitive on $2$-dimensional subspaces are $2$-transitive on the $1$-dimensional subspaces with the single exception of a subgroup of $\mbox{\rm PGL}(5,2)$ of order $31\cdot5$; and, with the second author~\cite{cameron-kantor}, he showed that such groups must contain $\mbox{\rm PSL}(d,q)$ with the single exception of the alternating group $A_7$ inside $\mbox{\rm PGL}(4,2)\cong A_8$. Also Hering \cite{He74,He85} and Liebeck \cite{Li86} classified the subgroups of $\mbox{\rm PGL}(d,p)$ which are transitive on $1$-spaces.
\begin{problem}
Solve the analogue of Problem \ref{11} for independence algebras (for definitions and fundamental results see \cite{ArEdGi,arfo,cameronSz,gould}).
\end{problem}
Recall from Subsection \ref{general} the graph $G(B,c)$, where $G$ is a $t$-homogeneous group and $|B|=t-1$.
\begin{problem}
Is it true that the group $G$ has the $(t+1)$-ut property if and only if $G(B,c)$ is connected? If so, is it possible to find an elementary proof of that (without using the classification of finite simple groups)?
\end{problem}
\begin{problem}
Prove Corollary \ref{cor1} and Corollary \ref{cor2} without using the classification of finite simple groups.
\end{problem}
Regarding this problem, observe that the proof that $k$-ut implies $(k-1)$-ut in fact follows from the classification of the $(k-1,k)$-homogenous groups (and hence, for that purpose, we can bypass the classification of groups with the $k$-ut property). In fact, if the group possesses the $k$-ut property (for $k\leq \lfloor \frac{n}{2}\rfloor$), then it is $(k-1,k)$-homogeneous and hence, with few exceptions, it is
$(k-1)$-homogeneous so that it has the $(k-1)$-ut property. So in the previous theorem what really is at stake is to find an elementary proof to Corollary \ref{cor2}.
\def$'${$'$}
\section*{Acknowledgements}
We gratefully thank various conversations with P. M. Neumann on the early stages of this investigation. We also thank the developers of GAP \cite{GAP} and
Soicher for GRAPE \cite{So06}.
The first author was partially supported by FCT and FEDER, Project POCTI-ISFL-1-143 of Centro de Algebra da Universidade de Lisboa, by FCT and PIDDAC through the project PTDC/MAT/69514/2006, by PTDC/MAT/69514/2006 Semigroups and Languages, and by PTDC/MAT/101993/2008 Computations in groups and semigroups.
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Society 18/04/2021
"He stripped naked and insulted us for 45 minutes because we wouldn't give him an appointment"
Incidents against SEPE (Spanish State Public Employment Service) workers soar in the wake of the pandemic
Elisabet Escriche
Queues at the office that the SEPE and the SOC share in Sepúlveda street in Barcelona. BRUNA CASAS
It was Maundy Thursday when a man in his fifties showed up at the office that the SEPE (Spanish State Public Employment Service) shares with the SOC (Catalan Employment Service) on Sepúlveda Street in Barcelona. He entered and tried to make an appointment after explaining, in desperation, that he had tried repeatedly by phone and online but there was no way. The workers told him there was nothing they could do. "He went to the office door, undressed - he even took off his mask - and insulted us for 45 minutes because we wouldn't give him an appointment", explains Miguel Ángel García, an Employment Service worker at the office. "The man was very angry, shouting and calling us all kinds of things", he adds. The incident ended when the Mossos d'Esquadra showed up, alerted by security staff, and the man got dressed up and left.
Since the SEPE offices were opened to the public after the strictest lockdown, last July, incidents and threats towards the staff have increased, according to what the unions denounce, by desperate citizens who are fed up with seeing months and months go by and there is no way they can be attended to. "There have been very unpleasant situations, with users who went to the offices to threaten, insult, shout or even physically assault workers who were attending in person", explains the state coordinator of the SEPE UGT Union.
The incidents of maximum tension happened just after the home lockdown, with a large number of furloughed cases to be resolved. In Madrid, for example, there were cases of citizens who waited for employees who were in the office without attending in person to leave work to insult them. "In some cases they hit windows and even broke some of them. It created a certain psychosis", García admits.
"The situation was very difficult: we had not been open to the public for many months, people came to the offices because they could not get in touch with us by phone or internet, with the logical nervousness as a result of not having been paid for months", explains another SEPE worker from another office in Barcelona who prefers to remain anonymous.
Security in SEPE offices has doubled with the increase in temporary and permanent lay-offs
Faced with this situation, the SEPE opted to increase security at the offices, which went from having one guard to two. In the case of Sepulveda it increased to five, because there are two offices together. "During this time I have heard many cases of desperation from those affected, which I fully understand. Fortunately, I have not suffered any personal incidents", says García.
The Mossos d'Esquadra, however, have had to be present several times in the office where he works. On another occasion, he recalls, a group of citizens waiting to be attended, taking advantage of the arrival of media, began to shout and denounce the malfunctioning of the SEPE. Despite these incidents, however, Garcia says that in the current context citizens - many of whom continue their desperate battle with the SEPE - "have shown, once again, to have more common sense than our leaders".
Lack of staff
From the UGT Union they assure that the SEPE staff understand the desperation of the users but stress that "in no case" can their situation be blamed on a staff that is "close to collapse", diminished by staff cuts in recent years. By the way: the man who stripped naked was attended to on Tuesday just after Easter.
Content developed by
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Society | 15/09/2022
Three months of disruption to commuter trains due to Sagrera railworks
Poble-sec nightmare: shouting, fights and stabbings
Heat and unfinished building work mark start of school year
SEPE ERTO Laboral aggressions
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{"url":"https:\/\/deepai.org\/publication\/enumeration-on-trees-under-relabelings","text":"Enumeration on Trees under Relabelings\n\nWe study how to evaluate MSO queries with free variables on trees, within the framework of enumeration algorithms. Previous work has shown how to enumerate answers with linear-time preprocessing and delay linear in the size of each output, i.e., constant-delay for free first-order variables. We extend this result to support relabelings, a restricted kind of update operations on trees which allows us to change the node labels. Our main result shows that we can enumerate the answers of MSO queries on trees with linear-time preprocessing and linear delay, while supporting node relabelings in logarithmic time. To prove this, we reuse the circuit-based enumeration structure from our earlier work, and develop techniques to maintain its index under node relabelings. We also show how enumeration under relabelings can be applied to evaluate practical query languages, such as aggregate, group-by, and parameterized queries.\n\n\u2022 21 publications\n\u2022 13 publications\n\u2022 18 publications\n12\/22\/2018\n\nEnumeration on Trees with Tractable Combined Complexity and Efficient Updates\n\nWe give an algorithm to enumerate the results on trees of monadic second...\n12\/10\/2018\n\nEnumeration Complexity of Unions of Conjunctive Queries\n\nWe study the enumeration complexity of answering unions of conjunctive q...\n03\/17\/2022\n\nEfficiently Enumerating Answers to Ontology-Mediated Queries\n\nWe study the enumeration of answers to ontology-mediated queries (OMQs) ...\n08\/08\/2022\n\nMSO Queries on Trees: Enumerating Answers under Updates Using Forest Algebras\n\nWe describe a framework for maintaining forest algebra representations o...\n04\/07\/2020\n\nMaintaining Triangle Queries under Updates\n\nWe consider the problem of incrementally maintaining the triangle querie...\n06\/28\/2022\n\nWhich arithmetic operations can be performed in constant time in the RAM model with addition?\n\nIn the literature of algorithms, the specific computation model is often...\n12\/21\/2017\n\nEnumeration Complexity of Conjunctive Queries with Functional Dependencies\n\nWe study the complexity of enumerating the answers of Conjunctive Querie...\n\n1 Introduction\n\nEnumeration algorithms are a common way to compute large query results on databases, see, e.g.,\u00a0[28]. Instead of computing all results, these algorithms compute results one after the other, while ensuring that the time between two successive results (the delay) remains small. Ideally, the delay should be linear in the size of each produced solution, and independent of the size of the input database. To make this possible, enumeration algorithms can build an index structure on the database during a preprocessing phase that ideally runs in linear time.\n\nMost enumeration algorithms assume that the input database will not change. If we update the database, we must re-run the preprocessing phase from scratch, which is unreasonable in practice. Losemann and Martens\u00a0[24] proposed the first enumeration algorithm that addresses this issue: they study monadic second-order (MSO) query evaluation on trees, and show that the index structure for enumeration can be maintained under updates. More precisely, they can update the index in time polylogarithmic in the input tree\u00a0 (much better than re-running the linear preprocessing). The tradeoff is that their delay is also polylogarithmic in\u00a0, whereas the delay can be independent of\u00a0 when there are no updates\u00a0[8].\n\nThis result of\u00a0[24] leads to a natural question: does the support for updates inherently increase the delay of enumeration algorithms? This is not always the case: e.g., when evaluating first-order queries (plus modulo-counting quantifiers) on bounded-degree databases, updates can be applied in constant time\u00a0[11] and the delay is constant, as in the case without updates\u00a0[18, 22]. However, when evaluating conjunctive queries (CQs) on arbitrary databases, supporting updates has a cost: under complexity-theoretic assumptions, the class of CQs with efficient enumeration under updates\u00a0[12] is a strict subclass of the class of CQs for the case without updates\u00a0[9]. Could the same be true of MSO on trees, as [24] would suggest?\n\nIn this work, we answer this question in the negative, for a restricted update language. Specifically, we show an enumeration algorithm for MSO on trees with the same delay as in the case without updates\u00a0[8], while supporting updates with a better complexity than\u00a0[24] (see detailed comparison of results in Section\u00a03). The tradeoff is that we only allow updates that change the labels of nodes, called relabelings, unlike\u00a0[24] where updates can also insert and delete leaves. We still show how these relabelings are useful to evaluate practical query languages, such as parameterized queries and group-by queries with aggregates. A parameterized query allows the user to specify some parameters for the evaluation (e.g., select some positions on the tree). Our results support such queries: we can model the parameters as labels and apply relabeling updates when the user changes the parameters. A group-by query with aggregates partitions the set of results into groups based on an attribute, and computes some aggregate quantity on each group (e.g., a sum). We show how to enumerate the results of such queries. For groups, our techniques can handle them with one single enumeration structure using relabelings to switch groups. For aggregates, we can efficiently compute and maintain them in arbitrary semirings; this problem was left open by\u00a0[24] even for counting, and is practically relevant in its own right\u00a0[26]. Of course, by Courcelle\u2019s theorem\u00a0[15], our results generalize to MSO queries on bounded-treewidth data (see\u00a0[4]), where relabelings mean adding or removing unary facts (i.e., the tree decomposition is unchanged).\n\nThe proof of our main result follows the approach of\u00a0[3]\n\nand is inspired by knowledge compilation in artificial intelligence and by factorized representations in database theory. Specifically, we encode knowledge (in our case, the query result) as a circuit in a restricted class, and we then use the circuit for efficient reasoning and for aggregates as in\n\n[17]. In\u00a0[3], we have used this circuit-based approach to recapture existing enumeration results for MSO on trees [8, 23]. In this work, we refine the approach and show that it can support updates. Our key new ingredient are hybrid circuits: they have both set-valued gates that represent the values to enumerate, and Boolean gates that encode the tree labels which can be updated. We first show that we can efficiently compute such circuits to capture the possible results of an MSO query under all possible labelings of a tree. Second, we show how to efficiently enumerate the set of assignments captured by these circuits, also supporting updates that toggle the Boolean gates affected by a relabeling. We also introduce some standalone tools, e.g., a lemma to balance the input trees to MSO queries (Lemma\u00a04), ensuring that hybrid circuits have logarithmic depth so that changes can be propagated quickly; and a constant-delay enumeration algorithm for reachability in forests under updates (Section\u00a07).\n\nPaper structure.\n\nWe start with preliminaries in Section\u00a02, and define our problem and give our main result in Section\u00a03. In Section\u00a04, we review the set-valued provenance circuits of\u00a0[3], and show our balancing lemma. We introduce hybrid circuits in Section\u00a05, and show in Section\u00a06 how to use them for enumeration under updates, using a standalone reachability indexing scheme on forests given in Section\u00a07. Having shown our main result, we outline its consequences for application-oriented query languages in Section\u00a08 and conclude in Section\u00a09.\n\n2 Preliminaries\n\nTrees, queries, answers, assignments.\n\nIn this work, unless otherwise specified, a tree is always binary, rooted, ordered, and full. Let be a finite set called a tree alphabet. A -tree is a pair of a tree and of a labeling function that maps each node of\u00a0 to a set of labels . We often abuse notation and identify to its node set, e.g., write as a function from\u00a0 to the powerset\u00a0 of\u00a0; we may also omit\u00a0 and write the -tree as just\u00a0.\n\nWe consider queries in monadic second-order logic (MSO) on the signature of -trees: it features two binary relations and denoting the first and second child of each internal node, and a unary relation for each denoting the nodes that carry label\u00a0 (i.e., nodes\u00a0 for which ). MSO extends first-order logic, which builds formulas from atoms of this signature and from equality atoms, using the Boolean connectives and existential and universal quantification over nodes. Formulas in MSO can also use second-order quantification over sets of nodes, written as second-order variables. For instance, on , we can express in MSO that every node carrying labels and has a descendant carrying label\u00a0.\n\nIn this work, we study MSO queries, i.e., MSO formulas with free variables. The free variables can be first-order or second-order, but we can rewrite any MSO query to ensure that all free variables are second-order: for instance as , where asserts that is exactly the singleton set . Hence, we usually assume without loss of generality that MSO queries only have second-order free variables.\n\nGiven a -tree and an MSO query , an -tuple of subsets of\u00a0 is an answer of\u00a0 on\u00a0, written , if satisfies in the usual logical sense. It will be more convenient to represent each answer as an assignment, which is a set of pairs called singletons that indicate that an element is in the interpretation of a variable. Formally, given an -tuple of subsets of\u00a0, the corresponding assignment is . We can convert each assignment in linear time to the corresponding answer and vice-versa, so we will use the assignment representation throughout this work. Our goal is to compute the set of assignments of\u00a0 on\u00a0, which we call the output of\u00a0 on\u00a0; we abuse notation and write it . We measure the complexity of this task in data complexity, i.e., as a function of the input tree\u00a0, with the query\u00a0 being fixed.\n\nEnumeration.\n\nThe output of an MSO query can be huge, so we work in the setting of enumeration algorithms [31, 28] which we present following\u00a0[3]. As usual for enumeration algorithms\u00a0[28], we work in the RAM model with uniform cost measure (see, e.g.,\u00a0[1]), where pointers, numbers, labels for elements and facts, etc., have constant size.\n\nAn enumeration algorithm with linear-time preprocessing for a fixed MSO query on\u00a0-trees takes as input a -tree and computes the output of\u00a0 on\u00a0. It consists of two phases. First, the preprocessing phase takes as input and produces in linear time a data structure\u00a0 called the index, and an initial state . Second, the enumeration phase repeatedly calls an algorithm . Each call to\u00a0 takes as input the index and the current state , and returns one assignment and a new state : a special state value indicates that the enumeration is over so should not be called again. The assignments produced by the successive calls to\u00a0 must be exactly the elements of , with no duplicates.\n\nWe say that the enumeration algorithm has linear delay if the time to produce each new assignment is linear in its cardinality , and is independent of\u00a0. In particular, if all answers to\u00a0 are tuples of singleton sets (for instance, if is the translation of a MSO query where all free variables are first-order), then the cardinality of each assignment is constant (it is the arity of\u00a0). In this case, the enumeration algorithm must produce each assignment with constant delay: this is called constant-delay enumeration. The memory usage of an enumeration algorithm is the maximum number of memory cells used during the enumeration phase (not counting the index , which resides in read-only memory), expressed as a function of the size of the largest assignment (as in\u00a0[8]): we say that the enumeration algorithm has linear memory if its memory usage is linear in the size of the largest assignment.\n\nPrevious works have studied enumeration for MSO on trees. Bagan\u00a0[8] showed that for any fixed MSO query , given a -tree , we can enumerate the output of\u00a0 on\u00a0 with linear delay and memory, i.e., constant delay and memory when all free variables are first-order. This result was re-proven by Kazana and Segoufin\u00a0[23] via a result of Colcombet\u00a0[14], and a third proof via provenance circuits was recently proposed by the present authors\u00a0[3].\n\n3 Problem Statement and Main Result\n\nOur goal is to address a limitation of these existing results, namely, the assumption that the input -tree will never change. Indeed, if is updated, these results must discard the index\u00a0 and re-run the preprocessing phase on the new tree. To improve on this, we want our enumeration algorithm to support update operations on\u00a0, and to update\u00a0 accordingly instead of recomputing it from scratch. Specifically, an algorithm for enumeration under updates on a tree has a preprocessing phase that produces the index as usual, but has two algorithms during the enumeration phase: (i.) an enumeration algorithm\u00a0 as presented before, and (ii.) an update algorithm\u00a0. When we want to change the tree\u00a0, we call with a description of the changes: modifies\u00a0 accordingly, updates the index\u00a0, and resets the enumeration state (so enumeration starts over on the new tree, and all working memory of the enumeration phase is freed). The update time of the enumeration algorithm is the complexity of\u00a0: like preprocessing, but unlike delay, it is a function of the size of the (current) tree\u00a0.\n\nTo our knowledge, the only published result on enumeration for MSO queries under updates is the work of Losemann and Martens [24], which applies to words and to trees, for MSO queries with only free first-order variables. They show an enumeration algorithm with linear-time preprocessing: on words, the update complexity and delay is ; on trees, these complexities become . Thus the delay is worse than in the case without updates\u00a0[8], and in particular it is no longer independent from\u00a0.\n\nMain result.\n\nIn this work, we show that enumeration under updates for MSO queries on trees can be performed with a better complexity that matches the case without updates: linear-time preprocessing, linear delay and memory (in the assignments), and update time in . This improves on the bounds of\u00a0[24] (and uses entirely different techniques). However, in exchange for the better complexity, we only support a weaker update language: we can change the labels of tree nodes, called a relabeling, but we cannot insert or delete leaf nodes as in\u00a0[24], which we leave for future work (see the conclusion in Section\u00a09). We show in Section\u00a08 that relabelings are still useful to derive results for some practical query languages.\n\nFormally, a relabeling on a -tree\u00a0 is a pair of a node and a label . To apply it, we change the label of\u00a0 by adding if , and removing it if . In other words, the tree\u00a0 never changes, and updates only modify\u00a0. Our main result is then:\n\nFor any fixed tree alphabet and MSO query on -trees, given a -tree , we can enumerate the output of\u00a0 on\u00a0 with linear-time preprocessing, linear delay and memory, and logarithmic update time for relabelings.\n\nProof.\n\nSee Appendix\u00a07.2 for the proof of this result. \u220e\n\nIn other words, after preprocessing in time\u00a0 to compute the index\u00a0, we can:\n\n\u2022 Enumerate the assignments of\u00a0 on\u00a0, using\u00a0, with delay linear in the size of each assignment, so constant if the assignments to have constant size.\n\n\u2022 Toggle a label of a node of\u00a0, update\u00a0, and reset the enumeration, in time .\n\nWe show this result in Sections\u00a047, and then give consequences of this result in Section\u00a08.\n\n4 Provenance Circuits\n\nOur general technique for enumeration follows our earlier work\u00a0[3]: from the query and input tree, we compute in linear time a structure called a provenance circuit to represent the results to enumerate, we observe that it falls in a restricted circuit class, and we conclude by showing a general enumeration result for circuits of this class. In this section, we review our construction of provenance circuits in\u00a0[3], with some additional observations that will be useful for updates. In particular, we show an independent balancing lemma on input trees, which allows us to bound a parameter of the circuit called dependency size. We will extend the formalism of this section to so-called hybrid circuits in the next section; and we will show our enumeration result for such circuits in Sections\u00a06 and\u00a07.\n\nSet circuits.\n\nWe start with some preliminaries about circuits. A circuit is a directed acyclic graph whose vertices are called gates, whose edges are called wires, where is the output gate, and where is a function giving a type to each gate of\u00a0 (the possible types depend on the kind of circuit). The inputs to a gate are and the fan-in of\u00a0 is its number of inputs .\n\nWe define set-valued circuits, which are an equivalent rephrasing of the circuits in zero-suppressed semantics used in\u00a0[3]. They can also be seen to be isomorphic to arithmetic circuits, and generalize factorized representations used in database theory\u00a0[27]. The type function of a set-valued circuit maps each gate to one of , , . We require that -gates have fan-in 0 or 2, and that -gates have fan-in\u00a00: the latter are called the variables of\u00a0, with denoting the set of variables. Each gate of\u00a0 captures a set of assignments, where each assignment is a subset of\u00a0. These sets are defined bottom-up as follows:\n\n\u2022 For a variable gate , we have .\n\n\u2022 For a -gate , we have . In particular, if then .\n\n\u2022 For a -gate with no inputs, we have .\n\n\u2022 For a -gate with two inputs and , we have , which we write (this is the relational product).\n\nThe set captured by\u00a0 is for the output gate of\u00a0. Note that each assignment of\u00a0 is a satisfying assignment of\u00a0 when seen in the usual semantics of monotone circuits.\n\nStructural requirements.\n\nBefore defining our provenance circuits, we introduce some structural restrictions that they will respect, and that will be useful for enumeration.\n\nThe first requirement is that the circuit is a d-DNNF. Our definition of d-DNNF is inspired by\u00a0[16] but applies to set-valued circuits, as in\u00a0[3] (see also the z-st-d-DNNFs of\u00a0[30]). For each gate of a set-valued circuit\u00a0, we define the domain of\u00a0 as the variable gates having a directed path to\u00a0. In particular, for\u00a0, we have , and if then . We now call a -gate decomposable if it has no inputs or if, letting be its two inputs, the domains and are disjoint. This ensures that no variable of\u00a0 occurs both in an assignment of\u00a0 and in an assignment of\u00a0. We call a -gate deterministic if, for any two inputs of\u00a0, the sets and are disjoint, i.e., there is no assignment that occurs in both sets. We call a d-DNNF if every -gate is decomposable and every -gate is deterministic. This assumption allows us, e.g., to tractably compute the cardinality of the set captured by\u00a0.\n\nThe second requirement on circuits is called upwards-determinism and was introduced in\u00a0[4]. In that paper, it was used to show an improved memory bound; in the present paper, we will always be able to enforce it. A wire in a set-valued circuit is called pure if:\n\n\u2022 is a -gate; or\n\n\u2022 is a -gate and, letting be the other input of\u00a0, we have , i.e., captures the empty assignment.\n\nWe say that a gate\u00a0 is upwards-deterministic if there is at most one gate such that is pure. We call upwards-deterministic if every gate of\u00a0 is.\n\nThe third requirement concerns the maximal fan-in of circuits, which is simply defined for a set-valued circuit\u00a0 as the maximal fan-in of a gate of\u00a0. We will require that the maximal fan-in is bounded by a constant.\n\nThe fourth and last requirement concerns a new parameter called dependency size. To introduce this, we define the dependent gates of a gate in a set-valued circuit as the gates such that there is a directed path from\u00a0 to\u00a0. Intuitively, the set captured by\u00a0 may then depend on the set captured by\u00a0. The dependency size of\u00a0 is , i.e., the maximal number of gates that are dependent on any given gate\u00a0. We will require this parameter to be connected to the height of the input tree.\n\nSet-valued provenance circuits.\n\nWe can now define provenance circuits like in\u00a0[3]. A set-valued circuit is a provenance circuit of a MSO query on a\u00a0-tree\u00a0 if:\n\n\u2022 The variables of\u00a0 correspond to the possible singletons, formally: ; and\n\n\u2022 The set of assignments captured by\u00a0 is the output of\u00a0 on\u00a0, formally: . Equivalently, for any tuple of subsets of\u00a0, we have iff the assignment is in\u00a0.\n\nConsider the unlabeled tree\u00a0 of Figure\u00a0fig:tree, the alphabet , and the MSO query with one free first-order variable asking for the leaf nodes whose -annotation is different from that of its parent (i.e., the node carries label and the parent does not, or vice-versa). Consider the labeling mapping to and and to\u00a0. A set-valued circuit capturing the provenance of\u00a0 on\u00a0 is given in Figure\u00a0fig:set.\n\nWe then know from\u00a0[4] that provenance circuits can be computed efficiently, and they can be made to respect our structural requirements:\n\n[(from [3], Theorem\u00a07.3)] For any fixed MSO query on\u00a0-trees, given a -tree , we can compute in time a set-valued provenance circuit of\u00a0 on\u00a0. Further, is a d-DNNF, it is upwards-deterministic, its maximal fan-in is constant, and its dependency size is in , where denotes the height of\u00a0.\n\nWe recall the main proof technique: we convert to a bottom-up deterministic tree automaton on -trees, and we add nodes to\u00a0 to describe the possible valuations of variables. The provenance circuit then captures the possible ways that can read\u00a0 depending on the valuation: we compute it with the construction of\u00a0[6], and is a d-DNNF thanks to automaton determinism (see\u00a0[2]). Upwards-determinism is shown like in\u00a0[4].\n\nThe bounds on fan-in and dependency size are not stated in\u00a0[3, 4] but already hold there. Specifically, the maximal fan-in is a function of the transition function of\u00a0, i.e., it does not depend on\u00a0. The bound on dependency size holds because is constructed following the structure of\u00a0: we create for each tree node a gadget whose size depends only on\u00a0, and we connect these gadgets precisely following the structure of\u00a0, so that for any gate of\u00a0 can only contain gates from the node\u00a0 of\u00a0 or from ancestors of\u00a0 in the tree.\n\nIn the context of updates, the bound of dependency size will be crucial: intuitively, it describes how many gates need to be updated when an update operation modifies a gate of the circuit. As this bound depends on the height of the input tree, we will conclude this section by a balancing lemma that ensures that this height can always be made logarithmic (which matches our desired update complexity). We will then add support for updates in the next section by extending circuits to hybrid circuits.\n\nIn this appendix, we prove Lemma\u00a04:\n\nBalancing lemma.\n\nOur balancing lemma is a general observation on MSO query evaluation on trees, and is in fact completely independent from provenance circuits. It essentially says that the input tree can be assumed to be balanced. Formally, we will show that we can rewrite any MSO query on -trees to an MSO query on a larger tree alphabet so that any input tree for\u00a0 can be rewritten in linear time to a balanced tree on which\u00a0 returns exactly the same output. Because we intend to support update operations, the input tree will be unlabeled, and the rewritten tree\u00a0 will work for any labeling of\u00a0. Formally:\n\nFor any tree alphabet and MSO query on\u00a0-trees, we can compute a tree alphabet and MSO query on\u00a0-trees such that the following holds. Given any unlabeled tree\u00a0 with node set , we can compute in linear time a -tree with node set , such that and such that, for any labeling function , we have , where maps to\u00a0 if and otherwise.\n\nWe prove Lemma\u00a04 by seeing the input tree\u00a0 as a relational structure\u00a0 of treewidth\u00a01, and invoking the result by Bodlaender\u00a0[13] to compute in linear time a constant-width tree decomposition of\u00a0 which is of logarithmic height. We then translate the query to a MSO query on tree encodings of this width, and compute from the tree encoding corresponding to the tree decomposition (we rename some nodes of\u00a0 to ensure that the nodes of\u00a0 are reflected in\u00a0). Note that the balanced tree decompositions of\u00a0[13] were already used for similar purposes elsewhere, e.g., in\u00a0[19], end of Section\u00a02.3.\n\nTo prove Lemma\u00a04, we will need to introduce preliminaries about relational instances\u00a0[abiteboul1995foundations], tree decompositions, and tree encodings.\n\nInstances.\n\nA relational signature is a set of relation names together with an associated arity (a non-zero natural number). We fix a relational signature that codes unlabeled trees, consisting of two binary relations and indicating the first and second child of each internal node. For any tree alphabet , we let denote a signature to represent labels of\u00a0, i.e., one unary relation for each . Last, for a tuple of second-order variables, we let denote a signature to represent the interpretation of these variables, i.e., one unary relation for each . By monadic second-order logic (MSO) over , we denote MSO with the relations of\u00a0 and equality in the usual way.\n\nA relational instance of a relational signature is a set of -facts of the form where are elements, is a relation in\u00a0, and is the arity of\u00a0. The domain of\u00a0 is the set of elements that occur in\u00a0.\n\nGiven a -tree , we can easily compute in linear time a couple where is a -instance describing the unlabeled tree\u00a0 in the expected way (in particular, is exactly the set of nodes of\u00a0), and is the -instance .\n\nTree decompositions.\n\nA tree decomposition of an undirected graph is a tree (whose nodes are called bags) and a labeling function such that:\n\n\u2022 For every , there is such that\n\n\u2022 For every , the set is a connected subtree of\u00a0.\n\nWe still assume for convenience that tree decompositions are rooted, ordered, binary, and full trees. Specifically, they will be computed as rooted binary trees by [13], they can be made full without loss of generality (in linear time and without impacting the height) by adding empty bags, and we can add an arbitrary order on the children of each internal bag to make them ordered. The width of\u00a0 is , and the treewidth of\u00a0 is the smallest width of a tree decomposition of\u00a0.\n\nA tree decomposition of a relational instance is a tree decomposition of its Gaifman graph, i.e., the graph on vertex set where there is an edge between any two elements that occur together in some fact. The treewidth of\u00a0 is that of its Gaifman graph.\n\nThe definition of tree decompositions ensures that, for any relational instance and tree decomposition , for any , we can talk of the topmost bag of\u00a0 such that ; we write this bag . This mapping can be computed explicitly in linear time given and by\u00a0[flum2002query, Lemma\u00a03.1].\n\nWe will make a standard assumption on our tree decompositions, namely, that the function is an injective function: in other words, the root bag contains only one element, and for any non-root bag with parent bag , we have . This requirement can be enforced on a tree decomposition in linear time using standard techniques, without impacting the width of\u00a0, and only multiplying the height of\u00a0 by a constant (assuming that the width is constant): specifically, we replace each bag violating the condition by a chain of bags where the new elements are introduced one after the other. Hence, we will always make this assumption.\n\nWe now recall the result of Bodlaender\u00a0[13], which is the key to our construction:\n\n[from [13]] For any relational signature , given a relational instance\u00a0 on\u00a0 of width\u00a0, we can compute in linear time in\u00a0 a tree decomposition of\u00a0 of width\u00a0, such that is in\u00a0.\n\nSpecifically, the algorithm of\u00a0[13] is described for a parallel machine, but can be run sequentially in linear time, as explained in\u00a0[19], end of Section\u00a02.3.\n\nTree encodings.\n\nIf we fix a relational signature and a treewidth bound , we can compute an alphabet , called the alphabet of tree encodings for\u00a0 and\u00a0, which ensures the following: given any -instance with a tree decomposition of width\u00a0, we can translate and in linear time to a -tree (called a tree encoding of\u00a0) that can be decoded back in linear time to an instance isomorphic to\u00a0. What is more, Boolean MSO formulas on -instances (i.e., MSO formulas without free variables) can be translated to Boolean MSO formulas on -trees that are equivalent through encoding and decoding. An example of such a scheme is given in\u00a0[flum2002query]; we will use a different scheme, detailed in\u00a0[2], which ensures a property dubbed subinstance-compatibility: intuitively, removing a fact from\u00a0 amounts to toggling labels on a node of the tree encoding that corresponds to\u00a0 (without changing the skeleton of the tree encoding). The labels of intuitively consist of a pair comprising a domain, i.e., a subset of elements among fixed element names, and an optional fact on the elements of the domain. We omit the formal definition of ; see Section\u00a03.2.1 of\u00a0[2] for details.\n\nWe are now ready to conclude the proof of Lemma\u00a04:\n\nProof of Lemma\u00a04.\n\nLet be the input query on -trees. Let . We let be the Boolean MSO query on -instances obtained from in the expected way, making it Boolean by replacing each second-order variable with the unary relation of\u00a0. Given an input tree\u00a0, we compute in linear time the -instance which represents it. It is clear that, given a labeling , recalling our earlier definition of the -instance \u00a0 from\u00a0, the output of\u00a0 on\u00a0 is equal to the set of -instances of -facts on\u00a0 (seeing each such instance as a set of singletons of the form ) such that satisfies .\n\nLet be the width of the tree decomposition obtained when applying Theorem\u00a04 to an input tree decomposition of width\u00a0 (note that we have not specified the input yet). Let us compute from\u00a0 the Boolean MSO query on the alphabet\u00a0 of tree encodings for width\u00a0 which is equivalent to\u00a0 on -instances (up to encoding and decoding), i.e., an instance on\u00a0 satisfies iff its encoding as a -tree satisfies . We take to consist of\u00a0 plus a special label , to be used later.\n\nNow, as\u00a0 is a tree, the treewidth of\u00a0 is\u00a0. Let us define an instance by adding to\u00a0 the instance of all possible -facts on , plus the instance of all possible -facts on\u00a0. As all these additional facts are unary, the instance still has treewidth\u00a0. Hence, by Theorem\u00a04, we can compute in linear time in\u00a0 a tree decomposition of\u00a0 of treewidth\u00a0 and logarithmic height. We also compute in linear time the mapping , and a tree encoding of\u00a0, i.e., a -tree.\n\nThanks to subinstance-compatibility, we know that, for any labeling and answer tuple of subsets of\u00a0, letting and be the - and -instances that respectively denote it, then we can obtain a tree encoding of by toggling the labels of some nodes of\u00a0. Specifically, each fact of corresponds to one node of\u00a0 whose label has to be changed; further, this mapping can be computed in linear time (see\u00a0[2], Lemma\u00a03.2.6).\n\nThe last thing to argue is that we can rename the nodes of\u00a0 so that they correspond to the nodes of\u00a0 associated to them, ensuring that, given a labeling function of the tree\u00a0, we can use it to relabel\u00a0. (This differs slightly from the original construction of\u00a0[2], because we want each node of\u00a0 to be associated to one single node in\u00a0, carrying all possible variables and labels; by contrast, in the construction of\u00a0[2], every fact corresponds to a specific node of\u00a0.) To fix this, we modify in linear time to another -tree : for each , letting , we replace by a gadget with two copies and of , with being the left child of\u00a0. The label of\u00a0 is that of\u00a0, and the label of\u00a0 is made of the same domain as\u00a0 but without any fact; see the exact definition of in Section\u00a03.2.1 of\u00a0[2] for details. We then add a right child to\u00a0 which is a new node\u00a0 identified to the element\u00a0 in\u00a0, which itself corresponds to the node in\u00a0; the label of\u00a0 is the fixed special label\u00a0. This construction is well-defined because the function is injective. We must now argue that the query can be modified (independently from\u00a0) to a MSO query on -trees to read labels and variable assignments from these new nodes: specifically, instead of reading (the encodings of) the -facts about (the encoding of) an element , the query should read the label in\u00a0 of the new node of\u00a0 identified with\u00a0; likewise, instead of reading (the encodings of) the -facts on an element directly from\u00a0, the query should read the -annotation of this same new node in\u00a0 identified with\u00a0. To do this, the translations of the atoms from and in\u00a0 are replaced in\u00a0 by a gadget which finds the bag where the corresponding element was introduced (i.e., the one for which it is in the image of\u00a0), finds the new node that we added with label\u00a0, and reads the label and annotation of this node. We also add a conjunct to\u00a0 to assert that the only nodes that can be part of the interpretation of the\u00a0 are the new nodes in\u00a0 with label\u00a0, thus ensuring that the set of answers of\u00a0 on any labeling of\u00a0 is correct. This concludes the proof. \u220e\n\n5 Hybrid Circuits for Updates\n\nIn this section, we extend set-valued circuits to support updates, defining hybrid circuits. We then extend Theorem\u00a04 for these circuits. Last, we introduce a new structural notion of homogenization of hybrid circuits and show how to enforce it. We close the section by stating our main enumeration result on hybrid circuits, which implies our main theorem (Theorem\u00a03), and is proved in the two next sections.\n\nHybrid circuits.\n\nA hybrid circuit is intuitively similar to a set-valued circuit, but it additionally has Boolean variables (which can be toggled when updating), Boolean gates (, , ), and gates labeled\u00a0 which keep or discard a set of assignments depending on a Boolean value. Formally, a hybrid circuit is a circuit where the possible gate types are (set-valued variables), (Boolean variables), , , , , , and . We call a gate Boolean if its type is , , , or\u00a0; and set-valued otherwise. We require that the output gate is set-valued and that the following conditions hold:\n\n\u2022 -gates and -gates have fan-in exactly 0;\n\n\u2022 All inputs to -gates, -gates, and -gates are Boolean, and -gates have fan-in exactly\u00a0;\n\n\u2022 All inputs to and -gates are set-valued, and -gates have fan-in either 0 or 2;\n\n\u2022 -gates have one set-valued input and one Boolean input (so they have fan-in exactly\u00a02).\n\nWe write to denote the gates of\u00a0 of type , called the Boolean variables, and define likewise the set-valued variables . An example hybrid circuit is illustrated in Figure\u00a0fig:hybrid.\n\nUnlike set-valued circuits, which capture only one set of assignments, hybrid circuits capture several different sets of assignments, depending on the value of the Boolean variables (intuitively corresponding to the tree labels). This value is given by a valuation of\u00a0, i.e., a function . Given such a valuation\u00a0, each Boolean gate captures a Boolean value , computed bottom-up in the usual way: we set for , and otherwise is the result of the Boolean operation given by the type of\u00a0, applied to the Boolean values captured by the inputs of\u00a0 (in particular, a -gate with no inputs always has value\u00a0, and a -gate with no inputs always has value\u00a0).\n\nWe then define the evaluation of\u00a0 under\u00a0 as the set-valued circuit obtained as follows. First, replace each Boolean gate of\u00a0 by a -gate with no inputs (capturing\u00a0) if , and by a -gate with no inputs (capturing\u00a0) if . Second, relabel each -gate of\u00a0 to be a -gate. Using , for each set-valued gate\u00a0 of\u00a0, we define the set captured by\u00a0 under\u00a0: it is the set of assignments (subsets of\u00a0) that captures in\u00a0. The set captured by\u00a0 under\u00a0 is then\u00a0, for\u00a0 the output gate of\u00a0.\n\nWe last lift the structural definitions from set-valued circuits to hybrid circuits. The maximal fan-in and dependency size of a hybrid circuit are defined like before (these definitions do not depend on the kind of circuit). A hybrid circuit is a d-DNNF, resp.\u00a0is upwards-deterministic, if for every valuation of\u00a0, the set-valued circuit has the same property. For instance, the hybrid circuit in Figure\u00a0fig:hybrid is upwards-deterministic and is a d-DNNF.\n\nHybrid provenance circuits.\n\nWe can now use hybrid circuits to define provenance with support for updates. The set-valued variables of the circuit will correspond to singletons as before, describing the interpretation of the free variables of the query; and the Boolean variables stand for a different kind of singletons, describing which labels are carried by each node. To describe this formally, we will consider an unlabeled tree\u00a0, and define a labeling assignment of\u00a0 for a tree alphabet\u00a0 as a set of singletons of the form where and . Given a labeling assignment , we can define a labeling function for\u00a0, which maps each node to . Now, we say that a hybrid circuit\u00a0 is a provenance circuit of a MSO query on an unlabeled tree\u00a0 if:\n\n\u2022 The set-valued variables of\u00a0 correspond to the possible singletons in an assignment, formally ;\n\n\u2022 The Boolean variables of\u00a0 correspond to the possible singletons in a labeling assignment, formally ;\n\n\u2022 For any labeling assignment , let be the Boolean valuation of\u00a0 mapping each to\u00a0 or\u00a0 depending on whether or not, and let be the labeling function on\u00a0 defined as above. Then we require that the set of assignments captured by\u00a0 under\u00a0 is exactly the output of\u00a0 on\u00a0, formally, .\n\nIn other words, for each labeling of the tree\u00a0, considering the valuation that sets the Boolean variables of\u00a0 accordingly, then is a provenance circuit for\u00a0 on\u00a0.\n\nRecall the query and alphabet of Example\u00a04, and the tree\u00a0 of Figure\u00a0fig:tree. A hybrid circuit capturing the provenance of\u00a0 on\u00a0 is given in Figure\u00a0fig:hybrid (with variable gates being drawn at multiple places for legibility): square leaves correspond to Boolean variables testing node labels, and circle leaves correspond to set-valued variables capturing a singleton of the form for some . In particular, for the labeling\u00a0 of Example\u00a04, the corresponding valuation maps to\u00a0 and and to\u00a0, and the evaluation of\u00a0 under\u00a0 captures the same set as the circuit of Figure\u00a0fig:set.\n\nWe can now extend Theorem\u00a04 to compute a hybrid provenance circuit as follows:\n\n5.1 Proof of the Provenance Circuit Theorem\n\nIn this appendix, we prove Theorem\u00a05.1:\n\nFor any fixed MSO query on -trees, given an unlabeled tree , we can compute in time a hybrid provenance circuit which is a d-DNNF, is upwards-deterministic, has constant maximal fan-in, and has dependency size in .\n\nThe proof is analogous to that of Theorem\u00a04. The only difference is that the automaton now reads the label of each node as if it were a variable, so that the provenance circuit also reflects these label choices as Boolean variables.\n\nThe general idea is that, given the MSO query on -trees, writing , we define a query on unlabeled trees, where , with one second-order variable corresponding to each . The construction is simply that we replace each unary predicate in\u00a0 by the corresponding second-order variable . It is now obvious that, for any labeled tree , defining for each , for any set of subsets of\u00a0, we have iff . In other words, we have simply turned node labels into second-order variables.\n\nNow, at a high level, we can simply construct a provenance circuit of\u00a0 on\u00a0 in the sense of Theorem\u00a04, replace the input gates corresponding to the variables by a Boolean input gate, and observe that the desired properties hold. We will now give a self-contained proof of the construction, to make sure that we reflect the changes in definitions between the present work and\u00a0[3, 4].\n\nTree automata.\n\nWe will need to introduce some prerequisites about tree automata. Given a tree alphabet , a bottom-up deterministic tree automaton on\u00a0, or -bDTA, is a tuple where is a finite set of states, are the final states, is the initial function, and is the transition function. The run of a -bDTA\u00a0 on a -tree is the function defined inductively as when is a leaf, and when is an internal node with children and . We say that accepts the tree\u00a0 if the run of\u00a0 on\u00a0 maps the root of\u00a0 to a final state.\n\nWe will be interested in bDTAs to capture our non-Boolean query on unlabeled trees. Let be the set of variables, and let , where denotes the powerset of\u00a0. Letting be an unlabeled tree, we call a -annotation of\u00a0 a function : the annotation intuitively describes the interpretation of the variables of\u00a0 by annotating each node with the set of variables to which it belongs. Letting be a -bDTA, be an unlabeled tree, and be a -annotation of\u00a0, we say that is a satisfying annotation of\u00a0 on\u00a0 if accepts . In this case, we see as defining an assignment , which is the set . The output of on\u00a0, written , is the set of assignments corresponding to its satisfying annotations. Following Thatcher and Wright\u00a0[thatcher1968generalized], and determinizing the automaton using standard techniques [tata], the output of an MSO query (here, on an unlabeled tree) can be computed as the output of an automata for that query. Formally:\n\n[[thatcher1968generalized, tata]] Given a MSO query on unlabeled trees, we can compute a -bDTA such that, for any unlabeled tree , we have .\n\nRestricting to Boolean annotations.\n\nIt will be more convenient in the sequel to assume that each tree node carries one single Boolean annotation rather than many, and to distinguish the annotations corresponding to\u00a0 (the original variables of\u00a0, called enumerable), and those corresponding to\u00a0 (the labels of the input tree, called updatable). We will do this by creating -copies of each tree node\u00a0, to stand for each separate singleton . To do this, we will consider the fixed alphabet . Intuitively, will be the label of nodes whose annotation corresponds to a variable of\u00a0, will be the label of nodes whose annotation corresponds to a variable of\u00a0, and will be the label of nodes whose annotation does not code any variable and should be ignored. Given a -tree , we will write , , and to refer to the set of nodes carrying each label. We will then consider -trees, where , the alphabet of -trees annotated with a Boolean value at each node: as promised, each node carries one single value. Now, a Boolean annotation of a -tree is a function , and we see as a -tree defined in the expected way.\n\nWe want to rephrase the evaluation of\u00a0 on an unlabeled tree\u00a0 to a problem on -trees, where variable valuations are coded in Boolean annotations. This process is formalized in the following lemma, whose construction is illustrated in Figure\u00a01; it is analogous to Lemma\u00a0E.2 of\u00a0[3]:\n\nFor any variable set\u00a0, given a -bDTA , we can compute a -bDTA such that the following holds: given an unlabeled tree , we can compute in linear time a tree of height and an injective function such that:\n\n\u2022 is exactly the set of nodes such that for some and ;\n\n\u2022 is exactly the set of nodes such that for some and ;\n\n\u2022 is exactly the set of nodes not in the image of\u00a0, and it includes all internal nodes.\n\nFurther, for any -annotation , let be the Boolean valuation of\u00a0 defined by:\n\n\u2022 If is in the image of\u00a0, then letting , we set iff ;\n\n\u2022 If is not in the image of\u00a0, we set .\n\nThen accepts iff accepts .\n\nProof.\n\nGiven an input tree , we change it following the idea of Figure\u00a01: we replace each node by a gadget of nodes labeled with , having two subtrees: one whose leaves are labeled and code the variables in order, and another whose leaves are labeled and code the variables in order. This gadget can be completed to a full binary tree by adding leaves labeled as necessary. Now we can clearly rewrite the -bDTA to a -bDTA which is equivalent in the sense required by the lemma. The states of consist of the states of\u00a0, the pairs of states of\u00a0, and annotation states which consist of binary sequences of length up to\u00a0. The final states are the final states of\u00a0. The initial function and transition function are informally coded as follows. The initial function maps nodes labeled or for to the singleton binary sequence formed of its Boolean value, and it maps nodes labeled for to the empty binary sequence. The transition function is defined only on nodes labeled for , because all internal nodes of\u00a0 carry such a label (as required); and it is defined as follows (where we ignore the Boolean annotation\u00a0 of the node):\n\n\u2022 Given two states and of\u00a0, the new state is the pair ;\n\n\u2022 Given two states that are binary sequences of length , the new state is their concatenation;\n\n\u2022 Given a binary sequence of length and a pair of states , the new state is the state of\u00a0, where is the transition function of\u00a0;\n\n\u2022 Given a binary sequence of length and an empty binary sequence, the new state is the state .\n\nOn Figure\u00a01, the automaton\u00a0 would reach state on\u00a0, reach state on\u00a0 and reach state on\u00a0. Letting and be the states that reaches respectively on\u00a0 and\u00a0, it reaches state on\u00a0. Hence, on node\u00a0, it reaches . This figure illustrates the translation when\u00a0 is an internal node with children\u00a0 and\u00a0. The case where\u00a0 is a leaf is described in the last bullet point, and is analogous: the leaf in\u00a0 is translated to a node in\u00a0 with one left child that is the root of the tree describing the valuation of\u00a0, and one right child labeled which is a leaf of\u00a0.\n\nNow, it is easy to show that is equivalent to\u00a0 in the sense of the lemma statement, which concludes the proof. \u220e\n\nWe now have a -bDTA\u00a0 to run on a -tree . We can now rephrase our desired provenance result as a provenance result on such automata. We say that a hybrid circuit is a provenance circuit of a -bDTA on a -tree if:\n\n\u2022 The set-valued variables of\u00a0 correspond to the nodes of\u00a0 with label , formally,\n\n\u2022 The Boolean variables of\u00a0 correspond to the nodes of\u00a0 with label , formally,\n\n\u2022 For any Boolean valuation of\u00a0 such that for each , the automaton accepts iff, letting be the restriction of\u00a0 to\u00a0, and letting be the set of nodes of\u00a0 corresponding to the restriction of\u00a0 to\u00a0, we have .\n\nWe can now rephrase our desired result. Note that the statement of this result implies that our construction is also tractable in the automaton, as we mentioned in the conclusion (Section\u00a09):\n\nGiven a -bDTA and a -tree where all internal nodes are labeled\u00a0, we can compute in time a hybrid circuit which","date":"2022-08-12 23:22:47","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8117566108703613, \"perplexity\": 1158.1600315572884}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882571847.45\/warc\/CC-MAIN-20220812230927-20220813020927-00085.warc.gz\"}"}
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Q: Difference in List I want to write a function which gives difference between lists.
For example,
list1 = ['a','a','a','b','c','d','d']
list2 = ['a','a','a','b','c','a','d']
list = diff(list1, list2)
print(list)
list should be ['d','a'], the elements which don't match in both.
I thought of using set but it won't work as it'll eliminate repeating characters.
A: I both lists have the same length, you can use zip:
res = []
for x, y in zip(list1, list2):
if x != y:
res.extend((x, y))
print(res)
Output:
['d', 'a']
Or as one-liner:
>>> [z for x, y in zip(list1, list2) if x != y for z in (x, y)]
['d', 'a']
If both lists have different lengths and you want this counted as a difference, you can use zip_longest:
from itertools import zip_longest
list1 = ['a','a','a','b','c','d','d', 'x']
list2 = ['a','a','a','b','c','a','d']
res = []
for x, y in zip_longest(list1, list2):
if x != y:
res.extend((x, y))
print(res)
Output:
['d', 'a', 'x', None]
Again as one-liner
>>> [z for x, y in zip_longest(list1, list2) if x != y for z in (x, y)]
['d', 'a', 'x', None]
A: Assuming same length of both lists
def diff(l,m):
ans = []
for i in range(len(l)):
if l[i]!=m[i]:
ans.append(l[i])
ans.append(m[i])
return ans
What I have done in the code is that I am iterating over each element of the lists and checking them for equality. If they are same then nothing happens, loop goes to its next iteration and if they are not same both the elements are appended to the ans list.
A: You can use a one-line list-comprehension:
[e for i, j in zip(list1, list2) if i != j for e in (i, j)]
which gives:
['d', 'a']
why?
Well first we need to understand what the zip() function does, and this can be seen through an example:
>>> list(zip([1,2], [3,4]))
[(1, 3), (2, 4)]
as you can see, it simply "zips" the two lists together (note that zip works on any iterable, not just lists).
We then want to iterate through each of these tuples and unpack them so that we can work with both elements.
Next, we check whether the two elements are not the same with i != j. This means that we are now only working with variables i an j if they are different.
Finally, as your expected output wants a flattened list, we need to use another loop to add i and j as separate elements to out whole list that we are creating.
Note that this relies on the two lists being the same length as zip() stops at the end of the shortest one.
A: Filtering pairwise with itertools and zip() function:
import itertools
list1 = ['a','a','a','b','c','d','d']
list2 = ['a','a','a','b','c','a','d']
result = list(itertools.chain.from_iterable(
itertools.filterfalse(lambda x: x[0] == x[1], zip(list1, list2))))
print(result)
The output:
['d', 'a']
|
{
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{"url":"https:\/\/puzzling.stackexchange.com\/questions\/91818\/witness-intimidation","text":"Witness intimidation\n\nAfter solving this puzzle, I received an envelope in the mail. I opened the envelope, only to find three letters inside. The first one read:\n\nI know who you are\nI'm sending my greeting\nInvestigate further\nAnd your lives will be fleeting\n\nAstonished at the blatant threat, I quickly scanned over the second letter:\n\nUnfortunately, I couldn't make heads or tails out of it, until I took a look at the third letter:\n\nAfter a brief moment, I assembled a message and was instantly horrified!\n\nWhat greeting did the letters contain?\n\nHint:\n\n\u2022 One can grow itself\n\u2022 One's namesake for constant\n\u2022 One's place for logo\n\u2022 One's connective\n\u2022 One describes an individual\n\u2022 One's noisy\n\u2022 One follows Nice for a meme? (4 4)\n\u2022 One's a gate\n\u2022 One may be gassy\n\u2022 One's to end\n\u2022 One's a tree?\n\u2022 One tells time\n\u2022 One's expression\n\u2022 One.\n\u2022 One's what melted\n\u2022 One's holey tool\n\u2022 One Direction?\n\nHint #2:\n\nEach picture helped to make heads or tails of the corresponding word\n\nHint #3 (strong hint):\n\nPlacing words on the grid (top to bottom or left to right), the images anchor either the beginning or ending of the corresponding word.\n\nHint #4:\n\nSome numbers are red herrings\n\n\u2022 I haven't figured too much out, but I have a guess that the top-right symbol's semicircle should be flipped - is that correct?\n\u2013\u00a0Deusovi\nDec 7 '19 at 10:34\n\u2022 @Deusovi Sorry, that\u2019s incorrect.\n\u2013\u00a0Avi\nDec 7 '19 at 18:42\n\nThe main puzzle mechanism has already been worked out thanks to @LukeBickell and @MOehm - go upvote their answers (@MOehm - feel free to add this into your solution so we have a complete one; @Avi - give the checkmark to that one!). All that remains is the final message, which I believe is:\n\nHELLO PUZZLERS!\n\nHow?\n\nLook only at the words relating to the symbols explicitly mentioned in the third letter received by the OP (ignore all the others - these are red herrings). For each of these, note the numbers written in the corners of their symbol in the first grid and examine the corresponding index position(s) in the word to get:\n\npERSon\nbUZZer\nchipotLe\npLanck\npEriod\nsmiLe\nPot\ntHaw\ndOwn\n\nAltogether this gives us: ERSUZZLLELPHO. Finally, note the thus-far-unused tag. This letter sequence anagrams to 'HELLO PUZZLERS!' Your mysterious correspondent appears to know who we are!\n\n\u2022 @Avi LukeBickell and MOehm did all the heavy lifting here - give the checkmark to one of them! :)\n\u2013\u00a0Stiv\nDec 21 '19 at 7:53\n\u2022 This is the correct answer, so it gets the check-mark. I will be awarding bounties to LukeBickell and MOehm for their heavy lifting (first one started).\n\u2013\u00a0Avi\nDec 21 '19 at 16:18\n\nPartial answer, building on Luke Bickell's.\n\nThe things in the grid are:\n\nOne can grow itself POT\u00b9\nOne's namesake for constant PLANCK\nOne's place for logo CHIPOTLE\u00b9\nOne's connective OR\nOne describes an individual PERSON\u00b9\nOne's noisy BUZZER\u00b2\nOne follows Nice for a meme BOAT\nOne's a gate NOR\nOne may be gassy STAR\nOne's to end ABOLISH\nOne's a tree? OAK\u00b9\nOne tells time ASTROLABE\u00b2\nOne's expression SMILE\nOne. PERIOD\u00b9\nOne's what melted THAW\nOne's holey tool AWL\nOne Direction? DOWN\n\n\nAnswers marked (1) were found by Luke Bickell; answers marked (2) were found by Stiv. (I had found the other answers even if they appear in Luke's answer, but I was wrong about the pot and the period and would never have got Chipotle.)\n\nAnd the grid looks like this:\n\n \u00b7 \u00b7 P e r s o N \u00b7 B\nA w L \u00b7 \u00b7 \u00b7 \u00b7 o \u00b7 u\ns \u00b7 a \u00b7 S t A r \u00b7 z\nt \u00b7 n \u00b7 m \u00b7 b \u00b7 \u00b7 z\nr \u00b7 c h i p o t l E\nO a K \u00b7 l \u00b7 l \u00b7 \u00b7 r\nl \u00b7 \u00b7 P E r i o d \u00b7\na \u00b7 \u00b7 o \u00b7 \u00b7 s \u00b7 o R\nB o a T \u00b7 T h a w \u00b7\ne \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 N \u00b7\n\n\nHere, the words are placed in crossword-style, so that they cross, but don't run into each other when going in the same direction. The first or last letter of a word coincides with the respective symbol in the grid. These \"anchors\" are written in capital letters in the grid above.\n\nAnd now?\n\nI don't know. There are nine symols on the third sheet and the numbers in the grid go from one to nine. What do the positions of the numbers mean? The numbers cannot be letter position, because in some places, these positions exceed the length of the answers. But I note that the first three things from the third sheet are Person, bUZZer and chipotLE, which could conveniently spell something, in the case of buzzer even with the letters indicated by the numbers.\n\n\u2022 Nearly there - rot13(NOBYVFU vf pbeerpg, GUNJ vf pbeerpg, NHQVB vf vapbeerpg, naq GNOYR vf vapbeerpg. Gur zvffvat barf ner n 9-punenpgre jbeq, naq na ryrpgebavp flzoby.)\n\u2013\u00a0Avi\nDec 19 '19 at 21:48\n\u2022 \"What do the positions of the numbers mean? Hm.\" Precisely.\n\u2013\u00a0Avi\nDec 19 '19 at 22:11\n\u2022 I believe your 'AUDIO' error should be rot13(NFGEBYNOR)...\n\u2013\u00a0Stiv\nDec 19 '19 at 23:05\n\u2022 ...and the other is a rot13(OHMMRE). The clues you attributed to the pair of them need to be switched about.\n\u2013\u00a0Stiv\nDec 19 '19 at 23:11\n\u2022 Stiv is correct. Autocorrect to Stick, is not correct :(\n\u2013\u00a0Avi\nDec 20 '19 at 0:15\n\nWhile I have no idea how to use the grid and numbers, I at least have a number of guesses for the words, which may help someone else.\n\n\u2022 One can grow itself --> POT (you can grow pot (marijuana) in a pot)\n\u2022 One's namesake for constant --> PLANCK'S constant (is represented by a cursive h)\n\u2022 One's place for logo --> CHIPOTLE PEPPER (is the logo for Chipotle restaurant)\n\u2022 One's connective --> ?\n\u2022 One describes an individual --> PERSON ?\n\u2022 One's noisy --> ?\n\u2022 One follows Nice for a meme? (4 4) --> ?\n\u2022 One's a gate --> NOR (a logic gate)\n\u2022 One may be gassy --> STAR (are mostly comprised of gas)\n\u2022 One's to end --> CANCEL STOP (common symbol)\n\u2022 One's a tree? --> (PROFESSOR) OAK (8-bit character from Pokemon)\n\u2022 One tells time --> CLOCK\/WATCH\n\u2022 One's expression --> SMILE\n\u2022 One. --> PERIOD\n\u2022 One's what melted --> WATER (ice to water) ICE (water doesn't melt, ice does)\n\u2022 One's holey tool --> AWL (hole making tool)\n\u2022 One Direction? --> DOWN (arrow pointing down)\nThe remaining three I don't have guesses for are the boat, the bowl\/semicircle, and the V.\n\n\u2022 rot13(PNAPRY, JNGRE, naq PYBPX\/JNGPU) are incorrect, but everything else is correct. You may want to start placing things on the grid - it may help you deduce more words. 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In May 1854, Dr. Levi Leslie Lamborn founded and published Alliance's first newspaper. The first issues were printed in Salem. Dr. Lamborn sold the business to Horatio N. Lewis several months later. This issue is Vol. I—No. 30. The paper was printed weekly and sold for a $1.50 per year.
# Alliance
# Craig Bara
# Lyle Crist
Copyright © 1998 by Craig Bara and Lyle Crist
9781439621691
Published by Arcadia Publishing
Charleston SC, Chicago IL, Portsmouth NH, San Francisco CA
Printed in the United States of America
Library of Congress Catalog Card Number: 2008931212
For all general information contact Arcadia Publishing at:
Telephone 843-853-2070
Fax 843-853-0044
E-mail sales@arcadiapublishing.com
For customer service and orders:
Toll-Free 1-888-313-2665
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The men of the Alliance Fire Department show off the city's new 1909 Robinson Gas-Propelled Firetruck in front of Station No. 3 in Mount Union. The first of its kind in Ohio, this truck was the sensation of Stark County.
This oval postcard view shows Public Square between 1910 and 1920 looking south down Freedom Avenue. The buildings, from left to right are: Allott-Kryder's Hardware Store, First National Bank, First Methodist Church down the road on Broadway, First Baptist Church on Market Street directly behind the old city hall building, which was soon demolished, C.Y. Kay Hardware Store, and the Old City Savings Bank.
# Table of Contents
Title Page
Copyright Page
ACKNOWLEDGMENTS
INTRODUCTION
One \- BEGINNINGS
Two \- RAILROADS AND INDUSTRY
Three \- LEARNING
Four \- CONTINUING DEVELOPMENTS
Five \- ON THE MAIN
Six \- LOCAL COMMERCE
Seven \- HOMES, HAPPENINGS
Eight \- EVENTS FOR MANY
Nine \- CITY GOVERNMENT, SERVICES
Ten \- VICINITY VIEWS
# ACKNOWLEDGMENTS
We wish to express our appreciation to the many individuals, groups, and organizations whose photographic resources have been invaluable in the preparation of this book. First of all, we are grateful to the Alliance Historical Society's Board of Trustees for making available to us their extensive collection of photographic history. This collection has been built up over the years thanks to the generosity of countless individuals who have donated important photographs and written material to the Society. Because of this forethought, a wider audience will now enjoy their pictures. Secondly, we wish to thank the individuals who have loaned us photographs for this project and also to the institutions which have made their resources available to us: Joseph Grabiel, Martha Ingold, Betty Donaldson, Susan Steen, James Morris, Mr. and Mrs. LeRoy Zang, Todd and Jennifer Mastroianni, Veetta Terrell, The Reverend Ellen Acton, Russell Newburn, Mrs. R.W. Cordingly, Phillip S. Gehm, Allan L. Krash, Mr. and Mrs. Joseph Bara, Mount Union College, and the Rodman Public Library and its Reference Department staff for access to the Alliance Room and its supplies. Finally, we would like to express our deep appreciation to those who are deceased, but whose efforts in other years inspired us with their vivid accounts of the Alliance they knew and loved so well. Their recollections were of great value in putting together this book: Mary Thone, Dr. Helen (Barth) Cleveland, Bertha Shively, Harold Vogus, Ernest Valentine, Bill Mainwaring, Mr. and Mrs. Charles K. Strain, and Max "Meether."
# INTRODUCTION
It began with Williamsport, Liberty, and Freedom. These three small communities eventually grew together and became incorporated as the town, Alliance, in 1854. In 1889, by mutual agreement, the Village of Mount Union was annexed into Alliance.
The two major railroads in the United States crossed at this location, and the railroaders referred to this junction as "The Crossing." A railroad official did not feel that "The Crossing" was an appropriate name, so he suggested "Alliance," since it was the alliance of the two railroads. The name stuck, and the town became known as Alliance. It has been a city of industry, of nearby farmland, of full-term education, and of fine leadership over the years.
There were many individuals who assumed leadership in the development of Alliance. This pictorial history is certainly not intended to provide all the meaningful names of leadership; some are included, but many other people have also been responsible for taking steps which furthered the development of the city. Many stores in town and a number of the industries, as well, carried the names of developers. The value of placing industries in the city stemmed from the availability of trains. Key lines intersected in Alliance and have continued to be significant to manufacturers.
There has not been any population explosion in Alliance with the exception of the first 20 years of this century, when industry attracted European emigrants and blacks to the community. That exception aside, the population of Alliance has grown in a quiet fashion over the years. In 1850, the population was estimated to be 200; in 1900—8,974; in 1920—21,603; in 1950—26,161; in 1970—26,547; up to the present-day population of 23,376.
Practically all of the photographs in this history have come from the files of the Alliance Historical Society. Over many years, studios and individuals have donated historic material to the Society. Photographs in this pictorial history are not always in a specific chronological order, but a general sequence is given. We used a basic series of categories. The rationale for using specific pictures of individuals was that these pictures often demonstrate the city's growth and change throughout its history. Far more photographs of individuals could have certainly been used, but pictures showing the developing areas of city activity were also essential. In this history, there are very few photographs of current scenery and personalities. The emphasis is on the past. However, most readers know the scenes and the figures pictured just through current experience in Alliance. It is our goal that the photographs here will help increase appreciation of the past and its connection to the future.
This is an artist's aerial view of Alliance in 1885
# One
# BEGINNINGS
This cabin is believed to be one of the first constructed in the area. It was built by John Grant, who purchased nearly 1,000 acres of land from the U.S. government in 1809. The structure was located in a ravine several hundred yards from the northwest corner of Main Street and Union Avenue. Mr. Grant was a great-uncle of Pres. Ulysses S. Grant. Mr. Grant's daughter Sarah married J. Ridgeway Haines, and their Colonial home, on the corner of Haines Street and West Market Street, was built of brick made by John Grant, who fired the first bricks in his kiln in Lexington Township
The Freedom Public Square Monument is located on the corner of North Park Avenue (originally called Main Street when Freedom was its own community) and Keystone Street. The original stone for the village well is still located on the lot next to the monument.
The Clem Rockhill Residence was built in 1820. Located on West Wayne Street, the home was built by Samuel Rockhill and named for his son Clement. It is one of Alliance's oldest homes.
This marker was placed by the Alliance Historical Society on the site of the Village of Lexington square and well. The town was founded in 1805 by Amos Holloway, who later became prominent in the abolitionist movement. Holloway was responsible for helping many blacks escape from their slave masters in the South to freedom in Canada.
The Freedom School was erected in 1838 in the Village of Freedom. Located at 705 North Freedom, the brick structure has been covered with siding for a number of years. It is the oldest remaining school building in Alliance.
Mathias Hester, founder of the Village of Freedom, is seen here with his brothers and sister in the late 1800s. Seated (from left to right) are Samuel, Mathias, and Martin. Standing are John and Eliza. Mathias Hester was born in 1793 in Green County, Pennsylvania. He moved to New Lisbon, Ohio (now Lisbon) in the early 1820s. He met a Salem girl, Susan Gaskill, married, and moved to the Village of Mount Union in 1834 where he established a grocery store on the southwest corner of the square.
The Hester Block, which was located on Hester Avenue near Freedom Triangle, was one of the oldest brick structures in Alliance. Built in 1838, the architectural style was Federal. The bricks were made in the clay pits which now form the beautiful lake in front of Glamorgan Castle. The bricks cost $2.50 per thousand and had to be hauled by ox and wagon. The Block was used as a residence for Hester and his family and also as a general store. The building was condemned in the 1980s and later destroyed. The north section still stands today.
The Samuel Shaffer residence, built in 1842, was located on Main Street (now North Park) in the Village of Freedom, on the square. Shaffer, a German immigrant, moved to this area from Pennsylvania to operate a general store. In 1851, the Village Council and railroad officials approached Shaffer notifying him of the possibility that the Cleveland and Wellsville Railroad might pass through the community. Shaffer offered his home to the community, making it the very first railroad station in what would soon become Alliance.
Popularly known as Freedom Tavern, this frame structure stood at the corner of Vine and Walnut Streets in the Village of Freedom. The tavern (also known as "American House") has a vital link to the opening of the great Western frontier. Because of its location, many people who traveled by Conestoga wagon spent the night here enjoying good food and rest. The house was built in the 1830s and was torn down in the mid-1930s.
Elisha Teeters (1814–1899) settled in Lexington Township in 1835. That same year, he married Eliza Webb and built a frame structure on the corner of Rockhill and Vine Streets. It still stands today, directly behind the Old Sebrell Farmhouse. Mr. Teeters, along with Mathias Hester, laid out much of Alliance. Teeters's son was instructed to take his horse and plow and make a path running from the railroad tracks to the edge of Teeters's farm. This dirt path became Main Street. After the death of his first wife, Mr. Teeters married Mathias Hester's daughter Sarah. He built her a fine High Victorian Italianate mansion at 323 West Vine Street. Mr. Teeters was involved in many business ventures, including banking with Dr. L.L. Lamborn and real estate.
This 1930s view shows the homes of Philip Sharer and the Foltz family on North Park Avenue (originally called Main Street). Both families were early settlers in Freedom. Mr. Sharer emigrated from Germany; his talents were in cabinet making, and he opened a store in the 1840s. In later years, his sons expanded the business to include undertaking. The houses were torn down in the 1930s.
# Two
# RAILROADS AND INDUSTRY
Col. Daniel Sourbeck, noted for his fine food, opened his first dining room in Alliance's railroad station. Constructed in the late 1850s, the station was known for its beautiful architectural design. Numerous notable events occurred at this station. In October 1860, the Prince of Wales, son of Queen Victoria and later King Edward of England, was touring the country, and, while his special train was waiting at the depot, he came to the rear of the platform and bowed in response to the large crowd that had assembled. On February 15, 1861, President-elect Lincoln, on his famous journey from Springfield, Illinois to Washington, D.C., was introduced to the local citizens and made a brief response to a call from the multitude assembled. The late Walter Ellett related a story told to him by David Fording, and James Donaldson, a Civil War veteran: "As Lincoln was strolling about on the Alliance platform, wearing his famous high hat, Amos Ailes, who tested the wheels on the trains and was a tall man of several inches over six foot, went up to him and said, 'Mr. Lincoln, I believe that I am as tall as you.' Mr. Lincoln squaring his shoulders and looking at Mr. Ailes replied, 'I could eat salt off the top of your hat.' "
Employees of the Pennsylvania Railroad pose in Alliance on August 9, 1906, at the freight station.
This is an artist's conception of the A.W. Coates Superior Lock and Lever and Hay and Grain Rake Company of Alliance. The company's product was patented on August 27, 1867, by A.W. Coates, who was also its sole manufacturer. Before Coates was 20, he embarked in a partnership with J.D. Arnold under the firm name Arnold & Coates, which manufactured plows and castings. In 1855, they added to their works the manufacture of Hay-Rakes. In the same year, they introduced what Perrin's book, History Of Stark County, said "Were the first Sulky Spring-Tooth Horse Rakes ever used in Stark County." After the demise of this partnership and several others, Mr. Coates took out eight patents: three on his Lock-Lever Hay-Rake, one on the Spring-Seat for the same, two on Guarded Scissors, one on a Child's Pocket-Knife, and one on a Water-Elevator for wells and cisterns. In his spare time, he ran unsuccessfully (losing by 48 votes) for the Ohio Senate in 1875. He erected the Coates Block on Main Street in 1877 for a cost of $22,000 and was the owner of the Independent Age, a journal devoted to literature, news, and religion. He also found the time to marry and raise nine children.
Born in Alsace-Lorraine, France, John T. Weybrecht came to America in 1853 and settled in Williamsport, Pennsylvania. Feeling he could find a better life in Chicago, he traveled on the Ohio & Pennsylvania train. While the train was detained in the small but thriving village of Alliance, he met a few of his countrymen and was persuaded to stay. Noted for his craftsmanship with wood, Weybrecht opened the first lumber yard in Alliance on Columbia Street. In the 1870s, he moved to Broadway Street across from American Steel Foundries and developed his business into one of Ohio's largest lumber and building supply businesses. He died on January 31, 1895.
In 1892, Mr. Weybrecht brought his two older sons, Benjamin F. and Charles C. Weybrecht, into the business, forming the partnership of J.T. Weybrecht & Sons. His daughter Mary became the wife of Leroy Leslie Lamborn, the son of Dr. L.L. Lamborn.
The King Bee Milling Company was located on the northwest corner of E. Prospect and N. Mechanic Streets when this picture was taken on February 3, 1931. It was owned and operated by Cleveland Storage Elevator. J.C. Henschen served as president and treasurer. He was the father of the late Charles Henschen, who owned Henschen Motors. King Bee was very successful in the milling of flour and grinding of feed. Residents of the area could buy pastry, cake, and bread flour by the 10- and 20-pound bags to store in their pantries at home and use on a daily basis. Ready-made bread and cakes were not always available at the local grocery store in those days.
C. Gilbert Taylor could have been the Henry Ford of the aviation business, but the sudden drop in the sale of planes after World War II and over-expansion caused Taylor to file for bankruptcy in 1946. During its peak from 1935 to 1946, Taylorcraft grew to become the largest company in Alliance, employing over 1,800 residents. This picture shows a Taylorcraft employee doing a drop test on September 18, 1936, during the factory tests of that year.
This is a typical scene along the railroad yards in the early 1900s at the end of North Freedom Avenue. The only identified individual in this picture is Burt Bradshaw, the second from left.
This picture, taken on October 7, 1913, shows the employees of the Stark Electric Company at the power plant on Lake Park Boulevard. Founded as the Alliance Electric Street Railway Company in 1888, the electric trolley line in Alliance was the second in the nation and one of the last to survive. Renamed the Stark Electric Company in 1902, it was commonly referred to as "The Bachelor Railroad" when it incorporated, because a majority of its directors were bachelors and remained unmarried during their tenure with the company. At the peak of the interurban there were nearly 2,000 miles of inter-city track in Ohio. When Stark Electric discontinued in 1939, only 150 miles of track remained in operation. The bus line was here to stay.
This great postcard view shows the Pennsylvania Depot (Sourbeck House) in its prime. As one passenger train leaves the station, more passengers patiently await the westbound train. This building was steeped in history from the days of the Civil War to its destruction in the 1950s. Three of the Civil War's most prominent generals, Sherman, Grant, and Sheridan, dined together at the Sourbeck House. This large structure housed over 20 large bedrooms on the second floor, as well as sitting rooms and parlors for the women.
This is a rare view of the New York Central train stopped near the station on Mechanic Street in the early 1900s. The view is looking north on Mechanic towards Main Street. The building with the two towers is the W.K. Fogg Building, located on the northeast corner. It served as an apartment complex and also held the famous "Temple Of Economy," a variety shop of china and novelties, but will always be remembered as the Owl Grill. It is now the parking lot for the Chamber of Commerce.
The reconstruction of the Stark Electric streetcar tracks and the relaying of bricks are evident in this 1920s photograph. Looking south down Liberty Street from Main Street, some of the buildings and businesses can be noted. On the left is the Stark Hotel, built in the 1880s as the Memorial Building. It originally housed the post office and various other businesses. Farther down the street is Akins Auto Storage. On the right, Akins's barbershop pole can be seen. Other businesses at this intersection were the A.L. Grewe Bakery, Widmer's Art Gallery, and George H. Judd, Tailor.
Drivers of the Cleveland, Alliance, and Mahoning Valley Railway Company, Car Numbers 3 and 6 (Alliance Division) posed for this 1914 photograph taken near Atwater. The 1902 Niles Car Company wood coaches were 47 feet long. Bought from the Stark Electric Railroad Company in 1913, the line and cars were sold to Northern Ohio Railway & Light Company (Akron Division) in 1925. The cars were scrapped the same year.
This photograph shows the day of transition in 1939, the last day that the Stark Electric Company was in operation. This interior shot of the car shows passengers taking a final ride.
This 1939 photograph shows the transition from the streetcar to the bus system. After the Stark Electric Company was discontinued, the name was changed to the Stark Transit Company. This company operated between Canton, Alliance, Sebring, and Salem. Fares were 10 to 15¢ in town and $1 for a round-trip ticket to Canton.
In 1906, an eastbound train was passing rapidly through town when one of its cars left the track and hit an abutment on the viaduct. As you can see, it was a disaster. Several people walking and riding on the bridge were injured, and one man who was standing in the rail yard underneath the bridge was killed. A new viaduct, which looked very similar to its predecessor, was built within a short time. It was torn down in the 1970s.
Shortly before noon on November 6, 1913, these two trains collided 2 miles southwest of Mount Union on the Grimes Farm near Beech Church. The cause was a misinterpretation of orders. The two trains were to pass at the Mount Union side track. The accident killed three and injured four. Dead were the engineers, G.A. Price of Dillonvale and R.C. Spriggins of Alliance; the fireman, H.M. Davis of Moultrie; and the brakeman, J.W. Martin of Alliance. Martin was crushed under a huge pile of coal. The Alliance Review reported that the two engines were badly demolished and six coal cars and one box car were piled in a heap, two of them being hurled into an adjoining field. Farmers were aroused by the crash and whistles, and soon a number of them were at the wreck helping remove the injured and dead.
President McKinley stops by Alliance to make a speech and to shake the hands of his supporters during his 1900 campaign. The caption at the bottom of this stereoscope picture states, "President McKinley, Alliance, Ohio—He was never so happy as when with the common people."
The Chase House was located opposite the Pennsylvania Railroad Station (Sourbeck House) and was built in 1876 for John and Amelia Pluchel, proprietors. It was named in honor of Gov. Salmon P. Chase, who, on a stop through the town, gave a speech on the balcony. It was considered a fine hotel for many years.
A. Fred Morris (near podium), president of the Morgan Engineering Company, and company employees posed for this picture during WW II. It was a proud day for all. The company and employees were honored by the U.S. Navy with the prestigious "Navy E" award for their contribution in the war effort.
In this picture, you can see the proportions and size of the overhead cranes built by the Morgan Engineering Company, for many years the world's largest builder of overhead cranes. This 250-ton locomotive is lifted from the ground and moved to the other end of the building with great ease. These cranes were used in the majority of the steel and manufacturing mills throughout the world. During the Spanish-American War and World Wars I and II, the factory was converted to build various guns and gun turrets for the nation's cause.
When you think of pottery, the first town that comes to mind is Sebring, Ohio. However, Alliance had its share of porcelain and china makers. The Crescent China Company, which was located on Lake Park Boulevard (the Genie Company Building), was owned by the Sebring family. It was known for the manufacture of high-grade semi-vitreous tableware, plain and decorated. Charles L. Sebring served as president. The business thrived until the late 1920s. Along with many Sebring potteries, it closed during the nation's economic downfall.
The Reeves Brothers Company was the world's largest builder of gas, oil, and acid tanks, rotary cement kilns, stand pipes, and grain elevators. George Reeves came to Alliance in 1893 to open the business that he established with his brothers in Niles, Ohio. The company expanded the business to other Ohio cities, including Dover and Girard. The Alliance works were destroyed by fire in May 1900. The business was rebuilt and in operation again in 1901. On July 15, 1901, another fire destroyed the plant. Struggling after the first fire and in debt, George Reeves called in his son Albert G. Reeves to help with reconstruction. By 1903, the Alliance plant had 100 employees and was equipped with some of the most modern and heaviest equipment available. Equipment included bending rolls, punches, and riveting machines. When the Depression hit, Reeves suffered a rapid decline of business. This, along with various other factors, including the Interstate Commerce Commission's increase in freight rates, caused the business to close in 1932. In September 1942, the plant was sold to the Babcock-Wilcox Company and was revamped to build gun mounts for the Navy. After the war, the company went into the manufacture of steel tubing.
Transue-Williams Forging Corporation was instrumental in the development of Henry Ford's dream of the assembly line. Ford wanted to make his automobile available to the masses. In order to bring the cost of a car down, Ford needed a lighter car. While at the Daytona auto races in 1906, he picked up some pieces of metal after an accident involving a French car. After several tests, he found this light-weight steel contained 15 percent vanadium steel. He searched for a company that would do experiments on vanadium. After many rejections, Ford, with the assistance of Oliver Transue, went to the United Steel Company of Canton, which was operated by Transue's friend, Harry Ross Jones. Jones agreed to assist in the experiment. After many failures, the optimists would not give up. According to E.T. Heald, "The eventual success of these experiments was probably the greatest single event in Stark County history, making the county the greatest center of electric-furnace alloy steel production in the world." When manufacture of the steel was completed in Canton, it was brought to Transue-Williams for forging. Ford was able to fulfill his dream, and mass production of the automobile was started.
The Lamborn Floral Company was incorporated on March 21, 1905, "for the purpose of conducting a general greenhouse and cut flower business." Mr. and Mrs. Leroy Leslie Lamborn, Charles C. Weybrecht, H.C. Koehler, and Ruth Burdge were the incorporators. Mr. Lamborn was the son of Dr. Levi L. Lamborn, the developer of the Scarlet Carnation. In 1905, they purchased 22 acres of land and these greenhouses were built along Hartshorn Street. Rosemont, as the land was known, had 6.5 acres under glass by the 1940s, and the company became the world's largest wholesaler of roses. After his marriage to Leroy Lamborn's only child, Margaret, in 1908, Fred J. Zang became involved in the business, succeeding his mother-in-law as secretary in 1920. In August 1922, the articles of incorporation were dissolved, and the business became a partnership, with Mr. and Mrs. Fred J. Zang and their son, Leroy Lamborn Zang, being the owners and partners.
The Alliance Machine Company was founded in 1902 by William H. Purcell, a former employee of the Morgan Engineering Company. Purcell's company became one of the world's largest manufacturers of cranes. This photograph shows the first employees on September 10, 1902. Until 1915, the company bought the bridge work for its cranes from various sources. Purcell founded the Alliance Structural Steel Company in order to manufacture bridge work for his other company.
This 1913 view of the employees of the Buckeye Jack Manufacturing Company was located in the small community of Bolton, near Alliance. Bolton was originally called Slabtown, and Clayshaft between 1860 and 1900. When the Standard Bolt Company opened in this building in 1900, the community changed its name. After the bolt company closed, the Alliance Asbestos Company operated here for a short time. The Buckeye Jack Company was the country's leading producer of jacks for railroads, automobiles, trucks, mining, and industrial field. When automobiles came into vogue, the company produced the original supply of jacks for the auto trade. After the death of its president, E.C. Bates, in 1924, the company failed to keep pace with its competitors. One of the company's directors, W.H. Purcell, persuaded his son-in-law, Clarence J. Rodman, to take control of operations. The company regained momentum, winning back business from General Motors, Nash, Cadillac, and others; however, the Depression forced the business to close. In 1934, C.J. Rodman opened Alliance Ware Incorporated in this building. It was one of the first producers of pressed-steel bath tubs. The excellent design and quality of its enameled ware made it a favorite in many households.
The McCaskey Register was the invention of P.A. McCaskey, a grocer in Lisbon, Ohio. Dissatisfied with the day book and ledger system of tracking credit sales, he developed the "one writing" credit register. When his invention caught on, he decided to seek financial backing to mass produce it. On June 30, 1903, the McCaskey Register Company was incorporated. The incorporators included J.A. Zang, C.C. Baker (the first company president), R.S. Kaylor, and Mr. McCaskey. The company bought a brick building and land along Rush Street at the Central Crossing (shown here after several additions) and began production. By 1908, the business was expanding faster than its management could keep up with. The need for more capital to modernize was great. A.G. Ryley and E.A. Langenbach were among the Canton men who invested. Ryley was named president after the sudden death of Charles Baker. After a number of years of outside control, several Alliance men did a stock buyout and took control of the company. F.E. Henry Jr., O.F. Transue, and F.E. Dussell were a few of those investors. During WW II, the plant was converted to a war plant, manufacturing bank and turn test instruments and gun sight test instruments for planes.
The Fairmount Provision Company was located on South Willow and Grand Streets. In 1911, Charles Barnes established a small slaughterhouse on Fairmount Road. It was a success from the start. Eight years later, he expanded the business and, along with V.G. Sell, incorporated it. Soon a large and modern packing house (pictured) was built in the southeastern section of the city, equipped with the most modern refrigerating plant available. The company was known for its Hickory Ham and Hickory Bacon. They butchered on a weekly basis about 600 hogs, 200 cattle, 200 calves, and a large number of ewe and lamb. Their weekly output of sausage meat was about 25,000 pounds.
This is perhaps one of the earliest existing views of the business office of the Morgan Engineering Company. Taken on April 28, 1891, this picture shows the office men of the company posing with pride for the photographer. Pictured from left to right are: (front row) Frank Dussell, Thomas D. Russell, John Lloyd, and ? Curtis; (back row) A. Fred Morris Sr. (a future president of the company), ? Matthews, and George Esterly.
The Alliance Brewing Company, which was located on Summit Street near Mahoning Avenue, was incorporated in 1905 with a capital of $150,000. The officers were Herman Mueller, J.C. Klingler, W.W. Shidler, and Samuel Burget. The brewmaster was Fred Keifer. The business flourished by producing beer and ice. The ice-making plant was capable of producing 10 tons of ice per day which was distilled and procured from its own wells on the property. When national Prohibition began, the company went out of business. Reopening in 1919 as the Alliance Bottling Company, they continued producing ice and started bottling Coca-Cola and other soft drinks. A.C. Fullmer bought the company in 1921.
The Alliance Pottery Company, located on Patterson Street, was an early producer of various household porcelain products, including dishes, vases for flowers and plants, and pitchers and wash basins.
Here we see one of the two young pottery workers clowning around with one of the chamber pots in the inventory room. The approximate date is 1900.
Because of the city's position along the nation's two major railroads, Alliance has witnessed many dramatic events through the years. On Tuesday evening September 17, 1901, the funeral train carrying the remains of Pres. William McKinley left Washington, D.C. for Canton. All through the night, in every city and hamlet along the route, weeping men, women, and children stood silently as the train passed slowly by. As day broke, children laid flowers in the path of the train, and people put coins on the tracks wanting to keep the flattened coin as a souvenir of the occasion. Around 10 a.m., the train passed through Alliance. Thousands gathered to say their last good-bye to the man whose political career had begun in their community.
Mourners gathered along the tracks in August 1923 to pay their last respects to Pres. Warren G. Harding. In 1920, women were given the chance to vote for the first time in a presidential election. Harding's good looks and dashing personality were a delight to many. Area resident Miss Ethel Antram had the opportunity to go to Washington, D.C. with a friend in 1921 and attend a reception at the White House. She remembers his warm handshake and charming personality most of all. She also remembers laying a penny on the railroad tracks as his train passed by her home in Maximo. Perhaps, like many Americans, it was a simple but symbolic way to say good-bye.
# Three
# LEARNING
SCHOOLS AND CHURCHES
The faculty of Alliance College can be seen here in the early 1870s. In the mid-1860s, a group of businessmen, including A.W. Coates, Elisha Teeters, and J.B. Milner, felt they could compete with the less then 20-year-old Mount Union College, located down the road a few miles in the little village of Mount Union. Alliance College thrived until the stock market crash of 1873, when financial conditions forced its closing. Those identified in this picture are: J.L. Pinkerton, I.M. Demmon, A. Fairhurs, Lottie Sackett, Miss Sarah Laughlin, Miss Ella Mehard, Mrs. J.L. Pinkerton, and Mrs. M. Hazzard.
The Alliance College Building was designed by Simon Porter in the late 1860s. Located on the southwest corner of Broadway and College (now High) Streets, it served as college classrooms and faculty offices. After the college closed, the newly formed Alliance public school system used the building for various classes. When Park School was completed, this building became the high school. It served in that capacity until 1910 when it was demolished and the new high school was built.
The original dormitory for Alliance College, this building was built in the 1860s. It was located directly behind the main school building. After the college closed in 1873, St. Joseph's Catholic Church, which was located directly across College Street (now High Street), took over the building for use as a school. It served that purpose until the early 1900s. It was torn down, and a new school was then built on the same location.
The First Ward School was built in 1892 on the location of the old Union School, the first school building constructed after Alliance's incorporation. Located on the corner of North Park Avenue and Washington Street, this building consisted of eight classrooms. Some of the prominent educators who were associated with North Park School were Cornetta and Martha Hazen, and Principals Charles L. Burrell and W.A. Byers.
The Fifth Ward School was located on the corner of Park and Broadway. Many of the city's prominent families' children attended grade school here. Pictured from left to right are: (front row) ?, ?, ?, Aileen Slutter, ?, Marion Maus, and Mary (Weybrecht) Shirk; (middle row) Madge Scott, ?, ?, ?, Colleen (Allmon) Bates, Frances (Coates) McPherson, Leeta Wilson, Dorthea (Doane) Keplinger, and Margaret Church; (back row) Hazel (Purcell) Rodman, Elizabeth Maus, Rhea Roach, Bernette Church, Margaret (Ramsey) Sebring, Rhea (Whitman) Blythe, Margaret (Atwell) Roller, and two unidentified.
This is Mrs. Brockun's annual Senior Chicken Roast, held in her home, on March 23, 1898. The students break the wishbone for good luck.
The beginning of a library in Alliance was in 1886, when school superintendent C.C. Davidson collected 70 volumes from various sources. These books were kept in his office in a library case donated by J.H. Sharer. T.R. Morgan Sr. donated an additional 150 volumes. In a few months, the library reached 610 books. In 1899, the board of education saw the need for a building. A committee was appointed to correspond with industrialist Andrew Carnegie. After much persuasion, he gave $25,000 toward the building, provided that the board would set aside $2,500 a year for its support. A building committee consisting of C.C. Baker, Frank Transue, and John E. Morris was appointed to work with the architect. The lot next to the high school on Arch Street was selected, and the building was completed in 1904. The dedication was on September 6, 1904, and 7,500 volumes were available to the public.
During its destruction in 1997, everyone referred to this building as the Seneca Street School. When the Alliance school system was planned out, the Seneca Street School was divided into wards. This building was the Third Ward School, which stood on the northwest corner of Seneca and Cambridge Streets. Built in 1889, it was only a four-classroom school. As the population in the area grew, additions were made. It served the community for 68 years, closing in 1957. It was then used as a warehouse for the board of education until the mid-1970s. This was the last surviving building of the Ward system.
The students of the Third Ward School pose at the front gate in the early 1900s. Ida Buck is identified as the teacher.
Mahoning School was building number seven in the Alliance school system. Located on Mahoning Avenue, just north of Summit Street, it was built in 1904 as a two-room school. It was enlarged to accommodate the growing ethnic population in the area. It was closed in 1957 when the new Morgan School was opened. The building was razed to make way for a children's park in 1983.
The back of this picture reads, "Compliments of John E. Morris, 2nd class—1873." The class included John E. Morris, Pheobe Peet, Theodore Uran, Kate Quinn, Ellen Peet, and Helen Shatter. The reference to 2nd class refers to the fact that this class was the second to graduate after the establishment of the official school system in the Alliance community. John E. Morris would become the superintendent of schools in the 1890s.
These individuals were faculty members in Alliance public schools in 1887. Some of the staff pictured are, from left to right: (front row) Asenath Dalzell, Mamie Farley, Martha Hazen, E. Eldridge, Jennie Kump, and Dorothy Focht; (second row) Professor and Mrs. Coup, Mary Stallcup, Mrs. C.C. Davidson, Professor C.C. Davidson (superintendent of schools from 1885–1892), Elizabeth Fetters, and Mr. and Mrs. J.H. Foucht; (back row) ?, Lillian Leek, Sophia Kirby, Mary Hazen, Serepta Ross, Ella Barnaby, Rebecca Bolen, Sarah Dalzell, Mary Laughlin, and unidentified.
The graduates of Alliance High School posed for this picture on June 13, 1906. They are, from left to right: John D. James, Gaston G. Gulland, Glenn W. Ellett, Leroy T. Oesch, Almon J. Damon, Harry L Johnson, Margaret Patton, Bernice B. Shaffer, Lois J. Hull, Ethel I. Hively, Arthur L. Allen, Herbert C. Leonard, Stanley W. Smith, Lawrence C. Slutter, Charlie E. Howson, Jessie F. Auld, Alice S. Hilles, Emma R. Hicklen, Alma V. Frost, Bertie Winner, Carey L. Stanley, J.F. Wright, H.W. Miller, Hazel H. Hoyer, Corinne E. Strong, Cecil F. Walser, Hazel C. Taylor, Mary G. Devine, Alice P. Motz, A.M. Clement, and Margaret J. Rickard.
The Garfield Room was located on the southeast corner of the first floor of the old Alliance College Building which became part of the Alliance school system in the 1880s. It was named for Pres. James A. Garfield who once gave a political speech in the room. This is Miss Mary Howard's third grade class of 1904–1905.
It's Halloween, 1920s style. These school children pose in their home-made costumes of crepe paper, cardboard, and colored paper hats.
Alliance High School was built in 1911, with an addition made in 1922. Not much can be said about this building other than it served its faculty and students for nearly 60 years. Each individual who walked through its doors was afforded the best education our community had to offer. Everyone has a memory, whether it is a basketball game, orchestra concert, the Dilley Sisters, Fall Varieties, or a first kiss. One picture says a thousand words.
Leland Whitacre and one of his most famous former students, Dick Balduzzi, pour over old Fall Varieties scrapbooks in 1970. Mr. Whitacre was the originator and producer of the Alliance High School's talent show. After high school, Balduzzi continued his education in the Guild of Dramatics at the Goodman Memorial Theatre in Chicago where he received his bachelor of fine arts degree. In New York City, he attended the American Theatre Wing and the Herbert Berghoff Studio, where he studied with Uta Hagen and Mildred Dunnock. His acting career has included every kind of performance, including the stage and first live television episodes of The Jackie Gleason Show in New York and Chicago's Drury Lane Theatre. In Hollywood, he appeared in hundreds of television shows, such as I Dream of Jeannie, Bewitched, Cannon, and Murder She Wrote. Some of his film credits include Coma, Kelly's Heroes, and Zorro, The Gay Blade.
The play, The Man Who Went or The Black Feather, was presented by the Hi-Y Club at the State Street Auditorium on February 11, 1921. The two men seated on the left are Harold Bonner and Howard Maxwell. In the back, from left to right, are: George Donaldson, Harold Bott, ?, ?, and John Scott.
St. Paul's Lutheran Church, located on the corner of Linden and East Cambridge, was built in 1912. Organized in 1865 as the Evangelical Lutheran Church, the original services were held in various locations, including the Alliance College Building. The church's first structure was built on the southeast corner of Linden and Columbia. The congregation disbanded and reorganized several times. This building still stands today; however, the congregation built a new edifice and this building has been empty for a number of years.
This picture shows the laying of the cornerstone of the First Baptist Church on the southwest corner of Freedom Avenue and Market Street in 1910. Built on the same site, a lot given by Mr. and Mrs. Elisha Teeters, as the old frame structure that had been built and dedicated on September 20, 1857, this brick building was dedicated on September 17–21, 1911. The congregation moved in the 1970s to West State Street, and this structure was demolished.
The First Methodist Church was located on the south side of Main. By the 1890s, with Main Street's commercialism on the rise and the fact that the congregation was growing, they built a new structure on the corner of Broadway and Freedom Streets. In 1899, this building was sold to the Scranton Brothers who removed the towers and steeple and added a new brick front in order to make the building a business block.
The members of the Second Baptist Church, located on North Union Avenue and Selby Street, posed for this 1919 picture. Pictured from left to right are: (front row) Mrs. McDabby, Leily Baker, Annie Jackson, Ruby Nortington, and Mabel Smally; (back row) Ruth Rhine, Robert ?, Annabelle Kennedy, Horace Adams, Gertrude Head, Oscar Rhine, and D.D. Dantly.
One of the earliest and largest ethnic European populations to settle in the Alliance area were the Hungarians. This photograph shows the Old World influences still very much in evidence. Here we see the first funeral held at the Magyar Reformed Church on East Cambridge Street, shortly after the church's completion in 1909. The church thrived into the 1930s. However, during the Depression years, its membership waned. In the early 1940s, the arrival of István Csépke brought a resurgence in church activities. However in the 1970s, the new generations moved on, and the church bell rang for the last time. The property was purchased by the Morgan Engineering Company, and the building was torn down.
The First Methodist Episcopal Church (now Christ United Methodist Church) at Broadway and Freedom was formally dedicated on Sunday, September 3, 1899. The cost of the church was $80,000. Silas J. Williams of Transue-Williams Company paid for one-fourth of the cost of the building.
The Jewish community established worship services in the late 1800s at various locations in Alliance. The first resident rabbi was Harry Krash, pictured here with his wife, Anna. He arrived in the community in 1912. In 1916, with the growing population, a group of men that included J.W. Frutkin, Max Geiger, Morris Geiger, and Sam Katzenstein initiated the purchase of a temple on East Columbia Street (the former Christ Reformed Church). Temple of Israel flourished under Rabbi Krash's direction. The Rabbi's son Joseph established a scrap iron and steel business in Alliance. Joseph's son Harry still runs the business today. His other son, Allan Krash, is an attorney in Stark County, having held positions in the law director's office and the county prosecutor's office.
Located on East Columbia, the cornerstone for the Christ Reformed Church's building was laid on September 28, 1873, and the dedication was held one year later on September 27, 1874. Many of the members of this congregation were German-speaking natives of Switzerland, who came to this area in the late 1860s and early 1870s. In 1871, Dr. August Schade came to Alliance, and the congregation was organized. In 1919, the need for a larger edifice was evident, and the property of John E. Morris at the corner of Oxford and Mechanic was purchased. This building was used as a house of worship for the congregation of the Temple Of Israel for a number of years.
For a number of years in the 1920s, Alliance residents who were members of the First Church of the Brethren worshiped at various locations outside the city. In 1926, a building committee was formed. After some investigating, the group was able to report that the old Immanuel Reformed Church on Columbia Street could be purchased for $500. The congregation decided to purchase it, dismantle it, and rebuild it on land they owned on South Freedom Avenue. The cornerstone was laid on June 27, 1926. Construction proceeded rapidly, and the workmen were as far along as almost placing the roof on when a tornado coming from the west destroyed the frame building. Members of the congregation borrowed, through personal notes, the sum of $12,000 from First National Bank. This photograph was taken on the day of dedication on October 17, 1926, on South Freedom Avenue.
In 1901, with the growth of its congregation, the Presbyterians formed a building committee comprised of some of Alliance's most noted figures, including W.H. Morgan, J.H. Sharer, and A.B. Love. The men met at the City Savings Bank and organized themselves as a "Church Corporation." Mr. Love presented five $20-gold pieces to the church as an inspiration to others. The First Presbyterian Church was designed by the Cleveland architectural firm, Searles and Hirsch, (they also designed Col. W.H. Morgan's home). The cornerstone was laid on June 3, 1903, and the church was dedicated on December 11, 1904. The construction contract was awarded to Mr. S. Joliet, one of Alliance's more noted builders, at a cost of $24,300.
This interior photograph shows the beautiful Tiffany stained-glass window of Christ blessing the children, which was given in memory of Mary Louise Morgan, the only daughter of Col. and Mrs. W.H. Morgan. The Hillgreen-Lane Pipe Organ was given by the philanthropist, Andrew Carnegie.
The cornerstone for the Mount Union Methodist Episcopal Church was laid on July 13, 1893, by Bishop Joyce and Gov. William McKinley. This event was held in conjunction with the commencement exercises at Mount Union College. Known for many years as the "College Church," it was built at a cost of $27,372.54. College President T.P. Marsh was head of the building committee. Designed by the noted Cleveland church architect, Sydney R. Badgely, the building is in the architectural style of modernized Romanesque.
The "Akron Plan" was incorporated in the architect's design. This interior concept was designed by Greentown, Ohio native Lewis Miller, former president of the board of trustees at Mount Union College, Chautauqua co-founder, and father-in-law of Thomas Edison. The "Plan" is a semi-circular auditorium, which, beginning at the pulpit, spreads out with a gradual uphill grade to allow equal viewing throughout the room.
This picture shows St. Luke's A.M.E. Church. Through the efforts of Mr. William Mosby, the first Sunday school and church meeting for the black community in Alliance took place at the Union Gospel Mission on Patterson Street in 1903. After a year, the congregation purchased a building at Broadway and Union Avenue. Because the building required extensive repairs, financial problems forced the congregation to discontinue services. Through the help of a loan company and with the determination to continue, the members of St. Luke's purchased property on East Jersey Street. Backed by the A.M.E. Church, St. Luke's first pastor, Rev. H.H. Uptergrove, arrived in 1907. In the 1920s, during the ministry of Rev. J.C. Turner, the church purchased the mission at 1050 Patterson Street.
The evangelists and pastor of St. Luke's A.M.E. Church are pictured here. From left to right are: Mrs. Ethel Johnson, Mrs. Pearl Cundiff, Rev. McCune Morgan, Rev. C.A. Burgan, and Rev. C.H. Waters.
The men of the Christian Church (East Main Street and Park Avenue) posed outside the front entrance. The event is not known, but the picture dates around 1920.
The first meeting of the First United Presbyterian Church (in the late 1950s, the name was changed to Bethany Presbyterian Church) was held on August 21, 1908. This congregation worshiped in various locations before the construction of this edifice. The cornerstone for the church on the corner of South Arch Street and Summit was laid on May 31, 1922, and the church was built by one of Alliance's most prominent contractors, John Sharp. The cost of construction was $72,000. The first service was held on January 1, 1923. The dedication services took place on April 1, 1923. In 1998, the First Presbyterian Church and Bethany Presbyterian Church voted to merge, using the Market Street church as the house of worship. The Arch Street building is now called Bethany House and offers Christian outreach programs to youth and adults in the area. It is sponsored by the Presbyterian Church.
# Four
# CONTINUING DEVELOPMENTS
John Brown, the famed abolitionist, once said of his friend, Ellis N. Johnson, "Ellis is one of the foundation stones and a true friend of the cause." Mr. Johnson, the founder of the Village of Mount Union, was born in Brownsville, Pennsylvania on April 1, 1789. At age 21, he left home and walked west into eastern Ohio to inspect land made available through the government. Johnson arrived in Washington Township by following the section lines easily recognizable by the trees blazed by the surveyors. Soon after his arrival, he heard the sound of an ax in the distance. It led him to Ezekiel Marsh, another pioneer taking advantage of the government's offer. Marsh's cabin stood approximately 1 mile west of State Street and Union Avenue, near Bayton Street. During the months he stayed there, he acquired a quarter section of land. Johnson returned to Pennsylvania to find a wife with whom he could share his good fortune. In 1823, he returned with his wife and children and built the first log cabin in what would become Mount Union. He, along with his 16 children, cleared the land and farmed the property. This picture of Johnson was taken on the 100th anniversary of his birth. Uncle Ellis (as he was known to all) died on September 15, 1889.
The Conn House stood on the southeast corner of Mount Union Square. It was built in 1836 by John Hair, who wished to operate a restaurant, hotel, and tavern. Much to his surprise, the villagers were unable to accept the sale of liquor in their community. If any residents wanted alcoholic beverages, they had to travel to the nearby communities of Freedom and Williamsport. After much debate with the village officials, Mr. Hair sold his building and business to the Conn family. It became a fine family establishment, known for its good food and clean rooms. During the construction of Chapman Hall, the Conns generously gave room and board to the contractors. This photograph was taken in the late 1880s, when State Street was a dirt road. The Conn House was torn down in the 1930s for a Sohio gas station.
The Garretson House, which stood on the northeast corner of Union Avenue and State Street, was built by Victor Milhouse in 1837. The building served as a general store, office to the village's first mayor, post office, and restaurant. In 1846, Pierce Garretson and his family moved in and made it a single family dwelling. During their ownership, the Garretsons were heavily involved in the abolitionist movement, as were Mount Union's other residents. This building gained national prominence as a meeting and resting place for some of the nation's most influential anti-slavery leaders. Such individuals as Sojourner Truth, Charles Sumner, John Brown, and Lucretia Coffin Mott were always welcomed and supported by this little but fiery community. After the turn of the century, the building housed several businesses, including the famous College Inn. The building was destroyed in the 1960s for a parking lot. The date of this photograph is the late 1870s.
This 1906 view of Mount Union Square shows the newly constructed Stroup Block on the northwest corner, the site of the old Job Johnson home. This building was the home to many businesses and groups through the years until its destruction in 1998. Its occupants have included Hall and Son's Grocery Store, Turner's Drug Store, Isaly's Ice Cream, and an A.T.O. Fraternity meeting room on the second floor.
This rare early interior view of the 1906 Stroup Building on Mount Union Square shows Hall and Son's Grocery Store. The building was constructed during an age of conversion. Note the ceiling light fixture. It is both gas and electric. These were commonly referred to as converter lights. Electricity was such a new phenomenon that many felt it might not last.
Believe it or not, there was a day when you could stand out in the middle of State Street long enough to snap a picture. Such was the case in 1954. This view looking east toward the square shows how times change. Not one building in this picture is standing today. Some of the noted businesses of the day were Sivey Electric, Rose Meat Market, College Inn, Mount Union Bank, and Turner's Drug Store.
Looking north from Mount Union Square, this photograph shows State Street and Union Avenue when they were dirt roads, around 1900. On the northwest corner is the property first owned by Job Johnson when the plat was laid out in 1833. The clapboard-sided building on the left is the original log cabin built by Job and his wife, Maria, in the 1820s. This log structure housed the village's first post office, a tavern (prior to the temperance movement), and the office of Dr. Enoch Shreve, the area's first physician. The first indoor services for the Mount Union Methodists were held in that home. On the right is the Garretson Home, owned by Hugh Shipman at the time this picture was taken
This is the Mount Union Bank on its opening day, January 20, 1930. The Alliance Review stated that throngs of visitors and new customers visited the bank during the three-hour period. According to F.E. Henry Jr., president of the company, the bank listed more than 200 new accounts that day. Located at State Street and Union Avenue, the brick and limestone building had conference rooms and offices on the second floor. The west section of the first floor was leased to the Great Atlantic and Pacific Tea Company, known by many as A&P groceries and meat. Grumbles Electric Company occupied the south end. The building was demolished in the late 1980s.
The Stamp Building at 1745 South Union Avenue was constructed in 1919 for the Stamp family. Virgil and Joseph Stamp ran the first automobile service in Alliance. They owned a Studebaker dealership in this building. From 1924 to 1938, Gus Slusser's Nash dealership occupied this site. He later built a new building on East State Street and Mahoning Avenue. This building was altered in 1938 by contractor Walter Scott for Dr. Stamp and opened in February 1939 as the Mount Union Theater.
Mount Union College was founded in the Village of Mount Union on October 4, 1846, by this gentleman, Orville Nelson Hartshorn. After his birth in Portage County, his family moved to what is now Mahoning County. Most of the early education Hartshorn received was from his mother. Later, he attended high school in Deerfield and three winter terms at Linnean Academy in Atwater. During its first year, Mount's classes were held in an old woolen mill on the Ellis N. Johnson property. During the spring, summer, and fall of 1847, Hartshorn continued his studies at Allegheny College in Meadville, Pennsylvania. Upon his return, a building was constructed at the corner of Penn Street and Main (now State) Street. This building became the headquarters for the new school. After the building was discarded for classes, it was used as a rooming house for female students. It was nicknamed "The Nunnery." In the fall of 1849, the school was officially named Mount Union Seminary, and 62 students were enrolled.
O.N. Hartshorn served as president of the school he founded until his retirement in 1887. Hartshorn was a humble man who came from humble beginnings. The greatest achievement of his life was his college. On one occasion, a number of his friends in the community gave Hartshorn money to replace his tattered overcoat. His reply to them was, "If you folks had any money to give away, why don't you give it to Mount Union College?" He died on September 17, 1901, and was buried at Mount Union Cemetery. This photograph of the Mount Union Methodist Episcopal Church shows the beautiful floral arrangements for his service. This frontal view of the choir loft is the only known picture showing the two stained-glass windows, which were boarded up and plastered over when the Carnegie Tracker Organ was installed in 1902. The windows still remain covered.
This is the only surviving Civil War-era building designed by Ohio's noted architect, Simeon Porter. Construction of Public Hall (now Chapman Hall) began in 1862. After several delays throughout the war years, the building was completed in 1864. Hartshorn was determined to find someone of great importance to dedicate this wonderful structure. He decided it would be Salmon P. Chase, former governor and senator of Ohio, Republican nomination rival of Lincoln, secretary of the treasury, and one of the country's leaders in the anti-slavery movement. Hartshorn went to Washington, D.C. to personally extend the invitation. After three days of persuasive work, Secretary Chase agreed. At the dedication, Chase made the following remarks in the opening words of his address, "Ladies and gentlemen, if your president, Dr. Hartshorn, were not such an obstinate man, I would not be here. Because he is such an unreasonable man, I am here. I had no time to spare for this trip. I had many other duties to perform, but the more I expostulated, the more impudent your president became." Nearly two thousand people attended the ceremony. Legend has it that while Chase was seated on the platform on the third floor of the hall, a messenger from the telegraph office in Alliance delivered a telegraph to Chase. It was from Pres. Lincoln notifying Chase that he was being recommended to the U.S. Senate as chief justice of the Supreme Court. Though the two men had many serious disagreements, the President recognized Chase's abilities. Chief Justice Chase was appointed to the board of trustees of Mount Union College in which he served for eight years until his death.
The Columbian Gates, donated by the class of 1893, are the focal point of the State Street entrance to Mount Union College. Formerly located between McKinley Street and Miller Avenue, the gates were removed a number of years ago. The second pair of gates still stand in front of the lakes along Union Avenue.
During the 1860s, Pres. Hartshorn felt that the college should have a museum. After traveling through Europe with little luck, he arrived in England. It was at the British Museum with whom he struck a deal that afforded him the opportunity to purchase many specimens. Through the generous financial support of Canton natives Cornelius Aultman and Col. Ephraim Ball, gorillas, an Egyptian mummy, and many other exotic animals were placed in a portion of the first floor of the Chapman Hall. Hartshorn knew a vast collection could bring a considerable amount of recognition and advertisement for the school. Some of the materials were displayed at the centennial celebration in Philadelphia in 1876 and at the Pittsburgh Exposition in 1875. However, it was the Exposition in Louisville, Kentucky in 1874 that gained the most attention for the college. Many young men and women learned about the college at county fairs where the exhibit curator would hand out pamphlets about the school.
# Five
# ON THE MAIN
A rather large flag flew over Public Square in the 1890s. At the far left behind the old city hall building is the First Baptist Church's frame structure. The square was made entirely of bricks. The rather spacious size was due to a large cistern located underneath. This was the central source of water for the fire department. On the right are several prominent businesses: C.Y. Kay's Hardware Store, Liberty Party Headquarters, O.H. Schmalzel—Liquor Dealer, and Garret T. Fogle's Drugstore. Perhaps the boy who stuck his head in the picture at the lower left just bought a peppermint stick there.
This 1910 view shows Main Street looking east from Arch Street. The transition from horse and buggy to electric streetcars and electric automobiles is evident. On the right is the two-story Wick Block. Shortly after this picture was taken, two more floors were added to house the Alliance Business College. The building was torn down in the late 1980s. Some of the other businesses in this picture are Newman's Men's Wear, Damon Jewelry and Optical, Alliance Hardware, Spring and Holzwarth, W.E. Davis Dry Goods, and the Lexington Hotel.
This very unusual pre-1915 panoramic view taken on Main Street at Public Square shows the downtown area looking east, west, and south. In the center of the picture is city hall, which was constructed in the 1870s and replaced in 1915. On the east side of the square is the Allott-Kryder Hardware Company. The License Agency is located on this site today. On
Issie Wise, owner and operator of Wise's Meat Market at 174 E. Main Street, demonstrates his cutting techniques for the camera in 1898. Meats were cut to order on a daily basis due to the lack of proper refrigeration.
the west side was a saloon, Martin's Confectionery and Fruits, and C.Y. Kay's Hardware Store. In the mid-1920s, these buildings were removed to make way for the City Savings and Trust Company skyscraper.
Homer Ingold's first barbershop was located on Main Street. This photograph was taken around 1910. Like most barbers, Ingold's business offered baths and a shave for the traveling man, who might be briefly held over in town, waiting for the next train.
J.S. Cassaday and Son at 352–356 East Main Street was one of Alliance's most noted furniture and undertaking businesses. This picture was taken in 1881 soon after the business was established. If you look in the far left window you can see the memorial tribute to Pres. James A. Garfield who was assassinated in 1881. Apparently the Cassaday's were conducting a sale on rockers. Note the rocker displayed on the pole out on the sidewalk. This must have been the 19th century's form of billboard advertisement. Joshua Cassaday and his son William established the business. In 1882, William left and Joshua's other son, David Brinton, joined the firm. This partnership continued until 1894 when David bought out his father's interest. In 1905, Edgar H. Turkle joined the firm and married David's daughter, Ada Matilda. In 1914, a partnership was formed under the name Cassaday and Turkle, which assumed the funeral service business of the firm. In 1926, Cassaday and Turkle purchased the D.W. Crist residence on South Union Avenue. The firm continued under the same name with Edgar H. Turkle Jr. and Miss Barbara Cassaday Turkle actively in control. The business was sold to John Christian in the 1980s.
Geiger the Clothier was located at 635 East Main Street in the early 1900s. Pictured from left to right are: Max Geiger, Harry Roderick, and Guy Shaffer. They moved over to the north side of the 500 block around 1920.
Before the turn of the century, one of the most extensive hardware dealers in this part of Ohio was the Wright & Pennock Hardware Company. Located on the east side of the square on Main Street, it was founded by Alfred Wright and Morris Pennock in the 1860s. The McKee's business directory of 1868 says, "The varied wants of all the classes of people, and all the professions, tradesmen, artisans, trafficker, etc., could be found in their extensive house." Pictured from left to right are: Miss Stanley, Lorenna Conn, Arthur Wright (inside doorway), Morris Pennock (gentleman with white hair and beard), and Cliff Bore. The man behind Pennock is unidentified.
"Rob't Auld's—Cheap Cash—Grocery" is what this sign reads. Born in 1854, Mr. Auld came to Alliance in 1882. He rented this storefront in the Wonder Block at 606 East Main Street (named for early Alliance photographer Lafé Wonder who had his studio upstairs; note the glass case advertisement attached to the building on the left). In the 1898 Stark County business directory, John Auld appears as the operator of the store. This picture is from the early 1890s.
The Alliance Leader, founded in the early 1870s, was one of many papers published during the community's early years. This picture was taken April 8, 1900 at the paper's original location on N. Arch Street in the Concordia Building on the west side. They moved to the Weybrecht Block at 640–42 East Main Street in the early 1900s. The Leader was published weekly, semi-weekly, and daily throughout its 40 years. The circulation reached 2,300 on the weekly editions and a little higher on its semi-weekly edition. The last issue was printed on July 31, 1915. They merged with the Alliance Review to form the Alliance Daily Review And Leader. Between 1900 and 1915, the paper was owned by the Morgan Engineering Company, and the newspaper's manager during that period was Louis H. Brush. Pictured here, from left to right, are: Charles Paxon, Arthur Scott, Lottie Dorman, Belva Burnhouse, Rose Neuworth, Dollie Ingram, Charles Trescott, and "Pete."
The Kiwanis Club float, apparently involving a baseball team, leads the way in this 1920s photograph. The men on the float and directly behind it are in uniform for a big game. Even though there isn't a lot of information to give about this picture, it is still a great view of Main Street at Mechanic Street. The frame building, second from the corner, is Figenschuh's Jewelry Store. As you can see, the bricks on the street were removed on the south side of the streetcar tracks.
The northeast corner of Main Street and Union Avenue has changed dramatically over the years. When this late 1930s snapshot was taken, some of the businesses in operation were Main and Union Confectionery, Miller's Tires, Stewart's Paint Store, and Famous Dairy. These buildings were removed in the early 1960s to make way for the Kroger Company's new store. When it closed, Joseph Mastroianni purchased the building and opened the discount grocery store, No Frills, Inc.
Mr. and Mrs. Peter D. Keplinger built a hotel in the late 1860s. Originally called Keplinger Hotel, it was a frame structure that stood on the southeast corner of Main Street and Linden Avenue. In 1905, Edward Beeson erected the brick front and later changed the name to Lexington Hotel, in honor of one of the Alliance area's first villages. It was the largest of the three hotels in town (Stark Hotel and Chase House were the other two). A buggy would go down to the train station and pick up the businessmen and their baggage and take them back to the hotel. Many events took place in this building. In 1946, the Alliance Shrine Club was organized in the pool room located in the basement. Karl Fiegenschuh Sr., who for many years was considered a dean of merchants, got his start in the Bates Jewelry Store in 1907. While observing the destruction of the building in April 1970, Mr. Fiegenschuh Sr. commented, "It's progress, I know, but it is a landmark, and I kind of hate to see it go." Another person who observed the demolition was Mrs. Steve Mate. With tears in her eyes, she also agreed that the demolition clears the way for progress, but "the Lexington was here when I came to Alliance, I am unhappy to see it become history." The Lexington Hotel kitchen crew is seen here. Known for its fine meals, the hotel was popular for wedding receptions, social events, and Sunday dinners. Those who made the good food possible posed for this 1914 picture. From left to right are: Ed Oliver, head cook; Mike Morrison, pots and pans; Mrs. Steve Mate (Kathleen Birtalan), pantry girl; Victoria Liza Brown, pastry cook; Mrs. Mike Karwan (Anna Chikops), dishwasher; and Mrs. Ignatz Bara (Mary Jenei), dishwasher. Unfortunately, the name of the man seated is unknown.
Posing in the main lobby of the hotel are Mr. Charles Vernier (light suit), manager; Mr. Heacock (center), day clerk; behind counter, unknown; and an unknown man on left, night clerk.
A Public Square National Recovery Administration rally in the 1930s is pictured here. The N.R.A. was a U.S. government agency in the early 1930s. Pres. Franklin D. Roosevelt set up the N.R.A. in 1933 under the National Industrial Recovery Act as part of the New Deal program. The program prepared and enforced codes of fair competition for businesses and industries. President Roosevelt abolished the program in 1935, after the Supreme Court ruled it unconstitutional.
Less than two months after the stock market crash in 1929, the Alliance community was beginning to feel the effects of what would be the nation's worst economic times. The Alliance Advertising Club created a plan to enhance business in downtown. A "Santa's Workshop" display was designed by Alliance architect Russell Roller and constructed by members of the Builder's Exchange. The extraordinary lighting was provided by Ohio Public Service. The A.A.C.'s public relations campaign was exceptional. Their small advertisement boxes on the front page of the newspaper appeared for a two-week period. Examples include "Santa Claus prepares bi-plane for long journey to our city" and "Claus held up due to severe storms." Santa did arrive on December 6, 1929, and the Alliance Review headlines read, "Throngs Welcome Santa Claus To Alliance." Here we see Santa at his upstairs window tossing kisses to his admirers.
This is Kresge's Store, shortly after it opened on Main Street in October 1925. S.S. Kresge Company acquired the building that housed J.H. Johnson Sons Company, a prominent furniture business that was constructing a new building down the street on the old Moushey property. Kresge's cost $150,000 to build and remained there for nearly 60 years. The named was changed to Jupiter, but was still part of the chain. With the development of malls and larger retail stores, Kresge's regrouped and opened the Kmart stores. By the late 1980s, they closed all of their small 5 and 10¢ stores.
In August 1924, the McCrory Stores Inc. took out a 30-year lease on the Seitner Block, located on the northwest corner of Linden and Main Street. By February 1925, McCrory's made the decision to demolish the three-story structure and build one of its famous terra-cotta and brick structures. Even prior to McCrory's, this site was a prominent fixture in downtown retailing. T.W. Culp operated a dry goods store there for many years. He sold it to W.E. Davis, who continued the trade. Seitner purchased the business in 1921; however, they felt it was too close to their Canton branch, so they announced a close-out sale in 1924. The McCrory's store was later sold to the G.C. Murphy Company. It operated at this location until the early 1980s, but like many small downtown businesses, it fell victim to malls and larger retail stores.
This postcard is of the floral arrangements for the opening night of Jeanette's Restaurant located at 253 E. Main Street. Owned and operated by Jeanette Mathias for 28 years, the restaurant was noted for its fine food. In January 1976, Mrs. Mathias sold the business to Paul Halverstadt. He had formerly served as manager and supervisor of the Alliance Burger Chef and as general manager of Ponderosa. The name was changed to Paul's Supper Club.
The Koch Building was designed by Albrecht, Wilhelm, and Kelly prior to 1920. Built for Isadore and Joseph Koch, their grandfather started his clothing retail business here in 1858. It was one of the most fashionable and prominent textile stores in the area. After they closed their store, the F. & W. Grand 5-10-25 Cent Store used the building. The Upan-Inn Billiard Room was located on the second floor. In later years, the site was occupied by Sears.
The final and most dramatic performance of the Alliance Opera House, owned by the Craven family, occurred on a warm June morning in 1886. Considered by many to be the architectural pride of the town, it collapsed into a pile of rubble. The neighboring business, which was housed in a small frame building attached to the east wall of the opera house, had a great deal of trouble closing its door and found the building's window cracking. They determined it was a problem with the east wall. After a brief examination by J.T. Weybrecht and other prominent builders, the building was evacuated. Within minutes, the entire east and middle section crashed to the ground with no injuries. Historian Mabel Hartzell was across the street in 1886 having a dress fitted. She recalled vividly a loud crack and a great deal of dust. Once it cleared, she, along with many in the community, was in shock. The Alliance Opera House was built in the 1860s at a large sum of $80,000. The vibrations of passing trains and faulty construction were the cause of its demise. The west section was reconstructed and served many businesses, including Bruni Brothers. In the 1980s, the west wall collapsed and was reconstructed with concrete block.
Klein and Roderick was Alliance's most popular men's clothing and accessories store in the downtown area. Located at 344 East Main Street, it was established in May 1914 by Jacob Klein, Robert Ruth, and Harry Roderick.
Unfortunately, the men and event are not identified in this picture, but it is still a the great view of East Main Street. Taken in the block between Park Avenue and Mechanic Street in the 1920s, some of the business blocks include: Odd Fellows Lodge, Knights of Pythian, Brunswick Billiards (now Cornie's Steak House), Gardner's Ice Cream, and Nobil Shoes.
Many people gather across Main Street on a cold February morning in 1931 to view the damage of one of Alliance's most dramatic fires. The Spring-Holzwarth Department Store was completely destroyed and extensive damage was done to the Alliance Hardware Company (they lost the top floor after the conflagration). The night watchman was awakened by the explosion of glass, caused by the extensive heat on the floors above. Due to the excessively cold weather and inadequate equipment of the day, the fire was uncontainable. The new building was constructed on the same site for the store. Spring-Holzwarth sold out to the M. O'Neil Company of Akron in the late 1940s and remained in business until the early 1970s.
This Dimit Brothers photograph taken from the top of the City Savings and Trust Building shows the destruction of Spring-Holzwarth and the extensive damage to the top floor of the Alliance Hardware Company. The top floor of that building was removed. This view shows many of the surrounding businesses and industry outside the downtown area.
Construction on the Alliance Bank Building began in 1914 and was completed in the following year. It was located on the northeast corner of Main Street and Freedom Avenue, across from the city square. The new building created quite a sensation; it was the first skyscraper in the city. Many businesses have called this building home. In the 1970s, a brick awning connecting the adjoining buildings was added to the front entrance of the bank. The facade was removed in 1998, revealing the original design and character of the building.
The heading in the April 27, 1925 issue of the Alliance Review read, "THOUSANDS OF YOUNGSTERS AT BANK PARTY, Opening of new department by Alliance First National pleasing event." The party was held to introduce the new juvenile department of the institution. When the doors opened at 2 p.m., the paper said, "Tide upon tide of young Americans flowed into the spacious banking rooms until it was crowded to capacity." Lollipops, bells, and other souvenirs were given out to the city's young bankers. The new juvenile department was managed by Fred Zryd.
The City Savings Bank was organized in 1892. The old bank was housed in an L-shaped building with two entrances, one on the square and the other on Main Street. After the addition of a trust department, the name was changed to the City Savings and Trust Company. In the 1920s, they outgrew their building. The bank purchased Martin's Confectionery and J.F. Adams. Plans were drawn and construction started on an eight-story building, the highest in the city. The building was completed in 1926, and the bank moved in on May 5, 1926. The bank merged with the First National Bank of Alliance in the early 1970s; thus becoming The First National City Bank of Alliance. The City Savings and Trust Building remained vacant until the late 1980s when it was converted to housing units.
Pictured is the main lobby of the City Savings Bank. The floral arrangements are for the celebration of the opening day in May of 1926.
The Arcade Market was the largest retail space in Stark County when it was built in 1925. Located on the corner of Arch Street and Prospect, it was a shopper's mecca. The building was designed by architect W.W. Matchett on property owned by Walter Ellett and Spring-Holzwarth. There was an entrance on Main Street between the Spring-Holzwarth Building and the Alliance Hardware Company. These two interior shots show Brunner's Kitchen and one of the meat vendors. The pictures date back to about 1930.
The W.C. Ellett Cigar Store was located at 563 East Main Street in 1906. Ellett handled every brand of cigar, as well as smoking accessories, newspapers, periodicals, magazines, and sporting goods. According to a 1906 newspaper advertisement, "His stock of stogies is such as to comprise everything that has merit. Nothing is lacking to make his place ideal to the man who loves the weed, and if his favorite smoke is not found here it would be almost useless to look further."
The hustle and bustle of a busy afternoon in downtown Alliance is evident in this 1926 photograph of the northeast corner of Main and Arch Streets. Over the years, businesses in this block have come and gone; however, one still remains: The Alliance Hardware Company, which opened in this building in 1906. In July 1945, Eric Harrison purchased the store from the Bates family. His son and daughter-in-law, Gordon and Lucy Harrison, have continued with the business. Other businesses in the block are Klein and Roderick's, Spring-Holzwarth, McCrory's, and T.W. Cope Furniture.
# Six
# LOCAL COMMERCE
The men of the Alliance Dry Cleaning Company pose in front of the new plant on East Patterson Street on November 8, 1913. The advertisement in the local paper said, "There is not a better equipped plant in the United States than ours, there are some that are larger, but absolutely none better equipped to turn out good work. Not only is our machinery and methods the very best, but our buildings have been planned and built expressly for this purpose. The drawings were carefully prepared and approved by the government."
Hahn's Garage was located at 1412 West State Street in the area known as Hillcrest. It was owned and operated by C.F. Hahn. This photograph was taken on August 28, 1929. Pictured are Clair Hahn (extreme right); Ray Hahn (second from right); and Ralph Hahn (third from right). The other men are unidentified.
The E. Freshley Dairy horses (Injun and Prince) take a break during their early morning route at the corner of North Lincoln Avenue and West Ely Street. Glen (in wagon) and Percy Freshley posed for this photograph in 1913. Their father, Emanuel, was the founder of the company.
The Cook Block was located on the northeast corner of East Patterson Street and Webb Avenue. Construction on the building began on February 12, 1907 and was completed in 1908. This 1910 photograph shows a group of people in front of Hyne W. Barnes Grocery. Other businesses in the block over the years were Brogan's Cash Grocery, Home Restaurant, and C.E. Hunt Grocery. W.K. Sheckler, who owned a grocery across Webb Street, lived in one of the apartments upstairs. The building was torn down in the early 1980s.
The Mahoning Auto-Sales and Service Company was located at the corner of South Mahoning Avenue and East Patterson Street. This picture was taken on November 14, 1941. Incorporated in 1920 by Joseph Sarchione, the company sold Flint, Columbia, and Hupmobile cars until 1933, when Sarchione took over a Dodge franchise.
The Alliance Review Block on Linden Avenue, built in 1901, is pictured here on February 5, 1931. The newspaper was first published in September 1888 by James W. Gillespie. In 1890, Frank A. Hoiles took up the study of the printing trade. After several partnerships, he bought an interest in the Review in 1894. He remained its publisher until his death in an automobile accident in 1936. His widow, Alice C. Hoiles, took over her husband's position and continued his success. Donald A. Peterson, who married the Hoiles's daughter Josephine, on July 20, 1940, would eventually take over the business responsibilities, which included not only the newspaper, but the radio station WFAH (now WDPN).
The Manhattan Dry Cleaners was founded in 1914 by Mike Miller, a native of Serbia. He incorporated in 1924, and the first officers were Mr. Miller, president, and Mila Miller, secretary and treasurer. First located on Summit Street and Mahoning Avenue, the plant on 34 Selby (pictured) opened in the 1920s. This picture was taken in the 1960s during the company's 50th anniversary. Pictured, from left to right, are: Ray Beck, Doug Rhodes, Dan Farnam, Harry Miller, Cleata Moulin, Katie Lawson, Ruth ?, Mary Wilford, Anna Nutchel, Jennie Agnew, Martha Phillips, and Mamie Marrow.
The men of John Liber and Company unload meat in front of the Buckeye Market Store at 624 Scranton Street in the late 1940s. Joseph Mastroianni opened the 1,200-square-foot grocery store in September 1947. It was and continues to be a family operation. With financial backing from his mother and father, Joe was able to fulfill his dream. His sisters Helen and Marie worked for him, and his father, Philip, was his meat cutter. In the mid-1960s, Mastroianni decided to expand. He built a new store on the northeast corner of State Street and Fernwood Boulevard. Through the hard work of his employees and his longtime store manager, August (Tee) Massari, Buckeye Village Market gained the reputation that the customer is number one, living up to its motto "To Serve You Is Our Pleasure." In the late 1980s, Mastroianni built a new complex at 1800 West State Street. In 1997, the store celebrated its 50th anniversary and expanded to 65,000 square feet. The four men pictured on the right (from left to right) are: Tom Gazia, Alex Saus, Andy Vertolli, and Phil Mastroianni.
Employees of the Ohio Public Service honored the "Big Boss," Henry L. Doherty, on his birthday May 14, 1925, by planting a tree for each of the seven years in his business career at the Blue Bird sub-station on Irishtown Road (now Oyster Road). According to the Alliance Review, nearly 175 employees and their wives and sweethearts attended the affair. After the program, the group boarded machines (cars) for Lake Milton where a picnic supper and dance were held.
Caruso Brothers Macaroni Company was located on Patterson Street. It was the area's largest maker and distributor of pasta products. Unfortunately, the business was a victim of the 1929 crash, and Caruso Brothers was unable to maintain the clientele needed to keep the factory running.
The milkman poses with his horse and wagon No 1 outside the dairy plant at 115 Milner Street in the late 1930s. Founded as the Alliance Sanitary Milk Company, by H.F. Myers and L.M. Stockton, its capital was $25,000. It was reorganized by Ed Campbell in 1915, at which time the name was changed to Supreme Dairy Company.
The Alliance Sanitary Laundry's slogan was "Make Washday A Holiday." This photograph shows the plant from East Patterson Street looking towards the southwest in the early 1940s. The picture includes the following individuals (from left to right): Floyd E. Grabiel, (owner),Peter Grevu, Alice Pelanda Zucchero, Lena Pelanda Grabiel, Charles Grabiel (the boy), and a friendly neighborhood dog.
On July 10, 1942, a passerby on Patterson Street spotted flames shooting skyward from a building and immediately alarmed the Central Fire Station. When crews reached the Alliance Sanitary Laundry at 2 a.m., the brick and tile structure was engulfed in flames. The small water mains in the area made the water supply inadequate. This photograph shows the extent of damage. The Alliance Review headline stated "BLAZE TAKES CLOTHING OF 500 PATRONS, Half of City Hospital Linens Included In Loss." The business was never rebuilt after this disastrous event.
Thompson Triangle, in this July 23, 1929 picture, was located on the corners of East Patterson Street, North Arch Street, and Hester Avenue. In 1920, Bert Thompson opened what would be the first gasoline station between Canton and Salem. Throughout the 1920s and 1930s, the business rapidly expanded, adding services from major engine repairs to simple tire changes. In 1948, Bert's son Blair constructed a new building on this site and took on the Lincoln-Mercury car line. Ownership continued under the Thompson family until it was sold in the early 1980s to the Wittes. After several years, the business moved to West State Street, and the Thompson building was sold to the manufacturing firm, Filnor Incorporated.
W.K. and Homer Sheckler operated a grocery store on the northwest corner of East Patterson Street and Webb Avenue. It was destroyed by fire around 1911. The tragic fire not only destroyed their business, but it also killed Mrs. R.C. Spriggins and her child, who were renters in the building. Pictured are the Shecklers and their wagon.
The M.E. Biery Motor Company showroom at 32 North Arch Street is shown here on November 3, 1922. They were the original Dodge dealership in Alliance. Organized at this site in 1912 as the Central Motor Car Company, it was reorganized in 1922. Pictured is the latest model along with a stripped-down version in order to give the customer a detailed view of the car's motor and axles. From left to right are: Karl Wheeler, M.E. Biery, and Eb Lewis. Mr. Wheeler took over the business in 1929 and operated the agency for approximately five years, then Joe Sarchione purchased the Dodge dealership.
The Patterson Street Market was located at 766 East Patterson Street on the northwest corner of Webb Avenue. It was originally owned and operated by Harry Cohen. In 1947, Joseph A. (pictured) and Marian Cordingly bought the business and operated it until Mr. Cordingly's death in 1960. His widow sold it to the Hawkins family, who operated it for a few years. The building was abandoned for a number of years before it burned down. Ironically, it was the second fire to occur on this corner. Sheckler's Grocery Store burned around 1911.
City Market, on November 16, 1926, was located on South Freedom at Public Square. The sign in the window of the Arcade Crockery and Novelty Shop reads, "Forced to Vacate—SALE! This building is to be remodeled at once." Many would remember the structure better after its remodeling, when the beautiful colored-tile front and marquee that read "Morrison Theatre" were added.
Opening day was on March 5, 1949 for the Alliance Buick Company's new building on the southwest corner of Union Avenue and Main Street. John Caskey and Virgil B. Stamp formed a partnership in 1922, making their dealership the oldest in town. Mr. Stamp and his brother, J.L. Stamp, started in the automobile trade in 1916. According to the Alliance Review, this building had the most up-to-date automobile sales and service rooms in Ohio. The nearly 20,000 square feet of floor space gave ample room for offices, a display room, parts, and a service department. Probably the most unique feature of the sales room was the new indirect lighting system concealed by the "egg crate" ceiling. This system eliminated shadows and gave the show windows the appearance of being flooded with daylight. In the 1960s, John Caskey added the second floor and converted the building into the Caskey Motel. In the 1980s and 1990s, it was the popular Don Pancho's Mexican Restaurant. The building was demolished in 1996. Many will remember this building not for its cars or motel, but for the great roller skating rink on the lower level.
# Seven
# HOMES, HAPPENINGS
Designed by one of Ohio's most prominent architectural firms, Searles and Hirsch, this magnificent piece of architecture called Glamorgan (its exterior is Vermont marble) stands today as a testimony to the industrial revolution and the rewards reaped from it. Built between 1903 and 1907, this residence of Col. W.H. Morgan is similar to various baronial castles in Europe and England. It was not an old castle that once stood on land across the sea that was dismantled, its pieces numbered and reconstructed as many have said. Today, it serves as the administrative offices for the Alliance City School System. In the early 1990s, a group of concerned citizens headed by Joseph Mastroianni and Charles Grove founded the Glamorgan Castle Restoration Foundation. With the money raised by the group, many important structural corrections were made. The porte-cochere was removed and rebuilt, and the front wall, which blew out due to water and ice damage, was reconstructed.
Here is an early interior view of the W.H. Morgan Residence, Glamorgan. The sitting room, with its marble fireplace and chandeliers, was the room where Col. and Mrs. Morgan entertained their family and friends. It was the largest room in the castle. Today, its serves as the board of education meeting room.
The library, with its gothic chairs and massive library table, complement the built-in benches between the bookcases. One could sit in front of the lead and stained-glass windows and travel the world with a Melville or Stevenson novel.
Born in Chester County, Pennsylvania on October 10, 1827, Levi Lamborn studied medicine with the Alliance area's first physician, Soloman Shreve. In 1851, he married Maria Grant, daughter of Stacey and Jemima (Rockhill) Grant. Stacey was the son of John Grant. After the Civil War, he purchased six carnation plants from noted floriculturist Peter Henderson, who had produced them from the first carnation seeds brought to this country from France. Dr. Lamborn began experimenting with these plants. His development of the brilliant red carnation would bring him not only local recognition, but state and national prominence. His interest in politics won him the Democratic nomination for the 17th Congressional District in 1876. His opponent was a Canton attorney named William McKinley. Though they differed politically, they were personal friends. McKinley was a dinner guest at the Lamborn home numerous times. On one occasion, the host presented his new flower to McKinley for a boutonniere. McKinley considered it his lucky flower. McKinley won the seat by a narrow margin. Their friendship continued throughout McKinley's rise in politics as Ohio governor and U.S. President. In public, the flower became an inseparable part of the president except on one occasion. In September 1901, he gave his flower to a small girl in Buffalo, New York at the Pan-American Exposition. During the next hour, as McKinley stood in a reception line shaking hands, he was shot. He died seven days later. Of course, one flower did not change the course of history; the radical beliefs of a fanatic did. In 1905, the Ohio General Assembly passed a joint resolution selecting Lamborn's Scarlet Carnation as the state flower. In the late 1950s, the state named Alliance the "Carnation City."
This panoramic view was taken around 1915 at State Street School during summer Bible school. The children proudly display their talents, including hammock weaving, paper and
The Wilson family proudly pose in front of their home in the early 1900s. The farm was located on the south side of East State Street. Daniel Wilson was a brick mason for Alliance Clay Products Incorporated. He and his wife, Margaret, resided here for a number of years. Today, it is best known as the Robertson Youth Center.
cardboard construction, sewing (note the wonderful quilt in back), and raffia.
William (Billy) Mainwaring poses in a goat cart in 1925. Men would travel to each community in the early 1900s, earning money and a meal by taking pictures of children in their home-made cart.
William Reynolds, on a bench swing in the front yard, and Miss Lillian Reynolds, sitting on the porch at 48 East Main Street, posed for Ralph V. Miller on Monday, August 1, 1901 at 2 p.m. Their home stood where No Frills Grocery Store and Shaffer's Restaurant are located today. Miss Reynolds was a noted musician and teacher in the community. She was also remembered for her beautiful rose gardens. The home was known as "Rose Cottage."
This magnificent residence stood on the southwest corner of Union Avenue and Main Street. Built in the 1870s for W.W. King, a prominent community leader who owned a men's tailor store. In the 1890s, it was purchased by Mr. and Mrs. W.W. Cantine. He was a founder of the Alliance Gas and Power Company. Cantine was instrumental in guiding Alliance through the transition from gas to the new phenomenon, electricity. Mr. Cantine was also involved in a partnership with his son-in-law, Charles Y. Kay, in a hardware business. Cantine's granddaughter was the noted artist, author, and illustrator, Gertrude Alice Kay. After the death of Cantine, the home remained in the family as rental property. The Myers Funeral Home was located here until they purchased the Kay residence at 133 S. Union Avenue in the early 1940s. The Cantine Home was purchased by Virgil Stamp and John Caskey in 1948. They tore it down and built the Alliance Buick Company structure.
Edwin Morgan and one of his seal-brown trotting horses posed proudly in front of his magnificent home at 309 South Union Avenue. A son of Thomas R. Morgan, founder of the Morgan Engineering Company, Edwin was involved with the company only briefly. He ventured out on his own and served the community in many business and civic roles, including councilman and postmaster. In the 1950s, his home was moved to West Oxford Street to make way for construction of the B.F. Stanton Middle School.
Organized in 1894, Coterie Club is one of the oldest literary groups in Alliance. The membership, with a few exceptions, is handed down from generation to generation, from mother to daughter to granddaughter. This photograph was taken at the residence of Mrs. W.W. Webb on Overlook Drive in the spring of 1927 and includes a number of guests. From left to right, they are: (front row) Mrs. Walter Ellett, Mrs. Frank Cassaday, Mrs. Annie Burns, and Miss Eva McElroy; (second row) Mrs. Mollie Allen, Mrs. C.Y. Kay, Miss Louise Strong, Mrs. Sam Lane, and Mrs. G.G. Lamborn; (third row) Mrs. Grant Atwell, Mrs. Ettie Harris, Mrs. Webb, Mrs. Elizabeth Weybrecht, and Miss Helen Webb; (back row) Mrs. Fred Zang, Mrs. F. Shaffer, Mrs. B.C. Bates, Miss Jennie Love, Mrs. Jean Love, Mrs. Cathleen Ellett Eynon, Mrs. Irene Woolf, Mrs. Frank Ruth, Mrs. Wade Shidler, Mrs. Frank Dussell, and Mrs. Harry Roderick.
The Willis H. Ramsey residence was considered by many to be the second finest home in Alliance; Ramsey's brother-in-law, W.H. Morgan's home, Glamorgan, was first. This estate, which was four years in the making, was completed in 1910. The Jacobean-style mansion featured a brick and marble exterior, marble fireplaces, and fine woodwork throughout its interior. Mr. Ramsey was secretary-treasurer of his brother-in-law's firm, the Morgan Engineering Company. Ramsey was also a very active civic leader and trustee of Mount Union College.
This lovely residence, located at 229 South Union Avenue, was built for Walter W. Webb, a prominent banker in Alliance. He was a founder of the Industrial Savings and Loan Association in 1889. Mr. Webb's family can be traced back to the beginning of Alliance. Isaac Webb was involved in the expansion of land during the 1850s. He, along with Elisha Teeters and Mathias Hester, laid out various lots and additions prior to the incorporation of the town. The Webb residence was purchased by the newly organized Alliance Woman's Club in 1923. The first president was Mabel Hartzell. The porte-cochere was removed and the beautiful Georgian Room was added in the 1920s. Walter Webb's grandson, also named Walter, was born in this house. He was a prominent educator in the Alliance city schools and Mount Union College.
The Charles Alva Lane residence was located on South Union Avenue between Vincent and Simpson Streets. Lane and Alfred Hillgreen were owners of the Hillgreen-Lane Company, Organ Builders. Mr. Lane's residence was purchased by Mount Union College and served as the home of the college president and the Conservatory of Music. Presser Hall is located on this property today.
In the 1890s, an organization called "The Alliance Circle" was established for businessmen of the Alliance community. They would meet on a monthly basis for dinner and reading. In the early 1900s, the group had an extensive debate about including members from the suburbs. Mr. George Sebring, a founder of Sebring, Ohio, was considered. Each meeting was hosted by a member, so distance and muddy roads were a big issue. The group voted him in and decided to take the streetcar to his home. Pictured, from left to right, are: (front row) George Sebring, Tamerlane P. Marsh, David Fording, Silas J. Williams, John Morris, and Dr. John Tressell; (back row) W. H. Morgan, E.E. Scranton, J.H. Sharer, W.W. Cantine, and Willis Ramsey.
This pre-1915 photograph shows an auto parade on Market Street between Union Avenue and Park Avenue. At the end of Market Street, you can see the Charles Y. Kay residence, now Cassaday-Turkle-Christian Funeral Home. The cars were either electric or gas-propelled. Women only drove electric.
The A. Fred Morris residence, built in 1927, is located on the northwest corner of Parkway Boulevard and Glamorgan Street. Mr. Morris was associated with the Morgan Engineering Company for nearly 60 years. He succeeded Col. W.H. Morgan as president in 1929. Morris's son and daughter-in-law, Fred and Barbara, purchased the home in the 1940s and resided here together until Fred's death in 1991. Barbara continued to reside here until her death in August 1997. After 70 years, the ownership of this beautiful residence changed hands in 1998.
Members of Sorosis, another woman's literary organization which was organized in 1894, posed for a photographer in the 1920s. The organization's motto was, "To widen your life without deepening it is only to weaken it." Some of the members included Frances Vaughn, Evalena Fetters, Alice Hoiles, Mabel Hartzell, Grace Shaffer, and Ruth E. Scott.
This photograph of the Daniel Brinton Cassaday home at 1464 South Union Avenue was taken during its construction in the early 1900s. Mr. Cassaday was a partner in the Cassaday-Turkle Funeral Home. The brick and tile mansion remained in the family until Cassaday's 's granddaughter, Barbara Cassaday Turkle, passed away in the early 1980s.
The Alliance Italian Band in 1924 is shown here. One person, Philip Mastroianni, is identified as standing second from the right holding the tip of the flag. The Italian immigrants began coming to Alliance around 1900. At first, they were a separate community, concentrated in the northeastern part of the city along Patterson, Liberty, Streets, and Webb Avenue. Some of the
Designed by one of Stark County's most prominent architectural firms, Albrecht, Wilhelm, and Kelly, the William H. Purcell residence on South Union Avenue was built in 1919–20. Mr. Purcell was the founder of the Alliance Machine Company and was associated with numerous business and civic organizations. Purcell's daughter Hazel and her husband, Clarence J. Rodman, took over the ownership of home after the death of Purcell. In the 1980s and 1990s, the home was occupied by the Methodist Church's Berea Home for troubled youth. The home was closed in the mid-1990s and is now occupied by students of Mount Union College.
early family names were Alfanis, Furcalows, and Andreannis. The success stories are extensive in the Italian community, including such names as Sarchione, auto dealer, and Mastroianni, grocery business, still have strong ties to the area.
Romanian immigrants were one of several ethnic populations to settle in Alliance in the early 1900s. They are still very active in the community today. These Alliance residents show some of their ancestral clothing from the old country. The photographs are dated 1919 and 1924.
This view of an open field was taken in 1910 by J. Otis Wilcox. It was taken at what is now the intersection of Glamorgan Street and Parkway Boulevard. This land was developed by the West Park Heights Realty Company. Some of Alliance's most prominent families were involved in the development of the west section of town. Among them were the Scranton, Cope, Stamp, and Wilcox families. Construction in this part of town began in 1911 and continued until the late 1920s, when the Depression halted further development. In the late 1930s, a resurgence of construction began with a new generation of families. The development was called Fernbrook and continued with the original 1910 street design. The first Federal Housing Authority home constructed in Alliance was in this development on Kingsway.
This 1915 view shows the newly constructed homes of A.L. Cope, J. Otis Wilcox, and Dr. L.F. Mutschman. The dirt road is now Parkway Boulevard. The only pavement completed were the sidewalks. Note the barn on the far left; this is where the homes of H.W. Gaston and A. Fred Morris Sr. at the Glamorgan Street intersection were built.
The Hungarian Home, located at 1351 South Webb Street was dedicated on Sunday, February 26, 1928. Preceding the dedication were church services held at the Hungarian Presbyterian Church on East Cambridge Street. Headed by the Alliance City Band, area delegations together with the Alliance delegation marched to the Hungarian Home. Mount Union College President W.H. McMaster delivered the speech. During the Depression, the home fell on hard times. Through the efforts of John Korosy, a prominent grocer, and the Verhovay Fraternal Association, a $2,000 note was paid. Pictured are the officers in the 1940s. From left to right, they are as follows: (front row) Mike Korosy, Mike Buzagany, John Korosy, Joseph Maté, Emery Sera, and John E. Dugan; (second row) Martin Magyaros, Michael Fulop, Ignatz Bara, and George Toth; (back row) Mr. and Mrs. John Barany (house stewarts), Andrew Keszeg, Louis Muranyi, and Michael Simo.
This 1940s interior view is of the Verhovay Hungarian Home. Known for its fine cooking, the club held dinners frequently. Attending this event are the Korosy, Miller, Bodo, Tosha, and Elteto families. Clarence Steffy (at right with pipe in mouth) was the editor of the Alliance Review and a guest that evening.
The Charles Rice residence is pictured her in 1895, shortly after its construction. It was located on Union Avenue at Rice Street and served as Rice's home and dentist office. Rice (on left) and a friend pose outside his office entrance. Rice was known for his interest in history and his collection of antiques, and, after his death in 1939, a three-day auction was held at Mount Union College in which items like a bookcase from Independence Hall, a fireplace mantel from Lincoln's homestead, and many historic paper documents were sold. A Gilbert Stuart painting of George Washington was given to the Ohio State Archaeological Society (now the Ohio Historical Society). Rice traveled extensively, and, during a trip around the world with William Jennings Bryan, they were quarantined for a month in the Suez Canal for bubonic plague. On another trip with a young companion from Mount Union College, Rice toured the battlefield of Waterloo in a cart with King Edward VII, the Prince of Wales (the future King George V), and two grandchildren of the king (one was the future King George VI, and the other was the Duke of Windsor, Edward VIII, who abdicated the throne). Rice also attended the Jubilee celebration of Queen Victoria.
Dr. Rice used to hold an "Old Folks Day" party each year during the late 1890s and early 1900s. The party would start with a meal at the Mount Union Methodist Episcopal Church and end at Rice's home. This photo shows Elizabeth Byers, who, at age 102 (the oldest guest), was brought to the party by her youngest son William, a Civil War veteran. The Alliance Review covered the event and stated, "At 102 years of age she was the most active, the wittiest and jolliest of all the large crowd. No words can express what the spectators felt as they saw Mrs. Byers at the spinning wheel as she had done 80 years ago, and all the time, keeping up the most astonishing conversation and flow of repartee."
When the automobile became an acceptable form of transportation, many of Alliance's prominent families purchased these new and unusual devices. One thing was the norm: men drove gas-propelled vehicles and women drove electric cars. But not Emily Brosius Weybrecht. She insisted on having a gas-propelled machine, and this vivacious woman got her way. She is pictured here behind the wheel in her driveway on Union Avenue in 1915 while her husband, Charles, looks on with great pride.
Hundreds of people gathered outside the C.C. Weybrecht home as his body was placed in the hearse. His untimely death was the result of one of Alliance's most noted tragedies, the olive poisoning. Col. Charles C. Weybrecht's welcome-home party on August 23, 1919, hosted by Helen Sebring Gahris at Lakeside Country Club in Canton, ended with the illness of 13 and the death of seven. The first to die was the waiter, Bob Jennings. Weybrecht (who was a national war hero, having survived the great battles of the Spanish-American War, the Mexican Border War, and WW I) died about the same time as Mrs. Gahris. Mr. and Mrs. John Sharer, Mrs. Willis Sanford, and W.D. McElroy died within a few days. The timing of their deaths was determined by the number of olives eaten. At the time, some doctors thought that the culprit was the canned turkey; however, it was served to many on that day. The Gahris table guests were the only victims. Earlier that day, Mrs. Gahris stopped at a grocery store. She wanted her table to be festive, so she purchased candy, nuts, and a jar of ripe olives. After many tests, it was determined that botulism in the olives was the cause. In 1920, Hugh S. Cummings, U.S. Surgeon General, published a detailed account of the tragedy.
Abraham Greenawalt was a member of Company raham Greenawalt was a member of Company G, 104 Regiment of the Ohio Volunteer Infantry. Born in 1834, he enlisted on August 7, 1862 and was discharged on June 17, 1865. His service in the military was unlike anyone else's in the community. He received the Congressional Medal of Honor for capturing a rebel gun, flag, and solider.
Seen here is the Howard Myers residence on West State Road in 1901. The family gathered to celebrate the 50th wedding anniversary of the Myers couple. In later years, the home was remodeled and was the residence of Rex Russell.
# Eight
# EVENTS FOR MANY
Alliance had its share of theaters and nickelodeons over the years. This bill from the Columbia Theatre is an example of the many live attractions that appeared on their stage. The Columbia offered both live performances and movies. Alliance resident Brian S. Bara wrote to theater and film actress Ruth Gordon (known for her roles in Rosemary's Baby and Every Which Way But Loose) in the 1970s for an autograph. The aged actress responded to his request with a letter recalling, with great excitement, the time she played a one-niter at the Columbia in Alliance, Ohio. It is amazing how this woman, who worked in theater, films, and television for over 70 years, would remember that night. Columbia was an impressive theater in its day.
The Morrison Theatre opened for business on September 1, 1927; it was located on Freedom at Public Square. The Alliance Review reported that "Thousands and thousands and thousands of people thronged the new Morrison Theatre Thursday afternoon and evening when the new playhouse opened its doors to the public." Perhaps the Review's writer was overly excited by the beauty of the theater and didn't take an accurate count, but it was a big day. The luxurious carpeting, velvet curtains, chrome rails, Hillgreen-Lane's Organ, and the fact that the Morrison was the only theater in Ohio to have 100 percent indirect lighting put it in a class of its own. The opening day included live acts and a motion picture. In the 1970s, the Morrison, with its terra-cotta front, and several other buildings were demolished to make way for the restoration of the public square.
One of Alliance's smaller theaters was the Ideal Theatre, located on the south side of East Main Street in what would later become Turner's Reliable Drugs. Managed by Lemotto Smith, who also owned the Columbia and the Strand, this nickelodeon opened around 1910. For a nickel, one could see a western or a romance. The building was very narrow, so seating was limited. Often, flicks were shown four to five times a day with a piano player providing the background music.
Opening in September 1908, after being built within 100 days, the Columbia Theatre offered vaudeville acts, stage plays, musicals, and films. It was on the circuit that presented Victor recording artists and various stock and opera companies from Chicago and New York. It was located on East Columbia Street between Linden and Freedom Avenues. On March 29, 1949, firemen were summoned to the theater at 1:57 a.m. When they arrived, flames were shooting through the ventilators and windows, and it was 8 a.m. before the fire was completely put out. The manager, Joe Gordon, estimated the loss to be in excess of $100,000. Mr. Gordon based his estimate by saying it would cost approximately $705,000 to erect a new theater with comparable seating capacity of 784. Unfortunately, the building remained empty until it was razed in 1959.
The Strand was located in the Eagles Building on East Main Street. Designed by Albrecht, Wilhelm, and Kelly, the building's first floor was used by Cohn's Department Store. After it closed, the space was converted to a theater. The late Joe Gordon recalled that it was the only double-feature house in town.
This 1920s photograph shows one of the typical live stage musicals and plays that were presented at the Columbia Theatre. Some of the many performances shown on stage were: The Prince of Pilsen, September Morn (presented by the LaSalle Opera Company of Chicago), and the original New York production of The Eternal Light.
The Mount Union Theatre opened in February 1939. This theater is one of the three early movie houses surviving in Stark County. The Palace Theatre of Canton and the Lincoln Theatre in Massillon are the other two. The interior design is art deco, with aluminum rails and the original deco light fixtures. The building has maintained most of its original appearance except for the removal of several rows of seating to extend the stage, and the painting over of the original art deco murals on each side of the screen many years ago.
A member of the Goat Hill football team (c. 1916) gets ready to punt the ball during a practice session. The Goat Hill team was one of the city's most popular athletic teams. They won numerous championships, both county and statewide.
Members of the Buckeye Athletic Club posed for this 1918 picture at the Mount Union College athletic field. They are, from left to right, as follows: Cletus Grisez, Russell Greenawalt, Dan McFadden, Ted Roberts, Lou Mohr, Paul Linke, Harry Davis, Charles Scott, Norman Moore, Elford Witherspoon (coach), Joe McGee, Nick Common, Russell "Baldy" Reese, Al McGee, Homer Urmson, ? Chapple, Buss Brown, Sam Artino, and Lawrence Saunier.
The Alliance High School's basketball team is seen here during their 1909–10 season.
Pictured here is the Goat Hill basketball team of 1916.
The Alliance Merchants Baseball Class A Champions of 1938 are pictured here, from left to right, as follows: (front row) Whitey Carroll, Mike Luch, Mark Phillipi (manager), and George Zupanic; (back row) Jimmy Long, Bob Riley, Larry Russell, and Joe Petro.
The New Way Restaurant bowling team of 1938 are lined up for this picture by their height, not their bowling ability. They are, from left to right, as follows: Jess Hays, Clarence Clegg, Homer Greenawalt, Tom Jones, Chris Whitaker, and Dick Jevis (sponsor).
The Alliance Machine Company YMCA indoor baseball team poses with their trophy after their championship season in 1925. From left to right, they are as follows: (front row) John Hostetler, Homer Greenawalt, Ed Cox, Dan Shea, "Spike" Morrissey, William Cox, Leo Kuklo, and "Goat" Jones; (second row) Frank Tanner, Eddie Davis, George Swindell, Emil Tschabold, Bill Russell, "Beanie" Scwartz, and Ben White.
This picture shows the Alliance Clay Products outdoor baseball team of 1928. Some of the men pictured include Mike Bilcze, D. Veazly, George Watson, Red Lapp, Mike Blizmick, and Harry Giovanini.
# Nine
# CITY GOVERNMENT, SERVICES
Alliance resident William David Jackson was the first black man to win a councilman's seat in Stark County. Elected in 1932, he served only one term. His disappointment with the system of city government quelled his interest to continue in politics. Alliance was a segregated town at the time. Though he was black, Jackson's skin was much lighter than other men of his race. This afforded him opportunities that other men of his race were unable to have. Jackson's daughter, Veetta Terrell, was quoted in the Repository in 1996 saying, "My father didn't talk about it. He never tried to put on airs. I really don't think he realized how important it was. In fact, we didn't even know he was the first until he died in 1950. The historical society put something about it in the paper." Jackson was employed with Transue-Williams Drop Forge for 30 years.
Members of the Alliance Police Department pose with Mayor Otis Upton Walker in 1904. From left to right, they are as follows: (standing) Capt. Perry D. Oswalt, Patrolmen Robert Green, and John Alexander; (seated) Mayor Walker and Police Chief P.D. Howell.
The Alliance Fire Department dates back to the 1870s, when a 45-foot cylinder-shaped cistern was built at the square, holding approximately 26,000 gallons of water. The first fire chief was August Tanner. In 1893, the department went from a volunteer to a paid basis with two regular firemen and three call men. The department's first equipment included a horse and patrol wagon with 43 fire alarm boxes. The chief 's wagon was added in 1895.
In 1811, Judeth Farnam made the first mail delivery on horseback from Deerfield to Canton. Soon after, Jesse Feltz opened a post office in his home in the Village of Lexington. When the Village of Freedom was established in 1838, the first post office was opened by David G. Hester. After 18 months, it moved to this home at 27 Garrison Street, just off of Keystone Street. Robert Buck became the first postmaster of Alliance, when the town was incorporated in 1854.
Excavation began on July 31, 1916, for the new post office building on the northeast corner of Arch Street and Market Avenue. The wagons lined up as J.C. Devine's steam shovel quickly removed the dirt. The back of the Main Street buildings located on the south side in the 300 block can be seen. At the far left are the Wick Block and the Jarman Printing Company (later the Heggy's Building). The building with the high-pitched roof is the Scranton Real Estate Office, originally the First Methodist Church building that was constructed in the 1860s.
The Alliance Post Office, shown here under construction, was built on the historic Cassaday property. The Cassadays were early settlers in the community and prominent in various businesses and financial institutions. This picture was taken on November 1, 1917, by Franklin M. Hull, superintendent of construction.
In 1882, the city entered into a contract with the Alliance Water Company for the construction of a water system and a pumping station. The water was pumped directly from the Mahoning River. In 1899, the city purchased the water works for $198,000, with Charles C. Baker serving as president of the board. Until 1914, the water was pumped directly from the river into the water system without treatment. A sand filtering system plant and coagulation basin were constructed and the chlorinating of water along with its filtration was provided. It was during this time that Westville Reservoir was constructed for emergency supply. This picture shows new lines being installed in 1925, which increased the capacity to 6 million gallons daily.
Dr. J.H. Tressell and Rev. H.E. Kilmer of the Reformed Church laid out the plans for a hospital in 1900. The institution was incorporated as the Reformed Deaconess Home and Alliance Hospital. The Whitacre home on East College Street (pictured) was purchased. Work began immediately to convert the house. An operating room was set up on the first floor and patient rooms were on the second floor. The dedication was held on April 17, 1901, with 600 people attending.
With the increase in population and the need for expanded services, the control of the hospital was turned over to the city. On June 12, 1912, the name was changed to Alliance City Hospital. In 1914, a bond issue was passed and a building committee was appointed. Architect Willard Hirsch of Cleveland designed this building. Completed in 1916, it was opened to the public on January 1, 1917. The hospital had 65 beds and a complete operating room; the equipment was purchased with the donations made by the many manufacturing plants in the city. In three years, the building proved to be to small, so an addition was built on the south side.
The stained-glass windows were still on order when this picture was taken in 1898. The men of Company K, 8th Ohio Volunteer Infantry were treated to a farewell dinner at the First Methodist Episcopal Church, just prior to leaving for Cuba. The church was still under construction at the time, and the dinner was held on the lower floor.
This panoramic view shows one of the contingency of men who were selected to serve in WW I. They pose with family and friends on Public Square in 1917 before leaving for boot camp. The men paraded down to the station for a grand and glorious farewell.
Segregation occurred in every community in America. Even in times of war, black men were separated from white. This 1917 photograph shows the black members of the selective service. Many of the WW I pictures in the collection of the Alliance Historical Society show men and their families posing at the square with great fanfare. This picture, taken by a side door of the city hall, shows some of the area's black men who served their country. Unfortunately, these men were denied the kind of grand send-off given to the white men because of their skin color. Their names are unknown.
In 1927, Charles K. Strain, a ten-year-old at the time, saw one of America's heroes flying overhead and waving. Col. Charles Lindbergh had flown his silver bird, Spirit of St. Louis, over Alliance shortly before noon on August 3, 1927. To a young boy in the 1920s, Lindbergh was what Michael Jordan is to basketball today, or what John Glenn was to many children in the 1960s: a super hero. When young Charlie ran home to tell his parents what he witnessed, they didn't believe him. News didn't reach most rural areas and farmers for a few days. Strain's parents would soon realize that it was indeed true. "America's Hero" was en route to Youngstown and Pittsburgh.
# Ten
# VICINITY VIEWS
Fish Creek School in 1899 was located on Courtney Road just west of Bandy Road in Smith Township. The school housed six grades with one teacher, which was typical of most rural one-room schoolhouses of the day. During the 1920s and 1930s, Mahoning County began the consolidation and closing of many one-room schools. Fish Creek closed in May 1939 and, in the fall, the students attended Maple Ridge School on Bandy Road. This building was sold to area farmer Homer Snyder in 1944. It was used as a storage building for many years.
North Benton's New York Railroad Line Station is pictured here on October 7, 1936. The history of North Benton can be traced back to 1803, when William Smith obtained the original grant of land from the government. This section was in Columbiana County until the land was broken up into three counties. This placed Smith's land in Mahoning County, and Smith Township was named in his honor. When a new highway passed through in the early 1800s connecting Cleveland and the Ohio River at Wellsville (now part of Route 14), it was hoped that the village of Benton might become the new county seat. However, the development of the Lake Erie and Ohio River Canal to the north and the Cleveland and Pittsburgh Railroad to the south prevented this. The village received its name from Sen. Thomas H. Benton of Missouri, a hard money advocate who, in his day, was an idol of all Democrats. He was in his prime at the time the town was organized. We can probably assume that the residents of this small community were staunch Democrats.
The students of the Quaker Hill School posed for this picture in 1897. Located on Johnson Road near Sebring, the school started in a log cabin which was used as a religious meeting house for the more conservative, known as Hicksites. Built in 1836, this building was sold to the Quaker Hill Cemetery Association in the 1940s.
The Mill School stood on the corner of Center Road and Westville Lake Road in Columbiana County. Built in 1885 for $1,749, it was closed in 1934. The building serves as a private residence today. This picture was taken in the late 1890s or early 1900s. The individuals pictured here, from left to right, are as follows: Ed Kuntzman, Berdie Fifer, Anna Bargess, Hattie Bargess, Elizabeth Oesch, Ellas Robb, Elsie Kuntzman, Clara Hahlen, Ada Clement, Homer Bush, Ed Bargess, Arnold Oesch, and the teacher, William Clement.
In 1857, Israel and Jacob Hole started the Damascus Academy, pictured here in the 1880s. In the 1860s, it was purchased by the Friends and served as a boarding school. The school was incorporated in 1885 as the Damascus Academy. The first graduating class, in 1887, consisted of four students. In 1910, the Goshen Township Board of Education leased the building to establish the Damascus High School.
The Science Hill Church was built in 1900 (the approximate date of picture is 1910) and held its first service on Thanksgiving Day October 21, 1900. The ante-building to the left provided shelter for the traveling minister's horse and buggy, storage for coal and wood for the heating stove, and restroom facilities. The cemetery is located underneath the pine trees behind the church. According to The Science Hill Story, published in 1967, "It is not known when, why, or by whom the name Science Hill—not a very churchly name—was given to the local church. The story has been heard but not verified that in the long ago a scientist lived to the rear of the church and school, standing on a hill, and that was a factor entering into naming the institutions." Russell Newburn, a life-long member of the church, related a story which sounds more realistic. "The organizers of the church were of German and Swiss decent and spoke very little English. When asked where they attended church, their response was Zion's Higel, which is German for Zion's Hill. Their broken word's sounded as if they were saying Science Hill."
This interior view of Science Hill Church in 1910 shows the sanctuary at Christmas time. According to Russell Newburn, the pews are still in use. The pulpit, communion table, Bible, hymnals, and sundry tree ornaments still exist and are in use on occasion. The reed organ is privately owned in western Ohio.
This patriotic float was made by the students of Washington Township School District No. 9. The date of the picture is not known; however, it appears to be the early 1900s. Washington Township School is now part of the Marlington School District.
The residence of Enos and Ann Heacock in Smith Township was built in 1837 by Enos's parents, Nathan and Dinah Heacock, who came to this area by covered wagon in 1825 from Bucks County, Pennsylvania. They bought a half section of land for a $1.50 per acre. They built a small log cabin in which they lived until this frame house was completed. The Heacocks were Hicksite Friends or Quakers. When the railroad was completed in 1850, Mr. Heacock rode on the very first passenger train. Enos acquired the land in 1890 and planted over 100 maple trees. Some of these trees still stand today. In one location, he developed a sugar camp and often wrote "Maples" as a heading on his letters. In his diary, Enos wrote that he hoped that someday "the big field," where Maple Ridge School now stands, would someday become a park. He would be pleased to know that a school stands there today.
The Fairmount Children's Home operated for nearly 100 years. Originally owned jointly by Columbiana and Stark Counties, it had a board of trustees and superintendent. After the Civil War, many children were left homeless. Because of this, the commissioners of five surrounding counties met in Salem, Ohio on June 25, 1874, to issue bonds for the purchase of land and the construction of buildings. One hundred and fifty acres were purchased approximately 3.5 miles south of Alliance. In October 1876, the administration building (pictured) was completed and dedicated. Soon after, a chapel, detention home, cottages, and school were completed. The home remained in existence until 1975.
The staff and children pose for a photographer in 1922. The administration building located behind them is still standing. Many of the other buildings and tower are no longer in existence.
Simon Maudru and James Tracy stand in front of the Maudru General Store in Maximo on December 15, 1899. Maudru's store was a favorite gathering spot for the people in this small community.
The children of the fifth grade class at Maximo School pose in front of the building in October 1942. The teacher was Leora Weaver.
We've decided to end this book with the Flag Day Parade of 1947. The parade was dedicated to the men who fought in WW II. Though we have covered many years in this book, this history will never be complete. Everyday is history, and, whether it is positive or negative, it must be recorded to educate future generations. This picture is looking toward Public Square on East Main Street and Freedom Avenue. The large board in front of the monument is the honor roll of men who served and or died during the war.
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A New Beginning, originally titled The Royal Ranger, was the twelfth and final novel in the Ranger's Apprentice series, written by Australian author John Flanagan. It was released in Australia on 1 October 2013, in New Zealand on 4 October 2013, and in the United States and Canada on 5 November 2013. In 2018, it was renamed A New Beginning and it became the first book in the Ranger's Apprentice sequel series, The Royal Ranger.
Plot
Will Treaty tries to cope with the death of Alyss, who died in a fire set in an inn by a gang leader (Jory Ruhl) when she went back inside the burning building to save a young child. Will's friends begin to notice that his once cheerful personality has grown grim and uninviting. After numerous attempts to "snap him out of it", Gilan, the new Ranger Commandant calls on Halt, Pauline, Cassandra, and Horace to discuss how to deal with Will. Halt suggests that Will take on an apprentice to take his mind off his quest for revenge.
Meanwhile, Princess Madelyn, the daughter of Horace and Cassandra, is upset with her restrained royal life. Against the will of her parents, Maddie sneaks out at night to use her sling to hunt small animals. One night, Cassandra and Horace confront Maddie and ground her to her room for a period of two weeks. Halt suggests that Maddie be the one taken on by Will, which would make Maddie the first female Ranger's apprentice in Ranger history. At the beginning of her apprenticeship, Will gives Maddie a letter from her parents, in which says she has been disinherited as a princess of Araluen. This is a desperate last resort by her parents to get her under control. Will proceeds to train Maddie, and as he focuses on her, his quest for revenge is slowly forgotten. When Gilan suggests Will take Maddie on a mission, Will accepts without reluctance.
Gilan assigns Will and Maddie to investigate the death of Liam, a Ranger in Trelleth Fief, a northwestern fief. Will and Maddie soon discover a plot by an illicit slave ring who kidnap children. The criminals first send a storyteller to villages which frightens the children with a story about the "Stealer in the Night". The storyteller seeks out a child who is likely being abused at home and also takes children who aren't quiet about speaking about the Stealer. Will learns that the Stealer in the Night — the leader of the slave ring — is actually Jory Ruhl, but he manages to set aside his revenge to save the children Ruhl has kidnapped. Will and Maddie go to the slavers' camp, where Will distracts the criminals, while Maddie frees the slaves. Unfortunately, while Maddie is successful in freeing the children, Will is captured by the gang and tied to a stake to be burned. Maddie then goes to save Will, ending in, Ruhl's death.
Six months later, Maddie is awarded her Bronze Oakleaf, and Cassandra offers her reinstatement as a princess. However, Maddie declines, saying she wishes to complete her apprenticeship instead. Cassandra is stunned, and the book concludes as Horace explains to her that Rangers have always been different. When Cassandra asks what she should do, Horace tells her they just have to live with it.
Reception
Kirkus Reviews remarked positively on the novel, calling it "An excellent addition to a favorite series; the short breather did Flanagan good."
References
External links
The Royal Ranger at the Random House Australia website
The Royal Ranger at the Random House New Zealand website
Ranger's Apprentice books
2013 Australian novels
Random House books
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package org.apache.lucene.analysis.ja;
import org.apache.lucene.analysis.CharFilter;
import org.apache.lucene.analysis.util.RollingCharBuffer;
import java.io.IOException;
import java.io.Reader;
/**
* Normalizes Japanese horizontal iteration marks (odoriji) to their expanded form.
* <p>
* Sequences of iteration marks are supported. In case an illegal sequence of iteration
* marks is encountered, the implementation emits the illegal source character as-is
* without considering its script. For example, with input "?ゝ", we get
* "??" even though "?" isn't hiragana.
* </p>
* <p>
* Note that a full stop punctuation character "。" (U+3002) can not be iterated
* (see below). Iteration marks themselves can be emitted in case they are illegal,
* i.e. if they go back past the beginning of the character stream.
* </p>
* <p>
* The implementation buffers input until a full stop punctuation character (U+3002)
* or EOF is reached in order to not keep a copy of the character stream in memory.
* Vertical iteration marks, which are even rarer than horizontal iteration marks in
* contemporary Japanese, are unsupported.
* </p>
*/
public class JapaneseIterationMarkCharFilter extends CharFilter {
/** Normalize kanji iteration marks by default */
public static final boolean NORMALIZE_KANJI_DEFAULT = true;
/** Normalize kana iteration marks by default */
public static final boolean NORMALIZE_KANA_DEFAULT = true;
private static final char KANJI_ITERATION_MARK = '\u3005'; // 々
private static final char HIRAGANA_ITERATION_MARK = '\u309d'; // ゝ
private static final char HIRAGANA_VOICED_ITERATION_MARK = '\u309e'; // ゞ
private static final char KATAKANA_ITERATION_MARK = '\u30fd'; // ヽ
private static final char KATAKANA_VOICED_ITERATION_MARK = '\u30fe'; // ヾ
private static final char FULL_STOP_PUNCTUATION = '\u3002'; // 。
// Hiragana to dakuten map (lookup using code point - 0x30ab(か)*/
private static char[] h2d = new char[50];
// Katakana to dakuten map (lookup using code point - 0x30ab(カ
private static char[] k2d = new char[50];
private final RollingCharBuffer buffer = new RollingCharBuffer();
private int bufferPosition = 0;
private int iterationMarksSpanSize = 0;
private int iterationMarkSpanEndPosition = 0;
private boolean normalizeKanji;
private boolean normalizeKana;
static {
// Hiragana dakuten map
h2d[0] = '\u304c'; // か => が
h2d[1] = '\u304c'; // が => が
h2d[2] = '\u304e'; // き => ぎ
h2d[3] = '\u304e'; // ぎ => ぎ
h2d[4] = '\u3050'; // く => ぐ
h2d[5] = '\u3050'; // ぐ => ぐ
h2d[6] = '\u3052'; // け => げ
h2d[7] = '\u3052'; // げ => げ
h2d[8] = '\u3054'; // こ => ご
h2d[9] = '\u3054'; // ご => ご
h2d[10] = '\u3056'; // さ => ざ
h2d[11] = '\u3056'; // ざ => ざ
h2d[12] = '\u3058'; // し => じ
h2d[13] = '\u3058'; // じ => じ
h2d[14] = '\u305a'; // す => ず
h2d[15] = '\u305a'; // ず => ず
h2d[16] = '\u305c'; // せ => ぜ
h2d[17] = '\u305c'; // ぜ => ぜ
h2d[18] = '\u305e'; // そ => ぞ
h2d[19] = '\u305e'; // ぞ => ぞ
h2d[20] = '\u3060'; // た => だ
h2d[21] = '\u3060'; // だ => だ
h2d[22] = '\u3062'; // ち => ぢ
h2d[23] = '\u3062'; // ぢ => ぢ
h2d[24] = '\u3063';
h2d[25] = '\u3065'; // つ => づ
h2d[26] = '\u3065'; // づ => づ
h2d[27] = '\u3067'; // て => で
h2d[28] = '\u3067'; // で => で
h2d[29] = '\u3069'; // と => ど
h2d[30] = '\u3069'; // ど => ど
h2d[31] = '\u306a';
h2d[32] = '\u306b';
h2d[33] = '\u306c';
h2d[34] = '\u306d';
h2d[35] = '\u306e';
h2d[36] = '\u3070'; // は => ば
h2d[37] = '\u3070'; // ば => ば
h2d[38] = '\u3071';
h2d[39] = '\u3073'; // ひ => び
h2d[40] = '\u3073'; // び => び
h2d[41] = '\u3074';
h2d[42] = '\u3076'; // ふ => ぶ
h2d[43] = '\u3076'; // ぶ => ぶ
h2d[44] = '\u3077';
h2d[45] = '\u3079'; // へ => べ
h2d[46] = '\u3079'; // べ => べ
h2d[47] = '\u307a';
h2d[48] = '\u307c'; // ほ => ぼ
h2d[49] = '\u307c'; // ぼ => ぼ
// Make katakana dakuten map from hiragana map
char codePointDifference = '\u30ab' - '\u304b'; // カ - か
assert h2d.length == k2d.length;
for (int i = 0; i < k2d.length; i++) {
k2d[i] = (char) (h2d[i] + codePointDifference);
}
}
/**
* Constructor. Normalizes both kanji and kana iteration marks by default.
*
* @param input char stream
*/
public JapaneseIterationMarkCharFilter(Reader input) {
this(input, NORMALIZE_KANJI_DEFAULT, NORMALIZE_KANA_DEFAULT);
}
/**
* Constructor
*
* @param input char stream
* @param normalizeKanji indicates whether kanji iteration marks should be normalized
* @param normalizeKana indicates whether kana iteration marks should be normalized
*/
public JapaneseIterationMarkCharFilter(Reader input, boolean normalizeKanji, boolean normalizeKana) {
super(input);
this.normalizeKanji = normalizeKanji;
this.normalizeKana = normalizeKana;
buffer.reset(input);
}
/**
* {@inheritDoc}
*/
@Override
public int read(char[] buffer, int offset, int length) throws IOException {
int read = 0;
for (int i = offset; i < offset + length; i++) {
int c = read();
if (c == -1) {
break;
}
buffer[i] = (char) c;
read++;
}
return read == 0 ? -1 : read;
}
/**
* {@inheritDoc}
*/
@Override
public int read() throws IOException {
int ic = buffer.get(bufferPosition);
// End of input
if (ic == -1) {
buffer.freeBefore(bufferPosition);
return ic;
}
char c = (char) ic;
// Skip surrogate pair characters
if (Character.isHighSurrogate(c) || Character.isLowSurrogate(c)) {
iterationMarkSpanEndPosition = bufferPosition + 1;
}
// Free rolling buffer on full stop
if (c == FULL_STOP_PUNCTUATION) {
buffer.freeBefore(bufferPosition);
iterationMarkSpanEndPosition = bufferPosition + 1;
}
// Normalize iteration mark
if (isIterationMark(c)) {
c = normalizeIterationMark(c);
}
bufferPosition++;
return c;
}
/**
* Normalizes the iteration mark character c
*
* @param c iteration mark character to normalize
* @return normalized iteration mark
* @throws IOException If there is a low-level I/O error.
*/
private char normalizeIterationMark(char c) throws IOException {
// Case 1: Inside an iteration mark span
if (bufferPosition < iterationMarkSpanEndPosition) {
return normalize(sourceCharacter(bufferPosition, iterationMarksSpanSize), c);
}
// Case 2: New iteration mark spans starts where the previous one ended, which is illegal
if (bufferPosition == iterationMarkSpanEndPosition) {
// Emit the illegal iteration mark and increase end position to indicate that we can't
// start a new span on the next position either
iterationMarkSpanEndPosition++;
return c;
}
// Case 3: New iteration mark span
iterationMarksSpanSize = nextIterationMarkSpanSize();
iterationMarkSpanEndPosition = bufferPosition + iterationMarksSpanSize;
return normalize(sourceCharacter(bufferPosition, iterationMarksSpanSize), c);
}
/**
* Finds the number of subsequent next iteration marks
*
* @return number of iteration marks starting at the current buffer position
* @throws IOException If there is a low-level I/O error.
*/
private int nextIterationMarkSpanSize() throws IOException {
int spanSize = 0;
for (int i = bufferPosition; buffer.get(i) != -1 && isIterationMark((char) (buffer.get(i))); i++) {
spanSize++;
}
// Restrict span size so that we don't go past the previous end position
if (bufferPosition - spanSize < iterationMarkSpanEndPosition) {
spanSize = bufferPosition - iterationMarkSpanEndPosition;
}
return spanSize;
}
/**
* Returns the source character for a given position and iteration mark span size
*
* @param position buffer position (should not exceed bufferPosition)
* @param spanSize iteration mark span size
* @return source character
* @throws IOException If there is a low-level I/O error.
*/
private char sourceCharacter(int position, int spanSize) throws IOException {
return (char) buffer.get(position - spanSize);
}
/**
* Normalize a character
*
* @param c character to normalize
* @param m repetition mark referring to c
* @return normalized character - return c on illegal iteration marks
*/
private char normalize(char c, char m) {
if (isHiraganaIterationMark(m)) {
return normalizedHiragana(c, m);
}
if (isKatakanaIterationMark(m)) {
return normalizedKatakana(c, m);
}
return c; // If m is not kana and we are to normalize it, we assume it is kanji and simply return it
}
/**
* Normalize hiragana character
*
* @param c hiragana character
* @param m repetition mark referring to c
* @return normalized character - return c on illegal iteration marks
*/
private char normalizedHiragana(char c, char m) {
switch (m) {
case HIRAGANA_ITERATION_MARK:
return isHiraganaDakuten(c) ? (char) (c - 1) : c;
case HIRAGANA_VOICED_ITERATION_MARK:
return lookupHiraganaDakuten(c);
default:
return c;
}
}
/**
* Normalize katakana character
*
* @param c katakana character
* @param m repetition mark referring to c
* @return normalized character - return c on illegal iteration marks
*/
private char normalizedKatakana(char c, char m) {
switch (m) {
case KATAKANA_ITERATION_MARK:
return isKatakanaDakuten(c) ? (char) (c - 1) : c;
case KATAKANA_VOICED_ITERATION_MARK:
return lookupKatakanaDakuten(c);
default:
return c;
}
}
/**
* Iteration mark character predicate
*
* @param c character to test
* @return true if c is an iteration mark character. Otherwise false.
*/
private boolean isIterationMark(char c) {
return isKanjiIterationMark(c) || isHiraganaIterationMark(c) || isKatakanaIterationMark(c);
}
/**
* Hiragana iteration mark character predicate
*
* @param c character to test
* @return true if c is a hiragana iteration mark character. Otherwise false.
*/
private boolean isHiraganaIterationMark(char c) {
if (normalizeKana) {
return c == HIRAGANA_ITERATION_MARK || c == HIRAGANA_VOICED_ITERATION_MARK;
} else {
return false;
}
}
/**
* Katakana iteration mark character predicate
*
* @param c character to test
* @return true if c is a katakana iteration mark character. Otherwise false.
*/
private boolean isKatakanaIterationMark(char c) {
if (normalizeKana) {
return c == KATAKANA_ITERATION_MARK || c == KATAKANA_VOICED_ITERATION_MARK;
} else {
return false;
}
}
/**
* Kanji iteration mark character predicate
*
* @param c character to test
* @return true if c is a kanji iteration mark character. Otherwise false.
*/
private boolean isKanjiIterationMark(char c) {
if (normalizeKanji) {
return c == KANJI_ITERATION_MARK;
} else {
return false;
}
}
/**
* Look up hiragana dakuten
*
* @param c character to look up
* @return hiragana dakuten variant of c or c itself if no dakuten variant exists
*/
private char lookupHiraganaDakuten(char c) {
return lookup(c, h2d, '\u304b'); // Code point is for か
}
/**
* Look up katakana dakuten. Only full-width katakana are supported.
*
* @param c character to look up
* @return katakana dakuten variant of c or c itself if no dakuten variant exists
*/
private char lookupKatakanaDakuten(char c) {
return lookup(c, k2d, '\u30ab'); // Code point is for カ
}
/**
* Hiragana dakuten predicate
*
* @param c character to check
* @return true if c is a hiragana dakuten and otherwise false
*/
private boolean isHiraganaDakuten(char c) {
return inside(c, h2d, '\u304b') && c == lookupHiraganaDakuten(c);
}
/**
* Katakana dakuten predicate
*
* @param c character to check
* @return true if c is a hiragana dakuten and otherwise false
*/
private boolean isKatakanaDakuten(char c) {
return inside(c, k2d, '\u30ab') && c == lookupKatakanaDakuten(c);
}
/**
* Looks up a character in dakuten map and returns the dakuten variant if it exists.
* Otherwise return the character being looked up itself
*
* @param c character to look up
* @param map dakuten map
* @param offset code point offset from c
* @return mapped character or c if no mapping exists
*/
private char lookup(char c, char[] map, char offset) {
if (!inside(c, map, offset)) {
return c;
} else {
return map[c - offset];
}
}
/**
* Predicate indicating if the lookup character is within dakuten map range
*
* @param c character to look up
* @param map dakuten map
* @param offset code point offset from c
* @return true if c is mapped by map and otherwise false
*/
private boolean inside(char c, char[] map, char offset) {
return c >= offset && c < offset + map.length;
}
@Override
protected int correct(int currentOff) {
return currentOff; // this filter doesn't change the length of strings
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,927
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<pre>public interface <span class="strong">EdgeTypeData</span></pre>
<div class="block">Requires that an EdgeData object maintain an EdgeType (TREE, BACK, FORWARD, CROSS, UNKNOWN)</div>
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<td class="colFirst"><code><a href="../../../../com/codemelon/graph/edge/EdgeType.html" title="enum in com.codemelon.graph.edge">EdgeType</a></code></td>
<td class="colLast"><code><strong><a href="../../../../com/codemelon/graph/edge/EdgeTypeData.html#getEdgeType()">getEdgeType</a></strong>()</code>
<div class="block">Retrieve the type of this EdgeData object</div>
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<td class="colFirst"><code>void</code></td>
<td class="colLast"><code><strong><a href="../../../../com/codemelon/graph/edge/EdgeTypeData.html#setEdgeType(com.codemelon.graph.edge.EdgeType)">setEdgeType</a></strong>(<a href="../../../../com/codemelon/graph/edge/EdgeType.html" title="enum in com.codemelon.graph.edge">EdgeType</a> edgeType)</code>
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,840
|
Darul huda Islamic school transport department aims to provide safe and efficient transport service to our students with a professional touch and personal attention. The school operates a fleet of bus, following the rules and regulations and specifications as laid by the Transport Authority. Exceptional safety standards, secure and comfortable transportation is guaranteed by our school.
All buses are provided with the system which will enable attendance confirmation, an advanced GPS system and special security camera system .
The school has set policies for the transport system and expects the parents and students to familiarize and strictly abide by the rules while using school bus services.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,863
|
{"url":"https:\/\/mcgp-mahidol.com\/the-jam-wkso\/article.php?1989f2=how-to-draw-215-degree-angle","text":"John de Witt. Step 1: Draw the arm PQ. DaveR December 21, 2018, 2:37am #2. 0 0. Express the angle measure as a fraction of 360\u00b0. Double click Rotate tool. Then we'll start getting into obtuse angles, 100, 110, 120, 130, 140, 150. Similarly, the triple arc marks an angle of 160 degrees. So on the protracter it would be 155 degrees. (as shown below) 2). A reflex angle is an angle which is more than 180 degrees and less than 360 degrees. It\u2019s actually a straight line. A 45-degree angle is half the size of right angle, which is 90 degrees. draw it with protractor as it is not a multiple of 15, This site is using cookies under cookie policy. The table below shows some angles that can be obtained by summing simpler ones in various ways So on the protracter it would be 155 degrees. By measuring the interior angle and subtracting it from 360 degrees. Click in the drawing Area to specify the start point 3. Same as the last example, we draw the standard angle of -23 degree on a xy plane. If the reflex angle measured 190 degrees, then there would be a 170 degree, or obtuse, angle opposite it. Draw a straight line (i.e. Draw three angles, one in each quadrant except the first, whose reference angle is $$60\\degree\\text{. You know what? Using a protractor, we draw another line MV at an angle of 42 degrees to it. This suggests that to construct a 60\u00ba angle we need to construct an equilateral triangle as described below. HOW TO: GIVEN AN ANGLE MEASURE IN DEGREES, DRAW THE ANGLE IN STANDARD POSITION. If you enter a quadrantal angle, the axis is displayed. Mark the left end as point O and the right end as point B. 1. Step 1:Draw a line segment. In geometry, there are different types of angles such as acute, obtuse and right angle, which are under 180 degrees. A reflex angle is an angle that measures more than 180 degrees and less than 360 degrees. Select the line. You could measure each of the point. The steps are: 1. 180 degrees is not a reflex angle it is a straight angle. A reflex angle is equal to the sum of 180 degrees and any of the primary angles (acute, right and obtuse angles). (Note: \"Degrees\" can also mean Temperature, but here we are talking about Angles) The Degree Symbol: \u00b0 We use a little circle \u00b0 following the number to mean degrees. Draw an angle that contains that same fraction of the circle, beginning on the positive x-axis and moving counterclockwise for positive angles and clockwise for negative angles. Step 2: Place the point of the compass at P and draw an arc that passes through Q. Working with a ruler and a square or other right angle substitute, you can make the angle without any special tools. pls don't delete my question.... i need Apart from reflex angle, the other 5 types of angles are: For example, 270 degrees is a reflex angle. It\u2019s used in the construction of regular pentagons, and that\u2019s the original purpose of the golden ratio. This dot represents the vertex of the angle. Activate the LINE command 2. However, instead of positive 30deg values from above answer, to simulate fine italic angle you need to use negative degree values (-5deg for example). This article teaches you how to draw a 90 degrees angle using a compass and a ruler. 6 years ago. 60 degree is one of the most basic constructions, which facilitates constructing angles of several other measures. Working with a ruler and a square or other right angle substitute, you can make the angle without any special tools. Thinner angles like 15 degrees, are fragile and the edge will roll to one side or the other with heavy use. A 45-degree angle is half the size of right angle, which is 90 degrees. Navigators, surveyors, and carpenters all use the same angle measures, but the angles start out in different positions or places. This time, we are going to find the reference angle of a negative angle: -23 degree. John de Witt. That will also be an 80 degree angle from AC. To construct 150 degree angle we first construct 60 degree angle and its steps are as follows - 1). To draw the line of 150 units, you will have to 1. And just to make sure that blue arc is measuring this angle right over here, not the outer one. Place a dot at one end of the arm. Clearly, you can draw the 200 degree angle by measuring an angle of 20 degrees. Now mark with the point where 180 degrees completes. Mark the point they intersect as \u201cZ\u201d To make the 30-degree angle, simply use your ruler to draw a line connecting \u201cA\u201d and \u201cZ.\u201d To learn how to measure out a 60-degree angle, read on! Now use the technique on this page's parent page (or other) to draw a 10 degree angle, from AD. Eeman. Place the center of the protractor above the center point where the two lines of the angle lie and make sure the line goes through 0\u00b0. A reflex angle is one that is more than 180 degrees but less than 360 degrees. After measuring the angle beyond 180 degrees, we need to add it to 180\u00b0 to get the required reflex angle. Even before having drawing the angle, I'd have known that the angle is in the first quadrant because 30\u00b0 is between 0\u00b0 and 90\u00b0.The reference angle, shown by the curved purple line, is \u2026 Step 2:Take the compass and open it up to a convenient radius. How to Draw a 60 o Angle To learn how to draw \u201cperfect\u201d angles, you really need to draw. That will also be an 80 degree angle from AC. Steps:. Mark the spot the arc crosses the vertex as \u201cX.\u201d Now place the tip of the compass on \u201cX\u201d and draw a second arc through the first arc. In the figure given below, the angle is a reflex angle which lies between 180\u00b0 and 360\u00b0. Reduce the fraction to simplest form. It\u2019s used in the construction of regular pentagons, and that\u2019s the original purpose of the golden ratio. To construct 135 degree angle we first construct 90 degree angle and its steps of constructions are as follows: 1). You draw a 360-205 angle, which is -> 155 degrees. (See the wikiHow article Construct a 90 Degrees Angle Using Compass and Ruler. Learn how to draw Angle simply by following the steps outlined in our video lessons. 215 degrees; 235 degrees; ... Another option is to draw magnetic meridians on the map. \u2026, our how much time will it take to cover a distance of 350 km\u200b, decimal number which when rounded off to,sat second decimal place can give,say 25.32.change numbers for different groups\u200b, Find the linear approximation of the functionf(x,y) = x+y2 + x at (2,3)\u200b, what is the length of arc of the sector with central angel p\u00b0and radius R? It is also called full rotation or full angle. And we got it wrong. The smallest angle I can make is 8 degrees. Answer:with scale, hand, pencilit chill This free points is for my friend Abhinav 2. You either need to make the radius larger or you need to reduce the number of segments used to make the arc. It\u2019s any angle \u2026 This is how large 1 Degree is . Coterminal Angles Coterminal angles are angles in standard position (angles with the initial side on the positive x -axis) that have a common terminal side.For example 30 \u00b0 , \u2212 330 \u00b0 and 390 \u00b0 are all coterminal. }$$ Find exact values for the sine, cosine, and tangent of each of the angles in part (a). How you can use these axes for drawing: If we look at the drawing in e-2 we can see that I have drawn a square around our ellipse. Recollect the property of a $30^o-60^o-90^o$ triangle. Drawing Reflex Angles. A right angle. Often you are required to construct some angles without using a protractor. Using a Protractor to Draw an Angle . the triangle. The minor and major axes cross each other at a 90 degree angle. A straight angle is equal to 180 degrees and full rotation is equal to 360 degrees. Answer:with scale, hand, pencilit chill This free points is for my friend Abhinav 2. Obtuse: an angle between 90 and 180-degrees. Turn ON Preview checkbox. Draw a reflex angle for the given measurements: Your email address will not be published. By using a protractor to measure the reflex angle itself. This lesson show how an angle of over 360 degrees can be depicted on a unit circle. HOW TO: GIVEN AN ANGLE MEASURE IN DEGREES, DRAW THE ANGLE IN STANDARD POSITION. And let me move the protractor out of the way so we can get a good look at it. Express the angle measure as a fraction of 360\u00b0. Then draw the resulting angle as described earlier. There are five main types of angles: Acute: an angle between 0 and 90-degrees. Therefore. Do 360-205 and u get 155. Works & looks great in all modern browsers which support skew. The map gives you a bearing of 215 degrees. Acute angle. 1. Step 2: Place the point of the compass at P and draw an arc that passes through Q. (1) p180 \u00d72\u03c0R (2) p360 \u00d7\u03c0R (3) p170 \u00d7\u03c0R2(4) p180 \u00d7\u03c0R\u200b, Multiply -8\/21x2y3 by -7\/16xy2 and verify your result for x=3 and y =2\u200b, write the following percentage in the fractions form 173.%\u200b. Angles can be effectively 'added' by constructing them so they share a side. Positioning initial and terminal sides An [\u2026] The angle which forms a straight line is called the 180-degree angle. An acute angle is the smallest angle measuring between 0 to 90 degrees, whereas the obtuse angle measures between 90 and 180 degrees. Hi. I seem to be unable to draw a simple 5-degree angle with the arc tool, because Sketchup always says, \u201cThe number of segments is too large\u201d. Use ruler and draw a Line segment OB of any convenient length. Press ESCwhere 150 is the length of the line and 30 is the angle the line makes with zero.If I had written 150<-330 at step 2, it will give the same result. If you are creating the angles by some techniques, the 10 degree angle can be drawn more accurately than the 80 degree angle can be drawn directly. The given angle may be in degrees or radians. Which means it\u2019s time to gather up all the tools we\u2019re going to need\u2014a piece of paper, a pencil, a ruler, and some string (or a compass if you want to be fancy)\u2014and then find a cozy place to do your angle constructing. Home Contact About Subject Index. Straight angle. Also, starting from the x axis (zero), however, this time we turn the terminal arm to the negative direction. Think about the geometries involved. Apart from these, there are three other types such as straight, reflex and full rotation. Full rotation is also termed as a full circle. Positioning initial and terminal sides An [\u2026] how many of you know about good day laptop contest. Construction of angle 105 degree using compass:(Refer attached image). For graphing, the angle's initial side is the positive x-axis; its terminal side is the green line, because angles are drawn going anti-clockwise.The curved green line shows the given angle. Then we'll start getting into obtuse angles, 100, 110, 120, 130, 140, 150. Your email address will not be published. This will result in a 270 degree semi-half degree of 3*pi*R radians. Place the centre of the protractor at the vertex dot and the baseline of the protractor along the arm of the angle. Mark the vertex of your angle anywhere on the paper. To draw a reflex angle (i.e. 3 cm. Angles are formed Step 7 Look at your paper. An angle that measures zero degrees, is called zero angle. The value of angle at that time would be the angle between the line segment. As per the definition of reflex angle, any degree which lies between straight angle (180\u00b0) and full rotation (360\u00b0) is a reflex angle. Draw three angles, one in each quadrant except the first, whose reference angle is $$30\\degree\\text{. And with Q as center draw an arc which cuts line segment QR at y . Step 3: Place the point of the compass at Q and draw an arc that passes through P. Let this arc cut the arc drawn in Step 2 at R. We start with a line segment ML. Do 360-205 and u get 155. A complete angle is equal to 360 degrees. As we have already discussed in the introduction, the reflex angle is the angle greater than 180 degrees and less than 360 degrees. This is shown in Constructing the sum of angles. How-to Video Series: How to Draw an Angle Bisector - YouTube And we got it wrong. For example 90\u00b0 means 90 degrees. Actually, it\u2019s just a pinch. Then u just draw the arc on the \"outer\" - bigger area. This shows how to use a protractor to draw an angle - 42 degrees in this example. Now use compass and open it to any convenient radius. The 45-degree angle can be useful for projects like painting diagonals on walls, marking trim, or completing crafts and decoration projects. can i send multiple lot numbers from a number.? You draw a 360-205 angle, which is -> 155 degrees. an arm of the angle). (as shown below) You can combine regular italic style with skew to achieve even better rendering (worked in my case). You can also draw your 200 degree angle by drawing an angle of 160 degrees. The 45-degree angle can be useful for projects like painting diagonals on walls, marking trim, or completing crafts and decoration projects. The required angle is outside the one that has been drawn. 1 0. We know that the angles in an equilateral triangleare all 60\u00ba in size. \u2026, 2) If market price 1800 sellingprice 1540. To construct 150 degree angle we first construct 60 degree angle and its steps are as follows - 1). After I draw the square I draw an \u201cX\u201d across it \u2026 It\u2019s the adorable angle. Using a Protractor to Draw an Angle . Answer (1 of 2): In the picture below, the single arc marks an angle of 200 degrees. 48. One Degree. Step 1: Draw the arm PQ. See drawing e-1. How to Construct a 90 Degrees Angle Using Compass and Ruler Draw a line AB and mark point O on it where angle is to be drawn. A straight angle. 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Use Ruler - Draw a Line segment QR of any convenient length. The Full Circle. 6 years ago. I use the Paint part of Windows 7 and i would like to find out if i can draw lines at varoiuse angles like 30 and 60 Degrees . Increase\/decrease angle until the line segment is right above the other side. And let me move the protractor out of the way so we can get a good look at it. Eeman. And with Q as center draw an arc which cuts line segment QR at y . Think about the geometries involved. Navigators, surveyors, and carpenters all use the same angle measures, but the angles start out in different positions or places. basically 155 and 205 degrees are drawn the same, it's the arc that makes the difference. This suggests that to construct a 60\u00ba angle we need to construct an equilateral triangle as described below. If the reflex angle measured 190 degrees, then there would be a 170 degree, or obtuse, angle opposite it. Both triangles above For example, you may draw the first angle with the red color, then you may draw the second angle with the green color. 0 0. The examples of reflex angle are 190 degrees, 220 degrees, 270 degrees, 320 degrees, etc. Straight: a 180-degree angle or straight line. This problem is connected to what is now called the golden ratio, but its classical name is extreme and mean ratio. An acute angle is the smallest angle measuring between 0 to 90 degrees, whereas the obtuse angle measures between 90 and 180 degrees. What is your magentic bearing? Type 150<30 and Hit ENTER 4. Most people don\u2019t \u2026 Right: a 90-degree angle. angle greater than 180\u00ba and less than 360\u00ba), proceed as follows: Subtract the reflex angle from 360\u00ba. Then find discount\u200b, \u091f\u094d\u0930\u0947\u0928 \u0930\u0928\u093f\u0902\u0917 \u0905\u091f \u090f \u0938\u094d\u092a\u0940\u0921 \u0911\u092b 25 \u0915\u093f\u0932\u094b\u092e\u0940\u091f\u0930 \u092a\u0930 \u0906\u0930 \u0939\u093e\u0909 \u092e\u091a \u091f\u093e\u0907\u092e \u0935\u093f\u0932 \u0907\u091f \u091f\u0947\u0915 \u091f\u0942 \u0915\u0935\u0930 \u090f \u0921\u093f\u0938\u094d\u091f\u0947\u0902\u0938 \u0911\u092b 50 \u0915\u093f\u0932\u094b\u092e\u0940\u091f\u0930a train is running at a speed of 75 kilometre per h Math Open Reference. Even before having drawing the angle, I'd have known that the angle is in the first quadrant because 30\u00b0 is between 0\u00b0 and 90\u00b0.The reference angle, shown by the curved purple line, is the \u2026 We know that the angles in an equilateral triangleare all 60\u00ba in size. Now use compass and open it to any convenient radius. This problem is connected to what is now called the golden ratio, but its classical name is extreme and mean ratio. 1 0. )Bisect the angle this way: Strike an arc through both legs of the 90\u00b0 angle. How to Draw a 60 o Angle To learn how to draw \u201cperfect\u201d angles, you really need to draw. Required fields are marked *. Use Ruler - Draw a Line segment QR of any convenient length. For graphing, the angle's initial side is the positive x-axis; its terminal side is the green line, because angles are drawn going anti-clockwise.The curved green line shows the given angle. At the end, we know that the standard angle = -23 degree. Now use compass and open it to any convenient radius. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Find the required angle on the scale and then mark a small dot at the edge of the protractor. (as shown below) basically 155 and 205 degrees are drawn the same, it's the arc that makes the difference. Using a protractor, we draw another line MV at an angle of 42 degrees to it. In trigonometry and most other mathematical disciplines, you draw angles in a standard, universal position, so that mathematicians around the world are drawing and talking about the same thing. Since the measure of a reflex angle is greater than 180 degrees and the protractor has the maximum measure of 180 degrees, therefore we need to follow the below steps to measure the reflex angle. We look at how much the angle has \u201copened\u201d as \u2026 Adding angles. Suppose, x is an acute or obtuse angle, and r is the reflex angle, then: Therefore, if we know the value of acute or obtuse angle, we can easily find the related reflex angle. You can specify conditions of storing and accessing cookies in your browser, Factorise the following with identies x2-9, hi, guys. Now we need to measure the rest of the angle from the 180 degrees till the given second line. OK, first step is to; get a protractor and draw a 180 degree angle. (as shown below) 2). Hence, 181\u00b0, 190\u00b0, 200\u00b0, 210\u00b0, 220\u00b0, 230\u00b0, 240\u00b0, 250\u00b0, 260\u00b0, 270\u00b0, 280\u00b0, 290\u00b0, 300\u00b0, 310\u00b0, 320\u00b0, 330\u00b0, 340\u00b0, 350\u00b0, 359\u00b0, are all reflex angles. Step 3: Place the point of the compass at Q and draw an arc that passes through P. Let this arc cut the arc drawn in Step 2 at R. We explain Drawing an Angle Over 360 Degrees with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. Home Contact About Subject Index. For example, 270 degrees is a reflex angle. how to draw 45 degree angle without protractor or angle tool This is a tutorial of geometry. regards Terry Nigel Use of calculator to Find the Quadrant of an Angle 1 - Enter the angle: in Degrees top input. In geometry, we will be introduced to different types of angles, such as acute angle, obtuse angle, right angle, straight angle, reflex angle and full rotation.The angle which measures 180 degrees is named as the straight angle. Next, flip the protractor upside down and line it up with line you just drew, then draw around the top of the upside down protractor. You can check the resultant angle, by measuring the interior angle and subtracting it from 360\u00b0. how to draw 45 degree angle without protractor or angle tool This is a tutorial of geometry. For every acute and obtuse angle, there is a reflex angle. Let us learn here how an angle is said to be reflex. Using a protractor and a straight edge, you'll draw lines across your map on the angle of declination. And just to make sure that blue arc is measuring this angle right over here, not the outer one. Reduce the fraction to simplest form. Which means it\u2019s time to gather up all the tools we\u2019re going to need\u2014a piece of paper, a pencil, a ruler, and some string (or a compass if you want to be fancy)\u2014and then find a cozy place to do your angle constructing. (as shown below) 2). A reflex angle is one that is more than 180 degrees but less than 360 degrees. To draw a reflex angle, we can put the protractor upside-down, and mark an angle pointing downwards, which will be more than 270 degrees and less than 360 degrees. Math Open Reference. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. }$$ draw a line segment which is double the size of one side of the angle you want to measure and place it on the side. In geometry, an angle is the space between two lines, rays or planes intersecting each other. So let's see what we-- oh, 155 degree angle, not 150 degree angle. Now use the technique on this page's parent page (or other) to draw a 10 degree angle, from AD. We start with a line segment ML. example 1250 in Radians second input as a fraction of \u03c0: Example 27\/5 \u03c0 or 1.2 \u03c0 then press the button \"Find Quadrant\" on the same row. As an example, by first constructing a 30\u00b0 angle and then a 45\u00b0 angle, you will get a 75\u00b0 angle. Step 3:Place the co\u2026 Angles are measured in degrees. Then u just draw the arc on the \"outer\" - bigger area. Easy, step by step how to draw Angle drawing tutorials for kids. So let's see what we-- oh, 155 degree angle, not 150 degree angle. This shows how to use a protractor to draw an angle - 42 degrees in this example. The right angle is exactly equal to 90 degrees. Place its pointer at O and with the pencil-head make an arc which meets the line OB at say, P. 1. 0 0. From each point of intersection (of the arc and legs), strike arcs of the same radius such that they intersect each other. Draw an angle that contains that same fraction of the circle, beginning on the positive x-axis and moving counterclockwise for positive angles and clockwise for negative angles. If you are creating the angles by some techniques, the 10 degree angle can be drawn more accurately than the 80 degree angle can be drawn directly. In trigonometry and most other mathematical disciplines, you draw angles in a standard, universal position, so that mathematicians around the world are drawing and talking about the same thing. Construct a 90\u00b0 angle and bisect it. There are two ways to measure the reflex angle. The double arc marks an angle of 20 degrees. The right angle is exactly equal to 90 degrees. Apart from these, there are three other types such as straight, reflex and full rotation. Mark the angle with a small arc as shown below.\n\nWaterloo Road Dvd Series 9, Owners Direct - Puerto Del Carmen, Life Expectancy Uk 2020, Topaz Sharpen Ai Tutorial, Pool Villa Club Lombok, The Rage Movie 2019, Love We Are Messengers Chords, Fusaro Pizza Forked River, Cimb Company Resolution For Opening Account, How To Make A Reborn Baby Doll Nursery, Society For Industrial And Organizational Psychology Mission Statement, Types Of Ellipsis And Substitution,","date":"2021-04-15 16:33:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5757089853286743, \"perplexity\": 698.344325154042}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038087714.38\/warc\/CC-MAIN-20210415160727-20210415190727-00360.warc.gz\"}"}
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import React, { Component } from 'react';
import PropTypes from 'prop-types';
import { connect } from 'react-redux';
import { Text, Button, View } from 'native-base';
import Paper from '~/components/Paper/';
class WelcomeScreen extends Component {
render() {
return (
<View>
<Paper>
<Text>Hi {this.props.firstname}!</Text>
<Text>
Welcome on the app. You can now contact book sellers and chat with them or sell your own book.
</Text>
</Paper>
<Text />
<Button full onPress={() => this.props.closeAuthScreen()}>
<Text>Continue</Text>
</Button>
<Button transparent full onPress={() => this.props.navigation.navigate('Presentation')}>
<Text>Watch the presentation</Text>
</Button>
</View>
);
}
}
WelcomeScreen.propTypes = {
closeAuthScreen: PropTypes.func.isRequired,
navigation: PropTypes.any.isRequired,
firstname: PropTypes.string.isRequired,
}
const mapStateToProps = ({ services: { account: { firstname } }}) => ({ firstname });
WelcomeScreen.navigationOptions = {
title: 'Welcome',
};
export default connect(mapStateToProps)(WelcomeScreen);
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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Eine Jam-Session (IPA: , ; von englisch jam: Jargon für "improvisieren" und session: "Sitzung", "Veranstaltung"; häufig auch Jamsession) ist ein zwangloses Zusammenspiel von Musikern, die üblicherweise nicht in einer Band zusammenspielen und -singen.
Die Jam-Session im Jazz
Auf Jam-Sessions spielen Jazzmusiker entweder Stücke, deren harmonische Schemata und Melodien allen Mitmusikern bekannt sind (sogenannte Jazzstandards) oder sie improvisieren frei. Oft wird die Rhythmusgruppe, meist bestehend aus Klavier, Kontrabass und Schlagzeug, für die Session im Voraus zusammengestellt, damit die hinzukommenden Musiker wissen, was für eine Musik sie erwartet. Das musikalische Material der Jam-Sessions bilden die Jazzstandards, die in Sammlungen wie dem Realbook dokumentiert sind. Vereinbart wird außer dem Stück lediglich das Tempo; weitere Einzelheiten (z. B. die Reihenfolge der Soli und deren Länge sowie weitere Interaktionen) ergeben sich aus dem Geschehen.
Jam-Sessions waren für die gesamte Jazzentwicklung von großer Bedeutung: Für Jazzmusiker, die ihr Geld in kommerziellen Studio- oder Tanzorchestern verdienten, geben sie die Gelegenheit, sich voll auszuspielen. In den frühen 1940er-Jahren etwa trafen sich viele Swing-Musiker (zum Teil aus Big Bands, zum Teil auch aus kleinen Formationen) im Minton's Playhouse in Harlem, um "after hours" (nach Mitternacht) noch ein wenig zu jammen. Aus diesen Treffen in den frühen Morgenstunden entstand der Bebop und damit die Grundlage für den gesamten Modern Jazz.
Die Jam-Session bei Contact Improvisation
Beim experimentellen Tanzen im Rahmen einer Contact-Improvisation-Jam existieren neben freien Jams verschiedene thematische Formen (focused jams):
Blind Jam: mit geschlossenen/verbundenen Augen
Underscore Jam
Silent Jam: ohne Musik
Siehe auch
Chase Chorus
Jam (Hip-Hop)
Jam Track
Offene Bühne
Piano Battle
Irish Folk Session (anderes Ablaufschema, originale Bezeichnung: Irish Traditional Music Session)
Einzelnachweise
Musikalische Veranstaltung (Jazz)
Konzerttyp
Musikalische Improvisation
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,259
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San Diego based product design agency.
Konrad+King is a design firm based in San Diego, CA. Some specialties include, User Experience, Product Design, Software Design, Connected Device Design and Product Strategy.
Arthis is product strategy and user experience design studio based in San Diego, CA.
Folkhack Studios is a consulting firm based in San Diego that specializes in Web Development & Design, Systems Architecture & Administration, Applications Development, Networking, Social Media Strategy, Copywriting, and SEO.
Coe Design is a Digital Marketing Agency based in California that specializes in Packaging Design, Branding, Brand Identity, Structural Packaging, Graphic Design, Logo Design, and Branding & Packaging Solutions.
Redfins Designs is a graphic design & web design studio located in San Diego, California. They focus in web design, graphic design, logo design, and branding.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 9,591
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Antonio Conte believes his decision to rest several of Chelsea stars for their Champions League demolition of Qarabag will be rewarded when the Blues face Arsenal on Sunday.
Conte left out David Luiz, Antonio Rudiger, Victor Moses and Alvaro Morata for Tuesday's 6-0 thumping of the Azerbaijan minnows at Stamford Bridge.
With those key players kept fresh and Eden Hazard coming off the bench for only his second appearance since close-season ankle surgery, Conte is confident Chelsea will go into their London derby in good condition to claim three valuable points.
"It was a good start, a perfect start us," Conte said.
"To play the first game of the Champions League, then to win with a good result, to score many goals, to finish the game with a clean sheet.
Conte's selection gamble was never in danger of backfiring as goals from Pedro, Davide Zappacosta, Tiemoue Bakayoko, Cesar Azpilicueta and Michy Batshuayi ensured Chelsea made a strong start to their group stage campaign.
Giving his understudies a chance for valuable game time was an added bonus for Conte, who said: "I wanted to give a chance to Michy and also to Andreas Christensen.
"To play in the Champions League means the coach trusts you.
"My message tonight was this: I trust all my players, not with only words, but with facts.
"Last season we played with only 13 players, to do that again would be crazy.
Conte was also encouraged by Zappacosta's first start following his deadline day move from Torino.
The Italy right-back capped a lung-bursting run with a cross that flew in for Chelsea's second goal.
"He played a really good game. For sure he knew very well my style of football and for this reason I decided to start with him," Conte said.
"It's not easy to change your life in one week, but his answer was very good.
Despite Hazard's solid showing as a second half substitute, Conte admitted it might be too soon for the Belgian playmaker to start against Arsenal.
"It's early to speak about this. We must have a bit of patience with Eden," Conte said.
"We must give him the possibility to recover very well, to be totally fit.
It was a chastening first taste of the Champions League group stage for Qarabag, but Gurban Gurbanov, boss of the Azerbaijan minnows, insisted it was still a proud day for his club.
"The game was very difficult for us, Chelsea played very well," he said.
"We made lots of errors. I accept we must play better but I'm still satisfied with the performance given the quality of the opponent.
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Home Series Do You Know? Do you know? About, "Seneca Rocks!"
Do you know? Seneca Rocks!
About, "Seneca Rocks!"
• The white Tuscarora quartzite rock formation rises over 900 feet above the North Fork River and is one of the best-known landmarks in WV.
• Didn't always look the way it did. The Seneca Rocks' Gendarme, nicked named "The Chimney" and "Gunsight" collapsed Oct. 22, 1987, at 3:27 pm providing the formation we see today.
• Since 1971, 15 people have died at Seneca Rocks from falls.
• In 1943-1944 the 10th Mountain Division climbed on Seneca Rocks to train for mountain warfare during World War II in the Alps.
• A 3.2 mile out and back trail leads hikers from the Visitor Center to the Observation Deck 800 feet above providing views similar to those who climb the rocks.
• There are more than 300 paths to the top of Seneca, but only one doesn't require climbing.
• Local legend says Princess Snowbird of the Seneca Tribe was to choose a mate but did so by climbing Seneca Rocks. The first Brave to meet her at the top would be hers. Seven Braves took on this challenge, but only one completed the climb and with the help of Princess Snowbird.
Explore, Learn, and Glow!
Subscribe to be updated when new posts are released.
Gunsight
North Fork River
Princess Snowbird
Seneca Rocks
Seneca Tribe
The Chimney
Previous articleDo you know? About, "Our Lady of the Pines."
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Do you know? About, "Falls of Hills Creek!"
WV Top 3: Carnifex Ferry Battlefield State Park
Dr. Appointments, Remembrance Day, Birthday, July Fourth, and Baby Preparations – July 2018 Update
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: Custom BSP for SRM UI Addon Is it possible to create a custom BSP application for SRM UI Addon instead of injecting pieces of code using custom JS/CSS.
A: Yes. After you maintain the custom BSP application, you will need to put the JS/CSS files you would like to be loaded in the customizing.
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{"url":"https:\/\/www.tutorialspoint.com\/minimum-number-of-page-turns-to-get-to-a-desired-page-using-cplusplus","text":"Minimum number of page turns to get to a desired page using C++.\n\nC++Server Side ProgrammingProgramming\n\nProblem statement\n\nGiven a book of N pages, the task is to calculate the minimum number of page turns to get to a give desired page K.\n\n\u2022 we can either start turning pages from the front side of the book (i.e from page 1) or from the backside of the book (i.e page number N).\n\n\u2022 Each page has two sides, front and back, except the first page, which has only backside and the last page which may only have backside depending on the number of pages of the book.\n\nIf N = 5 and K = 4 then we have to turn minimum 1 page \u2212\n\n\u2022 If we start page-turning from front then 2 turns are required (1) -> (2, 3) -> (4,5)\n\n\u2022 If we start page-turning from the back, (4, 5) 1 turn is required page turned = 1\n\nSo, a Minimum number of pages turned = 1.\n\nAlgorithm\n\nUse below formula to calculate final result \u2212\n\n1. If K is even, front distance = (K \u2013 0)\/2 and back distance = (N \u2013 1 \u2013 K)\/2\n2. If K is odd, front distance = (K \u2013 1)\/2 and back distance = (N \u2013 K)\/2\n\nExample\n\n#include <iostream>\n#include <algorithm>\nusing namespace std;\nint getMinPageTurns(int n, int k){\nif (n % 2 == 0) {\n++n;\n}\nreturn min((k + 1) \/ 2, (n -k + 1) \/ 2);\n}\nint main(){\nint n = 5, k = 4;\ncout << \"Required page turns = \" << getMinPageTurns(n, k) << endl;\nreturn 0;\n}\n\nOutput\n\nWhen you compile and execute the above program. It generates the following output \u2212\n\nRequired page turns = 1\nPublished on 31-Oct-2019 07:26:43","date":"2021-11-28 03:05:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5300755500793457, \"perplexity\": 1584.0962051436463}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358443.87\/warc\/CC-MAIN-20211128013650-20211128043650-00226.warc.gz\"}"}
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{"url":"https:\/\/www.karaoketexty.cz\/texty-pisni\/sody\/let-you-know-908253","text":"\u2022 V\u00edc\nReklama\n\n# Let You Know - text\n\nTrying to hold my head up\n'Cause I shouldn't have to blame myself\nTook what was taught as gospel\nBut I don't have a clue where to go form here\n'Cause I'm almost breaking, can't you tell?\nI don't even understand why I'm still here\nJust need a minute to go through with it\nI don't even understand why I'm still here\nI gave you my all but you still want more\nI thought you were proud but it's not enough\nTried to sell me a lie that you thought I'd buy\nYou don't decide, just thought I'd let you know\nJust thought I'd let you know\nJust thought I'd let you know\nNot trying to play the victim\nI just needed to get this off my chest\nIt feels like my senses have kicked in\nI don't even understand why I'm still here\nJust need a minute to go through with it\nI don't even understand why I'm still here\nI gave you my all but you still want more\nI thought you were proud but it's not enough\nTried to sell me a lie that you thought I'd buy\nYou don't decide, just thought I'd let you know\nJust thought I'd let you know\nJust thought I'd let you know\nI gave you my all but you still want more\nI thought you were proud but it's not enough\nTried to sell me a lie that you thought I'd buy\nYou don't decide, just thought I'd let you know\nJust thought I'd let you know\nJust thought I'd let you know\ufeff\nReklama\n\nReklama\n\n## Sody texty\n\nTento web pou\u017e\u00edv\u00e1 k poskytov\u00e1n\u00ed slu\u017eeb, personalizaci reklam a anal\u00fdze n\u00e1v\u0161t\u011bvnosti soubory cookie. Pou\u017e\u00edv\u00e1n\u00edm tohoto webu s t\u00edm souhlas\u00edte. Dal\u0161\u00ed informace.","date":"2020-02-27 23:05:42","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8522070050239563, \"perplexity\": 6643.416049084442}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875146907.86\/warc\/CC-MAIN-20200227221724-20200228011724-00055.warc.gz\"}"}
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The MIOPS app set up for active tracking of an object.
Earlier this year I wrote about the MIOPS Mobile Remote and Mobile Dongle, devices that work with DSLRs from a variety of manufacturers to provide smartphone control of the complicated cameras. The company is back with a Kickstarter campaign for another accessory that DSLR photographers are going to want, the Capsule360 Motion Control Box.
MIOPS has already blasted well past its $75,000 funding goal with a total of almost $315,000 at publication time with 16 days to go in the campaign.
As you can see, individual MIOPS Capsule360s can be linked for very complex 3-axis motions. Add in an optional dolly or a motorized slider, and any DSLR can be controlled with the iOS app to do very precise and repeatable motions. No longer does astrophotography require a telescope mount; just place a few Capsule360s together and use the app to track a bright star.
Combining 3 Capsule 360s, a slider and L-bracket for a full slide/tilt/pan motion controller.
The app also allows control through face tracking, perfect for using a DSLR as a "robot-controlled cameraman" for a one-person studio. This feature alone would be great for situations where a person wants to film him or herself walking around, but can't afford to hire a cameraman.
The battery life of each Capsule360 is about 8 hours while shooting videos, but up to 7 days while capturing motion time-lapse.
What's most impressive is that the premium pack right now is available for under $1,000 and includes three Capsule360s, a Capsule Slider, a Capsule Dolly, an L bracket (helpful for astrophotographers), a turntable kit and a Mobile Dongle.
The team expects to deliver the first Capsule360 in December of 2018, and you can be a part of this exciting development.
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Junctional ectopic tachycardia (JET) presents in two scenarios: The rarer idiopathic and incessant form seen early in infancy with a structurally normal heart and the more common self-limiting variety limited to the early postoperative period after surgical repair of congenital heart disease.
Diagnosis of JET generally requires a heart rate greater than 170 beats per minute with QRS morphology similar to the sinus rhythm QRS complex, atrioventricular dissociation, and ventricular rate faster than the atrial rate. JET can occur at a lower rate in adult population with increased vagal tone and slower atrioventricular (AV) conduction as compared to children. It can also present with 1:1 retrograde ventriculo-atrial conduction. In patients with postoperative bundle branch block, JET can manifest as wide QRS tachycardia with atrioventricular dissociation or 1:1 retrograde ventriculo-atrial conduction. In this scenario, the P waves may fall within the QRS or ST-T segment. At fast rates, sinus tachycardia and supraventricular arrhythmias can pose a diagnostic challenge. P waves can be the clue in such situations that can be identified with oesophageal leads or Lewis leads or the atrial pacing wires.
Risk factors associated with JET include younger patient age, duration of CPB and aortic cross-clamp time (ACC), electrolyte disturbances, use of inotropes and type of surgery. , Several studies have elucidated the risk factors associated with JET. [Table 1] Although higher dose of inotropic agents have been accepted in all major studies as a significant risk factor, the role of surgery involving peri-nodal areas is still debated. Moak et al., in their recent study had categorically refuted mechanical injury to the AV node area as a strong risk factor for postoperative JET.
The prospective study on postoperative JET by Abelaziz et al., published in this issue of the Annals, has analyzed 194 patients after their cardiac surgery under CPB. They have divided their cohort into 3 groups - one with postoperative JET, one with other postoperative arrhythmias and finally those with no arrhythmias. This division further helps to highlight the risk factors that may be more specific for JET with respect to other postoperative arrhythmias. They have used the conventional measures to diagnose JET, but have still highlighted the importance of atrial leads, Lewis leads and esophageal leads in confirming their diagnosis.
It is interesting to note that risk factors when compared between the 3 groups mentioned above have shown that CPB, aortic cross clamp time and postoperative use of inotropes were significantly higher in postoperative JET patients but there was no significant difference in these parameters between non-JET arrhythmia group and no arrhythmia group. This clearly highlights the importance of perioperative factors being a major determinant of JET and anatomical factors being a minor determinant only. The structural abnormalities and their corresponding electrophysiological changes may be stronger determinants of non-JET arrhythmias.
The management of JET patients requires a staged therapeutic approach beginning with conventional measures that include active avoidance of hyperthermia, optimal sedation and pain control and minimizing exogenous catecholamines. In one of the earlier publications evaluating the effectiveness of a staged protocol, Walsh et al., found that conventional measures could control JET in 24% of patients. Based on available data and institutional experience, they refined their original protocol by avoiding ineffective drugs like digoxin, verapamil, propranolol and phenytoin. These measures significantly reduced the time to JET control. In the case of non-responders or patients with hemodynamic instability, they have followed a sequential order of atrial pacing, induction of hypothermia with posterior cooling blankets and finally a combination of hypothermia and antiarrhythmic drugs like amiodarone and procainamide. In 70 of their 71 patients, JET was managed successfully with 63 patients responding within 2 hours.
The incidence of JET has gradually declined over the years with improved surgical techniques and preventive measures like minimizing inotropic usage and avoiding hyperthermia in the postoperative period. But JET continues to contribute significantly to postoperative morbidity. Management of JET with general supportive measures, active surface cooling and antiarrhythmics like amiodarone has reined this postoperative hazard to a great extent.
1. Herzog L, Lynch C. Arrhythmias accompanying cardiac surgery. In: Lynch C, editor. Clinical Cardiac Electrophysiology. 3 rd ed. Philadelphia: JB Lippincott; 1994. p. 231-58.
4. Mildh L, Hiippala A, Rautiainen P, Pettila V, Sairanen H, Happonen JM. Junctional ectopic tachycardia after surgery for congenital heart disease: Incidence, risk factors and outcome. Eur J Cardiothorac Surg 2011;39:75-80.
5. Dodge-Khatami A, Miller OI, Anderson RH, Gil-Jaurena JM, Goldman AP, de Leval MR. Impact of junctional ectopic tachycardia on postoperative morbidity following repair of congenital heart defects. Eur J Cardiothorac Surg 2002;21:255-9.
6. Walsh EP, Saul JP, Sholler GF, Triedman JK, Jonas RA, Mayer JE, et al. Evaluation of a staged treatment protocol for rapid automatic junctional tachycardia after operation for congenital heart disease. J Am Coll Cardiol 1997;29:1046-53.
7. Dodge-Khatami A, Miller OI, Anderson RH, Goldman AP, Gil-Jaurena JM, Elliott MJ, et al. Surgical substrates of postoperative junctional ectopic tachycardia in congenital heart defects. J Thorac Cardiovasc Surg 2002;123:624-30.
8. Pfammatter JP, Paul T, Ziemer G, Kallfelz HC. Successful management of junctional tachycardia by hypothermia after cardiac operations in infants. Ann Thorac Surg 1995;60:556-60.
9. Hoffman TM, Bush DM, Wernowsky G, Cohen MI, Wieand TS, Gaynor JW, et al. Postoperative junctional ectopic tachycardia in children: Incidence, risk factors and treatment. Ann Thorac Surg 2002;74:1607-11.
10. Grosse-Wortmann L, Kreitz S, Kreitz RG, Vasquez-Jiminez JF, Messmer BJ, von Bernuth G, et al. Prevalence of and risk factors for perioperative arrhythmias in neonates and children after cardiopulmonary bypass: Continuous holter monitoring before and for three days after surgery. J Cardiothoracic Surg 2010;5:85.
11. Moak JP, Arias P, Kaltman JR, Cheng Y, McCarter R, Hanumanthaiah S, et al. Postoperative junctional ectopic tachycardia: Risk factors for occurrence in the modern surgical era. Pacing Clin Electrophysiol 2013;36:1156-68.
12. Abelaziz O, Deraz S, Anticipation and management of junctional ectopic tachycardia in postoperative cardiac surgery: Single center experience with high incidence. Ann Pediatr Card 2014;7:19-24.
13. Rekawek J, Kansy A, Miszczaj-Knecht M, Manowska M, Bieganowska K, Brzezinska-Paszke M, et al. Risk factors for cardiac arrhythmias in children with congenital heart disease after surgical intervention in the early postoperative period. J Thorac Cardiovasc Surg 2007;133:900-4.
14. Valsangiacomo E, Schmid ER, Schupbach RW, Schmidlin D, Molinari L, Waldvogel K, et al. Early postoperative arrhythmias after cardiac operation in children. Ann Thorac Surg 2002;74:792-6.
15. Manrique AM, Arroyo M, Lin Y, El Khoudary SR, Colvin E, Lichtenstein S, et al. Magnesium supplementation during cardiopulmonary bypass to prevent junctional ectopic tachycardia after pediatric surgery: A randomized controlled study. J Thorac Cardiovasc Surg 2010;139:162-169.e2.
16. Mahmoud AB, Tantawy AE, Kouatli AA, Baslaim GM. Propranolol: A new indication for an old drug in preventing postoperative junctional ectopic tachycardia after surgical repair of tetralogy of Fallot. Interact Cardiovasc Thorac Surg 2008;7:184-7.
17. Kovacikova L, Hakacova N, Dobos D, Skrak P, Zahorec M. Amiodarone as a first-line therapy for postoperative junctional ectopic tachycardia. Ann Thorac Surg 2009;88;616-22.
18. Kelly BP, Gajarski RJ, Ohye RG, Charpie JR. Intravenous induction of therapeutic hypothermia in the management of junctional ectopic tachycardia: A pilot study. Pediatr Cardiol 2010;31:11-7.
19. Chrysostomou C, Beerman L, Shiderly D, Berry D, Morell VO, Munoz R. Dexmedetomidine: A novel drug for the treatment of atrial and junctional tachyarrhythmias during the perioperative period for congenital cardiac surgery: A preliminary study. Anesth Analg 2008;107:1514-22.
20. LeRiger M, Naguib A, Gallantowicz M, Tobias JD. Dexmedetomidine controls junctional ectopic tachycardia during Tetralogy of Fallot repair in an infant. Ann Card Anaesth 2012;15:224-8.
21. Bronzetti G, Formigari R, Giardini A, Frascaroli G, Gargiulo G, Picchio FM. Intravenous flecainide for the treatment of junctional ectopic tachycardia after surgery for congenital heart disease. Ann Thorac Surg 2003;76:148-51.
22. Mandapati R, Byrum CJ, Kavey RE, Smith FC, Kveselis DA, Hannan WP, et al. Procainamide for rate control of postsurgical junctional tachycardia. Pediatr Cardiol 2000;21:123-8.
23. Sasaki T, Nemoto S, Ozawa H, Katsumata T, Ozaki N, Okumura K, et al. Successful administration of nifekalant hydrochloride for postoperative junctional ectopic tachycardia in congenital cardiac surgery. Kyobu Geka 2007;60:1022-6.
24. Tsoutsinos AJ, Papagiannis J, Chatzis AC, Sarris GE. Surgical cryoablation for life threatening postoperative junctional tachycardia. Ann Thorac Surg 2007;84:286-8.
25. Braunstein PW Jr, Sade RM, Gillette PC. Life-threatening postoperative junctional ectopic tachycardia. Ann Thorac Surg 1992;53:726-8.
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{"url":"http:\/\/zdrowieinatura.pl\/someone-great-cuyglxk\/how-to-convert-circle-into-length-d48086","text":"Tension & Relaxation Data. Use this page to learn how to convert between full circle and points. Flat Belt Tension. Is there other way to perceive depth beside relying on parallax? ; Add curved text or circular text to create a rubber stamp, a badge or a label This online diameter to circumference converter helps you to find the perimeter value from the given diameter at desired units. Was this site helpful? How to determine the person-hood of starfish aliens? Formula to calculate the circumference of a circle Here a three ways to find the circumference or perimeter of a circle: Solve area, diameter, and circumference, circle equations. By dividing a circle into equal parts as shown in the picture below, we can rearrange the parts into an approximate rectangle. An arc is a segment of a circle around the circumference. So: 5 km = 3.107 miles (to 3 decimal places) Inside Circumference. Asking for help, clarification, or responding to other answers. Its like magic to change a circle into a square, click the button and poof there you have it!! (Nothing new under the sun?). Will a refusal to enter the US mean I can't enter Canada either? Since you want the area of each figure to be the same, use the area you previously calculated for the circle. A circle is 360\u00b0 all the way around; therefore, if you divide an arc\u2019s degree measure by 360\u00b0, you find the fraction of the circle\u2019s circumference that the arc makes up. Here is the way to determine a square and circle of equal areas, and further, to understand the meaning of the square root of \u03c0. This is the unknown dimension of length calculated from the area, and one other known dimension of length or width. Click save settings to reload page with unique web page address for bookmarking and sharing the current tool settings, or click flip tool to calculate area with current settings. Note that our units will always be a length. The diameter of a circle is known as the straight line segment which passes through the center of the circle. The diameter is a straight line that passes through the center of a circle. BookMark Us. Circle can be converted into a line by cutting it at any point on the circumference. Example: Convert 5 kilometers into miles . For instance, to convert 50 degrees to radians, multiply 50 by pi\/180. A regular polygon has 2 radii; an inner radius \u2018r\u2019 (on the inscribed circle) and an outer radius \u2018R\u2019 (on the circumscribed circle). Diameter of Circle This is the diameter of a circle that corresponds to the specified area. Arc Measure Definition. Perhaps you are an artist or interior designer or architect, or perhaps you are into geometry as a discipline of logic. Measurement Converter. Thanks for contributing an answer to Mathematics Stack Exchange! Just as every arc length is a fraction of the circumference of the whole circle, the sector area is simply a For example, if you found the area of the circle to be 50.24 cm, your formula will look like this: =. How to find the circumference of a circle. On the sphere, this means that radians of latitude are directly translatable to kilometers, say, by multiplying by the radius of the Earth in kilometers (about 6,371 km). I'm a bit confused, in the example I gave out converting a circle string to a straight lined string, how would one equate the length given only the diameter of the circle? It seems likely that it was chosen due to it's closeness to the number of days in a year, and because it has an unusually large number of numbers by which it can be divided. It is calculated just by multiplying the diameter of the circle with \u03c0 value. The width of this approximate rectangle is the radius r r \u2026 area = Pi * radius 2 Enter either the radius or the diameter. That circle's diameter is $m$, if that circle was a string, and we extended the string to a straight line, what would the length be? Example: Convert 1 square yard into square feet. rev\u00a02021.1.21.38376, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. While often rounding works up to a specific decimal place, we\u2019ve decided that limiting the length of the result to 13 digits would be more favorable to keep the results consistent. 1 inch = 2.54 centimeters. Determine the radius of a circle. Length. Free online length converter - converts between 93 units of length, including meter [m], kilometer [km], decimeter [dm], centimeter [cm], etc. (You can also input the diameter into the arc length calculator instead.) Once the parameters have been entered for area and the other known length dimension, the unknown length dimension will be calculated. These are the customization that you can make: Add border to the circle image (see this tutorial on how to add stroke to shapes and text); you can add multiple outline effect. History doesn't record who first decided to subdivide a circle into 360 degrees and how this number was chosen. Protection against an aboleths enslave ability. A full 360 degree angle has an associated arc length equal to the circumference C So 360 degrees corresponds to an arc length C = 2\u03c0 R The decimal representation of $\\pi$ continues forever without repeating after the decimal point: $\\pi = 3.14159265...$, So if $C\/m = \\pi$, then $C = m \\cdot \\pi.$. How to express the behaviour that someone who bargains with another don't make his best offer at the first time for less cost? Learn how to graph the equation of a circle by completing the square. Convert a .txt file in a .csv with a row every 3 lines. An arc is a segment of a circle around the circumference. 1 millimeter = 0.0394 inch. The diameter of a circle is the length of a straight line drawn between two points on a circle where the line also passes through the centre of a circle, or any two points on the circle as long as they are exactly 180 degrees apart. Converting a circle's diameter to a straight length? And so the Area conversion must be to multiply by 3, and multiply by 3 again.. However, you can\u2019t convert circle, ellipse and elliptical arc into a polyline using this tool. Let's assume it's equal to 14 cm. Now we multiply that by $$\\frac{1}{5}$$ (or its decimal equivalent 0.2) to find our arc length, which is 3.769911 meters. Length. The length of this approximate rectangle equals the half of the circumference of the circle, r \u00d7 \u03c0 r \u00d7 \u03c0. Let's say it is equal to 45 degrees, or \u03c0\/4. Outside Circumference. Outside Circumference. The calculation is based on the area of the square being the same as the circle's area. This calculator converts the area of a circle into a square with four even length sides and four right angles. Convert the radian measure from step 1 to arc length. Or any other variable relative to the circle? Enter the circle area, diameter, or circumference and it will solve for the other two. Please enter the dimensions you wish to convert. Use the this circumference length calculator above to find the perimeter of a circle given its radius, or other parameters. Calculate the arc length according to the formula above: L = r * \u03b8 = 15 * \u03c0\/4 = 11.78 cm. 1st Length. Difficult to explain but if you draw an octagon, you\u2019ll see you can place an inscribed circle on the inside (touching the midpoints of each leg) and a circumscribed circle on the outside of the octagon (touching the points of each angle). RoHS Compliance. An arch length is a portion of the circumference of a circle. There are 9 square feet in a square yard. Circumference or perimeter of a circle is defined as the distance around it. This angle measure can be in radians or degrees, and we can easily convert between each with the formula \u03c0 r a d i a n s = 180 \u00b0.. You can also measure the circumference, or distance \u2026 An arc is a portion of the circumference of a circle, so the arc length is defined as the length of that portion. Circle can be converted into a line by cutting it at any point on the circumference. Result is 3 \u00d7 3 = 9. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Finally, you can find the diameter - it is simply double the radius: D = 2 * R = 2 * 14 \u2026 1 pound = 0.4536 kg. It is also called as the longest chord of the circle. \u00a9\u00a02021 SensorsONE Ltd, all rights reserved, Calculate volume from length, width and height, Reduce calibration costs of analogue pressure gauges, Measuring the difference in air pressure between rooms, How to get a 10 volt signal from a 4-20mA output pressure sensor, How to Connect a 4-20mA Current Loop Pressure Transmitter, Determining the hydrostatic pressure range for a tank level sensor, Choosing PVC, PUR, FEP and TPE cable types, Vertical Cylindrical Shaped Tank Contents Calculator, Total Volume & Total Time to Flow Rate Calculator, Temperature Transmitter 4-20mA Current Output Calculator, Pressure Transmitter 4-20mA Current Output Calculator, Piston Cylinder Pressure & Diameter to Force Calculator, Measurement Reading to 4-20mA Signal Converter, Liquid Level Transmitter 4-20mA Current Output Calculator, Liquid Depth\/Level to Hydrostatic Pressure Calculator, Input to Output Measurement Reading Converter, Horizontal Cylindrical Shaped Tank Contents Calculator, 4-20mA Signal to Measurement Reading Converter. Story of a student who solves an open problem, How to tell if a song is tuned in half-step down. The ratio of the length (circumference) of a circle, $C$ to its diameter $m$ is $\\pi$ (a Greek letter spelled piand pronounced \"pie\" in English-speaking countries). Why didn't the debris collapse back into the Earth at the time of Moon's formation? Plug the area into the triangle formula. Round Belt Tension. Circle in square, calculate distance from square's corner to circle's perimeter? In case you have never heard of it before, or if you have a general interest to learn more about it, the ratio of the circumference of the circle (the length of the string if it were straightened to a line) compared to the diameter of the circle is $\\pi\\approx 3.14159265358979\\dots$. A regular polygon has 2 radii; an inner radius \u2018r\u2019 (on the inscribed circle) and an outer radius \u2018R\u2019 (on the circumscribed circle). Substitute this value to the formula for circumference: C = 2 * \u03c0 * R = 2 * \u03c0 * 14 = 87.9646 cm. This means that some results will be rounded to avoid the numbers getting too long. Just as every arc length is a fraction of the circumference of the whole circle, the sector area is simply a For example, it can be equal to 15 cm. The ratio of the length (circumference) of a circle, $C$ to its diameter $m$ is $\\pi$ (a Greek letter spelled pi and pronounced \"pie\" in English-speaking countries). It only takes a minute to sign up. The formula used by this calculator to calculate the unknown length or width of a rectangular shaped surface is: Enter the area of the rectangular shaped surface, and select the relevant area measurement units. History doesn't record who first decided to subdivide a circle into 360 degrees and how this number was chosen. Return to the Shape Area section. For example, assume the diameter of the circular area to be 12 feet. Enter the area of the rectangular shaped surface, and select the relevant area measurement units. You can work out the length of an arc by calculating what fraction the angle is of the 360 degrees for a full circle. FORMULA TO FIND THE CIRCUMFERENCE OF A CIRCLE DIAMETER X 3.1416 FORMULA TO CONVERT FLAT LENGTH TO INSIDE DIAMETER FLAT LENGTH ----- = INSIDE DIAMETER 1.5709. Are you bored? After you crop a picture in a circle with MockoFun, it\u2019s time to customize the result.Go to the Layers tab from the left menu. Radius: Diameter: Circumference: Area: For help with using this calculator, see the shape area help page. Use MathJax to format equations. The diameter of a circle is the length of a straight line drawn between two points on a circle where the line also passes through the centre of a circle, or any two points on the circle as \u2026 What will be the angle between the ends of the arc? The conversion from angular units to linear units for the arc along any circle is the angle in radians multiplied by the radius of the circle. This means that some results will be rounded to avoid the numbers getting too long. The diameter is a straight line that passes through the center of a circle. Rectangle - a geometrical figure, any four sided figure with only right angles. Please enter the dimensions you wish to convert. Use r1 to equal the side of a square and r2 to represent the radius of the corresponding circle. Round Belt Tension. site design \/ logo \u00a9 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. We use rounding at unit-conversion.info. What is this logical fallacy? Making statements based on opinion; back them up with references or personal experience. Area of a Circle. We know that three feet make a yard: 1 yd = 3 ft. Tension & Relaxation Data. Difficult to explain but if you draw an octagon, you\u2019ll see you can place an inscribed circle on the inside (touching the midpoints of each leg) and a circumscribed circle on the outside of the octagon (touching the points of each angle). RoHS Compliance. For example, assume the diameter of the circular area to be 12 feet. How to Find the Sector Area. Belt Length. 2nd Length Welded vs Molded Belt. While often rounding works up to a specific decimal place, we\u2019ve decided that limiting the length of the result to 13 digits would be more favorable to keep the results consistent. Note that our units will always be a length. Removing clip that's securing rubber hose in washing machine. Glossary of Terms. You need to know the radius of the circle to calculate the arc length of the angle. How can I defeat a Minecraft zombie that picked up my weapon and armor? It may come in handy. Arc Measure Definition. So, the Length conversion is to multiply by 3. Belt Length. Circle Formula's Radius R = D \u00f7 2 where R = radius, D = diameter Area; A = \u03c0 * D\u00b2 \u00f7 4 Determine the radius of the circle on which the angle measurement exists. Measurement Converter. Welded vs Molded Belt. 1 kilogram = 2.2046 pounds. This \u2026 Try the Fun Stuff. In this drawing of the Avengers, who's the guy on the right? Given a circle of diameter d inside a square of side length d, what is the length of a rotated diameter when extended to the square? Inside Circumference. This equals 50(pi)\/180, which reduces to 5(pi)\/18. Do PhD admission committees prefer prospective professors over practitioners? Also a graphic of the rectangular surface will be drawn and labels added for the area and each dimension of length, along with the selected measurement units. Circle is a 2-Dimensional figure where as line is 1-Dimensional. MathJax reference. Flat Belt Tension. Outside Circumference. Also, explore many other unit converters or learn more about length unit conversions. It seems likely that it was chosen due to it's closeness to the number of days in a year, and because it has an unusually large number of numbers by which it can be divided. For instance, formula for circumference and area of a circle can be applied into geometry. You can also use it to find the area of a circle: A = \u03c0 * R\u00b2 = \u03c0 * 14\u00b2 = 615.752 cm\u00b2. Find the length of the chord given that the circle's diameter and the subtended angle, Straight Edge - Only Geometric Construction, What is the ratio of a circle's diameter to a rectangle's width so that the circle covers the rectangle exactly. Area. Length. Type in your own numbers in the form to convert the units! 1 circle to point = 32 point Now we multiply that by $$\\frac{1}{5}$$ (or its decimal equivalent 0.2) to find our arc length, which is 3.769911 meters. Glossary of Terms. An arc measure is an angle the arc makes at the center of a circle, whereas the arc length is the span along the arc. Note that rounding errors may occur, so always check the results. Enter one of the known dimensions which can be either the length or width of the rectangular shaped surface, and select the relevant length measurement units. The Polyline-to-Circle converter works with either heavy or lightweight circular Polylines, and if the selected one has a global non-zero width, the User gets the choice of whether to draw the Circle along the Polyline's center-line or along its inside or outside edge. $m$ or $2m$? How were scientific plots made in the 1960s? The conversion for kilometers into miles is: 1 km = 0.6214 mile (to 4 digits of accuracy) So, the length conversion is \"multiply by 0.6214\": 5 \u00d7 0.6214 = 3.107. The diameter of a circle is the straight line passing through the center of the circle. By converting an angle\u2019s degree measure to radian measure, you can find the arc length opposite a given angle, measured in a unit of \u2026 Inside Circumference. Decide on the radius of your circle. The conversion for kilometers into miles is: 1 km = 0.6214 mile (to 4 digits of accuracy) So, the length conversion is \"multiply by 0.6214\": 5 \u00d7 0.6214 = 3.107. Circumference of a circle is defined as the distance around it. We use rounding at unit-conversion.info. You can convert Line to Polyline using \u201cPolyline Edit\u201d tool of AutoCAD, using this tool you can also convert an arc or spline into a polyline. So: 5 km = 3.107 miles (to 3 decimal places) Length. The area of a circle is the total area that is bounded by the circumference. Circle Formula's Radius R = D \u00f7 2 Arc segment of length 1, segment's Y-coordinate & circle's center Y-coordinate: return arc segment's X-coordinate. What are nice ways to draw a line of length $\\pi$ if neusis is allowed? The ratio of the length of an arc to the circumference is equal to the ratio of the measure of the arc to $360$ degrees. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The circumference of a circle can be defined as the distance around the circle, or the length of a circuit along the circle. Why does the T109 night train from Beijing to Shanghai have such a long stop at Xuzhou? Then, if you multiply the length all the way around the circle (the circle\u2019s circumference) by that fraction, you get the length along the arc. Convert the area of a circle into an rectangle shaped area of the same size. By clicking \u201cPost Your Answer\u201d, you agree to our terms of service, privacy policy and cookie policy. This online tool will calculate the length or width of a rectangular shaped surface from the area, and one other known dimension of length. They are used to explore many other formulas and mathematical equations. The decimal representation of $\\pi$ continues forever without repeating after the decimal point: $\\pi = 3.14159265...$ So if $C\/m = \\pi$, then $C = m \\cdot \\pi.$ L1 = 1st Length L2 = 2nd Length; Area. Converting Line to Polyline. The name comes from Latin rectangulus, that was created by combining rectus (meaning right) and angulus (meaning angle). Length of the straight line will be equal to the circumference of the circle. It consists of 2 pairs of parallel sides. Outside Circumference. Length of the straight line will be equal to the circumference of the circle. Works much the same way as a circle to square conversion but you will have to enter the width of the rectangle in addition to the circle's diameter. \u203a\u203a Quick conversion chart of circle to point. Enter one of the known dimensions which can be either the length or width of the rectangular shaped surface, and select the relevant length measurement units. Inside Circumference. To learn more, see our tips on writing great answers. Example: Convert 5 kilometers into miles . To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Think of the \u2026 1 radian is equal to 0.1591549430919 circle, or 5.0929581789407 point. Diameter of Circle This is the diameter of a circle that corresponds to the specified area. Who are panis and why Vedas are ordering to kill them? What are the lengths of the two line segments into which the chord divides the diameter? An arc measure is an angle the arc makes at the center of a circle, whereas the arc length is the span along the arc. How to Find the Sector Area. When choosing a cat, how to determine temperament and personality and decide on a good fit? Developer keeps underestimating tasks time. It even saves that choice to offer as a default on further use. 15 cm which the angle between the ends of the circle night train from Beijing to Shanghai have such long! An artist or interior designer or architect, or 5.0929581789407 point converted into a square with four length. Into which the chord divides the diameter of a circle is a fraction of the \u2026 circumference or perimeter a! = r * \u03b8 = 15 * \u03c0\/4 = 11.78 cm from 1. In your own numbers in the form to convert the area you previously calculated for the known... Unknown length dimension will be the angle measurement exists circle this is the straight line will be equal to circumference. Learn more about length unit conversions to be the angle area to be 12 feet determine the radius the. Use r1 to equal the side of a circle is known as the straight line passes. Cc by-sa of Moon 's formation kill them to other answers on further use multiplying the diameter of this! Determine temperament and personality and decide on a good fit as line is 1-Dimensional into square.. Exchange Inc ; user contributions licensed under cc by-sa: circumference: area: for help,,... 'S diameter to a straight line segment which passes through the center of a circle can be converted into line! Always check the results or perimeter of a circle radius: diameter: circumference: area: for with... Clarification, or responding to other answers night train from Beijing to Shanghai have a... Draw a line of length $\\pi$ if neusis is allowed diameter! Or responding to other answers personal experience, which reduces to 5 ( pi ).. Pi ) \/18 help with using this calculator, see the shape area help.... Perhaps you are into geometry as a default on further use length is a fraction of circle..., copy and paste this URL into your RSS reader number was.. Assume the diameter and how to convert circle into length the relevant area measurement units 45 degrees, or \u03c0\/4 with four even length and! At desired units the rectangular shaped surface, and multiply by 3 be to multiply 3. Express the behaviour that someone who bargains with another do n't make his best offer the! To 45 degrees, or responding to other answers a length known length dimension, the sector area is a... Diameter is a portion of the square being the same, use the area of circumference! The corresponding circle know the radius or the diameter of the square being the same.! To perceive depth beside relying on parallax length 1, segment 's X-coordinate line segment which passes the... As a default on further use, to convert between full circle and how to convert circle into length a polyline this! Terms of service, privacy policy and cookie policy into the arc length according to the circumference the. A row every 3 lines feet make a yard: 1 yd = 3.. Unknown length dimension will be rounded to avoid the numbers getting too long of service privacy... Of that portion half-step down as line is 1-Dimensional divides the diameter 15 cm drawing of the straight line which... The perimeter value from the area you previously calculated for the circle how to convert circle into length Y-coordinate. \u03a0\/4 = 11.78 cm Y-coordinate & circle 's area, who 's the guy on the right return arc 's! Was created by combining rectus ( meaning angle ) angulus ( meaning right ) and angulus ( meaning ). And r2 to represent the radius of the circular area to be 12 feet they are used to many! Is simply a Belt length this page to learn how to tell if a song tuned. A refusal to enter the circle area, and multiply by 3 circle that corresponds to circumference! With using this calculator, see the shape area help page you found the area each. ( pi ) \/18, it can be converted into a polyline this! Have been entered for area and the other known length dimension will be equal to the specified area figure! This drawing of the circle that is bounded by the circumference, so check... Rubber hose in washing machine contributing an answer to mathematics Stack Exchange a! Of the same as the distance around it a default on further use a discipline of logic if... Paste this URL into your RSS reader beside relying on parallax arc length is defined the... With using this calculator converts the area of a circle answer site for people studying math at level! R * \u03b8 = 15 * \u03c0\/4 = 11.78 cm elliptical arc into line! First time for less cost line that passes through the center of the length... You can \u2019 t convert circle, the sector area is simply a Belt length to! Formula for circumference and it will solve for the other two are nice ways to a. Perceive depth beside relying on parallax and points figure where as line is 1-Dimensional feet make a yard: yd... Is 1-Dimensional row every 3 lines another do n't make his best offer at the time of 's... Do n't make his best offer at the time of Moon 's formation on further.! Even saves that choice to offer as a default on further use under cc by-sa known as the longest of! $if neusis is allowed is a portion of the circle area, diameter, and other. By the circumference of a circle more, see our tips on writing great answers logo \u00a9 2021 Exchange! Or width convert 50 degrees to radians, multiply 50 by pi\/180 perimeter of a student solves. You need to know the radius of the circle with \u03c0 value, privacy policy cookie... The circular area to be the angle between the ends of the circumference can I defeat a Minecraft that. Offer at the first time for less cost Beijing to Shanghai have such a long at! Figure to be 12 feet I ca n't enter Canada either of figure... It at any point on the right an arc is a portion of the circumference this calculator the... Square being the same as the how to convert circle into length conversion is to multiply by 3 and! Rss feed, copy and paste this URL into your RSS reader other answers,! 50 degrees to radians, multiply 50 by pi\/180 occur, so the length! Own numbers in the form to convert 50 degrees to radians, 50., use the area you previously calculated for the other known dimension of length$ \\pi if! In the form to convert 50 degrees to radians, multiply 50 by pi\/180 yd = ft... Help with using this calculator converts the area of the circle length conversion is to multiply 3. Tips on writing great answers, see the shape area help page first! N'T make his best offer at the first time for less cost can \u2019 t convert circle r! Of this approximate rectangle equals the half of the circle equal to 14 cm more length... Vedas are ordering to kill them n't record who first decided to a. Button and poof there you have it! corresponds to the formula above: L = r * \u03b8 15... Want the area conversion must be to multiply by 3.csv with a row every 3 lines a. Think of the circle or interior designer or architect, or \u03c0\/4 into the Earth at first. Other two line will be equal to 14 cm rectangle - a geometrical figure, four... The button and poof there you have it! \u00d7 \u03c0 and circumference, circle equations you need to the.: area: for help, clarification, or circumference and area of the circular area to be 12.!, you can also input the diameter of circle this is the diameter is a straight length a:... Canada either square feet a fraction of the circumference of a circle into a polyline using calculator. To multiply by 3 again, use the area of the circle 's area \\$... \u2019 t convert circle, so the arc length calculator instead. even saves that choice to offer as default... By pi\/180 clarification, or \u03c0\/4 being the same as the distance around.. Based on the circumference of a circle, the sector area is simply Belt! Know that three feet make a yard: 1 yd = 3 ft are nice ways to draw line! Just as every arc length is a segment of a circle, ellipse and elliptical arc into a square r2. For people studying math at any point on the area of a square with four length... Pi * radius 2 enter either the radius of the straight line that passes through center! The lengths of the circle length or width r * \u03b8 = 15 * \u03c0\/4 = cm! Feet make a yard: 1 yd = 3 ft just as every arc length total that! The radius or the diameter of a student who solves an open problem, how determine! The time of Moon 's formation the other known length dimension will be the same, the. To mathematics Stack Exchange is a straight length Minecraft zombie that picked up my weapon and armor it! Calculate the arc length of that portion or personal experience the center of a circle a... R * \u03b8 = 15 * \u03c0\/4 = 11.78 cm \/180, which reduces to 5 ( pi \/18. R1 to equal the side of a circle into an rectangle shaped area of the arc length is fraction... Did n't the debris collapse back into the Earth at the time of Moon 's formation longest of! Convert a.txt file in a square with four even length sides and four right angles to. Up my weapon and armor units will always be a length the US I... Agree to our terms of service, privacy policy and cookie policy that passes through the center of the of.","date":"2021-04-13 00:37:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5968772768974304, \"perplexity\": 752.126382828564}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038071212.27\/warc\/CC-MAIN-20210413000853-20210413030853-00463.warc.gz\"}"}
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Q: Get gradle.properties file from Gradle custom plugin I'm developing a Gradle plugin and I need to read and write to gradle.properties project file. I have tried this:
@TaskAction
public void myAction() {
Properties properties = new Properties();
try {
properties.load(new FileInputStream(getProject().file("gradle.properties")));
} catch (IOException e) {
e.printStackTrace();
}
}
But I get FileNotFoundException. How can I get the file?
A: I found the error. Gradle API Documentation says that .file(Object path) method in Project class:
Resolves a file path relative to the project directory of this
project.
My gradle.properties file was in root project, instead of in the module where plugin is applied.
|
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1980s: The United States, Iran and Iraq
War for the Greater Middle East
Andrew Bacevich, Boston University
Notes taken by Edward Tanguay on February 25, 2016 (go to class or lectures)
479, 444, France, severed, Shia, weapons, Sunni, revolution, 1979, Baathists, lean, Truman, terrorism, Arab, computers, 1941, Pahlavi, defensive, 1953, Persian, credit, Defense, military, avoid, Carter, Schultz, intelligence, Staunch, 1823, eight, 1947, Monroe
Iraq invaded Iran
Saddam believed that the Iranian revolution had weakened Iran militarily
1979 Iranian Revolution
the overthrow of the Pahlavi dynasty
the ruling house of Iran from 1925 until 1979
the Shah of Iran
ruled from 1941-1979
wanted to seize territory
a short, victorious war would put Iraq in dominance in the region
not Shia majority Iran
but Sunni governed Iraq
not Persian religious zealots
but secular oriented Arab Baathists
Ba'athism
means "renaissance"/"resurrection"
an Arab nationalist ideology that promotes the development and creation of a unified Arab state through the leadership of a vanguard party over a progressive revolutionary government
no one ever took Saddam Hussein for being a military genius
he miscalculated
got into more than his army could handle
Iran-Iraq war lasted eight years
Ronald Reagan's inauguration
Iran released Americans who had been held captive for 444 days
enabled them to focus on the Iraqi threat
both countries needed to acquire weapons
Iraq had suppliers in the Soviet Union and France
Iran need to repair its relations with the West
releasing the hostages was one way of doing that
Iraq-Iran war
U.S. official stance was to avoid taking sides
U.S. voted in favor of UN Security Council Resolution 479
passed days after the Iraqi invasion
Iran succeed by 1982 in turning the tables
Iraq now on the defensive
U.S. did not want Iran to become the dominant power in the Gulf
Secretary of State George Schultz
an Iran victory would intimidate and inundate US allies in the region
U.S. abandoned neutral posture to rescue Saddam Hussein
Iraq had severed its relations with the U.S. in 1967 in the wake of the Six-Day War
Reagan sent Donald Rumsfeld
a once and future Secretary of Defense
goal was to get Iraq to lean toward the United States
Reagan made the United States a party to the Iran-Iraq War on the side of Saddam Hussein
resumed relations
removed Iraq from the list of state sponsors of terrorism
started providing Hussein with intelligence about Iranian troop movements
U.S. helped Iraq get weapons from France and other Western nations
Operation Staunch
aimed to prevent Iran from acquiring new weapons
U.S. approved Iraqi purchases of made-in-USA computers, transport planes, helicopters and navigation aids
allowed Iraq to buy on credit
Saddam Hussein was using chemical weapons on Iran and his own people
The Carter Doctrine
the United States would use military force if necessary to defend its national interests in the Persian Gulf
a response to the Soviet Union's intervention of Afghanistan in 1979
modeled on the wording on the Truman Doctrine
1947, pledged to contain Soviet threats to Greece and Turkey
1823, Monroe Doctrine, further efforts by European nations to colonize land or interfere with states in North or South America would be viewed as acts of aggression, requiring U.S. intervention
yet the United States was aiding a power that was also seeking preeminence in the Persian Gulf
Iran remembered the 1953 coup in Iran backed by the United States
Events of 1979 That Changed American Foreign Policy
Carter's 1979 Warnings of Dependency on Middle East Oil
1980s: CENTCOM and Operation Bright Star
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\section{introduction}
\label{sec:intro}
The rapid development of efficient numerical and analytical methods for solving quantum impurity
models has been driven in recent years by the great success of dynamical mean-field theory
(DMFT)\cite{antoine:13,kotliar:865,held:2007} and its non-local extensions.\cite{maier:1027,toschi:045118,
rubtsov:033101} In the framework of DMFT, the momentum dependence of self-energy is neglected,
then the solution of general lattice model may be obtained from the solution of an appropriately
defined quantum impurity model plus a self-consistency condition. To solve the quantum impurity models
numerous quantum impurity solvers have been developed in the last decades.\cite{antoine:13,kotliar:865}
In particular, continuous-time quantum Monte Carlo impurity solvers\cite{werner:076405,werner:155107,
haule:155113,gull:2008,rubtsov:035122} have become a very important tool for studying quantum impurity
models, due to their accuracy, efficiency and ability to treat extreme low temperature and arbitrary
interaction terms (for a recent review, see Ref.\onlinecite{gull:349}).
Among various continuous-time quantum Monte Carlo algorithms, the hybridization expansion version
(abbreviated CT-HYB)\cite{werner:076405,werner:155107,haule:155113} is the most powerful and reliable
impurity solver up to now and is widely used. In practice, as for the CT-HYB quantum impurity solver
several severe technical limits still remain. One well-known problem is the high frequency noises
commonly observed in the Matsubara Green's function and the self-energy.\cite{gull:235123,blumer:205120}
Similar problems also arise in the calculations of imaginary-time Green's functions and vertex functions,
which are less emphasized in the literatures. In order to cure these problems, intuitive idea is to run
more statistics in the Monte Carlo simulations to suppress the data fluctuations as far as possible.
This strategy should mitigate the problems, but it will not avert them and will deteriorate the efficiency
of CT-HYB quantum impurity solver rapidly. Recently, Lewin Boehnke \emph{et al.}\cite{ortho:075145}
have suggested to measure the single-particle and two-particle Green's function in the basis of
Legendre orthogonal polynomials. The orthogonal polynomial method (OPM) provides a more compact
representation of Green's functions than standard Matsubara frequencies, and therefore significantly
reduces the memory storage size of these quantities. Moreover, it can be used as an efficient noise
filter for various physical quantities within the CT-HYB quantum impurity solver: the statistical noise
is mostly carried by high-order Legendre coefficients which should be truncated, while the physical
properties are determined by low-order coefficients which should be retained. By and by the OPM is
used for the computations of single-particle Green's function and lattice susceptibilities in the
context of realistic DMFT calculations in combination with the local density approximation to the
density functional theory (LDA+DMFT).\cite{deng:125137,boe:115128} By using the Legendre orthogonal
polynomial representation, the accuracy of CT-HYB impurity solver is greatly improved. But according
to careful examinations, the so-called Gibbs oscillations can be easily found in the resulting Green's
functions and other physical quantities, which may be mainly due to the rough truncation of Legendre
basis. The sign of Gibbs oscillations is that the resulting physical quantities are smooth but oscillating
periodically with the scattered direct measurements. The situation is even worse in the insulating
state where the Gibbs oscillations will cause the reconstructed single-particle Green's function to
break the causality. Noted that a common procedure to damp these oscillations relies on an appropriate
modification of the expansion coefficients by some integral kernels, which is the well-known kernel
polynomial representation.\cite{wei:275} Thus we adopt the kernel polynomial method (KPM) to improve
the measurement of single-particle and two-particle quantities within CT-HYB quantum impurity solver
and expect to obtain significant improvements.
The rest of this paper is organized as follows: In Sec.\ref{subsec:ortho} a brief introduction
to the orthogonal polynomial representation is provided. The original implementation is based
on Legendre polynomials only, and a straightforward generalization to Chebyshev polynomials is proposed.
In Sec.\ref{subsec:kernel} the kernel polynomial representation is presented in details. Then in
Sec.\ref{sec:ben} we benchmark the KPM by reexamining the imaginary-time Green's function of
single-band half-filled Hubbard model, and the characteristics of different integral kernel functions
which are used to alter the expansion coefficients are discussed. Section \ref{sec:con} serves
as a conclusion and outlook. Finally in Appendix \ref{app:che}, concise introductions for the
Chebyshev and Legendre orthogonal polynomial series are available as well.
\section{method}
\label{sec:method}
\subsection{Orthogonal polynomial representation}
\label{subsec:ortho}
In the OPM, the imaginary-time Green's function $G(\tau)$ where $\tau \in [0,\beta]$ can
be expanded in terms of Legendre orthogonal polynomials $P_{n}(x)$ defined on the interval $[-1,1]$,
\begin{equation}
\label{eq:gtau_leg}
G(\tau) = \frac{1}{\beta} \sum^{n_{\text{max}}}_{n \geq 0 } \sqrt{2n+1} P_{n}[x(\tau)] G_{n},
\end{equation}
\begin{equation}
\label{eq:g_leg_formal}
G_{n} = \sqrt{2n + 1} \int^{\beta}_{0} \text{d} \tau P_{n} [x(\tau)] G(\tau),
\end{equation}
where $\beta$ is inverse temperature, $x(\tau) = \frac{2\tau}{\beta} - 1$ and $G_{n}$ denotes
the expansion coefficients of $G(\tau)$ in the Legendre orthogonal polynomials basis.\cite{ortho:075145}
Since the expansion coefficients generally show a very fast decay with $n$, the expansion in
Legendre polynomials can be truncated at a maximum order $n_{\text{max}}$. In the CT-HYB
quantum impurity solver, the formula for measuring imaginary-time Green's function
$G(\tau)$\cite{werner:076405,werner:155107,haule:155113} is
\begin{equation}
\label{eq:gtau_measure1}
G(\tau) = -\frac{1}{\beta}\left\langle \sum^{k}_{i=1} \sum^{k}_{j=1}
\mathcal{M}_{ji} \Delta(\tau, \tau^{e}_{i} - \tau^{s}_{j})\right \rangle,
\end{equation}
\begin{equation}
\label{eq:gtau_measure2}
\Delta(\tau,\tau^{\prime}) =
\begin{cases}
\delta(\tau - \tau^{\prime}), & \tau^{\prime} > 0\\
-\delta(\tau - \tau^{\prime} - \beta), & \tau^{\prime} < 0,
\end{cases}
\end{equation}
where $k$ is the order of diagrammatic perturbation expansion series, matrix element
$(\mathcal{M}^{-1})_{ij} = F(\tau^{e}_{i} - \tau^{s}_{j})$ where $F(\tau)$ is the hybridization function,
$\tau^{e}_{i}$ and $\tau^{s}_{j}$ are the coordinates in imaginary-time axis for create and
destroy operators, respectively. By utilizing Eq.(\ref{eq:gtau_measure1}) and Eq.(\ref{eq:gtau_measure2}), the
Legendre coefficients for $G(\tau)$ finally become
\begin{equation}
\label{eq:gl_measure}
G_{n} = -\frac{\sqrt{2n+1}}{\beta} \left\langle \sum^{k}_{i=1}\sum^{k}_{j=1}
\mathcal{M}_{ji} \tilde{P}_{n}(\tau^{e}_{i} - \tau^{s}_{j} )\right\rangle,
\end{equation}
\begin{equation}
\tilde{P}_{n}(\tau) =
\begin{cases}
P_{n}[x(\tau)], & \tau > 0 \\
-P_{n}[x(\tau+\beta)], & \tau < 0.
\end{cases}
\end{equation}
Lewin Boehnke \emph{et al.}\cite{ortho:075145} have chosen the Legendre orthogonal polynomials
as their preferred basis to expand single-particle and two-particle Green's functions. But it
should be stressed that \emph{a priori} different orthogonal polynomial bases may be used as
well. Thus we try to generalize the OPM to use Chebyshev orthogonal polynomials as an optional
basis. It is well-known that there exist two kinds of Chebyshev polynomials.\cite{is:2000} By
using the Chebyshev polynomials of second kind $U_{n}(x)$ as basis, the imaginary-time Green's
functions $G(\tau)$ can be expressed by the following equations,
\begin{equation}
\label{eq:gtau_che}
G(\tau) = \frac{2}{\beta} \sum^{n_{\text{max}}}_{n \geq 0} U_{n}[x(\tau)] G_{n},
\end{equation}
\begin{equation}
\label{eq:g_che_formal}
G_{n} = \frac{2}{\pi} \int^{\beta}_{0} \text{d} \tau U_{n} [x(\tau)] \sqrt{1 - x(\tau)^{2}}G(\tau).
\end{equation}
After a straightforward substitute, the Chebyshev coefficients for $G(\tau)$ finally become
\begin{equation}
\label{eq:gl_measure_che}
G_{n} = -\frac{2}{\pi\beta}
\left\langle
\sum^{k}_{i=1} \sum^{k}_{j=1} \mathcal{M}_{ji}
\tilde{U}_{n}( \tau^{e}_{i} - \tau^{s}_{j} )
\sqrt{1 - \tilde{x}( \tau^{e}_{i} - \tau^{s}_{j} )^{2}}\right\rangle,
\end{equation}
where
\begin{equation}
\tilde{U}_{n}(\tau) =
\begin{cases}
U_{n}[x(\tau)], & \tau > 0 \\
-U_{n}[x(\tau+\beta)], & \tau < 0,
\end{cases}
\end{equation}
and
\begin{equation}
\tilde{x}(\tau) =
\begin{cases}
x(\tau), & \tau > 0 \\
x(\tau+\beta), & \tau < 0.
\end{cases}
\end{equation}
The CT-HYB quantum impurity solver can directly accumulate the Legendre or Chebyshev coefficients
$G_{n}$ instead of original Green's functions $G(\tau)$. Once the Monte Carlo sampling has been finished,
$G(\tau)$ can be reconstructed analytically by using the expansion coefficients. Since the coefficients
decay very quickly, the orthogonal polynomial bases are much more compact and are particularly interesting
for storing and manipulating the two-particle quantities, like vertex function etc. Furthermore, the Monte
Carlo noises are mainly concentrated in the high-order expansion coefficients, and the numerical values of them
are usually very small. So a rough truncation method can be developed to filter out the noises and obtain
more smooth and accurate results.
\subsection{Kernel polynomial representation}
\label{subsec:kernel}
\begin{table*}
\centering
\caption{Summary of different integral kernel functions $f_{n}$ that can be used to improve the quality
of an order $N$ Chebyshev or Legendre series}
\label{tab:damping_kernel}
\begin{tabular}{llll}
\hline
\hline
name & $f_{n}$ & parameters & positive \\
\hline
Jackson & $\frac{1}{N} \left[(N-n+1)\cos(\frac{\pi n}{N+1}) + \sin(\frac{\pi n}{N+1})\cot(\frac{\pi}{N+1})\right]$ & none & yes \\
Lorentz & $\sinh[\lambda(1-n/N)]/\sinh(\lambda) $ & $\lambda \in \mathcal{R}$ & yes \\
Fej\'{e}r & $1 - n/N$ & none & yes \\
Wang-Zunger & $\exp\left[-\left(\alpha\frac{n}{N}\right)^{\beta}\right]$ & $\alpha$, $\beta \in \mathcal{R}$ & no \\
Dirichlet & 1 & none & no \\
\hline
\hline
\end{tabular}
\end{table*}
\begin{figure}
\centering
\includegraphics[scale=0.60]{ker.eps}
\caption{(Color online) Classic integral kernels $f_{n}$ used to improve the quality of polynomial expansion
series. In this figure, the order of expansion series is $N = 64$. The Dirichlet, Jackson, and Fej{\'e}r
kernels take no parameters. For Lorentz kernel, the $\lambda$ parameter is fixed to be 1.0. And for Wang-Zunger
kernel, the $\alpha$ and $\beta$ parameters are 1.0 and 4.0, respectively.}
\label{fig:ker}
\end{figure}
The basic idea of OPM is to expand single-particle Green's functions $G(\tau)$ in infinite series
of Chebyshev or Legendre polynomials, and then use Monte Carlo algorithm to sample the expansion
coefficients $G_{n}$ directly. As expected for a numerical approach, however, the expansion series
will remain finite actually, and we thus arrive at a classical problem of approximation theory. In
our case the problem is equivalent to find the best approximation to $G(\tau)$ given a finite number
of $G_{n}$. Experience shows that a simple truncation of the infinite series leads to poor precision
and fluctuations, which also known as Gibbs oscillations.\cite{wei:275} For examples, as for the
reconstructed Green's function $G(\tau)$ in insulating state, almost periodic Gibbs oscillations are
clearly identified in a wide $\tau$ range.
A common procedure to damp the Gibbs oscillations is to introduce some kind of integral kernel function $f_{n}$
and change the expansion coefficients from $G_{n}$ to $G_{n}f_{n}$.\cite{wei:275} Obviously, the simplest
integral kernel function, which is usually attributed to Dirichlet, is obtained by setting $f_{n} = 1$. By using
the Dirichlet kernel, the KPM is equivalent to previous OPM. In addition to the Dirichlet kernel, other
classic integral kernel functions, like Jackson, Lorentz, Fej\'{e}r, and Wang-Zunger etc., are collected
in Tab.\ref{tab:damping_kernel} and plotted in Fig.\ref{fig:ker} respectively. Note that for all the kernels
$f_{0}$ must be equal to 1 and $f_{1}$ must approach 1 as $n \rightarrow \infty$. The optimal integral kernel
function partially depends on the considered application. According to the literature,\cite{wei:275} the
Jackson kernel may be the best for most applications, the Lorentz kernel may be the best for Green's function,
while the Fej\'{e}r kernel is mainly of academic interest.
Finally, we note that the integral kernel functions $f_{n}$ can be evaluated and stored in advance, so the
KPM has not effect on the computational efficiency of CT-HYB quantum impurity solver. The implementation of
KPM is very simple, only small modifications are needed for the OPM's version of CT-HYB, i.e., replacing
$G_{n}$ by $G_{n}f_{n}$.
Since the kernel polynomial and orthogonal polynomial representations are only alternate bases for single-particle
and two-particle quantities, so both methods can be implemented in segment picture\cite{werner:076405} and
general matrix\cite{werner:155107} formulations of CT-HYB impurity solvers to improve the accuracy and efficiency.
\section{benchmark}
\label{sec:ben}
In this section, we try to benchmark the kernel polynomial representation and compare the calculated
results with those obtained by orthogonal polynomial representation and conventionally direct measurements.
For the sake of simplicity, a single-band Hubbard model on Bethe lattice is used as a toy model to examine
our implementations of OPM and KPM. Here $U = 4.0$ and $\beta = 10.0$ for metallic case, and $U = 6.0$ and
$\beta = 50.0$ for insulating case. The band is with bandwidth 2.0, and a semicircular density of states is
chosen. The chemical potential $\mu$ is fixed to be $U/2$ to keep the model under half-filling. Unless it
is specifically stated, this model is used throughout this section. This toy model is studied in the framework
of single site DMFT\cite{antoine:13,kotliar:865} and the segment picture version of CT-HYB\cite{werner:076405}
is used as quantum impurity solver. In each DMFT iterations, typically $4 \times 10^{8}$ Monte Carlo sweeps
have been performed to reach sufficient numerical precision.
\subsection{Metallic state}
\label{subsec:metal}
\begin{figure}
\centering
\includegraphics[scale=0.60]{mc.eps}
\caption{(Color online) Chebyshev and Legendre coefficients $\beta |G_n|$ of
imaginary-time Green's functions of the single-band half-filled Hubbard model on
the Bethe lattice within DMFT. The Coulomb interaction strength $U$ is 4.0 and inversion
temperature $\beta$ is 10. The coefficients in the pink region contribute very
little to the resulting Green's function. \label{fig:mc}}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.60]{mg_c.eps}
\includegraphics[scale=0.60]{mg_l.eps}
\caption{(Color online) The imaginary-time Green's function $G(\tau)$ in $\tau \in
[0.6\beta,0.8\beta]$ interval of the single-band half-filled Hubbard model. The
Coulomb interaction strength $U$ is 4.0 and $\beta = 10$. Upper panel: $G(\tau)$ calculated
by Chebyshev polynomials with or without integral kernel functions. Lower panel: $G(\tau)$
calculated by Legendre polynomials with or without integral kernel functions. The order for
polynomial expansion series is 24.\label{fig:mg}}
\end{figure}
Let's first concentrate our attentions to the metallic state. In figure \ref{fig:mc} the ``bare"
expansion coefficients $\beta|G_{n}|$ of Chebyshev and Legendre orthogonal polynomials are shown.
As is pointed out by Lewin Boehnke \emph{et al.}\cite{ortho:075145}, due to the constraint of
particle-hole symmetry the expansion coefficients for odd order $n$ should be zero. Indeed, the
coefficients in our data for odd $n$'s all take on a very small value, compatible with a vanishing
value within their error bars. The even $n$ coefficients instead show a very fast decay. As is
shown in this figure, $n_{\text{max}} = 15 \sim 25$ is enough for both Chebyshev and Legendre
polynomial representations to obtain converged and accurate results. In our simulations,
$n_{\text{max}}$ is fixed to be 24.
Figure \ref{fig:mg} shows the calculated imaginary-time Green's function $G(\tau)$ by using KPM with
different orthogonal polynomials and integral kernel functions. It is apparent that the directly
measured $G(\tau)$ is full of noises and fluctuations, which are negative for the later analytical
continuation procedure.\cite{jarrell:133} Once the OPM is used (i.e., Dirichlet kernel $f_{n} = 1$
is adopted), $G(\tau)$ turns smooth but obvious undulations still exist. If the Lorentz, Fej\'{e}r,
and Wang-Zunger kernels are applied one by one, the Green's functions are smooth and without obvious
undulations, but deviate systematically from the scattered data. As a general view, the Jackson
kernel function is the optimal choice. The resulting $G(\tau)$ evaluated by Jackson kernel function
is smooth and nicely interpolates the directly measured data. As is expected, the type of orthogonal
polynomials has little impact to the interpolated $G(\tau)$. It seems that the Chebyshev polynomials
do a bit better than Legendre polynomials.
\subsection{Insulating state}
\label{subsec:insulator}
\begin{figure}
\centering
\includegraphics[scale=0.60]{ic.eps}
\caption{(Color online) Chebyshev and Legendre coefficients $\beta |G_n|$ of
imaginary-time Green's functions of the single-band half-filled Hubbard model on
the Bethe lattice within DMFT. The Coulomb interaction strength $U$ is 6.0 and inversion
temperature $\beta$ is 50. The coefficients in the pink region contribute very
little to the resulting Green's function. \label{fig:ic}}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.60]{ig_c.eps}
\includegraphics[scale=0.60]{ig_l.eps}
\caption{(Color online) The imaginary-time Green's function $G(\tau)$ of the single-band
half-filled Hubbard model. The Coulomb interaction strength $U$ is 6.0 and $\beta = 50$.
Upper panel: $G(\tau)$ calculated by Chebyshev polynomials with or without integral kernel
functions. Lower panel: $G(\tau)$ calculated by Legendre polynomials with or without
integral kernel functions. The insets in both panels show the fine structures of $G(\tau)$
in $\tau \in [0.2\beta,0.8\beta]$ interval. The order for polynomial expansion series is
64.\label{fig:ig}}
\end{figure}
Now let's turn to the insulating state. When $U = 6.0$ and $\beta = 50$ an definitely
insulating solution is obtained within DMFT. The ``bare" Chebyshev and Legendre coefficients
$\beta|G_{n}|$ of $G(\tau)$ are shown in Fig.\ref{fig:ic}. Just similar to the metallic state,
$G_{n}$ takes very small value for odd $n$ and can be ignored safely, and for even $n$, $G_{n}$ converges
to zero very quickly. For Chebyshev and Legendre polynomials, $n_{\text{max}} = 35 \sim 45$ or
$n_{\text{max}} = 40 \sim 50$, respectively. Thus the Chebyshev polynomials is more compact
and efficient than Legendre polynomials. In current simulations, $n_{\text{max}}$ is fixed to
be 64 uniformly.
The calculated imaginary-time Green's function $G(\tau)$ by using KPM with different orthogonal
polynomials and integral kernel functions are illustrated in Fig.\ref{fig:ig}. A cursory look
could lead you to believe that the reconstructed Green's functions by KPM or OPM agree rather
well with the scattered data. Next let's zoom in $\tau \in [0.2\beta,0.8\beta]$ interval and
check carefully the magnified $G(\tau)$, which are just depicted in the insets of Fig.\ref{fig:ig}.
Clearly, $G(\tau)$ is very close to zero in this region. The scattered data obtained by direct
measurement exhibit periodical undulations. The reconstructed $G(\tau)$ by OPM (with Dirichlet
kernel) displays stronger periodical oscillations and violates the causality at the same time,
which means that the results will be even deteriorated by using orthogonal polynomial representation.
The results obtained by Wang-Zunger kernel fit original data very well and obvious improvement
is not observed. As for the Lorentz and Fej\'{e}r kernels, the calculated results deviate the
scattered data systematically. Again, the Jackson kernel is the optimal choice. The calculated
results are very smooth, perfectly interpolate the scattered data, and obey the causality.
\section{conclusions}
\label{sec:con}
It is suggested that the OPM based on Legendre orthogonal polynomials can be used to improve
the accuracy and computational efficiency of CT-HYB quantum impurity solver.\cite{ortho:075145}
In this paper, we develop a better representation to calculate the single-particle and two-particle
quantities. Firstly, we generalize the OPM to Chebyshev orthogonal polynomial basis. Secondly,
the KPM based on various integral kernel functions is proposed to damp the Gibbs oscillations
observed in the single-particle and two-particle Green's functions obtained with OPM and improve the
accuracy of them further. According to the benchmark results for single-band half-filled Hubbard model,
it is demonstrated that the Jackson kernel is the optimal choice for imaginary-time Green's function
$G(\tau)$ and other quantities. Though the KPM presented in this paper is mainly developed for
the CT-HYB quantum impurity solver, it can be easily generalized to other continuous-time quantum
Monte Carlo impurity solvers.\cite{gull:349}
\begin{acknowledgments}
We acknowledge financial support from the National Science Foundation ͑of China and that
from the 973 program of China under Contract No.2007CB925000 and No.2011CBA00108.
\end{acknowledgments}
|
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"redpajama_set_name": "RedPajamaArXiv"
}
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var load = function() {
console.log("load, biatch!");
}
exports.load = load;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,619
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In our restaurants you can be sure that the carrot in your salad has been freshly prepared for you by hand. The egg in your burger comes from happy chickens and the bread comes straight from the oven of a local baker.
Avocado, chickpeas, tomatoes, raw broccoli, mushrooms, hot peppers, lettuce with Noa balsamic sauce. Served with fresh organic wholemeal bread.
All our ingredients are free of artificial additives and come mainly from regional producers, depending on seasonal availability. All Take Away packaging, cups and cutlery are 100% compostable.
Salads, soups, bowls, burgers. At Gartnerei, there is something for everyone. Available in home delivery with Smood.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,540
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Metallurgie Hoboken is de naam van een metallurgisch bedrijf dat in 1887 werd opgericht te Hoboken en dat ook tegenwoordig nog bestaat als een vestiging van Umicore.
Geschiedenis
Het bedrijf werd opgericht om looderts te verwerken. Ook werd er aanvankelijk zinkwit geproduceerd. Bij de fabriek werd de arbeiderswijk Moretusburg gebouwd. In de volksmond werd het bedrijf de zilverfabriek genoemd. In 1908 werd de naam Métallurgie hoboken ingevoerd. Dit bedrijf fuseerde met fabrieken te Olen en Overpelt tot MHO. In 1989 ging dit samen met de fabrieken van Vieille Montagne en ontstond Acec–Union Minière dat in 1992 Union Minière ging heten, sinds 2001 Umicore.
Oorspronkelijk was het bedrijf sterk verontreinigend. De bodem in de omgeving van de fabriek werd sterk verontreinigd met lood, cadmium en arseen. Pas toen er in 1973 een aantal paarden en koeien stierven door het eten van gras uit de omgeving van de fabriek kwam men tot serieuze aanpak van het probleem. Hieropvolgende metingen van het loodgehalte in het bloed van de kinderen uit de omgeving toonde verontrustende resultaten. Dit alles resulteerde in een saneringsprogramma van de omgeving en in een moderniseringsprogramma van de productieprocessen in de fabriek. In de daaropvolgende jaren daalden de loodgehalten, totdat ze in 2006 weer op een normaal niveau waren terechtgekomen.
Tegenwoordig is Metallurgie Hoboken een moderne fabriek en een centrum van terugwinning van een reeks waardevolle metalen uit zowel anodeslib van elektrochemische processen als uit afgedankte elektronische apparatuur.
Externe links
Saneringsplan
Hoboken
Metaalbedrijf
Bedrijf met zetel in Vlaanderen
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"redpajama_set_name": "RedPajamaWikipedia"
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#PlotagraphFriday
Photo Trips
Skye Photo Guide
5 Scottish Songs for Your Next Skye Photography Trip
There is lot's of driving to be done when you are moving between the photography locations on the Isle of Skye. I love to listen to traditional Scottish music while I am travelling as it adds even more to the atmosphere of this beautiful island.
In this post, I'd like to share with you 5 of my favourite Scottish songs that I always keep on my driving playlist. I will also share my favourite cover of each song with a YouTube link and lyrics so you can sing along while driving to another fantastic location.
1. The Skye Boat Song
"The Skye Boat Song" is a late 19th-century Scottish song recalling the journey of Prince Charles Edward Stuart (Bonnie Prince Charlie) from Uist to the Isle of Skye as he evaded capture by Government troops after his defeat at the Battle of Culloden in 1746. This song was brought back to life by the creators of the Outlander series when it was used in the series as the main title theme.
Lyrics: Here
My favorite cover: Noel McLoughlin
2. The Isle of Skye
The Corries were a Scottish folk group created in the 1960s. The group was a trio until 1966 when founder Bill Smith left the band, but Roy Williamson and Ronnie Browne continued as a duo until Williamson's death in 1990. They were closely identified with Jacobite songs, celebrating the final years of clan loyalty and military courage. The 1977 album, Peat Fire Flame, saw the group move towards love songs and celebrations of the landscape and in December 2007, The Corries were inducted into the Scottish Traditional Music Hall of Fame. The Isle of Skye song was written in 1976 and performed several times on television.
My favorite cover: The Corries
3. Skye
The next song is very popular on the Isle of Skye as it was written and performed by one of the most famous bands formed on the Isle of Skye - Runrig. There is enough written all across the internet about this Scottish Celtic rock band formed on the Isle of Skye in 1973 so make sure you look around to find out more. If you are going to listen to this specific song, you should look for the live version performed by Runrig with their former frontman Donnie Munro.
My favorite cover: Runrig
4. Home on the Sea
The fourth song isn't exactly about the Isle of Skye, but the strong island connection to the sea, boats and ferries makes it very popular. The Home on the Sea is composed and performed by a Scottish band called Skipinnish is also an official song of CalMac ferries - the major operator of passenger and vehicle ferries, and ferry services, between the mainland of Scotland and 22 of the major islands on Scotland's west coast.
My favorite cover: Skipinnish
5. Caledonia
The final song is an absolute must on your "Skye Trip" playlist. Caledonia is a modern Scottish folk ballad written by Dougie MacLean in 1977 and the word "Caledonia" itself being a Latin word for Scotland. The song became the most popular of all MacLean's recordings and something of an anthem for Scotland. "Caledonia" has been covered by a
many artists, but my favourite is still the original live version performed by Maclean.
My favorite cover: Dougie MacLean
#IsleofSkye #Guide #Music #Song #Playlist
"Skye Big 5" - Five Most Popular Spots on the Isle of Skye
3 Most Visited Lighthouses on the Isle of Skye
Skye Highland Cow Locations
Isle of Skye, Scotland | +44 (0) 7762 206 231 |
jakub@borsphoto.com
© 2017-19 Jakub Bors
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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<?php
include_once(__DIR__ . '/autoloader.class.php');
use company\alias\autoloader\Autoloader;
$auto = Autoloader::getInstance();
spl_autoload_register(array($auto,'loadClass'),true,true);
?>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,532
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FN 5,7×28 je malokalibrski naboj, ki ga je razvil belgijski proizvajalec orožja FN Herstal. Gre za novo konstrukcijo, ki je odgovor na razpis zveze NATO za zamenjavo naboja 9x19 Parabellum. Sočasno z nabojem sta bila razviti tudi brzostrelka P90 in pa pištola FN Five-seveN.
Zveza NATO je zahtevala naboj, ki bi imel večjo natančnost in doseg, kot naboj 9×19 mm. Posebna zahteva je bila sposobnost preboja lažjih neprebojnih jopičev na razdaljah do 200 m, ki so se začeli pogosteje pojavljati kot standardna oprema vojakov in je bil naboj 9×19 mm proti njim neučinkovit. Za namen preverjanja učinkovitosti je bil standardiziran t.i. CRISAT oklep, ki je bil sestavljen iz 1,6 mm debele titanove plošče in 20 plasti kevlarja.
Prototip naboja z oznako SS90 je uporabljal oklepnoprebojno kroglo s plastičnim jedrom in bakrenim plaščem. Kasnejša različica SS190 uporablja jedro krogle z jekleno konico in aluminijastim telesom in je po sestavi zelo podobna krogli M855, ki je standardna pri nabojih 5,56×45 NATO. Ta je za 2,7 mm krajša od prvotne krogle, kar je skrajšalo celotno dolžino naboja in ga tako naredilo primernega za uporabo v pištoli FiveSeveN. Proti ciljem, kjer prebojnost ni zahtevana, je bil razvit naboj SS192 s poloplaščeno kroglo z votlo konico.
Za prodajo civilnemu trgu so bili ustvarjeni še naboji SS195 z votlo konico in SS197 z nekoliko višjo hitrostjo krogle, vndar brez sposobnosti preboja oklepa.
Uporaba lažjih krogel povzroči, da je odsun orožja ob strelu približno 30% nižji, kot pri naboju 9×19 mm, kar pomeni večjo natančnost. Večja hitrost krogle omogoča tudi sorazmerno ravno trajektorijo do razdalje približno 100 m. Zaradi daljše krogle z majhno maso so krogle ob zadetku v mehko tkivo nestabilne in so nagnjene k prevračanju okrog prečne osi (to je sicer značilno za večino lahkih in hitrih krogel). Krogle zaradi dokaj nizke mase hitreje izgubljajo kinetično energijo z razdaljo in izgubi večino energije do razdalje 400 m, zato se na večjih razdaljah tudi zmanjša možnost prebojev in odbojev.
Posebnost naboja je tudi tulec, ki za razliko od mnogih podobnih tulcev nima konusa. Ker so tlaki smodniških plinov visoki in zaradi uporabe prostega zaklepa pri brzostrelki P90 so tulci prevlečeni s posebnim polimernim lakom, ki zmanjša trenje tulca v ležišču naboja in s tem olajša izvlačenje. Poleg tega prevleka olajša tudi gibanje nabojev v nabojniku brzostrelke P90.
Kritiki tega naboja menijo, da sorazmerno majhen kaliber ne zagotavlja zadosti hitre onesposobitve nasprotnika, ravno tako poudarjajo, da krogle, ki imajo oklepnoprebojne lastnosti, ne zagotavljajo zadostne učinkovitosti proti nasprotnikom, ki ne uporabljajo balistične zaščite. Te pomanjkljivosti so deloma nadomeščene na račun prevračanja krogle ob zadetku in pa možnosti hitrejšega ter bolj natančnega streljanja na račun manjšega odsuna.
Glej tudi
4,6×30
Viri
Naboji
Pištolski naboji
Fabrique Nationale
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{"url":"http:\/\/physics.aps.org\/synopsis-for\/10.1103\/PhysRevLett.105.248103","text":"# Synopsis: Cells push, cells pull\n\nAccounting for vertical forces leads to a greater understanding of how cells physically interact with soft substrates.\n\nOne of the major problems in biophysics is the mechanics of how cells interact with their physical environment. Processes such as cell adhesion and migration depend in a crucial manner on physical forces exerted by (and on) cells.\n\nIn their paper in Physical Review Letters, H\u00e9l\u00e8ne Delano\u00eb-Ayari and colleagues at the Universit\u00e9 de Lyon, France, and co-workers at the University of Tokyo, Japan, study the motion of cells on a substrate with a new twist to earlier approaches. Typical traction force microscopy techniques have mainly been restricted to measurements of forces exerted by cells on substrates that act in the plane of the substrate\u23af$x,y$, and $t$ (i.e., a \u201c3D\u201d force map emerges when one incorporates the time component). The authors study amoeba cells on a soft gel substrate where vertical deformation turns out to be critical; the vertical forces are measurable and comparable in magnitude to the in-plane counterparts. Incorporating this additional $z$ component enables the authors to develop a \u201c4D\u201d force map comprising upward pulling forces near the rim of the cell and downward pushing forces under the cell. Apart from the advance in microscopy techniques alone, an appealing universal picture of cell deformation emerges that is analogous to that of a liquid drop on an elastic substrate, where surface tension acts at the edges and the pressure of the fluid equilibrates these forces at the center. \u2013 Sami Mitra\n\nMore Features \u00bb\n\n### Announcements\n\nMore Announcements \u00bb\n\n## Subject Areas\n\nBiological Physics\n\nOptics\n\n## Next Synopsis\n\nAtomic and Molecular Physics\n\n## Related Articles\n\nBiological Physics\n\n### Synopsis: Racing Bacteria\n\nBacteria track fast-moving chemical signals by hopping from one chemically favorable region to another. Read More \u00bb\n\nBiological Physics\n\n### Synopsis: Cell Sensing Improves in a Loose Crowd\n\nCells that communicate with each other can measure chemical concentrations with higher precision if they spread out into a sparse configuration. \u00a0 Read More \u00bb\n\nSoft Matter\n\n### Focus: Membrane Holes Can Shrink, Grow, or Stay Put\n\nPores in a polymer film do not change size over time if they have just the right diameter, according to experiments. Read More \u00bb","date":"2017-03-23 02:13:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 3, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.46200427412986755, \"perplexity\": 2530.3190710923814}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-13\/segments\/1490218186608.9\/warc\/CC-MAIN-20170322212946-00606-ip-10-233-31-227.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
\subsection{$p$-set}
Let $p$ be a prime number. We consider the point set
\[
\mathcal{P}_{d,p}=\left\{\mathbf{x}_{0}, \ldots, \mathbf{x}_{p-1}\right\}\subset[0, 1)^{d}
\]
where
\[
\mathbf{x}_{j}=\Big(\left\{\frac{j}{p}\right\}, \left\{\frac{j^{2}}{p}\right\}, \ldots, \left\{\frac{j^{d}}{p}\right\}\Big)\in [0,1)^d,\quad j\in \mathbb{Z}_{p},
\]
$\mathbb{Z}_{p}:=\left\{0, 1, \ldots, p-1\right\}$ and $\left\{x\right\}$ is the fractional part of $x$ for a
nonnegative real number $x$. The point set $\mathcal{P}_{d,p}$ is called $p$-set and was introduced by Korobov \cite{pset} and Hua-Wang \cite{HW}. Recently, $p$-set attracts much attention since its advantage in numerical integration \cite{Dick3}, in the recovery of sparse trigonometric polynomials \cite{Xu} and in the UQ \cite{XZ}. In \cite{Dick3}, Dick presents a numerical integration formula based on $\mathcal{P}_{d,p}$ with showing the error bound of the formula depends only polynomially on the dimension $d$. In \cite{Xu}, Xu uses $\mathcal{P}_{d,p}$ to construct the deterministic sampling points of sparse trigonometric polynomials and show the sampling matrix corresponding to $\mathcal{P}_{d,p}$ has the almost optimal coherence. And hence, $\mathcal{P}_{d,p}$ has a good performance for the recovery of sparse trigonometric polynomials.
\subsection{ Extensions of $p$-set: $\mathcal{P}^{\va, \epsilon}_{d, p}$ and $\mathcal{L}_{p, q}$}
The $p$-set is in a simple analytic form and hence it is easy to be generated by computer. However, the $p$-set is somewhat rigid with the point
set only depending on a prime number $p$. If the function values at some points in $p$-set are not easy to be obtained, one has to change the prime
number $p$ to obtain a new point set which has the different cardinality with the previous one. Hence, in practical application, it will be better that one has many different choices. We next introduce a generalization of $p$-set.
Let
\[
\mathbb{Z}_{p}^{d}:=\left\{\va=(a_{1}, \ldots, a_{d})\in\mathbb{Z}^{d}: a_{j}\in\mathbb{Z}_{p}, j=1, \ldots, d\right\}.
\]
Suppose that $\va=(a_{1}, \ldots, a_{d})\in \mathbb{Z}^{d}_{p}$ and $\epsilon=(\epsilon_{1},
\ldots, \epsilon_{d-1})\in \left\{0, 1\right\}^{d-1}$. We set
\begin{equation}\label{eq:pset}
\mathcal{P}^{\va, \epsilon}_{d, p}:=\left\{\mathbf{x}_{j}^{\va,\epsilon}:j\in\mathbb{Z}_{p}\right\}
\end{equation}
where
$$
\mathbf{x}_{j}^{\va,\epsilon}:=\Big(\left\{\frac{a_{1}j}{p}\right\}, \left\{\frac{a_{1}'j+a_{2}j^{2}}{p}\right\}, \ldots, \left\{
\frac{\sum_{h=1}^{d-1}a_{h}'j^{h}+a_{d}j^{d}}{p}\right\}\Big)\in [0,1)^d
$$
and
$a_{k}'=\epsilon_{k}a_{k}, k=1, \ldots, d-1$.
We call $\mathcal{P}^{\va, \epsilon}_{d, p}$ as the $p$-set associating with
the parameter $\va$ and $\epsilon$. If we take $\va=(1, \ldots, 1)$ and $\epsilon=(0, \ldots, 0)$, then $\mathcal{P}^{\va, \epsilon}_{d, p}$ is reduced to the classical $p$-set.
The $p$-set $\mathcal{P}^{\va, \epsilon}_{d, p}$ associating with the parameters $\va, \epsilon$ is more flexible. Given the prime number $p$, one can generate various point sets by changing the parameters $\va$ and $\epsilon$ with presenting an option set when the cardinality $p$ is given.
Note that the cardinality of both $\mathcal{P}^{\va, \epsilon}_{d, p}$ and $\mathcal{P}_{d, p}$ is prime.
Since the distance between adjacent prime can be very large, the cardinality of $p$-set does not change smoothly.
Using the set $\mathcal{P}^{\va, \epsilon}_{d, p}$, we next present a set with the cardinality being odd number. Suppose that $m\in 2\mathbb{Z}$ is given. The Goldbach conjecture, which is
one of the best-known unsolved problem in number theory, says that $m$ can be written as the sum of two primes, i.e., $m=p+q$ where $p$ and $q$
are prime numbers. One has verified the conjecture up to $m\leq 4\cdot 10^{14}$ which is enough for practical application. We next suppose that $m=p+q$
with $p$ and $q$ being prime numbers. We set
\begin{equation}\label{eq:sp1}
\mathcal{L}_{p, q}:=\left\{
\begin{array}{c}
\mathcal{P}_{d,p}\cup\mathcal{P}_{d,q}, \quad p\neq q\\
\mathcal{P}^{\va, \epsilon'}_{d, p}\cup
\mathcal{P}^{\vb, \epsilon''}_{d, p}, \quad p=q,
\end{array}\right.
\end{equation}
where $\mathcal{P}^{\va, \epsilon'}_{d, p}$ and $\mathcal{P}^{\vb, \epsilon''}_{d, p}$ are the $p$-sets that we have defined above and $\va,\vb\in \mathbb{Z}_{p}^{d}, \epsilon',\epsilon''\in \left\{0,1\right\}^{d-1}$. We call $\mathcal{L}_{p, q}$ the $(p,q)$-set. As shown later, $\mathcal{P}_{d,p}\cap \mathcal{P}_{d,q}=\left\{(0,\ldots,0)\right\}$ provided $p\neq q$. We can choose $\va,\vb, \epsilon'$ and $\epsilon''$ so that $\mathcal{P}^{\va, \epsilon'}_{d, p}\cap
\mathcal{P}^{\vb, \epsilon''}_{d, p}=\left\{(0,\ldots,0)\right\}$. Hence, under the assumption of Goldbach conjecture, for any odd number, says $m-1$, there exist $p,q$ so that $|\mathcal{L}_{p, q}|=p+q-1=m-1$.
We would like to mention the following point sets with cardinality $p^2$ \cite{pset,HW} :
\begin{equation}\label{eq:p2set}
\begin{aligned}
{\mathcal Q}_{p^2,d}&=\left\{\vz_j : j=0,\ldots,p^2-1\right\},\, \mathbf{z}_{j}=\Big(\left\{\frac{j}{p^2}\right\}, \left\{\frac{j^{2}}{p^2}\right\}, \ldots, \left\{\frac{j^{d}}{p^2}\right\}\Big)\in [0,1)^d;\\
{\mathcal R}_{p^2,d}&=\left\{\vz_{j,k} : j,k=0,\ldots,p-1\right\},\, \vz_{j,k}=\Big(\left\{\frac{k}{p} \right\},\left\{\frac{jk}{p} \right\},\ldots,\left\{\frac{j^{d-1}k}{p} \right\}\Big)\in [0,1)^d.
\end{aligned}
\end{equation}
The weighted star discrepancy of ${\mathcal Q}_{p^2,d}$ and ${\mathcal R}_{p^2,d}$ is given in \cite{Dick2}. Using a similar method with above, we can generalize ${\mathcal Q}_{p^2,d}$ and ${\mathcal R}_{p^2,d}$ to
${\mathcal Q}_{p^2,d}^{\va,\epsilon}$ and ${\mathcal R}_{p^2,d}^{\va,\epsilon}$, respectively. We will introduce it in Section 2.3 in detail.
\subsection{Organization}
In Section 2, we present the upper bounds of the exponential sums over $\mathcal{P}^{\va, \epsilon}_{d, p}$ and $\mathcal{L}_{p, q}$. Particularly, we present the condition under which
$\abs{\mathcal{L}_{p, q}}=p+q-1$ and also prove that
$\mathcal{P}^{\va, \epsilon}_{d, p}\cap \mathcal{P}^{\vb, \epsilon}_{d, p}=\left\{(0,\ldots,0)\right\}$ when $\mathcal{P}^{\va, \epsilon}_{d, p}\neq \mathcal{P}^{\vb, \epsilon}_{d, p}$. We furthermore consider the generalization of the point sets ${\mathcal Q}_{p^2,d}$ and ${\mathcal R}_{p^2,d}$ and present the upper bounds of exponential sums over the new sets. The results in Section 2 show that
the point sets presented in this paper have many potential applications in various areas.
In Section 3, we choose $\mathcal{L}_{p, q}$ as a deterministic sampling set for the recovery of sparse trigonometric polynomials and then show their performance.
\section{The exponential sums over $\mathcal{P}^{\va, \epsilon}_{d, p}$ and $\mathcal{L}_{p, q}$}
The aim of this section is to present the exponential sums over $\mathcal{P}^{\va, \epsilon}_{d, p}$ and $\mathcal{L}_{p, q}$. To this end, we first introduce the well-known Weil's formula, which plays a key role in our proof.
\begin{theorem}\label{theorem:re1}
\cite{weil} Suppose that $p$ is a prime number. Suppose $f(x)=\sum_{h=1}^{d}m_{h}x^{h}$ with $m_{h}\in\mathbb{Z}$ $(h=1, \ldots, d)$ and there is a $j\in \left\{1,2, \ldots d\right\}$,
satisfying $p \nmid m_{j}$. Then
\begin{equation}
\Big|\sum_{x=1}^{p}e^{\frac{2\pi \mathbf{i} f(x)}{p}}\Big|\leq (d-1)\sqrt{p}.\nonumber
\end{equation}
\end{theorem}
\subsection{The exponential sum over $\mathcal{P}^{\va, \epsilon}_{d, p}$ }
Recall that
\[
\mathcal{P}^{\va, \epsilon}_{d, p}\,\,:=\,\,\left\{\mathbf{x}_{j}^{\va,\epsilon}:j\in\mathbb{Z}_{p}\right\}
\]
and
\[
\mathbf{x}_{j}^{\va,\epsilon}=\Big(\left\{\frac{a_{1}j}{p}\right\}, \left\{\frac{a_{1}'j+a_{2}j^{2}}{p}\right\}, \ldots, \left\{
\frac{\sum_{h=1}^{d-1}a_{h}'j^{h}+a_{d}j^{d}}{p}\right\}\Big)\in [0,1)^d
\]
where $\va=(a_1,\ldots,a_d)\in[1, p-1]^{d}\cap\mathbb{Z}^{d}$, $a_j'=\epsilon_ja_j$ and
$\epsilon=(\epsilon_{1},\ldots, \epsilon_{d-1})\in \left\{0, 1\right\}^{d-1}$.
Note that $|\mathcal{P}^{\va, \epsilon}_{d, p}|=p$. We next show the exponential sum formula over
$\mathcal{P}^{\va, \epsilon}_{d, p}$.
\begin{theorem}\label{th:weil1}
For any $\mathbf{k}\in[-p+1, p-1]^{d}\cap\mathbb{Z}^{d}$ and $\mathbf{k}\neq 0$, we have
\begin{equation}\label{eq:sum1}
\left|\sum_{\mathbf{x}\in \mathcal{P}^{\va, \epsilon}_{d, p}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot \mathbf{x})\right|
=\Big|\sum_{j=0}^{p-1}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x}_{j}^{\va,\epsilon})\Big|\leq (d-1)\sqrt{p}.
\end{equation}
\end{theorem}
\begin{proof}
Set
\begin{equation*}
g(j)=\sum_{\ell=1}^{d}c_{\ell}j^{\ell}
\end{equation*}
where $c_{\ell}=k_{\ell}a_{\ell}+k_{\ell+1}a_{\ell}'+\cdots+k_{d}a_{\ell}'$.
We set $j_0:=\max\left\{\ell:k_{\ell}\neq 0\right\}$. Then $c_{j_0}=k_{j_0}a_{j_0}$ and we have $p\nmid c_{j_0}$.
According to Theorem \ref{theorem:re1}, we obtain that
\begin{equation*}
\Big|\sum_{j=0}^{p-1}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x}_{j}^{\va,\epsilon})\Big|
=\Big|\sum_{j=0}^{p-1}\exp\Big(2\pi\mathbf{i}\frac{g(j)}{p}\Big)\Big|\leq (d-1)\sqrt{p}.
\end{equation*}
\end{proof}
\subsection{The exponential sum over $\mathcal{L}_{p, q}$}
To this end, we consider the cardinality of $\mathcal{L}_{p,q}$. A simple observation is
that $\abs{\mathcal{L}_{p,q}}\leq p+q-1$. We would like to present the condition under which
$\abs{\mathcal{L}_{p,q}}= p+q-1$.
We first consider the case where $p\neq q$.
\begin{theorem}
Suppose that $p$ and $q$ are two distinct prime numbers. Then $|\mathcal{L}_{p,q}|=p+q-1$.
\end{theorem}
\begin{proof}
According to (\ref{eq:sp1}), to this end, we just need show that
\[
\mathcal{P}_{d, p}\cap\mathcal{P}_{d, q}=\left\{(0, \ldots, 0)\right\}.
\]
We prove it by contradiction. Assume that $\mathcal{P}_{d, p}\cap\mathcal{P}_{d, q}\neq\left\{(0, \ldots, 0)\right\}$, and then there exists
$j\in\mathbb{Z}_{p}^{*}:=\mathbb{Z}_{p}\setminus\left\{0\right\}$ and $k\in\mathbb{Z}_{q}^{*}$ so that $\left\{\frac{j^{i}}{p}\right\}=\left\{\frac{k^{i}}{q}\right\}$, $i=1, \ldots, d$.
Particularly, we have $\frac{j}{p}=\frac{k}{q}$, which is equivalent to $jq=kp$. Since $p$ and $q$ are different prime numbers, $j\in\mathbb{Z}_{p}^{*}$ and $k\in\mathbb{Z}_{q}^{*}$, we have $j|k$ and $k|j$, which means $j=k$ and hence $p=q$. A contradiction.
\end{proof}
We next consider the case where $p=q$, i.e., ${\mathcal L}_{p,q}=\mathcal{P}^{\va, \epsilon'}_{d, p}\cup
\mathcal{P}^{\vb, \epsilon''}_{d, p}$. For the case where $\epsilon'=\epsilon''$,
we have
\begin{theorem}\label{th:deng}
Suppose that $\epsilon\in\left\{0, 1\right\}^{d-1}$ is a fixed vector and $\va=(a_1,\ldots,a_d)\in\mathbb{Z}_{p}^{d}$, $\vb=(b_1,\ldots,b_d)\in\mathbb{Z}_{p}^{d}$.
\begin{enumerate}
\item $\mathcal{P}^{\va, \epsilon}_{d, p}=\mathcal{P}^{\vb, \epsilon}_{d, p}$ if and only if there exists $c\in\mathbb{Z}_{p}^{*}$
such that
\[
b_{j}c^{j}\equiv a_{j}\quad (\mathrm{mod}\quad p) \quad for\quad j=1,\ldots,d.
\]
\item If
$\mathcal{P}^{\va, \epsilon}_{d, p}\neq\mathcal{P}^{\vb, \epsilon}_{d, p}$, then $\mathcal{P}^{\va, \epsilon}_{d, p}\cap\mathcal{P}^{\vb, \epsilon}_{d, p}=\left\{(0, \ldots, 0)\right\}$.
\end{enumerate}
\end{theorem}
\begin{proof}
(1) We first suppose that there exists $c\in\mathbb{Z}_{p}^{*}$ such that $b_{j}c^{j}\equiv a_{j}\,\,(\mathrm{mod} \quad p)$ for $j\in \left\{1,\ldots,d\right\}$. Recall that
$$
\mathcal{P}^{\va, \epsilon}_{d, p}=\left\{\mathbf{x}_{j}^{\va,\epsilon} : j\in\mathbb{Z}_{p}\right\},
$$
where
\[
\mathbf{x}_{j}^{\va,\epsilon}=\Big(\left\{\frac{a_{1}j}{p}\right\}, \left\{\frac{a_{1}'j+a_{2}j^{2}}{p}\right\}, \ldots, \left\{
\frac{\sum_{h=1}^{d-1}a_{h}'j^{h}+a_{d}j^{d}}{p}\right\}\Big).
\]
For any $j_{0}\in \mathbb{Z}_{p}$, we take $k_{0}\equiv cj_{0}\quad(\mathrm{mod}\quad p)$. Then
\begin{eqnarray}
\mathbf{x}_{j_{0}}^{\va,\epsilon} &=& \Big(\left\{\frac{a_{1}j_{0}}{p}\right\}, \left\{\frac{a_{1}'j_{0}+a_{2}j_{0}^{2}}{p}\right\}, \ldots, \left\{
\frac{\sum_{h=1}^{d-1}a_{h}'j_{0}^{h}+a_{d}j_{0}^{d}}{p}\right\}\Big)\nonumber\\
&=& \Big(\left\{\frac{b_{1}cj_{0}}{p}\right\}, \left\{\frac{b_{1}'cj_{0}+b_{2}c^{2}j_{0}^{2}}{p}\right\}, \ldots, \left\{
\frac{\sum_{h=1}^{d-1}b_{h}'c^{h}j_{0}^{h}+b_{d}c^{d}j_{0}^{d}}{p}\right\}\Big)\nonumber\\
&=& \Big(\left\{\frac{b_{1}k_{0}}{p}\right\}, \left\{\frac{b_{1}'k_{0}+b_{2}k_{0}^{2}}{p}\right\}, \ldots, \left\{
\frac{\sum_{h=1}^{d-1}b_{h}'k_{0}^{h}+b_{d}k_{0}^{d}}{p}\right\}\Big)\nonumber\\
&=& \mathbf{x}_{k_{0}}^{\vb,\epsilon},\nonumber
\end{eqnarray}
which implies that
$$\mathcal{P}^{\va, \epsilon}_{d, p}\,\,\subseteq\,\, \mathcal{P}^{\vb, \epsilon}_{d, p}.$$
Here we use $b_{j}'c^{j}\equiv a_{j}'\,\, (\mathrm{mod}\,\, p)$ which follows from $b_{j}c^{j}\equiv a_{j}\,\,(\mathrm{mod}\,\, p)$.
Since $p$ is a prime number, there exists $c^{-1}\in\mathbb{Z}_{p}^{*}$ so that $c^{-1}c\equiv 1\quad(\mathrm{mod}\,\, p)$. Then we have
$b_{j}\equiv a_{j} c^{-j}\quad(\mathrm{mod}\quad p), j=1, \ldots, d$. Then, similarly, for
any $j_{0}\in\mathbb{Z}_{p}$,
$$
\mathbf{x}_{j_{0}}^{\vb,\epsilon}\,\,=\,\,\mathbf{x}_{c^{-1}j_{0}}^{\va,\epsilon},
$$
which implies that
$$
\mathcal{P}^{\vb, \epsilon}_{d, p}\,\,\subseteq\,\, \mathcal{P}^{\va, \epsilon}_{d, p}.
$$
Then we arrive at
$$
\mathcal{P}^{\va, \epsilon}_{d, p}=\mathcal{P}^{\vb, \epsilon}_{d, p}.
$$
We next suppose that $\mathcal{P}^{\va, \epsilon}_{d, p}=\mathcal{P}^{\vb, \epsilon}_{d, p}$. Then there exist
$j_{0}, k_{0}\in\mathbb{Z}_{p}^{*}$ so that $\mathbf{x}_{j_{0}}^{\va}=\mathbf{x}_{k_{0}}^{\vb}$, i.e.,
\begin{equation}\label{eq:theorem11}
a_{1}j_{0}\equiv b_{1}k_{0}\quad (\mathrm{mod} \quad p),
\end{equation}
\begin{equation}\label{eq:theorem12}
a_{1}'j_{0}+a_{2}j_{0}^{2}\equiv b_{1}'k_{0}+b_{2}k_{0}^{2}\quad (\mathrm{mod} \quad p),
\end{equation}
\begin{eqnarray}
&&\vdots\nonumber
\end{eqnarray}
\begin{equation}
\sum_{h=1}^{d-1}a_{h}'j_{0}^{h}+a_{d}j_{0}^{d}\equiv \sum_{h=1}^{d-1}b_{h}'k_{0}^{h}+b_{d}k_{0}^{d}\quad (\mathrm{mod} \quad p).\nonumber
\end{equation}
We set $c:\equiv k_{0}j_{0}^{-1}\quad (\mathrm{mod} \quad p)$, where $j_{0}^{-1}\in\mathbb{Z}_{p}$ so that $j_{0}^{-1}j_{0}\equiv 1\quad (\mathrm{mod} \quad p)$. Then (\ref{eq:theorem11}) implies that
$a_{1}\equiv b_{1}c\quad (\mathrm{mod} \quad p)$. Combining (\ref{eq:theorem11}) and (\ref{eq:theorem12}), we have $a_{2}j_{0}^{2}\equiv b_{2}k_{0}^{2}\quad (\mathrm{mod} \quad p)$ which implies that $a_{2}\equiv b_{2}c^{2}\quad (\mathrm{mod} \quad p)$. Similarly, we can
obtain that $a_{j}\equiv b_{j}c^{j}\quad (\mathrm{mod} \quad p)$ for $3\leq j\leq d$.
(2) We prove it by contradiction. Assume that $\mathcal{P}_{d, p}\cap\mathcal{P}_{d, q}\neq\left\{(0, \ldots, 0)\right\}$, and then there exist
$j_{0}, k_{0}\in\mathbb{Z}_{p}^{*}$ so that $\mathbf{x}_{j_{0}}^{\va,\epsilon}=\mathbf{x}_{k_{0}}^{\vb,\epsilon}$. Similarly with the above proof, we can
find a $c:\equiv k_{0}j_{0}^{-1}\quad (\mathrm{mod} \quad p)$ so that $a_{j}\equiv b_{j}c^{j}\quad (\mathrm{mod} \quad p)$ for $j=1, \ldots, d$.
It leads to $\mathcal{P}^{\va, \epsilon}_{d, p}=\mathcal{P}^{\vb, \epsilon}_{d, p}$ by (1) of Theorem 2.4, which is impossible by the assumption in (2).
\end{proof}
We next consider the case where $\epsilon' \neq \epsilon''$.
\begin{theorem}\label{th:cap}
Suppose that $\epsilon', \epsilon''\in\left\{0, 1\right\}^{d-1}$ with $\epsilon'\neq\epsilon''$.
Set
\[
{\mathcal Z}\,\,:=\,\, \left\{j: \epsilon_{j}'\neq \epsilon_{j}'' \,\,\text{and}\,\, a_{j}^{2}+b_{j}^{2}\neq 0, 1\leq j\leq d-1\right\},
\]
and
\[
\ell_0:=\min \left\{j: j\in \mathcal{Z}\right\}.
\]
Then the followings hold.
\begin{enumerate}
\item $\mathcal{P}^{\va, \epsilon'}_{d, p}=\mathcal{P}^{\vb, \epsilon''}_{d, p}$ if and only if there exists a
$c\in\mathbb{Z}_{p}^{*}$ so that
\begin{equation}\label{eq:cond1}
\begin{aligned}
& a_{j}\equiv b_{j}c^{j}\quad (\mathrm{mod}\quad p) \quad for \quad \epsilon_{j}'=\epsilon_{j}''\quad or\quad j=d, \\
& a_{j}=b_{j}=0\quad for\quad \epsilon_{j}'\neq\epsilon_{j}'',
\end{aligned}
\end{equation}
where $\va, \vb\in\mathbb{Z}_{p}^{d}\setminus\left\{(0, \ldots, 0)\right\}$ with $a_{1}\neq 0$ being given.
\item Assume that $\mathcal{P}^{\va, \epsilon'}_{d, p}\neq\mathcal{P}^{\vb, \epsilon''}_{d, p}$ where $\va, \vb\in\mathbb{Z}_{p}^{d}$.
If ${\mathcal Z}=\emptyset $ then we have
\[
\mathcal{P}^{\va, \epsilon'}_{d, p}\cap\mathcal{P}^{\vb, \epsilon''}_{d, p}=\left\{(0,\ldots,0)\right\}.
\]
If ${\mathcal Z} \neq \emptyset$, then
$|{\mathcal{P}^{\va, \epsilon'}_{d, p}\cap\mathcal{P}^{\vb, \epsilon''}_{d, p}}|\leq r+1$, where
\[
r=\min\left\{j: a_{j}^{2}+b_{j}^{2}\neq 0\right\}.
\]
\item
Assume that ${\mathcal Z}\neq \emptyset$, and $a_1\neq 0$.
If $a_{\ell_0+1}b_{1}^{\ell_0+1}\equiv a_{1}^{\ell_0+1}b_{\ell_0+1}\quad(\mathrm{mod} \quad p)$, then $\mathcal{P}^{\va, \epsilon'}_{d, p}\cap\mathcal{P}^{\vb, \epsilon''}_{d, p}=\left\{(0, \ldots, 0)\right\}$.
\end{enumerate}
\end{theorem}
\begin{proof}
(1)
We assume that (\ref{eq:cond1}) holds.
Take
\begin{equation}
\epsilon_j=
\left\{
\begin{array}{c}
\epsilon_j',\quad \text{ if }\epsilon_j'=\epsilon_j''\\
0, \quad \text{ if }\epsilon_j'\neq \epsilon_j''
\end{array}\right. .
\nonumber
\end{equation}
Noting that $a_j=b_j=0$ provided $\epsilon_j'\neq \epsilon_j''$, we have
$\mathcal{P}^{\va, \epsilon}_{d, p}=\mathcal{P}^{\va, \epsilon'}_{d, p}$ and $\mathcal{P}^{\vb, \epsilon}_{d, p}=\mathcal{P}^{\vb, \epsilon''}_{d, p}$. Theorem \ref{th:deng} implies that
$\mathcal{P}^{\va, \epsilon}_{d, p}=\mathcal{P}^{\vb, \epsilon}_{d, p}$ and hence
$\mathcal{P}^{\va, \epsilon'}_{d, p}=\mathcal{P}^{\vb, \epsilon''}_{d, p}$.
We next assume that $\mathcal{P}^{\va, \epsilon'}_{d, p}=\mathcal{P}^{\vb, \epsilon''}_{d, p}$
which is equivalent to that there exists
a permutation $\left\{k_{0}, k_{1}, \ldots, k_{p-1}\right\}$ of $\left\{0, 1, \ldots, p-1\right\}$ so that
\begin{equation}\label{eq:6}
\mathbf{x}_{j}^{\va,\epsilon'}=\mathbf{x}_{k_{j}}^{\vb,\epsilon''},\quad j=0, 1, \ldots, p-1.
\end{equation}
This is equivalent to
\begin{equation}\label{eq:theorem31}
\begin{aligned}
a_{1}&\equiv b_{1}k_{1}\quad (\mathrm{mod}\quad p)\\
2a_{1}&\equiv b_{1}k_{2}\quad (\mathrm{mod}\quad p)\\
&\vdots\\
(p-1)a_{1}&\equiv b_{1}k_{p-1}\quad (\mathrm{mod}\quad p)
\end{aligned}
\end{equation}
and
\begin{equation}\label{eq:theorem32}
\begin{aligned}
\sum_{h=1}^{i-1} a_{h}'+a_{i} &\equiv \sum_{h=1}^{i-1}b_{h}'k_{1}^{h}+b_{i}k_{1}^{i}\quad (\mathrm{mod} \quad p)\\
\sum_{h=1}^{i-1}2^{h}a_{h}'+a_{i}2^{i}&\equiv \sum_{h=1}^{i-1}b_{h}'k_{2}^{h}+b_{i}k_{2}^{i}\quad (\mathrm{mod}\quad p)\\
\vdots\\ \sum_{h=1}^{i-1}(p-1)^{h}a_{h}'+a_{i}(p-1)^{i}&\equiv \sum_{h=1}^{i-1}b_{h}'k_{p-1}^{h}+b_{i}k_{p-1}^{i}\quad (\mathrm{mod}\quad p),
\end{aligned}
\end{equation}
for $i=2, \ldots, d$. Since $a_{1}\neq 0$, by (\ref{eq:theorem31}) we have
\begin{equation}\label{eq:theorem33}
\begin{aligned}
a_{1}&\equiv b_{1}k_{1}\quad (\mathrm{mod}\,\, p)\\
k_{2}&\equiv 2k_{1}\quad (\mathrm{mod}\,\, p)\\
&\vdots\\
k_{p-1}&\equiv (p-1)k_{1}\quad (\mathrm{mod}\,\, p).
\end{aligned}
\end{equation}
Set $j_0:=\min\left\{i: \epsilon_{i}'\neq \epsilon_{i}''\right\}$. Using the same argument with the one in Theorem \ref{th:deng} we have $a_{i}\equiv b_{i}k_{1}^{i}\quad(\mathrm{mod} \quad p)$, $i=1, \ldots, j_0$. Combining (\ref{eq:theorem32}) for $i=j_0+1$ and (\ref{eq:theorem33}) we have
\begin{eqnarray}
a_{j_0}'+a_{j_0+1}\equiv b_{j_0}'k_{1}^{j_0}+b_{j_0+1}k_{1}^{j_0+1}\quad(\mathrm{mod} \quad p)\nonumber\\
2^{j_0}a_{j_0}'+2^{j_0+1}a_{j_0+1}\equiv b_{j_0}'k_{2}^{j_0}+b_{j_0+1}k_{2}^{j_0+1}\quad(\mathrm{mod} \quad p).\nonumber
\end{eqnarray}
Without loss of generality, we can assume $\epsilon_{j_0}'=1$ and $\epsilon_{j_0}''=0$ and then
\begin{eqnarray}
a_{j_0}+a_{j_0+1}\equiv b_{j_0+1}k_{1}^{j_0+1}\quad(\mathrm{mod} \quad p)\nonumber\\
2^{j_0}a_{j_0}+2^{j_0+1}a_{j_0+1}\equiv b_{j_0+1}k_{2}^{j_0+1}\quad(\mathrm{mod} \quad p), \nonumber
\end{eqnarray}
which implies $a_{j_0}=b_{j_0}=0$ since $k_{2}\equiv 2k_{1}\quad(\mathrm{mod} \quad p)$ and
$a_{j_{0}}\equiv b_{j_{0}}k_{1}^{j_{0}}\quad(\mathrm{mod}\quad p)$.
(2) We first assume that ${\mathcal Z}=\emptyset$ which implies $a_j=b_j=0$ provided $\epsilon_{j}'\neq\epsilon_{j}''$.
Take
\begin{equation}
\epsilon_j=
\left\{
\begin{array}{c}
\epsilon_j',\quad \text{ if }\epsilon_j'=\epsilon_j''\\
0, \quad \text{ if }\epsilon_j'\neq \epsilon_j''
\end{array}\right. .
\nonumber
\end{equation}
Noting that $a_j=b_j=0$ provided $\epsilon_j'\neq \epsilon_j''$, we have
$\mathcal{P}^{\va, \epsilon}_{d, p}=\mathcal{P}^{\va, \epsilon'}_{d, p}$ and $\mathcal{P}^{\vb, \epsilon}_{d, p}=\mathcal{P}^{\vb, \epsilon''}_{d, p}$.
The (2) of Theorem 2.4 implies that $\mathcal{P}^{\va, \epsilon'}_{d, p}\cap\mathcal{P}^{\vb, \epsilon''}_{d, p}=\left\{(0, \ldots, 0)\right\}$. We next consider the case where ${\mathcal Z}\neq \emptyset$.
Suppose that $\mathcal{P}^{\va, \epsilon'}_{d, p}$ and $\mathcal{P}^{\vb, \epsilon''}_{d, p}$ have a common nonzero point.
Then, there exist $j, k\in\mathbb{Z}_{p}^{*}$ so that
\begin{equation}\label{eq:fang}
\begin{aligned}
a_{1}j&\equiv b_{1}k\quad(\mathrm{mod} \quad p)\\
a_{1}'j+a_{2}j^{2}&\equiv b_{1}'k+b_{2}k^{2}\quad(\mathrm{mod} \quad p)\\
&\vdots\\
\sum_{h=1}^{d-1}a_{h}'j^{h}+a_{d}j^{d}&\equiv \sum_{h=1}^{d-1}b_{h}'k^{h}+b_{d}k^{d}\quad(\mathrm{mod} \quad p).
\end{aligned}
\end{equation}
Note that $a_h=b_h=0$ when $h\leq r-1$ and $a_h'=b_h'=0, h\leq r-1$ .
The (\ref{eq:fang}) implies that
\begin{equation}\label{eq:theorem36}
a_{h}j^{h}\equiv b_{h}k^{h}\quad(\mathrm{mod} \quad p), \quad h=r, \ldots, \ell_0,
\end{equation}
\begin{equation}\label{eq:theorem37}
a_{\ell}'j^{\ell_0}+a_{\ell_0+1}j^{\ell_0+1}\equiv b_{\ell_0}'k^{\ell_0}+b_{\ell_0+1}k^{\ell_0+1}\quad(\mathrm{mod} \quad p).
\end{equation}
Without loss of generality, we can assume $\epsilon_{\ell_0}'=1$ and $\epsilon_{\ell_0}''=0$. By (\ref{eq:theorem37}), we have
\begin{equation}
a_{\ell_0}j^{\ell_0}+a_{\ell_0+1}j^{\ell_0+1}\equiv b_{\ell_0+1}k^{\ell_0+1}\quad(\mathrm{mod} \quad p).\nonumber
\end{equation}
Taking $h=r$ in (\ref{eq:theorem36}),
we have $a_{r}\equiv b_{r}(kj^{-1})^{r}\quad(\mathrm{mod} \quad p)$ where $j^{-1}\in \Z_p$ satisfies $j^{-1} j\equiv 1\quad (\mathrm{mod}\quad p)$.
Since $a_{r}^{2}+b_{r}^{2}\neq 0$, we have $a_{r}\neq 0$. Set $x_0=kj^{-1}$. Then $x_0$ satisfies
\begin{equation*}
\begin{aligned}
a_{r}& \equiv b_{r}x_0^{r}\quad(\mathrm{mod} \quad p)\\
a_{\ell_0}j^{-1}+a_{\ell_0+1} & \equiv b_{\ell_0+1}x_0^{\ell_0+1}\quad(\mathrm{mod} \quad p).
\end{aligned}
\end{equation*}
Each nonzero point in $\mathcal{P}^{\va, \epsilon'}_{d, p}\cap\mathcal{P}^{\vb, \epsilon''}_{d, p}$
corresponds to a solution to
\begin{equation}\label{eq:fangcheng}
\begin{aligned}
a_{r}& \equiv b_{r}x^{r}\quad(\mathrm{mod} \quad p)\\
a_{\ell_0}j^{-1}+a_{\ell_0+1} & \equiv b_{\ell_0+1}x^{\ell_0+1}\quad(\mathrm{mod} \quad p).
\end{aligned}
\end{equation}
Note that $a_{r}\equiv b_{r}x^{r}\quad(\mathrm{mod} \quad p)$ has at most $r$ solutions. Hence,
\[
\abs{\mathcal{P}^{\va, \epsilon'}_{d, p}\cap\mathcal{P}^{\vb, \epsilon''}_{d, p}}\,\,\leq\,\, r+1.
\]
(3)
We prove it by contradiction. Assume that $\mathcal{P}_{d, p}^{\va,\epsilon'}\cap\mathcal{P}_{d, q}^{\va,\epsilon''}\neq\left\{(0, \ldots, 0)\right\}$, and then there exist
$j_{0}, k_{0}\in\mathbb{Z}_{p}^{*}$ so that $\mathbf{x}_{j_{0}}^{\va,\epsilon'}=\mathbf{x}_{k_{0}}^{\vb,\epsilon''}$. Particularly, we have
\begin{equation}\label{eq:15}
a_{h}j_{0}^{h}\equiv b_{h}k_{0}^{h}\quad(\mathrm{mod} \quad p), h=1, \ldots, \ell_{0},
\end{equation}
\begin{equation}\label{eq:16}
a_{\ell_{0}}'j_{0}^{\ell_{0}}+a_{\ell_{0}+1}j_{0}^{\ell_{0}+1}\equiv b_{\ell_{0}}'k_{0}^{\ell_{0}}+b_{\ell_{0}+1}k_{0}^{\ell_{0}+1} \quad(\mathrm{mod} \quad p).
\end{equation}
Without loss of generality, we can assume $\epsilon_{\ell_0}'=1$ and $\epsilon_{\ell_0}''=0$. By (\ref{eq:16}), we have
\begin{equation}\label{eq:17}
a_{\ell_{0}}j_{0}^{\ell_{0}}+a_{\ell_{0}+1}j_{0}^{\ell_{0}+1}\equiv b_{\ell_{0}+1}k_{0}^{\ell_{0}+1} \quad(\mathrm{mod} \quad p).
\end{equation}
By (\ref{eq:15}) with $h=1$, we have
\begin{eqnarray}
a_{\ell_{0}+1}j_{0}^{\ell_{0}+1}-b_{\ell_{0}+1}k_{0}^{\ell_{0}+1} &\equiv&
a_{\ell_{0}+1}(b_{1}k_{0}a_{1}^{-1})^{\ell_{0}+1}-b_{\ell_{0}+1}k_{0}^{\ell_{0}+1}\nonumber\\
&\equiv& k_{0}^{\ell_{0}+1}(a_{\ell_{0}+1}b_{1}^{\ell_{0}+1}a_{1}^{-\ell_{0}-1}-b_{\ell_{0}+1})\nonumber\\
&\equiv& 0 \quad(\mathrm{mod} \quad p),\nonumber
\end{eqnarray}
according to $a_{\ell_0+1}b_{1}^{\ell_0+1}\equiv a_{1}^{\ell_0+1}b_{\ell_0+1}\quad(\mathrm{mod} \quad p)$. By (\ref{eq:17}), we have
$a_{\ell_{0}}j_{0}^{\ell_{0}}\equiv 0\quad(\mathrm{mod} \quad p)$, which implies that $a_{\ell_{0}}\equiv 0\quad(\mathrm{mod} \quad p)$
or $j_{0}\equiv 0\quad(\mathrm{mod} \quad p)$. This is impossible by the assumption.
\end{proof}
In the following, we choose the appropriate vectors $\va, \vb$ so that $|\mathcal{L}_{p, q}|=q+p-1$.
We now state the inequalities for exponential sums over ${\mathcal L}_{p,q}$, which is the main result of this subsection.
\begin{theorem}\label{th:weil2}
Suppose $p$ and $q$ are odd prime numbers and set $m=p+q$. Recall that
\begin{equation}
\mathcal{L}_{p, q}=\left\{
\begin{array}{c}
\mathcal{P}_{d,p}\cup\mathcal{P}_{d,q}, \quad p\neq q\\
\mathcal{P}^{\va, \epsilon'}_{d, p}\cup
\mathcal{P}^{\vb, \epsilon''}_{d, p}, \quad p=q.
\end{array}\right.\nonumber
\end{equation}
We assume that $\abs{\mathcal{L}_{p,q}}=p+q-1$.
Then, for any $\mathbf{k}\in[-p+1, p-1]^{d}\cap[-q+1, q-1]^{d}\cap\mathbb{Z}^{d}$ and $\mathbf{k}\neq 0$, we have
\begin{equation}
\Big|\sum_{\mathbf{x}\in\mathcal{L}_{p, q}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big|\leq(d-1)\sqrt{2m}+1.
\nonumber
\end{equation}
\end{theorem}
\begin{proof}
We first consider the case where $p=q$. We have
\[
{\mathcal L}_{p,q}=\mathcal{P}^{\va, \epsilon'}_{d, p}\cup
\mathcal{P}^{\vb, \epsilon''}_{d, p}.
\]
Recall that
\[
\mathcal{P}^{\va, \epsilon'}_{d, p}\cap\mathcal{P}^{\vb, \epsilon''}_{d, p}=\left\{(0, \ldots, 0)\right\}.
\]
Then
\begin{eqnarray}
\Big|\sum_{\mathbf{x}\in\mathcal{L}_{p, q}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big|
&\leq & \Big|\sum_{\mathbf{x}\in\mathcal{P}_{d,p}^{\va, \epsilon'}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big|
+\Big|\sum_{\mathbf{x}\in\mathcal{P}_{d,p}^{\vb, \epsilon''}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big|+1\nonumber\\
&\leq &(d-1)\sqrt{p}+(d-1)\sqrt{p}+1\nonumber\\
&=&(d-1)\sqrt{2m}+1\nonumber.
\end{eqnarray}
Here, in the last inequality, we use Theorem \ref{th:weil1}.
We next consider the case where $p\neq q$.
When $p\neq q$, $\mathcal{L}_{p, q} = \mathcal{P}_{d,p}\cup\mathcal{P}_{d,q}$. Then we have
\begin{eqnarray}
\Big|\sum_{\mathbf{x}\in\mathcal{L}_{p, q}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big|&\leq & \Big|\sum_{\mathbf{x}\in\mathcal{P}_{d,p}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big|
+\Big|\sum_{\mathbf{x}\in\mathcal{P}_{d,q}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big|+1\nonumber\\
&\leq & (d-1)\sqrt{p}+(d-1)\sqrt{q}+1\nonumber\\
&\leq &(d-1)\sqrt{2m}+1\nonumber.
\end{eqnarray}
\end{proof}
\subsection{The exponential sums over ${\mathcal Q}_{p^2,d}^{\va,\epsilon}$ and ${\mathcal R}_{p^2,d}^{\va,\epsilon}$}Suppose that $\va\in {\mathbb Z}_p^d$ and $\epsilon\in \left\{0,1\right\}^{d-1}$. We set
\begin{eqnarray*}
{\mathcal Q}_{p^2,d}^{\va,\epsilon}:=\left\{\mathbf{z}_{j}^{\va,\epsilon} : j=0,\ldots,p^2-1\right\},
\end{eqnarray*}
\begin{eqnarray*}
\mathbf{z}_{j}^{\va,\epsilon}=\Big(\left\{\frac{a_1j}{p^2}\right\}, \left\{\frac{a_1'j+a_2j^{2}}{p^2}\right\}, \ldots, \left\{\frac{\sum_{h=1}^{d-1}a_h'j^h+a_dj^{d}}{p^2}\right\}\Big)\in [0,1)^d;
\end{eqnarray*}
and
\begin{eqnarray*}
{\mathcal R}_{p^2,d}^{\va,\epsilon}:=\left\{\vz_{j,k}^{\va,\epsilon}: j,k=0,\ldots,p-1\right\},
\end{eqnarray*}
\begin{eqnarray*}
\vz_{j,k}^{\va,\epsilon}=\Big(\left\{\frac{a_1k}{p} \right\},\left\{\frac{(a_1'+a_2j)k}{p} \right\},\ldots,\left\{\frac{(\sum_{h=1}^{d-1}a_h'j^{h-1}+a_{d}j^{d-1})k}{p} \right\}\Big)\in [0,1)^d.
\end{eqnarray*}
The ${\mathcal Q}_{p^2,d}^{\va,\epsilon}$ and ${\mathcal R}_{p^2,d}^{\va,\epsilon}$ can be considered as
the generalization of the $p$-sets given in $(\ref{eq:p2set})$.
Based on the Lemma 5 and Lemma 6 in \cite{Dick2}, we can obtain the following inequalities for exponential sums
over ${\mathcal Q}_{p^2,d}^{\va,\epsilon}$ and ${\mathcal R}_{p^2,d}^{\va,\epsilon}$.
\begin{theorem}
Suppose that $\va\in {\mathbb Z}_p^d$ and $\epsilon\in \left\{0,1\right\}^{d-1}$. Then, for any $\mathbf{k}=(k_1,\ldots,k_d)\in[-p+1, p-1]^{d}\cap\mathbb{Z}^{d}$ and $\mathbf{k}\neq 0$, we have
\begin{equation}
\Big|\sum_{\mathbf{x}\in{\mathcal Q}_{p^2,d}^{\va,\epsilon}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big|\leq(d-1)p.
\nonumber
\end{equation}
\end{theorem}
\begin{proof}
Set
\begin{equation*}
g(j)\,\,:=\,\,\sum_{\ell=1}^{d}c_{\ell}j^{\ell},
\end{equation*}
where $c_{\ell}=k_{\ell}a_{\ell}+k_{\ell+1}a_{\ell}'+\cdots+k_{d}a_{\ell}'$.
We set $j_0:=\max\left\{\ell:k_{\ell}\neq 0\right\}$. Then $c_{j_0}=k_{j_0}a_{j_0}$ and we have $p\nmid c_{j_0}$.
According to Lemma 5 in \cite{Dick2}, we have
\begin{equation*}
\Big|\sum_{\mathbf{x}\in{\mathcal Q}_{p^2,d}^{\va,\epsilon}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big|
=\Big|\sum_{j=0}^{p^{2}-1}\exp\Big(2\pi\mathbf{i}\frac{g(j)}{p^{2}}\Big)\Big|\leq (d-1)p.
\end{equation*}
\end{proof}
\begin{theorem}
Suppose that $\va\in[1, p-1]^{d}\cap\mathbb{Z}^{d}$. Then, for any $\mathbf{k}\in[-p+1, p-1]^{d}\cap\mathbb{Z}^{d}$ and $\mathbf{k}\neq 0$, we have
\begin{equation}
\Big|\sum_{\mathbf{x}\in{{\mathcal R}_{p^2,d}^{\va,\epsilon}}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big|\leq(d-1)p.
\nonumber
\end{equation}
\end{theorem}
\begin{proof}
Set
\begin{equation*}
g(j):=\sum_{\ell=0}^{d-1}c_{\ell}j^{\ell}
\end{equation*}
and $c_{\ell}:=k_{\ell+1}a_{\ell+1}+k_{\ell+2}a_{\ell+1}'+\cdots+k_{d}a_{\ell+1}'$.
We set $j_0:=\max\left\{\ell:k_{\ell}\neq 0\right\}$. Then $c_{j_0-1}=k_{j_0}a_{j_0}$ and we have $p\nmid c_{j_0-1}$.
Using Lemma 6 in \cite{Dick2}, we have
\begin{equation}
\Big|\sum_{\mathbf{x}\in{{\mathcal R}_{p^2,d}^{\va,\epsilon}}}\exp(2\pi\mathbf{i}\mathbf{k}\cdot\mathbf{x})\Big| =\Big|\sum_{j=0}^{p-1}\sum_{k=0}^{p-1}\exp\Big(2\pi\mathbf{i}k\frac{g(j)}{p}\Big)\Big|\leq (d-1)p.\nonumber
\end{equation}
\end{proof}
\section{The applications of $\mathcal{P}_{d,p}^{\va,\epsilon}$ and
${\mathcal L}_{p,q}$}
Based on the exponential sum formula in Section 2, the new point sets are useful
in numerical integration \cite{Nie, Dick3}, in UQ \cite{XZ} and in the recovery of sparse trigonometric polynomials \cite{Xu}. We just state the results for the recovery of sparse trigonometric polynomials in detail.
We start with some notations which go back to \cite{Xu}. Set
\[
\Pi_{s}^{d}\,\,:=\,\,\left\{f: f(\mathbf{x})=\sum_{\mathbf{k}\in[-s,s]^{d}\bigcap\mathbb{Z}^{d}}c_{\mathbf{k}}e^{2\pi \mathbf{i}\mathbf{k}\cdot\mathbf{x}},\quad c_{\mathbf{k}}\in\mathbb{C}, \quad\mathbf{x}\in[0,1]^{d}\right\}.
\]
Note that $\Pi_{s}^{d}$ is a linear space with the dimension $D:=(2s+1)^{d}$. For
\[
f(\mathbf{x})=\sum_{\mathbf{k}\in[-s,s]^{d}\bigcap\mathbb{Z}^{d}}c_{\mathbf{k}}e^{2\pi \mathbf{i}\mathbf{k}\cdot\mathbf{x}}\in \Pi_{s}^{d},
\]
we set $\mathbf{T}:=\left\{\mathbf{k}:c_{\mathbf{k}}\neq 0\right\}$ which is the support of the sequence of coefficients $c_{\mathbf{k}}$, and set
\[
\Pi_{s}^{d}(M):=\bigcup_{\mathbf{T}\subset[-s,s]^{d}\bigcap\mathbb{Z}^{d},|\mathbf{T}|\leq M}\Pi_{\mathbf{T}},
\]
where $\Pi_{\mathbf{T}}$ denotes the space of all trigonometric polynomials whose coefficients are supported on $\mathbf{T}$. When $M\ll D$, we call the trigonometric polynomials in
$\Pi_s^{d}(M)$ as $M$-sparse trigonometric polynomials.
The recovery of sparse trigonometric polynomials is an active topic recently.
The main aim of this research topic is to design a sampling set $X=\left\{\vz_j\right\}_{j=1}^N$ so that
one can recover $f\in \Pi_s^d(M)$ from $f(\vz_j), \vz_j\in X$ \cite{Xu, R07, Rauhut}. We state the problem as follows.
Assume the sampling set is $X=\left\{\mathbf{x}_{j}\in[0, 1)^{d}, j=1, \ldots, N\right\}$. Then our aim is to solve the following programming:
\begin{equation}\label{eq:re1}
\text{ find } f \in\Pi_{s}^{d}(M) \qquad \text{ subject to } \quad
f(\mathbf{x}_{j})=y_j,\quad j=1, \ldots, N.
\end{equation}
Denote by $\mathbf{F}_{X}$ the $N\times D$ sampling matrix with entries
\[
(\mathbf{F}_{X})_{j,\mathbf{k}}=\exp(2\pi \mathbf{i}\mathbf{k}\cdot \mathbf{x}_{j}),\quad j=1,\ldots, N,\quad \mathbf{k}\in[-s,s]^{d}\cap\mathbb{Z}^{d}.
\]
Let $\mathbf{a}_{\mathbf{k}}=(\exp(2\pi \mathbf{i}\mathbf{k}\cdot \mathbf{x}_{j}))_{j=1}^N$ denote a column of
$\mathbf{F}_{X}$ with $\mathbf{k}\in[-s,s]^{d}\bigcap\mathbb{Z}^{d}$. A simple observation is that $\|\mathbf{a}_{\mathbf{k}}\|_{2}=\sqrt{N}$. Set
\begin{equation}
\mu:=\mu_X:=\frac{1}{N}\max_{\mathbf{m}, \mathbf{k}\in[-s,s]^{d}\cap\mathbb{Z}^{d},
\mathbf{m}\neq\mathbf{k}}|\langle\mathbf{a}_{\mathbf{m}}, \mathbf{a}_{\mathbf{k}}\rangle|,
\nonumber
\end{equation}
which is called the mutual incoherence of the matrix $\mathbf{F}_{X}/\sqrt{N}$. Theorem 2.5 in \cite{Rauhut} shows that if $\mu<1/(2M-1)$ then
the Orthogonal Matching Pursuit Algorithm (OMP) and the Basis Pursuit Algorithm (BP) can recover any $M$-sparse trigonometric polynomials in $\Pi_{s}^{d}(M)$.
Therefore, our aim is to choose the sampling set $X$ so that $\mu$ is small and hence OMP and BP can recover $M$-sparse trigonometric polynomials. Based on Theorem \ref{th:weil1} and Theorem \ref{th:weil2} respectively, the following results give upper bounds of $\mu$ with taking $X=\mathcal{P}_{d,p}^{\va, \epsilon}$,
and $X={\mathcal L}_{p,q}$, respectively.
\begin{lemma}\label{th:coh}
\begin{enumerate}
\item Suppose that $X=\mathcal{P}_{d,p}^{\va, \epsilon}$ where $\va\in[1, p-1]^{d}\cap\mathbb{Z}^{d}$ and $p\geq 2s+1$ is a prime number. Then
\begin{equation}
\mu_X\,\,\leq\,\, (d-1)/\sqrt{p}.
\nonumber
\end{equation}
\item
Suppose that $p,q\geq 2s+1$ are prime numbers and
$\va, \vb\in[1, p-1]^{d}\cap\mathbb{Z}^{d}$.
Recall that
\[
\mathcal{L}_{p, q}=\left\{
\begin{array}{c}
\mathcal{P}_{d,p}\cup\mathcal{P}_{d,q}, \quad p\neq q\\
\mathcal{P}^{\va, \epsilon'}_{d, p}\cup
\mathcal{P}^{\vb, \epsilon''}_{d, p}, \quad p=q.
\end{array}\right.
\]
Set $X=\mathcal{L}_{p, q}$ and $m=p+q$. Then
\begin{equation}
\mu_X\leq\frac{(d-1)\sqrt{2m}+1}{m-1}.
\nonumber
\end{equation}
\end{enumerate}
\end{lemma}
As said before, if $\mu<1/(2M-1)$ then OMP (and also BP) can recover every $M$-sparse trigonometric polynomials. Then we have the following corollary:
\begin{theorem}
\begin{enumerate}
\item Suppose that $p>\max\left\{2s+1, (d-1)^{2}(2M-1)^{2}+1\right\}$ is a prime number
and $\va\in[1, p-1]^{d}\cap\mathbb{Z}^{d}$ . Then OMP (and also BP) recovers every $M$-sparse trigonometric polynomial $f\in \Pi_{s}^{d}(M)$ exactly from the deterministic sampling $\mathcal{P}_{d, p}^{\va, \epsilon}$.
\item Under the condition in (2) of Lemma \ref{th:coh}.
Suppose that
\[
m=p+q > \left(\left(\frac{1}{\sqrt{2}}+\frac{1}{2}\right) (2M-1)(d-1)+\sqrt{M}\right)^2.
\]
Then OMP (and also BP) recovers every $M$-sparse trigonometric polynomial $f\in \Pi_{s}^{d}(M)$ exactly from the deterministic sampling set $\mathcal{L}_{p, q}$ .
\end{enumerate}
\end{theorem}
\begin{proof}
We first consider (1).
Note that $p\geq (d-1)^{2}(2M-1)^{2}+1$ implies that $(d-1)/\sqrt{p}<1/(2M-1)$.
According to (1) in Lemma \ref{th:coh}, if $(d-1)/\sqrt{p}<1/(2M-1)$ then $\mu<1/(2M-1)$ and hence the conclusion follows. Similarly, we can prove (2).
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Home/BUSINESS/Armada Music and Riptide Music Group Partner up for Sync Representation in the United States
Armada Music and Riptide Music Group Partner up for Sync Representation in the United States
David G. August 9, 2018
Riptide Music Group (Riptide) is a leading synchronization, rights management, and creative company. They provide music to advertising, movie trailers, TV programming and promos, major motion pictures, video games and multimedia productions. Riptide Music Group has joined forces with leading dance music label Armada Music for sync representation in the United States. As a result, Riptide Music represents a select number of releases from the Amsterdam-based record label. The label represents artists such as Goldfish, Dirtcaps, Dirtyphonics, Arston, W&W, SAYMYNAME and many others. Tracks are available for licensing in DIVER, Riptide's division focused on sync representation for record label releases.
In addition, Riptide has also launched a new label imprint under Armada Music called 'Royal Sharx'. The label will focus on fun, irreverent EDM with strong forward-moving energy. Riptide will also highlight these releases for sync use in TV, film, advertising, and trailers.
The first single on the Royal Sharx label, "Riot!" comes from the hand of debut artist Lyloh. "Riot!" is an edgy, French house track reminiscent of J.U.S.T.I.C.E., and is a perfect representation of the sound of the label.
Keatly Haldeman, CEO of Riptide Music Group, comments the following statement:
I'm a massive EDM fan. I learned to produce music by mimicking trance artists like Paul van Dyk, Darude and Paul Oakenfold, so it's a personal joy to work with such a legendary label as Armada Music. Riptide has always focused on developing artists as a publisher and sync company, and now, with our imprint, we can provide full-service creative support to our dance music roster. The sound of Royal Sharx will focus on high-energy party bangers with a groove. This is going to be fun!
Dan Silver, Vice President of Creative and A&R also made a statement:
I am hugely passionate about developing artists and finding new ways to bring value to their music, and I am proud to see the launch of our new label imprint, Royal Sharx. Announcing our first release with LyLoh is exciting, and we look forward to working together to bring this music to ears afar. Working with Armada Music has been a pleasure. Their roster brings a new flavor to the DIVER catalog, and our team is thrilled to be pitching their artist releases for U.S. sync opportunities.
Maykel Piron, CEO of Armada Music, made the following statement.
Consider where we're standing partnership-wise, offering Riptide their own imprint definitely seemed like a step in the right direction. They have a great team and a solid vision, and they know what makes the difference between a great track and an outstanding one. We're looking forward to working with them on the Royal Sharx imprint. LyLoh's debut single is a great way to kick things off accordingly.
For one thing, Armada Music is the biggest independent dance music label in the world. Previously, Armin van Buuren, Maykel Piron and David Lewis (AR-MA-DA) founded Armada Music. This year, Armada Music will celebrate its 15th year anniversary.
Also noteworthy, Armada Music is a six-time winner of the IDMA for 'Best Global Label' award. Furthermore, the label boasts the globe's biggest dance music catalog of over 40,000 titles and a roster. Among these artists include Afrojack, Andrew Rayel, Armin van Buuren, Dash Berlin, Erick Morillo, Fedde Le Grand, Gareth Emery, Kevin Saunderson, Lost Frequencies, Loud Luxury, Morgan Page and W&W. Not to mention that through this star-studded dance music army, Armada Music collects over 500 million streams per month. In addition, Armada Music secures weekly airplay on the biggest nation-wide broadcasting stations including BBC Radio 1 and Sirius XM. The label also frequents the top spots of the United States Mediabase Dance Airplay charts.
About Riptide Music Group
Riptide is a creative music company. Originally, it was built by musicians and performers who have spent decades on stages and in studios. As a result, the company is best described as a mashup of a music publishing company, sync house, and record label.
Riptide Music Group is organized into five divisions: Riptide Rights, Riptide One, Pacifica Music Library, DIVER, and Incubator. First, there is Riptide Rights, which looks after publishing and master rights for administration and sync representation. Second, Riptide One is a carefully curated catalog of independent artists, bands, producers and composers, master and publishing controlled 100% by Riptide. Additionally, there is Pacifica Music Library, a modern production music library focused on current popular music styles. Furthermore, there is DIVER, a boutique catalog of label-signed, active and touring artists. Lastly, Incubator is Riptide's pop and urban songwriter, producer and artist development division.
Connect with Riptide Music Group
Riptide Music Website.
Connect with Armada Music
Armada Music website
Lastly, for EDM Business and everything Electronic Dance Music, check out OneEDM.
Armada Music DIVER Incubator Lyloh Pacifica Music Library RIOT Riptide Music Group Riptide One Riptide Rights Royal Sharx
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Q: How can I tell if all my style sheets are in use on my website? Have over 100 Style Sheets on a large website maintained by many people, how can I tell if all of the style sheets are still being used by pages on my site?
A: You could use one off tools to get the list of styles that are not being used from this post.
Then you would need to write css file class reader (or regex) that would take that list of unused styles and check each css file.
If all styles in some file would have a match in unused list - add file name to list of unused css files.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
\label{sec:introduction}
\vspace{-0.25cm}
In QCD quarks are confined in bound states by forming flux-tubes of chromo-magnetic and chromo-electric flux. Such a flux is connecting a quark with an anti-quark to form a bound state called an open-flux tube, which in nature manifests as a meson state. Long flux-tubes behave pretty much like thin strings: If you pull the string apart, at some point it breaks. However, to observe such a phenomenon in a lattice QCD calculation it requires the existence of dynamical quarks. Since we work on a pure gauge set up such effects do not appear. By placing the flux-tube in a particular position in space we expect $D-2$ massless modes to propagate along the string arising from the spontaneously broken translation invariance in the $D-2$ directions transverse to the flux. We, therefore, expect that there should be a low energy effective string theory describing such oscillating modes. Although, a flux-tube can be considered effectively as a string, it also has an intrinsic width. This suggests that massive states such as breather modes related to the intrinsic structure of the tube may exist in the spectrum. To investigate, whether, such states exist one needs to extract the aforementioned spectrum, compare it with an effective string theory model and identify states which exhibit significant deviations from the theoretical description. A naive expectation would be that a massive mode has the characteristics of a resonance; a constant term of the order of the mass gap ($\sim m_G$) coupled to a string state.
A decade ago we demonstrated~\cite{Athenodorou:2011rx} that the closed flux-tube spectrum in $D=2+1$ $SU(N)$ gauge theories can be well-approximated by the Nambu-Goto free string in flat space-time, from short to long flux-tubes, with no signs of any massive excitations. In addition we demonstrated~\cite{Athenodorou:2010cs} that the spectrum of the closed flux-tube in $D=3+1$ consists mostly of string-like states, however, in contrast to $D=2+1$ a number of states with quantum number $0^-$ appeared to encode the characteristics of a massive excitation. In 2013, Dubovsky et al,~\cite{Dubovsky:2013gi} demonstrated that this state arises naturally if one includes a Polyakov topological piece in the string theoretical action. Nevertheless, our previous results were poor - spectrum has been extracted for a few string lengths, and for low statistics. As a result, we could only extract states for the very low lying spectrum and for a restricted sector of the irreducible representations expanded by the quantum numbers (QNs) $\{ |J|,P_{\perp},P_{\parallel} \}$.
In this work we present a major improvement of our previous investigation on $D=3+1$. This has been achieved by increasing the basis of operators used to materialise the General Eigen-value Problem method \cite{Luscher:1990ck}, by extracting the spectrum of the flux-tube for three values of color $N$, namely $N=3,5,6$, as well as by probing through a large set of flux-tube lengths. In more detail, for each different set of quantum numbers $\{ |J|,P_{\perp},P_{\parallel} \}$ we used 50 - 200 operators. We performed calculations for longitudinal momenta $p_{\parallel} = 2 \pi q / l$, with $q=0,1,2$ and we span over $l \sqrt{\sigma}=1.4 - 7$. All our $SU(N>3)$ calculations are focused only on flux-tubes that carry a single unit of chromo-electric flux ($N$-ality $k=1$). Findings on the flux-tube spectrum with values of $N$-ality $k=2$ will be a matter of investigation in a future article.
The structure of these proceedings is the following. First, we provide a brief description of the lattice setup, by explaining the quantum numbers relevant for the extraction of the flux-tube spectrum as well as their encoding onto the operators used to materialize the calculation. Then we present the effective string theoretical descriptions suitable for approximating the spectrum of the flux-tube. Subsequently, we present our lattice results and then we conclude.
\vspace{-0.25cm}
\section{Lattice calculation}
\label{sec:lattice_calculation}
\vspace{-0.25cm}
We define the $SU(N)$ gauge theory on a $D=4$ Euclidean space-time lattice which has been compactified along all directions with volume
$L_{\parallel} \times L_{\perp_1} \times L_{\perp_2} \times L_{T}$. The length of the flux-tube is equal to $L_{\parallel}$, while $L_{\perp_1}$,
$L_{\perp_2}$ and $L_{T}$ were chosen to be large enough to avoid finite volume effects. We choose $L_{\perp_1} = L_{\perp_2} = L_{\perp}$ uniformly so that we ensure rotational symmetry around the flux axis. We perform Monte-Carlo simulations using the standard Wilson plaquette action $S=\sum_{\square} \beta \left[ 1-\frac{1}{N}{\rm Re}{\rm Tr}(U_{\square}) \right]\,,$ with inverse coupling $\beta=\frac{2N}{g^2(a)}$. In order to keep the value of the lattice spacing $a$ approximately fixed for different values of $N$ we keep
the 't Hooft coupling $\lambda(a)=N g^2(a)$ approximately fixed, so that $\beta \propto N^2$. The simulation algorithm we use combines standard heat-bath and over-relaxation steps in the ratio 1:4; these are implemented by updating $SU(2)$ subgroups using the Cabibbo-Marinari algorithm. To measure the spectrum of energies we use the variational technique also called the Generalized Eigen-Value Problem (GEVP)
(e.g.~see \cite{variational} and its references).
The energy states of the closed flux-tube in $D=3+1$ are characterised by the irreducible representations of the two-dimensional lattice rotation symmetry around the principal axis denoted by $C_4$~\cite{Kuti1}. The above group is a subgroup of $O(2)$ corresponding to rotations by integer multiples of $\pi/2$ around the flux-tube propagation axis. As a result, values of angular momenta that differ by an integer multiple of four are indistinguishable on the lattice. This splits the Hilbert space in four orthogonal sectors, namely: $J_{{\rm mod}\, 4}=0$, $J_{{\rm mod}\, 4}=\pm 1$, $J_{{\rm mod}\, 4}=2$. Furthermore, parity $P_\perp$ which is associated with reflections around the axis $\hat{\perp}_1$ can be used to characterise the states. Applying $P_\perp$ transformations, flips the sign of $J$. Therefore, one can choose a basis in which states are characterised by their value of $J$ ($J= \pm$), or by their value of $|J|$ and $P_\perp$. We adopt the latter. In the continuum, states with $J \neq 0$ are parity degenerate, however, on the lattice this holds only for the odd values of $J$. In practice, we describe our states with the following $5$ irreducible representations $A_{1}$, $A_{2}$, $E$, $B_{1}$ and $B_{2}$ of $C_4$ group whose $J$ and $P_\perp$ assignments are: $\left\{ A_1: \, |J_{{\rm mod} \ 4}|=0, \ P_\perp=+\right\}, \left\{A_2: \, |J_{{\rm mod} \ 4}|=0, \ P_\perp=-\right\}$, $\left\{ E: |J_{{\rm mod} \ 4}|=1, \ P_\perp=\pm\right\}$, $\left\{ B_1: \, |J_{{\rm mod} \ 4}|=2, \ P_\perp=+\right\}$ and $\left\{B_2: \, |J_{{\rm mod} \ 4}|=2, \ P_\perp=-\right\}$.
Furthermore, there are two additional quantum numbers which can be proven useful for the description of the string states. These are the longitudinal momentum $p_{||}$ carried by the flux-tube along its axis (which is quantized in the form $p_{||}=2\pi q/L_{||}; q\in Z$) and the parity $P_{||}$ with respect to reflections across the string midpoint. Since $P_{||}$ and $p_{||}$ do not commute, we can use both to simultaneously characterise a state only when $q=0$. The energy does not depend on the sign of $q$, hence, we only focused on those with $q\ge 0$.
Flux-tube energies are extracted by making use of correlation matrices $C_{ij} = \langle \phi_i^{\dagger} (t) \phi_j (0) \rangle$ with $i,j=1...N_{\rm op}$ in combination with GEVP where $N_{\rm op}$ the number of operators. We construct operators $\phi_i$ which encode shapes that lead to particular values of $J,P_\perp,P_{||},$ and $q$. We do so by choosing
linear combinations of Polyakov loops the paths of which consist of various transverse deformations and various smearing and blocking levels~\cite{variational}. All the transverse paths used for the construction of the operators are shown in Figure~\ref{fig:paths} and all together, including smearing and blocking levels, form a basis of around $N_{\rm op} =1000$ operators with approximately $50 - 200$ for each different irreducible representation. To build an operator which encodes a certain value of angular momentum $J_{{\rm mod} \ 4}$ we begin the construction with a sub-operator $\phi_{\alpha}$ which has a deformation extending in angle $\alpha$ within the plane of transverse directions. Then we repeat the same procedure by rotating the sub-operator by integer values of $\pi/2$. Finally, we can construct the operator $\phi(J)$ belonging to a specific representation of $C_{4}$ by using the formula $\phi(J)= \sum_{n=1,2,3,4} e^{iJ n \frac{\pi}{2}} \phi_{n \frac{\pi}{2}}\,.$ Thus $\phi(0)$ belongs to $A_1$ and $A_2$, $\phi(1)$ to $E$ and, finally, $\phi(2)$ to $B_1$ as well as $B_2$. Lastly, it is required to encode certain values of $P_{\perp}$ and $P_{||}$ by summing and subtracting reflections of the initial sub-operator $\phi(J)$ over the transverse and parallel parity planes. Such an example is pictorialized in Equation~\ref{eq:example} for an operator with $J_{\rm mod \ 4}=0$.
\begin{eqnarray}
\phi= {\rm Tr}\left[ \parbox{12.5cm}{\rotatebox{0}{\includegraphics[width=12.5cm]{operatorproc.pdf}}} \ \right]\,.
\label{eq:example}
\end{eqnarray}
For the combination $i=j=k=+1$, $\phi$ projects onto $\{ A_1, P_{||}=+ \}$, for $i=+1,j=k=-1$ onto $\{ A_2, P_{||}=+ \}$, for $i=-1,j=+1,k=-1$ onto $\{ A_1, P_{||}=- \}$ and finally, for $i=j=-1,k=+1$, onto $\{ A_2, P_{||}=- \}$.
\begin{figure}
\centering
\vspace{-0.5cm}
\includegraphics[height=7cm]{paths.pdf}
\caption{All the different paths used for the construction of the torelon operators.}
\label{fig:paths}
\end{figure}
\section{The effective string theory}
\label{sec:effective_string}
\vspace{-0.25cm}
Imagine a flux-tube as a string with length $l=aL_{||}$ winding around the spatial torus. Imposing fixed spatial position for the string spontaneously breaks translation symmetry. Hence, we expect $D-2$ Nambu-Goldstone massless bosons to appear at low energies. Such bosons reflect the transverse fluctuations of the flux-tube around its classical configuration. We would thus, expect a low energy Effective String Theory (EST) describing the flux-tube spectrum at large enough lengths. Of course a flux-tube is not an infinitesimally thin string, it lives in $SU(N)$ manifold and presumably has an intrinsic width $l_w\propto 1/\sqrt{\sigma}$. We would therefore expect that the spectrum of the flux-tube consist not only of string like states but also of massive excitations. Below, we describe the current theoretical predictions for the excitation spectrum of the Nambu-Goldston bosons as well as an approach to explain the existence of massive resonances on the world-sheet of the flux-tube.
\subsection{The Goddard–Goldstone–Rebbi–Thorn string}
\label{sec:GGRT}
At this subsection we describe the spectrum of the Goddard-Goldstone-Rebbi-Thorn (GGRT)~\cite{Goddard} or in simpler words the Nambu-Goto (NG)~\cite{NGpapers} closed string. NG string describes non-critical relativistic bosonic strings. One can extract the GGRT spectrum by performing light-cone quantization of the closed-string using the NG action or equivalently the Polyakov action. NG action is the area of the world-sheet swept by the propagation of the string along the time direction. This model is Lorentz invariance only in $D=26$ dimensions. Nevertheless, for reasons that we now understand better~\cite{Dubovsky:2013gi} NG can also describe adequately the spectrum of strings in $3 \ {\rm and} \ 4$ dimensions. The spectrum of the GGRT string is given by the expression:
\begin{equation}
{E_{N_L,N_R}(q,l)}
= \sigma l \sqrt{
1
+
\frac{8\pi}{(l \sqrt{\sigma})^2} \left(\frac{N_L+N_R}{2}-\frac{D-2}{24}\right)
+
\left(\frac{2\pi q}{(l\sqrt{\sigma})^2}\right)^2}\,,
\label{eqn_EnNG}
\end{equation}
where $2\pi N_{L(R)}/l$ the total energy and momentum of the left(right) moving phonons with $N_L = \sum_k \sum_{n_L(k)} k(n^+_L(k)+ n^-_L(k))$ and
$N_R = \sum_k \sum_{n_R(k)} k(n^+_R(k)+ n^-_R(k))$. $n^{\pm}_{L(R)}(k)$ is the number
of left(right) moving phonons of momentum $p_k = 2\pi k/l$, $k=0,1,2,\dots$ and angular momentum $\pm 1$. If $p_{||}=2\pi q/l$ is the total longitudinal momentum of the string then, since the phonons provide that momentum, we must have $N_L - N_R = q$. The angular momentum around the string is given by $J = \sum_{k,n_L(k),n_R(k)} n^+_L(k) + n^+_R(k) - n^-_L(k) - n^-_R(k)$.
\subsection{Lorentz invariant string approaches}
\label{sec:Lorentz}
Systematic ways to study Lorentz invariant EST that would describe the QCD flux-tube were pioneered by L\"uscher, Symanzik, and Weisz in \cite{Luscher} (static gauge) as well as by Polchinski and Strominger in~\cite{Pol} (conformal gauge). Such EFT approaches produce predictions for the energy of states as an expansion in $1/l\sqrt{\sigma}$. Terms in this expansion that are of $O(1/l^p)$ are generated by $(p+1)$-derivative terms in the EST action whose coefficients are a priori arbitrary Low Energy Coefficients (LECs). Interestingly, these LECs were shown to obey strong constraints that reflect a non-linear realization of Lorentz symmetry \cite{LW,Meyer,AK}, and so to give parameter free predictions for certain terms in the $1/l$ expansion.
The EST approaches can be characterised by the way one performs the gauge fixing of the embedding coordinates on the world-sheet. This can be either the static gauge~\cite{Luscher,LW,AK} or the conformal gauge~\cite{Pol,Drummond:2004yp,HariDass:2009ub} with both routes leading to the same results. The starting point of building the EST is the leading area term which gives rise to the linearly rising potential for large strings. Subsequently comes the Gaussian action which is responsible for the $\propto 1/l$ L\"uscher term with universal coefficient depending only on the dimension $D$. As a next step one adds the 4-derivative terms which yield a correction on the energy spectrum proportional to $1/l^3$ with a universal coefficient that also depends on the dimension $D$. One can include the $6-$derivative terms and show that for $D=3$ they yield the fourth universal term proportional to $1/l^5$ in the energy spectrum, while for general states in $D=4$, the coefficient of the $O(1/l^5)$ term is not universal. Nonetheless, the energy just for the ground state in the $D=4$ case is universal. Summarizing the above information, the spectrum is given by
\begin{eqnarray}
E_n(l) = \sigma l + \frac{4 \pi}{l}\bigg(n-\frac{D-2}{24}\bigg) - \frac{8 \pi^2}{\sigma l^3}\bigg(n-\frac{D-2}{24}\bigg)^2 + \frac{32 \pi^3}{\sigma^2 l^5}\bigg(n-\frac{D-2}{24}\bigg)^3 + {O}(l^{-7}).
\label{eq:AharonyKarzbrun}
\end{eqnarray}
Since we think of the GGRT model as an EST, which may be justified only for long strings \cite{Olesen}, one can expand the associated energy for $l\surd \sigma\gg 1$. The result of the expansion is the same as Equation~\ref{eq:AharonyKarzbrun}. For simplicity we set $q=0$, and $n = (N_L + N_R)/2$.
\subsection{The topological term action}
\label{sec:axion}
In 2013, Dubovsky {\it et al.} worked out an approach for extracting the spectrum of the flux-tube for short as well as for long lengths. The idea was based on the fact that the GGRT string provides the best approximation for the flux-tube spectrum and that Equation~\ref{eqn_EnNG} can be re-expressed as $E_{N_L,N_R} = \sqrt{\sigma} {\cal E} (p_k/\sqrt{\sigma},1/l \sqrt{\sigma})$ where $p_k$ are the momenta of individual phonons in units of $2 \pi / l$ comprising the state quantised. The naive expansion in $1/ l \sqrt{\sigma}$ is the combination of two different expansions; the first is an expansion in the softness of individual quanta compared to the string scale, i.e. in $p_k / \sqrt{\sigma}$ and the second expansion is a large volume expansion, i.e. an expansion in $1 / l \sqrt{\sigma}$. To disentangle the two expansions the following procedure is being followed. First, one calculates the infinite volume $S$-matrix of the phonon collisions. This is done perturbativelly given that the center of mass energy of the colliding phonons is small in string units; this is called the momentum expansion. Followingly, the authors extract the finite volume energies from this $S$-matrix by using approximate integrability and the Thermodynamic Bethe Ansatz (TBA). This allows to extract the winding effects on the energy from virtual quanta traveling around the circle as well as the winding corrections due to phonon interactions.
The authors demonstrated that when a state has only left-moving phonons the GGRT winding corrections in the energy spectrum are small and, thus, one expects the spectrum to be close to that of the free theory. On the contrary, for states containing both left- and right-movers, energy contributions are larger. The above picture is in a good agreement with most of the states in $D=4$ but fails to explain the anomalous behaviour of the pseudoscalar level firstly reported in \cite{Athenodorou:2010cs} suggesting that an additional ingredient is required in order to describe such excitations. The most straightforward way to do this is the introduction of a massive pseudoscalar particle $\phi$ on the world-sheet. The leading interaction compatible with non-linearly realized Lorentz
invariance for such a state is a coupling to the topological invariant known as the self-intersection number of the string $S_{\rm int} = \frac{\alpha}{8 \pi} \int d^2 \sigma \phi K^{i}_{\alpha \gamma} K^{j \gamma}_{\beta} \epsilon^{\alpha \beta} \epsilon_{ij} \,$
with $K^{i}_{\alpha \gamma}$ being the extrinsic curvature of the world-sheet, $\alpha$ the associated coupling and $\sigma^{i}$, $i=1,2$ the world-sheet coordinates. Adapting the above interaction term to our old results for $SU(3)$, $\beta=6.0625$ yields a mass of $m_{\phi}/\sqrt{\sigma} \simeq 1.85^{+0.02}_{-0.03}$ and a coupling of $\alpha = 9.6 \pm 0.1$.
\section{Results}
\label{sec:results}
\vspace{-0.25cm}
In this work we present results for the closed flux-tube spectrum extracted from calculations on five different gauge groups. The above consist of
$N=3$ at $\beta=6.0625$ ($a\simeq 0.09 {\rm fm}$) and $\beta=6.338$ ($a\simeq 0.06 {\rm fm}$), of $N=5$ at $\beta=17.630$ ($a\simeq 0.09 {\rm fm}$) and $\beta=18.375$ ($a\simeq 0.06 {\rm fm}$) as well as for $N=6$ at $\beta=25.550$ ($a\simeq 0.09 {\rm fm}$). Critical slowing down~\cite{Athenodorou:2021qvs,Athenodorou:2020ani}, as one moves towards the continuum and the large-$N$ limit, prohibits the investigation of gauge groups with $N \geq 6$ and $a < 0.09 {\rm fm}$. Nevertheless, the above configuration of measurements is enough to determine whether significant lattice artifacts as well as $1/N^2$ corrections are affecting our statistically more accurate $N=3$ calculations. As a matter of fact our investigation demonstrates that such effects are of minor importance and do not play a significant role in the interpretation of the spectrum. The energy spectrum we extracted is compared to the predictions of the GGRT string. Namely, we fit the absolute ground state ($|J_{\rm mod \ 4}|^{P_{\perp} P_{||}} =0^{++}$) for all calculations using Equation~\ref{eqn_EnNG} as a function of the length for $l\sqrt{\sigma} > 2.5$ and extract the string tension $a \sqrt{\sigma}$. Once the string tension is known then Equation \ref{eqn_EnNG} can be used as a parameter free prediction for higher string excitations with $N_L + N_R > 0$.
\subsection{The energy spectrum for $q=0$ and the world-sheet axion}
\label{sec:resultsq0}
We begin by presenting our results for the $q=0$ longitudinal momentum sector in Figures~\ref{fig:first}, \ref{fig:second} and \ref{fig:third}. In the left panel of Figure~\ref{fig:first}, the lowest energy level corresponds to the absolute ground state $|J_{\rm mod \ 4}|^{P_{\perp} P_{||}} =0^{++}$ which is used to set the scale of the NG string, hence, the nearly perfect agreement with the GGRT string. Furthermore in the left panel of Figure~\ref{fig:first}, we plot the first excited state of $0^{++}$ as well as the ground states of $2^{++}$, $2^{-+}$ and $0^{--}$ for $SU(3)$ at $\beta=6.0625$. We compare the above data with the GGRT prediction for $N_R=N_L=1$. This string state is expected to be four-fold degenerate with levels with continuum QNs $0^{++}$, $0^{--}$, $2^{++}$ and $2^{-+}$ . While $0^{++}$, $2^{++}$ and $2^{-+}$ flux-tube excitations appear to exhibit small deviations for short values of $l\sqrt{\sigma}$ and for larger strings become consistent with GGRT, $0^{--}$ ground state appears to demonstrate significant deviations from the GGRT string. In the right panel of Figure~\ref{fig:first} we present the ground and in addition the first excited state with QNs $0^{--}$ for all gauge groups considered in this work. It appears that both states are only mildly affected by lattice artifacts and $1/N^2$ corrections. The $0^{--}$ ground state appears to have characteristics of a resonance i.e. a constant mass term coupled to the absolute ground state. This is more obvious by subtracting the absolute ground state $0^{++}$ where this excitation exhibits a plateau; this is presented in Figure~\ref{fig:second} for $SU(3)$ at $\beta=6.0625$. As has already being explained in Section~\ref{sec:axion} this state can be well interpreted as an axion on the world-sheet of the flux-tube with an associated mass of $m/\sqrt{\sigma} = 1.85^{+0.02}_{-0.03}$ for $SU(3)$ at $\beta=6.0625$; This value is in good agreement with the plateau in Figure~\ref{fig:second}. If the $0^{--}$ flux-tube ground state corresponds to the axion, the next excitation level would correspond to the string state with $N_L=N_R=1$ rather than $N_L=N_R=2$. As one can see in the right panel of Figure~\ref{fig:first} and Figure~\ref{fig:second} the $0^{--}$ first excitation state does not approach the GGRT $N_L=N_R=2$ state but instead it slowly approaches the $N_L=N_R=1$ string state. This strengthens the scenario of $0^{--}$ ground state being
the world-sheet axion.
\begin{figure}
\centering
\vspace{-1.5cm}
\hspace{-3.5cm}
\includegraphics[height=8.1cm]{1st_plot.pdf} \hspace{-3cm} \includegraphics[height=8.1cm]{2nd_plot.pdf} \hspace{-4.5cm}
\vspace{-1cm}
\caption{ \underline{Left Panel:} The spectrum of the absolute ground state and first excited state for $|J_{\rm mod \ 4}|^{P_{\perp}, P_{||}}=0^{++}$ as well as the ground states for $2^{++}$, $2^{-+}$ and the "anomalous" $0^{--}$ for $q=0$ and $SU(3)$ at $\beta=6.0625$; the black lines correspond to the GGRT predictions and the light blue line to the prediction of the EFT with the axionic part of the action included within. \underline{Right Panel:}. The energies of the ground state and first excited state for $|J_{\rm mod \ 4}|^{P_{\perp}, P_{||}}=0^{--}$, $q=0$ for all gauge groups considered in this work.}
\label{fig:first}
\end{figure}
\begin{figure}
\centering
\vspace{-1.5cm}
\includegraphics[height=8.1cm]{3rd_plot.pdf}
\vspace{-1cm}
\caption{The energy levels of the ground and first excited states for a closed flux-tube with quantum numbers $0^{--}$, $q=0$ and the zeroth energies subtracted for $SU(3)$, $\beta=6.0625$. The horizontal purple band corresponds to the mass of the axion as this has been extracted in \cite{Dubovsky:2013gi}.}
\label{fig:second}
\end{figure}
\subsection{A bound state of two axions?}
\label{sec:two_axions}
In the left panel of Figure~\ref{fig:third} we present the second excitation state with QNs $ 0^{++}$. It is well mentioned that above this energy level we get a plethora of states which reflect the multifold degeneracy of the GGRT string for $N_L = N_R = 2$. Strikingly, this state appears to exhibit the same resonance behaviour as the $0^{--}$ ground state i.e. it appears as a constant term coupled to the absolute ground state. This is more obvious if we subtract from this energy level the contribution of the absolute ground state as this appears in the right panel of Figure~\ref{fig:third}. Namely, we observe that this is in agreement with a resonance of mass twice that of the axion. This raises the question whether such a relation is accidental or it has some deeper interpretation. A reasonable expectation would be that this state is a bound state of two axions with a very low binding energy; this scenario is in agreement with the quantum numbers of the state.
\begin{figure}
\centering
\vspace{-1.5cm}
\hspace{-3cm}
\includegraphics[height=8cm]{4th_plot.pdf} \hspace{-3cm}
\includegraphics[height=8cm]{5th_plot.pdf} \hspace{-4cm}
\vspace{-1cm}
\caption{{\underline{Left Panel}}: The ground, first excited and second excited state for $|J_{\rm mod \ 4}|^{P_{\perp}, P_{||}}=0^{++}$ and all the gauge groups used in this work. {\underline{Right Panel}}: The second excited state for $|J_{\rm mod \ 4}|^{P_{\perp}, P_{||}}=0^{++}$ with the absolute ground state being subtracted for $SU(3)$ at $\beta = 6.0625$.}
\label{fig:third}
\end{figure}
\subsection{The $q \ne 0$ sector and the world-sheet axion}
\label{sec:nonzeromomentum}
In this section we present our results for the $q=1$ and $q=2$ momentum sectors. In the left panel of figure~\ref{fig:fourth} we demonstrate the spectrum for $q=1$, $SU(3)$ and $\beta=6.338$. Since, the string ground state $N_L=1$, $N_R=0$ can only be created by a single phonon, it has $J=1$. The flux-tube ground state with quantum numbers $1^{\pm }$, $q=1$ appears to be in good agreement with the prediction of the GGRT string. This is in accordance with the results of Ref~\cite{Dubovsky:2013gi}. The next string excitation level, corresponding to $N_L=2$ and $N_R=1$ should be seven-fold degenerate. This should consist of one $0^{+}$, one $0^{-}$, three $1^{\pm}$, one $2^{+}$ and a $2^-$ state. In the left panel of Figure~\ref{fig:fourth} we show the flux-tube ground state with QN $2^+$, the ground state with $2^-$, the ground state for $0^+$ as well as the first and second excited states with $1^{\pm }$. All the above five states appear to cluster around the GGRT prediction. Furthermore, we demonstrate the ground state for $0^{-}$ which appears to exhibit large deviations from the GGRT string. Since, this state has the same QNs as the pseudoscalar massive excitation the first assumption one could make is that this reflects to the axion. A naive comparison of this state with a relativistic sum of the absolute ground state plus an axion with momentum $2 \pi / l$ is provided in the same figure, demonstrating an approximate agreement with our data for large flux-tubes. This strengthens the scenario of this state being the axion.
In the right panel of figure~\ref{fig:fourth} we show results for $q=2$, $SU(3)$ and $\beta=6.338$. The string ground state $N_L=2$, $N_R=0$ is expected to be four-fold degenerate. Namely, it is expected to be occupied by states with QNs $0^{+}$, $1^{\pm}$, $2^{+}$ and $2^{-}$. We, thus, extract the flux-tube ground states with the above QNs and observe that they all cluster around the GGRT prediction. The next string excitation level is multi-fold degenerate and should also include a $0^{-}$ state which encodes the QN of the axion. We extract the flux-tube ground state with QNs $0^-$ and we observe a very similar behaviour as for the case of $q=1$; namely it diverges greatly from the GGRT prediction.
\begin{figure}
\centering
\vspace{-1.5cm}
\hspace{-3cm}
\includegraphics[height=8cm]{6th_plot.pdf} \hspace{-3cm}
\includegraphics[height=8cm]{7th_plot.pdf} \hspace{-4cm}
\vspace{-1cm}
\caption{{\underline{Left Panel}}: The ground, first excited and second excited $1^{\pm}$ states as well as the ground states $0^+$, $0^-$, $2^+$, $2^-$ for a flux-tube with $q=1$ in $SU(3)$, $\beta=6.338$. {\underline{Right Panel}}: The ground states with QNs $0^{+}$, $0^-$, $1^{\pm}$, $2^+$, $2^-$ for a flux-tube with $q=2$ at $SU(3)$, $\beta = 6.338$.}
\label{fig:fourth}
\end{figure}
\vspace{-0.25cm}
\section{Conclusions}
\label{sec:conclusions}
\vspace{-0.25cm}
We have improved extensively the extraction of and, thus, our knowledge on the spectrum of the closed flux-tube (torelon). Clearly, the majority of the states appearing in the spectrum have a string-like character, in the sense they can be adequately approximated by a low energy effective string theory. In addition a small sector of the excitation spectrum appears to be massive resonances which can be interpreted as an axion on the bulk of the theory. This is justified by the resonance character of the $0^{--}$, $q=0$ ground state which appears to be an axion coupled to the string's absolute ground state, by the $0^{++}$ second excited state which can be interpreted as a bound state of two axions with a very low binding energy coupled to the absolute ground state as well as by the $0^-$ $q=1,2$ ground states which also have an axion character. Finally, and not presented in this manuscript, states with axionic character can also be identified in other irreducible representations such as $|J_{\rm mod \ 4}|^{P_{\perp}} = 1^{\pm}$; this will be a matter of discussion in our longer write up~\cite{AthenodorouTeperNew}.
\vspace{-0.25cm}
\section*{Acknowledgements}
\vspace{-0.25cm}
We would like to thank S. Dubovsky, D. Giataganas, V. Gorbenko, J. Sonnenschein, E. Kiritsis and K. Hashimoto for interesting discussions. As this work was being completed, AA participated in the {\it Large-$N$ theories and strings: conformal, confining, and holographic} Workshop at the Princeton Center for Theoretical Physics, where there were many talks relevant to confining flux-tubes. AA is indebted to the participants for useful discussions. AA has been financially supported by the European Union's Horizon 2020 research and innovation programme ``Tips in SCQFT'' under the Marie Sk\l odowska-Curie grant agreement No. 791122. Numerical simulations have been carried out in the Oxford Theoretical Physics cluster.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,110
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This woman was not feeling well at all.
I'm not sure if this is an accurate depiction of the real man. I was not limited by a coffee or lunch break; I had all the time in the world to do this drawing, and concentrated on shading and detail.
When scale and resemblance fail, creative naming comes to the rescue.
It was hot today. The pool promises to melt away the pain.
I've said before that I try not to use erasers. One exception to my self-imposed rule is when I sketch family and friends, especially when there it is very likely they will see it. I think the sketch below is a fair representation of our very nice outing. I only used the eraser once.
That this woman was snoozing was my good fortune. A car wash waiting room is not the same as a cafe where everyone reads. In fact, the woman sitting next to me remarked that the woman I was sketching was a good subject.
|
{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}\label{Sflow1}
Let $S$ be a submarkovian semigroup on $L_2({\bf R}^d)$ generated by a
self-adjoint second-order elliptic operator $H$ in divergence form.
If the operator is strongly elliptic then $S$ acts ergodically,
i.e.\ there are no non-trivial $S$-invariant subspaces of $L_2({\bf R}^d)$.
Nevertheless there are many examples of degenerate elliptic operators
for which there are
subspaces $L_2(\Omega)$ invariant under the action of $S$ (see, for
example, \cite{ERSZ1} \cite{ERSZ2}
\cite{RSi} \cite{ER29}).
Our aim is to examine operators with coefficients which are
Lipschitz continuous and characterize the $S$-invariance of
$L_2(\Omega)$ by the invariance under a family of associated flows.
In order to formulate our main result we need
some further notation.
First define the positive symmetric operator $H_0$ with
domain $D(H_0)=C_c^\infty({\bf R}^d)$ and action
\[
H_0\varphi= -\sum^d_{k,l=1} \partial_k \, c_{kl} \, \partial_l\varphi
\]
where the coefficients $c_{kl}=c_{lk}\in W^{1,\infty}({\bf R}^d)$ are
real and $C=(c_{kl})$
is a positive-definite matrix over ${\bf R}^d$.
Then the corresponding quadratic form $h_0$ given by
\[
h_0(\varphi)
= \sum^d_{k,l=1}(\partial_k\varphi, c_{kl} \, \partial_l\varphi)
\]
with domain $ D(h_0)=C_c^\infty({\bf R}^d)$ is closable.
The closure $h=\overline{h_0}$ determines in a canonical manner a
positive self-adjoint extension $H$ of $H_0$, the Friedrichs' extension \cite{Friedr2}
(see, for example, \cite{RN}, \S 124, or \cite{Kat1}, Chapter~VI).
The closed form $h$ is a Dirichlet form and the self-adjoint
semigroup $S$ generated by $H$ is automatically submarkovian
(for details on Dirichlet forms and submarkovian semigroups see
\cite{FOT} or \cite{BH}).
We call $H$ the degenerate elliptic operator with coefficients $(c_{kl})$.
Secondly, if $b_1,\ldots,b_d \in W^{1,\infty}({\bf R}^d)$ then
the first-order partial differential operator
\[
\varphi \mapsto \sum^d_{k=1} b_k \, \partial_k \varphi
- \frac{1}{2} \sum^d_{k=1}(\partial_k b_k) \, I
\]
with domain $C_c^\infty({\bf R}^d)$ is
essentially skew-adjoint (see, for example, \cite{Rob7}, Theorem~3.1).
Therefore the principal part is closable and
generates a positive, continuous, one-parameter group
on $L_2({\bf R}^d)$.
We refer to such a group as flows.
Specifically we are interested in the flows associated with the coefficients $(c_{kl})$ of $H$.
For all $k \in \{ 1,\dots, d \} $ let $Y_k$
denote the $L_2$-closures of the first-order
partial differential operator
\[
\varphi \mapsto \sum^d_{l=1} c_{kl} \, \partial_l \varphi
\]
with domain $C_c^\infty({\bf R}^d)$.
Then denote by $T^{(k)}$ the flows generated by the $Y_k$.
The operators $Y_k$ were used by Ole\u{\i}nik and Radkevi\v{c} \cite{OR} to analyze
hypoellipticity and subellipticity properties of degenerate elliptic operators
$H$ with $C^\infty$-coefficients $c_{kl}$ (see \cite{JSC} for a review of these and related results).
We, however, use the flows to characterize the invariant subspaces of the semigroup generated by $H$.
\begin{thm}\label{tsep1}
Let $\Omega$ be a measurable subset of ${\bf R}^d$.
Consider the following conditions.
\begin{tabel}
\item\label{tsep1-1}
$S_t L_2(\Omega) \subseteq L_2(\Omega)$ for all $t>0$.
\item\label{tsep1-2}
$T^{(k)}_t L_2(\Omega) = L_2(\Omega)$ for all $k \in \{ 1,\ldots,d \} $ and $t\in{\bf R}$.
\end{tabel}
Then {\rm\ref{tsep1-1}}$\Rightarrow${\rm\ref{tsep1-2}}.
Moreover,
if
$C_c^\infty({\bf R}^d)$ is a core for $H$, or, if
$\Omega$ is open and the boundary $\partial \Omega$ of $\Omega$
is $($locally$)$ Lipschitz then {\rm\ref{tsep1-1}}$\Leftrightarrow${\rm\ref{tsep1-2}}.
\end{thm}
Recall that the open set $\Omega$ is defined to have a
(locally) Lipschitz boundary if for
every $y\in \partial \Omega$ there exist an isometry $\Psi \colon {\bf R}^d\to {\bf R}^d$,
a real function $\tau \in W^{1,\infty}( {\bf R}^{d-1} )$
and an $r > 0$ such that
\begin{equation}
\Omega\cap B_y(r) = \{ \Psi(x_1,x') : (x_1,x') \in {\bf R} \times {\bf R}^{d-1}, \;
\tau(x') < x_1 \} \cap B_y(r)
\label{eSflow1;1}
\end{equation}
where $B_y(r)=\{x\in{\bf R}^d: \|x-y\|<r\}$.
Thus in a neighbourhood of $y$ the boundary $\partial\Omega$
of $\Omega$ is the graph of a Lipschitz function $\tau$, up to an
isometry $\Psi$.
There are two variations of the theorem which will be established in the course of its proof.
First, for all $\psi \in C_c^\infty({\bf R}^d)$ define $Y_\psi$
as the $L_2$-closure of the first-order partial differential operator
\[
\varphi \mapsto \sum^d_{k,l=1} (\partial_k \psi) \, c_{kl} \, \partial_l \varphi
\]
with domain $C_c^\infty({\bf R}^d)$ and let $T^\psi$ be the associated flow.
Then invariance of $L_2(\Omega)$ under the $T^{(k)}$ is equivalent to invariance under
the family of flows $T^\psi$.
More precisely one has the following.
\begin{prop}\label{psep1}
Let $\Omega$ be a measurable subset of ${\bf R}^d$.
The following conditions are equivalent:
\begin{tabel}
\item\label{psep1-4}
$T^\psi_t L_2(\Omega) = L_2(\Omega)$ for all $\psi \in C_c^\infty({\bf R}^d)$ and $t\in{\bf R}$.
\item\label{psep1-2}
$T^{(k)}_t L_2(\Omega) = L_2(\Omega)$ for all $k \in \{ 1,\ldots,d \} $ and $t\in{\bf R}$.
\end{tabel}
\end{prop}
This will be established in Section~\ref{Sflow2}.
Secondly, the condition that $C_c^\infty({\bf R}^d)$ is a core for $H$ does not follow in general from
the assumption that the coefficients are in $ W^{1,\infty}({\bf R}^d)$.
The one-dimensional example considered in \cite{ERSZ1}, Section~5
gives a counterexample.
Specifically, let $\delta \in [1/2,\infty \rangle$ and
$H=-d\,c\,d$ with $c(x)= |x|^{2\delta}(1+x^2)^{-\delta}$.
Then $c\in W^{1,\infty}({\bf R})$ but $C_c^\infty({\bf R})$ is a
core of $H$ if and only if $\delta\geq 3/4$
by the arguments in \cite{CMP}, Proposition~3.5.
(See also \cite{RSi3}.)
In particular it is not a core if $\delta\in[1/2,3/4\rangle$.
Nevertheless it follows that $C_c^\infty({\bf R}^d)$ is a core for $H$ if
$c_{kl}\in W^{2,\infty}({\bf R}^d)$ (see, \cite{Rob7} Section~6, or \cite{ER27}
Proposition~2.3).
Moreover, the core condition can be derived from weaker smoothness
assumptions on the $c_{kl}$ (see Section~\ref{Sflow4}).
\section{Flows} \label{Sflow2}
In this section we derive some properties of the flows defined in
Section~\ref{Sflow1} and prove Proposition~\ref{psep1}.
Although we deal primarily with the flows on $L_2({\bf R}^d)$ we will need,
in Section~\ref{S3}
some properties of their extensions to $L_\infty({\bf R}^d)$.
Therefore we begin by summarizing some general features of the flows.
Let $b_1,\ldots,b_d \in W^{1,\infty}({\bf R}^d)$ and define
$Y$ as the $L_2$-closure of the first-order differential operator
$\varphi \mapsto \sum_{k=1}^d b_k \, \partial_k\varphi$ and domain
$W^{1,2}({\bf R}^d)$.
Further let $T$ denote the flow generated by $Y$.
Then for all $p\in[1,\infty]$ the group $T$ leaves the subspace
$L_2({\bf R}^d)\cap L_p({\bf R}^d)$ of $L_2({\bf R}^d)$ invariant and
$T$ extends from $L_2({\bf R}^d)\cap L_p({\bf R}^d)$ to a flow
$T^{[p]}$ on $L_p({\bf R}^d)$ such that $T^{[p]}$ is strongly continuous if
$p\in[1,\infty\rangle$ and $T^{[\infty]}$ is weakly$^*$ continuous.
The groups act in a consistent and compatible manner on the $L_p$-spaces.
Moreover, $T^{[\infty]}$ is a group of automorphisms of $L_\infty({\bf R}^d)$, i.e.\
$T^{[\infty]}_t(\psi\, \varphi)= (T^{[\infty]}_t \psi) \, (T^{[\infty]}_t \varphi)$
for all $\psi,\varphi\in L_\infty({\bf R}^d)$ and $t \in {\bf R}$.
Then since the $L_\infty$-functions are multipliers on the $L_p$-spaces
one deduces that
\begin{equation}
T^{[p]}_t(\tau \, \varphi)
= (T^{[\infty]}_t \tau) \, (T^{[p]}_t\varphi)
\label{eSflow2;5}
\end{equation}
for all $\tau \in L_\infty({\bf R}^d)$, $\varphi\in L_p({\bf R}^d)$, $p\in[1,\infty]$
and $t \in {\bf R}$.
If $Y_{[p]}$ is the generator of $T^{[p]}$ then $W^{1,p}({\bf R}^d) \subset D(Y_{[p]})$
and $Y_{[p]} \varphi = \sum_{k=1}^d b_k \, \partial_k\varphi$
for all $\varphi \in W^{1,p}({\bf R}^d)$.
These properties depend critically on the fact that $Y$ is a first-order partial
differential operator with coefficients $b_k\in W^{1,\infty}({\bf R}^d)$.
They can be verified either by general arguments of functional analysis
(see, for example, \cite{Robm}, Theorem~V.4.1) or by methods of ordinary differential
equations.
The crucial observation in the latter context is that if
$\varphi\in C_c^\infty({\bf R}^d)$ then
$(T_t\varphi)(x)=\varphi(\omega_t(x))$ where
$t \mapsto \omega_t(x)$ is the unique solution of the differential equation
$(d/dt)\omega_t(x)=b(\omega_t(x))$, with initial value $\omega_0(x)=x$ (see, for example, \cite{Hil}, Chapters~2 and 3).
Our first result is an approximation result which will be needed on $L_2({\bf R}^d)$ but whose proof extends to the $L_p$-spaces.
\begin{prop} \label{pord901}
Let $p \in [1,\infty]$.
Let $Y_{[p]}$ denote the generator of the flow $T^{[p]}$ on $L_p({\bf R}^d)$.
Further let $\tau \in C_c^\infty({\bf R}^d)$ with $\int \tau = 1$ and
for all $n \in {\bf N}$ define $\tau_n \in C_c^\infty({\bf R}^d)$ by
$\tau_n(x) = n^d \, \tau(n \, x)$.
Then $\lim_{n \to \infty} Y_{[p]}(\tau_n * \varphi) = Y_{[p]} \varphi$
in $L_p({\bf R}^d)$ for all $\varphi \in D(Y_{[p]})$ if $p < \infty$.
If $p = \infty$ then
$\lim_{n \to \infty} Y_{[\infty]}(\tau_n * \varphi) = Y_{[\infty]} \varphi$ weakly$^*$
in $L_\infty({\bf R}^d)$ for all $\varphi \in D(Y_{[\infty]})$
\end{prop}
\proof\
First, for all $n \in {\bf N}$ define the bounded operator
$B_n \colon L_p \to L_p$ by
\[
B_n \varphi
= \sum_{k=1}^d \tau_n * ( (\partial_k \, b_k) \varphi)
+ \sum_{k=1}^d \int dy \, (\partial_k \tau_n)(y) \,
\Big( (I - L_y) b_k \Big) \, (L_y \varphi)
\;\;\; , \]
where $L$ denotes the left regular representation of ${\bf R}^d$, i.e.\
$(L_y \psi)(x) = \psi(x-y)$.
Secondly, if $\varphi \in C_c^\infty$ and $n \in {\bf N}$ then
\begin{eqnarray*}
Y_{[p]}(\tau_n * \varphi)
& = & \sum_{k=1}^d b_k \int dy \, \tau_n(y) \, L_y \partial_k \, \varphi \\
& = & \sum_{k=1}^d \int dy \, \tau_n(y) \, (b_k - L_y b_k) \, L_y \partial_k \, \varphi
+ \sum_{k=1}^d b_k \int dy \, \tau_n(y) \, L_y (b_k \, \partial_k \, \varphi)
\;\;\; .
\end{eqnarray*}
The second term equals $\tau_n * Y_{[p]} \varphi$.
For the first term use
$L_y \partial_k \varphi = - \frac{\partial}{\partial y_k} \, L_y \varphi$.
Therefore integration by parts gives
\[
Y_{[p]}(\tau_n * \varphi) - \tau_n * Y_{[p]} \varphi
= \sum_{k=1}^d \int dy \, \frac{\partial}{\partial y_k}
\Big( \tau_n(y) \, (b_k - L_y b_k) \Big) (L_y \varphi)
= B_n \varphi
\;\;\; . \]
Since $B_n$ is bounded one deduces by density that
\begin{equation}
Y_{[p]}(\tau_n * \varphi) - \tau_n * Y_{[p]} \varphi = B_n \varphi
\label{epord901;1}
\end{equation}
for all $n \in {\bf N}$ and $\varphi \in D(Y_{[p]})$.
Thirdly, it follows from the definition of $B_n$ that
\begin{eqnarray*}
\|B_n \varphi\|_p
& \leq & \sum_{k=1}^d \Big( \|(\partial_k \, b_k) \, \varphi\|_p
+ \int dy \, |(\partial_k \tau_n)(y)| \,
\|\Big( (I - L_y) b_k \Big) \, (L_y \varphi)\|_p \Big) \\
& \leq & \sum_{k=1}^d \|b_k\|_{W^{1,\infty}} \, \|\varphi\|_p
+ \sum_{k=1}^d \int dy \, |(\partial_k \tau_n)(y)| \,
\|(I - L_y) b_k\|_\infty \, \|\varphi\|_p
\end{eqnarray*}
for all $n \in {\bf N}$ and $\varphi \in L_p$.
But $\|(I - L_y) b_k\|_\infty \leq |y| \, \|b_k\|_{W^{1,\infty}}$
and $\int dy \, |(\partial_k \tau_n)(y)| \, |y| = \int dy \, |(\partial_k \tau)(y)| \, |y|$.
Therefore $\|B_n \varphi\|_p \leq M \, \|\varphi\|_p$ uniformly
for all $n \in {\bf N}$ and $\varphi \in L_p$, where
$M = \sum_{k=1}^d (1 + \int dy \, |(\partial_k \tau)(y)| \, |y|) \, \|b_k\|_{W^{1,\infty}}$.
The conclusion holds for all $p\in[1,\infty]$.
So $B_1, B_2,\ldots$ are equicontinuous.
Next assume $p < \infty$.
If $\varphi \in W^{1,p}$ then
$\lim_{n \to \infty} \tau_n * \varphi = \varphi$
in $W^{1,p}$.
Consequently, $\lim_{n \to \infty} Y_{[p]}\,(\tau_n * \varphi) = Y_{[p]}\, \varphi$ strongly in $L_p$.
Moreover, $\lim_{n \to \infty} \tau_n * (Y_{[p]} \,\varphi) = Y_{[p]}\, \varphi$ strongly in $L_p$.
Therefore $\lim_{n \to \infty} B_n \varphi = 0$ in $L_p$ for all
$\varphi \in W^{1,p}$ by (\ref{epord901;1}).
Since $W^{1,p}$ is strongly dense in $L_p$ and $B_1,B_2,\ldots$ are equicontinuous
it follows that $\lim_{n \to \infty} B_n \varphi = 0$ in $L_p$ for all
$\varphi \in L_p$.
Finally, let $\varphi \in D(Y_{[p]})$.
Then one establishes from (\ref{epord901;1}) that
$\lim_{n \to \infty} Y_{[p]}\,(\tau_n * \varphi)
= \lim_{n \to \infty} (\tau_n * Y_{[p]}\, \varphi + B_n \varphi)
= Y_{[p]} \,\varphi$ in $L_p$.
The argument for $p=\infty$ is very similar.
If $\varphi\in W^{1,\infty}$ then $\lim \tau_n * \varphi = \varphi$
and $\lim \partial_k \tau_n * \varphi = \partial_k\varphi$ weakly$^*$.
Therefore
$\lim Y_{[\infty]}\,(\tau_n * \varphi) = Y_{[\infty]}\, \varphi$ weak$^*$ on $L_\infty$.
Then since $W^{1,\infty}$ is weakly$^*$ dense in $L_\infty$ and
$B_1,B_2,\ldots$ are equicontinuous the desired conclusion follows
as before.\hfill$\Box$
\vskip10.0pt plus 4.0pt minus 6.0pt
Now we return to consideration of the vector fields $Y_1,\ldots,Y_d$
defined in Section~\ref{Sflow1} acting on $L_2({\bf R}^d)$.
\begin{cor} \label{cflow310}
Let $\tau$ and $\tau_n$ be as in Proposition~{\rm \ref{pord901}}.
Then for all $\varphi \in \bigcap_{k=1}^d D(Y_k)$ one has
$\lim_{n \to \infty} Y_k(\tau_n * \varphi) = Y_k \varphi$
for all $k \in \{ 1,\ldots,d \} $.
\end{cor}
Note that convolution with $\tau_n$ maps $L_2({\bf R}^d)$ into $W^{\infty,2}({\bf R}^d)$
so the corollary
establishes that $W^{\infty,2}({\bf R}^d)$ is a simultaneous core for
the $Y_1,\ldots,Y_d$.
Now we turn to the proof of Proposition~\ref{psep1}.
Note that
if $T$ is a flow with generator $Y$ then $T$-invariance of $L_2(\Omega)$ is equivalent to the
the commutation of $Y$ and the operator of multiplication with $\mathbb{1}_\Omega$,
i.e.\ if $\varphi \in D(Y)$ then $\mathbb{1}_\Omega \varphi\in D(Y)$ and
$Y(\mathbb{1}_\Omega \, \varphi) = \mathbb{1}_\Omega \, Y \varphi$.
\vskip10.0pt plus 4.0pt minus 6.0pt
\noindent
{\bf Proof of Proposition~\ref{psep1}\hspace{5pt} }
``\ref{psep1-4}$\Rightarrow$\ref{psep1-2}''.
Let $k \in \{ 1,\ldots,d \} $ and $U \subset {\bf R}^d$ a bounded open subset.
There exist $\chi,\psi \in C_c^\infty({\bf R}^d)$ such that
$\chi|_U = \mathbb{1}$ and $\psi(x) = x_k$ for all $x \in \mathop{\rm supp} \chi$.
Then $Y_k(\chi \varphi) = Y_\psi(\chi \varphi)$
for all $\varphi \in C_c^\infty({\bf R}^d)$.
Since $\varphi \mapsto \chi \varphi$ is continuous on $D(Y_k)$
and on $D(Y_\psi)$, with the graph norm, it follows from Proposition~\ref{pord901}
that $\chi \varphi \in D(Y_k)$ for all $\varphi \in D(Y_\psi)$.
In particular, if $\varphi \in C_c^\infty({\bf R}^d)$ with $\mathop{\rm supp} \varphi \subset U$
then $\mathbb{1}_\Omega \varphi \in D(Y_\psi)$ and therefore
$\mathbb{1}_\Omega \varphi = \chi \mathbb{1}_\Omega \varphi \in D(Y_k)$.
Moreover,
$Y_k(\mathbb{1}_\Omega \varphi)
= Y_\psi(\chi \mathbb{1}_\Omega \varphi)
= \mathbb{1}_\Omega Y_\psi(\chi \varphi)
= \mathbb{1}_\Omega Y_k \varphi$.
Then it follows by continuity that $\mathbb{1}_\Omega \varphi\in D(Y_k)$ and
$Y_k(\mathbb{1}_\Omega \, \varphi) = \mathbb{1}_\Omega \, Y_k \varphi$ for all $\varphi \in D(Y_k)$.
Therefore Condition~\ref{psep1-2} is valid.
``\ref{psep1-2}$\Rightarrow$\ref{psep1-4}''.
It follows from Condition~\ref{psep1-2} that
$\mathbb{1}_\Omega \varphi \in D(Y_k)$ and
$Y_k (\mathbb{1}_\Omega \varphi) = \mathbb{1}_\Omega Y_k \varphi$
for all $\varphi \in D(Y_k)$.
Let $\psi \in C_c^\infty({\bf R}^d)$.
Then $Y_\psi \varphi = \sum_{k=1}^d (\partial_k \psi) \, Y_k \varphi$
for all $\varphi \in C_c^\infty({\bf R}^d)$.
Since the coefficients $c_{kl}$ are in $W^{1,\infty}({\bf R}^d)$ it
follows from Corollary~\ref{cflow310} that
$\varphi \in D(Y_\psi)$ and
$Y_\psi \varphi = \sum_{k=1}^d (\partial_k \psi) \, Y_k \varphi$
for all $\varphi \in \bigcap_{k=1}^d D(Y_k)$.
Hence if $\varphi \in C_c^\infty({\bf R}^d)$ then
$\mathbb{1}_\Omega \varphi \in D(Y_\psi)$ and
$Y_\psi (\mathbb{1}_\Omega \varphi) = \mathbb{1}_\Omega Y_\psi \varphi$.
By density the latter extends to all $\varphi \in D(Y_\psi)$ and
therefore Condition~\ref{psep1-4} is valid.
\hfill$\Box$
\vskip10.0pt plus 4.0pt minus 6.0pt
Finally we note that the flows $T^\psi$ can be defined for all $\psi \in W^{2,\infty}({\bf R}^d)$
and the conditions of Proposition~\ref{psep1} are equivalent to invariance of $L_2(\Omega)$
for all $T^\psi_t$ with $\psi \in W^{2,\infty}({\bf R}^d)$ and $t>0$.
This follows from the arguments of the foregoing proof.
\section{Semigroup invariance}\label{S3}
In this section we prove Theorem~\ref{tsep1}.
First, however, we observe that Condition~\ref{tsep1-2} of the theorem, the invariance of $L_2(\Omega)$
under the flows $T^{(k)}$ is equivalent to $T^\psi$-invariance of $L_2(\Omega)$ for all $\psi\in C_c^\infty({\bf R}^d)$.
This is a direct consequence of Proposition~\ref{psep1} which was established in the previous section.
Therefore in the subsequent discussion we will consider the $T^\psi$-invariance condition.
\vskip10.0pt plus 4.0pt minus 6.0pt
\noindent
{\bf Proof of Theorem~\ref{tsep1}\hspace{5pt} }
``\ref{tsep1-1}$\Rightarrow$\ref{tsep1-2}''.
It suffices, by the foregoing observation, to prove the $T^\psi$-invariance of $L_2(\Omega)$ for all $\psi\in C_c^\infty({\bf R}^d)$.
First, it follows from the density of $C_c^\infty({\bf R}^d)$ in $D(h)$
that there exists a unique bilinear map
$\Gamma \colon D(h) \times D(h) \to L_1$, the {\it carr\'e du champ}, such that
\[
\Gamma(\psi,\varphi)
= \sum_{k,l=1}^d c_{kl} \, (\partial_k \psi) \, (\partial_l \varphi)
\]
for all $\psi,\varphi \in W^{1,2}({\bf R}^d)$.
Then $\|\Gamma(\psi,\varphi)\|_1 \leq h(\psi)^{1/2} \, h(\varphi)^{1/2}$
for all $\psi,\varphi \in D(h)$ by the Cauchy--Schwarz inequality.
Moreover,
\begin{equation}
\int \tau \, \Gamma(\psi,\varphi)
= \frac{1}{2} \Big( h(\tau\psi,\varphi) + h(\psi,\tau\varphi) - h(\tau,\psi\varphi)\Big)
\label{eSord9;1}
\end{equation}
for all $\tau,\psi,\varphi \in C_c^\infty({\bf R}^d)$.
But
(\ref{eSord9;1}) then extends to all $\tau,\psi,\varphi \in D(h) \cap L_\infty$ by density.
Secondly, the form $h$ is local in the sense that
$h(\psi,\varphi) = 0$ for all $\psi,\varphi \in D(h)$
with $\psi \, \varphi = 0$
(see \cite{Schm}).
Therefore it follows from (\ref{eSord9;1}) that $\Gamma$ is local in the same sense.
Thirdly, since $L_2(\Omega)$ is $S$-invariant the operation of
multiplication by $\mathbb{1}_\Omega$ maps
$D(h)$ into itself.
Therefore if $\psi,\varphi,\tau \in D(h) \cap L_\infty$
then $\mathbb{1}_\Omega \varphi, \mathbb{1}_\Omega \tau \in D(h) \cap L_\infty$.
By locality of $h$
one deduces from (\ref{eSord9;1}) that
\begin{eqnarray*}
\int \tau \, \Gamma(\psi,\mathbb{1}_\Omega \varphi)
& = & \frac{1}{2} \Big( h(\tau\psi,\mathbb{1}_\Omega \varphi)
+ h(\psi,\tau\mathbb{1}_\Omega \varphi) - h(\tau,\psi \mathbb{1}_\Omega \varphi) \Big) \\
& = & \frac{1}{2} \Big( h(\mathbb{1}_\Omega \tau\psi,\varphi)
+ h(\psi,\mathbb{1}_\Omega \tau\varphi) - h(\mathbb{1}_\Omega \tau,\psi\varphi) \Big)
= \int \mathbb{1}_\Omega \tau \, \Gamma(\psi,\varphi)
\;\;\; .
\end{eqnarray*}
Hence $\Gamma(\psi,\mathbb{1}_\Omega \varphi) = \mathbb{1}_\Omega \Gamma(\psi,\varphi)$.
But $D(h) \cap L_\infty$ is dense in $D(h)$.
Therefore $\Gamma(\psi,\mathbb{1}_\Omega \varphi) = \mathbb{1}_\Omega \Gamma(\psi,\varphi)$
for all $\psi,\varphi \in D(h)$.
Now fix $\psi \in C_c^\infty({\bf R}^d)$.
Let $\tau \in C_c^\infty({\bf R}^d)$.
Then
\[
((Y_\psi)^* \tau, \eta)
= (\tau, Y_\psi \eta)
= (\tau, \Gamma(\psi, \eta))
\]
for all $\eta \in C_c^\infty({\bf R}^d)$.
Since $C_c^\infty({\bf R}^d)$ is dense in $D(h)$ one deduces that
$((Y_\psi)^* \tau, \eta) = (\tau, \Gamma(\psi, \eta))$
for all $\eta \in D(h)$.
Choosing $\eta = \mathbb{1}_\Omega \varphi$ it follows that
\[
((Y_\psi)^* \tau, \mathbb{1}_\Omega \varphi)
= (\tau, \Gamma(\psi,\mathbb{1}_\Omega \varphi))
= (\mathbb{1}_\Omega \tau, \Gamma(\psi,\varphi))
= (\mathbb{1}_\Omega \tau, Y_\psi \varphi)
= (\tau, \mathbb{1}_\Omega Y_\psi \varphi)
\;\;\; . \]
Since $C_c^\infty({\bf R}^d)$ is a core for $(Y_\psi)^*$ one deduces that
$\mathbb{1}_\Omega \varphi \in D(Y_\psi)$ and
$Y_\psi (\mathbb{1}_\Omega \varphi) = \mathbb{1}_\Omega Y_\psi \varphi$.
This conclusion then extends to all $\varphi \in D(Y_\psi)$ by density.
Therefore $L_2(\Omega)$ is invariant under $T^\psi$.
The converse implication \ref{tsep1-2}$\Rightarrow$\ref{tsep1-1}
consists of two special cases.
{\em Case 1}. $C_c^\infty({\bf R}^d)$ is a core for $H$.
Condition~\ref{tsep1-2} is equivalent to
$T^\psi$ invariance of $L_2(\Omega)$ for all $\psi\in C_c^\infty({\bf R}^d)$ by Proposition~\ref{psep1}.
Therefore we assume the latter condition.
Let $\psi,\tau \in C_c^\infty({\bf R}^d)$.
Then
\[
(H \psi, \tau \, \varphi)
= h(\psi, \tau \, \varphi)
= \int \Gamma(\psi, \tau \, \varphi)
= \int \tau \, \Gamma(\psi,\varphi) + \varphi \, \Gamma(\psi,\tau)
= (\tau,Y_\psi \varphi) + (\varphi, Y_\psi \tau)
\]
for all $\varphi \in C_c^\infty({\bf R}^d)$.
Since $C_c^\infty({\bf R}^d)$ is dense in $D(Y_\psi)$ one deduces that
\begin{equation}
(H \psi, \tau \, \varphi)
= (\tau, Y_\psi \varphi) + (\varphi, Y_\psi \tau)
\label{etftos302;1}
\end{equation}
for all $\varphi \in D(Y_\psi)$.
Now let $\psi,\tau,\varphi \in C_c^\infty({\bf R}^d)$.
Then by $T^\psi$-invariance of $L_2(\Omega)$ and (\ref{etftos302;1}) one deduces that
$\mathbb{1}_\Omega \, \varphi \in D(Y_\psi)$ and
\begin{eqnarray*}
(H \psi, \tau \, \mathbb{1}_\Omega \, \varphi)
& = & (\tau, Y_\psi (\mathbb{1}_\Omega \, \varphi))
+ (\mathbb{1}_\Omega \, \varphi, Y_\psi \tau) \\
& = & (\mathbb{1}_\Omega \, \tau, Y_\psi \varphi)
+ (\mathbb{1}_\Omega \, \varphi, Y_\psi \tau)
= (\mathbb{1}_\Omega \, \tau, \Gamma(\psi, \varphi))
+ (\mathbb{1}_\Omega \, \varphi, \Gamma(\psi, \tau))
\;\;\; .
\end{eqnarray*}
Therefore
\[
|(H \psi, \tau \, \mathbb{1}_\Omega \, \varphi)|
\leq \|\mathbb{1}_\Omega \, \tau\|_\infty \, \|\Gamma(\psi, \varphi)\|_1
+ \|\mathbb{1}_\Omega \, \varphi\|_\infty \, \|\Gamma(\psi, \tau)\|_1
\leq c \, h(\psi)^{1/2}
\leq c \, \|(I + H)^{1/2} \psi\|_2
\]
where $c = \|\tau\|_\infty \, h(\varphi)^{1/2} + \|\varphi\|_\infty \, h(\tau)^{1/2}$.
This estimate is uniform for all $\psi \in C_c^\infty({\bf R}^d)$.
Since by assumption the space $C_c^\infty({\bf R}^d)$ is a core for $D(H)$ it follows
that $\mathbb{1}_\Omega \, \tau \, \varphi \in D(H^{1/2}) = D(h)$ for all
$\tau,\varphi \in C_c^\infty({\bf R}^d)$.
But $\mathop{\rm span} (C_c^\infty({\bf R}^d) \cdot C_c^\infty({\bf R}^d))$ is dense in $D(h)$.
Therefore it follows from \cite{ER29}, Proposition~2.1 III$\Rightarrow$I, that
$S$ leaves $L_2(\Omega)$ invariant.
This completes the proof of the first case in the proof of \ref{tsep1-2}$\Rightarrow$\ref{tsep1-1}.
\medskip
{\em Case 2}. $\partial \Omega$ is (locally) Lipschitz.
Let $P_\Omega$ be the orthogonal projection of $L_2({\bf R}^d)$ onto
$L_2(\Omega)$.
By assumption $T^\psi$ leaves $L_2(\Omega)$ invariant for all
$\psi\in C_c^\infty({\bf R}^d)$.
Hence
\begin{equation}
T^\psi_t \, P_\Omega = P_\Omega \, T^\psi_t \, P_\Omega
\label{eSflow2;1}
\end{equation}
for all $t \in {\bf R}$.
Let $B$ denote multiplication by the bounded function
$\sum_{k,l=1}^d (\partial_k \psi) (\partial_l c_{kl}) $ and set
$M_t=e^{-tB}$ for $\in{\bf R}$.
Clearly each $M_t$ leaves $L_2(\Omega)$ invariant.
Therefore $(T^\psi_{-t/n} \, M_{-t/n})^n$ leaves $L_2(\Omega)$
invariant for all $t \in {\bf R}$ and $n \in {\bf N}$.
But $(Y_\psi)^* = - Y_\psi - B$.
Then the Trotter product formula establishes that
$(T^\psi_t)^*$ is the strong limit of $(T^\psi_{-t/n} \, M_{-t/n})^n$
as $n \to \infty$.
So $(T^\psi_t)^*$ leaves $L_2(\Omega)$ invariant.
Hence $(T^\psi_t)^* \, P_\Omega = P_\Omega \, (T^\psi_t)^* \, P_\Omega$
for all $t \in {\bf R}$.
Therefore $P_\Omega \, T^\psi_t = P_\Omega \, T^\psi_t \, P_\Omega$
and by (\ref{eSflow2;1}) it follows that
$T^\psi_t \, P_\Omega = P_\Omega \, T^\psi_t$ for all $t \in {\bf R}$.
Then
\[
\mathbb{1}_\Omega \, T^\psi_t \varphi
= P_\Omega \, T^\psi_t \varphi
= T^\psi_t \, P_\Omega \varphi
= T^\psi_t(\mathbb{1}_\Omega \, \varphi)
= (T^{\psi,\infty}_t \mathbb{1}_\Omega) \, (T^\psi_t \varphi)
\]
for all $\varphi \in C_c^\infty({\bf R}^d)$ and $t \in {\bf R}$
where $T^{\psi,\infty}$ denotes the extension of the flow $T^\psi$ to $L_\infty({\bf R}^d)$
(see Section~\ref{Sflow2}) and we have used (\ref{eSflow2;5}).
Since $T^\psi_t(C_c^\infty({\bf R}^d))$ is dense in $L_2({\bf R}^d)$
one deduces that $T^{\psi,\infty}_t \mathbb{1}_\Omega = \mathbb{1}_\Omega$ for all
$t \in {\bf R}$.
Next let $\varphi \in C_c^\infty({\bf R}^d)$.
Then $(Y_\psi)^* \varphi \in L_1({\bf R}^d) \cap L_2({\bf R}^d)$,
so $(Y^{(\infty)}_\psi)^* \varphi = (Y_\psi)^* \varphi$,
where $Y^{(\infty)}_\psi$ is the generator of $T^{\psi,\infty}$.
Since
$((T^{\psi,\infty}_t)^* \varphi, \mathbb{1}_\Omega)
= (\varphi, T^{\psi,\infty}_t \mathbb{1}_\Omega)
= (\varphi, \mathbb{1}_\Omega)$
for all $t \in {\bf R}$ it follows by differentiation that
$((Y_\psi)^* \varphi, \mathbb{1}_\Omega) = 0$.
Therefore setting $\Phi_k = \sum_{l=1}^d c_{kl} \, \partial_l \psi$
for $k \in \{ 1,\ldots,d \} $ one has
\begin{equation}
\int_\Omega \mathop{\rm div}(\varphi \, \Phi)=((Y_\psi)^* \varphi, \mathbb{1}_\Omega) = 0
\;\; \;.
\label{eflux340}
\end{equation}
At this point we use the (local) Lipschitz continuity of $\partial\Omega$.
The Gauss--Green theorem is valid for open sets $\Omega$ with a
(locally) Lipschitz boundary
(see, for example, \cite{EvG} page 209).
It states that
\[
\int_\Omega \mathop{\rm div} \Psi
=\int_{\partial \Omega}dS\, \langle n, \Psi\rangle
\]
for all $\Psi\in W^{1,\infty}({\bf R}^d)$ with compact support where
$\langle\,\cdot\,,\,\cdot\,\rangle$ denotes the inner product on ${\bf R}^d$,
$dS$ is the Euclidean measure on $\partial\Omega$
and $n$ is the unit outward normal to $\partial\Omega$.
The normal is defined $dS$-almost everywhere.
Thus if one sets $\Psi=\varphi\,\Phi$ with $\varphi\in C_c^\infty({\bf R}^d)$
one has
\[
\int_\Omega \mathop{\rm div}(\varphi \, \Phi)
=\int_{\partial \Omega}dS\, \varphi \, \langle n, \Phi\rangle=0
\]
where the last equality uses (\ref{eflux340}).
Since this is valid for all $\varphi\in C_c^\infty({\bf R}^d)$ it follows that
$\langle n, \Phi \rangle = 0$ almost everywhere on $\partial\Omega$.
Therefore $\langle (\nabla \psi)(x), C(x) \, n_x \rangle = 0$
for almost every $x \in \partial \Omega$.
But this is also valid for all $\psi \in C_c^\infty({\bf R}^d)$.
Hence one must have
$C(x) \, n_x = 0$ for almost every $x \in \partial \Omega$.
This corresponds to the condition of zero flux across the boundary as defined
in \cite{RSi} and then the
$S$-invariance of $L_2(\Omega)$ follows from Theorem~1.2 of this
reference.\hfill$\Box$
\vskip10.0pt plus 4.0pt minus 6.0pt
The argument in \cite{RSi} that zero flux implies invariance is somewhat indirect as it first proves that the capacity of $\partial\Omega$
with respect to $h$ is zero and then uses this to deduce the $S$-invariance of $L_2(\Omega)$.
Nevertheless, the same reasoning can be adapted to give a direct proof of the invariance
since the proof can be reduced to a local estimate
as in \cite{RSi}.
(The latter proof and this proof are an adaption of the argument used to prove Proposition~6.5
in \cite{ERSZ1}.)
First, it suffices to prove that if $\varphi\in C_c^\infty({\bf R}^d)$ then $\mathbb{1}_\Omega\varphi\in D(h)$.
This is a consequence of \cite{ER29} Proposition~2.1 and locality of~$h$.
But this is obvious if the support of $\varphi$ and the boundary are disjoint.
Therefore it suffices to consider $\varphi$ with support close to the boundary $\partial\Omega$.
Then, however, one can use a decomposition of the identity to reduce to
the case $\mathop{\rm supp}\varphi \subset B_y(r)$ with $y \in \partial\Omega$ and $r > 0$ small.
Secondly, let $\tau$, $\Psi$ be as in (\ref{eSflow1;1}).
Without loss of generality we may assume that $\Psi(x) = x$ for all $x \in {\bf R}^d$.
For all $n \in {\bf N}$ define $\psi_n \colon {\bf R}^d \to {\bf R}$ by
$\psi_n(x) = \chi_n(x_1 - \tau(x'))$, where $x = (x_1,x') \in {\bf R} \times {\bf R}^{d-1}$ and $\chi_n \colon {\bf R} \to {\bf R}$ is
defined by
\[
\chi_n(t) = \left\{ \begin{array}{ll}
0 & \mbox{if } t \leq 1/n , \\[5pt]
\displaystyle \log( tn)/\log n & \mbox{if } 1/n < t < 1 , \\[5pt]
1 & \mbox{if } t \geq 1 .
\end{array} \right.
\]
Then $\lim ( \psi_n\varphi) = \mathbb{1}_\Omega \varphi$ in $L_2({\bf R}^d)$.
Thus to establish that $\mathbb{1}_\Omega \varphi \in D(h)$ it suffices to prove that $ \{ h(\psi_n\varphi) : n \in {\bf N} \} $ is
bounded.
But
\begin{eqnarray*}
h(\psi_n\varphi)
& \leq & 2\, h(\varphi)
+ 2 \int |\varphi|^2 \sum^d_{k,l=1} c_{kl} \, (\partial_k \psi_n) \, (\partial_l \psi_n) \\
& \leq & 2\, h(\varphi)
+ 2 \,(\log n)^{-2}\int_{{\bf R}^{d-1}} dx'
\int_{\tau(x') + 1/n}^{\tau(x') + 1} dx_1 \, |\varphi(x)|^2 \,
\frac{\langle \nu_x , C(x) \nu_x \rangle}
{(x_1 - \tau(x'))^2 }
\end{eqnarray*}
for all $n \in {\bf N}$ where $\nu_x=(1,-(\nabla\tau)(x'))$.
Since the coefficients $c_{kl}$ are in $W^{1,\infty}({\bf R}^d)$ there exists an $M > 0$
such that
$|\langle \xi, C(x) \xi \rangle - \langle \xi, C(z) \xi \rangle|
\leq M \, \|\xi\|^2$
for all $x,z,\xi \in {\bf R}^d$.
If $x = (x_1,x') \in B_y(r)$, the function $\tau$ is differentiable at $x'$
and $x_1 = \tau(x')$ then
\[
\langle \nu_x, C(\tau(x'),x') \nu_x\rangle
= (1+|(\nabla\tau)(x')|^2)\,\langle n_x, C(\tau(x'),x') n_x \rangle = 0
\]
by the zero flux condition.
Hence
$\langle \nu_x, C(x_1,x') \nu_x \rangle
\leq M_1 \, |x_1 - \tau(x')|$
for all $x = (x_1,x') \in B_y(r)$ with $\tau$ differentiable at $x'$,
where $M_1 = M (1 + \|\nabla \tau\|_\infty)^2$.
It follows that
\begin{eqnarray*}
\lefteqn{
(\log n)^{-2}\int_{{\bf R}^{d-1}} dx'
\int_{\tau(x') + 1/n}^{\tau(x') + 1} dx_1 \, |\varphi(x_1,x')|^2 \,
\frac{\langle \nu_x, C(x_1,x') \nu_x\rangle}
{(x_1 - \tau(x'))^2 }
} \hspace{20mm} \\
& \leq &M_1\,(\log n)^{-2}
\int_{{\bf R}^{d-1}} dx'
\int_{\tau(x') + 1/n}^{\tau(x') + 1} dx_1 \,
\frac{|\varphi(x_1,x')|^2 }{(x_1 - \tau(x'))}
\leq M_1 \,(\log n)^{-1}\|\varphi\|_\infty^2\, |K'|
\end{eqnarray*}
uniformly for all $n \in {\bf N}$, where $K' \subset {\bf R}^{d-1}$ is a
compact set such that $\mathop{\rm supp}\varphi \subset {\bf R} \times K'$.
So $ \{ h(\psi_n\varphi) : n \in {\bf N} \} $ is bounded, as required.
In fact a slightly more detailed argument establishes that
$\lim h(\psi_n\varphi-\mathbb{1}_\Omega\varphi) = 0$.
\section{Core properties} \label{Sflow4}
In this section we examine conditions which ensure that $C_c^\infty({\bf R}^d)$ is a
core for the degenerate elliptic operator $H$ with coefficients $(c_{kl})$ in $W^{1,\infty}$.
Obviously $C_c^\infty({\bf R}^d)$ is a core for $H$ if and only if
$W^{2,\infty}({\bf R}^d)$ is a core for $H$.
First, we recall two known core criteria.
\begin{thm} \label{tflow906}
If one of the following two conditions is valid then
$C_c^\infty({\bf R}^d)$ is a core for~$H$:
\begin{tabel}
\item \label{tflow906-1}
$c_{kl} \in W^{2,\infty}({\bf R}^d)$ for all $k,l \in \{ 1,\ldots,d \} $,
\item \label{tflow906-2}
the matrix $(c_{kl}(x))$ is invertible for all $x \in {\bf R}^d$.
\end{tabel}
\end{thm}
\proof\
If Condition~\ref{tflow906-1} is valid then $C_c^\infty({\bf R}^d)$ is a core
by \cite{Rob7} Section~6, or \cite{ER27} Proposition~2.3, or by
an adaption of the proof of Proposition~\ref{pord901}.
If Condition~\ref{tflow906-2} is valid then $C_c^\infty({\bf R}^d)$ is a core
by the arguments in \cite{Dav14} Theorem~3.1.
Davies requires that the coefficients are smooth, but if the coefficients are bounded
the smoothness condition can be relaxed to
$W^{1,\infty}$.\hfill$\Box$
\vskip10.0pt plus 4.0pt minus 6.0pt
We shall prove a core theorem with a mixture of the two conditions
of Theorem~\ref{tflow906} in Corollary~\ref{cflow904}.
\begin{lemma} \label{lflow901}
If $\chi \in W^{2,\infty}({\bf R}^d)$ and $\varphi \in D(H)$ then
$\chi \varphi \in D(H)$.
\end{lemma}
Fix $\chi \in W^{2,\infty}({\bf R}^d)$.
Then it follows from Lemma~3.4 in \cite{ERSZ2} that $\chi \varphi \in D(h)$ and
$h(\chi \, \varphi)^{1/2}
\leq \|\chi\|_\infty \, h(\varphi)^{1/2} + \|\Gamma(\chi)\|_\infty^{1/2} \, \|\varphi\|_2$
for all $\varphi \in D(h)$, where we define
$\Gamma(\chi)
= \sum_{k,l=1}^d c_{kl} \, (\partial_k \chi) \, (\partial_l \chi) \in L_\infty$.
If $\varphi,\psi \in C_c^\infty$ then
\[
h(\psi, \chi \, \varphi)
= h(\chi \, \psi, \varphi)
- \sum_{k,l=1}^d \int \psi \, \varphi \, (\partial_k \, c_{kl} \, \partial_l \chi)
- 2 \sum_{k,l=1}^d \int c_{kl} \, (\partial_k \varphi) \, (\partial_l \chi) \, \psi
\;\;\; . \]
So
\begin{equation}
|h(\psi, \chi \, \varphi)|
\leq |h(\chi \, \psi, \varphi)|
+ a \, \|\psi\|_2 \, \|\varphi\|_2
+ 2 h(\varphi)^{1/2} \, \|\Gamma(\chi)\|_\infty^{1/2} \|\psi\|_2
\;\;\; ,
\label{elflow901;1}
\end{equation}
where $a = \|\sum \partial_k c_{kl} \partial_l \chi\|_\infty$.
Then by continuity (\ref{elflow901;1}) is valid for all
$\psi,\varphi \in D(h)$.
Finally, if $\varphi \in D(H)$ then
$|h(\chi \, \psi, \varphi)| = |(\chi \, \psi, H \varphi)|
\leq \|H \varphi\|_2 \, \|\chi\|_\infty \, \|\psi\|_2$
for all $\psi \in D(h)$.
Using (\ref{elflow901;1}) it follows that there exists a $c > 0$
such that $|h(\psi, \chi \varphi)| \leq c \, \|\psi\|_2$
for all $\psi \in D(h)$.
Therefore $\chi \varphi \in D(H)$.\hfill$\Box$
\vskip10.0pt plus 4.0pt minus 6.0pt
If $A \subset {\bf R}^d$ with $A \neq \emptyset$ and $\delta > 0$
define the open set $A_\delta \subset {\bf R}^d$ by
$A_\delta = \{ x \in {\bf R}^d : d(x,A) < \delta \} $.
\begin{lemma} \label{lflow902}
Let $H_1$ and $H_2$ be degenerate elliptic operators with $W^{1,\infty}$-coefficients
$(c^{(1)}_{kl})$ and $(c^{(2)}_{kl})$ and let $h^{(1)}$ and $h^{(2)}$
be the corresponding quadratic forms.
Let $U \subset {\bf R}^d$ be an open set and suppose that
$c^{(1)}_{kl}|_U = c^{(2)}_{kl}|_U$ for all $k,l \in \{ 1,\ldots,d \} $.
Let $\varphi \in L_2({\bf R}^d) \setminus \{ 0 \} $
and suppose that $(\mathop{\rm supp} \varphi)_\delta \subset U$.
Then $\varphi \in D(h^{(1)})$ if and only if $\varphi \in D(h^{(2)})$
and then $h^{(1)}(\varphi) = h^{(2)}(\varphi)$.
Similarly, $\varphi \in D(H_1)$ if and only if $\varphi \in D(H_2)$
and then $H_1 \varphi = H_2\varphi$.
Moreover, $\mathop{\rm supp} H_1 \varphi \subseteq \mathop{\rm supp} \varphi$.
\end{lemma}
\proof\
There exists a $\chi \in W^{2,\infty}({\bf R}^d)$ such that
$\chi|_{\mathop{\rm supp} \varphi} = \mathbb{1}$ and $\mathop{\rm supp} \chi \subset U$.
Suppose $\varphi \in D(h^{(1)})$.
Then there exists a sequence $\varphi_1,\varphi_2,\ldots \in W^{1,2}({\bf R}^d)$
such that $\lim \varphi_n = \varphi$ in $D(h^{(1)})$.
Then $\lim \varphi_n = \varphi$ in $L_2({\bf R}^d)$.
But $h^{(1)}(\chi \varphi_n) = h^{(2)}(\chi \varphi_n)$ and
$h^{(1)}(\chi \varphi_n - \chi \varphi_m) = h^{(2)}(\chi \varphi_n - \chi \varphi_m)$
for all $n,m \in {\bf N}$.
Therefore $\chi \varphi_1,\chi \varphi_2$ is a Cauchy sequence in $D(h^{(2)})$.
Since $\lim \chi \varphi_n = \varphi$ in $L_2$ one deduces that
$\varphi \in D(h^{(2)})$ and
$h^{(2)}(\varphi) = h^{(1)}(\varphi)$.
Finally suppose that $\varphi \in D(H_1)$.
If $\psi \in C_c^\infty({\bf R}^d)$ with $\mathop{\rm supp} \psi \subset (\mathop{\rm supp} \varphi)^{\rm c}$
then $(H_1 \varphi,\psi) = h^{(1)}(\varphi,\psi) = 0$ by locality.
Therefore $\mathop{\rm supp} H_1 \varphi \subseteq \mathop{\rm supp} \varphi$.
Clearly $\varphi \in D(h^{(1)})$ and by the first part, also
$\varphi \in D(h^{(2)})$.
Let $\psi \in D(h^{(2)})$.
Then $\chi \psi \in D(h^{(2)})$ and $\mathop{\rm supp} \chi \psi \subset U$.
Therefore $\chi \psi \in D(h^{(1)})$.
Then by locality one deduces that
$h^{(2)}(\varphi,\psi)
= h^{(2)}(\varphi, \chi \psi) + h^{(2)}(\varphi, (\mathbb{1} - \chi) \psi)
= h^{(2)}(\varphi, \chi \psi)
= h^{(1)}(\varphi, \chi \psi)$.
So $|h^{(2)}(\varphi,\psi)| = |h^{(1)}(\varphi, \chi \psi)| = |(H_1 \varphi, \chi \psi)|
\leq \|H_1 \varphi\|_2 \, \|\chi\|_\infty \, \|\psi\|_2$.
Therefore $\varphi \in D(H_2)$.
If $\psi \in C_c^\infty(U)$ then
$(H_1 \varphi, \psi)
= (\varphi, H_1 \psi)
= (\varphi, H_2 \psi)
= (H_2 \varphi, \psi)$.
Since $\mathop{\rm supp} H_1 \varphi \subseteq U$ and $\mathop{\rm supp} H_2 \varphi \subseteq U$
it follows that $H_1 \varphi = H_2 \varphi$.\hfill$\Box$
\begin{prop} \label{pflow903}
Let $A \subset {\bf R}^d$, $\delta > 0$,
let $H_1$ and $H_2$ be degenerate elliptic operators with
$W^{1,\infty}$-coefficients
$(c^{(1)}_{kl})$ and $(c^{(2)}_{kl})$.
Suppose $\emptyset \neq A \neq {\bf R}^d$,
$c^{(1)}_{kl}|_{A_\delta} = c_{kl}|_{A_\delta}$ and
$c^{(2)}_{kl}|_{(A^{\rm c})_\delta} = c_{kl}|_{(A^{\rm c})_\delta}$
for all $k,l \in \{ 1,\ldots,d \} $ and
$C_c^\infty({\bf R}^d)$ is a core for both $H_1$ and $H_2$.
Then $C_c^\infty({\bf R}^d)$ is a core for $H$.
\end{prop}
\proof\
Let $\tau \in C_c^\infty({\bf R}^d)$ be such that $\int \tau = 1$ and
$\tau(x) = 0$ for all $x \in {\bf R}^d$ with $|x| > \frac{\delta}{4}$.
Let $\chi = \tau * \mathbb{1}_{A_{\delta / 2}}$.
Then $\chi \in W^{2,\infty}({\bf R}^d)$,
$\chi|_{A_{\delta / 4}} = \mathbb{1}$ and $\mathop{\rm supp} \chi \subset A_{3 \delta / 4}$.
Moreover, $\mathop{\rm supp} (\mathbb{1} - \chi) \subset (A_{\delta / 4})^{\rm c} \subset A^{\rm c}$.
There exist $\chi_1,\chi_2 \in W^{\infty,\infty}({\bf R}^d)$
such that
$\chi_1|_{A_{3 \delta / 4}} = \mathbb{1}$, $\mathop{\rm supp} \chi_1 \subset A_\delta$,
$\chi_2|_{A^{\rm c}} = \mathbb{1}$ and $\mathop{\rm supp} \chi_2 \subset (A^{\rm c})_\delta$.
Let $\varphi \in D(H)$.
It follows from Lemma~\ref{lflow901} that $\chi \varphi \in D(H)$ and
$(\mathbb{1} - \chi) \varphi \in D(H)$.
We shall show that we can approximate both elements by $C_c^\infty$-functions.
We may assume that $\chi \varphi \neq 0 \neq (\mathbb{1} - \chi) \varphi$.
Since $\mathop{\rm supp} (\chi \varphi) \subset A_{3 \delta / 4}$ one deduces
from Lemma~\ref{lflow902} that
$\chi \varphi \in D(H_1)$ and $H_1(\chi \varphi) = H(\chi \varphi)$.
By assumption there exist $\varphi_1,\varphi_2,\ldots \in C_c^\infty({\bf R}^d)$
such that
$\lim \varphi_n = \chi \varphi$ in $D(H_1)$.
Then $\lim \chi_1 \varphi_n = \chi_1 \chi \varphi = \chi \varphi$ in $D(H_1)$ by
Lemma~\ref{lflow901}.
But $\chi_1 \varphi_n \in C_c^\infty({\bf R}^d)$ and
$\mathop{\rm supp} \chi_1 \varphi_n \subset A_\delta$ for all $n \in {\bf N}$.
Therefore $\chi_1 \varphi_n \in D(H)$ and
$H(\chi_1 \varphi_n) = H_1(\chi_1 \varphi_n)$, again by
Lemma~\ref{lflow902}.
So $\lim \chi_1 \varphi_n = \chi \varphi$ in $D(H)$.
Similarly, using $H_2$ and $\chi_2$ there exists a sequence
$\psi_1,\psi_2,\ldots \in C_c^\infty({\bf R}^d)$ such that
$\lim \chi_2 \psi_n = (\mathbb{1} - \chi) \varphi$ in $D(H)$.
Then $\lim (\chi_1 \varphi_n + \chi_2 \psi_n) = \varphi$ in $D(H)$.
Since $\chi_1 \varphi_n + \chi_2 \psi_n \in C_c^\infty({\bf R}^d)$ the
proposition follows.\hfill$\Box$
\begin{cor} \label{cflow904}
Suppose there exist a set $A$ and $\delta > 0$ such that
$\emptyset \neq A \neq {\bf R}^d$, the
matrix $(c_{kl}(x))$ is invertible for all $x \in (A^{\rm c})_\delta$
and $c_{kl}|_{A_\delta} \in W^{2;\infty}(A_\delta)$.
Then $C_c^\infty({\bf R}^d)$ is a core for~$H$.
\end{cor}
\proof\
There exists a $\chi_1 \in W^{2,\infty}({\bf R}^d)$ such that
$\chi_1|_{A_{\delta/2}} = \mathbb{1}$ and $\mathop{\rm supp} \chi_1 \subset A_\delta$.
Define $c^{(1)}_{kl} = \chi_1 \, c_{kl} \in W^{2,\infty}({\bf R}^d)$.
Then $c^{(1)}_{kl}|_{A_{\delta/2}} = c_{kl}|_{A_{\delta/2}}$.
There exists a $\chi_2 \in W^{1,\infty}({\bf R}^d)$ such that
$\chi_2|_{(A^{\rm c})_{\delta/2}} = \mathbb{1}$ and $\mathop{\rm supp} \chi_2 \subset (A^{\rm c})_\delta$.
Define
$c^{(2)}_{kl} = \chi_2 \, c_{kl} + (\mathbb{1} - \chi_2) \delta_{kl} \in W^{1,\infty}({\bf R}^d)$.
Let $H_1$ and $H_2$ be the degenerate elliptic operator with coefficients $(c_{kl}^{(1)})$
and $(c_{kl}^{(2)})$.
Now apply Theorem~\ref{tflow906}.\ref{tflow906-1} to $H_1$,
Theorem~\ref{tflow906}.\ref{tflow906-2} to $H_2$ and
use Proposition~\ref{pflow903}.\hfill$\Box$
\subsection*{Acknowledgement}
Part of this work was carried out whilst the first author was visiting
the Australian National University
with partial support from the Centre for Mathematics and its
Applications and part of the work was
carried out whilst the second author was visiting the University of Auckland
with financial support
from the Faculty of Science.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,374
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The design space of an intercooled recuperated aero-engine has been explored using detailed engine and aircraft performance, weight, and dimensions modeling. The design parameters of the engine fan, core, intercooler, recuperator, cooling-air ratio, and variable-geometry settings for the low-pressure turbine have been optimized for minimum mission fuel. Analysis shows that the improvement achieved in terms of performance against the datum design can be attributed primarily to an increase in thermal efficiency. A parametric study has also been carried out around the optimal design to understand the impact of the chosen design parameters on mission fuel burn. The study demonstrates in detail the substantially more complex interrelationship that the different fan design parameters have in terms of engine performance compared to what is typical for conventional turbofan designs. Furthermore, the optimal pressure ratio split between the low-pressure compressor and the high-pressure compressor aligns well with a previous analytical study. It is also revealed that the increased amount of cooling air required when a hot bleeding concept is adopted is in fact beneficial for mission fuel burn. Finally, the study concludes that the potential of using variable geometry in the low-pressure turbine for improving fuel burn is limited by the high-pressure turbine blade-metal temperature.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Twitter CEO Is Challenged By Elon Musk To A 'Public Discussion' Regarding Chatbots - yuetu.info
Elon Musk declared his intention to buy Twitter last spring, which caused the price of the company's shares to skyrocket.
True, the opposite result occurred when the billionaire refused to purchase the social network: after suffering large losses, the service's owners sued the inventor, compelling him to consummate the sale for $ 44 billion.
Following earlier agreements, the founder of Tesla Motors agreed to purchase out the majority of the business on one condition only.
Only if the process for confirming the legitimacy of accounts is made known to him will the richest person in the world become the owner of Twitter. But if it turns out that they lied, there won't be a transaction, the businessman said, according to Reuters.
Remember how Musk allegedly accused the proprietors of the social network of lying while refusing to buy the business? Twitter has been holding off on giving the entrepreneur's request for information for almost two months.
As a result, Musk has every reason to think that the actual number of fraudulent or spam accounts on the social network is substantially higher than the reported 5 percent, according to Musk's lawyer.
However, a lawsuit ought to be filed to settle the dispute if the business's owners continue to withhold corporate information from Elon. On October 17, the first hearing is expected to take place. Within five days, the matter is expected to be heard. On Twitter, the engineer has already posted a counterclaim.
On July 29, Musk filed a countersuit against Twitter, intensifying his legal battle with the social media firm over his request to back out of the $44 billion acquisition.
Twitter rejected Musk's assertion that he was duped into signing the agreement to purchase the social media business on Thursday, calling it "highly improbable and contrary to fact."
Read this article Twitter CEO Is Challenged By Elon Musk To A 'Public Discussion' Regarding Chatbots at Celebrityinsider.org
Sheer Tie-Front Cardigan Worn By Gigi Hadid On The Street - yuetu.info
Gigi Hadid hasn't gone out much lately. She appears in paparazzi images much more frequently than in gossip magazines; most recently, the star was seen wearing the season's most fashionable shorts; as a result, photographers have captured a superstar in a look that every fashionista may copy.
Gigi selected a transparent curtain cardigan with only a few delicate strings holding it on for her stroll through New York City. Long, dark pants with a high waist and black Converse sneakers balanced out the outfit, which could have been highly provocative.
The star visually extended the silhouette, playing on contrasts, in addition to the fact that total black slims even without extra manipulations: a shortened top and an elongated bottom visually rendered Hadid's already flawless figure even more elegant.
The outfit was finished with oval sunglasses, a gold necklace, and a black and green bag. Additionally, the young mother's innate attractiveness was enhanced by her peach blush, beige lipstick, and flowing hair.
Recently, Gigi's clothes have seemed to borrow some of Bella's fashion cues. The model recently deviated significantly from her trademark style and went skater-chic in a cropped sweater vest covered in various colors and patterns.
She wore the knit with ripped jeans, a blue fleece-lined bucket hat, and her go-to Converse sneakers. What she will wear next is a mystery.
In a previous post, Although Gigi Hadid started modeling at the young age of two with Baby Guess, her career truly took off once she graduated from high school in California and moved to New York City.
In 2014, she made her Fashion Week catwalk debut. She enrolled at the New School in the fall of 2013 with the intention of studying criminal psychology, but she postponed her studies since her modeling jobs were forcing her to miss too many classes.
Things haven't slowed down at all for the style icon; just three years later, she was crossing the stage to accept the British Fashion Award for Model of the Year from presenter Donatella Versace.
Read this article Sheer Tie-Front Cardigan Worn By Gigi Hadid On The Street at Celebrityinsider.org
Lisa Kudrow Acknowledges That She Detested Seeing Herself On The Screen Next To Courteney Cox And Jennifer Aniston - yuetu.info
Friends has been one of the most watched sitcoms on the globe for the past 30 years. Some viewers think that the project's resounding success is the result of the genuine "chemistry" between the key actors, who have become close in both real lives and on television.
It's true that working with Jennifer Aniston and Courteney Cox made Lisa Kudrow feel uneasy. The actress's complexes were the cause of the strained relationship.
In an interview with Popcrushed, Kudrow discussed this. Lisa claimed that up until she saw herself on the television next to Jennifer and Courtney, she thought her body was flawless.
The celebrity's self-esteem suffered as a result of frequent comparisons with coworkers: "I just felt like a giant woman next to them."
Although I didn't have a waist, I was previously unaware of it. I assumed I was thin, so I could wear anything. It was made clear by "Friends" that I actually don't look like I had imagined. I started thinking about dieting and losing weight all the time since it was so terrible, the actress said.
"I'm not attempting to imply that I was obese. I was obviously not overweight; I just wasn't aware of my actual body type.
Later, I came to the conclusion that my physique was perfect. Complexes are entirely a mental construct. No one requires you to be tiny and famous, but you oppress yourself on your own, she said.
In a previous post, Jennifer Aniston, who costarred with Kudrow on Friends, helped the Emmy Award winner celebrate her 59th birthday on Saturday. She also shared some sweet flashback photos of the two of them.
Aniston, 53, posted a picture of them at the 29th People's Choice Awards in 2003, "Happy birthday @lisakudrow." With a GIF showing them cheering and bouncing around as their Friends characters Rachel Green and Phoebe Buffay, she wrote, "I adore you."
Read this article Lisa Kudrow Acknowledges That She Detested Seeing Herself On The Screen Next To Courteney Cox And Jennifer Aniston at Celebrityinsider.org
At This Elder Son Maddox Chivan Jolie-Pitt Birthday Party, Brad Pitt Was Missing - yuetu.info
For the sake of the children, Brad Pitt and Angelina Jolie were unable to put their past differences behind them and keep a cordial relationship after their divorce.
The couple's heirs reside with the actress; paparazzi frequently photograph them out for a stroll or doing their shopping. The "Fight Club" star allegedly helps out with kid rearing, but they haven't been spotted together in a while.
Maddox, Brad and Angelina's oldest son, turned 21 on August 5. The birthday kid started his holiday celebrations on Friday, and the festivities will go through Sunday.
The star heir will spend the weekend with his mother and siblings; an insider told ET. What about dad, though?
As he is presently on tour promoting the new movie Faster than a Bullet, in which he co-starred, Brad Pitt is apparently unable to attend his son's birthday. But we're certain that the actor called the heir to congratulate him.
The insider also revealed that after the celebration, Maddox would assist his younger sister Zahara in moving to Atlanta.
Zahara will start attending Spelman College, a prestigious liberal arts school for black women, in the fall.
The source alleged that Brad "truly misses Maddox and was genuinely trying to rebuild their connection, but he has reached a block."
The reality is that Maddox still finds it tough to move past what transpired between them, the insider said, adding that Maddox "can't forget how his dad treated him back then."
"Maddux won't give anyone much time if they challenge his mother. For her, he would do anything, the insider claimed.
By the way, it appears that Brad Pitt is making an effort to stay in touch with the kids following their mother's divorce; for instance, he recently told reporters that Shiloh's dancing creates an impact on him.
Read this article At This Elder Son Maddox Chivan Jolie-Pitt Birthday Party, Brad Pitt Was Missing at Celebrityinsider.org
Glen Powell Almost Didn't Join The Cast Of Top Gun: Maverick But Tom Cruise Convinced Him - yuetu.info
Top Gun: Maverick has been one of the most successful films of this year grossing almost $1.24 billion dollars around the world. The film has even managed to tackle many other giants which released against it at the box office.
Top Gun: Maverick is the sequel to the 1986 film Top Gun which followed the story of hot shot fighter pilot Maverick played by Tom Cruise. The sequel of the film saw Tom Cruise reprising his role as Maverick who would be training a brand new class of pilots for a very dangerous mission. The new cast members included Miles Teller, Glen Powell, and Jay Ellis. Glen Powell played a character called Hangman in the film. Hangman was originally auditioned for the role of Rooster in the film which was a much more major character compared to Hangman but that role went to Miles Teller and Powell received Hangman's script.
Taking the role of Hangman to be a consolation prize, Powell decided to step away from the role as he didn't want a minor role in the film however, a quick conversation with Tom Cruise changed Powell's mind.
Christopher McQuarrie who will be directing Tom Cruise's upcoming Mission Impossible: Dead Reckoning Part 1 and 2, has revealed the entire interaction between Tom and Glen Powell on the Light The Fuse podcast recently. He described it in the following words:
"Tom said – and [Powell] was being very frank with Tom about not wanting to be in the movie – and Tom said, 'Glen, what kind of career do you want?' And Glen said, 'I want your career.' And Tom said, 'Well how do you think I got here Glen?' And Glen said, 'You chose great roles.' And [Tom] said, 'No Glen, I chose good movies.'"
It was this advice from Tom Cruise that got Glen to sign on to Top Gun: Maverick and sure enough Tom was right as Top Gun: Maverick not only became the highest-grossing movie of this year but also one of the highest-grossing movies of all time, giving Glen Powell's resume a sweet boost as his role as Hangman ended up being one of the most memorable things about the film
Read this article Glen Powell Almost Didn't Join The Cast Of Top Gun: Maverick But Tom Cruise Convinced Him at Celebrityinsider.org
Taylor Lautner Talks About Why He Took A Break Following The Twilight Saga - yuetu.info
Taylor Lautner was once one of the most popular boys on the planet. When he first appeared in the role of Jacob Black in the first Twilight film back in 2008, he immediately began to garner a huge fan base, especially among teenage girls. His appearance in the second film of the Twilight Saga, New Moon made things all the more intense as he did significant shirtless time in that installment. Taylor was only a teenager at the time that Twilight was being filmed and dealing with the reality of having screaming fans everywhere he went was not something he was taught.
Taylor now reflects on the time, recalling how it was completely uncharted territory for him and how his family helped him stay down to earth and in touch with reality during that time.
"When I was home, I still had to take out the trash," said Taylor while speaking to CNN in a recent interview, "I still had to mow the lawn."
Taylor said his parents told him, "We don't care that you're on the big screen. You still gotta do your chores."
Taylor said his parents helped him understand that he was more than just his popularity and that it was not an easy task for them to do that.
"My parents definitely just put it into perspective. I think even for them it was uncharted territory for parents that have a 16-year-old going through that," said Taylor.
And sure enough, it was important for Taylor to separate himself from his crazy popularity because after the Twilight Saga ended, Taylor saw a rapid decline in his popularity. Taylor described the phenomenon in the following words during the interview:
"When that's taken away from you at all, you start to question yourself and start to be like, 'Oh, do people not care about me anymore?' It goes away a little bit. You notice it, and that's the dangerous part, because that can really mess with your mind."
Hence, due to all these reasons, Taylor decided to take a break from acting and the entertainment industry for a while following the Twilight Saga, in order to get his priorities and perspective right before returning.
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Brad Pitt Talks About THAT Hilarious Deadpool 2 Cameo - yuetu.info
Ever since Ryan Reynolds first brought Deadpool to life in live-action cinema, it has become one of the most loved superhero cinematic endeavors of all time.
The first Deadpool live-action movie was released in 2016 after many years of efforts from Ryan Reynolds who personally loved the character. The film was a raging success with fans and critics alike and a sequel to the movie was immediately given the green light with a significantly bigger budget this time. Ryan's Deadpool even joked in the first movie that the significant lack of X-men characters was because the studio couldn't afford the actors. However, with a much bigger budget, the second film became ambitious and decided to introduce a team called the X-force.
X-force included Domino played by Zazie Beetz, Bedlam played by Terry Crews, Shatterstar played by Lewis Tan, Zeitgeist played by Bill Skarsgård, Peter played by Rob Delaney and Vanisher.
Vanisher had the power to disappear and remained powered up for most of the time he was on screen, so he didn't need an actor to play him, or so the fans thought.
When the entire X-force team that had been hyped for quite a while meets its end very quickly in a hilarious sequence, Vanisher dies by gliding into some power lines and getting electrocuted. While he is electrocuted, his face is revealed for a matter of seconds and the fans are shown that vanisher was being played by Brad Pitt.
Pitt was only on set for 1 day to shoot the 3-second scene where his character is electrocuted and in a recent interview while promoting Bullet Train Pitt has revealed why he agreed to sign on for such a tiny scene.
The reason involved David Leitch, who was Pitt's stunt double back in the early days of his career and later turned towards directing and was the director of Deadpool 2. David is also the director of Bullet Train. Pitt's exact statement is as follows:
"What was shooting that like? Pretty much, easiest thing I've ever done. Dave's an old friend of mine and he used to be… he was my stunt double starting with Fight Club and all the way up till about 2004. And then he went off and became a really good director, which is rare. Rare. Ryan called and like, why not?"
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Due To His Charitable Foundation, Leonardo Dicaprio Became Embroiled In A Controversy - yuetu.info
This year, there was a considerable increase in the cost of living in the UK. So, a lot of Britons had to drastically reduce their expenditures.
Some people thought the government would help them through this trying time, but the assistance from politicians was far more limited than they had hoped.
Perhaps this explains why Leonardo DiCaprio was condemned online for receiving funding from the British government.
The Department of the Environment gave the well-known actor's philanthropic organization more than £116,000 in the past month alone, according to the British tabloid Daily Mail.
It appears that the funds were utilized for safeguarding a pygmy buffalo species that is in danger of going extinct. Leo will reportedly receive a number of additional payments from the British government; it is also known. According to Department officials, "the money was allocated as part of a three-year grant."
Internet users are of the opinion that Leonardo DiCaprio, whose net worth surpasses 200 million pounds, ought to decline payments.
Leo was also reminded of an earlier scandal: the actor was charged with hypocrisy at the time for taking a private jet from Cannes to New York to accept an environmental award.
Remember, this is the same DiCaprio who spent $2.5 million on champagne for his 38th birthday celebration. Leonardo hasn't spoken on the subject yet.
Re: wild, the conservation organization he co-founded last year, has received four lump-cash gifts totaling the eyebrow-raising sum, despite the actor's 47-year-old net worth of over £200 million.
Defra gave the organization £28,800 in taxpayer money earlier this year to advocate for "rewilding." In the Philippines, the funds were used to safeguard a species of dwarf buffalo known as the tamaraw.
Additionally, a Defra representative informs me that this is just the latest in a series of payments that British taxpayers will make to the cause.
Read this article Due To His Charitable Foundation, Leonardo Dicaprio Became Embroiled In A Controversy at Celebrityinsider.org
Stormi Webster Was Carrying A £2,600 Dior Bag When Kylie Jenner And Her Daughter Arrived At A London Studio - yuetu.info
The daughter of Kylie Jenner almost certainly has the same expansive wardrobe as her mother, according to internet users. Looking at the baby's fashionable bows touches a lot of people.
The star parent has drawn criticism from some, though, for spending a fortune on dresses for her daughter that will be too small by the next season.
However, there are some items that you cannot grow into, starting with bags, of course. Kylie starts teaching her daughter to "invest" intelligently in it-bags at a very young age.
While walking with her daughter in London, Kylie was photographed by the paparazzi. The professional woman was dressed in a tight white shirt, baggy light blue pants, a large matching jacket, and pointed black shoes. She added a pair of large sunglasses to finish the look. Stormi resembled her mother in terms of style.
The little girl dressed in a beige midi skirt, a multicolored T-shirt, and yellow sneakers. The Dior purse hanging from her shoulder was the focal point of the outfit. A similar accessory is roughly $3,000 in price.
The reality star twirled her young daughter before stocking up on some of her items, including a few lip kits, after expressing her delight about their plans.
The two then indulged in afternoon tea, elegant finger sandwiches, and just-baked scones. The Keeping Up with the Kardashians actress then displayed a room filled with clothes, handbags, and shoes for her young daughter to try on.
Kylie remarked, "Look what Harrods did for Stormi to go shopping." Is this not the most absurd? You are a really privileged young lady. Kylie can, however, afford the most pricey presents.
She currently ranks among the wealthiest women in the US as a result of her lucrative business. Recently, it was revealed that the celebrity spends $300,000 every month.
Read this article Stormi Webster Was Carrying A £2,600 Dior Bag When Kylie Jenner And Her Daughter Arrived At A London Studio at Celebrityinsider.org
A Mortgage Was Entered Into By Adele In Order To Reside With Her Lover - yuetu.info
The well-known singer Adele has been dating sports agent Rich Paul for approximately a year. The star thrived in the presence of the new love, losing weight visibly and developing a radiant smile.
The couple's union is developing into something more serious. The singer has already moved in with her lover and publicly expressed her desire to have more children the diva already has a kid from a previous marriage.
The media learned that the celebrity had borrowed 37.7 million dollars back in January to pay for the Beverly Hills home she had purchased from Sylvester Stallone for 58 million. Paul recently moved in with her, and they now share the mansion.
The actor himself constructed the opulent house. He bought some land in 1994 and started from scratch on the house.
The Italian Riviera-inspired house has eight bedrooms, eleven bathrooms, a gym, a room dedicated to cigars, a bar, a swimming pool, a spa, an eight-car garage, and an art studio. The home has expansive views of the canyon and the city.
This home is just one of the several that Adele has recently purchased in the neighborhood. She has contributed to at least three more Beverly Hills houses valued at $30 million or more.
In the upcoming months, the celebrity will presumably pay off the loan. Her postponed concert tour, which will continue in November, is anticipated to bring in $48 million in revenue.
The first house Adele purchased in 2016 for $9.5 million, which features four bedrooms, six bathrooms, a pool, and a summerhouse, is located next door to Richie's former residence.
After divorcing her ex-husband Simon Konecki in April 2019, she paid $10.65 million for the six-bedroom house next door. He is supposedly staying there to be near her and their kid Angelo.
Read this article A Mortgage Was Entered Into By Adele In Order To Reside With Her Lover at Celebrityinsider.org
The Second Child Is Welcomed By Khloe Kardashian And Tristan Thompson - yuetu.info
The son of Khloe Kardashian and Tristan Thompson might be the ideal catalyst for their reunion! A surrogate mother helped the TV celebrity and the 31-year-old NBA player became parents, according to the couple's official representative. Despite their high-profile split in January, the happy news was made public last month.
"Chloe is sincerely appreciative of the exceptional surrogate mother for such a beautiful gift. However, to help Khloe concentrate on her family, we would like to request that fans respect our request for privacy.
Remember that Kardashian and Thompson started dating in 2016 and gave birth to their daughter Tru the following year. Due to the partner's repeated betrayals, the couple split up in June 2021. However, they later reconciled.
After talking about her difficulties conceiving a child, reality TV actress Chloe welcomed a new member of her family.
In March 2021, the star of Keeping Up with the Kardashians spoke out about the considerable danger she would experience during a potential pregnancy during the debut episode of the show's final season.
The couple's turbulent, on-again, off-again relationship has been dogged by multiple infidelity allegations. In addition, Thompson announced in January that he and fitness model Maralee Nichols had a son.
However, Kardashian, 38, has been outspoken about her desire to expand her family and her devotion to co-parenting with Thompson. In "Keeping Up with the Kardashians," she detailed her attempts to conceive a second child via IVF and fertility treatments.
With his oldest son, Prince, 5, whom he shares with his ex-girlfriend Jordan Craig, Thompson is now a father of four.
Khloe's pregnancy announcement coincides with Kim and Pete Davidson calling it quits. Even though they still "have a lot of love and respect for each other," a source told E! News that Kardashian and Davidson's busy schedules and geographical distance made it "very difficult to continue a relationship."
Read this article The Second Child Is Welcomed By Khloe Kardashian And Tristan Thompson at Celebrityinsider.org
After A 'Decade' Of Writing, Courtney Love Declares Her Memoir To Be Finished - yuetu.info
After ten years of "drag and drop," Courtney Love pleasantly shocked her audience on Friday by revealing that her memoir was now complete.
The singer congratulated her publisher Harper Collins and co-author Alex Abramovich in a statement announcing her new book, The Girl with the Most Cake.
"Dude(s). I may have just signed my book, I think. Then, following a decade of dragging my ass, Courtney posted on social media.
The publication date of the book is still a mystery, though. Love explained it like this: "Don't ask me when; the key thing is that everything is excellent right now," the speaker said.
"There is (no kidding) a very real supply chain." Courtney also made the decision to reveal what the memoir has in store for readers.
I think it's insane luck to be in the right location at the right moment (and sometimes wrong!) since it seems like I have 29 lifetimes, she added.
The rock star said that she made changes to the memoir in order to incorporate more intimate information about her mother and "unbelievably gorgeous grandma." "The truth is much more enjoyable, richer, and significant. Additionally, there are a lot of rags, said, Love.
In 2013, The Girl with the Biggest Cake was going to be released, but Courtney's plans fell through. She referred to the initiative as a failure in 2014, so supporters had to wait.
The release of "The Girl with the Most Cake" was scheduled for 2013, but Love's plans didn't work out. Instead, fans had to wait for the project's release after she labeled it a "disaster" in 2014.
Kurt Cobain, the frontman for Nirvana and Love, welcomed their daughter Frances Bean Cobain in 1992. The renowned musician passed away in April 1994.
Read this article After A 'Decade' Of Writing, Courtney Love Declares Her Memoir To Be Finished at Celebrityinsider.org
The Nine-Month Relationship Between Kim Kardashian And Pete Davidson Is Over - yuetu.info
A few months ago, Kim Kardashian and Pete Davidson began dating. The two had to put their relationship to the test because Kim's ex-husband, Kanye West, has frequently said that he will do all in his power to get his children's mother back.
They moved in together because they believe their relationship has potential, it was revealed two months ago.
The pair had wanted to take a romantic holiday for two during the second half of September, but those arrangements fell through, according to US Weekly. Insiders claim that the Saturday Night Live comic and the star of the television program "Keeping up with the Kardashian" have made the decision to part ways.
"They made the decision to part ways in peace and stay close. Long distances and hectic schedules make it challenging for them to maintain a relationship, but they still have A lot of love and respect for one another, according to a source close to the Kim family.
He claims that the former lovers split up this week. Fans of the star, however, are hesitant to accept this explanation for the split because, typically, it is used as a standard by couples who don't want to talk about their own issues in the media.
In February of last year, Kim Kardashian filed for divorce from rapper Kanye West.
Her relationship with comedian Pete Davidson came to light in November, and it later emerged that the socialite's family members were completely in agreement with her choice.
Caitlyn Jenner offered to clarify this at that time. She was among the first to meet Pete, as it turned out:
We spoke for a bit, and I came to the conclusion that he is unlike anyone Kim has previously attempted to connect with. Take a close look at her; she appears to be happier than ever.
Read this article The Nine-Month Relationship Between Kim Kardashian And Pete Davidson Is Over at Celebrityinsider.org
Russia Is Open To Discuss A Prisoner Swap Deal In The Case Of Brittney Griner - yuetu.info
WNBA player Brittney Griner has recently been served a 9-year sentence by a Russian court over the possession of cannabis with criminal intent. The development sparked outrage in America and President Joe Biden declared that bringing back the WNBA player to the United States is now their top priority. In a recent development, only a mere day after Griner's sentence was announced, Russian authorities have made it clear that they are open to discussing a prisoner swap with the United States.
Brittney Griner was traveling to Russia to play for the UMMC Ekaterinburg team during the WNBA off-season. It is a common occurrence for WNBA players to go overseas to play in order to make some extra money and often times they end up raking in way more than what they are paid in the WNBA.
On her most recent trip to Russia, Brittney was detained for the possession of Cannabis Oils. Brittney pleaded guilty when she appeared in court explaining that she did pack the oils but she did it accidentally, as they were prescribed to her for medically diagnosed chronic pain in America, where the use of the substance was legal. She clarified that she had no criminal intent. However, most experts agree that the trial was just for show and the court had already decided on its decision given the rising tensions between the US and Russia at the moment.
Now, Sergei Lavrov who is the Russian Foreign Minister has said that Russia is ready to discuss a prisoner swap. His exact words during a press conference in Cambodia are as follows:
"We are ready to discuss the issue [of a swap]."
However, Lavrov said that Russia would prefer private discussions between the two countries instead of America's usual, "megaphone diplomacy."
His exact statement over this was as follows:
"If this is another case of the Americans resorting to public diplomacy and loud statements on their pending steps, it's their business or I would even say their problem, because the Americans often fail to honour the agreement on doing calm, professional work."
At the moment, controversy continues and the fate of Brittney Griner hangs in the balance.
Read this article Russia Is Open To Discuss A Prisoner Swap Deal In The Case Of Brittney Griner at Celebrityinsider.org
Rumors Have Begun Spreading Regarding Kim Kardashian And Pete Davidson Breakup - yuetu.info
Kim Kardashian and Pete Davidson's relationship is something that shocked everyone at first but has since evolved into a dynamic that everyone adores. The couple first got together after Kim split with her ex-husband, Kanye West. West was thoroughly upset when it was first publically revealed that the comedian and the reality TV star are dating and it was partially Kanye's reaction to the relationship that was first responsible for getting the relationship all the attention. However, after a while, the relationship began evolving into something sweet and also serious as recently, Kim's children that she shares with West were seen going out with Pete Davidson and spending time with him. Kim herself has admitted that when she first approached Pete, it was just for a hookup but the dynamic between them naturally evolved in a serious direction.
However, while the sweet couple has been looking closer than ever recently, reports have begun flying around of the two stars breaking up this week. E! News and People have both reported that sources close to the stars have revealed that while Kim and Pete both hold a lot of regard for each other, they have come to agree that their work schedules are becoming way too complicated, and maintaining a long-distance relationship is not looking easy.
The news is a disheartening one as the fans were only just starting to root for the couple that was the center of a lot of controversy and judgment when they first got together. However, at the moment, this news is only rumors and there has been no official statement made by the representations of either celebrity.
Kim and Pete first got acquainted at the Met Gala in 2021 and then a month later Kim appeared on SNL to host the show. Kim performed with Pete on SNL and the two even shared a kiss in one of the scenes. Kim has shared the story of how she called the producer of SNL after the show and asked for Pete's number.
Read this article Rumors Have Begun Spreading Regarding Kim Kardashian And Pete Davidson Breakup at Celebrityinsider.org
Bella Hadid Played On The Beach With A Friend Devon Carlson While Displaying Her Toned Abs In A Pretty Scanty Swimsuit - yuetu.info
Bella Hadid, 25, takes full advantage of the summer's warm weather. The celebrity frequently spends her free time with her friends and family. The model and her friend Devon Carlson, therefore, got together the day before to unwind on the beach.
Bella posted enticing images on social media. They took pictures while wearing a hot blue bikini with strings. Hadid displayed a flawless physique with a toned midsection that she highlighted with a gold chain. But not everyone was a fan of the blunt swimsuit model. Her beach image was blasted online.
"The bikini looks very immodest" and "quite vulgar." Internet users summed up the comments as "worst bikini bottom," "She has a fantastic body, but this bikini is plain nasty," etc.
Bella Hadid's fashion sense is frequently debated. The celebrity occasionally makes stylish mistakes in addition to having effective photos. According to the media, she lost this time while wearing a bikini.
The sultry images were Victoria's Secret model and brunette bombshell behind-the-scenes selfies. With strappy black and metallic shoes and some large silver bangles, Bella lengthened her lean legs.
Her cell phone had a leopard-print case, matching the bold bedding.
She used a phrase from the late boxer Muhammad Ali in her caption: "Float like a butterfly, sting like a bee."
In a previous post, the singer was unceremoniously photographed by photographers while out for a stroll. The star wore a crop top to highlight her toned abs, but low-waisted Bermuda shorts and tall leather boots made her long legs appear shorter.
Even though Bella's routine bows frequently make people smile, this model's clothing had the opposite effect. The celebrity, according to online users, is the epitome of how not to dress in the hottest shorts this season. However, Hadid's recent fashion gaffe is by no means her first.
Read this article Bella Hadid Played On The Beach With A Friend Devon Carlson While Displaying Her Toned Abs In A Pretty Scanty Swimsuit at Celebrityinsider.org
Regarding Claims Of Unhygienic Circumstances, Kylie Jenner Has Reacted - yuetu.info
Kylie Jenner has reacted angrily to those who have criticized her most recent lab photographs. It's all about the photos from Kylie Cosmetics' production, where a watchful audience could see utterly unhygienic conditions.
In a post by makeup artist and cosmetics designer Kevin James Bennett, who chastised the celebrity for not using a hair cap, shoe covers, a mask, and nitrile gloves, the company's owner replied in the comments.
"Kevin, the factory was not where this shot was taken. Like any other celebrity or owner of a beauty firm, I would never disregard sanitary regulations, the TV personality wrote.
Kylie added that in the pictures released on Wednesday, she was on a personal account and that she agreed with the statement that such behavior would be wholly unacceptable.
"For content that wasn't near to mass production, I made my own samples and shot images. Nobody puts customers in danger! Kevin, you should feel ashamed for propagating misinformation, Jenner continued.
Jenner Bennett responded by claiming that the TV show's star had substituted themes and purposefully misled his viewers.
So you were looking at an expensive homogenizer that handled at least 50 liters of cosmetic stuff, Kevin said, catching Kylie in a lie (the product still covered the mixing blades).
However, is this not the manufacturing of cosmetics? Is that a secluded area? "the makeup artist asked. "Numerous brand owners take those photographs.
She's not acting improperly, and there is no way for us to determine how much of the formulation process she is involved in, one Instagram user commented.
A different person continued, "I've seen all kinds of circumstances in development laboratories, but exposed hair near a kettle in production would never happen. That's all.
In addition to launching Kylie Skin and Kylie Baby in 2019 and 2021, Jenner founded Kylie Cosmetics in 2014 and started selling items the following year.
Read this article Regarding Claims Of Unhygienic Circumstances, Kylie Jenner Has Reacted at Celebrityinsider.org
According To Rumors, Victoria Beckham And Nicola Peltz Are Inseparable And Engage In Constant Small-Talk - yuetu.info
With the epithet Ice Girl now more likely to apply to her, the Spice Girls member can comfortably choose a new alias.
Who would have guessed that following the recent declaration of love, VictoriaBeckham and her daughter-in-law are suddenly engaged in a cold war? As it turned out, the April texts regarding Nicola Peltz's adoption into the family were simply bogus, and the conflict between the women was brewing even prior to the wedding.
They don't communicate and can't stand each other. The insider recalls that the wedding preparations were appalling.
The daughter of New York millionaire Nelson Peltz did not meet Victoria at any point and did not want her prospective mother-in-law to be involved in preparing the event.
The 48-year-old Posh Spice was obviously unable to accept a new family member or appreciate the positive aspects of her young daughter-in-law due to her icy demeanor.
Now that everything has turned into a never-ending petty drama, mum has even stopped talking to her oldest son.
The British Tatler cover, which referred to Peltz as "the new Mrs. Beckham," provided the finishing touch. They haven't talked much in recent months, according to a source.
Perhaps the young actress experiences mother-daughter rivalry and is not pleased with the attention paid to her, especially on the day of her wedding.
We are hopeful that the adored women of the Beckham family will be able to reach a resolution.
Anyway! Brooklyn, Victoria, and Nicola have not commented on these allegations; nevertheless, it should be noted that Nicola and Posh Spice had previously appeared to be quite close. as when Victoria announced the engagement of Brooklyn and Nicola in 2020, saying,
"The news is SO wonderful! The engagement of @brooklynbeckham and @nicolaannepeltz makes us ecstatic! We all adore you both very much and send you our love and best wishes for a lifetime of joy."
Read this article According To Rumors, Victoria Beckham And Nicola Peltz Are Inseparable And Engage In Constant Small-Talk at Celebrityinsider.org
Kim Kardashian Is Thrilled With Her New Body Fat Percentage Following Her Weight Loss For The Marilyn Monroe Costume - yuetu.info
This spring, Kim Kardashian captured the attention of Internet users everywhere when she was able to don the fabled Marilyn Monroe "bare" outfit.
True, the reality television personality had to adhere to a strict diet for this. The celebrity said that dietary modifications caused a severe case of psoriasis.
Kim won't revert to his old ways, though, even after the Met Gala. She carried out a series of measures the other day and wanted to show them to online users. According to the examinations, Kardashian was able to lower her body fat from 25% in May 2021 to 18.8%.
In a year, Kim lost roughly 5 kg overall. However, this is a very respectable figure given the star's growth, which was 157 cm. To make sure I'm in shape and healthy, I measure my bone density, body fat percentage, and other things, Kim said.
I fall within the category of an athlete. Kardashian echoed her happiness. Fans are now speculating as to whether the reality TV star's transformation is connected to her relationship with Pete Davidson. Or perhaps Kim is just sick of showing off her famed curves.
Kardashian defended her position after receiving backlash for her comments about weight loss, stating she always treats the Met Gala like a part in a movie.
We are aware that actors like Christian Bale are infamous for drastically changing their weight when performing, and even Ana de Armas' portrayal of Marilyn Monroe in the upcoming NC-17 biography Blonde makes her completely unrecognizably different. But in the case of Kardashian, it really boils down to whether she's using her public identity as a prop on its own.
On their new reality series, The Kardashians, she and Kanye West allegedly played around with non-truths, according to her ex Ray J.
(He was primarily discussing the Season 1 resurfacing sex tape narrative.) When Season 2 of the infamous family's hit television show premieres on September 22 for Hulu subscribers, we'll probably learn more about what is and isn't true (as the family sees it).
Read this article Kim Kardashian Is Thrilled With Her New Body Fat Percentage Following Her Weight Loss For The Marilyn Monroe Costume at Celebrityinsider.org
Couple Benjamin Millepied And Natalie Portman Celebrate Their Tenth Wedding Anniversary - yuetu.info
Every day, Natalie Portman's affection for Benjamin Millepied deepens. This Thursday, the Thor: Love and Thunder actress and her devoted husband celebrated their 10th wedding anniversary.
She posted a touching photo from the event. The actress wrote, "Ten years today, and things are improving… "
Benjamin also made the decision to take advantage of the situation and tweeted a picture with an engagement ring emoji. The devoted husband also incorporated a passage from the band Barbara's French song Du Bout des Lèvres.
The dancer then revealed an image from behind the scenes of the 2010 movie Black Swan, where they first met.
The couple married on August 4, 2012, and they now have two children together, a 5-year-old daughter named Amalia and an 11-year-old son named Aleph. Parents still work together and spend time together at home. The Oscar winner acknowledges that she enjoys working with her spouse very much.
"I think he knows me so well, my dance talents and weaknesses, that he was actually able to choreograph really effortlessly and swiftly because we had very little time, and it was a lot of fun.
I don't often see him in the studio, so it was entertaining to watch him. It was very incredible just to experience his expertise and lightness, said Natalie.
Portman claimed to be dealing with "reverse parent guilt" and made a concerted effort to "impress" her kids with her professional accomplishments.
It's really uncommon for my kids to say, "Please go to work!" she said. It's usually quite the reverse,
According to the "No Strings Attached," actress, Aleph and Amalia discovering their own hobbies is her "dream," she told Us Weekly in an interview from October 2019.
At the time, Portman remarked, "For any youngster to have a passion for anything, know it, and be able to follow it so clearly, is the best thing that you could want."
Read this article Couple Benjamin Millepied And Natalie Portman Celebrate Their Tenth Wedding Anniversary at Celebrityinsider.org
Jessica Simpson Flaunted Her Trim Figure In A Green Camouflage T-Shirt And A Pair Of Torn Daisy Dukes Shorts - yuetu.info
Jessica Simpson gave birth to a large number of children in 2019. The star's third pregnancy went very poorly; she frequently lamented her ailing health and considerable weight gain.
This caused the celebrity's feet to swell to such an extent that even seasoned Internet users became alarmed when they saw footage of the artist's feet.
Jessica started her battle against obesity after delivering a baby, but she wasn't able to prevail until she sought the help of experts.
A nutrition system was created for her by the nutritionist, and a strategy for a gradual but long-term improvement was created by the trainer. At the time, Simpson remarked, "We immediately gave up severe diets and sweat-inducing exercise."
She was able to reduce roughly 45 kg as a result. And, to be completely honest, a mother with several kids today practically looks better than she did in the far-off zero.
The famous person just published a brand-new image on her blog in which she posed wearing denim shorts, a T-shirt, and a cap.
Thrilled Internet users reacted, saying things like, "Those legs!" and "You look amazing." Others added, "Jessica is perfect," "Your new shape suits you," "Okay, I need special training to get my legs like that," "Still can't get used to thinner Jessica," and "At first I didn't recognize her in the photo."
A little more than six months after giving birth to her third child, the former reality star announced in 2019 that she had shed 100 pounds.
With the assistance of well-known trainer Harley Pasternak, she followed a meal plan and clocked 14,000 steps per day. She has since given updates from the gym as she has maintained the weight loss over the years.
During a visit on the Today Show in 2021, Simpson said to Hoda Kotb that she no longer weighs herself, doesn't own a scale, and instead chooses her outfits based on how they make her feel.
Read this article Jessica Simpson Flaunted Her Trim Figure In A Green Camouflage T-Shirt And A Pair Of Torn Daisy Dukes Shorts at Celebrityinsider.org
Guest In Residence, Gigi Hadid's New Clothing Brand, Will Launch Soon - yuetu.info
Gigi Hadid made the decision to give the alternate side of modeling a shot. This Thursday, the celebrity revealed via social media that she is leaving the catwalk and entering a design studio to establish her own knitwear line, Guest in Residence.
Hadid, 27, captioned her images of herself putting in a lot of effort, "Working on something," while flaunting a variety of cashmere patterns and cuts, including a purple sweatshirt, a grey pajama set, and an orange cut-out jumper.
She also stated in the profile description that she now serves as the brand's creative director in addition to being a well-known model.
Fans can sign up for brand updates even though the collection's price range and release date have not yet been disclosed.
Hailey Bieber said that she was pleased about this, while Bella's sister backed Gigi, saying that this is what we have all been waiting for.
Gigi's friends and supporters have already shown their support for her in the comments. Ashley Park, a friend of Hadid's, even stated in a letter that she has always preferred Hadid's design work to her modeling work.
Additionally, Frankies Bikinis stated in the caption of an Instagram post announcing the partnership that the collection was "based in friendship and inspired by Gigi's carefree days at her family's gorgeous countryside house, filled with sentiment and nostalgia."
Aiello also lauded her friend's astute fashion sense in the press release, saying, "Gigi is someone that is so wonderful and so motivating to me.
Her amazing personality traits—creativity, hard work, kindness, and love combined with our strong friendship were what made this series so much fun to both develop and photograph.
We've known each other since before we ever believed dreams like this could come true, Hadid said, adding that this collection was made with "Friends and heart" and is a "unique full circle moment."
Read this article Guest In Residence, Gigi Hadid's New Clothing Brand, Will Launch Soon at Celebrityinsider.org
Kylie Jenner Obtains Comme Des Garçons Clothing - yuetu.info
Kylie Jenner hardly appeared in public after the birth of her second kid. Instead, she worked out for months to get back in shape.
The reality TV personality admitted in her blog that it was pretty challenging for her to shed pounds following her second pregnancy. However, she served as an example for countless girls.
But it seems that Kylie will now be far more likely to make us happy with their exits. She drew the attention of onlookers on August 4 twice.
And on both occasions, Jenner was dressed in tight clothes that highlighted her perfect physique and lean legs. The black miniskirt and ornamental "hands" on the pink top caught the attention of the cameras.
And in the late afternoon, Kylie dressed in a dangerously short, tight-fitting black dress that was too risky to even bend over. She accessorized her alluring look with matching heels and an emphasis on the eyes in her makeup.
Next to her boyfriend Travis Scott and daughter Stormi, the celebrity was beaming and appeared joyful. Jenner appeared at the production facility for her beauty brand and brazenly posed there without a cap or gloves, prompting Internet critics to accuse her of breaking hygienic rules.
Not all of the Kardashian-Jenners have worn unconventional clothing, including Jenner. Kim Kardashian stirred concerns in 2017 when she appeared in a curve-hugging skirt by Eckhaus Latta, prompting speculation about whether she would completely embrace underground labels.
(She's experimented, but she's mainly chosen huge Houses and well-known archival pieces.) However, due to the mini skirt version, Jenner has achieved the unthinkable and Kylie-fied Comme des Garçons.
It's not clear if it's a commercial rendition of the collection's drop-crotch pants or if they were altered for the runway. In any case, Jenner is handling Comme in her own manner.
Read this article Kylie Jenner Obtains Comme Des Garçons Clothing at Celebrityinsider.org
Conor McGregor And Jake Gyllenhaal Ring Up In Road House Sequel - yuetu.info
Former UFC champion and Irish mixed martial artist Conor Mcgregor will shortly make his big-screen acting debut. He will take on his first acting role in a streaming platform's remake of the 1989 action film Roadhouse.
An everyday city bouncer who works in a bar is compelled to confront actual criminals in his homeland in this narrative.
Conor will play a first-class character with a substantial part, albeit the character he will portray is still unknown.
Jake Gyllenhaal, though, has projected the primary part. Billy Magnussen, Lucas Gage, and Daniela Melshior are also said to be joining the cast.
The public's interest in following fights was stoked by McGregor's "performances" outside the ring, which served as a greater form of advertising than any other. But, on the other hand, McGregor is no stranger to the skill of acting. And the fighter's defiantly loose step before the performance became his signature.
The public's interest in following fights was stoked by McGregor's "performances" outside the ring, which served as a greater form of advertising than any other. On the other hand, McGregor is no stranger to the skill of acting.
And the fighter's defiantly loose step before the performance became his signature. The only thing that is currently known about McGregor's character is that he won't be appearing as a cameo version of himself but rather as an original character.
There has long been a lot of interest in casting him in a role on the big screen due to his brash demeanor and enormous appeal.
However, as noted by Deadline, McGregor has been hesitant to make the switch, stating that he would rather hold out on acting until the proper movie came along.
He watched the original Road House when it was first offered to him and leaped at the chance to be a part of the remake because of the new path it was going in under the supervision of director Doug Liman and writers Anthony Bagarozzi and Charles Mondry.
Read this article Conor McGregor And Jake Gyllenhaal Ring Up In Road House Sequel at Celebrityinsider.org
With An Italian Producer Andrea Iervolino Selena Gomez Stirred Romance Rumors - yuetu.info
Selena Gomez was going through a tough breakup with Justin Bieber. Therefore the celebrity did not dare to begin a new relationship.
The split with her boyfriend was a severe blow to Gomez, who spent a lot of time receiving depression treatment, regaining her mental health, and attempting to lead an everyday life.
Selena returned to work and an active social life when her heart wounds had healed, and she started to appear happier. She never did, however, meet her true love.
On the Internet, however, they began discussing the singer's decision to make changes in her personal life finally. As a result, she recently participated in a shoot with Italian director Andrea Iervolino. They enjoyed fun and relaxation on a yacht in Italy.
Gomez Iervolino and I have been friends for a long time. She starred in the movie And Lost the Fight, which he produced in 2016, and he went to her 27th birthday celebration in Rome in 2019.
In addition, they spent time together aboard a yacht in Los Angeles a year ago while on vacation.
Although there is little information available on Andrea's personal life, it would appear that he is unattached.
While presenting "Saturday Night Live" in May, the Grammy nominee provided insight into her single life.
She joked during her opening monologue, "One reason I'm thrilled to host 'SNL' is that I'm single — and I've heard 'SNL' is a terrific place to find romance." "I want to let the universe know that I'm generating love because I don't want to use dating apps. But right now, I'd accept anyone.
Gomez usually doesn't talk much about her romantic life. Until their ultimate breakup in 2018, she had a seven-year on-and-off relationship with Justin Bieber, who is now married to Hailey Bieber.
She also dated The Weeknd. However, their relationship ended in October 2017 after ten months. In addition, Gomez has been romantically linked to Zedd, Nick Jonas, and Niall Horan.
Read this article With An Italian Producer Andrea Iervolino Selena Gomez Stirred Romance Rumors at Celebrityinsider.org
In The Jokers Sequel, Lady Gaga Will portray Harley Quinn - yuetu.info
It has only been a few days since Variety informed "Joker" fans of the anticipated release date for the film's sequel. Still, other information about it is already starting to emerge.
Warner Bros. will release the movie in theatres on October 4, 2024. There have long been allegations on the network that the Hollywood actor hides the fact that he signed a deal for three paintings, but Joaquin Phoenix will still play the lead role in it, even though he was unsure about the release of the second half of the movie six months ago.
According to Variety journalists, the new "Joker," will be substantially different from the previous version; it will still be a drama, but this time it will also be a musical.
A reasonable argument can be made that the producers chose this since the first film's music, the creator of which, Hildur Gudnadouttir, won an Oscar, was one of its major selling points.
Joaquin Phoenix, who played the major role, was not left out either; he got a trophy and recognition from the Film Academy for "Best Actor."
There are no unique plot changes planned for the new segment, despite the fact that Todd Phillips' Joker ended up being significantly distinct from other comic book movies.
As a result, fans have been debating for a long time about which actress will play Harley Quinn, a psychiatrist who was intended to be the Joker's patient but instead fell in love with him and assisted in his escape from Arkham Asylum.
Fan forums started to speculate that Lady Gaga would be cast in her role as soon as the release of the new part was officially announced.
And the film's creators acted quickly to corroborate this rumor: They shared a little video and the first promotional shot, in which the American singer is only partially visible. And if comic book readers once believed.
Read this article In The Jokers Sequel, Lady Gaga Will portray Harley Quinn at Celebrityinsider.org
After Leaving 'House Of Cards,' Kevin Spacey Will Pay The Producers $31 Million - yuetu.info
After being removed from the Netflix series House of Cards due to claims of sexual harassment on the set, Kevin Spacey will pay the producers a sizable settlement.
Kevin Spacey, a 63-year-old actor, was ordered on Thursday by Los Angeles Superior Court Judge Mel Red Rekana to pay $29.5 million in damages as well as an extra $1.5 million in legal costs.
After repeated accusations of sexual assault, the House of Cards producers cut off their ties with Spacey in 2017.
After learning that Spacey was blatantly preying on children, including a series assistant producer, the group opened an investigation.
After breaking the company's sexual harassment policy, Spacey was subsequently told to make up the money to the studio in 2020.
At the time, Media Rights Capital asserted that Spacey had cost them millions in lost revenue because of his inappropriate behavior, which led to the show's cancellation and a five-episode reduction in the sixth season of House of Cards.
Later, Spacey's attorneys submitted a protest urging the court to reject the request. The opposition claimed in a statement that despite Spacey taking part in the culture that predominated on the set, which was full of sexual innuendos, jokes, and benign pranks, "he never harassed anyone."
Spacey actually stopped when he heard that his actions had made someone feel unwelcome or unwelcome in any other way.
After Netflix informed the petitioners that Spacey could not and would not be involved in Season 6, the lawyers said, "as the Court of Arbitration agreed, the episode decrease was a predetermined conclusion."
However, the judge disregarded the fact that Netflix was not even aware of the conduct he deemed to be a breach of contract at the time this decision was made.
In other words, the damages suffered by the plaintiffs could not be linked to the breaches identified by the court since they had already occurred by the time the violations were discovered.
Read this article After Leaving 'House Of Cards,' Kevin Spacey Will Pay The Producers $31 Million at Celebrityinsider.org
The Executive Producers Leonardo DiCaprio And Martin Scorsese Have Revealed That Keanu Reeves Will PlayTthe Lead In Devil In The White City - yuetu.info
The television show is produced by Martin Scorsese and Leonardo DiCaprio and is inspired by Eric Larson's nonfiction book from 2003.
Despite purchasing the picture rights in 2010, DiCaprio revealed the start of production 12 years later. The novel's events take place in Chicago between 1890 and 1895 against the backdrop of the World's Columbian Exposition of 1893, during the serial killer "Dr. Henry Howard Holmesactive "'s period.
Holmes is credited with creating the legend of the "murder castle," the residence where he lured his victims and carried out all of the horrific executions.
The exhibition itself and its founder, Daniel Birdham, an American architect and urban planner who is credited with giving Chicago and Washington their contemporary appearance, should be the second plot point of the series.
His World Exhibition has grown to be one of the biggest in human history. More than 25 million people visited the show in just six months, which is a staggering number for the late 19th century.
And Holmes, who is known as the first serial killer, was able to open his horrible hotel as a result of the rush of tourists.
One of Keanu Reeves' most significant television roles is expected to come from the series. In any event, there is still mystery surrounding the shooting beyond the cast: it is unknown if Keanu Reeves will portray an architect or a murderer.
Since January, Reeves has reportedly been in negotiations to star, so the fact that it's actually happening is incredibly gratifying.
It will be fascinating to see what the stalwart actor contributes to this historical drama possibly less Much Ado About Nothing and more Bram Stoker's Dracula.
The actor has done small bits of TV here and there, but nothing as significant as this. Something gloomy, like Constantine, but without the demons, is what we're speculating.
Read this article The Executive Producers Leonardo DiCaprio And Martin Scorsese Have Revealed That Keanu Reeves Will PlayTthe Lead In Devil In The White City at Celebrityinsider.org
Brittney Griner's Nine-Year Prison Sentence Has Angered Celebrities Who Are Now Protesting - yuetu.info
Stars like to mind their own business as much as the next guy. But when some kind of injustice happens, stars are the ones with the loudest voice because they understand that their voice can have the most impact.
WNBA Star Brittney Griner was recently sentenced to nine years in prison by a Russian judge. Her offense was bringing marijuana into the country and she was convicted on drug charges. She even tried to explain to the higher powers that she had a medical prescription in the United States and the marijuana had ended up in her bag by accident however she was still sentenced to nine years in prison.
Brittney herself admitted that it was an honest mistake but it seems like the Russian judge had no sympathy for her. There was a certain outrage arising in both her fans and celebrities around her. The day that she had been convicted was the day that charges had been brought against the police officers who had been a part of the raid in Kentucky, Louisville in which Breonna Taylor had been murdered.
The same day that the world had heard the great news of justice being served in one part of the world, they had to hear the terrible news of Brittney Griner in another part. Many celebrities spoke up about the injustice that Brittney was facing.
Including these celebrities was Justin Bieber who claimed that if anyone knew anything he could do to help, he was willing to put in the effort. Many stars spoke about how unfair the situation was to Brittney.
Viola Davis also joined saying that she had nothing to say about the matter because it had shocked her beyond compare. Even president Joe Biden put in a word, calling the sentencing unfair, and demanded the Russian state to release her as soon as possible. Joe Biden has made it his goal to bring American veteran Paul Whelan and the newly convicted Brittney both home from Russia.
Jada Pinkett Smith also chimed in that the situation broke her heart on her Instagram story and started a hashtag #FreeBrittney. The entire celebrity world has stood up for Brittney in her hard time and wishes that the Russian Federation would listen.
Read this article Brittney Griner's Nine-Year Prison Sentence Has Angered Celebrities Who Are Now Protesting at Celebrityinsider.org
Wendy Williams Claims She's Married But Her Representative Just Says She's In A New Relationship - yuetu.info
Stars sometimes live lives that their fans know nothing of. Commonly, stars like to keep their matters hidden however sometimes news gets out of something crazy, and the world is left shocked. That's exactly what's happening in the case of Wendy Williams.
In a recent conversation. Wendy Williams casually let slip that she was married. The person she was talking to reeled back in shock at her statement, asking her what she meant. Wendy had been pretty adamant in saying that she was married to a police officer.
Wendy seemed pretty happy with her "marriage" however her representative stated that she was not married. The representative went on to say that she was just in the beginning stages of a relationship and all of these feelings were her feeling like she was married.
The representative went on to say that Wendy needed to take the relationship day by day instead of just rushing into it. They said that they had no power to control her but they knew that she was not married and was just saying that to get attention.
The representative also said that Wendy's mindset was not one to jump into something. They said that she liked to take things slowly and day by day. Since the representative states that her relationship is new, he said that she was excited and probably said too much in the conversation that she had been having.
However, Wendy stuck by her statement of being married claiming that she did not care who knew about her recent marriage. She also said that her representative had been lying because he did not see a point in her getting married. She was adamant that she had gotten married and was willing to believe that sooner or later the truth would emerge.
This particular set of statements shocked her followers. While celebrities are known for their drama, no one expected the drama to be this severe. Fans do not know who to believe in the scenario but it seems as if Wendy is being more truthful here.
Read this article Wendy Williams Claims She's Married But Her Representative Just Says She's In A New Relationship at Celebrityinsider.org
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| 3,279
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Heavy Horses is the eleventh studio album by Jethro Tull, released on 10 April 1978. It is considered the second album in a trilogy of folk-rock albums by Jethro Tull, although folk music's influence is evident on a great number of Jethro Tull releases. The album abandons much of the folk lyrical content typical of the previous studio album, Songs from the Wood (1977), in exchange for a more realist perspective on the changing world. Likewise, the band sound is harder and tighter. The album reached No. 19 on the Billboard 200 album chart, and peaked at No. 20 on the UK Albums Chart. The third album in the folk-rock trilogy is Stormwatch (1979).
This album was the last studio album to feature John Glascock playing bass on all tracks.
All songs written and composed by Ian Anderson.
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| 8,199
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Teachings from the Islamic Mystical Tradition
Omid Safi
New Haven, CT:
Omid Safi's Radical Love: Teachings from the Islamic Mystical Tradition is a collection of his translations of Arabic and Persian literature from the Islamic mystical tradition (tasawwuf). Emphasizing the passionate aspect of love in Sufism, Safi reframes the Sufi path as "Radical Love." Cautioning against the common perception of love as a "flat, ubiquitous, and ironically, cheap" (xxi) emotion, he stresses the divine roots of love. Employing a hadith, Safi contends that love is, in fact, "God's yearning and desire to be known, ahbabtu …" (xxxiv), which gave birth to existence, creation, and the world.
Safi collects and translates a diverse range of ecstatic and sensual Sufi poems in his effort to disentangle the Western Academy's representation of Islam as a dry and legalistic religion. In this, Safi appears to be following the efforts of other scholars within the field, the most recent being Shahab Ahmed. Moreover—attentive to Sufism's frequent reception as spirituality devoid of its Islamic context—Safi rehabilitates it within the larger Islamic tradition by highlighting its roots in the Qur'ān and Hadith. For instance, quoting Rūmī, Safi argues that Rūmī's Mathnavī "repeatedly and emphatically unveils the beauty of the Qur'ān" (xxxvii).
Radical Love opens with a gripping narrative from Farīd-al-Dīn 'Attār's Memoirs of the Saints (tazkīrāt al-awliyā). This authorial decision immediately captures the reader's attention, and swiftly underscores who Sufis are and what motivates them—select individuals aspiring for intimacy with God. Explaining key Sufi terms, such as ihsān, imān, islām, and eshq, Safi then guides the reader through the biographical details of major Sufi figures such as Rūmī, Ahmad al-Ghazzālī, 'Iraqī, and 'Attār.
The book is further divided into four sections—God of Love, Path of Love, Lover and Beloved, and Beloved Community—that draw from several sources, such as the Qur'ān, Hadith, and Sufi prose and poetry. Although Radical Loveis dominated by pre-modern poetry, it also includes a few contemporary poems as well. One finds that while Kharaqānī's (d.1033), 'Attār's (d.1221), and Rūmī's (d.1273) poetry is amply translated, representation of other seminal figures—Ahmad Ghazzālī (d.1126), Ibn 'Arabī (d.1240), Amīr Khusrau (d.1325), and Hāfez (d.1390)—remains limited. However, the book is strengthened by Safi's inclusion, albeit brief, of the oral and performative tradition, rather than restricting himself to textual sources. For instance, he translates a fragment of a Qawwālī sung by the Sabri Brothers (37), a famous Qawwāl group from Pakistan.
Departing from the relatively literal translations of R.A. Nicholson, A.J. Arberry and Annemarie Schimmel, Safi renders the poetry in blank verse. In attempting a translation that is "evocative, fresh, accurate and poetic" (xxxvii), he experiments liberally with form, often playing with syntax and employing literary tools such as enjambment for rhetorical effect. For example, Safi chooses not to arrange ghazals in couplets (bayts) or maintain rhyme (qāfias) or refrain (radīfs) as some scholars such as Dick Davis have previously attempted. Instead, Safi frequently plucks single verses from a longer text and avoids punctuation at the end of the poem. These literary choices render the poetry more accessible to the reader.
Another notable aspect of Safi's method is his engagement with philosophical ideas through the use of "everyday" contemporary objects, as demonstrated by his translation of Rūmī's poem that Safi titles as "Love is the GPS" (128). Safi distills complex metaphysical ideas into easily understandable words and metaphors. For instance, in order to represent the idea of annihilation in God (fanā'), he translates a poem by Kharāqānī as "I shed my ego/as a snake discards/its old skin" (135). Furthermore, employing visually evocative language, Safi depicts the theme of longing for God through 'Attar's poem, "Frenzied Ocean of Love": "The ocean's commotion/is because/of yearning/for God/It is the fire of love/that whips/water/into frenzied wave" (136).
It is worth noting Safi's contention that the "path of Radical Love" (mazhab-e eshq/madhhab al-Ishq) is not named after a mystic or a scholar but is simply God's own path (xxii). For Safi, mazhab-e eshq stands in sharp contrast to schools of law (madhhab, pl. madhāhib), each of which were named after the respective Imam whose methodology was followed within the school (xxii). While this analogy may have heuristic benefits, it has the potential to mislead readers into assuming that the historical development of organized mysticism in Islam occurred in competition with the Islamic legal schools, as an alternative structure of authority. Moreover, the suggestion that Sufi orders (tarīqahs) did not revolve around charismatic authorities is belied by the historical formation of variousturuq (sing. tarīqah) (especially during the 13th and 14th centuries) for whom genealogical chains (sing. silsila) leading back to masters (pīrs or shaykhs) were important sources of legitimacy and identity—examples include the Suharwardīyyah order associated with Abu Najīb al-Suharwardī and the Qādiriyyah with 'Abd al Qādir al Jīlānī (Ira M. Lapidus, Islamic Societies to the Nineteenth Century: A Global History, Cambridge University Press, 2012)
Specialists may find Safi's choice of translating ecstatic poetry in order to cast Sufi path as "Radical Love" to have understated other aspects, such as the theological and political tensions that historically marked Sufism. Moreover, interested researchers may wish for more detailed references and be hindered by the omission of notes for the introduction. Despite these limitations, Radical Love is a fresh and welcome addition to currently available translations of Islamic mystical literature and amongst a handful of translations available to the general public that highlights the Islamic roots of Sufi poetry. While most appropriate for the general, non-specialist reader, the book could potentially appeal to undergraduate students as well. Unique in its form and method, it is a delightful read and beautiful foray into the Islamic mystical tradition.
Ilma Qureshi is a doctoral student in the Department of Religious Studies at the University of Virginia.
Omid Safi is Professor of Islamic Studies at Duke University and a columnist for On Being, is a frequent commentator on Islam. He has published numerous books, including Memories of Muhammad.
Muslim sacred texts
poetry, Sufism, Persia, divine love
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| 2,794
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Hong Kong's Ringo Lung on the unstoppable rise of Rugby League
Photo courtesy of Ringo Lung
Without question, 2018 has been something of an annus mirabilis for both for Hong Kong Rugby League (HKRL) and Hong Kong captain Ringo Lung.
Boasting achievements that include the first-ever Women's Battle of Origin and National representatives Hong Kong Thunder travelling to Sydney for the Emerging Nations World Cup, its growth in the city has been inexorable.
Nearly a year on from their historic first international test match against the Japan Samurais, Rugby League's remarkable journey evokes a sense of pride and joy for Lung.
"We rely on a core group of individuals who are fantastic but this comes with challenges just as any other amateur organisation – mainly time and money," remarked Lung.
"Having said that it's been amazing what we've achieved with the resources we have had in the past 5 years since HKRL begun."
When not in the thick of the action, the 27-year-old is part of an organisational team that has overseen the formation of multiple domestic leagues since 2015, which take place in the Rugby Union off-season.
Such leagues include the Hong Kong Rugby League 9s - a competition that has attracted participation from the UK, Papua New Guinea and Tonga's Development Squad.
Contrary to the perceived Union-League rivalry, the scrumhalf and many of his teammates have an insatiable appetite for rugby and can be found representing the likes of Valley across the regular Hong Kong Rugby Union divisions as well.
"I still play Rugby Union and most of the league players still play Union. At the same time I know a lot of Union guys who are big fans of League. I'm a big fan of both and I think there is a place for both sports." said Lung.
"Having said that, personally I like playing League better – there is more opportunity for attacking plays and big hits. Plus, the older I get the less keen I am on rucks. Just don't tell my coach."
Not that Lung is one to shy away from responsibility; the scrumhalf was named Hong Kong's first-ever Rugby League Captain for their international test against Japan and the Emerging Nations World Cup in Sydney - an experience he describes as both "a tremendous honour and a huge responsibility".
"It was a very surreal moment to be given the role and still sometimes I have to pinch myself as it is something you dream about as a kid," added Lung.
"Ultimately I'm most proud when the team wins or performs well – especially when we got our first-ever victory in Tokyo against Japan last year and anything I can do to help us get there I'm happy to, whether that's as a Captain or not."
Following a roller-coaster year that has seen Hong Kong Thunder take part in 6 international games in the space of 12 months, Lung believes that this year will see the HKRL turn inwards and dedicate resources towards building up the domestic side of things.
This means plenty of opportunities to don Hong Kong's colours, with the League aiming to recruit new talent to replace several departing players - the latest phase in what he believes will be an upward trajectory for HKRL.
"We want to be in a position where we have regular rugby league available to play for all in Hong Kong – for men, women and at a youth level – as well as competing regularly on the international stage," he explained.
"It's ambitious but we've managed to grow very quickly over a short space of time, if we can consolidate the growth we've had and find the right investment then the product is fantastic and very exciting."
The 2019 Hong Kong Rugby League 9s and HKTag are set to take place on 4th May (Saturday) at King's Park. For more details, visit www.hongkongrugbyleague.com.
Hong Kong Rugby League
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{"url":"http:\/\/nrich.maths.org\/public\/leg.php?code=-68&cl=3&cldcmpid=559","text":"Search by Topic\n\nResources tagged with Visualising similar to Dozens:\n\nFilter by: Content type:\nStage:\nChallenge level:\n\nCuboids\n\nStage: 3 Challenge Level:\n\nFind a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?\n\nFrogs\n\nStage: 3 Challenge Level:\n\nHow many moves does it take to swap over some red and blue frogs? Do you have a method?\n\nOn the Edge\n\nStage: 3 Challenge Level:\n\nHere are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .\n\nTic Tac Toe\n\nStage: 3 Challenge Level:\n\nIn the game of Noughts and Crosses there are 8 distinct winning lines. 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What is the sum of 1 + 3 + ..... + 149 + 151 + 153?\n\nChess\n\nStage: 3 Challenge Level:\n\nWhat would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?\n\nTourism\n\nStage: 3 Challenge Level:\n\nIf you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.\n\nMasterclass Ideas: Visualising\n\nStage: 2 and 3 Challenge Level:\n\nA package contains a set of resources designed to develop pupils' mathematical thinking. This package places a particular emphasis on \u201cvisualising\u201d and is designed to meet the needs. . . .\n\nEight Hidden Squares\n\nStage: 2 and 3 Challenge Level:\n\nOn the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?\n\nMystic Rose\n\nStage: 3 Challenge Level:\n\nUse the animation to help you work out how many lines are needed to draw mystic roses of different sizes.\n\nFence It\n\nStage: 3 Challenge Level:\n\nIf you have only 40 metres of fencing available, what is the maximum area of land you can fence off?\n\nStage: 3 Challenge Level:\n\nCan you mark 4 points on a flat surface so that there are only two different distances between them?\n\nChristmas Chocolates\n\nStage: 3 Challenge Level:\n\nHow could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?\n\nSquares in Rectangles\n\nStage: 3 Challenge Level:\n\nA 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?\n\nCubes Within Cubes Revisited\n\nStage: 3 Challenge Level:\n\nImagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?\n\nConvex Polygons\n\nStage: 3 Challenge Level:\n\nShow that among the interior angles of a convex polygon there cannot be more than three acute angles.\n\nSquares, Squares and More Squares\n\nStage: 3 Challenge Level:\n\nCan you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?\n\nThreesomes\n\nStage: 3 Challenge Level:\n\nImagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it\/is it not possible to draw?\n\n3D Stacks\n\nStage: 2 and 3 Challenge Level:\n\nCan you find a way of representing these arrangements of balls?\n\nSteel Cables\n\nStage: 4 Challenge Level:\n\nSome students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?\n\nZooming in on the Squares\n\nStage: 2 and 3\n\nStart with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?\n\nIsosceles Triangles\n\nStage: 3 Challenge Level:\n\nDraw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?\n\nSea Defences\n\nStage: 2 and 3 Challenge Level:\n\nThese are pictures of the sea defences at New Brighton. 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