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Mamurio Veturio è un personaggio semi-mitico della storia di Roma, probabilmente di origine sabina ed appartenente all'antichissima Gens Veturia.
Secondo la tradizione Numa Pompilio ricevette dal dio Marte l'Ancile, uno scudo sacro disceso dal cielo. Re Numa turbato da quel prodigio chiese consiglio, come spesso faceva, alla ninfa Egeria, che gli spiegò che il dono del dio era molto prezioso, perché costituiva il pegno dell'eterna invincibilità di Roma, finché fosse rimasto presso l'Urbe. Allora Numa per impedire che potesse essere trafugato, chiamò il valido fabbro Mamurio Veturio, nel quale riponeva grande stima e fiducia, e gli affidò l'Ancile, affinché ne forgiasse undici copie identiche. Una volta conclusa la sua fatica, Mamurio consegnò tutti e dodici gli scudi a Numa Pompilio, che li affidò in custodia ad un collegio di altrettanti sacerdoti scelti fra i membri delle gentes originarie, le più antiche e nobili famiglie di Roma.
Venne così istituito il prestigioso collegio dei Salii, che nei mesi di marzo e di ottobre, sacri al dio Marte, portavano solennemente in processione i dodici scudi sacri, saltando (da cui il nome, dal verbo latino salire) ed intonando un canto particolare, il Carmen Saliare, del quale ci sono pervenuti alcuni frammenti. Numa Pompilio voleva ricompensare Mamurio per il suo ottimo lavoro, ma il buon artefice non volle essere pagato in denaro; chiese però di essere ricordato dal popolo Romano, e Numa lo accontentò, disponendo che i Salii lo invocassero nel loro canto, inneggiando anche a Mamurio.
Mamurio Veturio venne ricordato anche in altri modi dai Romani: in suo onore la festa del 14 marzo, detta degli Equirria e corrispondente al nostro capodanno, venne chiamata Mamuralia. In quella festa Mamurio Veturio, impersonato da un vecchio vestito di pelli rappresentava l'anno vecchio, e veniva scacciato tra grandi risate dai bambini con piccole verghe, per far posto all'anno nuovo.
Note
Bibliografia
Voci correlate
Gens Veturia
Gentes originarie
Numa Pompilio
Salii
Collegamenti esterni
Personaggi della mitologia romana
Fabbri immaginari
Mamurio | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,920 |
\section{Introduction}
Run one at LHC discovered a Higgs-like boson, and
beyond the Standard Model (BSM) particles where not discovered, for masses $\lesssim 1 \, { \rm TeV}$.
This has led to interest in effective field theory (EFT)
approaches to Standard Model (SM) processes.
In this paper we discuss a subtlety that is present when constraining higher dimensional operators in an EFT, using an operator basis reduced by the Equations of Motion (EoM).
We will illustrate this point with the Standard Model effective field theory (SMEFT), which assumes that $\rm SU(2) \times U(1)_Y$ is linearly realized in the scalar sector, and that this symmetry is spontaneously broken by the SM Higgs. The dimension six operators are suppressed by $1/\Lambda^2$. LHC results indicate
$\Lambda \gg v= 246\,\text{GeV}$, which provides a straightforward EFT expansion. The minimal classification of higher dimensional operators for this theory was given in
Ref.~\cite{Grzadkowski:2010es}, which further reduced the operator basis of a previous classification \cite{Buchmuller:1985jz}, by the classical EoM for the SM fields. Although the reduction of the basis is a useful step, when considering experimental constraints on the reduced basis, subtleties can appear.
Here we discuss one such subtlety. $S$ matrix elements correspond to physical quantities, but Wilson coefficients in a Lagrangian can be unphysical.
The EoM relate different operators, with completely different field content
in some cases, and yet $S$ matrix elements are unchanged by the EOM. One can remove an
operator entirely from a basis, and yet the physical effects present in the theory are not changed,
as $S$ matrix elements are not changed by the EOM. In this manner, the invariance of field theories under field redefinitions \cite{Weinberg:1968de,Callan:1969sn,Politzer:1980me}
shows that an operator basis is unphysical.
At the same time, when constraining the SMEFT at the Lagrangian level, there is a conservation of
constraints in changing basis.
The subtlety discussed in this paper corresponds to the case when observables are constructed from the data to determine such constraints, with a series of assumptions imposed about the nature of possible deviations in the SMEFT, i.e under the assumptions that certain parts of Feynman diagrams are as in the SM. These defining conditions can introduce subtle constraints onto
the field theory.\footnote{We avoid the use of the phrase pseudo-observable in this paper, due to its various historical definitions in the literature, see Ref \cite{Gonzalez-Alonso:2014eva} for a recent comprehensive discussion on pseudo-observables in the SMEFT.}The theory can be properly constrained
if the defining conditions of the observables are incorporated in a basis independent manner, in conjunction
with the constructed experimental bound. Failing to do so
can lead to a {\it functionally redundant} operator basis, in that the number of parameters present in the Lagrangian is inconsistent with the assumptions required
to incorporate the bound from a constructed observable.\footnote{It is important to distinguish the standard definition of operator redundancy, where a basis is being used that is not reduced fully with the EoM,
from a functional redundancy. The latter can still occur in a fully reduced basis if constructed observables are used in an inconsistent manner. The confusion
that results from either redundancy is the same.} A concrete example of a functional redundancy is given in Fig 1.
\begin{figure}
\begin{tikzpicture}[
decoration={
markings,
mark=at position 0.55 with {\arrow[scale=1.5]{stealth'}};
}]
\draw[postaction=decorate] (0,0) --(1,1) ;
\draw[postaction=decorate] (1,1) --(2,0) ;
\filldraw (1,1) circle (0.075);
\draw[decorate,decoration={snake}] (1,1) -- node [right] {$W^{\pm}$, Z} (1,2.5);
\filldraw (1,2.5) circle (0.075);
\draw[postaction=decorate] (1,2.5) -- (2,3.5) ;
\draw[postaction=decorate] (0,3.5) -- (1,2.5) ;
\end{tikzpicture}
\hspace{0.5cm}
\begin{tikzpicture}[
decoration={
markings,
mark=at position 0.55 with {\arrow[scale=1.5]{stealth'}};
}]
\draw[postaction=decorate] (0,0) --(1,1) ;
\draw[postaction=decorate] (1,1) -- node [left] {$\nu$}(1,2.5) ;
\draw[postaction=decorate] (1,2.5) --(0,3.5) ;
\draw[decorate,decoration={snake}] (1,1) -- node [below] {$W^+$} (3,1);
\filldraw (1,1) circle (0.075);
\filldraw (1,2.5) circle (0.075);
\draw[decorate,decoration={snake}] (1,2.5) -- node [above] {$W^-$} (3,2.5);
\end{tikzpicture}
\hspace{0.5cm}
\begin{tikzpicture}[
decoration={
markings,
mark=at position 0.55 with {\arrow[scale=1.5]{stealth'}};
}]
\draw[postaction=decorate] (0,0) --(1,1.75) ;
\draw[postaction=decorate] (1,1.75) --(0,3.5) ;
\filldraw (1,1.75) circle (0.075);
\draw[decorate,decoration={snake}] (1,1.75) -- node [below] {$Z^\star$} (3,1.75)
node [rectangle,draw=black,fill=black] {};
\draw[decorate,decoration={snake}] (3,1.75) -- node [left] {W} (4,3.5);
\draw[decorate,decoration={snake}] (3,1.75) -- node [left] {W} (4,0);
\end{tikzpicture}
\hspace{0.5cm}
\begin{tikzpicture}[
decoration={
markings,
mark=at position 0.55 with {\arrow[scale=1.5]{stealth'}};
}]
\draw [dashed] (0,2.4) --(1.75, 2.4) ;
\draw[decorate,decoration={snake}] (1.75, 2.4) -- node [below] {$W^\star$} (2.75, 2);
\draw[decorate,decoration={snake}] (1.75, 2.4) -- node [right] {$W$} (2.75,4);
\draw[postaction=decorate] (2.75, 2) --(3.75,2.75) ;
\draw[postaction=decorate] (3.75,0.75) --(2.75, 2) ;
\filldraw (2.75, 2) circle (0.075);
\end{tikzpicture}
\caption{\label{fig:1}
An example of a functional redundancy. An operator basis can be chosen that maps parameters characterizing
differences in the coupling of the $W$ and $Z$ to leptons (compared to the SM) into another sector of the field theory, where these parameters contribute to an anomalous TGC vertex.
(Parameters can be mapped from the dot in the diagrams above to the box with the EoM.)
Subsequently, using a TGC vertex bound, naively constrains these parameters in the SMEFT. Some of the parameters apparently constrained in this manner
are functionally redundant, as in the middle two diagrams the production and decay of the $W,Z$ is simultaneously assumed to be SM like.
(When experimental bounds are constructed on the parameters in the box, the dot is assumed to be SM like.)
This procedure is inconsistent and does not constrain a flat direction due to LEP $Z$ pole data that can modify $h \rightarrow V \mathcal{F}$ decay,
when $V = W$. Unphysical field redefinitions, or an operator basis choice, do not make this procedure consistent.}
\end{figure}
We illustrate the basic issues involved in Section \ref{defn}.
Constraints due to LEP data on the SMEFT are discussed in Section \ref{LEPdata}.
The impact of the defining conditions for the oblique electroweak precision data (EWPD) $\rm S$ parameter, and the off shell TGC verticies are discussed in Section \ref{pseudoobservablesection}.
We then show that reporting the relationship between the differential spectra in $h \rightarrow V \bar{f} \, f$ decay\footnote{In this paper, we will consistently use the notation $V$ for $W,Z$, a general massive gauge boson.}
and off shell TGC verticies has a potential basis dependence due to this issue, and how to resolve this problem
by taking into account constraints of this form in Section \ref{hvfspec}. We find that in the limit of strong constraints from LEP data (we define this limit precisely below), off shell TGC verticies are not related to $h \rightarrow V \bar{f} \, f$ decay spectra.
Our results make clear that data analyses can benefit from using (at least) two bases, with careful attention paid to the EoM mapping between them.\footnote{One might consider
it even more ideal to have no basis at all. However, combining constraints from multiple scales, and correlating such information with future higher energy measurements
requires the machinery of perturbative corrections.}
The subtlety discussed here is relevant to future efforts to obtain more precise constraints, from more complex final state studies at LHC.
In analyzing such processes, constructed observables will be extracted if simplifying assumptions that do not generate Ward identities are made about the nature of possible deviations from the SM.
\section{Operator relations due to the EoM}\label{defn}
We adopt notation for the linear SMEFT consistent with Ref \cite{Grojean:2013kd,Jenkins:2013zja,Jenkins:2013wua,Alonso:2013hga,Alonso:2014rga,Alonso:2014zka}.(With the shorthand $\bar{s}_\theta = \sin \bar{\theta},\bar{c}_\theta = \cos \bar{\theta}$. The notation is also summarized in the Appendix.)
The Lagrangian $ \mathcal{L}^{(6)} = \sum_i C_i \, Q_i$
consists of all dimension six operators that can be constructed preserving $\rm SU(3)_C \times SU(2)_L \times U(1)_Y$ (linearly), and assuming the conservation of baryon and lepton number.
Taking into account flavour indicies, there are $2499$ parameters to constrain in $ \mathcal{L}^{(6)}$, as shown in Ref.\cite{Alonso:2013hga}. Despite this large number, the EoM have been used extensively to reduce the number of parameters to a minimal set. The SM EoM are summarized in the Appendix.
It is well known that a choice of operator basis is arbitrary and cannot effect a physical conclusion, such as how strongly constrained an EFT is by an experimental measurement.
Considering the EoM makes clear the requirement of thinking of a Wilson coefficient as an ensemble parameter that can obtain experimental constraints from all possible measurements that can constrain the parameter in any basis.(So long as measurements are not reused.) The EoM can also make clear the consequences of defining conditions for constructed observables.
Careful use of the EoM is the easiest way to avoid a functional redundancy.
A simple example of the ensemble nature of the Wilson coefficient, and how the EoM can be useful, is afforded with the dimension six operator
\begin{align}
E_{H \Box} &= [H^\dagger H] [H^\dagger (D^2 H) + (D^2 H^\dagger) H],
\label{ebox}
\end{align}
this operator can be converted via Eqn \ref{eomHiggs} to
\begin{align}
\widetilde E_{H \Box} & = 2\lambda v^2 (H^\dagger H)^2 -4 \lambda Q_H -\left( [Y_u^\dagger]_{rs}\, Q_{\substack{u H \\ rs }} + [Y_d^\dagger]_{rs} \, Q_{\substack{d H \\ rs }}+ [Y_e^\dagger]_{rs}\, Q_{\substack{eH \\ rs }}+ \hbox{h.c.} \right).
\label{ebox2}
\end{align}
Note here the introduction of the operators $Q_{\substack{u H \\ rs }}$ etc, which are matricies in flavour space in general, with flavour indicies $r,s$ contracted with the SM
Yukawa matricies. The SM Yukawa matricies are defined in the Appendix.
Define $\mathcal{R} = \mathcal{L}^{(6)} + C_{E \Box} \, E_{H \Box}$ and
applying Eqn \ref{ebox2} to reduce $\mathcal{R}$ to $ \mathcal{L}^{(6)}$ gives the following parameter redefinitions at a chosen scale
\begin{align}
C'_H & = C_H - 4 \, \lambda C_{E\Box}, &
C'_{\substack{u H \\ rs }} & = C_{\substack{u H \\ rs }} - C_{E\Box} \, [Y_u^\dagger]_{rs}, \nonumber \\
C'_{\substack{d H \\ rs }} & = C_{\substack{d H \\ rs }} - C_{E\Box} \, [Y_d^\dagger]_{rs}, &
C'_{\substack{e H \\ rs }} & = C_{\substack{e H \\ rs }} - C_{E\Box} \, [Y_e^\dagger]_{rs}, \nonumber \\
\lambda' & = \lambda + 2 \lambda v^2\, C_{E\Box}.
\label{parametershift}
\end{align}
The hermetian conjugate Wilson coefficients of $C_{\substack{u H}},C_{\substack{d H}},C_{\substack{e H}}$ are similarly
shifted. Now consider two bases. In the first, one choses to remove $Q_H$ in favour of $E_{H \Box}$, in the second one choses to remove $E_{H \Box}$ in favour of $Q_H$.
The Wilson coefficients are identified when changing basis in this case:
$C_{E\Box} \equiv - 4 \, \lambda C_H$.
The same parameter in the field theory can obtain direct constraints from
measurements that constrain $C_{H \Box}$ in basis one, and $C_H$ in basis two, even though the field content present in the operators differ.
The constraints obtained in the two bases are related by the EoM, and the strongest constraint is relevant for the optimal basis independent bound on the EFT.
A functional redundancy would be present if the parameter $C_{E\Box}$ is retained, while {\it simultaneously} a constructed observable
was used to constrain the field theory that assumed $C_H = 0$.
This point holds for more complicated basis changes. Two bases of operators are of interest in the following sections, the basis of Ref.~\cite{Grzadkowski:2010es}, and the basis used in Ref.~\cite{Pomarol:2013zra}.
The former will also be referred to as the standard basis.\footnote{We emphasize that any basis is allowed, and no basis is superior to any other, if calculations are performed correctly
and functional redundancy is avoided. We refer to the standard basis here as
this basis was the first dimension six operator basis fully reduced by the SM EoM.} We will denote the operators in the later
basis with $\mathcal{O}$ labels to avoid confusion. Define the Wilson coefficients to be
\begin{align}\label{lagrangianequivalence}
\mathcal{L}^{(6)} &= \sum_i C_i \, Q_i = \sum_i \, \mathcal{P}_i \, \mathcal{O}_i.
\end{align}
$C_i$ and $\mathcal{P}_i$ have mass dimension $- 2$.
The operators that are present in the $Q_i$ but not the $\mathcal{O}_i$ are given
by
\begin{align}
Q_{HW} &= H^\dagger H \, W^{I}_{\mu\nu} \, W_{I}^{\mu\nu}, &
Q_{\substack{H\ell \\pr}}^{(1)} &= (H^\dagger \, i\overleftrightarrow D_\mu H) \, \bar{\ell}_p \, \gamma^\mu \, \ell_r, &
Q_{HWB} &= H^\dagger \, \tau_I \, H \, W^{I}_{\mu\nu} \, B^{\mu\nu}, \nonumber \\
Q_{\substack{H\ell \\pr}}^{(3)} &= (H^\dagger \, i\overleftrightarrow D^I_\mu H) \, \bar{\ell}_p \, \tau^I \, \gamma^\mu \, \ell_r, &
Q_{HD} &= (H^\dagger D^\mu H)^\star \, (H^\dagger D_\mu H).
\label{standardops}
\end{align}
The operators that are present in $\mathcal{O}_i$ and not in the $Q_i$ are given by
\begin{align}
\mathcal{O}_{HW} &= -i \, g_2 \, (D^\mu H)^\dagger \, \tau^I \, (D^\nu H) \, W^I_{\mu \, \nu}, & \hspace{1cm}
\mathcal{O}_{HB} &= -i \, g_1 \, (D^\mu H)^\dagger \, (D^\nu H) \, B_{\mu \, \nu},\nonumber \\
\mathcal{O}_{W} &= -\frac{i \, g_2}{2} \, (H^\dagger \, \overleftrightarrow D^I_\mu H) \, (D^\nu W^I_{\mu \, \nu}), & \hspace{1cm}
\mathcal{O}_{B} &= -\frac{i \, g_1}{2} \, (H^\dagger \, \overleftrightarrow{D}^\mu H) \, (D^\nu B_{\mu \, \nu}), \nonumber \\
\mathcal{O}_{T} &= (H^\dagger \, \overleftrightarrow{D}^\mu H) \, (H^\dagger \, \overleftrightarrow{D}^\mu H).
\label{SILHops}
\end{align}
The relevant relationships between the operators in these basis are completely given in Ref. \cite{Alonso:2013hga} (see Appendix B).\footnote{
Operator relations of this form were partially discussed in Refs \cite{SanchezColon:1998xg,Kilian:2003xt,
Grojean:2006nn} and many other works previously.} The transformation from the standard basis
to the $\mathcal{O}_i$ basis is derived using the SM EoM\footnote{In these relations only the flavour singlet component of the operators appears. This is indicated with the notation $Q_{\substack{H d \\ rr}}$ for example,
which explicitly corresponds to an operator that is proportional to a unit matrix in flavour space.
}, and found to be
\begin{align}
g_1 \, g_2 \, Q_{HWB} &= 4 \, \mathcal{O}_{B} - 4 \, \mathcal{O}_{HB} - 2 \, \mathsf{y}_H \, g_1^2 \, Q_{HB}, \nonumber \\
g_2^2 \, Q_{HW} &= 4 \, \mathcal{O}_{W} - 4 \, \mathcal{O}_{B} - 4 \, \mathcal{O}_{HW} + 4 \, \mathcal{O}_{HB} + 2 \, \mathsf{y}_H \, g_1^2 \, Q_{HB}, \nonumber \\
g_1^2 \, \mathsf{y}_\ell \, Q_{\substack{H l \\ tt}}^{(1)} &= 2 \, \mathcal{O}_{B} + \mathsf{y}_H \, g_1^2\, \mathcal{O}_T -
g_1^2\left[\mathsf{y}_e Q_{\substack{H e \\ rr}} + \mathsf{y}_q Q_{\substack{H q \\ rr}}^{(1)}+\mathsf{y}_u Q_{\substack{H u \\ rr}}+ \mathsf{y}_d Q_{\substack{H d \\ rr}} \right], \nonumber \\
g_2^2 \, Q_{\substack{H l \\ tt}}^{(3)} &= 4 \, \mathcal{O}_{W} - 3 \, g_2^2 \, Q_{H \Box} + 2 \, g_2^2 m_h^2 \, (H^\dagger \, H)^2 - 8 \, g_2^2 \, \lambda \, Q_H - g_2^2 \, Q_{H q}^{(3)}, \nonumber \\
&- 2 \, g_2^2\left( [Y_u^\dagger]_{rr} Q_{\substack{ uH \\ rr}} + [Y_d^\dagger]_{rr} Q_{\substack{dH \\ rr}}+ [Y_e^\dagger]_{rr} Q_{\substack{eH \\ rr}}+h.c. \right).
\label{inversetransform1}
\end{align}
Some parameters are only redefined changing basis,
and a constraint is lost in the arbitrariness of redefining parameters.
This is not always the case. Considering the case of interest, we find the mapping
\begin{align}\label{mappingparameters}
\mathcal{P}_B &\rightarrow \frac{4}{g_1 \, g_2} C_{HWB} - \frac{4}{g_2^2} \, C_{HW} + \frac{2}{g_1^2 \, \mathsf{y}_\ell} \, C_{\substack{H \ell \\tt}}^{(1)}, \quad \quad \quad \quad \mathcal{P}_W \rightarrow \frac{4}{g_2^2} \, C_{HW} + \frac{4}{g_2^2} \, \, C_{\substack{H \ell \\ tt}}^{(3)}, \nonumber \\
\mathcal{P}_{HB} &\rightarrow -\frac{4}{g_1 \ g_2} \, C_{HWB} + \frac{4}{g_2^2} \, \, C_{HW}, \hspace{2.85cm}
\mathcal{P}_{HW} \rightarrow -\frac{4}{g_2^2} \, C_{HW}.
\end{align}
This mapping is obtained by using Eqn \ref{inversetransform1} in Eqn \ref{lagrangianequivalence}.
These parameters in the $\mathcal{O}$ basis are identified with alternate parameters in the standard basis.\footnote{This is true at a fixed scale, when the RGE evolution of the theory is taken
into account, this relationship will be weakened by loop corrections.} The choice that has been made in constructing this basis is to remove operators directly related to $V$ decay and phenomenology, and to map possible differences in $Z$ and $W$ couplings to leptons in the SMEFT to a different sector of the field theory. When strong constraints on the parameters $C_{\substack{H \ell \\tt}}^{(1)},C_{\substack{H \ell \\tt}}^{(3)},C_{HWB}$ are present, this results in a large degree of non intuitive hidden correlations in the $\mathcal{P}_i$ Wilson coefficients. Of course the converse is also true, constraints on the $\mathcal{P}_i$ lead to
non intuitive hidden correlations on the $C_i$ Wilson coefficients. There is no intrinsically intuitive basis, as a basis choice is unphysical.
It is well known that setting an operator to zero for a measurement, and removing the same operator
with the EoM are not equivalent procedures. A consequence of this fact is that using field redefinitions to attempt to satisfy the defining condition of a constructed observable
corresponds to a poor choice of basis.
A defining condition is still present for the constructed observable in this case, consistency requires this {\it always} leads to a constraint on the field theory.
The constraint will simply be non intuitive and the resulting basis can be functionally redundant.
Another important consequence of this fact is that removing parameters by field redefinitions,
as they are considered to be strongly experimentally bounded and irrelevant for future experimental studies, can also lead to a functionally redundant basis.
Using field redefinitions in this manner is in general a mistake.
\section{LEP data}\label{LEPdata}
The discussion of the previous section is relevant to efforts to constrain the SMEFT with LHC and pre-LHC (LEP, Tevatron and other EW) data.
Considering pre-LHC data, we will take as input parameters the measured values
of the fine structure constant $\hat{\alpha}_{ew}$ (from the low energy limit of electron Compton scattering), the fermi decay constant in muon decays $\hat{G}_F$ and the measured $Z$ mass ($\hat{m}_Z$). It is convenient to relate observables in terms of the parameters $g_2, \sin^2 \theta = g_1^2/(g_1^2 + g_2^2)$ and the electroweak vev {\it{v}}.
Defining at tree level the effective {\it measured} mixing angle
\begin{eqnarray}\label{sinequation}
\sin^2 \hat{\theta} = \frac{1}{2} - \frac{1}{2}\sqrt{1 - \frac{4 \, \pi \hat{\alpha}_{ew}}{\sqrt{2} \, \hat{G}_F \, \hat{m}_Z^2}},
\end{eqnarray}
then the measured value of a gauge coupling can be inferred as
\begin{eqnarray}
\hat{g}_2 \, \sin \hat{\theta} = 2 \, \sqrt{\pi} \, \hat{\alpha}_{ew}^{1/2}.
\end{eqnarray}
The measured vev can be defined as $\hat{v}^2 = 1/\sqrt{2} \, \hat{G}_F$.
\subsection{Parameter counting and LEP data}
The number of parameters present to constrain in the lepton sector are two parameters corresponding to $C_{HWB},C_{HD}$, $(n_g^2 + n_g^4)/2$ parameters for the coefficient
$C_{\substack{ll \\ prst}}$ with $n_g=3$ generations of leptons, and $n_g^2$ parameters for each of $C^{(3)}_{\substack{Hl \\ pr}},C^{(1)}_{\substack{Hl \\ pr}},C_{He}$. Finally, the Wilson coefficient of the operator $(\bar{e}_p \gamma_{\mu} e_r)(\bar{e}_s \gamma^{\mu} e_t)$ corresponds to $n_g^2 (1+ n_g)^2/4$ parameters. The total number of parameters sums to 110 in the lepton sector in the standard basis.
In the $\mathcal{O}$ basis three of these parameters: $C_{HWB}, C^{(3)}_{\substack{Hl \\ tt}},C^{(1)}_{\substack{Hl \\ tt}}$, are chosen to be mapped to alternate parameters
using the EoM operator relations. $C_{HD}$ is exchanged for $\mathcal{P}_T$, and the operator $Q_{HW}$ is exchanged for
$O_{HW}$. This leads to a net reduction of two parameters $C^{(3)}_{\substack{Hl \\ tt}},C^{(1)}_{\substack{Hl \\ tt}}$ in some of the well
measured EWPD observables.
To constrain $ \mathcal{L}^{(6)}$, there are the lepton flavour specific
LEP observables $\mathcal{A}_\ell, R_\ell$, $\sigma_{had}^0, \Gamma_Z$, reported results on the $\rho$ parameter,
inferred constraints on the EWPD parameters from global fits, and TGC verticies.
EWPD and TGC verticies are not directly observable and are discussed in the following sections.
In both bases, there are not enough reported measurements to constrain all the parameters model independently.
As a result, simplifying assumptions are made. One can neglect the effects of some four fermion operators, assuming that there are no significant hierarchies
in the Wilson coefficients to counteract their relative $\Gamma_Z^2/M_Z^2 \sim 10^{-3}$ suppression, in this case 22 parameters are relevant.
Further neglecting parameters related to flavour violation reduces the number of parameters down to ten.
A simplified scenario where all flavour structure in BSM physics is assumed to be vanishingly small is sometimes also considered.
This corresponds to adopting a strict $\rm U(3)^5$ flavour symmetry assumption consistent with MFV \cite{DAmbrosio:2002ex} in the SMEFT. In this case, $n_g = 1$, and the number of free parameters is trivialized down to seven
in the standard basis. Flavour universality in the leptonic decays of the $V$ is the difference between the ten and seven parameters quoted. Further neglecting the $(\bar{e}_t \gamma_{\mu} e_t)(\bar{e}_t \gamma^{\mu} e_t)$ operator leaves 6 parameters to constrain with LEP data.
\subsection{Constraints due to LEP data}\label{constraints}
Predicting observables in the SMEFT,
each of the measured input parameters has been shifted from its theoretical value in the SM. This shift has been
absorbed into the measured value. To aid in simplifying results,\footnote{In the following discussion we largely follow the
analysis of Ref \cite{Pomarol:2013zra}.The main difference is the use of the standard basis, and considering the EOM relations in Eqn \ref{mappingparameters} when comparing results between bases.}, we introduce the parameters
\begin{eqnarray}
\mathcal{S} = \frac{v_T^2 \, C_{HBW}}{\bar{g}_1 \, \bar{g}_2}, \quad \quad \mathcal{T} = \frac{1}{2} v_T^2 \, C_{HD}.
\end{eqnarray}
To leading order in the standard basis, the input parameters are modified (compared to the usual definition of these parameters in the SM Lagrangian) by a shift given by
\begin{eqnarray}\label{parameterdefinition1}
\frac{\delta \alpha_{ew}}{(\alpha_{ew})_{SM}} &=& - 2 \, (s^{SM}_\theta)^2 \, \bar{g}_2^2 \, \mathcal{S}, \nonumber \\
\frac{\delta G_F}{(G_F)_{SM}} &=& - \frac{v_T^2}{2} \left(C_{\substack{ll \\ \mu ee \mu}} + C_{\substack{ll \\ e \mu\mu e}}\right) + v_T^2 \left(C^{(3)}_{\substack{Hl \\ ee }} + C^{(3)}_{\substack{Hl \\ \mu\mu }} \right), \nonumber \\
\frac{\delta m_Z^2}{(m_Z^2)_{SM}} &=& \mathcal{T} + 2 \, (s^{SM}_\theta)^2 \, \bar{g}_2^2 \, \mathcal{S}.
\end{eqnarray}
Parameterizing deviations in LEP data for $\Gamma_Z^L \equiv Z \rightarrow \bar{\ell}_L \, \ell_L, \Gamma_Z^R \equiv Z \rightarrow \bar{\ell}_R \, \ell_R, \Gamma_Z^\nu \equiv Z \rightarrow \bar{\nu} \, \nu$, and $m_W$, one finds
\begin{eqnarray}\label{LEPdata1}
\frac{\delta\Gamma_Z^{L(t)}}{\Gamma_Z^L} &=& \frac{1}{\bar{c}^2_{2 \, \theta}} \left(\mathcal{T} + \frac{\delta G_F}{(G_F)_{SM}} + 4 \bar{s}^2_\theta \, \bar{g}_2^2 \, \mathcal{S} \right) + \frac{2 \, v_T^2}{2 \, \bar{s}^2_\theta - 1} \left(C_{\substack{H \ell \\tt}}^{(1)} + C_{\substack{H \ell \\ tt}}^{(3)} \right), \\ \label{LEPdata2}
\frac{\delta\Gamma_Z^R}{\Gamma_Z^R} &=& - \frac{1}{\bar{c}_{2 \, \theta}} \left(\mathcal{T} + \frac{\delta G_F}{(G_F)_{SM}} + 2 \, \bar{g}_2^2 \, \mathcal{S} \right) - \frac{v_T^2 \, C_{He}}{\bar{s}^2_\theta}, \\ \label{LEPdata3}
\frac{\delta\Gamma_Z^\nu}{\Gamma_Z^L} &=& \mathcal{T} + \frac{\delta G_F}{(G_F)_{SM}} + 2 \, v_T^2 \left(C_{\substack{H \ell \\tt}}^{(1)} - C_{\substack{H \ell \\ tt}}^{(3)} \right), \\
\label{LEPdata4}
\frac{\delta m_W}{m_W} &=& \frac{1}{2 \, \bar{c}_{2 \, \theta}} \left( \bar{c}^2_{\theta} \mathcal{T} + \bar{s}^2_{\theta} \left(\frac{\delta G_F}{(G_F)_{SM}} + 2 \bar{g}_2^2 \, \mathcal{S} \right) \right).
\end{eqnarray}
The introduction of two extra parameters compared to the $\mathcal{O}$ basis leads to two purely unconstrained
parameters.\footnote{Note that other directions in the operator parameter space can be numerically less constrained due to accidental approximate cancelations in Eqn \ref{LEPdata1}-\ref{LEPdata4}. We refer to pure flat directions to make this distinction clear. The following discussion is consistent with a careful examination of the results of in Ref \cite{Han:2004az,Grojean:2006nn}. Note the distinction between pure flat directions and approximate flat directions due to numerical accidents
is relevant in this comparison. The t-channel $\nu$ exchange contribution to $\sigma(e^+ \, e^- \rightarrow W^+ \, W^-)$, was included in the fit in Ref \cite{Han:2004az}. This consistency does not extend to some subsequent literature.}
Despite $C_{\substack{H \ell \\tt}}^{(1)}, C_{\substack{H \ell \\ tt}}^{(3)}$ being present compared to the $\mathcal{O}$ basis
the pure flat directions do not have to involve these parameters. There is no special basis.
Consider the case of the multiple Wilson coefficients present in $\delta G_F$, or the relation
between $\mathcal{T}, \delta G_F$ and $C_{\substack{H \ell \\ tt}}^{(3)}$ in $\delta\Gamma_Z^{\nu}$ in the limit of the flavour trivialized SMEFT\footnote{It is interesting to note the nontrivial effects of the $\rm U(3)^5$ symmetry on this choice, and the difference in the 10 vs 7 parameters present.
In the case where flavour structure is not trivialized, each of the $\delta\Gamma_Z^{L(t)}$ for $t =e, \mu, \tau$ has an individual shift in Eqn \ref{LEPdata1}.
Conversely, in $\delta G_F$ the flavour specific sum $C^{(3)}_{\substack{Hl \\ ee }} + C^{(3)}_{\substack{Hl \\ \mu\mu }}$
appears. Flat directions in LEP data are sensitive to lepton flavour symmetry assumptions in this manner.}. One can always choose the accidental relation
\begin{align}\label{unconstrained}
2 \, C^{(3)}_{\substack{Hl}} &= C_{\substack{ll}}, & v_T^2 \, C_{\substack{H \ell}}^{(3)} &= \frac{\mathcal{T}}{2} + \frac{\delta G_F}{2 \, (G_F)_{SM}}.
\end{align}
With this choice the dependence on $C^{(3)}_{\substack{Hl}}$ and $\delta G_F$ is removed in Eqn \ref{LEPdata1}-\ref{LEPdata4}.
One can consider the remaining parameters constrained to have fixed relationships due to LEP measurements,
and then the above relations represent chosen pure flat directions (in this case $v_T^2 \, \mathcal{C}_{He} = \mathcal{T}$). This choice is arbitrary, as is any other in a system of unconstrained equations.
This choice is interesting to consider, when examining off-shell TGC vertex bounds, as in this case the coupling of the $W$ and $Z$ to leptons are physically allowed to differ.
The Wilson coefficient $C_{HWB}$ exactly canceling against the parameters $C_{\substack{H \ell \\tt}}^{(1)},C_{\substack{H \ell \\ tt}}^{(3)}$
which has been argued to be relevant to the definition of the $S$ parameter (see Section \ref{Sparameter}), need not correspond to a pure unconstrained direction. If two more measurements are made, all of the parameters appearing in the lepton sector are then constrained.
\subsection{Lifting flat directions through scale dependence}
LEP data is not blind to the pure unconstrained directions resulting from Eqn \ref{LEPdata1}-\ref{LEPdata4}, before considering TGC verticies, as the operators
are scale dependent quantities. The full renormalization of the dimension six operators in the
SMEFT (with nontrivial flavour structure) has been determined in Ref \cite{Jenkins:2013zja,Jenkins:2013wua,Alonso:2013hga,Alonso:2014zka}. Considering the chosen relations in Eqn \ref{unconstrained}, we find the leading scale dependence
\begin{eqnarray}
\mu \, \frac{d}{d \, \mu} (C_{HD} - 2 \, C^{(3)}_{\substack{Hl}}) = \frac{12 \, \lambda}{16 \, \pi^2} \, C_{HD} + \cdots
\end{eqnarray}
where $\lambda = M_h^2/2 \, v^2$. Here we have assumed $C_{HD} - 2 \, C^{(3)}_{\substack{Hl}}$ vanishes at the scale $\mu \sim m_Z$, and we have neglected mixing
with other operators for simplicity. The dependence due to the top Yukawa accidentally cancels.
Numerically, running from the Z pole to $\sim 200 \, {\rm GeV}$ for LEP II $Z$ phenomenology, at least a percent level breaking of
this relation is already present. The leading breaking of the $C_{HD} - C_{ll}$ chosen relation, neglecting mixing, is similarly
\begin{eqnarray}
\mu \, \frac{d}{d \, \mu} (C_{HD} - C_{ll}) = \frac{3}{4 \, \pi^2} \left(\lambda + y_t^2\right) \, C_{HD} + \cdots
\end{eqnarray}
There is some value in performing global EWPD fits, and not neglecting the scale dependence of the Wilson coefficients when considering flat directions.
Phenomenology involving $V$ bosons at LHC is also not identical to $V$ phenomenology at LEP in this manner.
\section{Constructed Observables and basis choice}\label{pseudoobservablesection}
To further constrain the SMEFT, one can consider bounds on constructed observables.\footnote{A somewhat similar concept to constructed observables was recently discussed in Ref \cite{Gupta:2014rxa}.} The challenges of constructed observables are well illustrated by the familiar oblique parameters, initially developed in Refs \cite{Kennedy:1988sn,Peskin:1990zt,Holdom:1990tc,Golden:1990ig,Altarelli:1990zd,Peskin:1991sw}. Using an oblique constraint as well as
Eqn \ref{LEPdata1}-\ref{LEPdata4} would be redundant. We first discuss oblique corrections as they illustrate the challenge
of constructed observables more directly than TGC verticies.
In both cases, these
quantities are constructed with the assumption that the direct coupling of the $V$ to leptons is SM-like.
Consider the consequences of this defining assumption for the effective axial and vector couplings in the SMEFT.
With the normalization
\begin{eqnarray}
\mathcal{L}_{chir} = (\bar{g}_1^2 + \bar{g}_2^2)^{1/2} J_\mu^0 Z^\mu + \frac{\bar{g}_2}{\sqrt{2} }J_\mu^\pm W_{\pm}^\mu,
\end{eqnarray}
where $J_\mu^0 = \bar{\ell}_p \, \gamma_\mu \left(\bar{g}_V^{pr}- \bar{g}_A^{pr} \, \gamma_5 \right) \ell_r$, the shift in
$\bar{g}_{V,A}$ in the standard basis (for charged leptons) are
\begin{eqnarray}\label{higherdgvga}
\bar{g}_V^{pr}&=& \left(\frac{1}{4} - \bar{s}^2_\theta \right) \delta{g_V^{pr}}, \nonumber \\
\delta{g_V^{pr}} &=& 1 + \frac{v_T^2}{4 \, g_V^{SM}} \left(C_{HWB} \, \bar{s}_\theta \, \bar{c}_\theta \left(1- 4 \bar{c}_\theta^2 \right) + C_{\substack{H e \\ pr}} + C_{\substack{H \ell \\ pr}}^{(1)} + C_{\substack{H \ell \\ pr}}^{(3)}\right), \nonumber \\
\bar{g}_A^{pr}&=& \frac{1}{4} \delta{g_A^{pr}}, \nonumber \\
\delta{g_A^{pr}} &=& 1 + \frac{v_T^2}{4 \, g_A^{SM}} \left( C_{HWB} \, \bar{s}_\theta \, \bar{c}_\theta + C_{\substack{H e\\ pr}}
+ \left(C_{\substack{H \ell \\pr}}^{(1)} + C_{\substack{H \ell \\ pr}}^{(3)}\right) \frac{\left(1 + 2 \,\bar{s}_\theta^2 \right)}{1 - 2 \,\bar{s}_\theta^2} \right).
\end{eqnarray}
Here $p.r$ are flavour labels.
If the direct coupling of the $Z$ to leptons is SM-like, this naively corresponds to assuming
\begin{eqnarray}\label{standardbasis-vanish}
C_{HWB} \, \bar{s}_\theta \, \bar{c}_\theta \left(1- 4 \bar{c}_\theta^2 \right) + C_{\substack{H e \\ pr}} + C_{\substack{H \ell \\ pr}}^{(1)} + C_{\substack{H \ell \\ pr}}^{(3)} \rightarrow 0,
\nonumber \\
C_{HWB} \, \bar{s}_\theta \, \bar{c}_\theta + C_{\substack{H e\\ pr}}
+\frac{\left(1 + 2 \,\bar{s}_\theta^2 \right)}{1 - 2 \,\bar{s}_\theta^2} \left(C_{\substack{H \ell \\pr}}^{(1)} + C_{\substack{H \ell \\ pr}}^{(3)}\right) \rightarrow 0,
\end{eqnarray}
while in the $\mathcal{O}$ basis this corresponds to assuming
\begin{eqnarray}\label{obasis-vanish}
\frac{\left(\mathcal{P}_{B} + \mathcal{P}_{W}\right) \, g_1 \, g_2}{4} \, \bar{s}_\theta \, \bar{c}_\theta \left(1- 4 \bar{c}_\theta^2 \right) + C_{H e}
+ \left[C_{\substack{H \ell \\ pr}}^{(1)} + C_{\substack{H \ell \\ pr}}^{(3)}\right]_{p \neq r}
\rightarrow 0,
\nonumber \\
\frac{\left(\mathcal{P}_{B} + \mathcal{P}_{W}\right) \, g_1 \, g_2}{4} \, \bar{s}_\theta \, \bar{c}_\theta + C_{H e}
+ \frac{\left(1 + 2 \,\bar{s}_\theta^2 \right)}{1 - 2 \,\bar{s}_\theta^2} \left[C_{\substack{H \ell \\pr}}^{(1)} + C_{\substack{H \ell \\ pr}}^{(3)}\right]_{p \neq r}
\rightarrow 0.
\end{eqnarray}
The resulting constraints on the field theory when bounds on the oblique parameters are incorporated -- derived from experiments -- can be basis dependent
if this assumption is not imposed in a basis independent manner.\footnote{Notice that the number of constraints that the assumption corresponds to
is not even consistent between the bases, when flavour indicies are not ignored.} Expressing an observable
in terms of other observables is basis independent. Naively relating an observable to constructed observables is not.
\subsection{The $S$ parameter}\label{Sparameter}
In the PDG \cite{Beringer:1900zz1} (Sec.10) the $S$ parameter is defined as
\begin{eqnarray}\label{Sdefn}
\frac{\hat{\alpha}(m_Z)}{4 \, \hat{s}_Z^2 \, \hat{c}_Z^2} \, S \equiv
\frac{\Pi_{ZZ}^{new}(m_Z^2) - \Pi_{ZZ}^{new}(0)}{m_Z^2}
- \frac{\hat{c}_Z^2-\hat{s}_Z^2}{\hat{c}_Z \, \hat{s}_Z} \frac{\Pi^{new}_{Z \, \gamma}(m_Z^2)}{m_Z^2} - \frac{\Pi^{new}_{\gamma \, \gamma}(m_Z^2)}{m_Z^2}.
\end{eqnarray}
The hatted parameters in Eqn \ref{Sdefn} are defined in the $\overline{\rm MS}$ scheme and conform with the PDG convention.
Assuming new physics is heavy enough for an operator interpretation, the $S$ parameter can be mapped to the Wilson coefficient of the operator $Q_{WB}$ \cite{Grinstein:1991cd}, as
\begin{eqnarray}
S_{\mathcal{Q}} = - \frac{16 \, \pi \, v_T^2}{g_1 \, g_2} \, C_{HWB}.
\end{eqnarray}
In the $\mathcal{O}$ basis, the kinetic mixing of the
photon and $Z$ due to higher dimensional operators is proportional to $\mathcal{P}_{B} + \mathcal{P}_{W}$; one finds
\begin{eqnarray}\label{PbasisS}
S_{\mathcal{O}} = - 4 \, \pi \, v_T^2 \, \left(\mathcal{P}_{B} + \mathcal{P}_{W}\right).
\end{eqnarray}
Using the EoM relations between the Wilson coefficients to change basis
\begin{eqnarray}\label{grojeansconfusion}
- 4 \, \pi \, v_T^2 \, \left(\mathcal{P}_{B} + \mathcal{P}_{W}\right) \rightarrow - \frac{16 \, \pi \, v_T^2}{g_1 \, g_2} \, C_{HWB} - \frac{8 \, \pi \, v_T^2}{g_1^2 \, \mathsf{y}_\ell} \, C_{\substack{H \ell \\ tt}}^{(1)} - \frac{16 \, \pi \, v_T^2}{g_2^2 \, \mathsf{y}_\ell} \, C_{\substack{H \ell \\ tt}}^{(3)}.
\end{eqnarray}
The idea of oblique parameters has an implicit challenge from field redefinitions, which is illustrated
by the above equation.
This is a point previously discussed, in part, in Ref \cite{SanchezColon:1998xg,Kilian:2003xt,Grojean:2006nn}.
At this stage it is important to note that even though the $S$ parameter in the two bases are related as in Eqn \ref{grojeansconfusion},
it {\it does not} directly follow that the $S$ parameter always has pure flat directions related to the operators $C_{H\ell}^{(1)},C_{H\ell}^{(3)}$ in the standard basis.
As explicitly demonstrated in Section \ref{constraints} the pure flat directions need not
be related to $C_{HWB}$.
Nevertheless, the definition of an oblique correction does have the defining assumption of a SM like $V$ coupling to leptons associated with it.\footnote{"Oblique" parameters are discussed in
Ref \cite{Peskin:1991sw} as "Vaccum polarizations affect the above interactions by modifying the gauge-boson propagators....
This is the reason why they are called "oblique" corrections as opposed to the
"direct" vertex and box corrections that modify the form of the interactions themselves". } In both bases the operator $Q_{He}$ is present, so some version of this
defining assumption must always be imposed.
One can consider a weak version of this defining condition, where only the combination of parameters present
in Eqn \ref{standardbasis-vanish} or Eqn \ref{obasis-vanish} are assumed to vanish.\footnote{This limit is fine tuned and is not invariant under the renormalization scale evolution of the theory.}
A strong
version of an oblique parameter defining condition is to impose that each of the Wilson coefficients in Eqn \ref{higherdgvga} (other than the $C_{HWB}$) individually vanish. For the standard basis
this implies
\begin{eqnarray}
C_{\substack{H e \\ pr}}, C_{\substack{H \ell \\ pr}}^{(1)}, C_{\substack{H \ell \\ pr}}^{(3)} \rightarrow 0.
\end{eqnarray}
The strong version of the oblique parameter defining assumption leads to the definitions in the bases being {\it identified}
\begin{eqnarray}
-4 \, \pi \, v_T^2 \, \left(\mathcal{P}_{B} + \mathcal{P}_{W}\right) \equiv - \frac{16 \, \pi \, v_T^2}{g_1 \, g_2} \, C_{HWB}.
\end{eqnarray}
So long as the standard definition of the oblique parameters \cite{Peskin:1991sw} is adhered to with the strong defining condition,
there is no issue with basis dependence. The weak version of this assumption results in the definitions still differing between bases.
This supports imposing the strong defining condition.
The PDG Higgs review \cite{Beringer:1900zz2}
currently defines the oblique parameter $\Delta S$ in a basis dependent manner, proportional to $\mathcal{P}_{B} + \mathcal{P}_{W}$. This
is equivalent to the definition in Eqn \ref{Sdefn} in the PDG EW review \cite{Beringer:1900zz1} only when the strong version of the defining condition is imposed.
Not imposing this condition changes the definition of this oblique parameter from its standard definition \cite{Kennedy:1988sn,Peskin:1990zt,Holdom:1990tc,Golden:1990ig,Altarelli:1990zd,Peskin:1991sw}, and introduces a basis dependent constructed observable. Using such a definition is inconsistent with basis independent bounds being obtained on the SMEFT.
Finally, we note that in the $\mathcal{O}$ basis, the strong LEP bound limit (including a strong constraint on the $S$ parameter) seems to correspond to $\mathcal{P}_{B} = - \mathcal{P}_{W}$ and $C_{H e} \rightarrow 0$. But this is an incomplete and basis dependent conclusion. Taking into account the EoM, and the strong LEP bound defining condition of the
$S$ parameter, $C_{\substack{H \ell \\ tt}}^{(1)},C_{\substack{H \ell \\ tt}}^{(3)} \rightarrow 0$ gives the non intuitive relationships
$- \mathcal{P}_{HW} = \mathcal{P}_{HB} = \mathcal{P}_W = - \mathcal{P}_B$ in the $\mathcal{O}$ basis. Not imposing this relationship
while assuming the strong LEP bound would lead to a functionally redundant operator basis.
\subsection{Triple Gauge coupling verticies}
Off shell TGC verticies are also not directly observable, like the oblique parameters, they are constructed observables.
The TGC vertex $Z \, W^+ \, W^-$ requires one of the massive gauge bosons to be off shell.
Leading experimental studies of this vertex result from measurements of
\begin{eqnarray}\label{TGCfinal}
\ell^+ \, \ell^- \rightarrow W^+ \, W^- \rightarrow jj \ell \, \nu, jjjj, jjX, \ell X,
\end{eqnarray}
where $j$, $\ell$ and $X$ are a jet, lepton and missing final state energy \cite{Abdallah:2010zj,ALEPH:2005ab}.
There are many ways to appreciate the distinction between the resulting constructed observable and the cross section measurement.
The kinematics of t and s channel exchange are distinct in $\sigma(e^+ \, e^- \rightarrow W^+ \, W^-)$.
The t-channel contribution dominates at threshold, however at high energies, the s-channel contribution related to the TGC vertex dominates \cite{Hagiwara:1986vm,Hagiwara:1992eh}.
The potential strength of TGC vertex bounds are directly related to the anomalous growth at high energies that results when the deviations from the
SM in the s-channel are introduced. Using a reported bound for a TGC vertex for $Z \, W^+ \, W^-$,
the possible effect of $\mathcal{L}_6$ on the $t$-channel $\ell^+ \, \ell^- \rightarrow W^+ \, W^- $ process is set to zero.
To obtain the numerical values for the TGC bounds \cite{Abdallah:2010zj,ALEPH:2005ab}, exclusive processes in Eqn \ref{TGCfinal} are assumed to have a SM like coupling of the $V$, and final states (including non-leptonic decays of the $W$) are combined, to improve statistics. This combination sets to zero possible modifications due to $ \mathcal{L}^{(6)}$ in the decay channels. TGC verticies are clearly reported under the assumption that the possible effects of $\mathcal{L}_6$ on the direct coupling of the $V$ to leptons are set to zero.
It is important to reiterate that setting these contributions due to $ \mathcal{L}^{(6)}$ to zero in constructing the observable is not equivalent to only removing the
parameters that lead to these effects by field redefinitions. The defining condition must be mapped to the field theory using the EoM.
\subsubsection{TGC results}
TGC verticies have recently come under renewed scrutiny for the SMEFT in Refs.~\cite{Eboli:2010qd,Corbett:2012ja,Corbett:2013pja,Buchalla:2013wpa}.
These analyses descend from the classic works on higher dimensional operators in TGC's \cite{Hagiwara:1986vm,DeRujula:1991se,Hagiwara:1992eh}
that introduced the standard notation \cite{Hagiwara:1986vm}
\begin{eqnarray}
(-\mathcal{L}_{TGC}) &=& i \, \bar{g}_2 \left[\delta g_1^Z \, \bar{c}_\theta \, \mathcal{Z}^\mu \left(\mathcal{W}^{+}_{\mu \nu} \, \mathcal{W}^{-\nu} - \mathcal{W}^{-}_{\mu \nu} \, \mathcal{W}^{+\nu}\right) \right]
+ i\, \bar{g}_2 \left[\delta g_1^\gamma \, \bar{s}_\theta \, \mathcal{A}^\mu \left(\mathcal{W}^{+}_{\mu \nu} \, \mathcal{W}^{-\nu} - \mathcal{W}^{-}_{\mu \nu} \, \mathcal{W}^{+\nu}\right) \right], \nonumber \\
&+& i \, \bar{g}_2 \left[\delta \kappa_Z \, \bar{c}_\theta \, \mathcal{Z}_{\mu \nu} \, \mathcal{W}^{+\nu} \, \mathcal{W}^{-\mu} + \delta \kappa_\gamma \, \bar{s}_\theta \mathcal{A}_{\mu \nu} \mathcal{W}^{+\nu} \, \mathcal{W}^{-\mu} \right], \nonumber \\
&+& i \, \frac{\bar{g}_2}{\bar{m}_W^2} \, \left(\lambda_Z \, \bar{c}_\theta \mathcal{Z}^{\mu \, \nu} +\lambda_\gamma \, \bar{s}_\theta \mathcal{A}^{\mu \, \nu} \right) \left(W^+_{\rho \, \mu} W^{- \rho}_{\nu} \right).
\end{eqnarray}
Here the field strengths for the massive gauge bosons are using the short hand notation $V_{\mu \, \nu} = \partial_\mu V_\nu -\partial_\nu V_\mu$,
and a number of other notational conventions are present. The mass eigenstate gauge bosons in $\mathcal{L}_{SM} + \mathcal{L}^{(6)}$ are denoted $\mathcal{Z},\mathcal{A},\mathcal{W}$. (See Ref \cite{Alonso:2013hga} Section 5.4 for the explicit definitions.)
Note that the lagrangian parameters in the canonically normalized SMEFT, $\bar{g}_2$, $\bar{c}_\theta$, $\bar{s}_\theta$ are present defining the anomalous parameters
$\delta g_1^{Z, \gamma}, \delta \kappa_{Z, \gamma}, \lambda_{Z, \gamma}$. The overall sign convention is consistent with Ref \cite{Hagiwara:1986vm}, indicated in the above equation with an explicit $-\mathcal{L}_{TGC}$, which is opposite the overall sign convention in Refs. \cite{Jenkins:2013zja,Alonso:2013hga}.
The $\mathcal{L}_{SM}+ \mathcal{L}^{(6)}$ TGC anomalous couplings in the standard basis are given by
\begin{align}\label{standardTGC2}
\delta g_1^{Z} &= \frac{\bar{s}_\theta \, v_T^2}{2 \, \bar{c}_\theta} \, C_{HWB}, &
\delta g_1^{\gamma} &= - \frac{\bar{c}_\theta \, v_T^2}{2 \, \bar{s}_\theta} \, C_{HWB}, \\
\delta \kappa_{Z} &= - \frac{\bar{s}_\theta \, v_T^2}{2 \, \bar{c}_\theta} \, C_{HWB}, &
\delta \kappa_{\gamma} &= \frac{\bar{c}_\theta \, v_T^2}{2 \, \bar{s}_\theta} \, C_{HWB}, \\
\delta \lambda_{Z} &=6 \, \frac{\bar{m}_W^2}{\Lambda^2} \, C_{W}, &\delta \lambda_{\gamma} &=6 \, \frac{\bar{m}_W^2}{\Lambda^2} \, C_{W}.
\end{align}
Note that these results are expressed in terms of the canonically normalized Lagrangian parameters, including $\bar{c}_\theta,\bar{s}_\theta$ as defined in Ref \cite{Alonso:2013hga}. A redefinition of the effective mixing angle, to absorb a shift due to $C_{HWB}$, has not yet been done.
The $\mathcal{L}_{SM}+ \mathcal{L}^{(6)}$ TGC anomalous couplings in the $\mathcal{O}$ basis are given by \cite{Pomarol:2013zra}
\begin{align}\label{PRresult}
\delta g_1^{Z} &= \frac{v_T^2 \, \bar{g}_1^2 \, \left(\mathcal{P}_{B} + \mathcal{P}_{W}\right)}{8} + \frac{(\bar{g}_1^2 + \bar{g}_2^2) \, v_T^2}{4} \left(\mathcal{P}_{HW} + \mathcal{P}_{W}\right), \\
\delta g_1^{\gamma} &= - \frac{v_T^2 \, \bar{g}_2^2 \, \left(\mathcal{P}_{B} + \mathcal{P}_{W}\right)}{8}, \\
\delta \kappa_{Z} &= \frac{v_T^2 \, \bar{g}_1^2 \, \left(\mathcal{P}_{B} + \mathcal{P}_{W}\right)}{8} + \frac{v_T^2}{4 } \, (\bar{g}_1^2 + \bar{g}_2^2) \, \left(\mathcal{P}_{HW} + \mathcal{P}_{W}\right) - \frac{v_T^2}{4 } \, g_1^2 \left(\mathcal{P}_{HW} + \mathcal{P}_{HB}\right), \\
\delta \kappa_{\gamma} &= - \frac{v_T^2 \, \bar{g}_2^2 \, \left(\mathcal{P}_{B} + \mathcal{P}_{W}\right)}{8} + \frac{g_2^2 \, v_T^2}{4} \, \left(\mathcal{P}_{HB} + \mathcal{P}_{HW}\right), \\
\delta \lambda_{Z} &=6 \, \frac{\bar{m}_W^2}{\Lambda^2} \, C_{W}, \\
\delta \lambda_{\gamma}&=6 \, \frac{\bar{m}_W^2}{\Lambda^2} \, C_{W}.
\end{align}
The operator $Q_W = \epsilon_{IJK} \, W_\mu^{I,\nu} \, W_\nu^{J,\rho} \, W_\rho^{K,\mu}$
is not to be confused with the operator $\mathcal{O}_W$ in the $\mathcal{O}$ basis. Including this operator,
leads to a flat direction in constraints derived from TGC verticies \cite{Brooijmans:2014eja,Ellis:2014dva}, as expected \cite{Alonso:2013hga}.\footnote{The PDG Higgs review \cite{Beringer:1900zz2} treatment of this flat direction is unspecified, and this operator is not included in the quoted TGC bounds in the PDG.} The
mixing angles have not been related to input observables as yet in Eqn \ref{PRresult}. Doing so the dependence on $\mathcal{P}_{B} + \mathcal{P}_{W}$ is removed and the
expressions satisfy $\delta \kappa_Z = \delta g^Z_1 - t_{\bar{\theta}}^2 \, \delta \kappa_\gamma$ in both bases, as expected \cite{Hagiwara:1986vm}.
\subsubsection{Relation to input observables}\label{TGCredefine}
One can absorb the redefinition of the mixing angles in the SMEFT in a finite renormalization. This takes into account how the dependence on
$C_{HWB}$ modifying the mixing angle cancels when relating TGC verticies to input observables.
Doing so, the deviations in $\delta g_1^Z, \delta g_1^{\gamma} $ due to $C_{HWB}$ are canceled, and
\begin{align}
\delta \tilde{\kappa}_{Z} &= 1 + \frac{\bar{s}_\theta \, v_T^2}{\bar{c}_\theta} \, C_{HWB}, & \delta \tilde{\kappa}_{\gamma} &=1 - \frac{\bar{c}_\theta \, v_T^2}{\bar{s}_\theta} \, C_{HWB}.
\end{align}
In Eqn. \ref{standardTGC2}, the Wilson coefficients $C_{HWB}$ and $C_W$ are present. The Wilson coefficient $C_{HWB}$ need not be related to a pure flat direction in the standard basis. Exchanging $\bar{g}_2$ in terms of $m_W$,
introduces the parameter shifts $\delta m_W$ and $\delta G_F$. However, the former is already used as a measurement in Eqn \ref{LEPdata4} and the flat direction can be chosen
to set $\delta G_F =0$, as demonstrated. Similarly
exchanging the mixing angles in terms of input parameters cancels the deviations in $\delta g_1^Z, \delta g_1^{\gamma}$ but does not introduce sensitivity
to the remaining flat directions.
The defining conditions of the off-shell TGC bounds are inconsistent with choices that can be made for flat directions present due to
LEP data. These directions can be chosen so that it is crucial to probe $C_{H\ell}^{(3)}$ to break the remaining degeneracy, see Eqn \ref{unconstrained}.
Breaking this degeneracy can be done by studying {\it exclusive} $W$ decay to leptonic final states, as
$ \delta \Gamma_W/\Gamma_W \propto \bar{g}_2 \, v_T^2 C_{\substack {H\ell \\ tt}}^{(3)}$. For example, a process that can
remove a degeneracy is exclusive $\sigma(e^+ \, e^- \rightarrow \ell \, \bar{\ell} X)$.
Inclusive $\sigma(e^+ \, e^- \rightarrow W^+ \, W^-)$ production that includes the $\nu$ t-channel exchange can also be used.
Bounds on the off-shell TGC vertex do not directly probe these effects, and their defining assumptions assume these effects
in the SMEFT are set to zero.
The conclusion that TGC verticies are limited in their utility holds in the $\mathcal{O}$ basis, but the reasoning is more subtle and involves a functional redundancy.
Examining the EoM relations one finds
\begin{align}
\mathcal{P}_{HW}+\mathcal{P}_{HB} &\rightarrow - \frac{4}{\bar{g}_1 \, \bar{g}_2} C_{HWB}, & \mathcal{P}_{HW}+\mathcal{P}_{W} &\rightarrow \frac{4}{\bar{g}_2^2} C_{\substack{H\ell \\ tt}}^{(3)}.
\end{align}
Using TGC constructed observables to bound a parameter equivalent to $C_{H\ell}^{(3)}$ is functionally redundant.
Analyses that use these constructed observables can constrain the field theory in a consistent manner, when the defining assumptions of the TGC verticies are imposed in a basis independent manner, avoiding a functional redundancy. In this case $ \mathcal{P}_{HW}+\mathcal{P}_{W} \rightarrow 0$.\footnote{One can always choose that the flat directions do correspond to the TGC constructed observables, and the $\mathcal{O}$ basis makes such a choice intuitive. However, this does not establish
the general utility of this constructed observable, but reinforces the inconsistency of incorrectly treating it as a measurement. How strongly constrained an EFT is cannot depend on an arbitrary operator basis, or flat direction, choice.} Measurements of $\sigma(e^+ \,e^- \rightarrow W^+ \, W^-)$,
are sensitive at the $\sim 1 \%$ level to deviations in the coupling of the $W$, so no pure flat directions are expected in a full analysis
using the observables that can lift the flat direction consistently. The nature of the exact numerical bound is worthy of future
study.\footnote{It is interesting to note that allowed deviations in the $h \rightarrow V \, F$ spectra are de-correlated in
the case of the nonlinear EFT from LEP measurements. This makes accurate and precise studies of the $h \rightarrow V \, F$ spectra, in light of consistent LEP constraints, particularly interesting \cite{Isidori:2013cga,Brivio:2013pma}.}
\section{Triple Gauge coupling verticies and $h \rightarrow V \mathcal{F}$}\label{hvfspec}
In this section, we reexamine the relationship between reported bounds on TCG verticies and the $h \rightarrow V \, F$ differential distributions.
We ensure that the defining condition of the TGC constructed observable is also imposed consistently when considering this relationship by adopting
the strong LEP limit. We demonstrate how accounting for the the subtlety of the functional redundancy, and considering the EoM
makes the connection between these observables vanish in the limit of strong LEP bounds.
The importance of the $h \rightarrow V \, F$ differential distributions has recently been studied in Refs.~\cite{DeRujula:2010ys,Gainer:2013rxa,Isidori:2013cla,Grinstein:2013vsa,Buchalla:2013mpa,Beneke:2014sba,Gonzalez-Alonso:2014rla}.
The relationship between these quantities has received some attention in Refs \cite{Pomarol:2013zra,Brooijmans:2014eja,Ellis:2014dva}. The arguments of
Ref.\cite{Pomarol:2013zra} have been influential and have lead to claims in the recent Higgs review of the
PDG \cite{Beringer:1900zz2}.
We focus on the case when $V = Z$, although the same arguments apply for $V = W$.
In the SM, the result for the offshell gauge boson invariant mass ($q^2$) distribution is given by
\begin{eqnarray}
\frac{d\Gamma_0(\hat{q}^2)}{d \hat{q}^2} = \frac{ (\bar{g}_1^2 + \bar{g}_2^2)^2 \, (g_A^2 + g_V^2) \, m_h \,\left[\lambda^2(\hat{q}^2,\rho) + 12 \rho \, \hat{q}^2 \right] }{256 \pi^3}
\frac{\lambda(\hat{q}^2,\rho)}{\left(\hat{q}^2 - \rho \right)^2}.
\end{eqnarray}
Here $\hat{q}^2 = q^2/m_h^2$ and $\rho = m_V^2/m_h^2$. The masses here are the physical (measurable)
on shell masses of the vector bosons and $\lambda(\hat{q}^2,\rho) = \sqrt{(1+ \hat{q}^2 - \rho)^2 - 4 \hat{q}^2}$. The modification of the $q^2$ distribution
due to $ \mathcal{L}^{(6)}$ is given by
\begin{eqnarray}\label{Zspectra2}
\frac{1}{v_T^2}\frac{ {\rm d} \Gamma }{{\rm d} q^2 } &=& \frac{1}{3} \frac{ {\rm d} \Gamma_0 }{{\rm d} \hat{q}^2 } \, \biggl\{\frac{1}{v_T^2} + 2 C_{H \Box} + C_{HD} + \frac{g_V + g_A}{(g_V)^2 + (g_A)^2} \frac{C_{He}}{2} + \frac{2 \, g_1 \, g_2}{g_1^2 + g_2^2} C_{HWB} \biggl\} , \nonumber \\
&+& \frac{1}{3} \frac{ {\rm d} \Gamma_0 }{{\rm d} \hat{q}^2 } \, \biggl\{ \frac{\bar{s}_\theta \, \bar{c}_\theta C_{HWB} }{2 \, (g_A^2 + g_V^2)} \left(g_A + g_V (1 - 4 \, \bar{c}_\theta^2) \right) + \frac{C_{H \ell}^{(1)}+ C_{H \ell}^{(3)}}{2 \, \left(g_V^2 + g_A^2 \right)} \left(g_V - g_A \, \frac{2 \, \bar{s}_\theta^2 + 1}{2 \, \bar{s}^2_\theta - 1} \right) \biggl\}, \nonumber \\
&+& 8 \, \hat{q}^2 \, \frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \, \left[ \bar{s}_\theta \, \bar{c}_\theta\, C_{HWB} \, + C_{HB} \, \bar{s}^2_\theta + C_{HW} \, \bar{c}^2_\theta \right] \, \left( \frac{\hat{q}^2 -1 + \rho}{\lambda^2(\hat{q}^2,\rho) + 12 \rho \, \hat{q}^2} \right), \nonumber \\
&+& \frac{256 \, \pi \, \alpha_{ew} \, \bar{s}_\theta \, \bar{c}_\theta}{\bar{g}_1^2 + \bar{g}_2^2} \, \frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \, \mathscr{C}_{\gamma Z} \,\left( \frac{g_V}{(g_V)^2 + (g_A)^2}\right) \,
\left(\frac{\rho \, (\rho - \hat{q}^2) \, (\hat{q}^2 -1 + \rho)}{\lambda^2(\hat{q}^2,\rho) + 12 \rho \, \hat{q}^2} \right), \nonumber \\
&-& \frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \, \frac{(\rho
- \hat{q}^2)}{6 \, \rho} \, \left(C_{He} \frac{\left(g_V - g_A \right)}{\left(g_V^2 + g_A^2 \right)} + (C_{H \ell}^{(1)}+ C_{H \ell}^{(3)})\frac{\left(g_V + g_A \right)}{\left(g_V^2 + g_A^2 \right)} \right).
\end{eqnarray}
We have explicitly labelled the term that comes from the photon pole exchange
with $\mathscr{C}_{\gamma Z}$.The
Wilson coefficients for $h \rightarrow \gamma \, Z, h \rightarrow \gamma \, \gamma$ are defined with the normalization in Ref \cite{Alonso:2013hga}.
A consistent scheme can include the squared photon pole contribution \cite{Beneke:2014sba}, however, for the sake of our illustrative discussion on the EoM effects, we neglect this term.
In the case of strong experimental LEP bounds, it has been argued that the $h \rightarrow V \, F$ offshell invariant mass ($q^2$) spectrum is not
a competitive source of information on higher dimensional operators due to their relationship with TGC verticies. In this limit,
\begin{eqnarray}\label{Zspectra6}
\frac{1}{v_T^2}\frac{ {\rm d} \Gamma }{{\rm d} q^2 } &=& \frac{ {\rm d} \Gamma_0 }{{\rm d} \hat{q}^2 } \, \mathcal{N} \left(0, C_{H \Box}, C_{HD},0,0, 0 \right), \nonumber \\
&+& 32 \, \pi \, \alpha_{ew} \, \frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \, \mathscr{C}_{\gamma \gamma} \, \hat{q}^2 \, \left(\frac{\hat{q}^2 -1 + \rho}{\lambda^2(\hat{q}^2,\rho) + 12 \rho \, \hat{q}^2} \right), \nonumber \\
&+& 32 \, \pi \, \alpha_{ew} (\cot_{\bar{\theta}} - \tan_{\bar{\theta}})\frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \,\mathscr{C}_{\gamma Z} \, \hat{q}^2 \, \left(\frac{\hat{q}^2 -1 + \rho}{\lambda^2(\hat{q}^2,\rho) + 12 \rho \, \hat{q}^2} \right), \nonumber \\
&+& \frac{256 \, \pi \, \alpha_{ew} \, \bar{s}_\theta \, \bar{c}_\theta}{\bar{g}_1^2 + \bar{g}_2^2} \, \frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \, \mathscr{C}_{\gamma Z} \,\left( \frac{g_V}{(g_V)^2 + (g_A)^2}\right) \,
\left(\frac{\rho \, (\rho - \hat{q}^2) \, (\hat{q}^2 -1 + \rho)}{\lambda^2(\hat{q}^2,\rho) + 12 \rho \, \hat{q}^2} \right),
\end{eqnarray}
where a normalization function,
$\mathcal{N}(C_{HWB}, C_{H \Box}, C_{HD},C_{He},C_{H \ell}^{(1)},C_{H \ell}^{(3)})$,
has been introduced. In the strong LEP limit, the BSM momentum dependence of this spectra is directly related to measurements of $\mathscr{C}_{\gamma \gamma},\mathscr{C}_{\gamma Z}$. However in this same limit, this spectrum is not related to TGC verticies. The functional form of the shape dependent deviation in the spectrum due to $C_{\gamma Z}$ is given in Eqn \ref{Zspectra6}, and can be fit for in dedicated searches.
Considering the relative experimental accessibility of $h \rightarrow \gamma Z$ and the $h \rightarrow V \, F$ spectra, the latter spectra can be thought of a leading indirect probe of $\mathscr{C}_{\gamma Z}$.
In the $\mathcal{O}$ basis the spectrum of interest is given by
\begin{eqnarray}\label{ZspectraP}
\frac{1}{v_T^2}\frac{ {\rm d} \Gamma }{{\rm d} q^2 } &=& \frac{1}{3} \frac{ {\rm d} \Gamma_0 }{{\rm d} \hat{q}^2 } \, \biggl\{\frac{1}{v_T^2} + 2 C_{H \Box} - \frac{\mathcal{P}_{T}}{4} + \frac{g_V + g_A}{(g_V)^2 + (g_A)^2} \frac{C_{He}}{2} + \frac{g_1^2 \, g_2^2}{2 (g_1^2 + g_2^2)} (\mathcal{P}_B + \mathcal{P}_W)\biggl\} , \nonumber \\
&+& \frac{1}{24} \frac{ {\rm d} \Gamma_0 }{{\rm d} \hat{q}^2 } \left( \mathcal{P}_{B} + \mathcal{P}_{W} \right) \frac{\bar{s}^2 \, \bar{c}^2 (g_1^2 + g_2^2) \left[g_A + g_V(1 - 4 \, \bar{c}^2) \right] }{\left(g_V^2 + g_A^2 \right)} , \nonumber \\
&+& 8 \, \hat{q}^2 \, \frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \, \left[ \bar{s}^2 \, C_{HB} \, - \frac{(\mathcal{P}_{HB} \, g_1^2 + \mathcal{P}_{HW} \, g_2^2)}{4}
\frac{\left(g_A^2 - g_V^2\right)}{\left(g_V^2 + g_A^2 \right)} \right] \, \left(\frac{\hat{q}^2 -1 + \rho}{\lambda^2(\hat{q}^2,\rho) + 12 \rho \, \hat{q}^2} \right), \nonumber \\
&-& 4 \, \frac{\bar{s}^2_\theta \, \bar{c}^2_\theta \, m_h^2}{v_T^2} \frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \, (\mathcal{P}_{HB} - \mathcal{P}_{HW}) \,\left( \frac{g_V}{(g_V)^2 + (g_A)^2}\right) \,
\left(\frac{\rho \, (\rho - \hat{q}^2) (\rho + \hat{q}^2 -1)}{\lambda^2(\hat{q}^2,\rho) + 12 \rho \, \hat{q}^2} \right), \nonumber \\
&-& \frac{2 \, \bar{s}^2 \, \bar{c}^2 \, m_h^2}{3 \, v_T^2} \frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \, (\rho - \hat{q}^2) \left( \frac{g_V}{(g_V)^2 + (g_A)^2}\right) \left( \mathcal{P}_{HB} +\mathcal{P}_{B} -\mathcal{P}_{HW} -\mathcal{P}_{W} \right), \nonumber \\
&+& \frac{2 \, m_h^2}{3 \, v_T^2} \frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \, (\rho + \hat{q}^2) \left( \bar{s}^2 (\mathcal{P}_{HB} +\mathcal{P}_{B}) + \bar{c}^2 (\mathcal{P}_{HW} +\mathcal{P}_{W})\right), \nonumber \\
&-& \frac{ {\rm d} \Gamma_0}{{\rm d} \hat{q}^2 } \, \frac{(\rho - \hat{q}^2)}{6 \, \rho} \, \left(C_{He} \frac{\left(g_V - g_A\right)}{\left(g_V^2 + g_A^2 \right)} \right).
\end{eqnarray}
Taking into account the EoM subtlety in the strong LEP limit, imposed to use constraints due to TGC vertex bounds, one finds
\begin{align}
\mathcal{P}_{B} + \mathcal{P}_{W} &\rightarrow 0, \\
\mathcal{P}_{HW} + \mathcal{P}_{W} &\rightarrow 0, \\
\mathcal{P}_{HB} + \mathcal{P}_{B} &\rightarrow 0, \\
\mathcal{P}_{HW} + \mathcal{P}_{HB} &\rightarrow 0, \\
\mathcal{P}_{HB} +\mathcal{P}_{B} -\mathcal{P}_{HW} -\mathcal{P}_{W} &\rightarrow 0.
\end{align}
Consistent between the bases, the TGC verticies are not related to $h \rightarrow V \, F$ measurements in this limit.
The combinations of Wilson coefficients that vanish in the strong LEP limit appear frequently in calculations using the $\mathcal{O}$ basis.
\section{Conclusions}\label{conclude}
There are 2499 free parameters in the dimension six operator corrections to the SM in the SMEFT. As such, it is inevitable that theoretical and experimental assumptions will be made to simplify the study of the SMEFT.
Although this can be done in a consistent manner using approximate symmetries that constrain the S matrix, it is likely that constructed
observables will also be used.
Any operator basis can be used to study the SMEFT and no basis is superior or inferior to any other.
At the same time, it is an unfortunate fact that the potential for a functional redundancy in the $\mathcal{O}$ basis
is directly related to imposing the assumption of a SM like $V$ coupling to leptons in future experimental studies, i.e the limit of strong LEP constraints
in constructed LHC observables.
We have illustrated the issues involved in avoiding the potential inconsistencies of constructed observables considering the oblique parameters,
TGC verticies, and the relation between the TGC verticies and the $h \rightarrow V \, F$ spectra. Using multiple bases, and keeping note of the EoM relations
between bases can make the non intuitive constraints, and defining conditions, of constructed observables transparent.
As the data set from LHC advances, ever more complicated final states will be studied, and derived constraints -- or deviations -- in such measurements
will be incorporated into the SMEFT. It is essential that such studies are performed in a consistent and basis independent manner when constructed observables are used.
\section*{Acknowledgements}
We thank Martin Gonzalez-Alonso, Cliff Burgess and particularly Gino Isidori for insightful and useful discussions related to this work. We also thank Rodrigo Alonso, Cliff Burgess, Ben Grinstein, Aneesh Manohar
and Gino Isidori for comments on the manuscript.
| {
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{"url":"http:\/\/funcall.blogspot.com\/2011\/09\/bit-of-problem.html","text":"## Friday, September 23, 2011\n\n### A bit of a problem\n\nOnce again, here is a plot of the actual garbage collection counts as a function of heap size:\nand we mention that we want to be \u201cbeyond the knee\u201d in the curve. So where is that \u201cknee\u201d?\n\nThe green line is the plot of `f(x) = 6100000\/x`, or, to put it another way, the plot of `x \u00d7 y = 6100000`. The knee in the curve ought to be where `x = y = \u221a6100000 \u2248 2470`. But look at the chart! Just by eyeballing it, we can see that 2470 is nowhere near what anyone would call the knee.","date":"2014-09-30 17:51:18","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\": 0.8620688319206238, \"perplexity\": 629.4866080224942}, \"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-2014-41\/segments\/1412037663060.18\/warc\/CC-MAIN-20140930004103-00331-ip-10-234-18-248.ec2.internal.warc.gz\"}"} | null | null |
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P272 half hourly metering systems will mean more accurate billing, and more information on energy usage for businesses. | {
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North Korea has opened a Russian-funded transhipment terminal in the north eastern port of Rajin, according to a report carried by the Korean Central News Agency (KCNA) on July 18.
The terminal cost over $250 million to construct and can facilitate the transfer of up to five million tonnes of freight per year.
KCNA said the wharf "provided another foundation for making a positive contribution to boosting the friendly and cooperative relations between the two countries and economic and trade relations among countries of Europe and Northeast Asia".
Before completion of the project the port's freight handling capacity was four million tonnes annually.
The wharf's location is strategically important for the project as Rajin is the furthest northern ice free port in the region, meaning that a continuous flow of trade can be maintained during winter.
The KCNA report also featured a picture of a Chinese cargo vessel already making use of the new facilities. At 44 thousand tonnes, the ship is nearly twice the size of the largest vessel in the North Korean fleet.
The new facility is part of the Rajin-Khasan Project, which also includes the reconstruction of a 54 km railway that links the port to Khasan on the Russian border.
"The successful reconstruction of the Rajin-Khasan railway section and the completion of Wharf No. 3 of Rajin Port are precious products of the vitality of the Moscow Declaration signed by leader Kim Jong Il and President V. [Putin]", said Jon Kil Su, Minister of Railways for North Korea, during the opening ceremony.
Russian shipping news website Port News reported that the work on the project was carried out by DPRK-Russian joint venture RasonKonTrans and the JSC Trading House RZD, a subsidiary of Russian Railways, and the local port authority of Rajin.
The Rajin-Khasan and Trans-Korean projects are part of the wider Moscow Declaration, an agreement to boost trade and economic cooperation between North Korea and Russia, signed in July 2000 by Vladimir Putin and Kim Jong Il.
President of the Russian Railways Company Yury Viktorovich Bochlkarev, the Consul General of the Russian Federation in Chongjin, and other dignitaries were also present at the ceremony.
Officials from South Korea's Korea Railroad Corporation (KORAIL), Hyundai Merchant Marine and the POSCO steel making company also visited Rason City near Rajin between June 15 and 22 to assess the project.
Subscribe to read the remaining 388 words of this article.
Hong Won Choi is an NK News contributor based in London. | {
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\section{Introduction} \label{sec:intro}
The methylidine cation $ \mathrm{CH}^+ $ was the first molecular ion detected in the interstellar medium \citep{Douglas1941}. In the cold and dilute environment of diffuse clouds it radiatively relaxes to an internal temperature of $ T_\mathrm{ex}\sim\SI{3}{K} $ and is predominantly found in its vibrational $ v=0 $ and rotational $J=0$ ground state \citep{Godard2013}.
Despite its early detection, the high abundance of $ \mathrm{CH}^+ $ in diffuse clouds still remains an astrochemical puzzle \citep{Indriolo2010}. This is because the only known production mechanism via C$ ^+ $,
\begin{equation}
\mathrm{C}^+ + \mathrm{H_2} \rightarrow \mathrm{CH}^+ + \mathrm{H} , \label{eq:production}
\end{equation}
is endothermic by $ \sim\SI{0.4}{eV} $ \citep{Gerlich1987} and cannot proceed at quiescent diffuse cloud kinetic temperatures\footnote[1]{We distinguish between the kinetic temperature $T_\mathrm{k}$ of the gas in the cloud, which characterizes the velocity distributions for all particles, and the excitation temperature $T_\mathrm{ex}$, which describes the internal state-population distribution of the CH$^+$ molecules. The kinetic temperature is sometimes also called the translational temperature.} of $ T_\mathrm{k}=\SI[parse-numbers=false]{40-130}{K}$ ($k_\mathrm{B}T_\mathrm{k}=\SI[parse-numbers=false]{3-11}{meV} $, where $k_\mathrm{B}$ is the Boltzmann constant; \citealt{Shull2021})\footnote[2]{In principle, CH$^+$ can also form via
\begin{displaymath}
\mathrm{C}^{2+} + \mathrm{H_2} \rightarrow \mathrm{CH}^+ + \mathrm{H}^+, \label{eq:production2+}
\end{displaymath}
but this reaction is negligible because of its experimentally-known low rate coefficient of $ \lesssim\SI{2e-12}{cm^3\,s^{-1}} $ and the low abundance of C$ ^{2+}$ compared to C$^+$ in diffuse clouds (\citealt{Plasil2021} and references therein).}.
A number of mechanisms have been proposed to address the CH$^+$ abundance puzzle, which drive Equation (\ref{eq:production}) by locally heating the gas or by producing a relative velocity between ions and neutrals that increase the effective temperature of the reaction. Such mechanisms include: Alfv\'{e}n waves \citep{Federman1996}, turbulent mixing \citep{Lesaffre2007, Valdivia2017}, magnetohydrodynamic shocks \citep{Lesaffre2013}, turbulent dissipation in vortices \citep{Godard2014}, ion-neutral drift \citep{Valdivia2017, Moseley2021}, and magnetohydrodynamic turbulence in clouds \citep{Moseley2021}. These processes are hypothesized to locally heat the gas or raise the effective temperature to greater than $ \SI{200}{K} $ ($k_\mathrm{B}T_\mathrm{k}=\SI{17}{meV}$) and predict CH$^+$ abundances that can differ from one another by several orders of magnitude. One aim of our work here is to reduce the uncertainties in the CH$^+$ chemical kinetics data so that remaining discrepancies between models and observations can be more reliably used to constrain the astrophysical properties of the diffuse gas and not be attributed to a lack of understanding of the underlying chemical kinetics.
Using $ \mathrm{CH}^+ $ observations to constrain which of these mechanisms is driving the observed abundances, requires an accurate understanding of the underlying $ \mathrm{CH}^+ $ chemistry. Reaction (\ref{eq:production}) has been studied both experimentally \citep{Hierl1997} and theoretically \citep{Wu2021}. The dominant destruction reactions are \citep{Myers2015}
\begin{eqnarray}
\mathrm{CH}^+ + \mathrm{H}\ \ \, &\rightarrow&\ \mathrm{C}^+ + \mathrm{H_2} , \label{eq:destructionH}\\
\mathrm{CH}^+ + \mathrm{H_2}\ &\rightarrow&\ \mathrm{CH_2}^+ + \mathrm{H} , \label{eq:destructionH2}\\
\mathrm{CH}^+ + \mathrm{e^-}\ &\rightarrow&\ \mathrm{C} + \mathrm{H} . \label{eq:dr}
\end{eqnarray}
The rate coefficients for hydrogen abstraction via Reaction (\ref{eq:destructionH}) and dissociative recombination (DR) in Reaction (\ref{eq:dr}) have been calculated theoretically (e.g., \citealt{Faure2017}). We are unaware of any theoretical calculations for Reaction (\ref{eq:destructionH2}) more sophisticated than the Langevin value \citep{Kim1975}. Experimentally, \citet{Plasil2011} and \citet{Gerlich2011} measured the rate coefficients for Reaction (\ref{eq:destructionH}) at $ \SI[parse-numbers=false]{12-100}{K} $ and Reaction (\ref{eq:destructionH2}) at $ \SI{50}{K} $, respectively, approximating diffuse cloud environments. However, we are unaware of any cryogenic measurement for the DR process, Reaction (\ref{eq:dr}).
In diffuse clouds, DR represents the largest uncertainty among all chemical reactions that form and destroy $ \mathrm{CH}^+ $. Calculations of the DR cross section are difficult due to the large number of intermediate states involved in the dissociation dynamics \citep{Chakrabarti2018,Mezei2019}. Experimentally, single-pass merged-beams experiments have been performed by \citet{Mitchell1978}, where the ions were likely to be electronically, vibrationally, and rotationally excited. The only DR rate-coefficient results for electronically and vibrationally relaxed ($ v=0 $) ions were obtained at the room-temperature Test Storage Ring (TSR; \citealt{Amitay1996}). However, the energy resolution in this measurement was $ \sim \SI{20}{meV} $, i.e., much higher than the collision energies in quiescent gas of a diffuse cloud. Additionally, the ions occupied a broad range of rotational states with only $\sim6\%$ in the $J=0$ state. Given that recent experimental work has found that the $J$-specific DR rate coefficient can vary by over an order of magnitude \citep{Novotny2019}, the uncertainty in the experimentally derived CH$^+$ rate coefficient from the above two CH$^+$ works is likely to be a previously unrecognized order of magnitude. This makes the DR process the most uncertain reaction in the chemistry forming and destroying $ \mathrm{CH}^+ $ in diffuse clouds.
With the development of the Cryogenic Storage Ring (CSR; \citealt{vonHahn2016}), it has become possible to perform electron-ion merged-beams collision studies with internally cold molecular ions. Recent CSR DR experiments with $ \mathrm{HeH}^+ $ revealed a strong $ J $-dependence of the rate coefficient \citep{Novotny2019}, demonstrating the importance of probing rotationally cold ions. In this work, we use the CSR to create an electronically and rovibrationally cold ($ v=0 $, $ J=0-2 $) ensemble of $ \mathrm{CH}^+ $ molecules, measure DR with an order-of-magnitude improved collision energy resolution of $ \sim\SI{2}{meV} $, and extract a rate coefficient for the rovibrational $ \mathrm{CH}^+ $ ground state appropriate for diffuse cloud conditions, thus refining the most uncertain reaction in the $ \mathrm{CH}^+ $ diffuse cloud chemistry.
\section{Experimental} \label{sec:exp}
\begin{figure*}[]
\epsscale{1}
\plotone{P1.pdf}
\caption{(a) Schematic of the CSR experimental setup for DR measurements of CH$^+$ and probing of its internal level populations. See text for details. (b) Measured relative $ J $-level populations in the CSR for storage times of $ \SI[parse-numbers=false]{60-80}{s} $ with statistical one-sigma error bars. (c) Relative $ J $-level populations predicted in TSR for a thermal equilibrium at $ T=\SI{300}{K} $.}
\label{fig:setup}
\end{figure*}
In order to determine the DR rate coefficient of the $ X^1\Sigma^+(v=0,J=0) $ ground state of $ \mathrm{CH}^+ $, we have used the experimental setup sketched in Figure \ref{fig:setup}(a). We stored and phase-space cooled a $ \mathrm{CH}^+ $ ion beam inside the CSR, prepared internally cold ions, utilized \emph{in situ} diagnostic methods to follow time-dependent changes of the internal level populations, and finally measured DR with well-defined mixtures of only a few $ J $ levels to infer the rate coefficient for the $ \mathrm{CH}^+ $ ground electronic, vibrational, and rotational state.
Phase-space cooling\footnote[3]{Commonly also called electron cooling in literature} of the $ \mathrm{CH}^+ $ ion beam was applied to reduce its diameter and energy spread \citep{Poth1990}. The stored ion beam was overlapped with a larger diameter, nearly mono-energetic electron beam in the CSR electron cooler section (Figure \ref{fig:setup}(a)). Phase-space cooling was reached by matching the electron and ion beam positions and velocities for the first $ \SI{21}{s} $ of storage, before the DR measurements began. Phase-space cooling was also regularly applied during the collision experiments to maintain the beneficial ion-beam properties.
Internal cooling of the initial electronically and rovibrationally hot ion beam, was achieved by radiative decay and by inelastic collisions with the merged, nearly monoenergetic, velocity-matched electron beam. Radiative cooling proceeded by interactions with the $ \sim\SI{20}{K} $ two-component blackbody radiation field in the CSR \citep{Kalosi2022}. This dominated the initial internal cooling of highly excited states. After $ \SI{21}{s} $ of storage, most ions were in their ground electronic state $ X^1\Sigma^+ $ with a rovibrationally cold distribution of $ v=0$ and $J=0-2 $. Further rotational cooling was driven by inelastic electron collisions and advanced towards an effective internal ion temperature of $ \sim\SI{26}{K} $ \citep{Kalosi2022}.
\begin{figure*}[]
\epsscale{0.8}
\plotone{P2.pdf}
\caption{Merged-beams DR rate coefficient (blue line) for the CH$^{+}$ ground state $X^1\Sigma^+ (v=0,J=0)$ with the statistical one-sigma uncertainties (blue shaded area). The one-sigma uncertainty of the absolute scale is $ \pm13 \% $. The room-temperature TSR results of \citet{Amitay1996}, which have a $ \pm50 \% $ absolute scaling systematic uncertainty, are shown by the gray squares. The symbols and lines on top of the figure indicate the full width at half maximum (FWHM) energy resolution $ \Delta E $ of the CSR and TSR measurements. The divergence of the CSR from the TSR data for $ E_\mathrm{d}\gtrsim\SI{10}{eV} $ is due to contribution from dissociative excitation (see text). The data behind the TSR and CSR graphs are both available. The latter are part of Figure \ref{fig:ratecoeffmeas}(b).}
\label{fig:ratecoeff}
\end{figure*}
We monitored the internal cooling \emph{in situ} by probing $ J $-level-resolved photodissociation of $ \mathrm{CH}^+ $ with an optical parametric oscillator (OPO) laser at a tunable wavelength near $ \sim\SI{305}{nm} $ and measuring the resulting H atoms on a neutral-fragment detector (Figure \ref{fig:setup}(a); \citealt{Kalosi2022}). A combined radiative and collisional equilibrium was reached by $ t\sim\SI{70}{s} $, with the $ J=0 $ and $ J=1 $ levels dominating the rotational populations (Figure \ref{fig:setup}(b)). In contrast to the room-temperature TSR rotational distribution (Figure \ref{fig:setup}(c)), in the present work over $ 40\% $ of the ions populate the $J=0 $ level. In addition to the $ X^1\Sigma^+ $ ground state, there is a small fraction of ions ($ <13\% $ at $ \SI{21}{s} $) in the metastable $ a^3\Pi $ state, which decays with a lifetime of $ \sim\SI{10}{s} $ and which we monitored by its unique DR fragmentation pattern. In this way, we obtained full knowledge about the time-dependent populations of all contributing rotational levels ($ p_J(t) [J=0-2] $) and the metastable state ($ p_\mathrm{m}(t) $), see Appendix \ref{app:rotational}.
DR rate-coefficient measurements on the internally cold $ \mathrm{CH}^+ $ ions used the electron beam from the electron cooler as a merged-beams collision target. Neutral C and H atoms resulting from DR in the interaction region of the electron cooler were collected downstream by our neutral-fragment detector. From the measured count rate we determined the merged-beams DR rate coefficient $ \alpha^\mathrm{mb} $, related to the DR cross section $\sigma$ via
\begin{equation} \label{eq:convolutionShort}
\alpha^\mathrm{mb}=\langle \sigma v_\mathrm{c} \rangle .
\end{equation}
Here, $\sigma(E)$ and the collision velocity $ v_\mathrm{c}(E) $ are functions of the collision energy $ E $ and $\langle...\rangle $ designates the mean value over the full distribution of $ v_\mathrm{c} $ inside the electron-ion overlap region. This distribution depends on the transverse and longitudinal temperatures in the rest frame of the electron beam. We achieved $ k_\mathrm{B}T_\perp=\SI[parse-numbers=false]{2.0^{+1.0}_{-0.5}}{meV} $ and $ k_\mathrm{B}T_\parallel\sim\SI{0.5}{meV} $, respectively, corresponding to a collision energy resolution of $ \sim\SI{2}{meV} $ at matched velocities. Here and throughout, all uncertainties are quoted at an one-sigma confidence level with $ ^{+}_{-}$ for asymmetric uncertainties and numbers in parenthesis for symmetric uncertainties.
In order to study the energy dependence of the DR process, electron energies were scanned by adjusting the potential of several drift tubes in the merged-beams interaction section (Figure \ref{fig:setup}(a)). This yielded $ \alpha^\mathrm{mb}(E_\mathrm{d}) $, where the detuning energy $ E_\mathrm{d} $ is defined as the collision energy of ideal, mono-energetic electron and ion beams, given by the drift-tube potentials. Details on the electron-ion collision setup and on the absolute scaling of $ \alpha^\mathrm{mb}(E_\mathrm{d}) $ are given in Appendices \ref{app:collSetup} and \ref{app:absRate}, respectively.
The astrophysically relevant $ \mathrm{CH}^+ $ DR rate coefficient was inferred from our storage-time-dependent measurements of $ \alpha^\mathrm{mb}(E_\mathrm{d},t) $. For storage times of $ t=\SI[parse-numbers=false]{21-100}{s} $, we used the experimentally determined internal level populations to extract the merged-beams rate coefficient $ \alpha^{\mathrm{mb}}_{J=0}(E_\mathrm{d}) $ for the pure $ J=0 $ state (Appendix \ref{app:stateResolved}). From this, we generated the DR kinetic rate coefficient $ \alpha^\mathrm{k}_{J=0}(T_\mathrm{k}) $ for the $ J=0 $ state (Appendix \ref{app:deconvolution}), needed for diffuse cloud chemistry at a gas kinetic temperature $ T_\mathrm{k} $.
\newpage
\section{Results} \label{sec:res}
\subsection{Merged-beams rate coefficient}
The rate coefficient $\alpha^{\mathrm{mb}}_{J=0}(E_\mathrm{d}) $ in Figure \ref{fig:ratecoeff} decays by two orders of magnitude in the energy range from $ \SI[parse-numbers=false]{10^{-5}}{eV} $ to $ \SI[parse-numbers=false]{10^{-2}}{eV} $ and shows various resonances in the region of $ \SI[parse-numbers=false]{0.03-2}{eV} $. These resonances can be attributed to capture into neutral Rydberg states of the CH molecule and subsequent coupling to a dissociative state (\citealt{Amitay1996}, \citealt{Mezei2019}). At energies beyond $ \sim\SI[parse-numbers=false]{3.6}{eV} $, a non-resonant DR channel opens up and leads to an increased DR rate coefficient (\citealt{Amitay1996}; \citealt{Chakrabarti2018}). In addition, $ \mathrm{CH}^+ $ dissociative excitation (DE) sets in at a threshold of $ \sim\SI{3.6}{eV} $ (\citealt{Bannister2003}; \citealt{Chakrabarti2017}), which we cannot distinguish from DR owing to our experimental setup.
Comparing our data to the room-temperature TSR data of \citet{Amitay1996}, which had rotational populations close to the simulation shown in Figure \ref{fig:setup}(c), we obtain information on the DR rotational level dependence at $ E_\mathrm{d}>\SI[parse-numbers=false]{0.03}{eV} $ and on measurement-related effects. At $ \SI[parse-numbers=false]{0.03-2}{eV} $, the resonant structures and absolute values agree well between the two data sets, despite the differing $ J $-level populations. This indicates the absence of major $ J $-dependent resonances at detuning energies above $ \SI[parse-numbers=false]{0.03}{eV} $. This conclusion is corroborated by the close match of the CSR results for the $ J=0 $ and $ J=1 $ merged-beams rate coefficients for $ E_\mathrm{d}>\SI{0.03}{eV} $ (Appendix \ref{app:stateResolved}). At $ E_\mathrm{d}\gtrsim\SI[parse-numbers=false]{3.6}{eV} $, due to a measurement-related effect, our data are enhanced compared to the TSR results, where it was possible to discriminate between DR and DE. In addition, our collision geometry imposes a rate coefficient baseline of $ \sim\SI{6e-9}{cm^3\,s^{-1}} $ to the data, which becomes visible at $ E_\mathrm{d}>\SI[parse-numbers=false]{2}{eV} $ (Appendix \ref{app:absRate}). Both effects are accounted for in the determination of the $ \mathrm{CH}^+ $ DR kinetic temperature rate coefficient (Appendix \ref{app:deconvolution}).
Due to the improved energy resolution of the CSR depicted on top of Figure \ref{fig:ratecoeff}, we were able to probe the diffuse cloud collision-energy regime ($\SI[parse-numbers=false]{0.001-0.02}{eV}$), where no experimental data have previously been available. Our results reveal a remarkable increase in the DR rate coefficient towards lower detuning energies. This feature is considerably stronger than the $ E_\mathrm{d}^{-1/2} $ behavior that would result from a typical $ E^{-1} $ cross section dependence of non-resonant DR \citep{Mitchell1990}. This can be attributed either to a constructive resonance near $ E_\mathrm{d}=\SI{0}{eV} $ or a destructive resonance near $ E_\mathrm{d}=\SI{0.01}{eV} $. We also note that at $ E_\mathrm{d}=\SI{0.012}{eV} $ the $ J=0 $ rate coefficient is lower by a factor of $ 5.4^{+5.0}_{-2.0}$ compared to $ J=1 $. However, we find no significant deviation for $ E_\mathrm{d}<\SI{0.008}{eV} $ and $ E_\mathrm{d}>\SI{0.02}{eV} $ (Appendix \ref{app:stateResolved}).
\begin{figure*}[]
\epsscale{1.15}
\plotone{P3.pdf}
\caption{CH$^{+}$ DR kinetic temperature rate coefficient and its relevance for the CH$^{+}$ destruction in diffuse interstellar clouds. (a) The thick solid line shows the experimentally determined $ J=0 $ kinetic temperature rate coefficient of this work. The one-sigma error band in gray represents the quadrature sum of all systematic uncertainties. The dashed lines indicate rate coefficients from the UMIST (\citealt{McElroy2013}, upper line) and KIDA (\citealt{Wakelam2012}, lower line) astrochemistry databases. The calculated $ J=0 $ rate coefficient of \citet{Mezei2019} is drawn as dotted line. (b) Diffuse cloud parameter ranges where DR is the dominant CH$^+$ destruction process (colored) for a standard electron fraction ($ \chi_{\mathrm{e}}=\SI{1.6e-4}{} $) and for enhanced values of $ \chi_{\mathrm{e}} $ as given. One-sigma error bars for $ f_\mathrm{H_2}=1 $ and $ f_\mathrm{H_2}=0 $ are shown at the top and bottom of the figure, respectively. They were calculated from the combined systematic uncertainties for the rate coefficients $ \alpha_\mathrm{H} $ ($\sim50\%$, \cite{Plasil2011}), $ \alpha_\mathrm{H_2} $ ($\sim20\%$, \cite{Gerlich2011}), and $ \alpha^\mathrm{k}_{J=0} $ ($ \sim20\% $, this work).}
\label{fig:plasmarate}
\end{figure*}
\subsection{Kinetic temperature rate coefficient}
To obtain the DR kinetic temperature rate coefficient $ \alpha^\mathrm{k}_{J=0}(T_\mathrm{k}) $ for the pure $ J=0 $ state, we converted the $\alpha^{\mathrm{mb}}_{J=0}(E_\mathrm{d}) $ data (Figure \ref{fig:ratecoeff}) as described in Appendix \ref{app:deconvolution}. The result in Figure \ref{fig:plasmarate}(a) shows that our improved experimental energy resolution enables us to reliably determine the kinetic rate coefficient down to $ T_\mathrm{k}\sim\SI{10}{K} $. With the broad range of collision energies probed, we can provide data up to $ T_\mathrm{k}\sim\SI{40000}{K} $, close to the temperature equivalent of the dissociation threshold $ D_\mathrm{0}/k_\mathrm{B}\sim\SI{47000}{K} $ \citep{Hechtfischer2002}. The kinetic rate-coefficient shape can be approximated by a $ T_\mathrm{k}^{-0.5} $ behavior with two broad peaks at low and high temperatures of $ T_\mathrm{k}\sim10 $ and $ \sim\SI{8000}{K} $, respectively. These can be attributed to the maxima in the merged-beams rate coefficient at $ E_\mathrm{d}\lesssim\SI{0.005}{eV} $ and $ E_\mathrm{d}=\SI[parse-numbers=false]{0.2-2}{eV} $ in Figure \ref{fig:ratecoeff}, respectively. A discussion about the uncertainties displayed as an error band is presented in Appendix \ref{app:deconvolution}.
Figure \ref{fig:plasmarate}(a) compares our experimental results to values in the UMIST \citep{McElroy2013} and KIDA \citep{Wakelam2012} astrochemistry databases. Here, the UMIST entry cites a review by \citet{Mitchell1990} reporting the results of a single-pass merged-beams experiment by \citet{Mitchell1978}, where multiple electronic, vibrational, and rotational states were likely to be involved. Thus, those results are not expected to represent diffuse cloud conditions. The KIDA rate coefficient is referred to as estimated. Both database rate coefficients follow a simple behavior near $ T_\mathrm{k}^{-0.5} $ and clearly miss the temperature-dependent features seen in our data. In the temperature regime for quiescent gas in diffuse clouds, the new CSR results exceed the database values by up to a factor of 6. Conversely, for gas heated by shocks or turbulence, our data are up to a factor of $2.5$ lower.
We also provide a comparison to multichannel quantum defect theory calculations for the CH$^{+} (J=0) $ DR cross section \citep{Mezei2019}, which we converted into a kinetic rate coefficient as shown in Appendix \ref{app:deconvolution}. Compared to their results, our data are significantly larger at low temperatures. In particular, in the temperature regime of diffuse clouds ($ T_\mathrm{k}=\SI[parse-numbers=false]{40-130}{K} $) theory underestimates the DR rate coefficient by up to a factor of $ 7 $. For heated gas at $ T_\mathrm{k}\sim\SI{1000}{K} $, the two results match but begin to diverge again for $ T_\mathrm{k}\gtrsim\SI{3000}{K} $, where our data exceed theory by up to a factor of 7.
\section{Astrochemical implications} \label{sec:astro}
Understanding the CH$^{+}$ abundance in diffuse clouds has been limited, in part, by uncertainties in the relevant chemistry. Our measurements of the CH$^{+}(J=0)$ DR rate coefficient reduce its uncertainty from above an order of magnitude to $ \sim20\% $ and thus provide a significant improvement in the reliability of the CH$^{+}$ chemistry. Remaining discrepancies between observations and models can now be more solidly used to constrain the physical properties of these clouds. In Appendix \ref{app:plasmaRateResults} we provide analytical representations of our kinetic temperature rate coefficient results for ready incorporation into astrochemical databases.
Based on our measured DR rate coefficient, we can determine the relative importance of the CH$^{+}$ destruction mechanisms given by Reactions (\ref{eq:destructionH})-(\ref{eq:dr}). A temperature-dependent rate coefficient $ \alpha_\mathrm{H}(T_\mathrm{k}) $ for Reaction (\ref{eq:destructionH}) with pure CH$^{+} (J=0)$ ions was derived from experimental data by \citet{Plasil2011} (Equation (6a) of their work), which we extrapolate up to $ T_\mathrm{k}=\SI{600}{K} $. For CH$^{+}$ destruction by molecular hydrogen (Reaction (\ref{eq:destructionH2})), we use the rate coefficient $ \alpha_\mathrm{H_2}=\SI{1.2e-9}{cm^3\,s^{-1}} $ from \citet{Smith1977} and \citet{Gerlich2011}, which were obtained for fully thermal temperatures, i.e., gas kinetic and excitation temperatures, of $T_\mathrm{k}=T_\mathrm{ex}=\SI{300}{} $ and $ \SI{50}{K} $, respectively. Given the match of the rate coefficients at different $ T_\mathrm{k} $, we assume that $ \alpha_\mathrm{H_2}(\SI{10}{K}\leq T_\mathrm{k}\leq\SI{600}{K})=\SI{1.2e-9}{cm^3\,s^{-1}}$.
To compare destruction by atomic and molecular hydrogen collisions to destruction by DR, we define a critical DR rate coefficient $ \alpha_\mathrm{cr,DR} $ at which the CH$^{+}$ destruction by both processes is balanced. Diffuse clouds are characterized by their electron densities $ n_\mathrm{e} $, their atomic and molecular hydrogen number densities, $n(\mathrm{H})$ and $n(\mathrm{H_2})$, and related quantities: the hydrogen nucleus number density $ n_\mathrm{H}=n(\mathrm{H})+2n(\mathrm{H_2}) $, the molecular fraction $f_\mathrm{H_2}=2n(\mathrm{H_2})/n_\mathrm{H}$, and the electron fraction $ \chi_\mathrm{e}=n_\mathrm{e}/n_\mathrm{H}$. The critical DR rate coefficient is then given by
\begin{equation} \label{eq:alphaCr}
\alpha_\mathrm{cr,DR}(T_\mathrm{k})=\frac{(1-f_\mathrm{H_2})\alpha_\mathrm{H}(T_\mathrm{k})+f_\mathrm{H_2}\alpha_\mathrm{H_2}(T_\mathrm{k})/2}{\chi_\mathrm{e}}.
\end{equation}
Destruction by DR dominates over that by hydrogen collisions if $ \alpha^\mathrm{k}_{J=0}>\alpha_\mathrm{cr,DR} $.
Combining Equation (\ref{eq:alphaCr}) with the condition $ \alpha^\mathrm{k}_{J=0}>\alpha_\mathrm{cr,DR} $, we calculated the diffuse cloud parameters ($ T_\mathrm{k} $, $ f_\mathrm{H_2}$), at which CH$^{+}$ destruction is mainly due to DR. The result is presented in Figure \ref{fig:plasmarate}(b). Assuming that $ \chi_\mathrm{e}=\SI{1.6e-4}{} $ matches the relative C$ ^+ $ abundance \citep{Sofia2004}, we find that for cold and mainly atomic gas ($ T_\mathrm{k}\lesssim\SI{50}{K} $, $ f_\mathrm{H_2}\lesssim0.05 $) DR is the dominant destruction process. Above this temperature, hydrogen abstraction becomes dominant. In case of higher molecular fractions ($ f_\mathrm{H_2}\gtrsim0.1 $), we find that for gas temperatures of $ T_\mathrm{k}>\SI{40}{K} $ destruction by DR is negligible compared to hydrogen abstraction. Given the assumptions made for $ \alpha_\mathrm{H} $ and $ \alpha_\mathrm{H_2} $, the uncertainties displayed in Figure \ref{fig:plasmarate}(b) are likely underestimated. For example, no lower limit for the $ J=0 $ rate coefficient $ \alpha_\mathrm{H} $ could be determined by \citet{Plasil2011}, which might lead to a higher relevance of DR in fully atomic clouds, beyond the results of Figure \ref{fig:plasmarate}(b).
The results in Figure \ref{fig:plasmarate}(b) can be put into the context of the CH$^{+}$ abundance puzzle in diffuse clouds. In quiescent gas the increased DR kinetic rate coefficient determined in our work leads to a factor of $\sim2$ enhancement in the destruction of CH$^{+}$ than previously assumed. However, because of the local heating mechanisms required to produce CH$^{+}$ and the high reactivity of the molecules, which prevents them from surviving long enough to flow into cold regions of the cloud, very little CH$^{+}$ abundance is expected in cold gas. Instead, the CH$^{+}$ chemistry is expected to take place at $ T_\mathrm{k}\gtrsim\SI{1000}{K} $ \citep{Moseley2021}, where our data indicate that DR can be neglected compared to destruction by hydrogen collisions, even when the electron fraction is increased to $\chi_\mathrm{e}=10^{-2}$.
Our measurements also indicate that destruction by DR can be enhanced in regions with higher cosmic ray ionization rates or lower than average gas density, where increased electron fractions are found. This is the case in the Galactic center \citep{LePetit2016} or in supernova remnants \citep{Priestley2017}. Figure \ref{fig:plasmarate}(b) contains the boundaries for the DR dominated CH$^{+}$ destruction regime for enhanced electron fractions. We emphasize that these boundaries depend on the behavior of $ \alpha_\mathrm{H} $ and $ \alpha_\mathrm{H_2} $ only for $ T_\mathrm{k}\lesssim\SI{600}{K} $, but not above. Figure \ref{fig:plasmarate}(b) demonstrates that increasing the electron fraction by factors of 6 and 60 compared to quiescent gas in diffuse clouds extends the DR dominated temperature regime to $ T_\mathrm{k}\sim100 $ and $ \SI{500}{K} $ respectively, even for fully molecular clouds. Thus, in environments with increased electron fractions and $ T_\mathrm{k}\lesssim\SI{500}{K} $, the results of this work are likely to be important for our understanding of the corresponding CH$^{+}$ chemistry. However, for heated regions in diffuse clouds at $T_\mathrm{k}\gtrsim\SI{1000}{K} $, destruction by hydrogen collisions remains the dominant process, even when considering an electron fraction of $ \chi_\mathrm{e}\sim10^{-2} $.
\section{Conclusions} \label{sec:disc}
With this work we have determined the rate coefficient for the most uncertain reaction in the CH$^+$ chemistry of diffuse clouds, namely the DR kinetic temperature rate coefficient. By mimicking diffuse cloud temperatures inside the CSR and conducting a high-resolution electron-ion merged-beams experiment, we have obtained the gas kinetic temperature dependence of the DR rate coefficient for the CH$ ^+(X^1\Sigma^+,v=0,J=0) $ state, involving all temperatures from quiescent to locally heated gas. We further characterized the implications of our experimental results on diffuse cloud chemistry, based on a simple reaction network that includes the experimental rate coefficients for the competing CH$^+$ reactions with atomic and molecular hydrogen. By reducing the uncertainty for the DR rate coefficient from above an order of magnitude to $ \sim\pm20\% $, our findings will improve the reliability of future studies for local heating mechanisms in diffuse clouds based on CH$^{+}$ abundance observations. DR experiments at the CSR with other key molecular ions are in development and could lead to further improvements in our understanding of chemical and physical properties of interstellar clouds.
\section*{Acknowledgements}
Financial support by the Max Planck Society is acknowledged. D.\ P., A.\ K., and D.\ W.\ S.\ were supported in part by the U.S.\ National Science Foundation Division of Astronomical Sciences Astronomy and Astrophysics Grants program under AST-1907188. We thank Zs.\ J.\ Mezei and I.\ F.\ Schneider for providing us their theoretical DR cross sections from \citet{Mezei2019} for comparison of kinetic temperature rate coefficients in this work.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 80 |
/* $This file is distributed under the terms of the license in LICENSE$ */
package edu.cornell.mannlib.vitro.webapp.utils.jena;
import java.util.HashMap;
import java.util.LinkedList;
import java.util.List;
import java.util.Map;
import org.apache.commons.logging.Log;
import org.apache.commons.logging.LogFactory;
import org.apache.jena.ontology.Individual;
import org.apache.jena.ontology.OntModel;
import org.apache.jena.ontology.OntModelSpec;
import org.apache.jena.ontology.OntResource;
import org.apache.jena.query.Query;
import org.apache.jena.query.QueryExecution;
import org.apache.jena.query.QueryExecutionFactory;
import org.apache.jena.query.QueryFactory;
import org.apache.jena.query.Syntax;
import org.apache.jena.rdf.model.Literal;
import org.apache.jena.rdf.model.Model;
import org.apache.jena.rdf.model.ModelFactory;
import org.apache.jena.rdf.model.ModelMaker;
import org.apache.jena.rdf.model.Property;
import org.apache.jena.rdf.model.RDFNode;
import org.apache.jena.rdf.model.Resource;
import org.apache.jena.rdf.model.ResourceFactory;
import org.apache.jena.shared.Lock;
public class JenaIngestWorkflowProcessor {
private static final Log log = LogFactory.getLog(JenaIngestWorkflowProcessor.class.getName());
private Individual workflowInd;
private ModelMaker modelMaker;
private Map<String,Literal> varMap;
private List<ActionHandler> actionHandlerList;
private JenaIngestUtils utils;
public JenaIngestWorkflowProcessor(Individual workflowInd, ModelMaker modelMaker) {
this.varMap = new HashMap<String,Literal>();
this.workflowInd = workflowInd;
this.modelMaker = modelMaker;
actionHandlerList = new LinkedList<ActionHandler>();
actionHandlerList.add(new ClearModelAction());
actionHandlerList.add(new AddModelsAction());
actionHandlerList.add(new SubtractModelsAction());
actionHandlerList.add(new ExecuteSparqlConstructAction());
actionHandlerList.add(new SplitPropertyValuesAction());
actionHandlerList.add(new ProcessPropertyValueStringsAction());
actionHandlerList.add(new SmushResourcesAction());
actionHandlerList.add(new NameBlankNodesAction());
this.utils = new JenaIngestUtils();
}
public void run() {
run(null);
}
/**
* Runs the workflow
*/
public void run(Individual startingWorkflowStep) {
for (Individual step : getWorkflowSteps(startingWorkflowStep)) {
Individual action = getAction(step);
log.debug("Executing workflow action "+action.getURI());
for (ActionHandler handler : actionHandlerList) {
ActionResult result = handler.handleAction(action);
if (result != null) {
break;
}
}
}
}
/*
* returns the Action related to the supplied WorkflowStep
*/
private Individual getAction(Individual stepInd) {
log.debug("Workflow step: "+stepInd.getURI());
RDFNode actionNode = stepInd.getPropertyValue(WorkflowOntology.action);
if (actionNode != null && actionNode.canAs(Individual.class)) {
return (Individual) actionNode.as(Individual.class);
}
return null;
}
public List<Individual> getWorkflowSteps(Individual startingWorkflowStep) {
List<Individual> workflowSteps = new LinkedList<Individual>();
Individual currentInd = (startingWorkflowStep == null) ? getWorkflowStep(workflowInd.getPropertyValue(WorkflowOntology.firstStep)) : startingWorkflowStep;
while (currentInd != null) {
workflowSteps.add(currentInd);
currentInd = getWorkflowStep(currentInd.getPropertyValue(WorkflowOntology.nextStep));
}
return workflowSteps;
}
private Individual getWorkflowStep(RDFNode stepNode) {
if (stepNode == null) {
return null;
}
if ( (stepNode != null) && (stepNode.canAs(Individual.class)) ) {
Individual nextStepInd = (Individual) stepNode.as(Individual.class);
if (instanceOf(nextStepInd,WorkflowOntology.WorkflowStep)) {
return nextStepInd;
}
}
return null;
}
private boolean instanceOf(Individual ind, Resource type) {
for ( Resource typeRes : (List<Resource>) ind.listRDFTypes(false).toList() ) {
if (!typeRes.isAnon() && typeRes.getURI().equals(type.getURI())) {
return true;
}
}
return false;
}
/*
* gets the appropriate Jena Literal for a Value individual in the model,
* depending on whether the Value is a Variable or a Literal
* At some point
*/
private Literal getValue(RDFNode valueIndNode) {
Individual valueInd = (Individual) valueIndNode.as(Individual.class);
if (instanceOf(valueInd,WorkflowOntology.Literal)) {
RDFNode valueNode = valueInd.getPropertyValue(WorkflowOntology.literalValue);
if ( (valueNode != null) && (valueNode.isLiteral()) ) {
return (Literal) valueNode.as(Literal.class);
}
} else if (instanceOf(valueInd,WorkflowOntology.Variable)){
RDFNode variableNameNode = valueInd.getPropertyValue(WorkflowOntology.variableName);
if ( (variableNameNode != null) && (variableNameNode.isLiteral())) {
return varMap.get( ((Literal)variableNameNode.as(Literal.class)).getLexicalForm() );
}
}
return null;
}
/*
* returns the model represented by the given Node, which is expected to be an Individual of type Model
*/
private Model getModel(RDFNode modelNode) {
if (modelNode == null) {
return null;
}
Individual modelInd = (Individual) modelNode.as(Individual.class);
String modelNameStr = ((Literal)modelInd.getPropertyValue(WorkflowOntology.modelName).as(Literal.class)).getLexicalForm();
// false = strict mode off, i.e.,
// if a model already exists of the given name, return it. Otherwise, create a new one.
return modelMaker.createModel(modelNameStr,false);
}
private interface ActionResult {}
private class ActionResultImpl implements ActionResult {}
private interface ActionHandler {
public ActionResult handleAction(Individual actionInd);
}
// ALL THE DIFFERENT ACTION HANDLERS
private class ClearModelAction implements ActionHandler {
public ActionResult handleAction(Individual actionInd) {
if (instanceOf(actionInd,WorkflowOntology.ClearModelAction)) {
Model sourceModel = getModel(actionInd.getPropertyValue(WorkflowOntology.sourceModel));
sourceModel.enterCriticalSection(Lock.WRITE);
try{
// this method is used so that any listeners can see each statement removed
sourceModel.removeAll((Resource)null,(Property)null,(RDFNode)null);
} finally {
sourceModel.leaveCriticalSection();
}
return new ActionResultImpl();
} else {
return null;
}
}
}
private class AddModelsAction implements ActionHandler {
public ActionResult handleAction(Individual actionInd) {
if (instanceOf(actionInd,WorkflowOntology.AddModelAction)) {
Model sourceModel = getModel(actionInd.getPropertyValue(WorkflowOntology.sourceModel));
Model modelToAdd = getModel(actionInd.getPropertyValue(WorkflowOntology.modelToAdd));
Model destinationModel = getModel(actionInd.getPropertyValue(WorkflowOntology.destinationModel));
Boolean applyChangesDirectlyToSource = false;
RDFNode valueNode = actionInd.getPropertyValue(WorkflowOntology.applyChangesDirectlyToSource);
if ((valueNode != null) && (valueNode.isLiteral())) {
applyChangesDirectlyToSource = ((Literal)valueNode.as(Literal.class)).getBoolean();
}
sourceModel.enterCriticalSection(Lock.WRITE);
try {
modelToAdd.enterCriticalSection(Lock.READ);
try {
if (applyChangesDirectlyToSource) {
// TODO: are all listeners notified this way?
sourceModel.add(modelToAdd);
} else {
destinationModel.enterCriticalSection(Lock.WRITE);
try{
destinationModel.add(modelToAdd);
} finally {
destinationModel.leaveCriticalSection();
}
}
} finally {
modelToAdd.leaveCriticalSection();
}
} finally {
sourceModel.leaveCriticalSection();
}
return new ActionResultImpl();
} else {
return null;
}
}
}
private class SubtractModelsAction implements ActionHandler {
public ActionResult handleAction(Individual actionInd) {
if (instanceOf(actionInd,WorkflowOntology.SubtractModelAction)) {
Model sourceModel = getModel(actionInd.getPropertyValue(WorkflowOntology.sourceModel));
Model modelToSubtract = getModel(actionInd.getPropertyValue(WorkflowOntology.modelToSubtract));
Model destinationModel = getModel(actionInd.getPropertyValue(WorkflowOntology.destinationModel));
Boolean applyChangesDirectlyToSource = false;
RDFNode valueNode = actionInd.getPropertyValue(WorkflowOntology.applyChangesDirectlyToSource);
if ((valueNode != null) && (valueNode.isLiteral())) {
applyChangesDirectlyToSource = ((Literal)valueNode.as(Literal.class)).getBoolean();
}
sourceModel.enterCriticalSection(Lock.WRITE);
try {
modelToSubtract.enterCriticalSection(Lock.READ);
try {
if (applyChangesDirectlyToSource) {
// TODO: are all listeners notified this way?
sourceModel.remove(modelToSubtract);
} else {
destinationModel.enterCriticalSection(Lock.WRITE);
try{
destinationModel.add(sourceModel.difference(modelToSubtract));
} finally {
destinationModel.leaveCriticalSection();
}
}
} finally {
modelToSubtract.leaveCriticalSection();
}
} finally {
sourceModel.leaveCriticalSection();
}
return new ActionResultImpl();
} else {
return null;
}
}
}
private class ExecuteSparqlConstructAction implements ActionHandler {
private static final String QUERY_STR_PROPERTY = "http://vitro.mannlib.cornell.edu/ns/vitro/0.7/sparql#queryStr";
public ActionResult handleAction(Individual actionInd) {
if (instanceOf(actionInd,WorkflowOntology.SPARQLCONSTRUCTAction)) {
OntModel sourceModel = ModelFactory.createOntologyModel(OntModelSpec.OWL_MEM);
for (RDFNode node : (List<RDFNode>) actionInd.listPropertyValues(WorkflowOntology.sourceModel).toList()) {
log.debug("SPARQL: adding submodel ");
sourceModel.addSubModel(getModel(node));
}
if (actionInd.getPropertyValue(WorkflowOntology.destinationModel) == null) {
log.debug("Error: destination model for SPARQL Construct action not specified for this action");
return null;
}
Model destinationModel = getModel(actionInd.getPropertyValue(WorkflowOntology.destinationModel));
Model tempModel = ModelFactory.createDefaultModel();
OntResource sparqlQuery = (OntResource) actionInd.getPropertyValue(WorkflowOntology.sparqlQuery);
String queryStr = ((Literal)sparqlQuery.getPropertyValue(ResourceFactory.createProperty(QUERY_STR_PROPERTY))).getLexicalForm();
log.debug("SPARQL query: \n" + queryStr);
Query query = QueryFactory.create(queryStr,Syntax.syntaxARQ);
QueryExecution qexec = QueryExecutionFactory.create(query,sourceModel);
qexec.execConstruct(tempModel);
destinationModel.add(tempModel);
return new ActionResultImpl();
}
return null;
}
}
private class SmushResourcesAction implements ActionHandler {
public ActionResult handleAction(Individual actionInd) {
if (instanceOf(actionInd,WorkflowOntology.SmushResourcesAction)) {
OntModel sourceModel = ModelFactory.createOntologyModel(OntModelSpec.OWL_MEM);
for (RDFNode node : (List<RDFNode>) actionInd.listPropertyValues(WorkflowOntology.sourceModel).toList()) {
sourceModel.addSubModel(getModel(node));
}
Model destinationModel = getModel(actionInd.getPropertyValue(WorkflowOntology.destinationModel));
String smushPropertyURI = getValue(actionInd.getPropertyValue(WorkflowOntology.smushOnProperty)).getLexicalForm();
destinationModel.enterCriticalSection(Lock.WRITE);
try {
destinationModel.add(utils.smushResources(sourceModel, ResourceFactory.createProperty(smushPropertyURI)));
} finally {
destinationModel.leaveCriticalSection();
}
return new ActionResultImpl();
}
return null;
}
}
private class NameBlankNodesAction implements ActionHandler {
public ActionResult handleAction(Individual actionInd) {
if (instanceOf(actionInd,WorkflowOntology.NameBlankNodesAction)) {
OntModel sourceModel = ModelFactory.createOntologyModel(OntModelSpec.OWL_MEM);
for (RDFNode node : (List<RDFNode>) actionInd.listPropertyValues(WorkflowOntology.sourceModel).toList()) {
sourceModel.addSubModel(getModel(node));
}
Model destinationModel = getModel(actionInd.getPropertyValue(WorkflowOntology.destinationModel));
String uriPrefix = getValue(actionInd.getPropertyValue(WorkflowOntology.uriPrefix)).getLexicalForm();
destinationModel.add(utils.renameBNodes(sourceModel, uriPrefix));
return new ActionResultImpl();
}
return null;
}
}
private class SplitPropertyValuesAction implements ActionHandler {
public ActionResult handleAction(Individual actionInd) {
if (instanceOf(actionInd,WorkflowOntology.SplitPropertyValuesAction)) {
// We use an OntModel here because this API supports submodels
OntModel sourceModel = ModelFactory.createOntologyModel(OntModelSpec.OWL_MEM);
for (RDFNode node : (List<RDFNode>) actionInd.listPropertyValues(WorkflowOntology.sourceModel).toList()) {
sourceModel.addSubModel(getModel(node));
}
Model destinationModel = getModel(actionInd.getPropertyValue(WorkflowOntology.destinationModel));
String propertyURI = getValue(actionInd.getPropertyValue(WorkflowOntology.originalProperty)).getLexicalForm();
String newPropertyURI = getValue(actionInd.getPropertyValue(WorkflowOntology.newProperty)).getLexicalForm();
String splitRegex = getValue(actionInd.getPropertyValue(WorkflowOntology.splitRegex)).getLexicalForm();
boolean trim = true;
try {
trim = getValue(actionInd.getPropertyValue(WorkflowOntology.trim)).getBoolean();
} catch (Exception e) {}
destinationModel.enterCriticalSection(Lock.WRITE);
try {
destinationModel.add(utils.splitPropertyValues(sourceModel, propertyURI, splitRegex, newPropertyURI, trim));
} finally {
destinationModel.leaveCriticalSection();
}
return new ActionResultImpl();
} else {
return null;
}
}
}
private class ProcessPropertyValueStringsAction implements ActionHandler {
public ActionResult handleAction(Individual actionInd) {
if (instanceOf(actionInd,WorkflowOntology.ProcessPropertyValueStringsAction)) {
// We use an OntModel here because this API supports submodels
OntModel sourceModel = ModelFactory.createOntologyModel(OntModelSpec.OWL_MEM);
for (RDFNode node : (List<RDFNode>) actionInd.listPropertyValues(WorkflowOntology.sourceModel).toList()) {
sourceModel.addSubModel(getModel(node));
}
Model destinationModel = null;
try {
destinationModel = getModel(actionInd.getPropertyValue(WorkflowOntology.destinationModel));
} catch (Exception e) {}
Model additionsModel = null;
try {
additionsModel = getModel(actionInd.getPropertyValue(WorkflowOntology.additionsModel));
} catch (Exception e) {}
Model retractionsModel = null;
try {
retractionsModel = getModel(actionInd.getPropertyValue(WorkflowOntology.retractionsModel));
} catch (Exception e) {}
String processorClass = getValue(actionInd.getPropertyValue(WorkflowOntology.processorClass)).getLexicalForm();
String processorMethod = getValue(actionInd.getPropertyValue(WorkflowOntology.processorMethod)).getLexicalForm();
String propertyURI = getValue(actionInd.getPropertyValue(WorkflowOntology.originalProperty)).getLexicalForm();
String newPropertyURI = getValue(actionInd.getPropertyValue(WorkflowOntology.newProperty)).getLexicalForm();
destinationModel.enterCriticalSection(Lock.WRITE);
try {
if (log.isDebugEnabled()) {
log.debug("calling processPropertyValueStrings ...");
}
utils.processPropertyValueStrings(sourceModel, destinationModel, additionsModel, retractionsModel, processorClass, processorMethod, propertyURI, newPropertyURI);
} finally {
destinationModel.leaveCriticalSection();
}
return new ActionResultImpl();
} else {
return null;
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,945 |
University College Cork Coláiste na hOllscoile Corcaigh
UCC News and Views
UCC News Archive
Plain Speaking
UCC features in Financial Times magazine
The inaugural issue of the Global Entrepreneurship Network (GEN) Magazine distirubted by the Financial Times worldwide highlighted Ireland and UCC as a hub for innovation, new ideas, start-ups and creativity.
International students happiest in Ireland
An online study choice portal platform has revealed that international students studying in Europe are the most satisfied with their university experience in Ireland.
UCC historians media pundits for Thomas Kent funeral
The remains of Thomas Kent were taken from Cork Prison and buried in Castlelyons after the general public paid their respects in Collins Barracks.
Sounds from a Safe Harbour
Science and music embrace one another as some of the world's most celebrated musicians and composers come face-to-face with the finest minds in neuroscience and physiology at 'Playing Your Heart Out'.
UCC holding its own in QS Rankings
UCC has performed creditably in the QS World University Rankings 2015, where it is ranked at 233. Its scores for academic reputation, employer reputation and international faculty have all improved.
UCC President delivers O'Connell lecture
President of UCC Dr Michael Murphy delivered this year's Daniel O'Connell Lecture at The Library, Cahersiveen, Co. Kerry.
Mental health experiences shared in new book
A powerful and inspiring collection of short stories written by UCC students and staff has been released, proving that even the darkest times can be overcome.
Cork Local Government
UCC looks forward to considering the report of the Cork Local Government Committee to see how the recommendations impact on the University and higher education in Cork generally.
Pregnancy research in Cork wins international award
Ground-breaking work on preeclampsia, a life threatening complication of late pregnancy, has earned researchers in Cork a major award from the American Heart Association.
Boole 200: Lord Mayor of Cork visits Lincoln
Relatives of renowned mathematician George Boole will today (September 4) join a delegation from Cork visiting Lincoln to celebrate the life and legacy of one of the city's most influential sons.
<< < 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 > >>
Coláiste na hOllscoile Corcaigh
College Road, Cork T12 K8AF | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,053 |
{"url":"http:\/\/mathhelpforum.com\/number-theory\/172548-primitive-characters-print.html","text":"# Primitive Characters\n\nLet $\\chi$ be the primitive character modulo 12.\nI need to find all possible values of $\\underset{m\\leq x}{\\sum}\\chi(m)$ for varying $x$ (not necessarily an integer).","date":"2014-08-30 05:20:17","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\": 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\": 0.7133680582046509, \"perplexity\": 180.808410350114}, \"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-2014-35\/segments\/1408500834258.45\/warc\/CC-MAIN-20140820021354-00444-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
typedef boost::mpl::vector<dynamicgraph::sot::FeatureVisualPoint> entities_t;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,588 |
{"url":"https:\/\/socratic.org\/questions\/how-do-i-find-the-slope-of-an-equation-in-the-form-y-mx-b","text":"# How do I find the slope of an equation in the form y = mx + b?\n\nSep 21, 2014\n\nIf the equation is in the form of y=mx+b, your slope is represented by the variable m.\n\nExample 1 :\nIf you have an equation like y = - 3x + 5, then your slope is - 3 since m = - 3.\n\nExample 2:\nIf you have an equation like 4x + 2y = 6, you would need to solve for y.\nFirst step you would do is subtract 4x to both sides.\n4x + 2y - 4x= 6 - 4x\n2y = -4x + 6\nSecond step you divide by 2.\n$\\frac{2 y}{2}$ = $\\frac{- 4 x}{2}$ + $\\frac{6}{2}$\ny = -2x + 3\nTherefore the slope of the equation is -2, since m = -2.","date":"2020-07-14 16:08:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 3, \"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\": 0.6546922326087952, \"perplexity\": 384.6180521244734}, \"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-2020-29\/segments\/1593655897168.4\/warc\/CC-MAIN-20200714145953-20200714175953-00572.warc.gz\"}"} | null | null |
package org.apache.pulsar.broker.service;
import org.apache.pulsar.PulsarStandaloneStarter;
import org.apache.pulsar.broker.auth.MockedPulsarServiceBaseTest;
import org.testng.annotations.Test;
import static org.testng.AssertJUnit.assertEquals;
import static org.testng.AssertJUnit.assertNotNull;
import static org.testng.AssertJUnit.assertNull;
@Test(groups = "broker")
public class StandaloneTest extends MockedPulsarServiceBaseTest {
@Override
protected void setup() throws Exception {
}
@Override
protected void cleanup() throws Exception {
}
@Test
public void testWithoutMetadataStoreUrlInConfFile() throws Exception {
String[] args = new String[]{"--config",
"../conf/standalone.conf"};
PulsarStandaloneStarter standalone = new PulsarStandaloneStarter(args);
assertNotNull(standalone.getConfig().getProperties().getProperty("metadataStoreUrl"));
assertNotNull(standalone.getConfig().getProperties().getProperty("configurationMetadataStoreUrl"));
}
@Test
public void testAdvertised() throws Exception {
String[] args = new String[]{"--config",
"./src/test/resources/configurations/pulsar_broker_test_standalone.conf"};
PulsarStandaloneStarter standalone = new PulsarStandaloneStarter(args);
assertNull(standalone.getConfig().getAdvertisedAddress());
assertEquals(standalone.getConfig().getAdvertisedListeners(),
"internal:pulsar://192.168.1.11:6660,internal:pulsar+ssl://192.168.1.11:6651");
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,004 |
\section{Introduction}
\paragraph{}
Observation of gravitational waves (GWs) within the last six years is transforming the field of astronomy and our understanding of compact objects \cite{VIRGO:2014yos, LIGOScientific:2016aoc, LIGOScientific:2016dsl, LIGOScientific:2020ibl, LIGOScientific:2020kqk, LIGOScientific:2021djp}. The LIGO and Virgo experiments have so far detected around {90} binary compact object mergers composed of black holes (BHs) and neutron stars (NSs) \cite{LIGOScientific:2021djp}, with a few events challenging the models of stellar evolution. LIGO/Virgo's third observing run \cite{LIGOScientific:2020ibl} announced the most massive binary BH (BBH) merger (GW190521) \cite{LIGOScientific:2020iuh} with the primary component mass of $\sim 85\,M_{\odot}$. This observation challenges the formation and evolution mechanisms of stellar BHs since models predict no objects larger than about $65\,M_{\odot}$ due to the pulsational pair-instability (PI) process \cite{Heger:2002by}. This sets a mass gap for BHs between $70 - 150\,M_\odot$ \cite{Belczynski:2016jno}. Primordial black holes (PBHs) formed from the collapse of large density fluctuations in the early Universe \cite{Zeldovich:1967lct, 1971MNRAS.152...75H, 1974MNRAS.168..399C, Carr:1975qj}, could provide an alternative origin for such events.
Another challenge is to model detected pairs which present a large asymmetry in the progenitors masses ({\em e.g.~} GW190412), with mass ratios of nearly 4--to--1 ($30\,M_{\odot} + 8\,M_{\odot}$) \cite{LIGOScientific:2020stg}. A scenario where more than one species of stars or BHs is confined to a small volume could be explained by a sequence of mergers.
Several mechanisms for the origin of the progenitors in BBHs have been considered in the literature; for example, 1)~binary star evolution through common envelope \cite{1973NInfo..27...70T, Bethe:1998bn, Belczynski:2001uc}, 2)~dynamical process in triples \cite{Arca-Sedda:2018qgq}, 3)~pairing of PBHs \cite{Carr:2016drx, Ali-Haimoud:2017rtz}, or 4)~as a product of hierarchical mergers \cite{Fishbach:2017dwv, Gerosa:2017kvu, Liu:2019rnx, Doctor:2019ruh, Wu:2020drm, Kimball:2020qyd, Gerosa:2021mno}.
In this article we propose a combination of two mechanisms (3 \& 4); {\em i.e.~} PBHs of a common mass pair to form a binary and the ``product" of their merger will pair with another (P)BH, and so on. We explore the possibility of repetition of this process in a ``hierarchical merger" sequence throughout the age of the Universe, considering the radiation of GWs, as they could potentially provide a detectable population of binaries with one or both component masses in the mass gap of stellar BHs.
The astrophysical environments suitable for hierarchical mergers, where multiple GW events will be produced, require a large escape velocity so that merger remnants are efficiently retained. A compact (dense) region will prevent merger remnants from being ejected by GW recoils. Scenarios previously considered for the efficient production of hierarchical mergers include globular clusters \cite{Rodriguez:2016kxx, Askar:2016jwt, Samsing:2017xmd, Mapelli:2021syv}, nuclear star clusters \cite{Kimball:2020opk, Fragione:2020nib, Kritos:2020wcl, Fragione:2021nhb}, and accretion disks surrounding active galactic nuclei \cite{Yang:2019cbr, Tagawa:2020jnc, Tagawa:2020qll}. Here we propose a hierarchical merger of PBHs in Dwarf Galaxies (DGs) for two reasons: First, DGs are dominated by DM \cite{2019ARA&A..57..375S}, which we assume to be in the form of PBHs\footnote{PBHs with a mass larger than $10^{15}$ g are candidates for Dark Matter (DM) \cite{Hawking:1974sw}.}, and second, these environments are dense enough for the hierarchical merger of PBHs \cite{2019ARA&A..57..375S}.
We explore the possible formation of systems like GW190412 and GW190521 as a result of our model (hierarchical merger is also considered in \cite{Rodriguez:2020viw}). We also investigate the formation of BHs in the mass gap and also merged pairs with a large mass ratio \cite{Gerosa:2019zmo}.
The paper is organized as follows: in section~\ref{sec2} we discuss DGs as hosts of PBHs. We also summarize the basic equations of PBH as a candidate for DM \cite{Carr:2020gox}. In section~\ref{sec3}, we explain our model and in section~\ref{sec4} we present our results. Finally, section~\ref{sec5} is devoted to our conclusions.
\section{Dwarf Galaxies and Primordial Black Holes}\label{sec2}
\paragraph{}
Dwarf Galaxies (DGs) are the oldest, and least chemically evolved stellar systems known \cite{2019ARA&A..57..375S}. These galaxies resemble globular clusters in density profiles, however, they are distinguished from star clusters since they present a dynamical mass that is substantially larger than the mass inferred from the luminous stellar population \cite{DES:2015zwj}. The stellar kinematics of DGs confirm that they contain a dominant component of DM which is in contrast with the star clusters. Altogether, these properties indicate that DGs are the astrophysical systems where DM is most abundant.
Since these galaxies are highly DM dominated, with mass-to-light ratios $\sim 1000\,M_{\odot}/L_{\odot}$ \cite{Simon:2007dq}, they are extremely valuable laboratories for characterizing the nature of DM, mostly as probes of DM on the smaller end of the large scale structure spectrum \cite{2018MNRAS.481.5073E}.
Dozens of DGs have recently been discovered by several surveys like the SDSS \cite{Willman:2004kk, Martin:2007ic, 2018PASJ...70S..18H} as satellites of the Milky Way, Andromeda, and as members of the Local Group \cite{Laevens:2015kla}\footnote{Recall that DGs have been a source of a tension known as missing satellites problem since predictions for the abundance of massive satellite galaxies in simulations present significantly larger numbers than the observed objects \cite{1999ApJ...522...82K, 1999ApJ...524L..19M}.}. The ultra-faint Milky Way satellites have masses ranging from just over $10^6 M_{\odot}$ (Coma Berenices) up to $2.8\times10^7 M_{\odot}$ (Canes Venatici I) \cite{Simon:2007dq}. Discovered DGs have Plummer (half-light) radii as small as $\sim 20$ pc \cite{1911MNRAS..71..460P}, with the total mass inside this radius in the range of one to three orders of magnitude larger than their stellar masses \cite{2012AJ....144....4M}.
Consequently, the measured central densities, and the density profiles are essentially those of DM halo. The central densities of DGs range from $\sim 0.08\,M_{\odot}$/pc$^{3}$ up to $\sim 2.1\, M_{\odot}$/pc$^{3}$ \cite{Simon:2007dq}.
The identity of the dominant DM in these environments, however, remains a mystery \cite{Planck:2018vyg}. One appealing possibility is that the DM consists of black holes formed in the early Universe, known as Primordial Black Holes (PBHs) \cite{Carr:2016drx, Green:2020jor}. Constraints on their abundance imposed in DG environments are reported in Refs.~\cite{Zhu:2017plg, Stegmann:2019wyz}\footnote{Recently, some studies examined whether DM candidates in the form of PBHs can solve the cusp-core problem in low-mass DGs \cite{Boldrini:2019isx}.}. In the following, we will briefly explain the properties of PBHs and their abundances.
The most common mechanism for PBHs formation is the collapse of density fluctuations larger than a threshold that reenters the horizon after inflation in the early Universe \cite{Zeldovich:1967lct, 1971MNRAS.152...75H, 1974MNRAS.168..399C}\footnote{See however \cite{Padilla:2021zgm} for an alternative, more likely mechanism during a long-lasting period of reheating.}. Thus the mass of PBH is roughly the mass of the horizon, $M_{\rm H}$
\begin{equation}} \newcommand{\ee}{\end{equation}\label{mass}
M_{\rm PBH} \approx M_{\rm H} \sim 10^{15} \left(\dfrac{t}{10^{-23}\,{\rm s}}\right) {\rm g}\,.
\ee
PBHs radiate thermally due to the Hawking radiation \cite{Hawking:1974sw}, but massive enough PBHs survive for more than a Hubble time. In consequence, their mass spectrum spans many orders of magnitude, from $\sim 10^{15}$ g to well over $10^{50}$ g.
A parameter which represents the abundance (the energy density fraction) of PBHs in the epoch of their formation is
\begin{equation}} \newcommand{\ee}{\end{equation}\label{beta1}
\beta \equiv \frac{\rho_{\rm PBH}(t)}{\rho(t)}\,.
\ee
Assuming an adiabatic expansion of the Universe, the above can be related to the present abundance mass fraction of PBHs, $f_{\rm PBH}\equiv\Omega_{\rm PBH}/\Omega_{\rm DM}$, by \cite{Carr:2020gox}
\begin{equation}} \newcommand{\ee}{\end{equation}\label{beta2}
\beta \simeq 3.7\times10^{-9}\,\left(\frac{g_*}{10.75}\right)^{1/4}\left(\frac{M_{\rm PBH}}{M_{\odot}}\right)^{1/2} \,f_{\rm PBH}\,,
\ee
where $g_{*}$ is a number of relativistic degree of freedom at the time of PBHs' formation.
A population of PBHs with a specific mass is subject to observational constraints which in many cases reduces the contribution of $f_{\rm PBH}$ to Dark Matter. However, DM can be completely formed of PBHs in the mass window around $(5\times10^{-16}-2\times10^{-14})\,M_{\odot}$ \cite{Carr:2020gox, Franciolini:2022htd}. The recent detection of GWs by LIGO/Virgo also imposes constraints on the abundance of PBHs with masses of order $10$ to $50\,M_{\odot}$. Searches for compact objects of sub-Solar masses also impose constraints on PBH abundance \cite{Authors:2019qbw}. There are also constraints on sub-Solar masses by gravitational microlensing observations \cite{Niikura:2017zjd}. These constraints are imposed without any assumption about the distribution of PBHs at their formation, their clustering, or the environment, which can alter the bounds significantly \cite{Clesse:2016vqa, Belotsky:2018wph}. In the next section, in light of the observational constraints, we will consider DM PBHs with four different masses.
\section{Model for the Merger Rate}\label{sec3}
\paragraph{}
In what follows we describe the details of the model employed to produce a succession of mergers of BHs in a DG. We consider a population of PBHs at the core of the DG, with $N_{\rm PBH}$ elements, which we call a {\it cluster}. We assume PBHs are the only component of DM, with a single initial mass (monochromatic population). We test four different initial masses of PBHs, where $N_{\rm PBH}$ is determined by the mass of DG. Throughout this work we consider a DG with the total mass, $M_{\rm DG}=10^{9}\,M_{\odot}$, and radius, $R_{\rm DG}\sim 10$ pc.
\begin{figure*}[h!]
\centering
\includegraphics[width=14cm, height=11cm]{heretichal.png}
\caption{The hierarchical tree of (P)BH mergers from an initial population of PBHs with mass $10\,M_{\odot}$. In the age of the Universe, collisions reach up to a fourth generation of BHs. Different colors indicate different generations, and approximate masses, in units of $M_{\odot}$, consider the mass-loss due to GW radiation.}
\label{tree}
\end{figure*}
In order for PBHs to merge more than once, a dense environment is required. We thus focus on core of DGs with mass, $M_{\rm c} = 10^5\,M_{\odot}$ confined in a core radius, $R_{\rm c} = 0.9$ pc, which is admitted by observations \cite{Simon:2007dq}. The number of PBHs is proportional to the density of DM, $\rho_{\rm DM}$,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{number}
N_{\rm PBH} = \dfrac{4\pi}{3}\dfrac{\rho_{\rm DM}\,R_{\rm c}^3}{m_{\rm PBH}}\,.
\ee
In our analysis we consider four different initial masses for PBHs, which we define as {\it first generation} (1G); $10^{-14}\,M_{\odot},\,10^{-2}\,M_{\odot},\,1\,M_{\odot}$ and 10 $M_{\odot}$. We choose $10^{-14}\,M_{\odot}$ since as mentioned there is no confirmed observational constraints for this mass and PBHs can be whole DM. Our interest in other mass values comes from the GWs observations. From Eq.~\eqref{number}, we get the number of PBHs at the core (see Table~\ref{table} below).
According to our assumption, the components of the first merger will have equal masses. Therefore, the merger of 1G+1G will produce the second generation BHs (2G) with roughly double the mass. Subsequently, from the third generation (3G) onward, BHs can form from 1G+2G and 2G+2G; two populations of BHs will form, and therefore, a variety of masses is possible (see Figure~\ref{tree}).
For estimating the merger rate, we know that the cross-section of two objects with masses $m_{i}$ and $m_{j}$ which merge after first being captured into binaries by the emission of gravitational radiation is \cite{1989ApJ...343..725Q, Mouri:2002mc}
\begin{equation}} \newcommand{\ee}{\end{equation}\label{crosssection}
\sigma= \dfrac{\sigma(m_i,m_j)}{|v_{i}-v_{j}|^{18/7}}\,,
\ee
with
\begin{equation}} \newcommand{\ee}{\end{equation}\label{sigmaij}
\sigma(m_i,m_j) = 2 \pi \left(\frac{85\pi}{6\sqrt{2}}\right)^{2/7}\frac{G^{2}(m_{i} + m_{j})^{10/7}\left( m_{i}m_{j}\right)^{2/7}}{c^{10/7}}\,.
\ee
\noindent Here $v_{i}$ represents the velocity of the components in the binary, which is given by the root mean-squared (rms) velocity, $\overline{v_i} \equiv \sqrt{\overline{v_i^2}}$
\begin{equation}} \newcommand{\ee}{\end{equation}\label{velocity}
\overline{v_{i}^{2}}(r) = \frac{4\pi}{n_{i}(r)}\int_{0}^{\phi(r)}\,f_{i}(E) \lb2\left(\phi(r)-E\right)\right]^{3/2}\,dE\,,
\ee
where $n_{i}(r) = \dfrac{\rho_{i}(r)}{m_{i}}$ is the number density, and the DM density is modelled by a Plummer sphere \cite{1911MNRAS..71..460P} at all times. This prescribes the following expressions for the density and the gravitational potential, respectively \cite{1911MNRAS..71..460P}
\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}
\rho_{i}(r) &=& {\rho}_i^{\rm c}\left( 1 + \frac{r^{2}}{R_{\rm c}^{2}}\right)^{-5/2} = \frac{3\,m_{i}\,N_{i}}{4\pi\,R_{\rm c}^{3}}\left( 1 + \frac{r^{2}}{R_{\rm c}^{2}}\right)^{-5/2}\,,\\
\label{gravpot}
\phi(r) &=& {\phi}^{\rm c}\left( 1 + \dfrac{r^{2}}{R_{\rm c}^{2}}\right)^{-1/2} = \dfrac{GM_{\rm c}}{R_{\rm c}}\left( 1 + \dfrac{r^{2}}{R_{\rm c}^{2}}\right)^{-1/2}\,,
\eea
where we defined the core density of the $i$-th species, ${\rho}_i^{\rm c} \equiv \frac{3\,m_{i}\,N_{i}}{4\pi\,R_{\rm c}^{3}}$, with $N_i$ representing the total number of the BHs with mass $m_i$, as well as a \textit{core potential}, ${\phi}^{\rm c} \equiv \frac{GM_{\rm c}}{R_{\rm c}}$. The distribution function of each BH population, $f_i(E)$ is given by \cite{1989ApJ...343..725Q}
\begin{equation}} \newcommand{\ee}{\end{equation}\label{DF}
f_{i}(E) = \frac{32\sqrt{2}}{7\pi^2}\,(\phi^{c})^{-5}\,n_i^c\,E_i^{7/2}\,,
\ee
where $n_i^c \equiv n_i(r=R_c)$, and according to the virial theorem, the energy of each species of the {\it cluster} is given by
\begin{equation}} \newcommand{\ee}{\end{equation}\label{energy}
E_{i} = \dfrac{G\,m_{i}^2\,N_{i}^{2}}{2\,R_{\rm c}}\,.
\ee
An approximate analytic expression for the merger rate in this case is estimated from the Fokker-Planck equation, in particular, from the terms that account for the loss and gain of BHs due to mergers with other BHs \cite{1989ApJ...343..725Q} (see also \cite{Stasenko:2021vmm}). Once the velocities take the rms value, this can be expressed as
\begin{equation}} \newcommand{\ee}{\end{equation}\label{rate}
\Gamma_{j} = \frac{14\pi}{3} \sum_{i}\sigma(m_{i},\,m_{j})\int\;dr\,r^{2}\,\frac{n_{i}}{\overline{v}_{i}}\frac{n_{j}}{\overline{v}_{j}} \left[\left(\overline{v}_{i} + \overline{v}_{j}\right)^{3/7} - |\overline{v}_{i} - \overline{v}_{j}|^{3/7}\right]\,.
\ee
The rms velocity is the solution of Eq.~\eqref{velocity}, which in light of the above definitions, can be expressed as
\begin{equation}} \newcommand{\ee}{\end{equation}
\overline{v_{i}}^2(r) = \frac{1}{2}\phi(r) \left(\frac{\phi(r)}{\phi^c}\right)^5 \left( \frac{n_i(r)}{n^c_i}\right)^{-1}\,.
\ee
In order to define the merging components, we take as different elements the species, $i$ or $j$, defined by their masses, and we also divide the galaxy core in shells which components will collide with a distance-dependent merger rate. From the expression in Eq.~\eqref{rate}, the merger rate in region between shells at radii $r_A$ and $r_B$ is given by
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{split}
\Gamma&(r_A,\,r_B) = \frac{14\pi}{3} \sum_{i}\sigma(m_{i},\,m_{j})\,\frac{n_j(r_{B}) n_i^c}{\overline{v_{j}}(r_{B})}\,\left(\frac{\phi^c}{2}\right)^{-2/7} \int\,dr_{A} \,r^{2}_{A}\,(1+A)^{-9/4}\\
& \times \left\{\left[\left(\frac{1}{(1+B)^{1/4}}\right) + \left(\frac{1}{(1+A)^{1/4}}\right) \right]^{3/7} - \left|\left(\frac{1}{(1+B)^{1/4}}\right) - \left(\frac{1}{(1+A)^{1/4}}\right)\right|^{3/7}\right\}\,,
\end{split}
\label{final:gamma}
\ee
where $A=(r_{A}/R_{\rm c})^2$ and $B=(r_{B}/R_{\rm c})^2$, label the shell to which each of the merging components belong.
For each example, we have sliced the galaxy core in 10 shells and we proceed to count the number of mergers by dividing the cosmic time in \textit{merger epochs} of constant $\Gamma(r_A,\,r_B)$. The lasting time for such epochs, $\tau_e$ is defined as one tenth times the inverse of the largest merging rate value among the pairs of shells. Mathematically,
\begin{equation}} \newcommand{\ee}{\end{equation}
\tau_e = \frac{1}{10} \mathrm{min}\left\{\Gamma(r_A,\,r_B)^{-1}\right\}\,.
\ee
We count the number of mergers, starting with a single population of PBHs at redshift $z = 20$. This initial condition sets an early enough time to discard the stellar origin for the initial BHs, and late enough for the formation of DGs. For the first iteration, the initial single-mass population is divided into two identical sets, taken as two different species as an input of our merging algorithm. In every epoch, we evaluate the probability of encounters from the merger rate in Eq.~\eqref{final:gamma} for a given combination of two species at two given radii ($r_A$ and $r_B$), verifying that the merger time is shorter than the period each merger epoch lasts. We thus count the number of mergers by considering the number of BHs available in each shell for both species. For the next epoch, we reset the merging rates at time $t_{i} + \tau_e$ for the updated population of each species. Thus the above procedure is repeated to compute the mergers at each shell. In Figures~\ref{numbers} and \ref{relative}, we report results for the total number of BHs formed in the whole DG core (after integrating all shells).
In each merger, we also consider the mass loss due to the emission of gravitational radiation. The binary is formed through gravitational capture where the energy is released through the emission of GW. Such dissipative binary formation is followed by the coalescence of the binary with a well-known rate of energy loss. The time-averaged energy loss rate of the binary in the Keplerian orbit is given by \cite{PhysRev.136.B1224}
\begin{equation}} \newcommand{\ee}{\end{equation}
\bigg\langle\dfrac{dE}{dt}\bigg\rangle = -\dfrac{32}{5}\dfrac{G^4\,(m_im_j)^2\,(m_i+m_j)}{a^5} F(e)\,,
\ee
where $e$ and $a$ are eccentricity and semi-major axis of the orbit of binary, respectively, and the explicit dependence on the eccentricity is given by
\begin{equation}} \newcommand{\ee}{\end{equation}
F(e) = \frac{1}{(1-e^2)^{7/2}}\left( 1+\dfrac{73}{24\,}\,e^2 + \dfrac{37}{96}\,e^4\right)\,.
\ee
\noindent Since the orbits follow initially a parabolic path, which is also required by the cross-section expressed in Eq.~\eqref{crosssection}, we take the eccentricity as $e = 0.99$ as an approximation to the parabola and the semi-major axis as the initial (periastron) separation. The energy emission rate can be expressed as a function of the semi-major axis decrease \cite{PhysRev.136.B1224}
\begin{equation}} \newcommand{\ee}{\end{equation}
\frac{da}{dt} = -\frac{64}{5}\frac{G^{3}}{c^{5}}\frac{m_{i}m_{j}\,(m_{i}+m_{j})}{a^{3}}F(e)\,,
\ee
\noindent Therefore the energy loss in GWs throughout the merger is given by
\begin{equation}} \newcommand{\ee}{\end{equation}\label{Eq:mass-loss}
\Delta E = \int_{a_{i}}^{a_{\rm merge}}\frac{dE}{da}\, \frac{da}{dt}\,dt = \frac{1}{2}m_{i}m_{j}\left(\frac{1}{a_{\rm merge}} - \frac{1}{a_i}\right)\,,
\ee
where the semi-major axis is integrated from an initial separation, $a_i$ up to the sum of the Schwarzschild radii of the progenitors, $a_{\rm merge}$. The energy emitted by gravitational radiation is accounted for by the effective mass loss. This results in the merger product with a mass smaller than the sum of the progenitors' masses, as illustrated in Figure~\ref{tree}. The accumulated effect on the total mass of the {\it cluster} is also taken into account in our evaluation of merger rates (see Table~\ref{table}). Our prescription for the mass loss is in agreement with events observed by the LIGO/Virgo collaboration\footnote{The computed mass via Eq.~\eqref{Eq:mass-loss} from the progenitor components deviates in average only 2\% from the final mass estimated for the events in the three runs of LIGO/Virgo (well within the determined errors).}. Finally, it is worth mentioning that we do not consider the disruption of BHs binaries by encounters with other compact objects (BHs and stars). We also neglect the ejection of BHs due to the recoil kicks of GW radiation. Note that even if we take these effects into account, they will only affect an insignificant fraction of the total number of merger products.
\section{Results}\label{sec4}
\paragraph{}
In this section, we present the results of a succession of merger epochs according to the model described above. We consider DG cores dominated by DM PBHs, with four cases of a single initial PBH mass given by $m_{\rm PBH}/M_{\odot} = \left( 10^{-14},\,10^{-2},\,1,\,10\right)$. We keep the initial mass of the core as a constant, $M_{\rm c}(t_i) = 10^5 M_{\odot}$, which implies that the initial number of PBHs and number density is sensitive to the initial mass of (P)BHs (see Table~\ref{table}). For illustration purposes, in Figure~\ref{tree} we show the merger tree of (P)BHs, for the specific case of initial mass, $m_{\rm PBH} = 10\,M_{\odot}$ (blue circles). Each merger gives way to a BH of the $n$-th generation if at least one of its progenitors belongs to the $n-1$-th generation. Thereby, the second generation is constituted by the 1G+1G progenitors (yellow circles), while mergers of 1G+2G and 2G+2G BHs constitute the third generation (red circles), and so forth. In our approach, only four epochs fit within the age of the Universe and thus, BHs merge up to the fourth generation, because the $n$-th epoch will show mergers from the first up to the $n$-th generation. As stated earlier, due to the GW radiation in each merger, the mass of each merger product is smaller than the sum of the components of the binary.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$m_{\rm PBH}$ & $N_{\rm PBH}(t_{i})$ & $M_{\rm c}(t_{0}) / M_{\rm c}(t_{i})$ &
$\sum m_{\rm PBH}(t_{0})/M_{\rm c}(t_{0})$ \\ \hline \hline
$10^{-14}\,M_{\odot}$ & $10^{19}$ & $0.953$ & $0.5897$ \\ \hline
$10^{-2}\,M_{\odot}$ & $10^{7}$ & $0.952$ & $0.5897$ \\ \hline
$1\,M_{\odot}$ & $10^{5}$ & $0.952$ & $0.5896$ \\ \hline
$10\,M_{\odot}$ & $10^{4}$ & $0.953$ & $0.5891$ \\ \hline
\end{tabular}
\caption{The first and second columns show the mass and number of the original population of PBHs (1G), respectively. The third column represents the remaining mass of the core after four epochs. The final column shows the final (present) percentage of the original PBH population in terms of mass.}
\label{table}
\end{center}
\end{table}
Table~\ref{table} indicates the number of PBHs in the initial population (at time $t_i$) and the remaining mass of the core after the four merger epochs at the present time, $t_0$, ($M_{\rm c}(t_0) / M_{\rm c}(t_i)$)--third column. The last column shows the mass fraction of the initial PBH population which did not collide with another BH up to $t_0$. It is worth noting that these results are practically independent of the initial mass of PBHs.
Figure~\ref{numbers} shows the number count of merger products at each epoch for the different initial populations of PBHs considered. Note that for smaller PBH masses, merger products of mass up to $8\, m_{\rm PBH}$ are possible. On the other hand, for large initial PBH masses, only BHs of mass $4.3\, m_{\rm PBH}$ are significantly produced. This is due to the number of PBHs initially present, as listed in Table~\ref{table}. Note also that in Figure~\ref{numbers}, the dominant product populations in numbers are those of the first and second generation of mergers. This is true for all investigated cases and is due to the fact that the relative merger rates are independent of the PBH mass but depend on the number of PBHs.
\begin{figure*}[h!]
\centering
\subfloat[]{\includegraphics[scale=0.5]{Mass_population_from_10M_sun.png}}
\qquad
\subfloat[]{\includegraphics[scale=0.5]{Mass_population_from_1M_sun.png}}
\quad
\subfloat[]{\includegraphics[scale=0.5]{Mass_population_from_1e-02M_sun.png}}
\qquad
\subfloat[]{\includegraphics[scale=0.5]{Mass_population_from_1e-14M_sun.png}}
\caption{Number of BHs formed from mergers after each epoch, shown as a function of the age of the Universe, starting from $t = 0.18~\mathrm{Gyr}$ ($z = 20$). Each epoch is marked with a vertical dashed line. The panels correspond to the following masses for the initial populations: (a) $10\,M_{\odot}$, (b) $1\,M_{\odot}$, (c) $10^{-2}\,M_{\odot}$, (d) $10^{-14}\,M_{\odot}$.}
\label{numbers}
\end{figure*}
Focusing on the galaxy core mass fraction of the merger products, we plot their contribution in Figure~\ref{relative}, where we show the relative mass abundance of each population in a DG core initially formed of PBHs with mass $m_{\rm PBH}$. We find that the resulting mass fractions are independent of the value of the original PBH mass (see Table~\ref{table}). The left plot of this figure shows that the fraction of PBHs decreases steadily with time, increasing the mass fraction of merger products after each merger epoch. After the four iterations, we find that more than 40\% of the PBHs have formed at least one binary and merged to form larger BHs. The right panel of Figure~\ref{relative} shows the mass fraction in merger products. Note that the mass fraction is dominated by the more massive species even when the number count of such species is subdominant (see the cyan component in Figures~\ref{numbers}~and~\ref{relative}).\\
\begin{figure*}[h!]
\centering
\subfloat[]{\includegraphics[scale=0.5]{Masses_percent_from_10M_sun.png}}
\qquad
\subfloat[]{\includegraphics[scale=0.5]{Masses_percent_from_10M_sun_b.png}}
\caption{Percentage of the total mass in each population (of a given mass) for the most abundant populations. This is plotted for the case where $m_{\rm PBH} = 10\,M_{\odot}$ but the fractions are largely independent of the PBH mass. The original population is included in Panel (a), where part of the missing mass is in smaller species, and only $4.7\%$ is lost in the radiation of gravitational waves. As shown in Panel (b), the percentage of mass in the three most numerous populations generated is of order $10\%$ each, while the most massive population constitutes more than $15\%$ of the final mass. Note that these percentages are the same for a range of initial masses of PBHs in the {\it cluster}. Finally, note that the plots account only for the mass contribution of each population, and therefore we make no distinction among the generation in which they are formed.}
\label{relative}
\end{figure*}
\section{Summary and Discussion}\label{sec5}
\paragraph{}
Primordial black holes formed in the early Universe, before the formation of stars, can exist as dark matter and also contribute to the black hole merger events observed through gravitational waves. GW observations have demonstrated that BHs mergers may be more frequent than expected. If merger products form new binaries, they may subsequently merge as detectable GW sources. For sequential mergers of BHs, a dense, DM-dominated environment is required. Dwarf galaxies are ideal scenarios to host a hierarchical merger of BHs.
In this paper, we studied sequence mergers of BH starting with a monochromatic {\it cluster} of PBHs at the core of DGs. Our study, featuring stages of constant merger rate, represents a first step towards the full numerical analysis of the Fokker-Planck equation. Since PBHs span several decades of mass, we considered four different initial masses. Starting the evolution of the system at redshift $z = 20$, we find that in cores with massive PBHs ($\sim 10\,M_\odot$), BHs with up to four times the initial mass can be formed significantly, while in the small PBH mass limit the masses of products are up to eight times the initial PBH mass. Our results also show that the total mass loss at the DG core from GW emission, and the mass fraction of BH that undergo collisions are mostly independent of the initial PBH mass. Figure~\ref{relative} in particular shows that the original population of PBHs is reduced to little more than 50\% of the total mass. The proportion of larger mass BH populations may be tested by future GW surveys.
Our results are not only relevant to the rate and spectrum of GW events. We have shown that binary formation in the dense cores of DM-dominated systems can give way to more than one population of PBHs. Our results are also in agreement with studies investigating systems of multiple stars/BHs in the central regions of clusters with an accretion disk \cite{McKernan:2014oxa, Miralda-Escude:2000kqv}, and without it \cite{Mapelli:2021syv, Rodriguez:2020viw, Martinelli:2022elq}.
There are important differences in the populations of BH binaries that may distinguish our merger scenario (see {\em e.g.~} \cite{Gerosa:2017kvu}).
Ultimately, it is important to assess if the stochastic GWs of the proposed mechanism yield a detectable signal (see {\em e.g.~} \cite{Garcia-Bellido:2021jlq}). There is, however, room for important complements to our assumptions before producing accurate forecasts. For example, since the DGs are faint with low stellar mass, we have neglected the effect of mergers on stars, but a more detailed study contemplating such collisions would include electromagnetic signals which will constrain the parameters of the model.
The recoil velocity of the product BH after a merger event is also an ingredient to be included in future studies \cite{Fitchett:1983, Gonzalez:2006md, Varma:2022pld}. In the meantime, our results indicate that a series of sequential mergers may take place at the cores of DGs.
\section*{Acknowledgments}
We are grateful to Luis Padilla for his insightful comments on the manuscript.
EE thanks the TWAS 2021 Fellowship for Research and Advanced Training. This work is sponsored by CONACyT grant CB-2016-282569 and by Program UNAM-PAPIIT Grant IN107521 ``Sector Oscuro y Agujeros Negros Primordiales''.
| {
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{"url":"https:\/\/www.shaalaa.com\/question-bank-solutions\/mirror-formula-magnification-write-down-magnification-formula-lens-terms-object-distance-image-distance-how-does-this-magnification-formula-lens-differ-corresponding-formula-mirror_27321","text":"Share\n\n# Write Down the Magnification Formula for a Lens in Terms of Object Distance and Image Distance. How Does this Magnification Formula for a Lens Differ from the Corresponding Formula for a Mirror? - CBSE Class 10 - Science\n\nConceptMirror Formula and Magnification\n\n#### Question\n\nWrite down the magnification formula for a lens in terms of object distance and image distance. How does this magnification formula for a lens differ from the corresponding formula for a mirror?\n\n#### Solution\n\nMagnification formula for a lens is given by:\n\nm=v\/u\n\nMagnification\u00a0formula\u00a0for\u00a0a\u00a0mirror\u00a0is\u00a0given\u00a0by\n\nm=-v\/u\n\nfor both the forulas,\n\nm= Magnifaction\n\nv =Image distance\n\nu= Object distance\n\nThere is a difference of negative sign between the lens formula and the mirror formula. In mirror magnification formula, negative sign is present, whereas in lens magnification formula, this negative sign is not present.\n\nIs there an error in this question or solution?\n\n#### Video TutorialsVIEW ALL [1]\n\nSolution Write Down the Magnification Formula for a Lens in Terms of Object Distance and Image Distance. How Does this Magnification Formula for a Lens Differ from the Corresponding Formula for a Mirror? Concept: Mirror Formula and Magnification.\nS","date":"2019-12-08 03:32:46","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\": 0.7449598908424377, \"perplexity\": 1052.343363601737}, \"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-2019-51\/segments\/1575540504338.31\/warc\/CC-MAIN-20191208021121-20191208045121-00244.warc.gz\"}"} | null | null |
Andrew Johnson (Bedford, Inglaterra, 10 de febrero de 1981) es un exfutbolista inglés. Jugaba de delantero. Su último club fue el Crystal Palace F. C. de Inglaterra, equipo al que regresó como embajador una vez ya retirado.
Selección nacional
Fue internacional con la selección de fútbol de Inglaterra en 8 ocasiones.
Clubes
Referencias
Enlaces externos
Personas de Bedford
Futbolistas del Birmingham City Football Club
Futbolistas del Crystal Palace Football Club
Futbolistas del Everton Football Club
Futbolistas del Fulham Football Club
Futbolistas del Queens Park Rangers Football Club
Futbolistas de la selección de fútbol de Inglaterra en los años 2000 | {
"redpajama_set_name": "RedPajamaWikipedia"
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Sign up to the new What Hi-Fi? Deals newsletter
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The draw to win Canton DM55 soundbase closes 11.59pm Sunday 10th July 2016. Open to UK residents aged 18 or over. No cash alternative. Prizes are non-transferable. Only one entry per person. For full terms and conditions see here. | {
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package me.hatter.tools.hostsmanager.hosts;
import java.util.ArrayList;
import java.util.List;
public class Hosts {
private List<Group> groups = new ArrayList<Group>();
public List<Group> getGroups() {
return groups;
}
public void setGroups(List<Group> groups) {
this.groups = groups;
}
public static Hosts parse(List<String> lines) {
Hosts hosts = new Hosts();
List<Line> ll = parseLines(lines);
Group group = new Group();
for (Line l : ll) {
if (l.isGroupStart()) {
if (!group.getLines().isEmpty()) {
hosts.getGroups().add(group);
group = new Group();
}
group.setGroup(l.getGroup());
group.getLines().add(l);
} else {
group.getLines().add(l);
if (l.isGroupEnd()) {
hosts.getGroups().add(group);
group = new Group();
}
}
}
if (!group.getLines().isEmpty()) {
hosts.getGroups().add(group);
}
return hosts;
}
public List<String> toLines() {
List<String> lines = new ArrayList<String>();
for (Group g : getGroups()) {
if (g.getGroup() == null) {
for (Line l : g.getLines()) {
lines.add(l.getLine());
}
} else {
lines.add(Line.GROUP_START + " " + g.getGroup());
for (Line l : g.getLines()) {
if (!l.isGroup()) {
lines.add(l.getLine());
}
}
lines.add(Line.GROUP_END);
}
}
return lines;
}
private static List<Line> parseLines(List<String> lines) {
List<Line> result = new ArrayList<Line>();
for (String l : lines) {
result.add(Line.parse(l));
}
return result;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
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Q: Extract file content with ifstream here is a little description of what I am trying to achieve:
I have a text file from which I have to extract data. This data represents basic information about 8 person and for every person, 6 lines of information is used in the text file:
Ex:
1st line: name, 2nd line: last name, ..... 7th line: name, 8th line: last name, etc...
After all this information, there is more data formatted in a different way.
The second part of the data is divided by the character '$'.
I succeeded in extracting all the person's data, but how do I change the way I extract the second part of the text file's data? I thought I could change my extracting method where the comment is (see code)
Here's my code to extract the data for the first part:
if (inputFile.is_open()) {
for (int i = 0; i < 8; i++) {
string name, last_name, dob, number, street, city;
getline(inputFile, name);
getline(inputFile, last_name);
getline(inputFile, dob);
getline(inputFile, number);
getline(inputFile, street);
getline(inputFile, city);
Person p = Person(name, last_name, atoi(dob.c_str()));
Address a = Address(atoi(number.c_str()), street, city);
Directory::register(p, a);
}
//Code for 2nd part of data should be here no?
} else {
throw logic_error("File can't be opened");
}
A: I managed to make it work using another for loop and ignoring the character for every data set inputFile.ignore('$', '\n');
Thanks everyone for the help
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{"url":"https:\/\/blog.bigsmoke.us\/tag\/plugin","text":"Smokes your problems, coughs fresh air.\n\n# Tag: plugin\n\nDuring most of my bachelor, I\u2019ve used paper and pen or pencil to take notes. Halfway my second minor [Okasys], though, I switched to my laptop and LaTeX, which I preferred, because typing is faster than writing and reworking my notes into a halfway decent summary usually proved too time-consuming with hand-written notes. Admittedly, though, although reorganizing my notes became easier with LaTeX, I still didn\u2019t really get to the finished summary stage, because I\u2019m still way too obsessive-compulsive about the whole thing, most of the time. Now, since I figured I use my blog for all sorts of notes, I can just as well let WordPress and Google do some of the organizing for me, while taking notes for my present course. I just have to be a bit more careful about copyright issues (but, if the need strikes, I can always set a post to private).\n\nThere are various WordPress plugins for keeping track of academic references. I\u2019m now experimenting with papercite [documentation. From the feature list, I was more interested in the AcademicPress plugin, but the former seems te be more actively developed. However, I\u2019m thinking of switching to a simple footnotes and\/or endnotes plug-in, since my use of papercite so far actually doesn\u2019t include maintaining biography files shared by more than one post, and papercite doesn\u2019t support author-year citations anyway. I\u2019m surrounding the text with [bibshow file=custom:\/\/data][\/bibshow] shortcodes, which references a BibTeX biography stored in a custom field called papercite_data.\n\nSoon, I wish to document some statistical issues I\u2019ve been running into lately due to the lack of understanding maintained by my recipe-level statistics training. Also, I\u2019d like to document some of the things I did learn over the years, and, hopefully, the things I find out while working myself out of the modelling mountain that I currently find so difficult to mount. For this I will need to use some mathematical language, which is why I just installed the MathJaX-LaTeX WordPress plugin. MathJax-LaTeX uses the MathJax JavaScript library to support LaTeX and MathML math equations in WordPress without requiring the browser to have MathML support.\n\nAs for testing it, my knowledge ($$K()$$) of MathML ($$M$$) is pretty much nonexistant, while I\u2019m quite comfortable with LaTeX ($$L$$) math exations, which is why I\u2019m typing the LaTeX code \u201cK(M) \\ll K(L)\u201d to generate the following simple equation:\n\n$$K(M) \\ll K(L)$$\n\nWordPress does automatic paragraph formatting using the wpautop filter, some PHP code originally developed by Matt Mullenweg. For most of the time that this blog has existed, I\u2019ve disabled the wpautop filter using the following two lines in my theme\u2019s functions.php file:\n\nremove_filter('the_content', 'wpautop');\nremove_filter('the_excerpt', 'wpautop');\n\nI\u2019m not the only one to do this. But, I\u2019m tired of having to manually type <p>s for every paragraph in every post that I write.\n\nI\u2019d like to be able have it back on, but preferably a bit smarter so that you don\u2019t get all that crap (also common on bulletin boards with empty paragraphs around undetected block-level elements, especially if these are non-standard, related to plugins.\n\nAt the very least I want to be able to turn it off for when it does annoy me, on these moments that I don\u2019t want auto-anything. Also, I need this if I don\u2019t want to break the nearly 300 posts that I formatted manually. So, I want to use a custom field to turn the filter on or off.\n\nI found an unpublished plugin (unpublished in the sense that the source isn\u2019t hosted somewhere proper such as wordpress.org\/extend\/plugins\/) which does some of what I want in a somewhat messy unmaintained manner. After years of just entering those damn <p>s (and over a year since this draft was in the making), I decided to do my own plugin for post-by-post (and global) control of the wpautop filter. It\u2019s called wpautop-control:\n\n<?php\n\/*\nPlugin Name: wpautop-control\nPlugin URI: http:\/\/blog.bigsmoke.us\/tag\/wpautop-control\/\nDescription: This plugin allows you fine control of when and when not to enable the wpautop filter on posts.\nAuthor: Rowan Rodrik van der Molen\nAuthor URI: http:\/\/blog.bigsmoke.us\/\nVersion: 1.0\n*\/\n\n}\n\nfunction wpautop_control_options() {\nif (!current_user_can('manage_options')) {\nwp_die( __('You do not have sufficient permissions to access this page.') );\n}\n\n?>\n<div class=\"wrap\">\n<h2>wpautop control options<\/h2>\n\n<form method=\"post\" action=\"options.php\">\n<?php settings_fields('wpautop-control') ?>\n<table class=\"form-table\">\n<tr valign=\"top\">\n<th scope=\"row\">wpautop filter on by default?<\/th>\n<td>\n<label><input type=\"radio\" name=\"wpautop_on_by_default\" value=\"1\" <?php if ( get_option('wpautop_on_by_default') == '1' ) echo 'checked=\"1\"' ?>> yes<\/label>\n<label><input type=\"radio\" name=\"wpautop_on_by_default\" value=\"0\" <?php if ( get_option('wpautop_on_by_default') == '0' ) echo 'checked=\"1\"' ?>> no<\/label>\n<\/td>\n<\/table>\n\n<p class=\"submit\">\n<input type=\"submit\" class=\"button-primary\" value=\"Save Changes\" \/>\n<\/p>\n<\/form>\n<\/div>\n<?php\n}\n\nfunction wpautop_control_settings() {\nregister_setting('wpautop-control', 'wpautop_on_by_default', 'intval');\n}\n}\n\nfunction wpautop_control_filter($content) { global$post;\n\n\/\/ Get the keys and values of the custom fields:\n$post_wpautop_value = get_post_meta($post->ID, 'wpautop', true);\n\n$default_wpautop_value = get_option('wpautop_on_by_default');$remove_filter = false;\nif ( empty($post_wpautop_value) )$remove_filter = ! $default_wpautop_value; elseif ($post_wpautop_value == 'true')\n$remove_filter = false; elseif ($post_wpautop_value == 'false')\n$remove_filter = true; if ($remove_filter ) {\nremove_filter('the_content', 'wpautop');\nremove_filter('the_excerpt', 'wpautop');\n}\n\nreturn \\$content;\n}\n}\n\n?>\n\n(I\u2019ve requested the plugin to be added to the WordPress plugin repository, so that it won\u2019t have to be reinvented another 20 times in the next year or so.) \ud83d\ude09\n\nAfter installing the plugin, you can choose whether to enable or disable wpautop by default. Then, for every post where you want to deviate from the default, you can set the wpautop custom field to \u2018true\u2019 or \u2018false\u2019.\n\nI want the new default to be to enable the filter, but since all my old posts have been manually formatted, I want all these to have the wpautop field added and set to \u2018false\u2019.\n\nAdding the appropriate custom field values to all existing posts is easy thanks to MySQL\u2019s INSERT \u2026 SELECT syntax:\n\nINSERT INTO wp_postmeta (post_id, meta_key, meta_value)\nSELECT wp_posts.ID, 'wpautop', 'false'\nFROM wp_posts\nWHERE post_type = 'post';\n\nDuring the recent redesign of my blog, I decided that I wanted to have pretty pagination with numbers instead of the WordPress default Older\/Newer Posts links. The plugin I decided to use was WP Page Numbers by Jens T\u00f6rnell.\n\nThis is how the pagination for page one of my home looks now:\n\nFor page six you can see more of what the plugin can do:\n\nAnd finally\u2026\n\nDo I have twenty pages of posts already?\n\nYesterday night, after mucking around with my Subversion repo for this blog for way too long, I finally stopped annoying the designer of my new theme and uploaded it, one and a half year after the last major redesign. Anyway, while implementing the new design for the comment list , I decided it was time to have comment previews.\n\nAt some time, I had already installed (but not activated) the Live Comment Preview plugin, but that\u2019s client-side only. I removed it because I want the comment to show as it would after being piped through all the hooks and filters that comments normally get piped through. Enter the AJAX Comment Preview plugin:\n\nOther preview plugins don\u2019t know what sort of changes WordPress will make to a visitor\u2019s comment, but this plugin uses AJAX and other buzzwords to send each previewed comment through WordPress\u2019 inner voodoo.\n\nThe result? With the click of a button, your site\u2019s visitors can preview their comments exactly as they will appear when they submit them for realies.\n\nYou just gotta love their phrasing. \ud83d\ude42 Enjoy the new preview feature.\n\nI like GeSHi (enough even to have written a language file for it). For ages now, I\u2019ve used a WordPress plugin by Dan Peverill. But for as long as I\u2019ve been using the plugin, I\u2019ve been looking to get rid of it.\n\nDan Peverill\u2019s GeSHI plugin sucks for two reasons:\n\n1. It\u2019s no longer being maintained. It doesn\u2019t even seem to justify a page on Dan\u2019s website anymore (for which reason I\u2019m not going to give him any link-juice).\n2. It breaks HTML. With the plugin enabled I can no longer use the <code> tag to mark in-line elements as being code. Frankly, this is annoying and I find myself typing <tt> often when I mean <code>.\n\nA search for WordPress plugins tagged GeSHi reveals a number of results: Sniplets, CodeColorer, Developer Formatter, and WP-SynHighlight. WP-Syntax is a plugin that is missing from the tag search.\n\nSniplets seems much too generic to my taste. I just want a GeSHi highlighter, period.\n\nCodeColorer says it does what I want, but if I ever want to use the TinyMCE editor again, I won\u2019t be able to with this plugin. Shouldn\u2019t be too much of a problem, but still\u2026\n\nDeveloper Formatter is very thoroughly written and even sports a TinyMCE plug-in for copying\/pasting the code. It is pretty big, though, and, as a rule, I tend to avoid plug-ins that complicate the database schema. I also don\u2019t really see how these extra tables are an advantage feature-wise.\n\nWP-SynHighlight uses a custom BBCode-style tag, [codesyntax] I like this (if you\u2019re going to use pointy brackets, at least keep out of the HTML namespace), though I don\u2019t like the attempt at a generic name; what\u2019s wrong with calling the tag [geshi]? Seriously\u2026 I\u2019m sure I\u2019m going to forget this name billions of times if I\u2019ll use this plug-in.\n\nWP-Syntax uses the <pre> tag with a few custom attributes. This at least is better than the officially inline <code> tag that my current plugin uses, because most of the time that I\u2019d use a <pre> tag I really do want syntax highlighting. Just wondering: will it also allow my to use it normally for that other rare occasion? Sadly, the plugin will doubtlessly wreak havoc with the visual (TinyMCE) editor.\n\nSo, which plugin will I choose? I am somehow inclined to want a plugin that can play nice with the visual editor because I keep telling myself how much nicer it would be to switch to the visual editor for all my posting. (That this will be difficult because I disabled WP\u2019s \u2018wpautop\u2018 filter to rid myself of its eagerness is a story for some later time.) This requirement rules out CodeColorer and WP-Syntax.\n\nThat leaves Developer Formatter and WP-SynHighlight. Both seem to fit my purpose. Developer Formatter sports a nice TinyMCE plugin for inserting code, but I don\u2019t think that switching to TinyMCE will suddenly and unexpectedly make me afraid of typing. Besides, I really don\u2019t want the extra tables in my database without a very good reason, so, for now, I will try WP-SynHighlight.\n\nAfter upgrading to WordPress 2.5.x, I had to fall back on a stock theme because my old customization of the Sandbox theme no longer worked with the upgrade. But, then, it was time to redo my theme anyway. So here you\u2019re looking at the first version of my new theme. I might have let it stabilize some more before putting it on-line, but who cares? My reader maybe? Let\u2019s just hope he or she doesn\u2019t use IE. \ud83d\ude09\n\nEver since the first time that I saw a blog which featured vertical navigation instead of the typical columns, I\u2019ve wanted to implement this for myself. Well, finally\u2026\n\nSite-wide elements use the complete width of the page. The page content is centered in the middle at 87.5%. The identity stuff in the header and the navigation in the footer sits against a back blackground while the content area has the proven black on white for easy reading. I hope that the strong color-contrast as well as the clear difference in with between site-wide elements and page content makes it easy to keep focused on either reading or navigating without distractions.\n\n## \u2026 and a talkative footer\n\nWith this theme, I didn\u2019t want another footer which consist of the odd logo and some loose copyright statements. I wanted a footer which you can actually read, even understand. And who cares if it takes up a little space? It\u2019s at the bottom of the page.\n\n## Related posts\n\nI\u2019ve written an (unpublished, unpolished) plug-in which can generate a list of posts that are chronologically related. Traditionally, most blogs have a next\/previous post link at the top and bottom of each post. This works very well if you limit your blog to one subject (which is really a very good idea anyway), but if, like mine, your blog is a little bit messy, you could say that someone who stumbled here searching for an article about Subversion is not necessarily interested in the next post if this is a photo of my baby niece.\n\nHence the chronologically related posts plugin. With this plugin I can say wether I want a link to the first, previous and next post in the blog, within the same category, or matching a given number of tags. (The tag matching isn\u2019t implemented yet, though. Also, matching on meta fields would be a kick-ass ass way to support explicit sequences.)\n\nI put the list generated by this plug-in on top of a blue background besides the various context links of the post.\n\n## Issues left\n\nI hope to have the first major revision of my theme ready soon. Here\u2019s a list of some issues that I might address:\n\n\u2022 The CSS renders a bit psychedelically in MSIE 6 (only version I tested) at the moment. Sigh\u2026 Let\u2019s just hope that IE 7 will give better results. Then I\u2019ll gladly drop the IE 6 support.\n\u2022 When viewing a category, the tag cloud in the navigation panel at the bottom only shows tags for that category. This has to do with the use with me calling the st_tag_cloud() from within the category template.\n\u2022 Some of the elements that I just showed to you don\u2019t really look that good and most elements that I didn\u2019t can be said to be \u2026 hideously ugly. \ud83d\ude15 Some highlights: the header (should really be a few cool images), the comment form, and the Next\/Previous Page links.\n\n## Comment!\n\nI\u2019d almost forget all about the clean, new look of the comment list. And, if you register a Gravatar, your comments will be accompanied by your avatar. Try it. Please!\n\nI was once again annoyed by the fact that WordPress doesn\u2019t allow dots in post slugs. Luckily, this time I hadn\u2019t published the post with a botched URL yet. (I don\u2019t like changing permalinks because they\u2019re meant to be permanent; cool URLs don\u2019t change.) A quick googling pointed me to a post in the WordPress support forum with a reference to the Periods in Titles WordPress plugin.\n\nThe plugin works great and allowed me to post http:\/\/\/2007\/05\/30\/jeroen-dekker.com with dots and without problems.","date":"2023-02-04 05:22:15","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\": 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\": 0.3225921094417572, \"perplexity\": 2927.8355606770624}, \"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-2023-06\/segments\/1674764500094.26\/warc\/CC-MAIN-20230204044030-20230204074030-00870.warc.gz\"}"} | null | null |
Posts Tagged /ACC/
Pitt Basketball Staff Update
Multiple reports, including one from the Pittsburgh Post Gazette have confirmed that new Pitt head coach Jeff Capel will add Tim O'Toole and Jason Capel to his staff. O'Toole has most recently served as the associate head coach at Cal. He also served as an assistant at Stanford, Syracuse and Duke (when Capel played there),
Miami, Larrañaga Agree in Principle to Two-Year Contract Extension
The University of Miami has agreed in principle to a two-year contract extension with men's basketball coach Jim Larrañaga, UM Director of Athletics Blake James announced Tuesday. The extension will stretch Larrañaga's contract through May 31, 2024. "Coach Larrañaga is an integral part of our athletics department's success, and he continues to demonstrate year after
Notre Dame's Mike Brey Signs Contract Extension Through 2024-25 Season
Mike Brey, the all-time winningest coach in Notre Dame men's basketball history, has signed a contract extension that will keep him on the Irish sidelines as the Glenn and Stacey Murphy Head Men's Basketball Coach through the 2024-25 season. "Mike Brey has built one of the most consistently successful programs in the country," University Vice
Louisville Basketball Staff Update
Three experienced assistant men's basketball coaches have joined head coach Chris Mack's staff at the University of Louisville, including Dino Gaudio, Luke Murray and Mike Pegues. Pegues has worked as an assistant coach with Mack for the past six seasons at Xavier, Murray has been an assistant with Mack for the past three seasons at
Duke Basketball Staff Update
Duke University men's basketball coach Mike Krzyzewski announced the hiring of former Blue Devil Chris Carrawell as an assistant coach. Earlier this week, Krzyzewski promoted Nate James and Jon Scheyer to associate head coaches after Jeff Capel was named the head coach at Pittsburgh. To complete his restructured staff, Coach K has also announced that
OFFICIAL: Chris Mack Chosen to Lead Louisville Men's Basketball Program
Chris Mack, whose teams have participated in the NCAA Tournament in eight of his nine seasons as a head coach, has been selected as the head coach of the University of Louisville men's basketball team. In nine seasons as the head coach at Xavier, Mack guided the Musketeers to a 215-97 record and eight NCAA
WATCH LIVE: Chris Mack Introductory Press Conference at Louisville – 4PM ET
Chris Mack will be introduced as Louisville's new head basketball coach at a 4pm ET press conference today. Watch the event live HERE.
OFFICIAL: Jeff Capel Named Head Basketball Coach at Pitt
Jeff Capel has been named the head coach of the University of Pittsburgh men's basketball program as announced Tuesday afternoon by Pitt Director of Athletics Heather Lyke. "We are excited to announce Jeff Capel as our head men's basketball coach at the University of Pittsburgh," said Lyke. "Coach Capel is a high-energy leader committed to
REPORT: Louisville, Chris Mack to meet this weekend
According to a report by the Courier Journal, Officials from Louisville are scheduled to meet with Xavier head coach Chris Mack this weekend. The Cardinals officially parted ways with interim head coach Scott Padgett yesterday. Louisville will not be using a search firm, but will instead on a group of "informal" advisors surrounding Interim AD
OFFICIAL: Padgett Will Not Continue as Louisville's Head Men's Basketball Coach
David Padgett, who has served as the University of Louisville's interim head coach for the past six months, will not be retained as head coach moving forward with the Cardinals. Padgett guided the Cardinals to a 22-14 record and the quarterfinals of the National Invitation Tournament as UofL finished in a tie for eighth in | {
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Nationally, people with mental health needs have the lowest employment rate of any disabled group, with only 24% securing long-term work. Fewer than four in ten employers will consider recruiting someone with a history of mental health problems. Whilst people suffering mental illness have the lowest employment rates of any group of disabled people, they are more vulnerable to the negative effects of unemployment.
One in four people in Surrey suffers a psychiatric disorder at some point in their lives. Surrey is perceived as an affluent area, and yet, there are hidden pockets of deprivation where there are high levels of child poverty, low income and poor mental health. Surrey residents undoubtedly face significant challenges to their mental well-being; the county has the highest average house price in any UK county, high levels of debt and people living in Surrey work many hours longer than average.
Oakleaf is the only charity offering a comprehensive vocational training and support service for mentally ill people in south west Surrey. We currently support approximately 400 people in both social inclusion activities and the following vocational training.
Upholstery: A very therapeutic practice involving restoring modern and traditional furniture and re-covering furniture pieces such as sofas, armchairs and dining chairs in new fabric.
Horticulture: The therapeutic nature of gardening, and scope for self-employment, is particularly suited to many clients. Working in a team encourages punctuality, commitment and motivation and promotes social cohesion.
IT: Providing clients with computer-training in the most commonly used Microsoft Office packages, making re-entry into the workplace closer and easier. | {
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Sherrie Levine, August Sander at Paula Cooper
Artist: Sherrie Levine, August Sander
Venue: Paula Cooper, New York
Date: February 16 – March 16, 2013
Click here to view slideshow
Full gallery of images, press release and link available after the jump.
August Sander. Installation view from Sherrie Levine / August Sander Paula Cooper Gallery, New York, NY (2/16 – 3/16/13). © Die Photographische Sammlung/SK Stiftung Kultur – August Sander Archive, Cologne; ARS, New York, 2013. Courtesy Paula Cooper Gallery, New York; and Galerie Priska Pasquer, Cologne. Installation photo: Steven Probert.
Sherrie Levine and August Sander. Installation view from Sherrie Levine / August Sander, Paula Cooper Gallery, New York, NY (2/16 – 3/16/13)
Sherrie Levine. Installation view from Sherrie Levine / August Sander , Paula Cooper Gallery, New York, NY (2/16 – 3/16/13).Courtesy Sherrie Levine and Paula Cooper Gallery, New York. Installation photo: Steven Probert.
Sherrie Levine. Installation view of President Profile (1978), Paula Cooper Gallery, New York, NY (2/16 – 3/16/13)
Sherrie Levine. After August Sander (detail), 2012 18 Lambda prints in artist frames each: 9 5/8 x 6 7/8 in. overall dimensions variable August Sander: Rural Bride , ca. 1925-30. Courtesy Sherrie Levine and Paula Cooper Gallery, New Y ork.
Images courtesy of Paula Cooper Gallery, New York
The German photographer August Sander (1876–1964) is remembered most notably for his monumental documentary project People of the 20th Century, a series of more than 600 individual portraits shot between 1910 and the early 1950s and whose organization into classes and professional groups form a veritable archive of German society in the first half of the 20th century. While only published in excerpts during Sander's lifetime, the project was finally released in book form by Sander's son Gunther in 1980. Prized for its aesthetic quality, philosophical scope and steadfast pursuit of documentary accuracy, the series is a cornerstone of 20th-century photography whose influence on contemporary notions of portraiture continues to be felt.
The thirty-six photographs on view were printed by Gunther Sander following his father's criteria and technique. They were selected by Gerd Sander, August's grandson, and depict an equal number of men and women. Grouped along gender lines rather than sociological categories, they provide a transversal look into the original project and include both well-known and rarely seen images.
An important part of Sherrie Levine's artistic practice since the 1970s has taken the form of photographic works based on other artists' photographs. "After August Sander: 1-18" is another distillation of Sander's original opus into a selection of eighteen pictures of men and women. This act of "looking back" not only holds the potential to reveal latent themes or motifs in Sander's work; it also implicitly and crucially differs from August, Gunther and Gerd Sander's in that the images have become wholly Levine's. Through this barely perceptible and yet immeasurable difference, this transfer of authorship across genders, continents and personal histories, the portraits' expressive powers resonate anew.
Link: Sherrie Levine, August Sander at Paula Cooper
Tags: August Sander, Group Show, New York, Paula Cooper, Sherrie Levine, United States
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Zhang Liang (; ur. 14 stycznia 1987 r. w Jinzhou) – chiński wioślarz, mistrz świata, trzykrotny złoty medalista igrzysk azjatyckich, uczestnik dwóch igrzysk olimpijskich.
Igrzyska olimpijskie
Zyskał sławę w 2008 roku, kiedy podczas igrzysk olimpijskich w Pekinie nie pojawił się na eliminacjach w konkurencji jedynek i tym samym został zdyskwalifikowany. Zgodnie z międzynarodowymi zasadami wioślarstwa został zdyskwalifikowany także w konkurencji dwójki podwójnej, w której miał wystąpić razem z Su Hui.
Cztery lata później w Londynie zajął piąte miejsce w finale B jedynek i ostatecznie został sklasyfikowany na jedenastej pozycji.
Przypisy
Linki zewnętrzne
Chińscy wioślarze
Wioślarze na Letnich Igrzyskach Olimpijskich 2008
Uczestnicy Letnich Igrzysk Olimpijskich 2012
Medaliści Igrzysk Azjatyckich 2010
Medaliści Igrzysk Azjatyckich 2014
Medaliści Igrzysk Azjatyckich 2018
Uczestnicy Mistrzostw Świata w Wioślarstwie 2007
Urodzeni w 1987 | {
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{"url":"http:\/\/copaseticflow.blogspot.com.ar\/2012\/09\/an-intuitive-way-to-spherical-gradient.html","text":"## Monday, September 10, 2012\n\n### An Intuitive Way to the Spherical Gradient and Laplacian\n\nIt's that time of year again when physics students everywhere are deriving the spherical and cylindrical del, nabla, gradient, or Laplacian operators. \u00a0Every derivation I saw prior to this week involved lots of algebra and the chain rule... even mine. \u00a0Fortunately for me, a comment on my derivation, and a homework assignment from Rutgers\u00a0[pdf] led me to a far simpler and more intuitive way of doing things. \u00a0You just start from the differential displacement in a given coordinate system and go from there.\n\nThe differential displacement in spherical coordinates is:\n\nThe element in the r direction is easy to understand. \u00a0A small displacement along the r direction is represented as dr. \u00a0The theta and phi displacements might not be as obvious. \u00a0The graphic to the left illustrates what's going on. \u00a0With small displacement along the theta direction you're moving along a circle with radius r. \u00a0The distance you've moved is equal to the length of the arc which is equal to the radius of the circle, r times the small angle, hence,\u00a0. \u00a0The element in the in the phi direction is similar to the element in the theta direction in that you're moving along a circle. \u00a0However, this time you have to keep in mind that the radius of the circle is not r. \u00a0The r vector is propped up by angle of theta and so, the projection of r on the r\/phi plane is\u00a0, and the distance\u00a0traveled\u00a0along the phi displacement is\u00a0.\nOnce we have our differential displacement element, the rest is kind of easy. \u00a0First, we get the gradient operator. \u00a0The gradient is just the derivative of a function with respect to the differential displacement, so we get:\n\nThe next step is to derive the divergence in spherical coordinates. \u00a0First, we need the differential surface area element in all three dimensions. \u00a0I don't have a great picture for this yet, so you'll just have to visualize a small square on the surface of a sphere. \u00a0The surface area element in the r direction is a rectangle on the surface of a sphere with a length of\u00a0\u00a0on one side and a length of\u00a0\u00a0on the other. \u00a0So, the element of area is the product of the sides or\u00a0\u00a0. \u00a0In the theta direction, the sides are a small displacement in the r direction and a small displacement in the phi direction giving us\u00a0. \u00a0Finally, in the phi direction the sides of the rectangle are a small displacement along r and a small displacement along theta giving us , (.\n\nYou might be wondering what all the surface elements had to do with divergence. \u00a0This is where the tip I mentioned at the start of the post comes in. \u00a0Divergence is defined to be the flux of a vector field, (the vector field dot producted with the surface area), per unit volume. \u00a0Now that we have a differential surface area vector, we can dot product our vector of interest, (say A), with it to get the flux through a differential surface area. \u00a0We wind up with:\n\nNow, we need the differential element of volume for spherical coordinates. \u00a0Once again, we can use purely geometric considerations to get where we want to go. \u00a0Visualize making a small cube out of the small rectangles we defined above for the surface area differentials. \u00a0If you took the volume of that cube, you'd just multiply the three sides:\n\nNow to get the divergence, we just divide each term of the flux by the differential volume element. \u00a0For the r term we get:\n\nthe theta and phi differentials cancel out. \u00a0You can bring the sin theta term outside of the derivative and cancel it since it doesn't depend on r.\n\nFor the theta term we get:\n\nThe r and phi differentials cancel. \u00a0The r can be taken outside the derivative since it is constant with respect to the remaining theta differential.\n\nFinally, for the phi term we get:\n\nThe r and theta differentials cancel and we can bring the r outside the remaining phi derivative.\n\nAll together, we now have the three terms of the divergence in spherical coordinates:\n\nWe didn't use the chain rule once, just simple geometry!\n\nThe final task is to derive the Laplacian. \u00a0The Laplacian is the divergence of the gradient. \u00a0We simply substitute our gradient result in place of the vector A in our divergence experession to get the following three terms:\nr:\n\ntheta:\n\nphi:\n\nWillemHekman said...\n\nDear Blogger, first of thanks for this post, it got me started in the right direction.\n\nBut, unfortunately your way of deriving had two connected oversimplifications which I think to have solved:\n\nFirst of, you need to account for the flux going through the box. To do so you dot the surface of ONE side with the vector. But here you do not account for the other side. A constant vector in the dot product with both sides would yield no net flux...\n\nSolution: You actually skip a step where you should say that the difference in the vector A in the direction of the surface on opposite sides of the box is the infinitesimal difference dA between A(r) and A(r + dr) for instance.\n\nThe other problem was to me a manifestation of the previous simplification:\n\nIn calculating the end result you do the following (1\/dr) x r^2 x Ar = (d\/dr) x r^2 x Ar. This is a big oversimplication\/ error to make.\n\nSolution: Use the dA instead from before this way you`ll get 1\/dr x r^2 x dA = d(r^2 x A)\/dr.\n\nI would really appreciate if you responded.\n\nHamilton Carter said...\n\nWillem,\nThanks very much for all the info. I think you're correct and it clears up a bit of confusion I had regarding some of the steps. Thanks very much! I'm going to write more when I have time to review what you've said more completely. Thanks again!","date":"2015-07-05 09:25:25","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\": 0.848878800868988, \"perplexity\": 546.4457672560869}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2015-27\/segments\/1435375097396.10\/warc\/CC-MAIN-20150627031817-00020-ip-10-179-60-89.ec2.internal.warc.gz\"}"} | null | null |
Specifies the file outputs of a custom build command or rule.
```lua
buildoutputs { "output" }
```
### Parameters ###
`output` is the file that is created or updated by the custom build command or rule.
### Applies To ###
Project configurations and rules.
### Availability ###
Premake 5.0 or later.
### See Also ###
* [Custom Build Commands](Custom-Build-Commands.md)
* [Custom Rules](Custom-Rules.md)
* [buildcommands](buildcommands.md)
* [builddependencies](builddependencies.md)
* [buildinputs](buildinputs.md)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,898 |
The Old Faithful Museum of Thermal Activity was one of a series of four "trailside" museums built in Yellowstone National Park in 1929. Funded by a grant of $118,000 from Laura Spelman Rockefeller, the museums interpreted park features for visitors, and represented an early version of the visitor information center concept that became widespread throughout the National Park Service. The four museums were notable examples of the National Park Service Rustic style, and all were designed by Park Service architect Herbert Maier. The surviving Norris Museum, Fishing Bridge Museum and the Madison Museum are collectively listed as National Historic Landmarks.
The Old Faithful museum, the first of the series, was built at a cost of $8,500 and was completed in 1929. The museum was a low T-shaped single-story structure of rustic log and stone construction. Two stepped sections of roof dominated the main portion of the building with deep overhangs supported by angled log brackets resting on a raised stone foundation sill. A perpendicular wing extended in the direction of the parking lot. The building resembled the Madison and Fishing Bridge museums. The museum's surroundings featured an amphitheater for ranger talks and a small garden of native botanical specimens.
The Old Faithful museum was demolished in 1971 to make way for a full-scale Mission 66 visitor center on the site, midway between the Old Faithful Inn and the Old Faithful Lodge, facing Old Faithful geyser. This visitor center was in turn demolished in 2006 and was replaced by the Old Faithful Visitor Education Center, opened in August 2010.
References
External links
Norris, Madison and Fishing Bridge Museums at the National Park Service
Buildings and structures demolished in 1971
Rustic architecture in Wyoming
Buildings and structures in Yellowstone National Park in Wyoming
Defunct museums in the United States
Natural history museums in Wyoming
1929 establishments in Wyoming
Museums established in 1971
1971 disestablishments in Wyoming
Museums disestablished in 1971 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 637 |
By the time you get to be an adult with CF, the need to maintain a good body weight is something you will have heard over and over again.
Diet definitely does matter. Achieving and maintaining a good weight helps to maintain good lung function, which can result in less time in hospital.
A healthy body mass index (BMI) for someone with CF is between 22-25.
A person with CF may need up to 120%-150% of the calories needed by someone without CF. This is due to an increased energy expenditure and a number of different reasons.
Why can a high calorie diet be hard to achieve?
Forgetting or taking too few enzymes will result in some fat (energy) being wasted in your stools.
Even if you are taking your enzymes appropriately you will continue to lose some fat in your stools and this contributes to your increased energy requirements.
In the case of maintaining a good body weight, the answer is very simple. If you want to stay well for longer and you want to stay out of hospital, then work at keeping up your weight so that your body can fight off the infections that are bound to hit you from time to time.
A gastrostomy tube (PEG) is used by people with CF (PWCF) to supplement their food intake. This can help to maintain or improve your weight and achieve good nutrition.
Maintaining or gaining weight to reach a healthy BMI.
A PEG can be used during the day or for overnight feeding while you sleep. It is used to supplement your food intake, not to replace what you usually eat.
How are PEG tubes put in?
PEG tubes are usually put in by a specialist team in the Gastroscopy Day Unit. While you are sedated, the tube is placed into your stomach through the abdominal wall. Initially you will have a longer tube in place until the incision site (stoma/tract) has healed completely (6-8 weeks).
The long tube is then replaced with a low profile "button" as shown in the picture. Using a connecting tube, supplement drinks/feeds can go straight into your stomach.
Most people will use their PEG tube on a daily basis, or several times a week, however some people who have achieved their target weight may only use it as a back up if they get sick or lose weight.
If you think a PEG may be helpful for you, or are interested in finding out more, please ask one of the CF team for more information.
If you would like to find out more about CF and nutrition, the websites below have some useful information. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,528 |
\section{Introduction}
Bayesian neural networks (BNNs) have achieved significantly less success than their deterministic NN counterparts.
Despite recent progress \citep[e.g.,][]{khan18a, maddox2019, osawa2019, dusenberry20a, daxberger21a, izmailov2021bayesian},
(a)~our theoretical understanding of BNNs remains limited, and (b)~practical applicability is hindered by high computational demands.
We make progress on both these fronts.
Our work revolves around a reparametrisation $\theta = T(\phi)$ of the (flattened) NN weights $\theta \in \R{d}$.
Its form comes from a
theorem in which we establish that the Kullback--Leibler (KL) divergence between the standard normal distribution $\mathcal{N}(0, I_d)$ and the reparametrised \emph{posterior} $
\dens{p}(\phi \given \mathcal{D})
=
\dens{p}\pp{T(\phi) \given \mathcal{D}}
\left| \det \partial_\phi T(\phi)\right|$
converges to zero as the BNN layers grow wide.\footnote{Case distinguishes distributions ($\distr{P}$) and their densities ($\dens{p}$). Densities are used in place of distributions where convenient.}
This closes a
gap in the theory of overparametrised NNs by providing a rigorous characterisation of the BNN \emph{parameter space} behaviour (\Cref{fig:wide_net_research} and \Cref{sect:reparam}), and enables faster posterior sampling (\Cref{sect:sampling}).
\tcbset{width=2cm}
\begin{figure*}
\centering
\begin{tabular}{l @{\hspace{0.25em}}|@{\hspace{0.25em}}c@{\hspace{-1em}}c}
& \tabhead{parameter space} & \tabhead{function space} \\[0.2em]
\hline \\[-0.5em]
\centeredintabularfixedwidth{%
\tabhead{Bayesian\\\ inference}
} &
\centeredintabular{ \begin{tcolorbox}[width=2.7in,height=1.3in,colback=tabblue!75,boxrule=0pt,colframe=white,coltext=white,arc=0pt,outer arc=0pt
\centering\footnotesize
\tabinner{\color{white} \bf
{Repriorisation}: posterior $\rightarrow$ prior} \\
{\tiny \hypersetup{linkcolor=white}
\Cref{eq:full_reparam}, \Cref{stm:zero_kl} (this paper)}\\[-1.1em]
{\begin{align*}
&\phi^{\ell}
\coloneqq
\begin{cases}
\,\Sigma^{-1/2} (\theta^{\ell} - \mu )
&
\qquad \ell = L + 1,
\\
\,\theta^{\ell}
&
\qquad \text{otherwise.}
\end{cases}
\\[1em]
&\mathrm{KL}\pp{\, \mathcal{N}(0, I_{d}) \, \| \, \distr{P}_{\priorParam \given X, y \,} \to 0
\text{ as } \dimParam_{\min} \to \infty
\end{align*}}
\end{tcolorbox} } &
\centeredintabular{ \begin{tcolorbox}[width=2.7in,height=1.3in,boxrule=0pt,colframe=white,colback=gray!8,arc=0pt,outer arc=0pt]
\centering\footnotesize
\tabinner{\bf Neural Network Gaussian Process (NNGP)} \\
{\tiny \citep{matthews2018gaussian,lee2018deep,hron2020exact}} \\[-0.5em]
{
\begin{gather*}
\theta \sim \distr{P}_\theta
\, \,
\overset{\dimParam_{\min}}{\longrightarrow}
\, \,
f_{\theta} \sim \mathrm{GP} (0, k)
\\
\theta \sim \distr{P}_{\theta \given \mathcal{D}}
\, \,
\overset{\dimParam_{\min}}{\longrightarrow}
\, \,
f_{\theta} \sim \mathrm{GP} (m_{k; \mathcal{D}}, S_{k; \mathcal{D}})
\end{gather*}
}
\end{tcolorbox} } \\
\centeredintabularfixedwidth{%
\tabhead{gradient\\\ descent\\\ training}
} &
\centeredintabular{ \begin{tcolorbox}[width=2.7in,height=1.3in,boxrule=0pt,colframe=white,colback=gray!8,arc=0pt,outer arc=0pt]
\centering\footnotesize
\tabinner{\bf Linearisation} \\
{\tiny \citep{lee2019wide,chizat2019lazy}}\\[-0.5em]
{
\begin{gather*}
\partial_t \theta_t
=
-\eta\, \nabla_{\theta_t} \,\mathcal{L}\pp{y, f_{\theta_t}(X)}
\\[1em]
f_{t}\pp{x} =
f_{0}\pp{x} +
\tfrac{
\partial f_{0} (x)
}{
\partial \theta_0}
\pp{\theta_t - \theta_0}
+
\mathcal{O}\pp{
\dimParam_{\min}^{-1/2}
}
\end{gather*}
}
\end{tcolorbox}
} &
\centeredintabular{ \begin{tcolorbox}[width=2.7in,height=1.3in,boxrule=0pt,colframe=white,colback=gray!8,arc=0pt,outer arc=0pt]
\centering\footnotesize
\tabinner{\bf Neural Tangent Kernel (NTK)} \\
{\tiny \citep{jacot2018neural,allen2019convergence,du2019gradient}}\\[-0.5em]
{
\begin{gather*}
\partial_t f_t(x)
=
-\eta\, \hat{\Theta}_{x, X} \nabla_{f_t(X)} \,\mathcal{L}\pp{y, f_t(X)}
\\[1em]
f_t \mid \mathcal{D}
\sim
\mathrm{GP}\pp{
m_{t; \mathcal{D}}, S_{t; D}
}
\text{ as } \dimParam_{\min} \to \infty
\end{gather*}
}
\end{tcolorbox} }
\end{tabular}
\caption{\textbf{As neural networks are made wide, their behaviour often becomes simple.}
Past results
examine wide NNs in either function space or parameter space, and
either under gradient descent training or Bayesian inference. The central result for each condition is stated,
using this paper's formalism where applicable.
The behaviour of wide Bayesian NNs in parameter space is largely unexplored.
We address this gap,
by reparametrising the weight posterior so that its KL divergence from the prior $\mathcal{N} (0, I_{d})$ vanishes with increasing width.
}
\label{fig:wide_net_research}
\end{figure*}
In \emph{function space}, wide BNN \emph{priors} converge (weakly) to a so-called neural network Gaussian process (NNGP) limit \citep{matthews2018gaussian,lee2018deep,alonso2018deep,novak2019bayesian,yang2019scaling,hron2020infinite},
and the \emph{posteriors} converge (weakly) to that of the corresponding NNGP \citep[assuming the likelihood is a bounded continuous function of the NN outputs;][]{hron2020exact}.
\emph{Parameter space} behaviour is less understood \citep{hron2020exact}.
The main exception is the work of \citet{matthews2017sample}
who showed that randomly initialising a NN, and then optimising only the last layer with respect to a mean squared error loss, is equivalent to drawing a sample from the \emph{conditional last layer posterior}.
This result holds for a Gaussian likelihood and zero observation noise.
Our reparametrisation $T$ can be seen as a non-zero noise generalisation of the underlying map from the initial to the optimised weights, and provides samples from the \emph{joint posterior} over all layers in the infinite width limit (\Cref{stm:zero_kl}).
Since sampling from the `KL-limit' $\mathcal{N}(0 , I)$ is trivial relative to the notoriously complex BNN posterior, our theory motivates applying Markov chain Monte Carlo (MCMC) to the reparametrised density.
To apply MCMC, two concerns must be addressed:
(i)~understanding if the KL-closeness to $\mathcal{N}(0 , I_d)$ actually simplifies sampling, and
(ii)~computational efficiency.
In \Cref{sect:gradients}, we partially answer (i) by proving that
the gradient $\nabla_{\phi} \log \dens{p} (\phi \given \mathcal{D})$
concentrates around the log density gradient of $\mathcal{N} ( 0 , I)$ in wide BNNs.
For (ii), we propose a simple-to-implement and
efficient way of computing both $T(\phi)$ and the corresponding density at the same time using Cholesky decomposition (\Cref{sect:cholesky}).
Applying Langevin Monte Carlo (LMC), we empirically demonstrate up to 50x improved mixing speed, as measured by effective sample size (ESS).
The phenomenon occurs both for residual and fully-connected networks (FCNs), and for a variety of dataset sizes and hyperparameter configurations.
The improvement over no reparametrisation increases with layer width, but
is observed
even far from the NNGP regime.
For example,
we find a 10x improvement on \texttt{cifar-10} for a 3-hidden layer FCN with 1024 units per layer.
However, for ResNet-20, a 10x improvement occurs
only when the top-layer width $\dimParam^{\nLayers}$ is similar to or larger than the number of observations, illustrating that gains are possible but not guaranteed outside of the NNGP regime.
\subsection{Assumptions and notation}
\label{sect:notation_assumptions}
A BNN models a mapping from inputs $x \in \mathcal{X}$ to outputs $y \in \mathcal{Y}$ using a parametric function $f \coloneqq f_\theta$.
An example is an $L$-hidden layer fully-connected network $f_\theta = f^{L + 1}$ with
\begin{align}\label{eq:fcn}
f^{\ell} (x)
=
\tfrac{\sigma_\weight^{\ell}}{\sqrt{d^{\ell-1}}}
h^{\ell - 1}(x) W^{\ell}
\, ,
\quad
h^{\ell}(x) = \psi(f^{\ell}(x))
\, ,
\end{align}
with $\psi$ the nonlinearity, $h^0 (x) \mathrel{\mathop{:}}= x$, and $W^{\ell} \in \R{d^{\ell-1} \times d^{\ell}}$
(bias terms are wrapped into $W^{\ell}$ by adding a constant entry dimension to $h^{\ell - 1}(x)$).
$\theta^{\ell}$ will denote the flattened $\ell$\textsuperscript{th} layer parameters,
and $\theta \coloneqq [ \theta^\ell]_{\ell=1}^{L + 1} \in \R{d}$ their concatenation.
Later on, the vector of readout weights $\param^{\nLayers + 1}$ will be of special importance, as will the \textbf{minimum hidden layer width} $\boldsymbol{\dimParam_{\min} \coloneqq \min_{1 \leq \ell \leq L} d^{\ell}}$.
While we will study how the behaviour of $f$ and $\theta$ changes with layer width, we suppress this dependence in our notation to reduce clutter.
The factor
$\nicefrac{\sigma_\weight^{\ell}}{\sqrt{d^{\ell-1}}}$
in \Cref{eq:fcn} is more commonly part of the weight prior.
We use this so-called `NTK parametrisation' as it allows taking $\mathcal{N}(0, I)$ as the prior regardless of the network width, which simplifies our notation without changing the implied function space distribution.
Our claims hold under the standard parametrisation as well, by making
multiplication by
$\nicefrac{\sigma_\weight^{\ell}}{\sqrt{d^{\ell-1}}}$
a part of the reparametrisation.
We assume the final readout layer is linear, and that the likelihood is Gaussian $\dens{p}(y \given X, \theta) \propto \exp \{ -\tfrac{1}{2 \sigma^2}\sum_{i=1}^n (y_i - f_{\theta}(x_i))^2 \}$ with observation variance $\sigma^2 > 0$.
In contrast, the rest of the network need not be an FCN, but can contain any layers for which NNGP behaviour is known, including convolutional and multi-head attention layers, and skip connections \citep[see, e.g.,][for an overview]{yang2020tensor}.
\section{Repriorisation: \texorpdfstring{posterior $\to$ prior}{mapping posterior to prior}}
\label{sect:reparam}
\subsection{Repriorisation of linear models}
\label{sect:lin_models}
To begin,
we consider a Bayesian linear model
$y \given \theta , x \sim \mathcal{N} (x^\top \theta, \sigma^2)$.
For a standard normal prior $\theta \sim \mathcal{N}(0, I_d)$,
the Bayesian posterior after observing $n$ points, $\mathcal{D} \coloneqq (X, y) \in \R{n \times d} \times \R{n}$, is a Gaussian $\theta \given \mathcal{D} \sim \mathcal{N} (\mu , \Sigma)$ with
\begin{align}\label{eq:lin_posterior_params}
\Sigma
=
(I_d + \sigma^{-2}X^\top X)^{-1}
, \, \text{and }
\mu
=
\sigma^{-2} \Sigma X^\top y
\, .
\end{align}
We can thus \emph{transform a posterior into a prior sample} using the data-dependent reparameterisation $\phi = T^{-1}\pp{\theta} = \Sigma^{-1/2}(\theta - \mu) \sim \mathcal{N}(0, I_d)$ as illustrated in \Cref{fig:lin_reparam}.
\begin{figure}[tbp]
\centering
\begin{tabular}[t]{@{\hspace{0em}}c @{\hspace{1.4em}} c @{\hspace{0.9em}} c}
\begin{overpic}[width=0.8in,trim=7 8 6 6]{FIGURES/lin_theta.pdf}
\put (90,0) {\huge $\theta$}
\end{overpic}
&
{\begin{minipage}[b]{1.2in}%
\centering%
\noindent%
\unskip\parfillskip 0pt \par%
\begin{overpic}[width=1in,trim=3 7 3 2]{FIGURES/lin_legend.pdf}
\end{overpic} \\[0.5em]
\scriptsize
$\phi = T_\mathcal{D}^{-1}\pp{\theta }$ \\[0.15em]
\begin{tikzpicture}[scale=1, transform shape]
\draw[>=triangle 45, <-] (0,0) -- (3,0);
\draw[>=triangle 45, ->] (0,0.2) -- (3,0.2);
\end{tikzpicture}
$\theta = T_\mathcal{D}\pp{\phi}$
\\[1em]
$\hat{y} = x^\top \theta = x^\top T_\mathcal{D} \pp{\phi}$
\end{minipage}}%
&
\begin{overpic}[width=0.8in,trim=7 8 6 6]{FIGURES/lin_phi.pdf}
\put (90,0) {\huge $\phi$}
\end{overpic}
\end{tabular}
\caption{
\textbf{Reparameterisation of Bayesian \emph{linear} regression maps the posterior to the prior.}
\textbf{Left:} Shows the parameter $\theta$ prior, and the posterior for a single datapoint $X = [ 0.9, 0.5 ], \ y = [ 2 ]$, with $\sigma^2 = 0.1$.
\textbf{Right:} After a data-dependent affine reparameterisation, described in \Cref{sect:lin_models},
the posterior becomes identical to the prior in terms of new parameters $\phi$.
MCMC sampling from the parameter posterior is easier in $\phi$ than in $\theta$.
A similar reparameterisation can be applied to deep BNNs.}
\label{fig:lin_reparam}
\end{figure}
\subsection{Repriorisation of Bayesian neural networks}
\label{sect:reparam_nonlinear}
A similar insight applies to BNNs with the assumed Gaussian prior and likelihood.
Specifically,
the posterior of readout (top-layer) weights \emph{conditioned} on the other parameters is $\param^{\nLayers + 1} \given \param^{\leq \nLayers}, \mathcal{D} \sim \mathcal{N}(\mu, \Sigma)$ with $\mu$ and $\Sigma$ as in \Cref{eq:lin_posterior_params}, except $X$ is replaced by the scaled top-layer embeddings
$
\Psi \coloneqq
\nicefrac{
\sigma_\weight^{L + 1}
h^{L}(X)
}{
\sqrt{\dimParam^{\nLayers}}
}
\in \R{n \times \dimParam^{\nLayers}}
\!$.
Our reparametrisation $\phi = T^{-1}(\theta)$ can thus be defined analogously:%
\footnote{
See \Cref{app:alt_reparam} for discussion of alternative reparametrisations which modify parameter values in layers beyond readout.
}
\begin{align}\label{eq:full_reparam}
\phi^{\ell}
=
\begin{cases}
\Sigma^{-1/2} (\theta^{\ell} - \mu )
&
\qquad \ell = L + 1,
\\
\theta^{\ell}
&
\qquad \text{otherwise.}
\end{cases}
\end{align}
This ensures $\priorParam^{\nLayers + 1} \given \priorParam^{\leq \nLayers}, \mathcal{D} \sim \mathcal{N}(0, I_{\dimParam^{\nLayers}} )$ for any fixed value of $\priorParam^{\leq \nLayers} = \param^{\leq \nLayers}$.
We omit the dependence of $\mu$, $\Sigma$ on the dataset $\mathcal{D}$ and the pre-readout parameters $\param^{\leq \nLayers}$ to reduce clutter, but emphasise that $T$ depends on both.
Since $T$ is a differentiable bijection, the full reparametrised density is $\dens{p} (\phi \given \mathcal{D}) = \dens{p} (\theta \given \mathcal{D}) \left| \det \partial_\phi \theta \right|$
where
\begin{align}\label{eq:determinant}
\det (\partial_\phi \theta)
=
\det
\begin{pmatrix}
\Sigma^{1/2} &
\tfrac{\partial \param^{\nLayers + 1}}{\partial \priorParam^{\leq \nLayers}}
\\
0 &
I_{(d - \dimParam^{\nLayers})}
\end{pmatrix}
=
\sqrt{\det(\Sigma)}
\, .
\end{align}
\subsection{Interpreting the reparametrised distribution}
To better understand the reparametrisation, note
$\dens{p} (\phi \given \mathcal{D}) = \dens{p}(\priorParam^{\nLayers + 1} \given \priorParam^{\leq \nLayers}, \mathcal{D}) \, \dens{p}(\priorParam^{\leq \nLayers} \given \mathcal{D})$
where we already know the first term is equivalent to $\mathcal{N}(0 , I_{\dimParam^{\nLayers}})$.
Since $\priorParam^{\leq \nLayers} = \param^{\leq \nLayers}$, the latter term is equal to the \emph{marginal} posterior over $\param^{\leq \nLayers}$
\begin{align}\label{eq:remainder_posterior}
&\dens{p}(\priorParam^{\leq \nLayers} \given \mathcal{D})
\propto
\dens{p} (\param^{\leq \nLayers})
\, \E_{\param^{\nLayers + 1} \sim \mathcal{N}(0, I_{\dimParam^{\nLayers}})} \left[
\dens{p} (y \given \theta, X)
\right]
\nonumber
\\
&\overset{\text{(i)}}{\propto}
\, \dens{p}(\param^{\leq \nLayers})
\, \sqrt{\det \Sigma}
\, \exp \bigl\{
\tfrac{1}{2} \,
y^\top (\sigma^2 I_n + \Psi \Psi^\top)^{-1} y
\bigr\}
\end{align}
where
(i)~comes from completing the square $\| \param^{\nLayers + 1} - \mu \|_{\Sigma^{-1}}^2$ ($\|v\|_A^2 \coloneqq v^\top A v$),
and applying the Woodbury identity.
Using the Weinstein--Aronszajn identity
\begin{align*}
\det (\Sigma)
=
\det (I_{\dimParam^{\nLayers}} + \sigma^{-2} \Psi^\top \Psi)
\propto
\det (\sigma^2 I_{n} + \Psi \Psi^\top)
\, ,
\end{align*}
the readers familiar with the GP literature may recognise $\dens{p}(\priorParam^{\leq \nLayers} \given \mathcal{D})$ as $\dens{p}(\param^{\leq \nLayers})$ weighted by the marginal likelihood of a centred GP with the \emph{empirical NNGP kernel} $\regnngp[\sigma^2] \coloneqq \sigma^2 I_n + \Psi \Psi^\top$ \citep{rasmussen2005gps}.
The marginal likelihood is often used for optimisation of kernel hyperparameters.
Here the role of the `hyperparameters' is played by $\param^{\leq \nLayers}$, i.e., all but the readout weights.
This relation to the empirical NNGP kernel is crucial in the next section where we exploit the known convergence of $\regnngp[\sigma^2]$ to a constant \emph{independent} of $\param^{\leq \nLayers}$ in wide BNNs, to prove that the reparametrised posterior converges to the prior distribution at large width.
\Cref{fig:posterior_to_prior} provides an informal argument motivating that same convergence, using the language of probabilistic graphical models (PGMs).
\begin{figure}[tbp]
\centering
\includegraphics[keepaspectratio,width=0.74\linewidth]{FIGURES/fcn_reparam_convergence}
\vspace*{-1em}
\caption{
\textbf{The functions parametrised by $T(\phi)$, $\phi \sim \mathcal{N}(0, I)$, converge to the true posterior.}
The distribution over functions $f_{T(\phi)}$
is shown for a 3-hidden layer FCN with 1D inputs and outputs, and 3 datapoints (dark circles).
As layer width increases (legend), the BNN posterior converges to the NNGP posterior \citep{hron2020exact}, and so does $f_{T(\phi)}$ (our \Cref{stm:kl_functions}).
}
\label{fig:reparam_convergence}
\vspace{-0.625\baselineskip}
\end{figure}
\subsection{Asymptotic normality under reparametrisation}
\label{sect:asympt_normality}
\input{pgm}
Our reparametrisation guarantees normality of the reparametrised posterior over $\priorParam^{\nLayers + 1}$ for \emph{any} width.
\Cref{eq:remainder_posterior} suggests
$
\priorParam^{\leq \nLayers}
$
may exhibit Gaussian behaviour as well in wide BNNs, since---outside of $\dens{p}(\param^{\leq \nLayers})$---its posterior depends on $\priorParam^{\leq \nLayers} = \param^{\leq \nLayers}$ only through the
empirical NNGP kernel
which is known to converge a constant $K \in \R{n \times n}$ as the layer width grows \citep[see][for an overview]{yang2020tensor}.
This is crucial in establishing our first major result: convergence of the $\mathrm{KL}$ divergence between the $\mathcal{N}(0, I_d)$ prior and the reparametrised posterior to zero as the number of hidden units in each layer goes to infinity.\footnote{Please refer to \Cref{app:proofs} for all proofs.}
\begin{restatable}{theorem}{zeroKL}
\label{stm:zero_kl}
Let $\distr{P}_{\priorParam \given X, y$ be the reparametrised posterior distribution defined by the density
$\dens{p} (\phi \given \mathcal{D})
$.
Assume the Gaussian prior and likelihood with $\sigma > 0$, and that $\smash{\Hat{\nngp} \overset{\Prob}{\to} K}$
and $\E [ \Hat{\nngp} ] \to K$
under the prior as $\dimParam_{\min} \to \infty$.
Then $\mathrm{KL}\pp{\mathcal{N}(0, I_{d}) \, \| \, \distr{P}_{\priorParam \given X, y} \to 0$ as
$\dimParam_{\min} \to \infty$.
\end{restatable}
The assumptions ($\smash{\Hat{\nngp} \overset{\Prob}{\to} K}$, $\E [ \Hat{\nngp} ] \to K$, as $\dimParam_{\min} \to \infty$) hold for most common architectures,
including FCNs, CNNs, and attention networks \citep{matthews2018gaussian,alonso2018deep,hron2020infinite,yang2020tensor}.
The fact that KL divergence does not increase under measurable transformations \citep[e.g.,][corollary~5.2.2]{gray2011entropy}, combined with \Cref{stm:zero_kl},
implies that the KL divergence between the distribution of $T(\phi)$ with $\phi \sim \mathcal{N}(0, I_d)$ and the posterior $\distr{P}_{\param \given X, y}$
also converges to zero.
This provides a simple way of approximating wide BNN posteriors as illustrated by \Cref{fig:reparam_convergence}.
We use the same argument to establish convergence in \emph{function space} by showing measurability of the $\theta \mapsto f_{\theta}$ mapping
w.r.t.\ a suitable $\sigma$-algebra.
\begin{restatable}{proposition}{klFunctions}
\label{stm:kl_functions}
Let $\phi \sim \mathcal{N}( 0 , I_{d})$, $\theta \sim \distr{P}_{\param \given X, y}$, and denote the functions they parametrise by $f_{T(\phi)}$ and $f_\theta$.
Assume all nonlinearities are continuous, and each layer's outputs are jointly continuous in the layer parameters and inputs.
Then $\mathrm{KL} ( \distr{P}_{f_{T(\phi)}} \, \| \, \distr{P}_{f_{\theta} \given \mathcal{D}} ) \to 0$ as
$\dimParam_{\min} \to \infty$, where the $\mathrm{KL}$ is defined w.r.t.\ the usual product $\sigma$-algebra on $\R{\mathcal{X}}$.
\end{restatable}
\citet{hron2020exact} showed that for continuous bounded likelihoods, $\distr{P}_{f_\theta \given \mathcal{D}}$ converges (weakly) to the NNGP posterior whenever $\distr{P}_{f_\theta}$ with $\theta \sim \distr{P}_{\theta}$ converges to the NNGP prior.
This implies that the $\distr{P}_{f_{T(\phi)}}$ from \Cref{stm:kl_functions} converges (weakly) to the NNGP posterior whenever $\distr{P}_{f_{\theta} \given \mathcal{D}}$ does:
by Pinsker's inequality, we have convergence in total variation which implies convergence of expectations of all bounded measurable (incl.\ bounded continuous) functions.
Our \Cref{stm:zero_kl} may moreover be seen as an appealing answer to the issue of finding a useful notion of `convergence' in parameter space of an \emph{increasing} dimension from \citep[section~4]{hron2020exact}.
As the authors themselves point out, their approach of embedding the weights $\theta$ in $\R{\mathbb{N}}$, and studying weak convergence w.r.t.\ its usual product $\sigma$-algebra, `tells us little about behaviour of \emph{finite} BNNs'.
This is most clearly visible in their proposition~2, which establishes asymptotic reversion of the parameter space posterior to the \emph{prior} (without any reparametrisation), which we know does not induce the correct function space limit.
We note that our \Cref{stm:zero_kl} does not contradict the \citeauthor{hron2020exact}'s result
exactly because their
definition of convergence essentially only captures the \emph{marginal} behaviour of finite weight subsets, whereas our $\mathrm{KL}$ divergence approach characterises the \emph{joint} behaviour of all the weights.
Indeed, our next proposition shows that reversion to the prior does not occur under our stronger notion of convergence.
\begin{restatable}{proposition}{klPrior}
\label{stm:kl_prior}
Under the assumptions of \Cref{stm:zero_kl}
\begin{align*}
&\mathrm{KL} \pp{\, \mathcal{N}(0, I_{d}) \, \| \, \distr{P}_{\param \given X, y} \,}
\\
&\to \tfrac{1}{2} [
\sigma^{-2} \mathrm{Tr}(K)
+ \| y \|_{\sigma^{-2} I_n - K_{\sigma^2}^{-1}}^2
- \log \det (\sigma^{-2}K_{\sigma^2})
]
\nonumber
\, ,
\end{align*}
as $\dimParam_{\min} \to \infty$.
The limit is equal to the KL divergence between the NNGP prior and posterior in function space,
defined w.r.t.\ the usual product $\sigma$-algebra on $\R{\mathcal{X}}$.
\end{restatable}
\begin{figure*}[tbp]
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[keepaspectratio,width=0.48\textwidth]{FIGURES/contours/fcn_width_contours_equal.pdf}};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\node at (0.55, 1.025) {{\footnotesize \textbf{width}}};
\end{scope}
\end{tikzpicture}
\hfill
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[keepaspectratio,width=0.48\textwidth]{FIGURES/contours/fcn_train_contours_equal.pdf}};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\node at (0.55, 1.025) {{\footnotesize \textbf{dataset size}}};
\end{scope}
\end{tikzpicture}
\caption{
\textbf{The posterior energy landscape is smoother after repriorisation.}
Plots show 2D slices of the log posterior on \texttt{cifar-10}
in terms of the original parameters $\theta$ (top row) and the reparametrised $\phi = T^{-1}\pp{\theta}$ (bottom row),
for a 1-hidden layer FCN.
The 2D slices are obtained by spherical linear interpolation between three parameter values found by gradient descent training on CIFAR-10 after random initialisation; norms may differ, and are therefore interpolated linearly.
\textbf{Left:} Columns represent varying hidden layer width. The number of observations is fixed at $n = 128$.
The vanishing structure of the reparametrised posterior as width increases, making mixing between the normally separated modes easier.
\textbf{Right:} Columns represent varying dataset size, width is fixed at $d^1 = 128$. As the number of datapoints becomes large compared to $d^1$, the reparametrised posterior gradually grows less smooth.
}
\label{fig:posterior_contour}
\end{figure*}
\section{Faster mixing via repriorisation}
\label{sect:sampling}
We now have a reparametrisation which \emph{provably} makes the reparametrised posterior close to the standard normal prior $\mathcal{N}(0, I_d)$.
Since sampling from the BNN posterior is notoriously difficult, the closeness to $\mathcal{N}(0, I_d)$ suggests sampling from $\distr{P}_{\priorParam \given X, y$ may be significantly easier.
However closeness in KL divergence does not guarantee that the gradients of $\dens{p}(\phi \given \mathcal{D})$ are as well behaved as those of the standard normal distribution,
which is crucial for gradient-guided methods like Langevin Monte Carlo (LMC).
In \Cref{sect:gradients}, we provide partial reassurance by proving that the $\ell^2$-norm of the differences between the two gradients vanishes in probability as the layers grow wide.
\Cref{sect:cholesky} then provides a simple-to-implement way of computing both the reparametrised $\theta = T(\phi)$ and the corresponding density $\dens{p}(\phi \given \mathcal{D})$ at a fraction of cost of the naive implementation.
\subsection{Convergence of density gradients}
\label{sect:gradients}
\looseness=-1
Our second major result proves that the \emph{posterior} mass of the region where $\nabla_{\phi} \log \dens{p}(\phi \given \mathcal{D})$ significantly differs from the gradient of the $\mathcal{N}(0, I)$ density
vanishes in wide BNNs.
\begin{restatable}{proposition}{gradientConvergence}
\label{stm:gradient_convergence}
Let $\Delta_\phi \coloneqq - \phi - \nabla_{\phi} \log \dens{p}(\phi \given \mathcal{D})$ where $- \phi$ is the gradient of the log density of $\mathcal{N}(0, I)$.
Assume the conditions in \Cref{stm:zero_kl}, and continuous nonlinearities.
Then $\distr{P}_{\priorParam \given X, y(\| \Delta_{\phi} \|_2 > \epsilon) \to 0$ as $\dimParam_{\min} \to \infty$, $\forall \epsilon > 0$.
\end{restatable}
Two more technical assumptions are in \Cref{app:grad_convergence}.
Albeit useful,
\Cref{stm:gradient_convergence} does not ensure
gradient-guided samplers like LMC
will not dawdle in
regions with ill-behaved gradients.
This behaviour did not occur in our experiments,
but the caveat is worth remembering.
\looseness=-1
To provide a rough quantitative intuition for the scale of the speed up, we show our reparametrisation enables $\nicefrac{\sqrt{n}}{\sigma}$-times higher LMC stepsize relative to no reparametrisation in \emph{linear} models, without any drop in the integrator accuracy (see \Cref{app:lmc_speedup_lin}).
\subsection{Practical implementation}\label{sect:cholesky}
To apply LMC, we need an efficient way of computing both $\theta = T(\phi)$ and the corresponding gradient of $\log \dens{p}(\phi \given \mathcal{D})$.
Since our largest experiments require millions of steps, the naive approach of computing each of $\mu$, $\Sigma^{1/2}$, and $\det (\Sigma^{1/2})$ separately---see \Cref{eq:determinant,eq:full_reparam}---
for the cost of \emph{three} $\mathcal{O}[(\dimParam^{\nLayers})^3]$ operations is worth improving upon.
We propose a three-times more efficient alternative.
The trick is to first compute the Cholesky decomposition
$U^\top U = \sigma^2 I_{\dimParam^{\nLayers}} + \Psi^\top \Psi$, and then reuse $U$ in computation of all three terms.
This can be done by observing $\Sigma = \sigma^2 U^{-1} (U^{-1})^\top$, which implies $\Var[\sigma U^{-1} \varepsilon ] = \Sigma$ for $\varepsilon \sim \mathcal{N} (0, I)$.
We can thus use the \emph{alternative} reparametrisation
\begin{align*}
\param^{\nLayers + 1}
=
U^{-1}
[
(U^\top)^{-1}
\Psi^\top
y
+
\sigma
\priorParam^{\nLayers + 1}
]
\, ,
\end{align*}
which uses the fact $\mu = (\sigma^2 I + \Psi^\top \Psi)^{-1} \Psi^\top y$ together with the above definition of $U$.
The determinant can then be computed essentially for free since
\begin{align*}
\det (\partial_\phi \theta)
\overset{\text{(i)}}{=}
\det (\sigma U^{-1})
\overset{\text{(ii)}}{=}
\frac{\sigma^{\dimParam^{\nLayers}}}{\det(U)}
\overset{\text{(iii)}}{=}
\frac{\sigma^{\dimParam^{\nLayers}}}{\prod_i U_{ii}}
\, ,
\end{align*}
where (i) is as in \Cref{eq:determinant}, (ii) uses standard determinant identities, and (iii) exploits that $U$ is triangular.
Because $U$ is upper-triangular, $U^{-1} v$ (resp.\ $(U^\top)^{-1} v$) can be efficiently computed by back (resp.\ forward) substitution for any $v$ \citep{voevodov}.
The sequential application of forward and backward substitution is known as the Cholesky solver algorithm \citep{cholesky}.
Since we would need to apply the Cholesky solver to compute $\mu = (\sigma^2 I + \Psi^\top \Psi)^{-1} \Psi^\top y$ anyway,
we obtain
both the
reparametrisation
the density
at essentially the same cost.
\Cref{fig:speed} demonstrates our approach introduces little wall-time clock overhead on modern accelerators relative to no reparametrisation when combined with LMC.
Since the per-step computational overhead is never larger than 1.5x in \Cref{fig:speed} and we observe $\geq$10x faster mixing in \Cref{sect:experiments}, the cost is more than offset by the sampling speed up.
\looseness=-1
The above (`feature-space') reparametrisation
scales as $\mathcal{O}[(\dimParam^{\nLayers})^3]$ in computation and $\mathcal{O}[(\dimParam^{\nLayers})^2]$ in memory.
This is typically much better than exact (NN)GP inference which scales as $\mathcal{O}(n^3)$ in computation and $\mathcal{O}(n^2)$ in memory.
To enable future study of very wide BNNs ($\dimParam^{\nLayers} \gg n$), we provide an alternative `data-space' formulation with $\mathcal{O}(n^3)$ computation and $\mathcal{O}(n^2)$ memory scaling in \Cref{app:data_space_reparametrisation}.
\subsubsection*{Interpolating between parametrisations}
We can swap the likelihood variance $\sigma^2$ for a general regulariser $\lambda > 0$ in the definition of $U^\top U \coloneqq \lambda I_{\dimParam^{\nLayers}} + \Psi^\top \Psi$.
Clearly, $T \to \mathrm{Id}$ (identity mapping) as $\lambda \to \infty$.
This allows interpolation between our proposed reparametrisation at $\lambda = \sigma^2$, and the standard parametrisation at $\lambda = \infty$, which may be useful when far from the NNGP regime.
Proper tuning of the hyperparameter $\lambda$ thus ensures the algorithm never performs worse than under the standard parametrisation.
\section{Experiments}\label{sect:experiments}
Armed with theoretical understanding and an efficient implementation, we now
demonstrate that our reparametrisation can dramatically improve BNN posterior sampling, as quantified by per-step effective sample size \citep[ESS;][]{ripley1989}, and the $\hat{R}$ statistics \citep{gelman1992inference}, over a range of architectural choices and dataset sizes.
\subsection{Setup}\label{sect:experimental_setup}
\subsubsection{\texorpdfstring{$\hat{R}$}{R-hat} diagnostic}
\label{sect:rhat}
$\hat{R}$ is a standard tool for testing non-convergence of MCMC samplers.
It takes $M$ collections of samples $\{ z_{mi} \}_{i=1}^S \subset \R{}$ produced by \emph{independent} chains, and computes
\begin{align}\label{eq:rhat_definition}
\hat{R}^2
=
\frac{
\hat{\E} \bigl[ \hat{\Var}(z_{mi} \given m) \bigr]
+
\hat{\Var} \bigl[ \hat{\E}(z_{mi} \given m) \bigr]
}{
\hat{\E} \bigl[ \hat{\Var}(z_{mi} \given m) \bigr]
}
\, ,
\end{align}
where $\hat{\E}$ and $\hat{\Var}$ compute expectation and variance w.r.t.\ the empirical distribution which assigns a $\tfrac{1}{MS}$ weight to each $z_{mi}$ (conditioning on $m$ yields within-chain statistics).
The numerator is just $\hat{\Var}(z_{mi})$ by the law of total variance, where $\hat{\Var} \bigl[ \hat{\E}(z_{mi} \given m) \bigr]$ vanishes when all chains sample from the same distribution.
Values significantly larger than one thus indicate non-convergence.
We follow \citet{izmailov2021bayesian} in reporting $\hat{R}^2$ (denoted by $\hat{R}$ in their paper), instead of the square root of the above quantity.
For multi-dimensional random variables, we estimate $\hat{R}^2$ for each dimension, and report the resulting distribution over dimensions.
\begin{figure}[tbp]
\centering
\includegraphics[keepaspectratio,width=\linewidth]{FIGURES/speed_n_units_duration_n_train=1_horizontal.pdf}
\caption{\looseness=-1
\textbf{Sampling with repriorisation is comparable to the baseline in wall-clock time.} While our reparametrisation cost is cubic in top layer width $\dimParam^{\nLayers}$ (\cref{sect:cholesky}),
it is notable only for very large FCNs (\textbf{left}), and even larger CNNs (\textbf{right}) where the quadratic cost of the forward pass dominates due to a large multiplicative constant.
See \Cref{app:data_space_reparametrisation} for an alternative formulation which scales only quadratically with $d^L$ but cubically with $n$.
}
\label{fig:speed}\vspace{-1em}
\end{figure}
\subsubsection{Effective sample size}\label{sect:ess}
ESS is a measure of sampling speed.
It estimates lag $k$ autocorrelations $\hat{\rho}_k \coloneqq \hat{\Covar} (z_{mi}, z_{m (i+k)}) / \, \hat{\Var}(z_{mi})$ for $k = 1, \ldots, S - 1$ ($\hat{\Covar}$ is the empirical covariance), and computes
\begin{align}\label{eq:ess_definition}
\widehat{\text{ESS}}
\coloneqq
\frac{S}{
1 +
2
\sum_{k=1}^{S - 1}
(1 - \tfrac{k}{S}) \hat{\rho}_k
}
\, ,
\end{align}
The per-step ESS is then $\nicefrac{\widehat{\text{ESS}}}{S}$.
Since $\hat{\rho}_k$ estimates for high $k$ are based on only $M (S - k)$ samples, we denoise by stopping the sum in \Cref{eq:ess_definition} at the first $\hat{\rho}_k < 0$, which is the default in Tensorflow Probability's \href{https://www.tensorflow.org/probability/api_docs/python/tfp/substrates/jax/mcmc/effective_sample_size}{\texttt{effective\_sample\_size}} \citep{dillon2017tensorflow}.
\Cref{eq:ess_definition} is based on the $\Var(\bar{z}_S) = \nicefrac{\Var(z_1)}{S}$ identity for (population) variance of the empirical mean $\bar{z}_S$ of $S$ \emph{i.i.d.}\ variables $z_i$
Since MCMC chains produce dependent samples, $\smash{\widehat{\text{ESS}}}$ estimates the
number which would satisfy the above equality when substituted for $S$ under the assumption of stationarity.
Because we need to estimate ESS in the very high-dimensional parameter space,
we randomly choose \emph{100 one-dimensional subspaces}, project the samples, estimate ESS in each of them,
and report their distribution.
\begin{figure*}[htbp]
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\includegraphics[keepaspectratio,width=0.45\textwidth]{FIGURES/new_ess/fcn_width_ess_threshold-loglog.pdf}
};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\node at (0.55, 1.075) {{\footnotesize \textbf{width}}};
\end{scope}
\end{tikzpicture}
\hfill
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\includegraphics[keepaspectratio,width=0.45\textwidth]{FIGURES/new_ess/fcn_train_ess_threshold-loglog.pdf}
};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\node at (0.55, 1.075) {{\footnotesize \textbf{dataset size}}};
\end{scope}
\end{tikzpicture}
\vspace*{-0.75em}
\caption{\textbf{Repriorisation makes mixing speed faster, and the performance benefits increase as the width to dataset size ratio increases.} Per-step ESS as a function of layer width (left) and dataset size (right)
for a 3-hidden layer FCN.
The line indicates the mean, and the bands the \emph{minimum} and \emph{maximum}, ESS over the $100$ projection subspaces (see \Cref{sect:ess}).
In all cases, reparametrisation achieves significantly better mixing speed (note the log-log scale of the axes).
\texttt{3M} samples collected for each configuration.
Hyperparameters were tuned for each configuration separately (see \Cref{sect:width_dataset_experiments}).
\textbf{Left:} Dataset size fixed at $n = 256$.
The benefit of reparametrisation increases with width, especially so when width becomes larger than dataset size where reparametrisation yields 50x higher ESS.
\textbf{Right:} Width fixed at $512$.
Note that reparameterisation leads to a higher ESS for all configurations.}
\label{fig:ess_width_dataset}
\end{figure*}
\subsubsection{Data and hyperparameters}\label{sect:data_hypers}
\looseness=-1
The underdamped LMC sampler \citep{rossky1978} and \texttt{cifar-10} dataset \citep{krizhevsky2009learning} are used in all experiments.
We use regression on one-hot encoding of the labels shifted by $-\tfrac{1}{10}$ so that the label mean of each example is zero, matching the BNN prior (extension to multi-dimensional outputs described in \Cref{app:multi_dim}).
Gaussian likelihood with $\sigma = 0.1$ is used throughout, which we selected to ensure that (a)~positive and negative labels are separated by at least $4\sigma$, but (b)~$\sigma$ is not too small to give ourselves an unfair advantage based on the linear case intuition (\Cref{app:lmc_speedup_lin}), or cause numerical issues.
Sample thinning is applied in all experiments,
and ESS and $\hat{R}$ are always computed based on the thinned sample.
However, \Cref{fig:ess_three_subsets,fig:rhat_three_subsets} visualise ESS evolution as a function of the number of LMC steps which is equal the \emph{unthinned} sample size.
\looseness=-1
As common, we omit the Metropolis-Hastings correction of LMC \citep{Neal93probabilisticinference}, and instead tune hyperparameters so that average acceptance probability after burn-in stays above 98\%.
We observed significant impact of numerical errors on both the proposal and the calculation of the acceptance probability;
we use \texttt{float32} which offers significant speed advantages over \texttt{float64}, but can still suffer 2-5 percentage point changes in acceptance probability compared to \texttt{float64} computation with the same candidate sample.
Our experiments are implemented in JAX \citep{jax2018github}, and rely on the Neural Tangents library \citep{neuraltangents2020} for both implementation of the NN model logic, and evaluation of the NNGP predictions (e.g., in \Cref{fig:reparam_convergence}).\footnote{Code: \href{https://github.com/google/wide_bnn_sampling}{github.com/google/wide\_bnn\_sampling}.}
\begin{figure*}[htb]
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\includegraphics[keepaspectratio,width=0.48\textwidth]{FIGURES/three_subsets/fcn_c3_ess_evolution.pdf}
};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\node at (0.55, 1.075) {{\footnotesize \textbf{3-hidden layer FCN}}};
\end{scope}
\end{tikzpicture}
\hfill
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {
\includegraphics[keepaspectratio,width=0.48\textwidth]{FIGURES/three_subsets/resnet_c3_ess_evolution_fixed_titles.pdf}
};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\node at (0.55, 1.075) {{\footnotesize \textbf{Wide ResNet-20}}};
\end{scope}
\end{tikzpicture}
\caption{\textbf{Repriorisation results in faster mixing across architectures.}
Per-step ESS estimates are plotted as a function of the number of sampler steps for three random subsets of \texttt{cifar-10} of size $n = 1024, 10240, 40960$.
The line indicates the mean, and the bands the \emph{minimum} and \emph{maximum}, ESS over the $100$ projection subspaces (see \Cref{sect:ess}).
The initial downward trend of the curves is due to underestimation of long range autocorrelations at earlier steps where sample size is too small. \textbf{Left:} Results for a FCN with three width 1024 hidden layers.
The non-reparametrised BNNs mix around 10x slower for all three dataset sizes,
and up to 200x slower if the worst mixing projections (bottom of shaded regions) are compared to each other for each case.
\textbf{Right:} Results for a normaliser-free Wide ResNet-20 with 128 and 512 units (channels) in the narrowest and widest layers (see \Cref{sect:architecture_experiments}).
For $n =$ \texttt{1K}, reparameterisation performs more than 10x better.
For $n =$ \texttt{10K}, both chains have very similar ESS / step values.
This could be because the dataset size is now two orders of magnitude larger than the width, a regime in which our theory does not apply.
It could also be because, even after 3 million sampling steps, the chains are so far from equilibrium and the ESS estimates are dominated by the initial transient effects.
}
\label{fig:ess_three_subsets}
\end{figure*}
\subsection{Results}
In Figure \ref{fig:posterior_contour} we visualize 2D slices through the weight posterior, for the original and reparameterized network for a 1-hidden layer FCN. The log posterior is far smoother after reparametrization, and its relative smoothness improves as network width is increased relative to dataset size. As we quantify in the following sections, this enables dramatic improvements in sampling efficiency.
\subsubsection{Dependence on width and dataset size}
\label{sect:width_dataset_experiments}
In \Cref{fig:ess_width_dataset}, we explore the dependence of mixing speed of the LMC sampler on network width and dataset size as measured by the per-step ESS.
We use a single chain run of \texttt{3M} steps with a \texttt{20K} burn-in per each width/training set size.
The \emph{results are computed on a sample thinned by a factor of 25}.
We use a FCN with three equal width hidden layers and GELU nonlinearities \citep{hendrycks2016gaussian}, and the weight scaling described in \Cref{sect:notation_assumptions}.
The weight and bias scaling factors are set to $\sigma_\weight^2 = 2$, $\sigma_\bias^2 = 0.01$ everywhere except in the readout layer where $\sigma_\weight^2 = 1$ to achieve approximately unit average variance of the network predictions under the prior (at initialisation).
For the width experiments (\Cref{fig:ess_width_dataset}, left), the dataset size was fixed at $n = 256$, and the layer width varied over $\{ 32, 64, 128, 384, 512, 768, 1024 \}$.
For each width and both parametrisations, we \emph{separately} tuned the LMC stepsize and damping factor to maximise the ESS of $\theta$ while satisfying the $>98\%$ mean acceptance probability criterion.
For the reparametrised sampler, we also tried to tune the regularisation parameter $\lambda$ (see \Cref{sect:cholesky}), but found that the value dictated by our wide BNN theory ($\lambda = \sigma^2$) worked best even for the smallest width models.
The distribution of ESS over the random projections of $\theta$ and $f(X^\text{test})$ on \texttt{5K} test points is shown, with its mean, minimum, and maximum values.
Our reparametrisation achieves significantly faster mixing in all conditions (note the log-log scale).
The improvement becomes larger as the BNN grows wider (left), achieving near-perfect values for the widest BNNs ($10^0=1$ is maximum) where the width ($>\! 2^{8}$) is larger than the dataset ($256 = 2^8$), with a 50x speed up at width 1024.
For the dataset size plots (right), the performance is again best when the ratio of width to the number of observations is the smallest (here on the left).
The ESS of both $f(X^\text{test})$ and $\theta$ decays with dataset size as expected, since the posterior becomes more complicated, and the width to dataset size ratio dips below one, i.e., we are leaving the NNGP regime.
The reparametrisation however maintains higher ESS even outside of the conditions where our theory holds.
\subsubsection{Dependence on architecture}
\label{sect:architecture_experiments}
While our theory does not cover the cases where the ratio of dataset size to width is large, the consistent advantage of reparametrisation even in this regime is intriguing (\Cref{fig:ess_width_dataset}, right).
In this section, we investigate the dependence of this phenomenon on the architecture and dataset size.
For the architecture, we compare the FCN from \Cref{sect:width_dataset_experiments} with a normaliser-free Wide ResNet-20 with 128, 256, and 512 channels arranged from bottom to the top layer in the usual three groups of residual layer blocks \citep{he2016resnet,zagoruyko2016wideresnet}.
GELU is used for both.
As in \citep{izmailov2021bayesian}, we remove batch normalisation since it makes predictions depend on the mini-batch, which interferes with Bayesian interpretation.
We however use a different alternative in its place, namely the proposal of \citet{shao2020normalisation}, which replaces each residual sum $f_k^\text{skip}(x) + f_k^\text{resid}(x)$ by $\sqrt{\nicefrac{k - 1 + c}{k + c}} \, f_k^\text{skip}(x) + \sqrt{\nicefrac{1}{k + c}} \, f_k^\text{resid}(x)$ where $k$ is the index of the skip connection, and $c$ is a hyperparameter we set to $1$.
Since \citep{izmailov2021bayesian} is the closest comparison for this section, we note the authors use a categorical instead of a Gaussian likelihood\footnote{Our reparameterization also enables accelerated sampling for categorical likelihood -- see Appendix \ref{app:non-gauss_likelihood}.}, and an i.i.d.\ $\mathcal{N}(0, \alpha)$ prior for all parameters, but \emph{without} scaling the weights by $\nicefrac{\sigma_\weight^\ell}{\sqrt{d^\ell}}$ as we do (see \Cref{sect:notation_assumptions}).
For the scaling factors, FCN is as in \Cref{sect:width_dataset_experiments}, whereas the ResNet uses $\sigma_\weight^2 = 2.2$ and $\sigma_\bias^2 = 0$, except in the last layer where $\sigma_\weight^2 = 1$ and $\sigma_\bias^2 = 0.01$ is again employed to ensure reasonable scale of the network predictions under the prior.
As a sanity check, using this prior as initialiser, and optimising with Adam, this normaliser-free ResNet achieves a decent $95.6\%$ validation accuracy on full \texttt{cifar-10}.
We tuned the LMC stepsize and damping factor separately for every configuration to maximise ESS of $\theta$ while satisfying $> 98$\% mean acceptance rate after burn-in.
For the reparametrised chain, the regulariser $\lambda$ was $\sigma^2$ and $n \sigma^2$ respectively for the FCN and the ResNet, with $\sigma^2 = 0.01$ the output variance.
The \emph{sample thinning factor is 100}.
\begin{figure}[tbp]
\centering
\includegraphics[keepaspectratio,width=0.925\linewidth]{FIGURES/three_subsets/fcn_c3_rhat_evolution-meanlb.pdf}
\caption{\textbf{Convergence of $\hat{R}$ with varying dataset size.} Complements the left side of \Cref{fig:ess_three_subsets}.
Line represents mean, band the maximum value.
Estimates based on three independent chains. Note the log-log scale, and
that $\hat{R}^2$ is plotted relative to
its unit lower bound.
High $\hat{R}^2$ indicate non-convergence so lower is better.}
\label{fig:rhat_three_subsets}\vspace{-1em}
\end{figure}
As can be seen in \Cref{fig:ess_three_subsets}, the mixing speed advantage of reparametrisation persists far from the NNGP regime in the case of the FCN (left), where even a factor of forty in dataset to width ratio did not erase the surprisingly consistent 10x higher (average) ESS.
This observation is supported by \Cref{fig:rhat_three_subsets}, in which $\hat{R}^2$ is already near one (optimum) when the standard parametrisation still exhibits $\mathcal{O}(10^2)$ values.
For the Wide ResNet (right), we observed a similar $\geq$10x benefit from reparameterisation at \texttt{1K} datapoints.
For the $n=$\texttt{10K} case, no benefit was observed.
This may be because both chains are far from equilibrium even after \texttt{3M} steps, and the per-step ESS measurement reflects initial transients.
It may also be because the dataset size (10,240) was much larger than smallest channel count (128), exceeding by two orders of magnitude the scale where the model is expected to approximate the NNGP limit.
\section{Other related work}
For ensembles of infinitely-wide NNs trained with gradient descent, \citet{shwartz2020information} find the KL divergence between the posterior and the prior to diverge linearly with training time, an interesting departure from the finite value we find for Bayesian inference in \Cref{stm:kl_prior}. The information content of infinitely-wide NNs was analysed by \citet{bernstein2021computing} using the NNGP posterior, leading to non-vacuous PAC-Bayes generalisation bounds, adding to an active line of work focusing on generalisation bounds for overparametrised models \citep[see, e.g.,][]{dziugaite2017computing,valle2018deep,vakili2021uniform}.
If some hidden layers are narrow and remain finite, the resulting model is a type of Deep Gaussian process~\cite{damianou2013deep,agrawal2020wide} or Deep kernel process~\cite{aitchison2021deep}. \citet{aitchison2020bigger} argues that such models may exhibit improved performance relative to uniformly wide BNNs owing to a form of representation learning. While we do not investigate such questions in this work, the favourable computational properties of our LMC algorithm could facilitate such analyses in the future.
Moving away from the infinite width limit, \citet{yaida2020non} and \citet{roberts2021principles} show how to compute the Bayesian prior and posterior systematically in powers of $1/\text{width}$, with the leading-order term given by the NNGP, though the subleading corrections pose computational challenges. For some architectures, a Gaussian prior in parameter space leads to closed-form expressions for the prior in function space, even at finite width \cite{zavatone2021exact}. Less is known about the posterior for finite BNNs, since computational considerations often require approximate inference, whose quality in real-world problems has
recently been investigated in \citep{izmailov2021bayesian}.
Our work is partially inspired by a thread of papers which accelerate MCMC sampling by applying an invertible transformation to a distribution's variables, and then running sampling chains in the transformed space
\citep[e.g.,][]{el2012bayesian,marzouk2014transport,parno2015transport,marzouk2016introduction,titsias2017learning,hoffman2019neutra}.
\section{Conclusion}
We introduced \emph{repriorisation}, a reparameterisation that (1)~provides a rigorous characterisation of wide BNN parameter space, and (2)~enables a more efficient BNN sampling algorithm which mixes \emph{faster} as the network is made wider.
Our theory shows
BNN posteriors exhibit non-negligible interactions between parameters in different layers,
even at large width.
In contrast, many popular BNN posterior approximations (e.g., mean-field methods) assume independence between parameters in different layers.
Beyond MCMC,
algorithms which incorporate between-layer interactions---from Laplace \citep{tierney1986accurate} to more recent ones like \citep{ober2021global}---are therefore better positioned to capture the true BNN posterior behaviour.
In fairness, approximation fidelity and downstream performance are not the same thing though, as clearly demonstrated by the success of non-Bayesian approaches.
\looseness=-1
Our sampling results parallel the interplay between optimisation and model size in deterministic NNs.
In particular, first-order methods used to be considered ill-suited for the non-convex loss landscapes of NNs, yet experimental and later theoretical results showed they can be highly effective, especially for large NNs \citep[e.g.,][]{allen2019convergence,du2019gradient}.
Similarly, MCMC theory tells us that sampling is often harder in high dimensions, yet our results show that a simple reparametrisation exploiting the particular structure of wide BNN posteriors makes sampling much easier.
While the gap between deterministic and Bayesian NNs remains considerable, we hope our work enables further progress for practical large-scale BNNs.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,758 |
var assert = require('assert');
var bzVoter = require('../../biz/voterlogic.js');
var dbmodel = require('../../model');
var mongoose = require('mongoose');
var mocha = require('mocha');
var bizVoter = new bzVoter();
var VOTED_VOTERID = mongoose.Types.ObjectId();
var VOTED_EXPIRED_VOTERID = mongoose.Types.ObjectId();
var DID_NOT_VOTE_VOTERID = mongoose.Types.ObjectId();
var currentDate = new Date();
var currentDateStr = currentDate.toString();
describe('voterlogic', function() {
describe('voterCanVote', function() {
beforeEach(function(done) {
var electionId;
var e = new mongoose.models.election( {
"name": "testElection",
winThreshhold: 100,
voteSustainDuration: 5,
voterDormancyDuration: 8
});
e.save(function (err, result) {
electionId = result._id;
var c = new mongoose.models.candidate( {
"name": "Dee Ellis",
"candidateElections": [
{
electionId: electionId,
value: 0,
"votes": [
{
"startTime": currentDate.toString(),
"value": 8,
"voterId": VOTED_VOTERID,
"endTime": new Date(currentDate.toString()).setMinutes(new Date(currentDate.toString()).getMinutes() + 8).toString(),
"endDormancyTime": new Date(currentDate.toString()).setMinutes(new Date(currentDate.toString()).getMinutes() + 8).toString(),
"expired": false,
"voterIsDormant": true
},
{
"startTime": new Date(currentDate.toString()).setMinutes(new Date(currentDate.toString()).getMinutes() - 30).toString(),
"value": 8,
"voterId": VOTED_EXPIRED_VOTERID,
"endTime": new Date(currentDate.toString()).setMinutes(new Date(currentDate.toString()).getMinutes() - 22).toString(),
"endDormancyTime": new Date(currentDate.toString()).setMinutes(new Date(currentDate.toString()).getMinutes() -22).toString(),
"expired": true,
"voterIsDormant": false
}
]
}
]
});
c.save(done);
});
});
afterEach(function(done) {
mongoose.models.candidate.remove ( { "name" : "Dee Ellis" } , function (error, result) {
mongoose.models.election.remove ({"name": "testElection"}, function(error, result) {
done();
})
});
});
it('cannot vote if a vote is not expired', function(done) {
bizVoter.voterCanVote(VOTED_VOTERID, function(error, result) {
if (error == undefined || error == null) {
try {
assert.equal(result, false);
} catch(err) {
return done(err);
}
return done();
}
});
});
it('can vote if there is no vote', function(done) {
bizVoter.voterCanVote(DID_NOT_VOTE_VOTERID, function(error, result) {
if (error == undefined || error == null) {
try {
assert.equal(result, true);
} catch(err) {
return done(err);
}
return done();
}
});
});
it('can vote if a vote is expired', function(done) {
bizVoter.voterCanVote(VOTED_EXPIRED_VOTERID, function(error, result) {
if (error == undefined || error == null) {
try {
assert.equal(result, true);
} catch(err) {
return done(err);
}
return done();
}
});
});
});
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,578 |
This Mitten Ornament is our sixth pattern in the 12 Days of Christmas Ornament collection! This is such a cute little crocheted gift to adorn your tree or use it instead of a bow on a package for anyone on your gift giving list!
Rnd 10: SC in each st around, fasten off and change color to white.
Rnd 13: SC in each st around, sl st to first st. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,262 |
Bergamot, Red – Monarda didyma also known as Bee Balm. Scarlet flowers spectacular when grown en masse. Scented leaves are used to make Oswego Tea. Add a leaf to your everyday tea to get that 'Earl Grey' flavour! | {
"redpajama_set_name": "RedPajamaC4"
} | 6,727 |
Top 10 Highest-Paid Footballers in the World
No doubt, the highest-paid footballers in the world are people you no doubt hear of their tremendous talents and exploits on the football pitch almost every week. Football has transcended beyond just a sport over the years and has produced super-rich individuals that make up the list of the richest athletes in the world.
With incredible talents at their disposal, the ability to dazzle opponents and mesmerize opposition and defenders with superb dribbling skill, speed, incredible pace, amazing shooting power, and amazing foresight to read their opponents, it is no wonder that these footballers are the best at what they do and top European clubs will do anything to keep them from their arch-rivals across Europe.
In this post, we detail the top 10 highest-earning footballers in the world, their clubs, and what they earn weekly with their club.
These are the top 10 highest-paid footballers in the world and how much they earn per week:
10. Robert Lewandowski – £350,000-per-week (₦245 million million)
Bayern Munich lethal finisher and Polish striker is the top 10th highest paid footballer in the world, earning £350,000 per week, which is about N245 million Nigerian Naira weekly! He is considered to be one of the most lethal and most dangerous strikers in modern-day football and has remained consistent for years now.
9. David De Gea – £375,000-per-week (N263 million)
Manchester United and Spanish goalkeeper is the 9th highest-paid footballer in the world and the only goalkeeper on the list. This means De Gea is automatically the highest-paid goalkeeper in the world, earning £375,000 (about N263 million) per week. While De Gea has slipped in performance compared to what he was some years ago, he is still one of the notable goalkeepers in modern-day football.
8. Kevin De Bruyne – £385,000-per-week (₦270 million)
Belgian and Manchester City midfielder, Kevin De Bruyne is no doubt one of the world's finest midfielders in recent years and this somewhat translates to why he is on this list. Manchester City currently pays the Belgian £385,000 (about ₦270 million) every week! De Bruyne is the highest-paid Belgian player in the world raking in a staggering £385,000 weekly.
7. Kylian Mbappe – £410,000-per-week (₦287 million)
French forward, Kylian Mbappe is the 7th highest-paid footballer in the world. The French star who plays for Paris Saint-Germain, also known as PSG currently earns £410,000 every week. He is considered to be one of the most dangerous players in the world, all thanks to his outrageous speed and strength. Mbappe is the third highest-paid PSG player after Neymar and Messi. Unfortunately, this might not be for long as the French striker is seeking to move away from the club to Real Madrid—the club of his dream since childhood.
6. Gareth Bale – £500,000-per-week (₦350 million)
Despite being out of the Real Madrid squad for a very long time, no thanks to the bad blood between himself and Zidane, former Real Madrid manager, Gareth Bale who is still a Real Madrid player but currently on loan playing for Tottenham Hotspur clinches the spot as the 6th highest-paid footballer in the world. He is currently paid an unbelievable amount of £500,000 (₦350 million) as weekly wages at Tottenham. However, Real Madrid still contributes to the huge chunk! Bale is the highest-paid Welsh player in the world.
5. Luis Suarez – £575,000-per-week (₦402 million)
Uruguayan forward, Luis Suarez is the world's 5th highest-paid footballer with a weekly salary of £575,000. He currently plays for Atletico Madrid following his exit at Barcelona. Suarez was one of the players released by Barcelona manager, Ronald Koeman for being 'too old'. Unfortunately for Barcelona, Suarez led Atletico Madrid to a league win while Barcelona ended up as the third-best team in the league behind Atletico Madrid and Real Madrid respectively. Suarez is the highest-paid Uruguayan footballer in the world.
4. Antoine Griezmann – £575,000-per-week (₦402 million)
French forward, Antoine Griezmann is currently the 4th highest-paid footballer in the world and his take-home every week is £5757,000; about ₦402 million in Nigerian currency. Fans and critics have argued that Griezmann is one of Barcelona's worst signings as he has struggled to replicate his magic while playing for Atletico Madrid at Camp Nou. Despite the criticisms, Griezmann remains the highest-paid French footballer in the world, earning £575,000 weekly!
3. Neymar – £606,000-per-week (₦424 million)
Brazilian winger, Neymar Jr. currently plays for Paris Saint-Germain and earns a mouth-watering £606,000 (about ₦424 million) as weekly wages, making him the highest-paid Brazillian footballer in the world and the third highest-paid footballer in the world. Neymar was recently reunited with former teammate, Lionel Messi, who left Barcelona after over two decades.
2. Cristiano Ronaldo – £900,000-per-week (₦630 million)
Cristiano Ronaldo is the world's second-highest-paid footballer with an unbelievable weekly wage of £900,000 (about ₦630 million)! The Portugal legend currently plays for Juventus and remains one the most notable footballers in the world. Ronaldo is considered the most complete footballer in the world and he still continues to perform and remain consistent despite being in his late thirties. Ronaldo is the world's most marketable and most bankable footballer. He is also the highest-paid Portugal player in the world!
1. Lionel Messi – £960,000-per-week (₦672 million)
Lionel Messi retains the spot as the highest-paid footballer in the world despite moving to PSG from Barcelona. He currently earns a staggering £960,000 weekly despite being in his mid-thirties. The Argentine who switched sides recently moved to Paris as a free agent from Barcelona after the club could not reach an agreement with him. This means Messi is not just the highest-paid footballer in the world but the highest-paid footballer in Argentina!
Weekly Wage (GBP)
Weekly Wage (in Naira)
Lewandoski Poland £350,000 ₦245 million
De Gea Spain £375,000 ₦263 million
De Bruyne Belgium £385,000 ₦270 million
Mbappe France £410,000 ₦287 million
Gareth Bale Wales £500,000 ₦350 million
Luis Suarez Uruguay £575,000 ₦402 million
Antoine Griezmann France £575,000 ₦402 million
Neymar Jr. Brazil £606,000 ₦424 million
Cristiano Ronaldo Portugal £900,000 ₦630 million
Lionel Messi Argentina £960,000 ₦672 million
PSG Players Salaries 2021/22 (Weekly Wages)
The 10 Richest Football Clubs in The World | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,771 |
{"url":"https:\/\/pigeonsupermodel.com\/DLC_StudentGuide.html","text":"# DeepLabCut - A Student\u2019s Quick Start Guide\u00b6\n\nThis guide adresses beginner DeepLabCut users (e.g. undergraduate students with little to no experience with operating python) and serves as an introduction to using DeepLabCut for video-based pose estimation. It follows the standard DeepLabCut workflow which was introduced by Nath et al. (2019) in one of the DeepLabCut key publications.\n\nThe tutorial is meant to give an impression of what a simple DeepLabCut project might look like and to serve as a template for the creation of new projects. It covers the steps of\n\nNote\n\nFor a more extensive insight into the functions of DeepLabCut, I point to the official DeepLabCut Documentation that is maintained by the DeepLabCut developer team.\n\n## What is DeepLabCut?\u00b6\n\nDeepLabCut is a python toolbox designed for markerless pose estimation of animals and is developed by the team of the Mathis lab of adaptive motor control. It is a video-based approach, meaning that users input video files and DeepLabCut outputs the positions of all tracked body parts for every frame of the video.\n\nTo do so, DeepLabCut utilizes a supervised machine learning algorithm. In practice, this means that frames are extracted from a video and manually labeled (= the user marks the points of interest to be tracked on each frame). These labeled frames are then used to train a neural network that can eventually recognize and label the defined points of interest on its own.\n\nThe resulting network (namely a DeepLabCut model) can be used to analyze new videos with respect to tracking the same points of interest.\n\n## Installation\u00b6\n\nThe installation of DeepLabCut can be tricky when you are doing it for the first time. Since DeeplabCut is not yet one of the end-user optimized programs as one is used to, it has to be installed via the command line.\n\nThe easiest way to do so is by following the step by step instructions provided by the DeepLabCut developers. Simply start with the CONDA section and follow along beginning with Step 1.\n\nIn general, you will have to execute the following steps:\n\n1. Install Anaconda. (\ud83d\udca1 That\u2019s a distribution platform for the Python programming language.)\n\n2. Download the DeepLabCut Installation File.\n\n3. Open Anaconda Prompt. (\ud83d\udca1 That\u2019s a command line application which was installed together with Anaconda.)\n\n4. Change directory to the directory where the installation file was saved (e.g. in Anaconda Prompt type cd Downloads).\n\n5. Create a virtual environment from the installation file. (Type conda env create -f DEEPLABCUT.yaml; \ud83d\udca1 This will install DeepLabCut and all its dependencies.)\n\n6. Activate the environment. (Type conda activate DEEPLABCUT)\n\nNote, that whenever you are working with DeepLabCut, the environment in which DeepLabCut is installed must be activated.\n\nThat\u2019s it. After completing the installation, the real DeepLabCutting can begin.\n\n### Do I need a GPU?\u00b6\n\nTraining an artificial neural network (ANN) using DeepLabCut recquires a lot of computational power. Therefore, the DeepLabCut developers strongly recommend using a GPU for training the model but also for video analysis. Although it is technically possible to train and analyze on a CPU, in practice this isn\u2019t practible, since even on a GPU the analysis can take several days to weeks (depending on the number and size of videos).\n\nSo yes, you may need a GPU in order to train an ANN and to analyze videos using a trained network.\n\nIt is possible to run DeepLabCut on a cloud-based computing platform, such as Google colab (that provides free GPU access). See the DeepLabCut GitHub repository for more information on this option. However, in my experience, the amount of time you get access to a GPU using this option is often not enough to train a model sufficiently.\n\n## Jupyter Notebooks\u00b6\n\nFrom this stage on, I find it easiest to create and manage DeepLabCut projects by using a Jupyter Notebook, as this will automatically log the project\u2019s progress:\n\nIn Anaconda Prompt type Jupyter Notebook (note that your DeepLabCut environment should be activated).\n\nFrom the Notebook Dashboard (the site that opens in your browser after running the above command), create a Jupyter Notebook by clicking on \u201cNew\u201d and selecting your DeeplabCut environment (in Jupyter Notebooks, virtual environments are called \u201ckernel\u201d).\n\nNow, you can start creating your DeepLabCut project by editing your Notebook!\n\nNote\n\nWhen you have downloaded this website in the .ipynb format and moved it to the directory from where you launched Jupyter Notebook, you can now open & edit it and use it as a template for your own DeepLabCut project.\n\nAlternatively, you can write your own notebook from scratch and simply copy & paste the code snippets listed below.\n\n## Create a DeepLabCut Project\u00b6\n\n### Import DeepLabCut\u00b6\n\nTo have access to the numerous DeepLabCut functions, you must first import DeepLabCut into your python interpreter. In your Jupyter Notebook add a python code cell and runn the following function:\n\nimport deeplabcut\n\n# Note: To get some feedback for the successful import, you can add the print function:\n\nprint(f'Successfully imported DeepLabCut version: {deeplabcut. __version__}')\n\n\n### Create Videolist\u00b6\n\nCreate a variable named \u201cvideolist\u201d that holds the paths to all the videos you want to include into the project. Simply write down all the file paths for the videos you want to analyze as a list:\n\nvideolist = ['Full\/path\/of\/video\/1', 'Full\/path\/of\/video\/2', 'Full\/path\/of\/video\/3']\n\n# Note: In Windows, the file paths must be formated as r'C:\\Full\\path\\of\\video.MP4'\n\n\n### Create Project\u00b6\n\nTo create a new DeepLabCut project, you simply have to run the function deeplabcut.create_new_project after customizing all its required parameters:\n\nconfig_path = deeplabcut.create_new_project('Name of the project', 'Name of the experimenter', videolist, working_directory='Full path of the working directory', copy_videos=True\/False, multianimal=True\/False)\n\n\nIt is easiest to define a variable config path when creating the project (as shown above). This will make it easier to use other DeepLabCut functions during later stages of the project, as it will often require you to indicate your config.yaml file. However, you can also do this in an independent step after creating your project.\n\nThis is also how you can \u201creload\u201d your project after closing and reopening the jupyter notebook. Simply run:\n\nconfig_path = 'full\/path\/to\/your\/projects\/config.yaml\/file'\n\nNote\n\nFor all DeepLabCut functions, you can always run deeplabcut.name_of_function? to call the embedded help function. This will show you e.g. a list with information on all the parameters of the respective function.\n\n### Customize the config.yaml file\u00b6\n\nAfter running deeplabcut.create_new_project, DeepLabCut created a new project directory within your selected working directory. Inside this project directory you find four prepared subdirectories (dlc-models, labeled-data, training-datasets, videos) and a configuration file named config.yaml.\n\nThe next step is now to customize the configuration file according to your requirements:\n\nTo do so, you first need to open the config.yaml file in the text editor of your choice (e.g. \u201cNotepad++\u201d, \u201cVisual Studio Code\u201d).\n\nThe first parameters in the config.yaml file (Task, scorer, date, multianimalproject) have been automatically filled in, when the project was created. The same is true for the project_path parameter and the list of your included videos in video_sets.\n\nWhilst the first four parameters should not be changed throughout the project, the project_path parameter should be updated, when you move the project to another directory.\n\nAll other parameters in the configuration file are currently set to the DeepLabCut default and may need to be changed throughout the project.\n\nNow, for the most important changes you have to make to the parameters in the config.yaml file in the beginning:\n\n1. bodyparts: The whole point of DeepLabCut is to track different bodyparts on a given animal. Therefore, in this parameter you have to define the key features you want to track throughout your video files. Use the notation that is already given by the dummy entries in the config.file (e.g. - NameOfBodypart).\n\n2. numframes2pick: This parameter specifies, how many frames will be extracted from each of your video files in the following step for manual labeling. The total number of labeled frames will later make up the size of your training dataset. There is no fixed specification on how large the training dataset should be (as this largely depends on your amount of videos, video quality, number of included animals, \u2026). However, the general rule is: the bigger, the better. For a start, I would aim for a size of at least 200 frames. Therefore, only adapt this parameter if you have a whole lot of videos. Otherwise, just leave it as it is for now.\n\n3. skeleton: Here, you can specify connections between your tracked bodyparts. This will become relevant after the next step, when you plot your manually annotated frames to check whether all labels were positioned correctly. As with the bodyparts, take care to use the exact syntax that is specified in the dummy entries (e.g. tabs & spaces!).\n\nThese are the most important adjustments to the configuration file that you need to make when you start a new project. In the dropdown entry below, you find a summary of all the parameters in the configuration file, taken from the DeepLabCut documentation. Most of them will become relevant later, during model training.\n\n## Create a training dataset\u00b6\n\n### Extract frames\u00b6\n\nThe core of a DeepLabCut project is the training dataset that is used to train the resulting neural network. The training dataset is a set of manually labeled frames that were extracted from the video files. The quality of the training dataset will determine the performance (= the tracking accuracy) of the trained model.\n\nAs stated above, there is no fixed specification on how many frames should be extracted. A sufficient training dataset should include frames from all shown behaviors across all individuals, lightning conditions & settings depicted in the videos. Therefore, a relatively large data set (consisting of a large number of frames) is desirable.\n\nThe following function will extract the amount of frames per video specified in the configuration file (numframes2pick) and store them in individual subdirectories within the labeled-data folder:\n\ndeeplabcut.extract_frames(config_path, mode='automatic\/manual', algo='uniform\/kmeans', userfeedback=False, crop=True\/False)\n\n\nNote, that you will have to adapt the parameters according to your whishes:\n\n\u2022 mode: Here, you specify whether you want to manually pick the frames to be included into your training dataset (\u00b4manual\u00b4) or if you want DeepLabCut to do it for you according to a specific algorithm you can define in the next parameter (automatic).\n\n\u2022 algo: When you set the mode parameter to automatic, here, you specify the algorithm that will be used to extract frames. Setting this parameter to uniform will cause DeepLabCut to select frames in a temporally uniformly distributed way (e.g. \u201cone frame per 1000 frames\u201d). Normally, I recommend setting this parameter to kmeans. This will select frames by clustering based on visual appereance - meaning, it will select frames that depict different forms of behavior even if they occur only occasionally throughout the video. However, slecting frames by clutering is also more time consuming than selecting them uniformly.\n\n\u2022 userfeedback: This parameter is set to False per default. Setting this to True will cause DeepLabCut to check for user approval for every frame throughout the extraction (very time consuming!).\n\n\u2022 crop: Setting this parameter to True will cause a GUI to pop up and ask the user to draw a box onto the frame, selecting the area of interest. The extracted frames will then be cropped in into this are. (This usually only makes sense, when your camera captures a much larger space than were the behavior of interest took place.)\n\n### Label frames\u00b6\n\nThe next step is now to manually label all the extracted frames. This can take up to several hours (depending on the size of your dataset and the number of key features) and should be done very carefully. As said before - the quality of the training dataset determines the tracking performance of the trained model.\n\nTo open the labeling GUI, I recommend using the main DeepLabCut GUI. To do so,\n\n\u2022 open a new window of Anaconda Prompt.\n\n\u2022 activate your DeepLabCut environment.\n\n\u2022 type python -m deeplabcut.\n\nA new window will open (this might take some time).\n\nClick on \u201cManage Project\u201d, then choose \u201cLoad existing project\u201d. Click on \u201cBrowse\u201d to select your project\u2019s config.yaml file and confirm.\n\nAfter confirming, your project is loaded into the DeepLabCut GUI.\n\nNext, change to \u201cLabel Data\u201d. The path to your configuration file is already filled in, so you can directly launch the labeling GUI by clicking on \u201cLabel Frames\u201d.\n\nTo open the labeling GUI, click on \u201cLabel Frames\u201d.\n\nThis will open the DeepLabCut labeling GUI, a tool that allows for the manual annotation of the extracted frames.\n\nIn the GUI, click on \u201cLoad Frames\u201d, then select one of the subdirectories with the extracted frames (you find them in \u201clabeled-data\u201d) and click \u201cok\u201d.\n\nIt can take a few moments until the frames are fully loaded into the GUI.\n\nNow, you can start to label your keypoints of interest. Check out the box below for more information on how to label frames efficiently.\n\n### Check the labeled frames\u00b6\n\nAfter manual labeling is complete, the labeled frames should be checked for any errors. Particularly serious are errors where, for example, the sides of the body parts to be labeled have been systematically swapped (e.g., right and left hands mixed up). Such errors in the training dataset can lead to systematic label errors in the final model and should therefore be avoided.\n\nTo check for labeling errors, DeepLabCut provides the deeplabcut.check_labels function. The function plots all extracted frames together with the manually set labels and the user-defined skeleton:\n\ndeeplabcut.check_labels(config_path, visualizeindividuals=True\/False)\n\n\nTo detect possible errors, all labeled frames should now be opened and the position of the labels checked. In order to be able to quickly detect errors (such as swapped label positions), it is recommended to have previously created a comprehensive skeleton in the config.yaml file.\n\nThe manually set labels will be plotted directly onto the extracted frames. If you detect any errors while checking the labels, you can simply correct them by reopening the labeling GUI and changing the respective labels positions. A: Discovered labeling error with mixed up labels. B: Corrected Frame.\n\n### Create training dataset\u00b6\n\nWhen you are happy with all your set labels, you can combine them into a training dataset that can be used to train your DeepLabCut model.\n\nTo do so, use the function:\n\ndeeplabcut.create_training_dataset(config_path)\n\n\nThis function takes all the files with the user-set labels positions and combines them into one big training (and a smaller test) dataset.\n\nAs the name suggests, the training dataset is used to train the model. The model generated in this way is then tested on the test dataset for its performance on previously \u201cunseen\u201d frames.\n\nThe division of the frames into training and test data set is defined in the config.yaml file in the TrainingFraction parameter. By default, this parameter is set to 0.95, which means that 95% of the manually labeled frames are used for training and 5% for testing the trained model.\n\nThere are a number of additional parameters that can be adjusted when creating a training dataset. For a complete list of parameters, please see the DeepLabCut documentation.\n\n## Train the Network\u00b6\n\nAfter a training dataset is created, the next step is to train a DeepLabCut model.\n\nThis is as simple as executing the following function (Note that a GPU is highly recommended for this step, as training a neural network on a CPU will take very long):\n\ndeeplabcut.train_network(config_path)\n\n\nRunning the function as depicted above, will lead to DeepLabCut using all the default settings for the function parameters.\n\nHowever, as the trained model will be the major outcome of a DeepLabCut project, you might want to dive deeper into the different options for model training.\n\nYou can for example specify what type of neural network is trained (e.g. residual neural networks (resnet) with different numbers of layers).\n\nI highly recommend to check out the DeepLabCut Documentation to make yourself familiar with the different options on model training.\n\n### Evaluate network\u00b6\n\nTo determine the quality of the trained model, you can evaluate the model\u2019s labeling performance by running:\n\ndeeplabcut.evaluate_network(config_path, plotting=True)\n\n\nThis will create a new subdirectory in your project directory where the evaluation results are stored. Evaluation results (namely the test- and training error) are then written into a .csv file that can be inspected with any common text editor.\n\nSetting the plotting parameter to True will plot all the labels of the training & test dataset together with the label predictions made by the model and the manually set labels. Manually set labels are displayed as a plus symbol \u201c+\u201d and the model\u2019s predictions either as a dot (for predictions with a high likelyhood) or as an \u201cx\u201d (for predictions with a low likelyhood)\n\nNote\n\nThe likelihood of the model\u2019s predictions is displayed with respect to the parameter p-cutoff that is set in the config.yaml file. The default value is 0.7, meaning that predictions with a likelihood of < 0.7 will be displayed as uncertain predictions.\n\nIf you find that the model\u2019s labeling accuracy is not yet sufficient, you should extract and label more frames from the video files, create a novel training dataset and retrain the model.\n\n## Analyze videos\u00b6\n\nWhen you are happy with your model\u2019s labeling accuracy on the test and training frames, you can start to analyze the original video footage. Note that this step can be time consuming (even more than the model training), depending on the amount and size of your video data.\n\nTo start the analysis, run:\n\ndeeplabcut.analyze_videos(config_path, videolist, save_as_csv=True)\n\n\nThe analysis results will be stored in the \u201cvideos\u201d subdirectory. For each video, there will be a .csv and a .h5 file created, that holds the model\u2019s label predictions for every frame of the video.\n\nAs the raw tracking results are often messy, it is recommended to filter the model\u2019s predictions. This can be done by applying a median filter, using the function:\n\ndeeplabcut.filterpredictions(config_path, videolist)\n\n\nThere is also an option to generate video files with the model set labels:\n\ndeeplabcut.plot_trajectories(config_path, ['fullpath\/TestProject\/videos\/videofilename.mp4'])\n\n\nHowever, I would recommend not to plot the labels for all analyzed videos, since these then naturally also take up twice as much storage space as the original videos. (Therefore, replace the variable \u201cvideolist\u201d by a list of one or few exemplary videos.)\n\n## Refine the network\u00b6\n\nAfter training & evaulating the model and seeing first analysis results, you might conclude that your model\u2019s labeling accuracy is not yet to your satisfaction.\n\nDon\u2019t worry, there is an easy way to fix it: Refine your model.\nIn particular, this means that you should label more data to create a more comprehensive training dataset.\n\nTo do so, you can extract outlier frames, meaning that you extract frames with a particuarly low overall labeling likelihood.\n\nFirst, run the function:\n\ndeeplabcut.extract_outlier_frames(config_path, videolist)\n\n\nNow, you can start to relabel the extracted frames. You can do so by either launching the DeepLabCut labeling GUI (using the main GUI as described above) or by launching a separate \u201crefinement\u201d GUI. This will plot all machine-set labels on the extracted frames and allow you to drag & draw them to their correct positions.\n\nAs with the \u201cnormal\u201d labeling GUI, I recommend to launch the refinement GUI using the main DeepLabCut GUI, as this will prevent any complications that can arise from using the Jupyter Notebook for this part.\n\nFirst, open your project in the main DeepLabCut GUI.\n\nThen, change to \u201cRefine Labels\u201d and click on \u201cLabel Frames\u201d.\n\nThe refinement GUI will open in front of you, much as the labeling GUI did earlier. The only real difference is, that you will have to insert a specific p-cutoff in the beginning, specifying how the machine-set labels are displayed in the GUI.\n\nLabels with a likelihood lower than the specified p-cutoff will be displayed as circles, whilst labels with a higher likelihood are displayed as larger dots on the frame.\n\nNote\n\nI recommend to use the \u2018normal\u2019 labeling GUI for sets frames with very poor labeling accuracy, as I find it very time consuming to \u2018search\u2019 for the individual labels throughout every frame. In the \u201cnormal\u201d labeling GUI, the extracted oulier frames will be displayed with no labels in contrast to the already manually labeled frames of the original training dataset.\n\nIf the overall labeling accuracy isn\u2019t too bad, it can however be faster to use the refinement GUI.\n\nWhen you are satisfied with the positions of the labels, you can merge the refined labels to your original training dataset by running:\n\ndeeplabcut.merge_datasets(config_path)\n\n\nAfterwards, you can refine the actual model by training it with the extended training dataset.\n\nTo do so, simply run deeplabcut.train_network again (see above).\n\nIf the results are satisfactory, you can now reanalyze new videos as described above. Otherwise, you can extract and label new outlier frames and continue to retrain the model, until it reaches a satisfactory labeling performance.\n\n## References\u00b6\n\nDeepLabCut Documentation. https:\/\/deeplabcut.github.io\/DeepLabCut\/docs\/intro.html\n\nNath, T., Mathis, A., Chen, A. C., Patel, A., Bethge, M., & Mathis, M. W. (2019). Using DeepLabCut for 3D markerless pose estimation across species and behaviors. Nature Protocols, 14(7), 2152\u20132176. https:\/\/doi.org\/10.1038\/s41596-019-0176-0","date":"2022-10-05 09:26:40","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\": 0.3561929762363434, \"perplexity\": 1738.49186777078}, \"config\": {\"markdown_headings\": true, \"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-2022-40\/segments\/1664030337595.1\/warc\/CC-MAIN-20221005073953-20221005103953-00556.warc.gz\"}"} | null | null |
package lodVader.exceptions.api;
public class DynamicLODAPINoLinksFoundException extends Exception{
public DynamicLODAPINoLinksFoundException() { super(); }
public DynamicLODAPINoLinksFoundException(String message) { super(message); }
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,127 |
\section{Introduction}\label{int}
A description of the processes of productions and decays of
fundamental unstable particles to satisfy the up-to-date
requirements must provide, on the one hand, gauge cancellations
and unitarity and, on the other hand, enough high accuracy of
calculation of resonant contributions of unstable particles.
Unfortunately, in the framework of conventional perturbation
theory (PT) a simultaneous fulfilling of these requirements is
obstructed by divergences caused by resonant contributions. For
this reason in the propagators of unstable particles the Dyson
resummation is usually applied, which shifts the resonant
singularities out of the region of physical momenta. However, a
resummation mixes the PT orders, which generally leads to
violation of the gauge cancellations. So simultaneously with using
the Dyson resummation an application of additional tricks is
required.
Among various approaches that include such tricks, the most known
one is based on the Laurent expansion of the amplitude around the
complex poles of the resonant propagators. Each term of this
expansion is considered expanded in the framework of the
conventional PT, as well, but a certain portion of the self-energy
is not involved in the latter expansion as having been absorbed by
the shift of the point of singularity (the remnant of the Dyson
resummation). The gauge cancellations are completely maintained in
this approach. However, the precision of the description vastly
falls at the increasing of a distance from the resonant region,
and an uncertainty arises at calculating the residues in the
complex-poles. Nevertheless, in the vicinity of the resonant
region the pole expansion in many cases is suitable for
applications. In particular, at LEP2 the loop corrections to the
$W$-pair production were calculated in the double pole
approximation (DPA) \cite{LEP2a,LEP2b}, the leading approximation
in the pole expansion. Unfortunately, at international linear
collider (ILC) \cite{ILC} the accuracy of DPA is no longer
sufficient \cite{CMS1}, and the higher-order corrections in the
pole expansion unlikely can save the situation. Therefore the
pinch-technique method and the method based on the
background-field formalism move forward to foreground, which, in
principle, can provide the necessary precision (see \cite{pinch}
and \cite{BFM}, and the references therein). However, the
consecutive application of the mentioned methods implies a
calculation of a huge volume of additional contributions that
formally appear outside the limits of required precision, which is
impractical \cite{Ditt}. So at present hopes are pinning on the
approach of ``complex-mass scheme'' (CMS), which avoids mentioned
difficulties \cite{CMS1,CMS2}. Nevertheless, in the CMS another
problem related to the unitarity arises. The point is that the CMS
uses the complex-valued renormalized masses for unstable particles
and this requires an introduction of the complex-valued
counterterms, which violates unitarity. For this reason the CMS
cannot be considered as a rigorous procedure \cite{CMS2}. The
problem becomes especially topical at calculating the
contributions in the next-to-next-to-leading order (NNLO). Thus to
make the calculations up to the NNLO alternative approaches are
required.
A promising candidate for this role is a modified perturbation
theory (MPT), first proposed in \cite{Tkach} and then elaborated
in \cite{EPJC} and \cite{MPT}. For determining the resonant
contributions the distribution theory is applied in this approach
instead of the Dyson resummation in whatever form. In essence, the
MPT implies a systematic expansion in powers of the coupling
constant directly of the probability instead of the amplitude.
This mode allows one to impart the sense of distributions to the
propagators squared of unstable particles, and on this basis to
asymptotically expand the propagators squared without the
appearance of the divergences in the cross-section. Since the
object to be expanded (the cross-section) is gauge invariant and
the expansion is made in powers of the coupling constant, the
result of the expansion must automatically be gauge-invariant.
This implies that the gauge cancellations in the MPT must be
automatically maintained. Of course, this should be so if the MPT
exists, i.e. if it is a well-determined method. In the case of
pair production of unstable particles this property was proved and
an algorithm of the calculation of each order of the MPT expansion
was elaborated \cite{MPT}.
The aim of the given paper is to perform numerical analysis of the
convergence properties of the MPT series in the case of pair
production of unstable particles. At once we should notice that in
the qualitative sense the outcomes should weakly depend on the
model under consideration because the choice of a model implies
mainly a definition of the test function in the presence of which
the relevant distributions (the propagators squared) are
MPT-expanded. So it is reasonable to carry out the examination in
the framework of a model possessing an exact solution. As such a
model, we consider the improved Born approximation for the process
$e^{+} e^{-} \to \gamma,Z \to t\bar t \to W^{+} b\:W^{-}\bar b$.
For simplicity we consider $W$ bosons and $b$ quarks to be stable
particles, with $W$ being massive and $b$ being massless. At the
same time we consider realistic corrections to the width of the
top quark. This should allow us to get an information about the
rapidity of convergence in the realistic case. A similar model has
actually been considered in \cite{N-tt} by examining the MPT
within the next-to-leading order (NLO). However, the contribution
from the soft massless particles to the process of production of
unstable particles have been omitted in that work. This was a
serious omission because the mentioned contributions include
Coulomb singularities \cite{Coulomb1}-\cite{Fadin1} appreciably
affecting the cross-section. In this paper we improve the
calculations of \cite{N-tt} (in particular eliminate some bug in
the calculations), and carry out numerical calculations further up
to the NNLO with taking into consideration universal Coulomb
singular contributions.
In the next section, we present the basic information about the
MPT and detail the model in the framework of which we carry out
computations. In Sect.~\ref{num} we present outcomes. In
Sect.~\ref{Conclusion} we discuss the results.
\section{MPT and a model for its examination}\label{mod}
The observable cross-section of production and decay of unstable
particles, for example in $e^+ e^-$ annihilation, has the form of
a convolution of the hard-scattering cross-section with the flux
function \cite{LEP2a},
\begin{equation}\label{not1}
\sigma (s) = \int\limits_{s_{\mbox{\tiny min}}}^s \frac{\mbox{d}
s'}{s} \: \phi(s'/s;s) \> \hat\sigma(s')\,.
\end{equation}
Here $s$ is the energy squared in the center-of-mass system,
$\hat\sigma$ is the hard-scattering cross-sections, $\phi$ is the
flux function describing contributions of nonregistered photons
emitted in the initial state. The $s'/s$ characterizes a fraction
of the energy expended on the production of unstable particles.
For our purposes it is sufficient to take $\phi$ in the
leading-log approximation. So we put
\begin{equation}\label{not2}
\phi (z;s) = \beta_e (1-z)^{(\beta_e -1)}
- \frac{1}{2} \beta_e (1+z), \qquad
\beta_e = \frac{2 \alpha }{\pi}\left(\ln \frac{s}{m_e^2}
-1\right).
\end{equation}
In the case of pair production of unstable particles the
double-resonant contributions are most crucial. Bearing this in
mind we write down the hard-scattering cross-section in the form
\begin{equation}\label{not3}
\hat\sigma (s) \; = \!\!\!\!
\int\limits_{\quad {\displaystyle\mbox{\scriptsize $s$}}_
{ 1 \mbox{\tiny min} } \atop
\quad {\displaystyle\mbox{\scriptsize $s$}}_
{ 2 \mbox{\tiny min} }}
^{\!\!\infty} \!\!\!\!\!\!\!\!
\int\limits^{\infty} \mbox{d} s_1 \, \mbox{d} s_2 \;\;
\hat\sigma(s\,;s_1,s_2) \left(1\!+\!\delta_{c}\right).
\end{equation}
In this formula $\hat\sigma(s\,;s_1,s_2)$ is an exclusive
cross-section, $\delta_{c}$ stands for soft massless-particles
contributions, $s_1$ and $s_2$ are virtualities of unstable
particles. In the case of the process $e^+e^- \to \gamma,Z \to
t\bar t \to W^{+}b\:W^{-}\bar b$ with massive $W$ and massless
$b$, we have $s_{1\,\mbox{\scriptsize min}} =
s_{2\,\mbox{\scriptsize min}} = M_{W}^2$, and
$s_{\mbox{\scriptsize min}}=4M_W^2$. In $\hat\sigma(s\,;s_1,s_2)$
we extract kinematic and Breit-Wigner (BW) factors,
\begin{equation}\label{not4}
\hat\sigma(s\,;s_1,s_2) = \frac{1}{s^2} \,
\theta(\sqrt{s}-\!\sqrt{s_1}-\!\sqrt{s_2}\,)
\sqrt{\lambda(s,s_{1},s_{2})}\;\Phi(s;s_1,s_2)
\> \rho(s_{1}) \> \rho(s_{2})\,.\quad
\vspace*{0.4\baselineskip}
\end{equation}
Here $\lambda (s,s_{1},s_{2}) = [s \!-\!(\sqrt{s_1} \!+\!
\sqrt{s_2}\,)^2] [s \!-\!(\sqrt{s_1} \!- \! \sqrt{s_2}\,)^2]$ is
the kinematic function, $\rho(s_{1})$ and $\rho(s_{2})$ are BW
factors. Function $\Phi(s;s_1,s_2)$ is the rest of the amplitude
squared. Below we consider $\Phi$ in the Born approximation, and
thus only the BW factors are subject to the MPT expansion. In the
general case, we define the BW factors as
\begin{equation}\label{not5}
\rho (s) = \frac{M \Gamma_{0}}{\pi}\times |\Delta(s)|^2\,.
\end{equation}
Here $M$ is the renormalized mass of the top quark, $\Gamma_{0}$
is its Born width, $\Delta(s)$ is a scalar part of the
Dyson-resummed propagator (the spin factor is referred to $\Phi$),
\begin{equation}\label{not6}
\Delta^{-1}(s) =
s - M^2 + \Re \Sigma(s) + \mbox{i}\:\Im \Sigma(s)\,,
\end{equation}
$\Re\Sigma(s)$ and $\Im\Sigma(s)$ are the real and imaginary parts
of the renormalized self-energy.
In the case of a smooth weight, an isolated BW factor may be
represented in the form of an asymptotic expansion in the sense of
distributions in powers of the coupling constant (generating thus
the MPT expansion of isolated BW factor). Up to and including the
NNLO this expansion looks as follows \cite{Tkach}:
\begin{eqnarray}\label{not7}
&{\displaystyle \rho(s) \;=\; \delta(s\!-\!M^2) \,+\,
\frac{M \Gamma_{0}}{\pi} \, PV \! \left\{\frac{1}{(s-M^2)^2} -
\frac{2\alpha\,\Re\Sigma_1(s)}{(s\!-\!M^2)^3}\right\}}&
\\
&\displaystyle +\;\sum\limits_{n\,=\,0}^2 c_{n}(\alpha)\,
\frac{(-)^{n}}{n!}\,\delta^{(n)}(s\!-\!M^2) + O(\alpha^3)\,.&\nonumber
\end{eqnarray}
Here $\alpha$ is the coupling constant, $\delta(\cdots)$ is the
$\delta$-function, $\delta^{(n)}$ is its $n$th derivative, $PV$
means the principal-value prescription. The leading term in
(\ref{not7}) defines the narrow-width approximation. The
contributions in the curly brackets appear as a result of the
naive expansion of the propagator squared; $PV$ makes the poles in
this expansion integrable. The contributions under the sum-sign
correct the contributions of the $PV$ poles (in the singular
point) so that the expansion becomes asymptotic. The coefficients
$c_{n}(\alpha)$ are polynomials in $\alpha$, determined by the
self-energy contributions of the unstable particle. In an
arbitrary UV-renormalization scheme the completely explicit
expressions for $c_{n}(\alpha)$ may be found in \cite{EPJC}. In
the case of the on-mass-shell (OMS) type scheme, they are found in
\cite{MPT}. In the latter case the coefficients $c_n$ within the
NNLO are determined by $I_1$, $I_2$, $I_3$, $I'_1$, $I''_1$, where
$I_n=\Im\Sigma_n(M^2)$, $I'_n=\Im\Sigma'_n(M^2)$,
$I''_n=\Im\Sigma''_n(M^2)$, and by $R_2$, $R'_2$, where
$R_n=\Re\Sigma_n(M^2)$, $R'_n=\Re\Sigma'_n(M^2)$. Here $\Sigma_n$
is the $n$-loop self-energy defined in accordance with relation
$\Sigma = \alpha \Sigma_1 + \alpha^2 \Sigma_2 + \cdots$.
Unfortunately, the weight in our case is not smooth because of the
kinematic factor in formula (\ref{not4}). A solution to this
problem is found on the basis of analytic regularization via the
substitution $[\lambda (s,s_{1},s_{2})]^{1/2} \to [\lambda
(s,s_{1},s_{2})]^{\nu}$. Furthermore, the weight
$\Phi\,(1+\delta_{c})$ may be expanded in powers of $s_1$ and
$s_2$ around $s_1=M^2$ and $s_2=M^2$. Then, it becomes possible to
analytically calculate singular integrals irrespective of details
of the definition of the weight. After calculating singular
integrals and removing the regularization the outcomes remain
finite and the expansion remains asymptotic \cite{MPT}. In
principle, this salvages the applicability of the approach, and
the problem is reduced to numerical calculations only.
Now we turn to the definition of the model in the framework of
which we will carry out calculations. At first we notice that
within the NNLO the MPT expansion of the general BW factor based
on propagator (\ref{not6}) coincides with that based on the
following ``minimal'' propagator:
\begin{eqnarray}\label{not8}
\Delta^{-1}_{NNLO}(s) &=& s - M^2 + \alpha \, \Re \Sigma_1(s) +
\mbox{i}\,\alpha \left[I_1 + (s \!-\! M^2)\,I'_1
+ \frac{1}{2}(s \!-\! M^2)^2 I''_1 \right]\nonumber\\
& + & \alpha^2 \left[ R_2 + \mbox{i}\, I_2 + (s \!-\! M^2)\,R'_2
\right] + \mbox{i}\,\alpha^3 I_3\,.
\end{eqnarray}
In fact, after the MPT expansion any contribution not included in
(\ref{not8}) appears outside the NNLO. However, under the
consideration in the conventional-function sense, the $R_3$ and
$I'_2$ in the region $s-M^2 \sim O(\alpha)$ may be assigned to the
NNLO, as well. So we start from the following modeling propagator:
\begin{eqnarray}\label{not9}
\Delta^{-1}_{NNLO}(s) & = & s \, - \, M^2 \, + \, \alpha \, \Re
\Sigma_1(s) \, + \, \mbox{i}\,\alpha \, \Im \Sigma_1(s)
\nonumber\\[0.5\baselineskip]
&+& \alpha^2 \left[R_2 + \mbox{i} \, I_2 + (s-M^2) (R'_2 +
\mbox{i} \, I'_2)\, \right] \: + \: \alpha^3 \left(R_3 + \mbox{i}
\, I_3 \right).
\end{eqnarray}
Propagator (\ref{not9}) ensures the NNLO precision from the point
of view of both the MPT and conventional functions. For uniformity
we consider $\Im\Sigma_1$ off-shell as well as in the case of
$\Re\Sigma_1$.
Now let us direct our attention to the definition of the two- and
three-loop contributions to propagator (\ref{not9}). Actually they
may be determined by using the only fact that they are the
on-shell contributions. Specifically, the real parts may be
determined by basing on the UV-renormalization conditions. But one
should remember that in the unstable-particles case the OMS scheme
may be determined in different fashions. In particular, the
conventional OMS scheme is determined by the conditions $R_n=0$
and $R'_n=0$ \cite{OMS}. However, it is inconvenient for the
calculations in the higher-orders, because the renormalized mass
$M$ in this scheme beginning with the two-loops is different from
the observable mass, and beginning with the two-loops generally is
gauge-dependent \cite{Sirlin1}. The problem is eliminated at
considering the first renormalization condition in the form $M^2 =
\Re \, s_p\,$, where $s_p$ is the pole of the propagator,
$\Delta^{-1}(s_p) = 0$, which means the equating of the
renormalized mass to the observable mass. The second
renormalization condition may be determined by equating the
imaginary part of the on-shell self-energy to the imaginary part
of $s_p\,$. As a result the equality $s_p = M^2 - \mbox{i} \, I$
is established by means of the UV-renormalization conditions,
where $I =\Im \Sigma(M^2)$. So both the renormalized mass and the
imaginary part of the on-shell self-energy become
gauge-independent. This scheme of the UV renormalization was
called the $\overline{\mbox{OMS}}$ scheme in \cite{OMS-bar} and
the ``pole scheme'' in \cite{Sirlin2}. In this scheme the $R_2$,
$R'_2$ and $R_3$ are determined as
\begin{equation}\label{not10}
R_2 = -I_1 I'_1 \,,
\qquad R'_2= - I_1 I''_1 / 2 \,, \qquad
R_3 = - I_2 I'_1 - I_1 I'_2 + I^2_1 R''_1 / 2\,.
\end{equation}
The imaginary contributions to the on-shell self-energy are
determined, in effect, by the unitarity condition. For $I_1$ and
$I_2$ the appropriate relations are
\begin{equation}\label{not11}
\alpha I_1 = M\Gamma_0\,, \qquad\quad \alpha^2 I_2 = M \alpha
\Gamma_1 \, .
\end{equation}
Here $\Gamma_0$ and $\alpha\Gamma_1$ are the Born and the one-loop
contributions to the width. The $I_3$, in the general case, is
nontrivially related with the two-loop contribution
$\alpha^2\Gamma_2$. In the $\overline{\mbox{OMS}}$ scheme this
relation is \cite{OMS-bar}
\begin{equation}\label{not12}
\alpha^3 I_3 = M \alpha^2 \Gamma_2 + \Gamma_0^3/(8M) \,.
\end{equation}
Unfortunately, the derivatives of $\Im \Sigma$ cannot be
determined by similar means. However the $I'_2$, which we need, is
a facultative quantity from the point of view of the MPT expansion
(see above). So we may determine $I'_2$ by using the approximate
relationships $\alpha^2 \Im\Sigma_2(s) = \sqrt{s}\,\alpha
\Gamma_1(s)$, $\Gamma_1(s) = \Gamma_1 \times
\Gamma_0(s)/\Gamma_0$, $\Gamma_0(s) = \alpha
\Im\Sigma_1(s)/\!\sqrt{s}$. This yields
\begin{equation}\label{not13}
\alpha^2 I'_2 = \frac{\alpha \Gamma_1}{\Gamma_0} \;\, \alpha I'_1
\,.
\end{equation}
Now we determine the one-loop self-energy $\Sigma_1(s)$. In the
framework of the model, we determine it with contributions of the
$W$ boson and $b$ quark only. In this way we avoid the IR
divergences generally arising at determining $\Re\Sigma_1(s)$.
Standard calculation in t'Hooft-Feynman gauge\footnote{The gauge
independence should be restored after including the higher-order
corrections in $\Phi$, and after including the single- and
non-resonant contributions in the cross-section \cite{MPT}.} gives
\begin{equation}\label{not14}
\alpha \, \Sigma_1(s) = A(s) - \Re A(M^2) - (s-M^2) \Re A'(M^2)\,,
\end{equation}
\begin{equation}\label{not15}
A (s) = - \frac{G_F M_W^2}{4\sqrt{2}\,\pi^2}\,s\left[
\left(2+\frac{M^2}{M_W^2}\right) B_1(s;0,M_{W}) + 1\right].
\end{equation}
Here $B_1(s;m_1,m_2)$ is the Passarino-Veltman function \cite{BP}.
Thus, we have determined all contributions to the propagator
(\ref{not9}) and thereby the BW factors in formula (\ref{not4}).
Further, by virtue of (\ref{not7}) we can determine the MPT
expansion of the BW factors. The coefficients $c_n$ on account of
(\ref{not10})--(\ref{not12}), (\ref{not14}), (\ref{not15}) and
\cite{MPT} are as follows:
\begin{eqnarray}\label{not16}
c_0 & = & - \, \alpha \, \frac{\Gamma_1}{\Gamma_0} + \alpha^2
\left[\frac{\Gamma_1^2}{\Gamma_0^2} -
\frac{\Gamma_2}{\Gamma_0}\right] - \frac{\Gamma_0^2}{8 M^2} -
\left(\frac{\Gamma_0}{M} \, \frac{M^2+M_W^2}{M^2-M_W^2}\right)^2,
\nonumber \\[0.5\baselineskip]
c_1 &=& 0, \qquad\qquad c_2 \;\;=\; -\, M^2 \Gamma_0^2 \,.
\end{eqnarray}
Recall that each $\Gamma_n$ includes an additional factor
$\alpha$, which is conditioned by the vertex origin of the width.
To complete definition of the model, we must determine also the
factor $(1+\delta_{c})$ in formula (\ref{not3}). Let us remember
that we have ignored in the self-energy all massless-particles
contributions that lead to IR divergences. For this reason we have
to ignore all other soft-massless-particles contributions whose
IR-divergent contributions are to be cancelled in the
cross-section. So, there should remain only the Coulomb singular
contributions that are not cancelled. Recall that they have the
meaning of universal corrections arising due to exchanges by soft
massless particles (photons, gluons) between outgoing massive
particles in the limit of small relative velocities. In the case
of strong-interacting top quarks, it is reasonable to ignore the
exchanges by photons and to take into account only gluon
exchanges. We also restrict our consideration to the one-gluon
approximation. Then, with taking into account the off-shell and
finite-width effects, we have \cite{Bardin,Fadin1}
\begin{equation}\label{not17}
{\delta_c} = \frac{\kappa \, \alpha_s \pi}{2\beta}\left[ 1 -
\frac{2}{\pi}\,\arctan \! \left( \frac{|\beta_{M}|^2 -
\beta^2}{2\beta \, \Im\beta_{M}} \right)\right] .
\end{equation}
Here $\kappa = 4/3$ is the group factor, $\alpha_s$ is the strong
coupling constant, $\beta = s^{-1}\sqrt{\lambda (s,s_{1},s_{2})}$
is the velocity of the unstable particles in the c.m.f.,
$\beta_{M} = \sqrt{1-4 (M^2\!- \mbox{i} M \Gamma)/s}$. Further we
put $\Gamma = \Gamma_0$ in the latter formula. The energy-scale
dependence in $\alpha_s$, we take into consideration as described
in \cite{Fadin2}.
So, now the model is completely determined. The cross-section in
the model may be straightforwardly calculated. We call the result,
the ``exact'' solution. Simultaneously we can calculate the MPT
expansion of the cross-section and compare the outcome with the
``exact'' result. Ultimately the expansion should have the form
\begin{equation}\label{not18}
\sigma(s) = \sigma_{0}(s) \,+\, \alpha\,\sigma_{1}(s) \,+\,
\alpha^2 \sigma_{2}(s) \,+\, \cdots \,.
\end{equation}
Here $\sigma_{0}$ means the cross-section in the LO approximation,
$\alpha\,\sigma_{1}$ and $\alpha^2 \sigma_{2}$ mean the NLO and
NNLO corrections, respectively. So, the $\sigma_{01} = \sigma_{0}
+ \alpha\,\sigma_{1}$ and $\sigma_{012} = \sigma_{0} +
\alpha\,\sigma_{1} + \alpha^2 \sigma_{2}$ determine the NLO and
NNLO approximations. Similarly we denote the contributions to the
hard-scattering cross-section $\hat\sigma(s)$.
\section{Results of numerical calculations}\label{num}
Parameters of the model we determine as follows: $M = 175 \:
\mbox{GeV}$, $M_W = 80.4 \: \mbox{GeV}$, and we use the following
previously calculated input-data for the width \cite{Top}:
\begin{eqnarray}\label{not19}
\Gamma_0 &=& 1.56 \; \mbox{GeV} \,, \nonumber \\
\Gamma_0 + \alpha \Gamma_1 &=& 1.45 \; \mbox{GeV} \,, \nonumber \\
\Gamma_0 + \alpha \Gamma_1 + \alpha^2 \Gamma_2 &=&
1.42 \; \mbox{GeV} \,.
\end{eqnarray}
The $\Gamma = \Gamma_0 + \alpha \Gamma_1 + \alpha^2 \Gamma_2$ we
consider as the total width of the top quark. From (\ref{not19}),
we get $\alpha \Gamma_1=-0.11$~GeV, and $\alpha^2
\Gamma_2=-0.03$~GeV. The $\alpha$ in $\Phi(s;s_1,s_2)$ and
$\phi(z;s)$, we set equal $1/137$. All calculations are carried
out on the basis of rather general FORTRAN code with double
precision written in accordance with the formulas and instructions
described in \cite{MPT}.
\begin{figure}[t]
\hbox{
\epsfxsize=420pt \epsfbox{Fig1.eps}}
\caption{\small Total cross-section $\sigma(s)$. The exact result
in the model, we show by thick~curve. Dotted, short-dashed, and
continuous thin curves mean the LO, NLO, and NNLO approximations
in the MPT, respectively. The results are presented in pb (a) and
in percents to the exact result (b). In panel (b), we show by the
long-dashed curve the result in the DPA.\label{Fig1}}
\end{figure}
\begin{figure}[t]
\hbox{
\epsfxsize=420pt \epsfbox{Fig2.eps}}
\caption{\small Hard-scattering cross-section $\hat\sigma(s)$. The
notation is the same that in Fig.~\ref{Fig1}.\protect\label{Fig2}}
\end{figure}
In Fig.~\ref{Fig1}(a) we present the results of the calculation of
the total cross-section $\sigma(s)$ above the threshold. The
results in percentages with respect to the exact solution are
shown in Fig.~\ref{Fig1}(b). In the latter figure we place also a
result in DPA, where $\sigma_{DPA}(s)$ is determined by the same
formulas as in the case of $\sigma(s)$ but by substituting
$\Phi(s;M^2,M^2)(1+\delta_{c}(s;M^2,M^2))$ for
$\Phi(s;s_1,s_2)(1+\delta_{c}(s;s_1,s_2))$ and $s - M^2 +
\mbox{i}\,\Gamma$ for $\Delta^{-1}(s)$. The similar results for
the hard-scattering cross-section $\hat\sigma(s)$ are presented by
Fig.~\ref{Fig2}(a,b). Let us remember that $\hat\sigma(s)$ is
responsible for the distribution over the invariant mass of the $t
\bar{t}$ system and therefore is of interest, as well
\cite{tt-mass}. In Fig.~\ref{Fig3} we show the results separately
for the NLO and NNLO corrections to $\sigma$. In Table~\ref{tabl}
the results are represented in the numerical form at the
characteristic energies accessible at the planned $e^{+}e^{-}$
colliders. In the last column the numbers in parenthesis represent
the uncertainties in the last digits. (See discussion of their
determination in \cite{ACAT}.) In the other columns the
uncertainties are omitted as they appear in the digits that are
not shown. In the lower positions in the Table the results are
presented in percentages with respect to the exact result in the
model.
\begin{figure}[t]
\hbox{ \hspace*{80pt}
\epsfxsize=260pt \epsfbox{Fig3.eps}}
\caption{\small Corrections $\sigma_1$ and $\sigma_2$ (dashed and
continuous curves, respectively).\protect\label{Fig3}}
\end{figure}
\begin{table}[t]
\begin{center}
\caption{\small The results of the calculation of the total
cross-section in pb.}
\begin{tabular} {ccccc}\hline
\hline\noalign{\medskip} \\[-6mm]
$\quad \sqrt{s}$ (TeV) $\qquad$ & $\qquad \sigma \qquad\;\;$ &
$\qquad \sigma_{0} \qquad\;\;$ & $\qquad \sigma_{01} \qquad\;\;$ &
$\qquad \sigma_{012} \qquad\quad$ \\
\hline\noalign{\medskip} \\[-6mm]
0.5 & 0.6724 &
0.5687 & 0.6344 & 0.6698(7) \\
& {\small 100\%} &
{\small 84.6\%} & {\small 94.3\%} & {\small 99.6(1)\%} \\
\hline\noalign{\medskip} \\[-6mm]
1 & 0.2255 &
0.1821 & 0.2124 & 0.2240(2) \\
& {\small 100\%} &
{\small 80.8\%} & {\small 94.2\%} & {\small 99.3(1)\%} \\
\hline\noalign{\medskip} \\[-6mm]
3 & 0.03697 &
0.02363 & 0.03377 & 0.03653(3) \\
& {\small 100\%} &
{\small 63.9\%} & {\small 91.4\%} & {\small 98.8(1)\%} \\
\hline\noalign{\medskip} \\[-6mm]
5 & 0.02032 &
0.00904 & 0.01705 & 0.01991(2) \\
& {\small 100\%} &
{\small 45.5\%} & {\small 83.9\%} & {\small 98.0(1)\%} \\
\noalign{\smallskip}\hline\hline
\end{tabular}\label{tabl}
\end{center}
\end{table}
The above outcomes exhibit very stable behavior of the NLO and
NNLO approximations in the energy region beginning with
approximately 400~GeV. (In this region simultaneously the right
hierarchy of the corrections is established, $\sigma_0 < \sigma_1
< \sigma_2$.) The accuracy of the NNLO approximation is
established greatly high in this region. In particular, at 400~GeV
$< \sqrt{s}\, <$ 600~GeV it is within $\pm 0.5 \%$. At increasing
energy the accuracy in relative units is slightly decreasing, but
the cross-section is decreasing, too, so that the effective
precision of the description remains approximately the same
(because the ratio of the discrepancy to the quantity
$\sqrt{\sigma}$, which characterizes statistical error, is
approximately constant). In contrast to the above picture, the DPA
exhibits greatly unstable behavior; its discrepancy varies from
+3.0\% to -7.4\% in the energy region 400 GeV $<\sqrt{s}\,<$
1500~GeV. On the whole, such a behavior coincides with the
expected one for DPA, including the order of magnitude of the
discrepancy in the Born approximation for $\Phi$ \cite{LEP2b}.
To conclude this section three important remarks are in order.
First we note that the results expressed in relative units are
almost insensitive to the choice of the test function $\Phi$. In
particular, the turning-on/off of the Coulomb factor has very
small effect. For instance, at $\sqrt{s} = 500$~GeV this leads to
0.7\%-modification of the ratio $\sigma_{012}/\sigma$ and at
$\sqrt{s} = 1500$~GeV does less than 0.1\%. (Although, the
variation of the absolute value of the cross-section is
considerable in both cases: about 35\% and 20\%, respectively.)
The second remark concerns a large value of the correction
$\sigma_2$ in comparison with the discrepancy
$\sigma\!-\!\sigma_{012}$. For example, at $\sqrt{s} = 500$~GeV
they constitute 5.3\% and 0.4\% of $\sigma$, respectively. However
we think that this is an incidental unbalance as $\sigma_2$ gains
its value mainly due to the correction $\alpha^2 \Gamma_2$ to the
width, which in our case exhausts the corrections, and
simultaneously $\alpha^2 \Gamma_2$ is quite large (approximately
2\% of $\Gamma$). If we put everywhere $\alpha^2 \Gamma_2=0$, then
at $\sqrt{s} = 500$~GeV the $\sigma_2$ decreases to 2.1\% with the
discrepancy remaining within the 0.5\%-interval. On the other
hand, if we put $\alpha^2 \Gamma_2=0$ only at calculating the
coefficients $c_n$ without the change of the model itself, then
the $\sigma_2$ becomes almost the same as in the latter case, but
the discrepancy increases to 3.6\%. The third remark concerns the
ill-convergent property of the MPT in the near-threshold region.
With the energy approaching the threshold the accuracy and
stability of MPT rapidly become worse. This manifests itself in
the violation of the hierarchy of the corrections and then in the
blowing up of the corrections. Actually, this behavior was
predicted in \cite{MPT}. To prevent this difficulty another mode
of the MPT near threshold must be applied \cite{MPT} that implies
Taylor expansion of $\sigma(s)$ both in powers of $\alpha$ and in
powers of $s-4M^2$, where $4M^2$ is the threshold. An alternative
method implies a secondary Dyson resummation in the framework of
the MPT approach \cite{EPJC,Proc}.
\section{Discussion}\label{Conclusion}
Although above calculations have been carried out in the framework
of a model, the obtained outcomes expressed in relative units to a
large extent are model-independent. By this we mean that the
outcomes are weakly sensitive to the choice of the test function
determined by the model under consideration. We verified this
property by carrying out calculations with various test functions
and found that the influence of the test function manifests itself
mainly in a factor common for different contributions to the
cross-section. In particular, even very large variation in the
test function that appear at the turning-on/off the Coulomb
factor, in relative units leads to small modifications of the
outcomes.
On this basis we suppose that the loop corrections to the test
function will lead in relative units to small modifications of the
outcomes, too. In particular, our result about the 0.5\%-accuracy
of the NNLO approximation near the maximum of the cross-section,
should remain in force at turning-on the loop corrections.
Moreover, one can further improve the results if applying the MPT
on the background of the loop corrections only, and considering
the Born contribution in the old fashion with the
Dyson-resummation in the unstable-particles propagators --- on
analogy of actual practice of application of DPA
\cite{LEP2a,LEP2b}. In this case the discrepancy in the MPT
description will be diminished by a factor $O(\alpha)$.
Another aspect of the problem of model-dependence of our results
concerns the corrections to the width of unstable particles.
Recall that these corrections determine coefficients $c_n$, which
are crucial for the definition of MPT expansion. We have
considered the case with rather large corrections to the width
(7\% and 2\% in the NLO and NNLO, respectively). At diminishing
these corrections, the MPT corrections to the cross-section should
diminish, too. At least, we have observed this property in the
framework of the model under consideration. On this ground we can
expect the improving of the precision of description at transiting
from the top quarks to EW-only interacting particles, for instance
to the $W$-bosons, because the corrections to the width are lesser
in the latter case.
As regards the application of our results to the description of
realistic processes with the top-quark pair production, we should
stress that our calculations simulate the main contribution to the
cross-section as they cover the double-resonant contributions. So
on the basis of our results we can judge about the precision that
must be achieved in realistic calculations. Fortunately, the
accuracy of the NNLO approximation detected in our analysis, is
satisfactory from the point of view of the ILC requirements.
Really, assuming that at the ILC several hundred thousands of the
$t\bar t$ events is expected, we conclude that the calculation of
the cross-section is needed with a few per mille accuracy. As we
have seen above, this, in general, is ensured by the NNLO in the
MPT.
In summary, we have shown that the MPT stably works at the
energies near the maximum of the cross-section and above at the
description of the total cross-section for the pair production and
decay of fundamental unstable particles. We have found also that
in the mentioned energy region the MPT provides very good
precision within the NNLO. In particular, at the ILC energies in
the case of the top-quark pair production the NNLO approximation
provides 0.5\%-precision of the description. The further increase
of the precision is possible at the proceeding to the NNNLO,
possible on the basis of the results of \cite{MPT}, or at the
proceeding to the compound use of the MPT, when the loop
corrections are treated completely in the framework of the MPT
while the Born contribution to the cross-section is taken into
consideration in the old fashion with the Dyson resummation in the
unstable-particles propagators. On the whole, the MPT method is a
real candidate for carrying out high-precision calculations needed
for ILC.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,948 |
{"url":"https:\/\/www.zbmath.org\/?q=an%3A1351.76182","text":"# zbMATH \u2014 the first resource for mathematics\n\nBoundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier-Stokes equations. (English) Zbl\u00a01351.76182\nSummary: Harlow and Welch [Phys. Fluids 8 (1965) 2182-2189] introduced a discretization method for the incompressible Navier-Stokes equations conserving the secondary quantities kinetic energy and vorticity, besides the primary quantities mass and momentum. This method was extended to fourth order accuracy by several researchers [25,14,21]. In this paper we propose a new consistent boundary treatment for this method, which is such that continuous integration-by-parts identities (including boundary contributions) are mimicked in a discrete sense. In this way kinetic energy is exactly conserved even in case of non-zero tangential boundary conditions. We show that requiring energy conservation at the boundary conflicts with order of accuracy conditions, and that the global accuracy of the fourth order method is limited to second order in the presence of boundaries. We indicate how non-uniform grids can be employed to obtain full fourth order accuracy.\n\n##### MSC:\n 76M20 Finite difference methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids\nFull Text:\n##### References:\n [1] Axelsson, O.; Kolotilina, L., Monotonicity and discretization error estimates, SIAM J. Numer. Anal., 27, 6, 1591-1611, (1990) \u00b7 Zbl\u00a00719.65036 [2] Botella, O.; Peyret, R., Benchmark spectral results on the lid-driven cavity flow, Comput. 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Fluids, 8, 2182-2189, (1965) \u00b7 Zbl\u00a01180.76043 [7] Hundsdorfer, W.; Verwer, J., Numerical solution of time-dependent advection-diffusion-reaction equations, (2007), Springer [8] Lorenz, J., Zur inversmonotonie diskreter probleme, Numer. Math., 27, 227-238, (1977) \u00b7 Zbl\u00a00325.65014 [9] Mahesh, K.; Constantinescu, G.; Moin, P., A numerical method for large-eddy simulation in complex geometries, J. Comput. Phys., 197, 215-240, (2004) \u00b7 Zbl\u00a01059.76033 [10] Manteuffel, T. A.; White, A. B., The numerical solution of second-order boundary value problems on non-uniform meshes, Math. Comput., 47, 176, 511-535, (1986) \u00b7 Zbl\u00a00635.65092 [11] Mattheij, R. M.M.; Rienstra, S. W.; ten Thije Boonkkamp, J. H.M., Partial differential equations: modeling, analysis, computation, (2005), SIAM \u00b7 Zbl\u00a01090.35001 [12] Mattson, K.; Nordstr\u00f8m, J., Summation by parts operators for finite difference approximations of second derivatives, J. Comput. Phys., 199, 503-540, (2004) \u00b7 Zbl\u00a01071.65025 [13] Mittal, R.; Moin, P., Suitability of upwind-biased finite difference schemes for large-eddy simulation of turbulent flows, AIAA J., 35, 8, (1997) \u00b7 Zbl\u00a00900.76336 [14] Morinishi, Y.; Lund, T. S.; Vasilyev, O. V.; Moin, P., Fully conservative higher order finite difference schemes for incompressible flows, J. Comput. Phys., 143, 90-124, (1998) \u00b7 Zbl\u00a00932.76054 [15] Morton, K. W.; Mayers, D. F., Numerical solution of partial differential equations, (2005), Cambridge University Press \u00b7 Zbl\u00a01126.65077 [16] Moser, R. D.; Kim, J.; Mansour, N. N., Direct numerical simulation of turbulent channel flow up to $$\\mathit{Re}_\\tau = 590$$, Phys. Fluids, 11, 943-945, (1999) \u00b7 Zbl\u00a01147.76463 [17] Perot, B., Conservation properties of unstructured staggered mesh schemes, J. Comput. Phys., 159, 58-89, (2000) \u00b7 Zbl\u00a00972.76068 [18] Sanderse, B., Energy-conserving Runge-Kutta methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 233, 100-131, (2012) \u00b7 Zbl\u00a01286.76034 [19] Sani, R. L.; Shen, J.; Pironneau, O.; Gresho, P. M., Pressure boundary condition for the time-dependent incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 50, 673-682, (2006) \u00b7 Zbl\u00a01155.76322 [20] Strand, B., Summation by parts for finite difference approximations for $$\\operatorname{d} \/ \\operatorname{dx}$$, J. Comput. Phys., 110, 47-67, (1994) \u00b7 Zbl\u00a00792.65011 [21] Vasilyev, O. V., High order finite difference schemes on non-uniform meshes with good conservation properties, J. Comput. Phys., 157, 746-761, (2000) \u00b7 Zbl\u00a00959.76063 [22] Veldman, A. E.P., \u201cmissing\u201d boundary conditions? discretize first, substitute next, and combine later, SIAM J. Sci. Stat. Comput., 11, 82-91, (1990) \u00b7 Zbl\u00a00683.76028 [23] Veldman, A. E.P., High-order symmetry-preserving discretization of convection-diffusion equations on strongly stretched grids, (Lube, G.; Rapin, G., Proceedings of the Int. Conf. on Boundary and Interior Layers, (2006)) [24] Verstappen, R. W.C. P., When does eddy viscosity damp subfilter scales sufficiently?, J. Sci. Comput., 49, 94-110, (2011) \u00b7 Zbl\u00a01432.76129 [25] Verstappen, R. W.C. P.; Veldman, A. E.P., Direct numerical simulation of turbulence at lower costs, J. Eng. Math., 32, 143-159, (1997) \u00b7 Zbl\u00a00911.76072 [26] Verstappen, R. W.C. P.; Veldman, A. E.P., Symmetry-preserving discretization of turbulent flow, J. Comput. Phys., 187, 343-368, (2003) \u00b7 Zbl\u00a01062.76542 [27] Wesseling, P., Principles of computational fluid dynamics, (2001), Springer\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. 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Far too many start out in pursuit of a business or idea because they believe they have something to contribute, or something to say. Then little by little for some unexplained reasons they believe customers, fans, or others no longer find them as unique, (or whatever word fits) and they can't understand why. I will assert it's because they haven't noticed the drummer they were marching too has changed the beat. Yes they're still marching, but not in the same rhythm or cadence they began with.
Steve Jobs is known for his disinterest in customer surveys. He felt he knew what people would want better than they did themselves. He did, and delivered ground breaking products people still clamor to buy. There are of course far many more that loose this sense of their vision and begin to transform their products to fit everybody.
We see this happen with companies that were once considered cutting edge before they decide to go public i.e., launch an IPO. (initial public offering on Wall Street) Then suddenly it seems the company has changed its tempo from what they wanted to build for customers – to what they believe they need to build for customers that shareholders like just as much. The company feels it's still pounding out the same beat, but to anyone listening and watching their tune has definitely changed.
Facebook® is a current example of this. The stated beat they don't care about the money march has noticeably changed to have a rhythm more in line with what they want (or need) to hear at a Wall St. parade.
This happens everywhere, in all businesses. Wall Street can show it with glaring clarity so it makes for great examples. However it happens subtly but with the same affects all the way down to the solo entrepreneur. And just as hard as it is to remain disciplined at the Fortune 500 level, so too is it just as hard at the smallest companies. Sometimes even more so.
Imagine for the moment you were a baker of artisan breads. You decide for whatever the reasons your mission is to bring this style of bread to markets. You open your bakery in some location and begin selling your wares. As far as you believe no one offers what you do – except you. You feel a need, or calling, or whatever, so you start.
Almost out of the gate you're endeavor takes off. It seems like your thinking was spot on and you try to improve daily. Then you begin to do something that seems innocuous but can change that beat you listened to if you're not careful: You start asking for, and reacting to all suggestions people might request.
It starts with a simple suggestions such as, "Why don't you offer any Pepperidge Farm® styled bread?" At first you laugh at such a thing. Then you begin to think, and say "Well maybe if I made one PF version in my style…" And you do. You think you're giving people what they want. Maybe that's true. However if you're not careful the beat of this drummer just might be changing.
As you continue more requests or suggestions come in. They suggest if it tasted a little more like this, or shaped a little more like that they would enjoy it even more. You begin to oblige such requests because this new bread is now selling rather well. A good thing you believe because what you've noticed is the sales of your artisan breads have slowed. More customers are now coming in for the PF styled bread while it seems your artisan customers are falling off.
Business remains good. You attribute this to your listening skills of understanding the beats of others. You might begin to contemplate "It was a good thing you listened" because as you look over your cases the artisan breads you started with are no longer even a definitive part of your offerings. Those customers no longer visit your store. Now your bakery is full of what everyone said they wanted: Pepperidge Farm styled breads.
Your drum beat changed, and you never noticed. You were doing everything as far as you thought correct because as it was pounding, you were marching. You were going through all the moves but that beat was not the one you originally chose to listen to.
In an effort to please everyone what you did was displease the very customers you went into business for. The ones who needed you. The one who can't get, find, or just enjoy your artisan offerings. Now they don't seek you out. They don't even come in. They know they aren't your customer any longer. The masses are. They never understood why an artisan bakery would even offer a version of PF. They won't return any time soon – if ever.
You're now out of sync with the rhythms of your original intended customers. You no longer produce what they march for. But you shake it off because you think: "Hey I'm selling a ton of the PF styled bread, and making a fair profit to boot. Maybe I was right to change my thinking." Then a real Pepperidge Farm bakery opens on your very street.
This scenario happens far more often than people realize. Yet it's easy to see how. This is why it's so important to stay true to your original vision, listen for clues in a change of rhythm. Yes you'll make changes along the way. You'll try to improve your products, try to offer products that meet the needs of more, and more customers. However it's a very slippery slope that you must heed the warnings of before you venture down. Because sometimes you just don't notice the pounding is no longer your drummer, but some parade you didn't realize you became part of. | {
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Design · News · Science
New periodic table sorts 3,700 known exoplanets into 18 categories
Written by Greg Beach
Environment Science
A new Periodic Table of Exoplanets guides scientists and science fans alike through over 3,700 known exoplanets, including those that may host life. To organize the thousands of worlds identified since the first exoplanet was discovered in 1992, astronomer Abel Méndez of the Planetary Habitability Laboratory at the University of Puerto Rico created a chart that sorts the exoplanets into 18 distinct categories. "We know of over 3,700 planets around other stars. They are very diverse," Méndez said in an interview with Gizmodo. "We can roughly classify them by their size and temperature. Only warm planets with the right size, similar to Earth, might provide some of the conditions for extraterrestrial life."
At the most general level, exoplanets, or planets beyond our solar system, are categorized based on distance from the star around which they orbit and their temperature. This places them in one of three zones: Hot Zone, Warm Zone, or Cold Zone. The exoplanets are also distinguished by size and composition (rocky "terran" planets vs. gas giants like Neptune and Jupiter). As in the actual periodic table, each exoplanet category has a number assigned to it, which indicates how many of a particular kind of planet have been discovered.
Related: Scientists discover new Earth-like planet only 11 light years away
According to the Periodic Table of Exoplanets, there are 53 known exoplanets with the appropriate size, temperature and features such as liquid water and a stable atmosphere to potentially host life as we know it. "Unfortunately, we don't know yet if they also have the right amount of water (e.g. oceans) or the right atmosphere for life too," said Méndez. As for the disproportionate number of hot planets on the Table, Méndez explained that this is due to the relative ease of discovery for hot planets and not necessarily because there are more of them.
At the top right corner of the Table, a chart indicates the number of stellar systems and the number of known exoplanets for each system. To Méndez, the possibilities are endless. "I'm overwhelmed by the number and diversity of planets in the stars around us. So many places to explore in our own Solar System, but much more is waiting for us beyond," Méndez told Gizmodo. "I won't be very surprised by another planet with life, Earth is the example that this is possible. I will be more surprised by something we haven't seen before."
Via Gizmodo
Images via Planetary Habitability Laboratory/University of Puerto Rico at Aricebo and NASA
Periodic Table of Exoplanets
The Periodic Table of Exoplanets offers a simple visualization to explore the known universe.
Kepler 20e
Kepler-20e was one of the first Earth-size planets discovered by scientists. | {
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So you work at a great company, and your company delivers an exceptional product. Even so, it may be no simple task to convince users to engage with your software on a consistent basis. Sure, in some cases the product speaks for itself.
But most of the time, people don't quickly change their daily routines. They need to be shown that investing time in your product is worth their while.
It takes more than a great marketing plan to make this happen. It takes features built into the software or website that subtly encourage users to come back again and again. This, in a nutshell, is how user onboarding happens.
But what are these features? In this article, we'll look at four key factors in increasing user engagement and thereby speeding up the onboarding process.
The first time a user encounters your product is critical, because their user experience will often decide whether they'll ever return to your SaaS. So strive to make engagement with your software immediate, simple, and fun.
What are some ways to do this? For one, you might want to introduce your product with a how-to app. Include a highly visible Help section in case they get confused, as well as a toll-free customer service number. Or, instead of all these, you can use an online guidance platform, such as WalkMe.
If possible, let them customize aspects of the service right from the beginning. This will give users a sense of agency and creative control.
Finally, introduce your product in a way that tells the reader a story. Giving users a narrative to latch onto, rather than just offering technology, will help them engage with the product and increase the likelihood of onboarding.
Don't hesitate to turn to other companies to see what they've done to engage users. There are numerous success stories out there – Twitter, Tumblr, Google, the list goes on – and insights can be gleaned from all of them.
For example, the meteoric rise of Twitter was fueled by an important Aha! moment.
As one recent article reports, Twitter began to take off when its executives discovered that users who followed enough people on the site (more than 5-10) were much more likely to become regular users.
Studying other companies' breakthroughs regarding user engagement will help you to achieve your own.
Another cue you can take from successful SaaS companies is the way their sites look. Compare the websites of major success stories in the SaaS industry. Notice any similar features?
Most successful websites are relatively minimalist, without much visual clutter. Clean lines are attractive and aesthetically pleasing, and make your SaaS look professional. Don't be afraid to use negative space (think of Google's homepage!).
In general, you don't want to scare off a potential user by confusing them. Your product/website should make it easy and intuitive for them to sign in and begin engaging with the product. In this regard, large, bright logos and simple, easy-to-find click-throughs will work to your advantage.
Another way to simplify your website is by incorporating social logins from other sites rather than forcing users to come up with a new username and password for your product. This process always entails the danger that the user will be lost. According to one recent study, 92 percent of users who forgot their logins to a site ended up not returning to the site.
There are more reasons to use social logins, however. The same study cited above also indicates that most users (52 percent) find that using social logins results in an improved online experience. Social logins also allow you to personalize your SaaS for each user. Customized content generally tends to make users feel good about their online experience.
One study found that 6 in 10 users believed that they knew more about, and had positive feelings toward, SaaS companies that provided customized content than those that did not.
Social logins not only let you personalize UX, they potentially provide access to each user's list of friends or followers, thereby expanding the reach of your product.
There are many other ways to increase user onboarding via user engagement, but these four are all critical aspects of the process of improving in onboarding, and should be taken into account.
If handled well and given the time they deserve, these steps will combine to help create an engaging and memorable user experience – which is the most important ingredient in the long-term success of any SaaS. | {
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Unpredictable Disaster
Larry O'Hanson, The furious storm: one wild hurricane could drown a major American City. Can scientist prevent the disaster in time, Scholastic, Inc., Science World (October 18, 2002):
Here's a tip from the experts: If you're in New Orleans when the "Big One" hits, have a lifeboat handy. Some scientist[s] warn that the right hurricane--a tropical cyclone with at least 74-mile-per-hour winds--could strike the Gulf Coast in a way that would hurl millions of gallons of water to turn the city known as the Big Easy into the Big Soup Bowl....
A major flood could submerge much of central New Orleans beneath 20 feet of water, leaving many of the metropolitan area's 1.3 million residents clinging to rooftops--a prospect that has engineers and city planners scrambling for defensive strategies. "It's the luck of the draw," says hurricane expect Hugh Willoughby at the National Oceanic and Atmospheric Administration (NDAA). He thinks it's a matter of when--not if--the Big One will pound New Orleans During some annual hurricane season between June and November.
The perfect storm could ... strike New Orleans east of the city, with gale-force winds blowing south, shoveling water from Lake Pontchartrain over the lake levees ....
Anticipate that these statements will be made during the next 100,000 or 1,000,000,000 years
1. "No one anticipated that an asteroid would hit the earth."
2. "No one anticipated that the sun would swell thousands of times."
3. "No one could have anticipated that a volcanic eruption would take place directly below Gotham City."
4. "No one could have anticipated that the fusion reactor would go awry."
5. "No one could have anticipated that a massive hurricane would hit New York City."
6. "No one could have anticipated that a cluster of rapidly-moving neutron stars would rip the planets out of their orbits around the Sun."
7. "No one could have anticipated that rebels would commandeer a star ship and steer it into our country's capital city."
8. "No one could have anticipated that glaciers would start to grow at a rapid rate."
9. "No one could have anticipated that the sun would begin to cool at a rapid rate."
10. "We can anticipate that many public officials will say in years to come that colossal disasters were not anticipated and could not have been anticipated."
Natural Disasters & Price Gouging
Mark P. Gergen, A Priest Responds to the Bean Counters: Leo Katz on Evasion, Blackmail, Fraud, and Kindred Puzzles of the Law (review essay), 22 Law & Soc. Inquiry 879, 890-891 (1997)(footnote omitted):
Consider the case of price gouging for food, water, or other essentials during an emergency. Price gouging is not blackmail (or even common law duress), no matter how extreme the price demand or desperate the needs of the victim. This is a difficult case for [Leo] Katz, for the price gouger can be made to seem a swine if we make the facts extreme enough. On the other hand, the case of the price gouger is an easy one to explain under some other theories. James Lindgren would explain that the price gouger is not a blackmailer because he is bargaining with his own rights. Robert Nozick would explain that price gouging is not blackmail because, on balance, the victim is better off for having the opportunity to buy necessities from the price gouger. But Katz's theory better describes our feelings about the price gouger than do these other two theories because it expresses our unease about the price gouger's behavior. Katz tells us that the question finally turns on whether the price gouger is committing a sufficiently grave immorality, and while from some moral perspectives he is not acting immorally (Lindgren's and Nozick's theories are evocative of some of the reasons), from other moral perspectives he is.
See also Gregory R. Kirsch, Hurricanes and Windfalls: Takings and Price Controls in Emergencies (Note), 79 Va. L. Rev. 1235, 1236-1237 (1993)(footnotes omitted):
... Hurricane Andrew provides a "disaster context" for this Note's inquiry into the rules of price controls, emergency takings, and just compensation, and whether the rules encourage and facilitate effective disaster relief. Andrew struck South Florida on August 24, 1992, with 150 mph winds, destroying more than 60,000 homes and leaving as many as a quarter of a million people homeless. The victims' need for food, water, home-repair materials, and other necessities exceeded available supplies. The destruction of much of the public and private infrastructure exacerbated the shortages. As microeconomic price theory predicts, established merchants and opportunistic entrepreneurs began charging sharply higher prices. For example, sheets of plywood, each priced at eight or nine dollars before the hurricane, were selling for as much as sixty dollars per sheet after the hurricane and milk was selling for up to six dollars per gallon. To combat this "price gouging," state and local officials enacted emergency laws prohibiting sellers from charging more than prehurricane prices. Four days after Andrew passed over South Florida, the federal government sent in troops to provide food and shelter.
When disasters such as hurricanes strike, most Americans expect the government to intervene where markets fail to provide basic necessities. In this century, the federal government has enacted emergency price controls during three wars, during a period of high inflation, and during periods of turmoil in the petroleum markets. The federal government also conducted major takings programs during both World Wars to obtain the materiel necessary for the prosecution of war.
When prices rise quickly, price controls are an appealing "quick fix." But it is well known that price controls are likely to result in shortages, queues, and black markets. Because price controls may not be an effective solution to market failures, the government may choose instead to buy or take (i.e., requisition) needed goods and distribute them to the public (or, as in wartime, consume them itself). Thus, takings and price controls are alternative modes of emergency market intervention.
Commenting on the release of Larry Peterson, one of Peterson's attorneys told a television reporter that witnesses lie but that DNA doesn't.
Do you suppose the attorney will say the same thing when the prosecution offers DNA evidence against a client represented by that attorney?
The statement is a nice example of rhetoric that trades on its literal accuracy but misleads. To wit: It is literally true that DNA cannot (as far as we know) intend to affirm the truth of a statement that it believes to be false -- or vice versa; but it is not true that DNA evidence is incapable of falsely pointing to innocence -- or, for that matter, guilt. See, e.g., Tillers on Evidence and Inference August 27, 2005.
Anticipate that these statements will be made duri... | {
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Hydrating Water
Vice confirms the importance of the judiciary reform and the elimination of corruption and corrupt it
Said the Integrity Commission, the parliamentary member Adel Nouri, Monday, on the importance of the judiciary reform and the elimination of corruption and corrupt it, noting that there are ministers and agents been sentenced by the court and when to change the prime minister a month later and the advent of the Prime Minister last dropped them all the verdicts. Said Nuri's / scales News / "The judicial power of one of the important joints in the state administration, and if you mess up the judiciary leads to the absence of any balance so it has been noted to corruption in the judiciary so much," adding that he "told the Chief Justice that the judicial system in Iraq politicized There is evidence conclusive that the judiciary and judicial rulings politicized ". He added that "corruption in the judiciary does not cover all judges, but there is evidence conclusive that the judiciary and judicial rulings march," noting that "there are arbitrary measures against certain things within the judiciary but there overlook so when sneaking corruption to the judicial system is rampant corruption in all aspects of the state." . Nuri demanded "the reform of the judicial system and the elimination of corruption and the corrupt," stressing "on joining the demonstrators who came out in front of the Supreme Judicial Council to demand the resignation of Judge Medhat al-Mahmoud."
He noted the "need for full and comprehensive change to eliminate," noting that "there is no corrupted files to Judge Mahmoud But there is jurisprudence from the elimination of politicized and there are provisions occurred under the pressure of favoritism and clientelism."
He said that "There was a person by the seven criminal convictions of major crimes but when arranged and put it in the era of former Prime Minister Nuri al-Maliki in the investigation did not last for minutes dropped him all the verdicts and seven in three days was sentenced each release," stressing that "with the Court of the Court and the judiciary and the law themselves. "
He said Nuri that "there are ministers and citizens, agents been sentenced by the court but when the change of prime minister after a month and the coming of the Prime Minister last dropped them all the verdicts," and wondered "why when the top of the pyramid is changing fall accused the charges." It ended 29 quarters e SOURCE
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West Indies levelled the series – 2nd ODI vs. India
Zia Rana November 24, 2013
West Indies 8-289 (Darren Sammy 63*, Lendl Simmons 62, Kieran Powell 59, Ravichandran Ashwin 2-37) beat India 7-288 (Virat Kohli 99, MS Dhoni 51*, Ravi Rampaul 4-60) by two wickets.
West Indies bounced back in the limited over game with their strong batting display, well led by Darren Sammy and won the 2nd ODI vs. India at Visakhapatnam.
The Blue Shirts managed 288 runs after losing seven wickets in 50 overs and the Caribbeans smashed 289 with two wickets remaining and three balls to spare.
Darren Sammy (West Indies) was declared 'Player of the match' for his match winning unbeaten knock of 63 off 45 mere balls.
The second One Day International of the three match series was played on Sunday, November 24, 2013, at Y.S. Rajasekhara Reddy ACA-VDCA Cricket Stadium, Visakhapatnam.
Darren Sammy – A match winning unbeaten innings of 63
Earlier, Dwayne Bravo, the skipper of the Windies, won the match and offered the batting to the hosts. The openers were gone for 69 as Yuvraj Singh and Virat Kohli joined at the wickets. They consolidated the innings while sending the ball out of the fence regularly and the latter dispatched his 28th ODI fifty. The partnership was dislodged at 138 when Yuvraj Singh lost his wicket for 28 and was replaced by Suresh Raina.
Raina was caught behind by Dwayne Bravo off Ravi Rampaul for 23 and Virat Kohli missed his ton by a single. MS Dhoni played an offensive knock of 51 and secured his end as the Men in Blue scored 7-288.
Ravi Rampaul was the star performer with four wickets whereas Jason Holder, Veerasammy Permaul and Darren Sammy shared one wicket each.
The Windies went down with a couple of wickets at 23 as Darren Bravo got together with Kieran Powell. The batsmen opted to attack, hit 12 fours in the coming overs and Bravo surrendered for 50. The consolidation process continued and Powell became the next one to grasp another half century to his credit. He was caught behind by Dhoni off Ravichandran Ashwin for 59 and the visitors were struggling at 5-185 in 34.4 overs.
Darren Sammy associated with Lendl Simmons who was sailing smoothly at the other end. The latter went past his fifty with a six while Sammy was punishing the bowlers at will. Simmons was declared leg before wicket off Ravindra Jadeja for 62 and Sammy powered his 6th ODI fifty in the process. Darren Sammy hit the winning run on the third delivery of the last over and secured his end for 63 off just 45 balls with 4 towering sixes and even fours. The Caribbeans hammered 289 runs for the loss of eight wickets and reached the target.
Bhuvneshwar Kumar, Mohammed Shami and Ravichandran Ashwin claimed two wickets each whereas Mohit Mishra and Ravindra Jadeja got one each.
West Indies triumphed in the interesting encounter with a margin of two wickets and squared the series at 1-1.
Tags: Darren Sammy, India, Kieran Powell, Lendl Simmons, MS Dhoni, ODI, Ravi Rampaul, Ravichandran Ashwin, Virat Kohli, West Indies
Pillars to level the ODI series for West Indies
Virat Kohli and Virender Sehwag powered India to easy win – 1st ODI vs. Sri Lanka
India outplayed West Indies – 1st Test
George Bailey powered Australia to another win – 2nd ODI vs. West Indies
All rounders won the game for Pakistan – 1st ODI vs. South Africa
Mitchell Johnson completed the demolition project- 1st Test vs. England
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Catch up on all the LATEST NEWS at Aspired Futures!
Come to our Charity Christmas Market - 24th November!
A successful bid for Comic Relief funding via the Community Foundation for Lancashire enabled Aspired Futures to establish its own gymnastics club GYMNASTICS. | {
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Lucrative outlook of automotive to resonate high demands is accounted to fetch concomitant growth in automotive lubricant market. Automotive lubricants play a defining role in ensuring optimal capabilities of engines. Innovative combustion engine designs also leverage considerable innovations, encouraging market participants to offer enhanced viscosity to enable improved functional attributes of engines. Superlative engine quality remains crucial for low vehicle emissions. Stringent emission control norms are likely to further augment greater adoption of vehicle lubricants, thereby entailing tremendous growth potential in automotive lubricant market.
The report is poised to equip readers with reliable details on market developments, in the ambit of competitive foresight and cues on market entry barriers. Based on such decisive insights, aspiring entrants as well as established players in automotive lubricant space can employ lucrative investment discretion to ascertain sustainable revenue pools amidst staggering competition. To provide readers with superlative understanding, the report is systematically clustered into coherent chapters. A dedicated chapter on market segmentation has been pinned in the trailing sections of the report to include veritable details on segments' historic as well as upcoming growth estimations.
This section of the report offers a detailed section on key contributors in automotive lubricant market. A dashboard view of each of the mentioned profiles complete with detailed insights on their respective SWOT analysis along with detailed assessment of their product portfolio, market contribution, as well as recent developments have been slated to aid readers' understanding about the competition spectrum. | {
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{"url":"http:\/\/rwanjohi.rbind.io\/2018\/04\/14\/generative-adversarial-networks\/","text":"$\\color{red}{\\text{ Draft: Work in Progress..... }}$\n\n\u2022 GAN consist of two competing models striving to outdo each other: the Generator and Discriminator models.\n\n\u2022 The Generator takes in random input and tries to generate real data (curves, images, texts, ).\n\n\u2022 The Discriminator is a binary classifier. It takes, as its input, the fake data generated by the Generator, and the real dataset and learn to tell whether the data is real or fake.\n\n\u2022 Through back propagation, the Generator updates its parameters using the output from the Discriminator.\n\n\u2022 The two models are trained simultaneously with the aim that the continuous competition will help produce data that is indistinguishable from the real data.\n\nDiagramatically\u2026\n\nGiven the objective function:\n\n$\\min_{G} \\max_{D} F(G, D) = \\log(D(x; \\phi)) + \\log(1- D( x^{'}; \\phi))$\n\nwhere:\n\n\u2022 $z \\sim p_{z}(z)$ is a random sample from distribution $p_{z}(z)$\n\n\u2022 $x^{\u2019} = G(z; )$\n\u2022 $x$ is the real data from distribution $p_{d}(x)$\n\n\u2022 Generator minimizes by maximazing log $p(y= true|x^{\u2019})$\n\u2022 Discriminator maximizes by maximizing log $p(y= fake|x^{'})$ and $p(y= true|x)$\n\nThe models are trained simultaneously with the goal of obtaining a Nash equilibrium.\n\nShare","date":"2020-02-21 22:12:00","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\": 3, \"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\": 0.4406163692474365, \"perplexity\": 1057.4084025925238}, \"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-2020-10\/segments\/1581875145538.32\/warc\/CC-MAIN-20200221203000-20200221233000-00462.warc.gz\"}"} | null | null |
Lenox es un lugar designado por el censo ubicado en el condado de Berkshire en el estado estadounidense de Massachusetts. En el Censo de 2010 tenía una población de 1.675 habitantes y una densidad poblacional de 355,73 personas por km².
Geografía
Lenox se encuentra ubicado en las coordenadas . Según la Oficina del Censo de los Estados Unidos, Lenox tiene una superficie total de 4.71 km², de la cual 4.68 km² corresponden a tierra firme y (0.55%) 0.03 km² es agua.
Demografía
Según el censo de 2010, había 1.675 personas residiendo en Lenox. La densidad de población era de 355,73 hab./km². De los 1.675 habitantes, Lenox estaba compuesto por el 96.6% blancos, el 0.36% eran afroamericanos, el 0.12% eran amerindios, el 0.6% eran asiáticos, el 0.12% eran isleños del Pacífico, el 0.9% eran de otras razas y el 1.31% pertenecían a dos o más razas. Del total de la población el 2.81% eran hispanos o latinos de cualquier raza.
Referencias
Enlaces externos
Lugares designados por el censo en Massachusetts
Localidades del condado de Berkshire | {
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\section{Introduction}
In this paper we are interested in possible origins of high-frequency
quasi-periodic oscillations (QPOs)
in neutron-star and black-hole low-mass X-ray binaries (LMXBs),
and additionally
in horizontal-branch QPOs in luminous neutron-star LMXBs (Z sources).
One of the important characteristics of kHz QPOs in neutron-star LMXBs
is that they usually occur in a pair, and their frequencies change
with time (see, e.g., van der Klis 2000, 2004 for reviews).
The time change occurs in such a way as their frequency ratio decreases
with an increase of frequencies.
In the case of black-hole LMXBs, high-frequency QPOs usually also occur
in a pair, but their frequencies change little with time,
the ratio being kept close to 2 : 3 (see van der Klis 2004 for a review).
In Z sources, another important characteristic of QPOs is present.
It is a strong correlation between kHz QPOs and horizontal branch QPOs
in their frequencies (Psaltis et al. 1999).
That is, the frequency of the lower
kHz QPO, $\nu_{\rm L}$, and that of the horizontal branch QPOs,
$\nu_{\rm HBO}$, are correlated
in each object so that $\nu_{\rm HBO}\sim 0.08
\nu_{\rm L}$.
Paying attention to the closeness to 2 : 3 of the frequency ratio in
high-frequency QPOs in balck hole LMXBs, Abramowicz and Klu{\' z}niak (2001)
and Klu{\' z}niak and Abramowicz (2001) presented a resonance model of
disk oscillations to explain the QPOs.
On the other hand, the importance of an external deformation of disks on
resonance processes
has been emphasized by Kato (2003, 2004a, 2004b), Klu{\' z}niak et
al. (2004), and Lee, Abramowicz, and Klu{\' z}niak (2004).
Kato (2003, 2004a, 2004b) especially considered models of resonant
oscillations on warped disks.
The models were extended to a case where the warp had precession
(Kato 2005).
In these warp models, the temperature is assumed to always
distribute isothermally in the vertical direction.
In real situations, however, this will not be the case.
The vertical structure of disks will change by the
difference of the disk state resulting from difference of the
mass-accretion rate.
Considering this, we assume in this paper that the pressure, density,
and temperature distributions in the vertical direction are polytropic, and
that the polytropic index changes with time.
By adopting this model, we can easily obtain the time variations
of the resonant oscillations as a result of changes of the polytropic
index.\footnote{In Kato (2005), time variations of QPOs was interpreted
as a result of time change of precession of warps.}
The purpose of this paper is to demonstrate this generalization and
to apply the results to QPOs.
In warped disks there are four types of resonant oscillations due to the
possible four combinations of two types of oscillations (g-mode and
p-mode oscillations)
and two types of resonances (vertical and horizontal resonances).
Without considering all cases, we consider here only a case where
the time variations of QPOs can be well explained.
This is the case where disk oscillations are g-modes,
and the resonances occur through vertical
motions.\footnote{ In Kato (2005), horizontal resonances of g-mode oscillations
were considered.}
\section{Vertical Resonances of G-Mode Oscillations in Warped Disks}
Details of the non-linear resonant oscillations on warped disks are
presented by Kato (2003, 2004a, 2004b).
An overview of the model is sketched in figure 1
of Kato (2004a).
We consider geometrically thin relativistic disks rotating with
angular velocity, $\Omega(r)$.
The epicyclic frequency on the disk is denoted by $\kappa(r)$.
The oscillations on geometrically thin disks are classified into g-mode
and p-mode oscillations (see, e.g., Kato et al. 1998; Kato 2001).
In simplified disks the oscillations are further sub-classified by the
set ($m$, $n$), where
$m=$(0, 1, 2...) is the number of nodes in the azimuthal direction, and
$n=$(0, 1, 2...) is the number related to nodes in the vertical direction.
That is, $n$ represents the number of nodes that $u_r$ (the radial
component of velocity associated with oscillations) has in
the vertical direction.
It is noted, however, that $u_z$ (the vertical component of velocity
associated with oscillations) has ($n-1$) nodes in the vertical direction.
In the case of $n=0$, $u_z=0$ and
a series of g-modes start from $n=1$.
A warp is generally a global pattern on disks with $m=n=1$.
The warp may have precession, but here we assume that it has no
precession in order to consider idealized cases.
On a disk deformed by the warp, we superpose g-mode oscillations with
arbitrary $m$ and $n$ ($n$ is taken to be 1 or 2, later).
A g-mode oscillation with the frequency $\omega$ and ($m$, $n$) has
a relatively large amplitude, global pattern only around the radius where
\begin{equation}
(\omega-m\Omega)^2-\kappa^2 = 0
\label{1.1}
\end{equation}
is satisfied.
This can be understood if the dispersion
relation for local perturbations is considered.
That is, the dispersion relation shows that the region of
$(\omega-m\Omega)^2-\kappa^2 >0$ is an
evanescent region of the g-mode oscillations.
In the region where
$(\omega-m\Omega)^2$ is smaller than $\kappa^2$, on the other hand,
the oscillations have very short wavelengths in the radial direction
when geometrically thin disks are considered.
A non-linear interaction of a g-mode oscillation with the warp
produces an oscillation with $\omega$, ${\tilde m}$,
and ${\tilde n}$, where ${\tilde m}=m\pm 1$ and ${\tilde n}=n\pm 1$
(these oscillations are called hereafter intermediate oscillations).
These intermediate oscillations resonantly interact with the disk at
the radius where
the dispersion relation for these intermediate oscillations is
satisfied [see Kato (2004b) for detailed discussions].
There are two types of resonances, corresponding two types of wave modes.
One is resonances that occur through motions in the vertical direction
(vertical resonances);
the other is those through motions in the radial direction
(horizontal resonances)(see, e.g., Kato 2004a,b).
In this paper we are interested in vertical resonances.
They occur
around the radius where
\begin{equation}
(\omega-{\tilde m}\Omega)^2-\Psi_{\tilde n} \Omega^2 \sim 0
\label{1.2}
\end{equation}
is satisfied (see e.g., Kato 2004a, b, and the Appendix in this paper), where
$\Psi_{\tilde n}$ is a number related to
${\tilde n}$ and $N$ (a polytropic index specifying the vertical
structure of disks, see the Appendix).
Hereafter, we are particularly interested in cases where ${\tilde n}=2$
(assuming $n=1$) and ${\tilde n}=3$ (assuming $n=2$).
The quantities $\Psi_{\tilde n}$ in these cases are
\begin{equation}
\Psi_2=2+{1\over N} \quad {\rm and}\quad \Psi_3 =3+{3\over N},
\label{1.3}
\end{equation}
as shown in the Appendix.
In the limit where the disk is isothermal in the vertical direction,
$N=\infty$ and we have $\Psi_2=2$ and $\Psi_3=3$.
They are cases that have been considered in previous papers (Kato 2003,
2004a, 2004b).
Combining equations (\ref{1.1}) and (\ref{1.2}), we find that the resonances
occur at radii of
\begin{equation}
\kappa=(\Psi_2^{1/2}-1)\Omega \quad {\rm and} \quad
\kappa=(\Psi_3^{1/2}-1)\Omega,
\label{1.4}
\end{equation}
for ${\tilde n}=2$ and ${\tilde n}=3$, respectively.
After this resonance the intermediate oscillations feedback to the
original oscillations by non-linear interaction with the original
oscillations themselves.
The results are amplification or dampening of the original oscillations,
depending on the types of oscillations and resonances (Kato 2004b).
Detailed examination shows that when ${\tilde m}=m-1$,
the frequencies of the g-mode oscillations that resonantly
interact with the disk at $\kappa=(\Psi_{\tilde n}^{1/2}-1)\Omega$ are
$m\Omega-\kappa$ at the radius.
On the other hand, when ${\tilde m}=m+1$ the frequencies of the g-mode
oscillations that resonantly interact with the
disk at $\kappa=(\Psi_{\tilde n}^{1/2}-1)\Omega$ are
$m\Omega+\kappa$ there (e.g., Kato 2004a).
The most observable oscillations are those with small $m$'s;
otherwise, they are phase-mixed.
Hence, the frequencies, $\omega$, of
typical non-axisymmetric resonant oscillations are
$\Omega-\kappa$ (i.e., $m=1, {\tilde m}=0)$, $\Omega+\kappa$
(i.e., $m=1, {\tilde m}=2)$, and $2\Omega-\kappa$
(i.e., $m=2, {\tilde m}=1)$.
As axisymmetric resonant oscillations we have $\kappa$
(i.e., $m=0, {\tilde m}=1)$.
Axisymmetric oscillations, however, will be less observable.
Hence, we think that the above non-axisymmetric oscillations
are related to the horizontal branch QPOs, upper and lower kHz QPOs.
Considering this, we introduce
\begin{equation}
\omega_{\rm H}=\Omega+\kappa, \quad \omega_{\rm L}=2\Omega-\kappa, \quad
{\rm and} \quad \omega_{\rm HBO}=\Omega-\kappa,
\label{1.5}
\end{equation}
and examine their frequencies at the resonance radius.
\section{Frequencies and Their Variations of Resonant Oscillations}
In the limit of isothermal disks in the vertical direction, i.e.,
$N=\infty$, the resonance occurs at $\kappa=(\sqrt{2}-1)\Omega$
when ${\tilde n}=2$ and at $\kappa=(\sqrt{3}-1)\Omega$ when
${\tilde n}=3$.
These radii are $3.62r_g$ and $6.46r_g$, respectively (e.g., Kato 2004a).
The temperature distribution in the vertical direction in disks is, however,
generally not isothermal.
Furthermore, it changes with time, depending on the state of the disks.
In figure 1, the $r/r_g$--$\Psi$ relation given by
$\kappa=(\Psi^{1/2}-1)\Omega$ is shown for the range of
$\Psi=2.0$--$4.0$.
In the case of ${\tilde n}=2$, $\Psi$ (i.e., $\Psi_2$) changes from 2
(for $\gamma=1$ or $N=\infty$)
to 2.67 ($\gamma=5/3$ or $N=1.5$), while $\Psi_3$ changes from 3
($\gamma=1$) to 5 (for $\gamma=5/3$) in the case of ${\tilde n}=3$.
The ranges of variations of $\Psi_2$ and $\Psi_3$ are also shown in
figure 1.
If the value of $\Psi$ is specified, the resonant radius is obtained.
Then, from equation (\ref{1.5}), the frequencies of the resonant oscillations,
$\omega_{\rm H}$, $\omega_{\rm L}$, and
$\omega_{\rm HBO}$ are derived as functions of $\Psi$.
The relations among these frequencies are shown in figure 2 as
functions of $\omega_{\rm H}$.
The relation between $2\omega_{\rm HBO}$ and $\omega_{\rm H}$ is
also shown.
Important points are that: i) figure 2 is free from the mass of
the central source, if the both horizontal and vertical scales are
normalized by $(M/M_\odot)^{-1}$, and
ii) the curves in figure 2 are universal as long as g-mode oscillations
are concerned.
They are free from $N$ (and $\Psi$), and even free from the presence or
absence of precession (see Kato 2005).
That is, the frequencies of the resonant oscillations change along
the curves when $N$ (and thus $\Psi$) is changed.
The relation between $\omega_{\rm H}$ and $\Psi$ is shown in figure 3.
In general, an increase of $\Psi$ decreases $\omega_{\rm H}$.
The range of the variation of $\omega_{\rm H}$ by a change of $\Psi$ is also
shown in figure 2.
\section{Summary and Numerical Estimate}
The main results of this paper are summarized in figure 2.
The figure should be compared with figure 2.6 of van der Klis (2004),
which summarizes observational data concerning QPO frequencies
on a frequency--frequency diagram.
This comparison suggests that the resonant oscillations specified by
$\omega_{\rm H}$, $\omega_{\rm L}$, and $\omega_{\rm HBO}$ well
correspond to the upper and lower kHz QPOs and the horizontal branch QPOs
in neutron-star LMXBs, respectively.
Adopting these identification, we qualitatively explain
some basic observational characteristics of the QPOs.
The kHz QPOs in neutron-star LMXBs usually appear in a pair, and
the separation frequency of the twin peaks decreases as the peak frequencies
increase.
This observational characteristic is derived in our model, as
shown in figure 2 (see the curves of $\omega_{\rm L}$ and $\omega_{\rm H}$).
In our model the frequency change of QPOs is the result of a change of
the temperature distribution in the vertical direction.
As the temperature distribution in the vertical direction approaches to
an isothermal one [i.e., $\Psi_2$ (the case of ${\tilde n}=2$) tends to
2.0 or $\Psi_3$ (the case of ${\tilde n}=3$) tends to 3.0],
the resonance radius becomes smaller (see figure 1) and
the frequencies of resonant oscillations increase.
Observations show that the frequencies of the pair QPOs increase
with an increase of the
mass-accretion rate (van der Klis et al. 1997).
Hence, if our model is correct, it suggests that the temperature
distribution in the vertical direction approaches to the isothermal one
as the mass-accretion rate increases.
Here, let us make a quantitative estimate.
Figure 2 shows that the 3 : 2 ratio of $\omega_{\rm H}$ and
$\omega_{\rm L}$ is realized when
log$[\omega_{\rm H}(M/M_\odot)]\sim 2.93$,
i.e., $\omega_{\rm H}=851(M/M_\odot)^{-1}$.
This occurs for $\Psi_3=3.24$ (i.e., ${\tilde n}=3$), as shown in figure 3.
This gives $1/N=0.08$ and $\gamma=1.08$.
The resonance radius is then at $8.33r_g$, as shown in figure 1.
One of another prominent correlations among QPO frequencies in
neutron-star LMXBs is that the frequencies of the horizontal-branch QPOs,
$\nu_{\rm HBO}$,
are correlated with the frequencies of lower kHz QPOs, $\nu_{\rm L}$,
by $\nu_{\rm HBO}\sim 0.08\nu_{\rm L}$.
In order to compare our rersults with the observations,
the curve of the $0.08\omega_{\rm L}$--$\omega_{\rm H}$
relation is drawn in figure 2.
The curve crosses the curve labelled by $\omega_{\rm HBO}$ around
log$[\omega_{\rm H}(M/M_\odot)]\sim 2.46$,
i.e., $\omega_{\rm H}=288(M/M_\odot)^{-1}$.
Figure 3 then shows $\Psi_3\sim 3.66$, which means
$3/N\sim 0.66$ or $\gamma\sim 1.22$.
This implies that the observed correlation can be explained if the
temperature distribution in the vertical direction is, on average,
around $\gamma\sim 1.2$.
From figure 1 we see that the resonance occurs around
$r=18.0r_g$ when $\Psi_3=3.66$.
As shown above, both $\omega_{\rm H}$ : $\omega_{\rm L} =$ 3 : 2 and
$\omega_{\rm HBO}=0.08\omega_{\rm L}$ cannot be simultaneously
satisfied in a rigorous sense.
A slight larger coefficient than 0.08, say 0.09, is compatible with
the ratio 3 : 2.
In the case of black-hole LMXBs, observations show that the
frequencies of pair QPOs change little, and their ratio is kept to be
close to 3 : 2, unlike the case of neutron-star kHz QPOs.
As discussed in the case of neutron-star QPOs, one possibility
of explaining the
observed 3 : 2 is that it represents $\omega_{\rm H}$ :
$\omega_{\rm L}$.
Another one, which is better, is to consider the resonances of
${\tilde n}=2$ and to regard the ratio as $\omega_{\rm L}$ :
$2\omega_{\rm HBO}$.
As discussed in Kato (2004b), if we consider the resonance at $4.0r_g$,
$\omega_{\rm H}$ is equal to $\omega_{\rm L}$ and the ratio
$\omega_{\rm L}$(or $\omega_{\rm H}$) : $2\omega_{\rm HBO}$ is just
3 : 2.
Figure 1 shows that the resonance at $4.0r_g$ is realized
for ${\tilde n}=2$ when $\Psi_2=2.25$,
which means $1/N=0.25$ (i.e., $\gamma=4/3$) in the present model.
Figure 2 (see also figure 3) shows that this occurs when
log$[\omega_{\rm H}(M/M_\odot)]=3.33$, i.e., $\omega_{\rm H}=
2.14\times 10^3(M/M_\odot)^{-1}$.
It is noted that in the case where the resonance occurs through
intermediate oscillations
of ${\tilde n}=2$ (not ${\tilde n}=3$), the frequency variation for a
change of $N$ (or $\gamma$) is weak (see the variation range of $\Psi_2$
in figure 2).
Furthermore, $\omega_{\rm H}=2.14\times 10^3(M/M_\odot)^{-1}$
is compatible with observed results derived by McClintock and
Remillard (2003).
It is, however, unclear why the balck-hole QPOs are
the resonances of ${\tilde n}=2$ $(n=1)$, while the neutron-star
QPOs are the resonances of ${\tilde n}=3$ $(n=2)$.
More quantitative comparisons of our model with observations would be
premature at the present stage, since the frequencies of vertical
resonances are sensitive to the vertical structure of the disks
(the vertical structure is not always polytropic).
In other words, if our present model is correct, a comparison of our results
with observations would give good information concerning the vertical
structure of disks.
\section{Discussion}
The present resonance model naturally explains some basic characteristics
of the kHz QPOs of neutron-star LMXBs and the high-frequency QPOs of
black-hole LMXBs.
It is especially noted that the observed frequency change of kHz QPOs,
which is observationally related to a change of the mass-accretion rate, is
naturally explained if the vertical structure of disks changes with
the change of the mass-accretion rate.
In this sense, the present model seems to be superior to a precession
model of warps (Kato 2005).
In the latter model, a time variation of precession is required to
explain the time variation of the observed kHz QPO frequencies, and
it is unclear whether such a variation of precession is generally expected
theoretically.
The latter model with precession, however, can naturally explain the
observed hectohertz QPOs as a manifestation of the precession of warps.
In the present model of the vertical resonances, on the other hand,
there is a problem concerning excitation.
The vertical resonances do not excite the g-mode
oscillations, but rather dampen them in the limit of $N=\infty$
(Kato 2004b).
We should carefully study in the future whether our previous results concerning
the stability of resonancs are correct and relevant, even when $N\not=\infty$.
Comparisons of the characteristics of the present model with those of the
precession model are given in table 1.
Many QPO models have been proposed so far.
Some of them connect the observed QPO frequencies with the characteristic
frequencies of disks, such as the orbital frequency, radial and vertical
epicyclic frequencies and others, or their combinations.
For example, in their precession model, Stella and Vietri (1998)
identify the periastron precession frequency, $\Omega-\kappa$
[see the third equation in (\ref{1.5})], with the lower frequency of
the kHz QPO.
An important issue in such models is where preferred radii to produce
specific frequencies exist.
Concerning this point, these models are classified into various types,
e.g., precession model (Stella, Vietri 1998), resonance model
(Klu{\' z}niak, Abramowicz 2001; Abramowicz, Klu{\' z}niak 2001), and
beat-frequency model (Miller et al. 1998).
Our warp model is based on hydrodynamical wave phenomena and different from
the models mentioned above.
However, if we discuss the present model in connection with them, a warp
can be regarded as a process to select preferred radii.
Finally, the applicability of the present model to cataclysmic variables
(CVs) is briefly discussed.
Mauche (2002) pointed out that the frequency correlation $\nu_{\rm
HBO}\sim 0.08 \nu_{\rm L}$ can be extended to CVs.
This has recently been confirmed by Warner and Woudt (2004).
That is, the DNO (dwarf novae oscillation)--QPO relation in
CVs is on the line of extension of the HBO--kHz QPO relation in X-ray
stars.
Furthermore, it is known that the DNOs in CVs change their frequencies,
accompaning harmonics (Warner, Woudt 2004).
The disks of CVs are Keplerian and $\Omega$ and $\kappa$ are almost
equal.
Hence, at a glance, the resonance conditions, i.e., equations (\ref{1.4}),
seem not to be realized anymore.
This is, however, not the case.
The observations show that DNOs usually occur in the phase of outbursts.
Near the transition front of outbursts, the disk thickness changes
sharply.
In such region, the derivation
of the resonance condition, $\kappa=(\Psi_3^{1/2}-1)\Omega$, will be
inaccurate, since the assumption of slow radial change of $H/r$ is
involved in the derivation.
If we assume, however, that the resonance condition,
$\kappa=(\Psi_3^{1/2}-1)\Omega$,
is still valid, the resonances occur even in the Newtonian Kepler disks, if
$\Psi_3\sim 4$.
Since $\Psi_3=3+3/N$, $\Psi_3=4$ is realized when $1/N=1/3$, i.e.,
$\gamma=4/3$.
(As mentioned before, $\Psi_3=3$ for $\gamma=1$ and $\Psi_3=5$ for
$\gamma=5/3$.)
A vertical disk structure with $N=3$ will not be unrealistic.
This consideration suggests that the harmonic structure of DNOs
can be interpreted as the oscillations of $2\Omega+\kappa$,
$\Omega+\kappa$, and $\kappa$, at the resonant radius.
Their frequency ratios are 3 : 2 : 1 in the Newtonian Kepler disks.
The observed frequency changes of DNOs are interpreted as being the result
of a change of the
resonance radius by a change of the disk structure.
[See also Klu{\'z}niak et al. (2005) for an interpretation of DNOs.]
The observed correlation between DNOs and QPOs in CVs, however, cannot be
explained by the present model.
Further considerations are needed.
\begin{longtable}{ccc}
\caption{Comparison of two resonant models of g-mode oscillations.}
\label{tab:LTsample}
\hline\hline
& Horizontal resonances (Kato 2005) & Vertical resonances
(present paper) \\
\hline
\endhead
\hline
\endfoot
\hline
\endlastfoot
Time variation & change of precession & change of vertical disk structure \\
Hectohertz QPOs & precession & precession ? \\
Excitation & $\bigcirc$(probably) & ? \\
\end{longtable}
| {
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An April 21st letter penned by David Balto, a Senior Fellow at American Progress who focuses on competition policy, intellectual property law and health care, and Michael Carrier, a professor at the Rutgers University School of Law-Camden, and sent to five U.S. Senators – the same cohort of Senators who signed a March 9th letter to FDA Commissioner Hamburg concerning generic LIPITOR (atorvastatin calcium) Tablets (Sens. Tom Harkin (D-IA), Jay Rockefeller (D-WV), Charles Schumer (D-NY), Debbie Stabenow (D-MI) and Sherrod Brown (D-OH) – takes issue with the "use it or lose it" 180-day exclusivity forfeiture system established by the 2003 Medicare Modernization Act ("MMA").
The letter does not concern 180-day exclusivity for generic LIPITOR, which is governed by the pre-MMA version of the FDC Act, but rather uses two sentences in the letter – i.e., "In 2003, Congress amended the [FDC Act] to change provisions concerning 180-day generic drug exclusivity. Specifically, the law was changed to a "use it or lose it" system that prevents a first filer's exclusivity from being indefinitely "parked" and creating a bottleneck to generic competition" – to springboard into a discussion of the effectiveness of post-MMA 180-day exclusivity forfeiture, and specifically the so-called "failure-to-market" provisions at FDC Act § 505(j)(5)(D)(i)(I), under which the "later of" date of the two bookend events described at FDC Act § 505(j)(5)(D)(i)(I)(aa) and (bb) controls the date of 180-day exclusivity forfeiture.
The widely held view by many policymakers working to end the "pay for delay" problem is that "reverse payments" from the brand to the generic in these arrangements are the cause of the problem. That is not the case. The cause of the problem is the ability of the first generic challenger to retain exclusivity even if it settles its patent challenge, and the related lack of incentive that subsequent generic challengers have to continue the patent fight when the first filer has settled and "parked" its exclusivity.
Inherent in the structure of the "failure to market" forfeiture provisions is the possibility that a first applicant would be able to enter into a settlement agreement with the NDA holder or patent owner in which a court does not enter a final judgment of invalidity or non-infringement (i.e., without a forfeiture event under subpart (bb) occurring), and that subsequent applicants would be unable to initiate a forfeiture with a declaratory judgment action. This inability to force a forfeiture of 180-day exclusivity could result in delays in the approval of otherwise approvable ANDAs owned by applicants that would market their generic drugs if they could but obtain approval. This potential scenario is not one for which the statute currently provides a remedy.
stems from the MMA's requirement that subsequent generic filers must prevail through the appellate court level on the same set of patents that the first-filer has filed against qualifying it for exclusivity in order for the first-filer to be put in the "use it or lose it" position of having to launch its product or forfeit its exclusivity. Thus, to preserve bottlenecks created by settlements in which the first-filer parks its exclusivity, brand companies either do not sue subsequent generic filers at all, or sue them on some but not all patents. This forces subsequent generic challengers to pursue DJs in order to litigate the patents they haven't been sued on but must win a judgment against in order to put the first-filer in the "use it or lose it" position. If subsequent generic filers cannot win the needed judgment(s), they cannot achieve the very action that the "use it or lose it" system hinges upon, ensuring the market will remain blocked by the "parked" exclusivity.
generic challengers are by no means guaranteed to get DJs today when brand companies do not sue. There are numerous post-MedImmune cases in which subsequent generic filers have been denied DJs. Moreover, even if generics do get DJs, there is no guarantee that a subsequent filer who is willing to pursue the case to conclusion can achieve a court victory quickly enough to open the market anyway. This process leaves subsequent filers at the mercy of brand companies whose sole objective is to sustain the bottleneck for as long as possible. The inadequate nature of the MMA's DJ provision ensures that the legal process will take so long that the clock simply runs out on subsequent generic filers fighting to open the market earlier than the date agreed to by the first filer in its "parked" exclusivity settlement.
Balto and Carrier do not discuss in the letter the ways in which the issues they raise can or should be addressed by Congress. Instead, they note articles (see e.g., here and here) they have published on these issues and offer their assistance to the Senators.
We note that in 2009 and 2010 legislation was introduced that appears to have been intended to address, at least in part, issues raised in the Balto/Carrier letter. Specifically, the Drug Price Competition Act (see our previous post here) would have amended the definition of "first applicant" at FDC Act § 505(j)(5)(B)(iv)(II)(bb) with respect to 180-day exclusivity eligibility so that certain subsequent ANDA applicants could trigger and also be eligible for such exclusivity. We are not aware of any current efforts to resurrect this legislation. | {
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Counters in CSS
============
PL: Liczniki dla CSS
http://webroad.pl/html5-css3/640-liczniki-w-css
| {
"redpajama_set_name": "RedPajamaGithub"
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"I Don't Belong" is a song by Irish band Fontaines D.C. from their third studio album, A Hero's Death (2020). The song was written by Carlos O'Connell, Conor Curley, Conor Deegan III, Grian Chatten and Tom Coll, while production was handled by Dan Carey. It was released on 9 June 2020 by Rough Trade and Partisan Records as the second single from the album.
A mid-tempo post-punk song, "I Don't Belong", is about dismissing expectations imposed by other people onto themselves. The production of the song features needeled electric guitars, a bass guitar, and drums, with elements of psychedelic and surf rock. Thematically, "I Don't Belong" discusses various scenarios that trace back to the message of the track. The opening is about soldier who has his military service commended by his country, but rejects the medal as the soldier sees the acceptance of the medal as compromising their personal principles. The second verse focuses about a fight breaking out in a bar, due to onself refusing to be enticed by friendliness or kindness.
Upon the single's release, "I Don't Belong" received critical acclaim from contemporary music critics, with some in retrospect, calling it one of the most musically mature singles on the album. Praise was directed at Chatten's singing and the overall song structure, and Carey's production of the track.
Commercially, "I Don't Belong" was the band's second single to chart, the first being the self-titled track for the album, which was released earlier that spring. The song peaked at 86 in the Irish Singles Chart. One music video was made to accompany the track, which was directed by Deegan III and produced by Hugh Mulhern, was released on 10 June. The visual depicts lead singer, Chattan, singing across the Irish countryside while going on a walk.
Release and promotion
Live performances
The song was released during the worldwide COVID-19 pandemic, limiting the band's ability to perform the song live or in front of a live audience. Due to the pandemic and related vaccination rollout, the first live performance of the song was during a livestream on 23 November 2020 at the O2 Academy Brixton. The band did not perform the song in front of a live audience until 2 August 2021, when they performed at the Pryzm nightclub in Kingston upon Thames, United Kingdom.
An acoustic version of the song premiered on 16 November 2020. It was filmed in October 2020 at La Blogothèque in Paris.
Commercial performance
"I Don't Belong" was the band's second single to chart in a commercial chart, reaching 86 in their home nation of Ireland in the Irish Singles Chart.
Music video
A corresponding music video for the song was released on 10 June 2020. The music video was shot in March and April 2020 with in Skerries, Dublin, Ireland, the hometown of lead singer, Grain Chatten. The video, directed by bassist Conor Deegan III, shows Chatten singing walking past numerous sites in Skerries before ultimately being submerged in the water on the coast. The music video, originally planned to be shot in London, was shot in Skerries due to the onset of the COVID-19 pandemic. Basssit Connor Deegan III explained that the band "had a whole different video shoot planned in London" for the track. The purpose of the filming in Skerries was to "use the limitations to our benefit. To try to match the emotional feel of the song but without being heavy-handed. Telling a visual story parallel to the lyrics."
Deegan further told NME that Chatten's character "goes on a walk, passing by all these places. He's telling us this dramatic story that slowly grows as the verses go on. At the start you do and you don't really get a sense of the anxiety or negativity to come. It's buried under the surface. At the end he submerges himself, completely consumed by his inner thoughts finally coming out into the world."
Charts
Credits and personnel
Credits adapted from A Hero's Death album liner notes.
Grian Chatten vocals, tambourine
Carlos O'Connell guitar
Conor Curley guitar, piano
Tom Coll drums, percussion
Conor Deegan bass guitar
Dan Carey production, mixing
References
2020 songs
2020 singles
Partisan Records singles
Fontaines D.C. songs | {
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Incorporated in 1986, SiTEST Ltd has steadily grown from a dedicated service provider of back-end test equipment into one of the leading microelectronic and semiconductor equipment distributors in the UK & Ireland.
Located in Dorset on the edge of the New Forest, we are fortunate to own Industrial Units, which have been converted into comfortable offices, giving us easy access to move equipment from Suppliers out to our Customers.
With all our equipment that we supply, we fully support not only in terms of Installing, commissioning, training and warranty but also Customer Engineering using the latest CAD/CAM software and local machine shops. | {
"redpajama_set_name": "RedPajamaC4"
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Topolo falu Horvátországban, Dubrovnik-Neretva megyében. Közigazgatásilag Dubrovačko primorje községhez tartozik.
Fekvése
A Dubrovnik városától légvonalban 42, közúton 59 km-re, községközpontjától légvonalban 20, közúton 25 km-re északnyugatra a tengermelléken, a hercegovinai határ közelében fekszik. Fontosabb településrészei: Donja és Gornja Banda, Klačina és Matići. Ránézésre tipikus tengermelléki település kőházakkal, kőlépcsőkkel, mezején juhok legelnek és évszázados fenyőerdő övezi.
Története
Topolo területe már ősidők óta lakott. Az itt élt első ismert nép az illírek voltak, akik az i. e. 2. évezredtől fogva éltek itt magaslatokon épített erődített településeken és kövekből rakott halomsírokba temetkeztek. Illír erődített település állt egykor a településtől északnyugatra emelkedő Veliki Lukovac nevű magaslaton. Halomsírjaikból jó néhány megtalálható Topolo terültén, többek között a Veliki Lukovacon, Rajdušán, Sokolova Grudán és több helyen a határában. Az illírek i. e. 35-ig uralták a térséget, amikor Octavianus hadai végső győzelmet arattak felettük. A Nyugatrómai Birodalom bukása után 493-tól a keleti gótok uralták a területet. 535-ben Dalmáciával együtt a Bizánci Császárság uralma alá került. A horvátok ősei a 7. században érkeztek Dalmáciába és csakhamar megalapították első településeiket. Ezek elsőként a termékeny mező melletti, ivóvízzel rendelkező helyeken alakultak ki. A Dubrovniki tengermellék hét plébániájával Zahumljéhez tartozott.
1399-ben a Dubrovniki tengermellék a Raguzai Köztársaság része lett, amely 1500 aranydukátért megvásárolta Ostoja bosnyák királytól. Topolo plébániáját 1620-ban alapították, ezt a rangját 1850-ben elveszítette, de 1951-től újra önálló plébánia. A Raguzai Köztársaság bukása után 1806-ban Dalmáciával együtt ez a térség is a köztársaságot legyőző franciák uralma alá került, de Napóleon bukása után 1815-ben a berlini kongresszus Dalmáciával együtt a Habsburgoknak ítélte. 1857-ben 237, 1910-ben 332 lakosa volt. 1918-ban az új szerb-horvát-szlovén állam, majd később Jugoszlávia része lett. A délszláv háború során 1991. október 1-jén kezdődött a jugoszláv hadsereg (JNA) támadása a Dubrovniki tengermellék ellen, mely kifosztotta és felgyújtotta a települést. A háború után rögtön elkezdődött az újjáépítés. 1996-ban újabb csapásként földrengés okozott súlyos károkat. 1997-ben megalakult Dubrovačko primorje község, melynek Topolo is része lett. A településnek 2011-ben 154 lakosa volt, akik főként mezőgazdaságból, állattartásból, szőlőtermesztésből éltek.
Népesség
Nevezetességei
A Kisboldogasszony plébániatemplom a 17. században épült barokk stílusban. Egyhajós épület, kapuzata felett lunettával, homlokzatán körablakkal, legfelül három harang számára épített harangdúccal. Az 1996-os földrengés után helyre kellett állítani. A templom mellett ősi, 15. századi temető található, amely még ma is használatban van.
A Szent Lujo (Alojzije Gonzaga, azaz Gonzaga Szent Alajos) templom 1882-ben épült fogadalmi kápolnaként. Mellette középkori temető maradványai találhatók.
A kuti Szent István templom a 9. – 11. században épült ószláv templomként. 1671-ben említik először és még a 17. században átépítették, de falain láthatók az ószláv ornamentika elemei. 2011/12-ben megújították. Egyhajós, kelet-nyugati tájolású épület, félkör alakú apszissal. A templom nyugati homlokzatát háromszögű oromzat tagolja, amely fölött egyszerű kő harangdúc áll, tetején íves áthidalóval. A portáltól délre egy egyszerű szenteltvíztartó van beépítve. A templom északi homlokzatát középen téglalap alakú ablaknyílás töri át. A templom minden homlokzata vakolt, a tetőzete nyeregtető. A templom belseje dongaboltozatos, durva faragású kőlapokkal burkolt. Térszerkezete, stilisztikai jellemzői és a liturgikus kőbútorzat alapján nagyjából a templom mai formájában a 17-18. századi időszakra tehető.
Római villa rustica maradványai
Topolotól északnyugatra Veliki Lukovacon ókori erődített település romjai találhatók.
Ókori halomsírok Veliki Lukovacon, Rajdušán, Sokolova Grudán.
Gazdaság
A helyi gazdaság a mezőgazdaságon alapul, mellette az állattartás és a szőlőtermesztés jelentős.
Oktatás
A település alsó tagozatos iskolája már nem működik, a helyi tanulók Smokovljanira járnak iskolába.
Jegyzetek
Források
Dubrovačko primorje község hivatalos oldala
A dubrovniki püspökség honlapja – Župa Male Gospe Topolo
A község turisztikai irodájának honlapja
Registar kulturnih dobara DNŽ Dubrovnik, 2017.
További információk
A dubrovniki püspökség honlapja
Dubrovniki turistakalauz
Dubrovnik-Neretva megye települései | {
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Wholesome dwelling begins right now. It could be extra appealing to comply with a trendy food plan or choose up the latest buzzy health e-book on what to eat, but the reality? The very best thing you are able to do for your body is to eat mostly crops. It is not an attractive reply by any means, however they're the holy grail of well being: You will be getting a wholesome dose of protein, fiber, wholesome fat, nutritional vitamins, and nutrients—and you won't be consuming empty energy in the course of.
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In early October, 46 adults and students met at a central location as an alternative of heading straight to high school. The thrill was evident as law enforcement officials turned on their lights, the native newspaper snapped photos and Clarendon's first Stroll to College Day kicked off with a mayoral proclamation. Two college students even rode house on model new bikes that day. The occasion spurred a movement in the school and group. The Clarendon Healthy Consuming, Energetic Living (HEAL) coalition worked with the town to calm visitors alongside roads with paint crosswalks and pace bumps. Later that yr, one coalition member stopped to speak to a railroad repairman at work. That same day, the shoulder was widened for youth to cross the railroad tracks safely. With these enhancements, now all 2500 residents can stroll and bike safely by means of the area.
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\section{Introduction} \label{sec:Introduction}
Antiferromagnets (AFMs) have promising applications in high-density spintronic devices due to their excellent properties, such as the absence of stray fields, robustness to magnetic field perturbation, and ultrafast spin dynamics~\cite{Jungwirth2016,Baltz2018}. Except for common collinear AFMs, the noncollinear magnetic orders are also widely seen in AFMs. The noncollinear AFMs can be further classified into coplanar and noncoplanar ones, which are characterized by vector and scalar spin chiralities~\cite{Kawamura2001},
\begin{eqnarray}
\boldsymbol{\kappa} &=& \sum_{<ij>}\mathbf{S}_{i}\times\mathbf{S}_{j}, \label{eq:kappa}\\
\chi &=& \sum_{<ijk>}\mathbf{S}_{i}\cdot\left(\mathbf{S}_{j}\times\mathbf{S}_{k}\right), \label{eq:chi}
\end{eqnarray}
respectively, where $\mathbf{S}_{i}$, $\mathbf{S}_{j}$, and $\mathbf{S}_{k}$ are the spins on neighboring sublattices. Recently, various unexpected physical phenomena were reported in noncollinear chiral AFMs. For example, the magneto-optical Kerr and Faraday effects (MOKE and MOFE), which are usually known to be linear in magnetization $\mathbf{M}$, are believed non-existent in AFMs due to their zero net magnetization. Nevertheless, with the allowance of special magnetic symmetries, MOKE and MOFE have been theoretically predicted or experimentally observed in coplanar noncollinear AFMs Mn$_3X$ ($X$ = Rh, Ir, Pt)~\cite{WX-Feng2015}, Mn$_3Y$ ($Y$ = Ge, Ga, Sn)~\cite{Higo2018,Balk2019,MX-Wu2020}, and Mn$_3Z$N ($Z$ = Ga, Zn, Ag, Ni)~\cite{XD-Zhou2019a} as well as in noncoplanar AFMs $\gamma$-Fe$_{0.5}$Mn$_{0.5}$~\cite{WX-Feng2020} and K$_{0.5}$RhO$_{2}$~\cite{WX-Feng2020}. The MOKE and MOFE unveiled in noncoplanar AFMs were termed \textit{topological} magneto-optical effects as they originate from scalar spin chirality instead of spin-orbit coupling and band exchange splitting, fundamentally distinguishing them from conventional magneto-optical effects~\cite{WX-Feng2020}.
The MOKE and MOFE that appeared in AFMs with zero net magnetization can still be regarded as the linear (first-order) magneto-optical effects since they are odd in $\hat{\mathbf{N}}=\mathbf{N}/|\mathbf{N}|$, where $\mathbf{N}$ is the N\'{e}el vector. That means, the signs of MOKE and MOFE will change if the direction of $\hat{\mathbf{N}}$ reverses. One would naturally concern about whether the nonlinear (e.g., second-order) magneto-optical effects exist in noncollinear chiral AFMs. Among the second-order magneto-optical effects, the Voigt~\cite{Voigt1908} and Sch\"{a}fer-Hubert~\cite{Schafer1990} effects are two representative ones, which refer to the rotation of the polarization plane of a linearly polarized light normally transmitted through and reflected from an in-plane magnetized material, respectively (Fig.~\ref{fig:model}). It should be noted that the terminologies used for magneto-optical Voigt and Sch\"{a}fer-Hubert effects (MOVE and MOSHE) in the literature are discrepant --- some measurements in a reflection geometry are called MOVE but not MOSHE and the MOVE is sometimes denoted as Cotton-Mouton effect or magnetic linear birefringence or magnetic linear dichroism. Very recently, the time-resolved MOVE (actually measured in a reflection geometry) has been observed in a coplanar noncollinear AFM Mn$_3$Sn using the pump-probe experimental technique, and the modulated Voigt angle is surprisingly one order of magnitude larger than the Kerr angle~\cite{HC-Zhao2021}. However, so far, little is known about the MOVE and MOSHE in noncoplanar AFMs.
The scalar spin chirality $\chi$ is a critical quantity in noncoplanar AFMs, which brings many exciting physics. Considering an electron moves along a close path connected by three noncoplanar spin sublattices ($\mathbf{S}_{i}$, $\mathbf{S}_{j}$, and $\mathbf{S}_{k}$), the electron shall experience a fictitious magnetic flux that is proportional to the scalar spin chirality, $\mathbf{b}_{f}\propto t_{3}\chi_{ijk}\mathbf{\hat{n}}$, where $t_{3}=t_{ij}t_{jk}t_{ki}$ is the successive transfer integral along the path ($i\rightarrow j\rightarrow k\rightarrow i$) and $\mathbf{\hat{n}}$ is a unit vector normal to the plane formed by the three sublattices. This is the so-called real space Berry phase effect, which is responsible for the topological Hall effect~\cite{Bruno2004,Franz2014} and its quantization~\cite{J-Zhou2016}, topological orbital magnetism~\cite{Hanke2017}, and the first-order topological magneto-optical effects~\cite{WX-Feng2020} and their quantization~\cite{WX-Feng2020}. It is reasonable to speculate that the second-order topological magneto-optical effects (such as topological MOVE and MOSHE), originating from scalar spin chirality and being even in $\hat{\mathbf{N}}$, can exist in noncoplanar AFMs as well. In other words, the topological magneto-optical effects should be able to generalize from the first-order cases (MOKE and MOFE) to the second-order cases (MOVE and MOSHE). The topological MOVE and MOSHE in noncoplanar AFMs are expected to differ from conventional MOVE and MOSHE in ferromagnets (FMs) ~\cite{Mertins2001,Valencia2010} and collinear AFMs~\cite{Saidl2017}, which are dependent on spin-orbit coupling and band exchange splitting.
In this work, using the first-principles calculations together with magnetic group analysis, we explore the topological MOVE and MOSHE in noncoplanar antiferromagnetic $\gamma$-Fe$_{x}$Mn$_{1-x}$ alloy. The theory and computational methods for the second-order magneto-optical effects are addressed detailedly in Sec.~\ref{sec:Theory}. The optical geometries for observing the MOVE and MOSHE are proposed for the collinear 1Q and 2Q spin states as well as the noncoplanar 3Q spin state. The magnetic group theory is then applied to determine the shape of the permittivity tensor, which is a crucial step in calculating the MOVE and MOSHE. In Sec.~\ref{sec:1Q}, the conventional MOVE and MOSHE effects are found in collinear 1Q and 2Q states of $\gamma$-Fe$_{x}$Mn$_{1-x}$. The calculated Voigt and Sch\"{a}fer-Hubert angles are comparable or even larger than that of the famous collinear AFM CuMnAs. The natural linear birefringence (NLB) due to crystal anisotropy in the strained 1Q and 2Q states are at least one order of magnitude smaller than the MOVE and MOSHE. In Sec.~\ref{sec:3Q}, we reveal the topological MOVE and MOSHE originating from scalar spin chirality in the noncoplanar 3Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$. Since the strain simultaneously induces scalar spin chirality and crystal anisotropy in the strained 3Q state, the topological MOVE/MOSHE and NLB are mixed together. We find that the magnitudes of topological MOVE/MOSHE and NLB are in the same order of magnitude, but their signs are opposite. A unique fingerprint for experimentally distinguishing the topological MOVE/MOSHE and NLB is identified. Finally, a brief summary is given in Sec.~\ref{sec:Summary}. Our work suggests that the antiferromagnetic $\gamma$-Fe$_{x}$Mn$_{1-x}$ alloy is an interesting material platform for exploring the novel high-order topological light-matter interactions.
\begin{figure}
\includegraphics[width=1\columnwidth]{fig_model.pdf}
\caption{Schematic illustration of magneto-optical Voigt and Sch\"{a}fer-Hubert effects. The incident linearly polarized light propagating along the $z$-axis normally shines on the surface ($xy$-plane) of an in-plane magnetized material (assuming $x$-axis). There exists an angle ($\alpha$) between the electric field of incident light ($\mathbf{E}_\textnormal{I}$) and the magnetization direction of materials ($\hat{\mathbf{N}}$ for AFMs or $\hat{\mathbf{M}}$ for FMs). The transmitted and reflected lights become elliptically polarized accompanying by the rotations of polarization planes. The rotation angles, $\theta_\textnormal{V}$ and $\theta_\textnormal{SH}$, describe the deflections of $\mathbf{E}_\textnormal{T}$ and $\mathbf{E}_\textnormal{R}$ with respect to $\mathbf{E}_\textnormal{I}$. The ellipticities, $\varepsilon_\textnormal{V}$ and $\varepsilon_\textnormal{SH}$, are the quotient of the short and long axes of the ellipse.}
\label{fig:model}
\end{figure}
\section{Theory and computational details} \label{sec:Theory}
For crystallographically-isotropic nonmagnetic materials, the application of an external magnetic field lowers the symmetry of systems, leading to a change in the shape of the permittivity tensor. Specifically, if a linearly polarized light normally shines on the surface of materials and the magnetic field lies on the surface (i.e., normal to the propagating direction of incident light), the light passing through the materials shall have two different indices of refraction, $n_{\parallel}$ and $n_{\perp}$, which are parallel to and perpendicular to the external magnetic field, respectively. Once the angle between the electric field of incident light and the external magnetic field does not equal 0$^\circ$ or 90$^\circ$, the transmitted and reflected lights have to become elliptically polarized, accompanying by the rotations of polarization planes. If the external magnetic field is replaced by spontaneous magnetic orders (FMs or AFMs), the resultant magneto-optical phenomena resemble, named Voigt~\cite{Voigt1908} and Sch\"{a}fer-Hubert effects~\cite{Schafer1990} building in transmission and reflection geometries, respectively (Fig.~\ref{fig:model}). It should be mentioned that for crystallographically-anisotropic nonmagnetic materials, the above optical phenomena still exist if the electric field of incident light is not parallel or perpendicular to the optical axis, but they are usually called NLB. In the following, we shall discuss how to separate the NLB from the magnetism-induced Voigt and Sch\"{a}fer-Hubert effects.
For the MOVE, the rotation angle ($\theta_{\textnormal{V}}$) and ellipticity ($\varepsilon_{\textnormal{V}}$) of elliptically-polarized transmitted light are usually combined into the so-called complex Voigt angle,
\begin{equation}\label{eq:Voigt-alpha}
\Phi_{\textnormal{V}}(\alpha)= \left(\varepsilon_{\textnormal{V}} + i \theta_{\textnormal{V}}\right) \sin(2\alpha), \\
\end{equation}
where $\alpha$ is the angle between the electric field of incident light ($\textbf{E}_\textnormal{I}$) and the magnetization direction ($\hat{\mathbf{N}}$ for AFMs or $\hat{\mathbf{M}}=\mathbf{M}/|\mathbf{M}|$ for FMs). It is clear that $\Phi_{\textnormal{V}}(\alpha)$ reaches its maximum when $\alpha = 45^{\circ}$, which corresponds to the standard Voigt geometry often employed in experiments. And $\Phi_{\textnormal{V}}(\alpha)$ is certainly zero if $\alpha = 0^{\circ}$ or $90^{\circ}$. Since the evolution of $\Phi_{\textnormal{V}}(\alpha)$ with $\alpha$ is just a sinusoidal curve with a period of $\pi$, hereafter, we shall only discuss the maximal value of $\Phi_{\textnormal{V}}(\alpha)$, which is expressed as~\cite{Mertins2001},
\begin{equation}\label{eq:Voigt}
\Phi_{\textnormal{V}}^{\textnormal{max}} = \varepsilon_{\textnormal{V}} + i \theta_{\textnormal{V}} \approx \frac{\omega d}{2c} (n_{\parallel}-n_{\perp}),
\end{equation}
where $\omega$ is light frequency, $c$ is the speed of light in vacuum, and $d$ is the thickness of materials. In our actual calculations, the Voigt angles are accounted in the unit of deg/cm, and therefore the thickness $d$ is not specified. Without loss of generality, we assume that $\hat{\mathbf{N}}$ is parallel to the $x$-axis and the incident light propagates along the $z$-axis. By solving the Fresnel equation, the refractive indices that are parallel to and perpendicular to $\hat{\mathbf{N}}$ can be written as,
\begin{eqnarray}
n_{\parallel} &=& \sqrt{\epsilon_{xx}}, \label{eq:n_parallel} \\
n_{\perp} &=& \sqrt{\epsilon_{yy} + \epsilon_{yz}^{2} / \epsilon_{zz}}, \label{eq:n_perp}
\end{eqnarray}
where $\epsilon_{\mu\nu}$ with $\mu,\nu \in \{x,y,z\}$ is the permittivity tensor. We can further obtain
\begin{eqnarray}
n_{\parallel} n_{\perp} &\approx& \frac{1}{2}(\epsilon_{xx} + \epsilon_{yy} + \epsilon_{yz}^2/\epsilon_{zz}), \label{eq:n_pn_p} \\
n_{\parallel} - n_{\perp} &\approx& \frac{1}{2\bar{n}}(\epsilon_{xx} - \epsilon_{yy} - \epsilon_{yz}^2/\epsilon_{zz}), \label{eq:n_p-n_p}
\end{eqnarray}
where $\bar{n} = \frac{1}{2}(n_{\parallel}+n_{\perp}) $ is the average of the refractive indices.
For the MOSHE, the rotation angle ($\theta_{\textnormal{SH}}$) and ellipticity ($\varepsilon_{\textnormal{SH}}$) of elliptically-polarized reflected light can be similarly written as the complex Sch\"{a}fer-Hubert angle,
\begin{equation}\label{eq:SH-alpha}
\Phi_{\textnormal{SH}}(\alpha)= \left(\theta_{\textnormal{SH}} + i\varepsilon_{\textnormal{SH}}\right) \sin(2\alpha), \\
\end{equation}
which also displays a period of $\pi$ with respect to the angle $\alpha$. The maximal value of $\Phi_{\textnormal{SH}}(\alpha)$ occurs at $\alpha=45^\circ$, that is~\cite{Valencia2010},
\begin{equation}\label{eq:SH}
\Phi_{\textnormal{SH}}^{\textnormal{max}} = \theta_{\textnormal{SH}} + i \varepsilon_{\textnormal{SH}} \approx \frac{n_{\parallel}-n_{\perp}}{1 - n_{\parallel}n_{\perp}}.
\end{equation}
One can read from Eqs.~\eqref{eq:Voigt} and~\eqref{eq:SH} that the birefringence, $\Delta n= n_{\parallel}-n_{\perp}$, is the dominating quantity for both MOVE and MOSHE. If a magnetic material presents in-plane crystal anisotropy (e.g., a $C^{z}_{2}$ symmetry lives on the $xy$-plane), a nonzero component of $\Delta n$ contributing from the NLB should be expected. This part has to be separated from total birefringence if one wishes to find out the magnetism-induced MOVE and MOSHE, just like what have done in crystallographically-anisotropic Cd$_{1-x}$Mn$_{x}$Se~\cite{Oh1991}. To do this, we expand the permittivity tensor into a Taylor series in powers of magnetization~\cite{Tesarova2014,Zelezny2017,XX-Yang2022},
\begin{equation}\label{eq:permittivity}
\epsilon_{\mu\nu}(\hat{\mathbf{N}})=\epsilon^{(0)}_{\mu\nu}+\epsilon^{(1)}_{\mu\nu} \hat{\mathbf{N}}+\epsilon^{(2)}_{\mu\nu} \hat{\mathbf{N}}^{2} + \cdots,
\end{equation}
where $\epsilon^{(0)}_{\mu\nu}$ is the part independent on magnetization, $\epsilon^{(1)}_{\mu\nu}$ and $\epsilon^{(2)}_{\mu\nu}$ are linearly and quadratically dependent on magnetization, respectively. In principle, we can calculate $\epsilon_{\mu\nu}$ for a given magnetic material and calculate $\epsilon^{(0)}_{\mu\nu}$ by artificially removing the magnetic order. Since $\epsilon^{(0)}_{\mu\nu}$ is responsible for the NLB due to crystal anisotropy, subtracting $\epsilon^{(0)}_{\mu\nu}$ from $\epsilon_{\mu\nu}$ is the part purely originating from magnetism, being responsible for the MOVE and MOSHE.
According to the Onsager relation, $\epsilon_{\mu\nu} (\hat{\mathbf{N}})=\epsilon_{\nu\mu} (-\hat{\mathbf{N}})$, we know $\epsilon^{(1)}_{\mu\mu}=0$, $\epsilon^{(0)}_{\mu\nu} = \epsilon^{(0)}_{\nu\mu}$, $\epsilon^{(1)}_{\mu\nu} = -\epsilon^{(1)}_{\nu\mu}$, and $\epsilon^{(2)}_{\mu\nu} = \epsilon^{(2)}_{\nu\mu}$. Omitting the magnetism-independent parts, the diagonal terms of the permittivity tensor are square of $\hat{\mathbf{N}}$, i.e., $\tilde{\epsilon}_{\mu\mu}(\hat{\mathbf{N}}) = \epsilon^{(2)}_{\mu\mu} \hat{\mathbf{N}}^{2}$, while the off-diagonal terms of the permittivity tensor have first-order antisymmetric and second-order symmetric parts, i.e., $\tilde{\epsilon}_{\mu\nu}(\hat{\mathbf{N}})=\epsilon^{(1)}_{\mu\nu} \hat{\mathbf{N}}+\epsilon^{(2)}_{\mu\nu} \hat{\mathbf{N}}^{2}$. Since $n_{\parallel} - n_{\perp}$ and $n_{\parallel}n_{\perp}$ are linearly (quadratically) dependent on the diagonal (off-diagonal) terms of the permittivity tensor (Eqs.~\eqref{eq:n_pn_p} and~\eqref{eq:n_p-n_p}), $\Phi_{\textnormal{V}}^{\textnormal{max}}$ and $\Phi_{\textnormal{SH}}^{\textnormal{max}}$ must be even in $\hat{\mathbf{N}}$ and to the lowest-order quadratic to $\hat{\mathbf{N}}$. It is in sharp contrast to the MOKE and MOFE, which are odd in $\hat{\mathbf{N}}$ and solely depend on $\epsilon^{(1)}_{\mu\nu}$. In many AFMs, the MOKE and MOFE are forbidden because the off-diagonal terms of the permittivity tensor are forced to be zero under certain symmetries (e.g., the $\mathcal{PT}$ symmetry, $\mathcal{P}$ and $\mathcal{T}$ are space inversion and time-reversal operations, respectively). Conversely, the MOVE and MOSHE can exist in AFMs due to the inequality of two diagonal terms, $\epsilon_{\mu\mu} \neq \epsilon_{\nu\nu}$, even though the off-diagonal term $\epsilon_{\mu\nu}$ is possibly zero, refer to Eq.~\eqref{eq:n_p-n_p}. Therefore, the MOVE and MOSHE take a great advantage over the MOKE and MOFE for studying AFMs.
The key ingredient for calculating the MOVE and MOSHE is the permittivity tensor or the optical conductivity in an equivalent way, since $\epsilon_{\mu\nu}=\delta_{\mu\nu} + \frac{4\pi i}{\omega} \sigma_{\mu\nu}$, where $\delta_{\mu\nu}$ equals 1 if $\mu =\nu$ and 0 otherwise. The optical conductivity can be evaluated using the Kubo formula~\cite{Yates2007} implemented in \textsc{wannier90} package~\cite{Pizzi2020},
\begin{eqnarray}\label{eq:OPC}
\sigma_{\mu\nu}&=& \frac{ie^2\hbar}{N_k V}\sum_{\textbf{k}}\sum_{n, m}\frac{f_{m\textbf{k}}-f_{n\textbf{k}}}{E_{m\textbf{k}}-E_{n\textbf{k}}} \nonumber\\
&&\times\frac{\langle\psi_{n\textbf{k}}|\hat{\upsilon}_{\mu}|\psi_{m\textbf{k}}\rangle\langle\psi_{m\textbf{k}}|\hat{\upsilon}_{\nu}|\psi_{n\textbf{k}}\rangle}{E_{m\textbf{k}}-E_{n\textbf{k}}-(\hbar\omega+i\eta)},
\end{eqnarray}
where $f_{n\textbf{k}}$, $V$, $N_k$, $\hat{\upsilon}_{\mu(\nu)}$, $\hbar\omega$, and $\eta$ are the Fermi-Dirac distribution function, volume of the unit cell, total number of $k$-points for sampling the Brillouin zone, velocity operators, photon energy, and energy smearing parameter, respectively. $\psi_{n\textbf{k}}$ and $E_{n\textbf{k}}$ are the Bloch wavefunction and interpolated energy at the band index $n$ and momentum $\textbf{k}$, respectively. A $k$-points mesh of $100 \times 100 \times 100$ is enough to converge the optical conductivity, and $\eta$ is set to be 0.1 eV. Because of the metallic nature of $\gamma$-Fe$_{x}$Mn$_{1-x}$, the intraband transitions can not be ignored in the low energy range (e.g., $<$ 1.0 eV), and the Drude term $\sigma_{\textnormal{D}} = \sigma_{0} /\left(1-i \omega \tau_{\textnormal{D}}\right)$ is added into the diagonal terms of optical conductivity. The Drude parameters $\sigma_{0}$ and $\tau_{\textnormal{D}}$ are obtained by linearly interpolating the experimental data of pure Fe ($\sigma_{0} = 6.40\times10^{15}$ s$^{-1}$ and $\tau_{\textnormal{D}} = 9.12\times10^{-15}$ s)~\cite{Lenham1966} and pure Mn ($\sigma_{0} = 4.00\times10^{15}$ s$^{-1}$ and $\tau_{\textnormal{D}} = 0.33\times10^{-15}$ s)~\cite{Lenham1966}.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{fig_crystal.pdf}
\caption{The 1Q (a), 2Q (b), and 3Q (c) states of $\gamma$-Fe$_{x}$Mn$_{1-x}$ as well as the optical geometries for calculating the magneto-optical Voigt and Sch\"{a}fer-Hubert effects. The incident light propagates along the $z$-axis, and its electric field lies on the $xy$-plane with a angle of $\alpha=45^\circ$ away from the $x$-axis. For 1Q state, the magnetization direction (i.e., the N\'{e}el vector $\mathbf{N}$) is along the $x$-axis; for 2Q state, the ``effective'' magnetization direction (i.e., $\mathbf{N}_{y}$, see the main text for details) is along the $y$-axis; for the 3Q state strained along the [111] direction, the fictitious magnetic field $\mathbf{B}=\sum_{f=1}^{4} \mathbf{b}_{f}$ is along the $x$-axis, where $\mathbf{b}_{f}$ is the fictitious magnetic flux on each face of the tetrahedron formed by four magnetic atoms in the unit cell.}
\label{fig:crystal}
\end{figure}
The electronic structure calculations of $\gamma$-Fe$_{x}$Mn$_{1-x}$ alloy are performed using the full-potential linearized augmented-plane-wave code FLEUR~\cite{Fleur}. The exchange-correlation effect is treated by the generalized gradient approximation with the Perdew-Burke-Ernzerhof (GGA-PBE) parameterization~\cite{Perdew1996}. To describe disordered alloys, the virtual crystal approximation is used by adapting the nuclear numbers under conservation of charge neutrality. The lattice constant of unstrained fcc $\gamma$-Fe$_{x}$Mn$_{1-x}$ is 6.86 $a_{0}$ (3.63 \r{A})~\cite{Hanke2017} and the muffin-tin radii of Fe and Mn atoms are chosen to be 2.29 $a_{0}$ ($a_{0}$ is Bohr radius). A compressive or tensile strain is applied to explore how the strain influences the MOVE and MOSHE. The strain is quantified by the distortion ratio $\delta = l'/l$, where $l'$ and $l$ are the interatomic distances along the distorted direction in strained and unstrained structures, respectively. The constant volume approximation is adopted on account of the Poisson effect. The plane-wave cutoff is chosen to be 3.8 $a_{0}^{-1}$ and the Brillouin zone is sampled using a $k$-points mesh of $12\times12\times12$. Spin-orbit coupling is not included in our calculations. After obtaining electronic ground states, a total of 72 maximally-localized Wannier functions are constructed by projecting the $s$-, $p$-, and $d$-orbitals on four magnetic atoms in the unit cell, using \textsc{wannier90} package~\cite{Pizzi2020}.
\section{Results and Discussion} \label{sec:Results}
By varying the alloying ratio $x$, $\gamma$-Fe$_{x}$Mn$_{1-x}$ hosts three different antiferromagnetic orders (Fig.~\ref{fig:crystal})~\cite{Kouvel1963,Endoh1971,Kubler1988,Schulthess1999,Sakuma2000}, including the collinear 1Q ($x<0.4$) and 2Q states ($x>0.8$) as well as the noncoplanar 3Q state ($0.4<x<0.8$). In this work, we investigate the MOVE and MOSHE for 1Q, 2Q, and 3Q states by typically choosing $x$ = 0.2, 0.9, and 0.5, respectively. In Sec.~\ref{sec:1Q}, we first discuss the conventional MOVE and MOSHE that appear in 1Q and 2Q states. Following in Sec.~\ref{sec:3Q}, we introduce the main discovery of this work --- the topological MOVE and MOSHE, which only exists in 3Q state with a noncoplanar spin texture.
\subsection{Conventional MOVE and MOSHE in 1Q and 2Q states} \label{sec:1Q}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig_1Q.pdf}
\caption{The Voigt and Sch\"{a}fer-Hubert rotation angles ($\theta_\textnormal{V}$ and $\theta_\textnormal{SH}$) and ellipticities ($\varepsilon_\textnormal{V}$ and $\varepsilon_\textnormal{SH}$) for the 1Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$ with $x$ = 0.2. The compressive ($\delta$ = 0.99) and tensile ($\delta$ = 1.01) strains are applied along the [100] direction. The 1Q$^\prime$ state (cyan dot-dashed lines) is the time-reversal counterpart of 1Q state. The natural linear birefringence (green and pink dashed lines) are calculated by removing the 1Q magnetic order on top of the two strained structures.}
\label{fig:1Q}
\end{figure}
The 1Q state is a collinear AFM with the N\'{e}el vector, defined by $\mathbf{N}=\frac{1}{4}(\mathbf{m}_{1}+\mathbf{m}_{2}-\mathbf{m}_{3}-\mathbf{m}_{4})$ where $\mathbf{m}_{1-4}$ are spin magnetic moments on four magnetic atoms, pointing to the $x$-axis (Fig.~\ref{fig:crystal}(a)). We first consider the perfect fcc lattice without any distortion. The magnetic space and point groups computed by \textsc{isotropy} code~\cite{isotropy} are P$_\textnormal{I}$4/mnc (BNS setting) and 4/mmm1$^\prime$, respectively. The symmetry-imposed shape of permittivity tensor can be identified by the linear response symmetry code \textsc{symmetr}~\cite{Zelezny2017a,Zelezny2018a},
\begin{equation}\label{eq:permittivity_1Q}
\epsilon^{\textnormal{1Q}(\textnormal{2Q})}=
\left(\begin{array}{ccc}
\epsilon_{xx} & 0 & 0 \\
0 & \epsilon_{yy} & 0 \\
0 & 0 & \epsilon_{yy}
\end{array}\right).
\end{equation}
One can see that all the off-diagonal terms are zero and there are two independent diagonal terms, which can be simply interpreted below. Since the time-reversal symmetry $\mathcal{T}$ is broken in antiferromagnetic 1Q state, only the antisymmetric part of off-diagonal terms ($\epsilon^{(1)}_{\mu\nu}$) is potentially zero, while the symmetric parts are always zero ($\epsilon^{(0)}_{\mu\nu}=\epsilon^{(2)}_{\mu\nu}=0$). In addition, the magnetic point group 4/mmm1$^\prime$ contains the space-time inversion symmetry $\mathcal{PT}$, which forces $\epsilon^{(1)}_{\mu\nu}=0$. Hence, all the off-diagonal terms have to be vanished. For the diagonal terms, the magnetism-independent parts equal to each other in the fcc lattice, i.e., $\epsilon^{(0)}_{xx}=\epsilon^{(0)}_{yy}=\epsilon^{(0)}_{zz}$, while magnetism-dependent parts should be $\epsilon^{(2)}_{xx}\neq\epsilon^{(2)}_{yy}=\epsilon^{(2)}_{zz}$ due to the collinear antiferromagnetic order along the $x$-axis. Accordingly, two independent diagonal terms, $\epsilon_{xx}\neq\epsilon_{yy}=\epsilon_{zz}$, appear in Eq.~\eqref{eq:permittivity_1Q}, which eventually results in the appearance of the conventional MOVE and MOSHE (refer to Eqs.~\eqref{eq:n_p-n_p},~\eqref{eq:Voigt}, and~\eqref{eq:SH}).
Figure~\ref{fig:1Q} plots the conventional MOVE and MOSHE for the 1Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$ with $x$ = 0.2. In the unstrained case ($\delta= 1.0$), $\theta_{\textnormal{V}}$ reaches up to $-4.0 \times 10^6 $ deg/cm at 2.1 eV and $6.1 \times 10^6 $ deg/cm at 2.9 eV (Fig.~\ref{fig:1Q}(a)). The spectral structure of MOSHE is similar to that of MOVE, except for a minus sign for the rotation angles (Figs.~\ref{fig:1Q}(a) and~\ref{fig:1Q}(c)). For example, the positive and negative maximums of $\theta_{\textnormal{SH}}$ angle appear to be 2.1 deg at 2.2 eV and -2.6 deg at 3.3 eV (Fig.~\ref{fig:1Q}(c)). The $\theta_{\textnormal{V}}$ and $\theta_{\textnormal{SH}}$ of 1Q state are larger than that of other collinear AFMs, such as CuMnAs ($\theta_\textnormal{V} \sim 2.3\times10^4 $ deg/cm)~\cite{Saidl2017}, CoO thin film ($\theta_\textnormal{SH} \sim 0.168 $ deg)~\cite{J-Xu2020}. If we apply the time-reversal symmetry for 1Q state (named 1Q$^\prime$ state), all the spin magnetic moments ($\mathbf{m}_{1-4}$) change their signs such that the N\'{e}el vector $\mathbf{N}$ reverses its direction. Nevertheless, $\theta_{\textnormal{V}}$ and $\theta_{\textnormal{SH}}$ change nothing (see cyan dot-dashed lines). It indicates that the observed MOVE and MOSHE are essentially even in $\mathbf{N}$, differing from the linear MOKE and MOFE that are odd in $\mathbf{N}$.
In practice, a small tetragonal distortion along the direction of $\mathbf{N}$ possibly occurs in 1Q state according to the previous works~\cite{Ekholm2011}, similarly to pure $\gamma$-Fe~\cite{Ehrhart1980} and pure $\gamma$-Mn~\cite{Oguchi1984}. Here, 1\% compressive ($\delta=0.99$) and tensile ($\delta=1.01$) strains along the $x$-axis (i.e., [100] direction) are taken into account. The magnetic space and point groups remain unchanged, that is, P$_\textnormal{I}$4/mnc (BNS setting) and 4/mmm1$^\prime$, respectively. Since the $\mathcal{PT}$ symmetry is reserved under the strains, all the off-diagonal terms of permittivity tensor are still zero. The shape of permittivity tensor thus keeps invariant, see Eq.~\eqref{eq:permittivity_1Q}. The strains only affect the magnetism-independent diagonal terms, leading to $\epsilon^{(0)}_{xx}\neq\epsilon^{(0)}_{yy}=\epsilon^{(0)}_{zz}$, which induces the NLB. Figure~\ref{fig:1Q} clearly shows that the influence of strains on the MOVE and MOSHE for 1Q state is rather weak (comparing the lines for $\delta=0.99$, $1.0$, and $1.01$). The difference between the strained and unstrained results is just the contribution from NLB, which can also be calculated by directly plugging $\epsilon^{(0)}_{xx}$ and $\epsilon^{(0)}_{yy}$ into Eqs.~\eqref{eq:n_pn_p} and~\eqref{eq:n_p-n_p} (see green and pink dashed lines in Fig.~\ref{fig:1Q}). The NLB under compressive and tensile strains along the $x$-axis have almost the same spectral structure but differ by a minus sign, which reflects the dominated absorption of the linearly polarized light along either $x$-axis or $y$-axis. Nevertheless, the NLB for both compressive and tensile cases are considerably weak, at least one order of magnitude smaller than the MOVE and MOSHE.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig_2Q.pdf}
\caption{The Voigt and Sch\"{a}fer-Hubert rotation angles ($\theta_\textnormal{V}$ and $\theta_\textnormal{SH}$) and ellipticities ($\varepsilon_\textnormal{V}$ and $\varepsilon_\textnormal{SH}$) for the 2Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$ with $x$ = 0.9. All the labels are the same as the 1Q state presented in Fig.~\ref{fig:1Q}.}
\label{fig:2Q}
\end{figure}
Next, we turn to the MOVE and MOSHE for 2Q state, which is also a collinear AFM since both the vector and scalar spin chiralities are zero ($\boldsymbol{\kappa}=0$ and $\chi=0$). The 2Q state has two collinear antiferromagnetic orders along the [0$\bar{1}$1] and [011] directions, which are orthogonal to each other. The calculated magnetic space and point groups of 2Q state are P$_\textnormal{C}$4$_2$/nnm (BNS setting) and 4/mmm1$^\prime$, respectively. The shape of the permittivity tensor for 2Q state is identical to 1Q state, as given in Eq.~\eqref{eq:permittivity_1Q}. The vanishing off-diagonal terms suggest that the first-order MOKE and MOFE must be absent, while the unequal diagonal terms imply the existence of second-order MOVE and MOSHE. However, at first glance, the magnetization direction of 2Q state does not satisfy the standard Voigt geometry (Fig.~\ref{fig:crystal}(b) and Fig.~\ref{fig:model}). Here we decompose the spin magnetic moment of each atom to the $y$- and $z$-axes. The system is still a compensated AFM as the net magnetization along both the $y$- and $z$-axes are zero. The collinear antiferromagnetic order along the $z$-axis, $\mathbf{N}_{z}=\frac{1}{4}(\mathbf{m}_{1,z}-\mathbf{m}_{2,z}-\mathbf{m}_{3,z}+\mathbf{m}_{4,z})$, plays no role in the first- and second-order magneto-optical effects, while the one along the $y$-axis, $\mathbf{N}_{y}=\frac{1}{4}(-\mathbf{m}_{1,y}+\mathbf{m}_{2,y}-\mathbf{m}_{3,y}+\mathbf{m}_{4,y})$, activates the second-order MOVE and MOSHE. The angle $\alpha$ between the electric field direction of incident light ($\mathbf{E}_\textnormal{I}$) and the ``effective" magnetization direction ($\mathbf{N}_{y}$) is 45$^\circ$ (Fig.~\ref{fig:crystal}(b)), giving rise to the maximal values of $\theta_{\textnormal{V}}$ and $\theta_{\textnormal{SH}}$.
Figure~\ref{fig:2Q} plots the conventional MOVE and MOSHE for the 2Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$ with $x$ = 0.9. Similarly to 1Q state, the spectral structures of MOVE and MOSHE resemble each other, only a minus sign differs for the rotation angles (Figs.~\ref{fig:2Q}(a) and~\ref{fig:2Q}(c)). On the other hand, in contrast to 1Q state, $\theta_{\textnormal{V(SH)}}$ and $\varepsilon_{\textnormal{V(SH)}}$ oscillate frequently as a function of photon energy. The positive and negative maximums of $\theta_{\textnormal{V}}$ ($\theta_{\textnormal{SH}}$) are $2.1\times 10^6$ deg/cm at 0.4 eV and $-1.7\times 10^6$ deg/cm at 2.9 eV ($1.0^{\circ}$ at 3.1 eV and $-1.5^{\circ}$ at 0.4 eV), respectively. The MOVE and MOSHE of 2Q state are overall smaller than that of 1Q state. It can be understood from the fact that the effective antiferromagnetic order of 2Q state (along the $y$-axis) is weaker than the true antiferromagnetic order of 1Q state (along the $x$-axis) because the spin magnetic moments of 2Q state projected onto the plane perpendicular to the incident light ($xy$-plane) are actually reduced. This is very similar to the case of CuMnAs when rotating $\mathbf{N}$ around an axis perpendicular to the direction of the spin magnetic moments~\cite{Saidl2017}. If the time-reversal symmetry is applied on 2Q state (named 2Q$^\prime$ state), all the moments will be reversed such that the effective antiferromagnetic order $\mathbf{N}_{y}$ changes its direction. By comparing to 2Q state, the unchanged MOVE and MOSHE of 2Q$^\prime$ state demonstrate that the magneto-optical effects being even in $\mathbf{N}_{y}$ are indeed second-order. The effect of strain on the MOVE and MOSHE for 2Q state is also considered by applying a 1\% compressive ($\delta=0.99$) or tensile ($\delta=1.01$) strain along the $x$-axis. The resultant NLB is significantly smaller than the magnetism-induced MOVE and MOSHE, similarly to 1Q state. The strain-insensitive character is also similar to the optical linear dichroism in two-dimensional zigzag-AFM FePS$_{3}$~\cite{Q-Zhang2021}.
\begin{figure*}
\centering
\includegraphics[width=2\columnwidth]{fig_3Q.pdf}
\caption{The Voigt and Sch\"{a}fer-Hubert rotation angles ($\theta_\textnormal{V}$ and $\theta_\textnormal{SH}$) and ellipticities ($\varepsilon_\textnormal{V}$ and $\varepsilon_\textnormal{SH}$) for the strained 3Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$ with $x$ = 0.5. (a)(b) The total angles contributed from both crystal anisotropy and scalar spin chirality. (c)(d) The contribution solely from crystal anisotropy, that is, natural linear birefringence. (e)(f) The contribution solely from scalar spin chirality, that is, topological Voigt and Sch\"{a}fer-Hubert effects. The compressive ($\delta$ = 0.96, 0.98) and tensile ($\delta$ = 1.02, 1.04) strains are applied along the [111] direction. The 3Q$^\prime$ state (blue dashed lines) is the time-reversal counterpart of 3Q state.}
\label{fig:3Q}
\end{figure*}
\subsection{Topological MOVE and MOSHE in 3Q state}\label{sec:3Q}
The 3Q state is a fully compensated noncoplanar AFM, in which all spin magnetic moments point to the center of the tetrahedron formed by four magnetic atoms in the fcc lattice (Fig.~\ref{fig:crystal}(c)). Considering one face of the tetrahedron, the three noncoplanar spins generate a nonzero scalar spin chirality, $\chi_{ijk} = \mathbf{S}_{i}\cdot\left(\mathbf{S}_{j}\times\mathbf{S}_{k}\right) \neq 0$, which in turn provides a fictitious magnetic flux, $\mathbf{b}_{f}\propto t_{3}\chi_{ijk}\mathbf{\hat{n}}$ (here, $t_{3}=t_{ij}t_{jk}t_{ki}$ is the successive transfer integral along the path $i\rightarrow j \rightarrow k \rightarrow i$ and $\mathbf{\hat{n}}$ is the surface normal vector)~\cite{WX-Feng2020}. The total fictitious magnetic field in the unit cell is the vector sum of the magnetic fluxes on the four faces of the tetrahedron, i.e., $\mathbf{B}=\sum_{f=1}^{4}\mathbf{b}_{f}$. It is clear that $\mathbf{B}$ is zero for the unstrained fcc lattice because the four fluxes cancel each other exactly (Fig.~\ref{fig:crystal}(c)). Accordingly, both the first- and second-order magneto-optical effects are not active in the unstrained 3Q state. This can be further understood from the symmetry point of view. For the unstrained 3Q state, the magnetic space and point groups are $Pn\bar{3}m^{\prime}$ and $m\bar{3}m^{\prime}$, respectively. The resultant permittivity tensor shows vanishing off-diagonal terms and equivalent diagonal terms,
\begin{equation}\label{eq:permittivity_3Q}
\epsilon^{\textnormal{3Q}}=
\left(\begin{array}{ccc}
\epsilon_{xx} & 0 & 0 \\
0 & \epsilon_{xx} & 0 \\
0 & 0 & \epsilon_{xx}
\end{array}\right),
\end{equation}
which forbids both the first- and second-order magneto-optical effects.
The scalar spin chirality in 3Q state plays its role once a strain is applied on the fcc lattice. For example, the topological orbital magnetization and topological Hall effect, originating from scalar spin chirality, were reported in strained $\gamma$-Fe$_{x}$Mn$_{1-x}$ with the 3Q spin texture~\cite{Hanke2017,Shiomi2018}. Moreover, the first-order topological magneto-optical effects (by taking MOKE and MOFE as prototypes) and their quantization were predicted in our previous work~\cite{WX-Feng2020}. One can rationally speculate that the second-order topological magneto-optical effects, i.e., topological MOVE and MOSHE, exist also in the strained 3Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$. To confirm this idea, a uniaxial strain is applied along the [111] direction such that a nonzero fictitious magnetic field, $\mathbf{B}=B\mathbf{\hat{n}}_{[111]}$ with $B\neq0$, emerges. Further assuming that the light is incident along the [$\bar{1}$10] direction, and its electric field is 45$^\circ$ away from the [111] direction. The optical geometry for calculating the topological MOVE and MOSHE of the strained 3Q state is schematically displayed in Fig.~\ref{fig:crystal}(c). The magnetic space and point groups for the strained 3Q state are $R\bar{3}m^{\prime}$ and $\bar{3}1m^{\prime}$, respectively, giving rise to the permittivity tensor,
\begin{equation}\label{eq:permittivity_3Q_strained}
\epsilon^{\textnormal{3Q,strained}}=
\left(\begin{array}{ccc}
\epsilon_{xx} & 0 & 0 \\
0 & \epsilon_{yy} & \epsilon_{yz} \\
0 & -\epsilon_{yz} & \epsilon_{yy}
\end{array}\right).
\end{equation}
Due to the unequal diagonal terms ($\epsilon_{xx}\neq\epsilon_{yy}$) and nonzero off-diagonal terms ($\epsilon_{yz}\neq0$), the $\theta_{\textnormal{V}}$ and $\theta_{\textnormal{SH}}$ are certainly nonvanishing for the strained 3Q state (refer to Eqs.~\eqref{eq:n_p-n_p},~\eqref{eq:Voigt}, and~\eqref{eq:SH}). However, the topological component originating from scalar spin chirality is mixed to the NLB that is due to crystal anisotropy. To obtain the topological MOVE and MOSHE, the NLB has to be separated from total birefringence.
Figure~\ref{fig:3Q} illustrates the Voigt and Sch\"{a}fer-Hubert spectra for the 3Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$ with $x$ = 0.5. The total angles ($\theta_{\textnormal{V}}^{\textnormal{Total}}$, $\varepsilon_{\textnormal{V}}^{\textnormal{Total}}$, $\theta_{\textnormal{SH}}^{\textnormal{Total}}$, and $\varepsilon_{\textnormal{SH}}^{\textnormal{Total}}$) contributed from both crystal anisotropy and scalar spin chirality are shown in Figs.~\ref{fig:3Q}(a) and~\ref{fig:3Q}(b). In the unstrained case ($\delta$ = 1.0), the Voigt/Sch\"{a}fer-Hubert rotation angles and ellipticities are definitely zero, which is consistent with the above symmetry analysis, referring to Eq.~\eqref{eq:permittivity_3Q}. Once a strain, for example, along the [111] direction, is applied ($\delta\neq$ 1.0), the rotation angles and ellipticities turn to be nonzero and are roughly proportional to the strain. Moreover, the direction of the angles reverses if the applied strain changes from tension ($\delta>$ 1.0) to compression ($\delta<$ 1.0) and vice versa. As mentioned above, the strain brings crystal anisotropy and finite fictitious magnetic field simultaneously in the fcc lattice, and therefore, the observed total Voigt and Sch\"{a}fer-Hubert angles should be a superposition of the NLB and topological MOVE/MOSHE.
If we remove the 3Q magnetic order on top of the strained fcc lattices, the off-diagonal terms of the permittivity tensor should be zero ($\epsilon_{yz}=\epsilon_{zy}=0$), while the diagonal terms retain only the magnetism-independent parts ($\epsilon_{xx}^{(0)}\neq\epsilon_{yy}^{(0)}\neq 0$). In such a case, the calculated Voigt and Sch\"{a}fer-Hubert angles can be completely ascribed to the NLB ($\theta_{\textnormal{V}}^{\textnormal{NLB}}$, $\varepsilon_{\textnormal{V}}^{\textnormal{NLB}}$, $\theta_{\textnormal{SH}}^{\textnormal{NLB}}$, and $\varepsilon_{\textnormal{SH}}^{\textnormal{NLB}}$) that is due to crystal anisotropy, as shown in Figs.~\ref{fig:3Q}(c) and~\ref{fig:3Q}(d). The NLB generated by the strain along the [111] direction is three to four times larger than that along the [100] direction (comparing Figs.~\ref{fig:3Q}(c,d) with Figs.~\ref{fig:1Q} and~\ref{fig:2Q}). On the other hand, the topological components of Voigt and Sch\"{a}fer-Hubert angles ($\theta_{\textnormal{V}}^{\textnormal{Topo}}$, $\varepsilon_{\textnormal{V}}^{\textnormal{Topo}}$, $\theta_{\textnormal{SH}}^{\textnormal{Topo}}$, and $\varepsilon_{\textnormal{SH}}^{\textnormal{Topo}}$), i.e., topological MOVE and MOSHE that root in scalar spin chirality, can be obtained by solely taking into account the magnetism-dependent terms of permittivity tensor (including $\epsilon_{xx}^{(2)}$, $\epsilon_{yy}^{(2)}$, and $\epsilon_{yz}^{(1)}$), as shown in Figs.~\ref{fig:3Q}(e) and~\ref{fig:3Q}(f). One should note that the magnitudes of topological MOVE and MOSHE are in the same order of the NLB. In addition, the Voigt and Sch\"{a}fer-Hubert angles originated from crystal anisotropy and scalar spin chirality own opposite signs (Figs.~\ref{fig:3Q}(c-f)), reflecting the contrary deflection of the polarization plane when a linearly polarized light goes through the strained 3Q state.
By applying the time-reversal operation $\mathcal{T}$, all the spin magnetic moments of 3Q state are inverted, giving the 3Q$^{\prime}$ state, in which the fictitious magnetic fluxes $\mathbf{b}_{f}$ on each face of the tetrahedron reverse their directions. As a result, the total fictitious magnetic filed $\mathbf{B}$ in the unit cell of 3Q$^{\prime}$ state also reverses its direction. One can see that the topological MOVE and MOSHE for 3Q and 3Q$^{\prime}$ states are fully identical to each other, as plotted in Figs.~\ref{fig:3Q}(e) and~\ref{fig:3Q}(f), by taking the strain $\delta$ = 1.02 as an example (others are not shown). It demonstrates clearly that the topological MOVE and MOSHE belong to the second-order magneto-optical effects as they are even in $\mathbf{b}_{f}$ or even in $\mathcal{T}$ equivalently. Here, it is not safe to say that the topological MOVE and MOSHE are even in the fictitious magnetic filed $\mathbf{B}$ since the direction of $\mathbf{B}$ can also be inverted by applying the strain from tension to compression and vice versa, while during this process the signs of topological MOVE and MOSHE follow with $\mathbf{B}$. The strain does not change the direction of $\mathbf{b}_{f}$ but creates an imbalance among the $\mathbf{b}_{f}$ on the four faces of the tetrahedron, which in turn reverses the direction of $\mathbf{B}$. Hence, such a change in the direction of $\mathbf{B}$ by strain is irrelevant to time-reversal symmetry. While the advantage is that the strain provides a practical way to control the signs of topological MOVE and MOSHE, which can not be realized in conventional MOVE and MOSHE. Finally, one can also find that the total Voigt and Sch\"{a}fer-Hubert angles of 3Q and 3Q$^{\prime}$ states are in a complete agreement, as shown in Figs.~\ref{fig:3Q}(a) and~\ref{fig:3Q}(b), because $\mathcal{T}$ plays no role in the NLB.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{fig_3Q_weight.pdf}
\caption{The relative weight of topological component and natural linear birefringence for total Voigt and Sch\"{a}fer-Hubert rotation angles ($W_{\textnormal{V}}$ and $W_{\textnormal{SH}}$) of the strained 3Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$ ($x$ = 0.5 and $\delta$ = 0.96, 0.98, 1.02, 1.04). The photon energies are marked when $W_{\textnormal{V}}$ and $W_{\textnormal{SH}}$ approach to $\pm1$.}
\label{fig:3Q_weight}
\end{figure}
In the above theoretical calculations, total Voigt and Sch\"{a}fer-Hubert rotation angles ($\theta_{\textnormal{V}}^{\textnormal{Total}}$ and $\theta_{\textnormal{SH}}^{\textnormal{Total}}$) contributed from topological origin ($\theta_{\textnormal{V}}^{\textnormal{Topo}}$ and $\theta_{\textnormal{SH}}^{\textnormal{Topo}}$) and NLB ($\theta_{\textnormal{V}}^{\textnormal{NLB}}$ and $\theta_{\textnormal{SH}}^{\textnormal{NLB}}$) have been explicitly separated. On the other hand, for experiments, the NLB part can also be captured by using, for example, the ultrafast pump-probe technique, which can induce demagnetization in AFMs~\cite{HC-Zhao2021,Saidl2017}. To describe the relative weight of topological origin and NLB quantitatively, we define such a quantity,
\begin{equation}\label{eq:3Q_weigt}
W_{\textnormal{V(SH)}} = \frac{|\theta_{\textnormal{V(SH)}}^{\textnormal{Topo}}| - |\theta_{\textnormal{V(SH)}}^{\textnormal{NLB}}|}{|\theta_{\textnormal{V(SH)}}^{\textnormal{Topo}}| + |\theta_{\textnormal{V(SH)}}^{\textnormal{NLB}}|},
\end{equation}
where $W_{\textnormal{V}}$ and $W_{\textnormal{SH}}$ approaching to 1 (-1) means that the topological (NLB) component dominates. Figure~\ref{fig:3Q_weight} depicts the $W_{\textnormal{V}}$ and $W_{\textnormal{SH}}$ for the 3Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$ ($x$ = 0.5) under four different strains ($\delta$ = 0.96, 0.98, 1.02, and 1.04). Taking $W_{\textnormal{V}}$ at $\delta$ = 0.96 as an example, the first positive peak appears at 0.94 eV, at where $\theta_{\textnormal{V}}^{\textnormal{Total}}$, $\theta_{\textnormal{V}}^{\textnormal{Topo}}$, $\theta_{\textnormal{V}}^{\textnormal{NLB}}$ are $1.52\times10^{6}$ deg/cm, $1.53\times10^{6}$ deg/cm, and $-0.02\times10^{6}$ deg/cm, respectively (see Fig.~\ref{fig:3Q}(a,c,e), left panels); the first negative pear appears at 0.25 eV, at where $\theta_{\textnormal{V}}^{\textnormal{Total}}$, $\theta_{\textnormal{V}}^{\textnormal{Topo}}$, $\theta_{\textnormal{V}}^{\textnormal{NLB}}$ are $-0.32\times10^{6}$ deg/cm, $-0.31\times10^{6}$ deg/cm, and $-0.01\times10^{6}$ deg/cm, respectively (see Fig.~\ref{fig:3Q}(a,c,e), left panels). It confirms that $W_{\textnormal{V}}$ and $W_{\textnormal{SH}}$ are precise fingerprints for experimentally probing the topological MOVE and MOSHE in the strained 3Q state of $\gamma$-Fe$_{x}$Mn$_{1-x}$.
\section{Summary} \label{sec:Summary}
In summary, the second-order magneto-optical effects have been investigated in the collinear 1Q and 2Q states as well as the noncoplanar 3Q state of antiferromagnetic $\gamma$-Fe$_{x}$Mn$_{1-x}$ alloy, using the first-principles calculations and group theory analysis. The conventional Voigt and Sch\"{a}fer-Hubert effects were found in the collinear 1Q and 2Q states, just like other common antiferromagnets. While the Voigt and Sch\"{a}fer-Hubert rotation angles emerged in the 1Q state reach up to $6.1 \times 10^6 $ deg/cm and 2.6 deg, respectively, which are much larger than that of some famous collinear antiferromagnets, e.g., CuMnAs. On the other hand, the Voigt and Sch\"{a}fer-Hubert rotation angles emerged in the 2Q state are relatively small since the effective magnetization perpendicular to the incident light is reduced. The natural linear birefringence originating from crystal anisotropy was observed in the strained 1Q and 2Q states, but their magnitudes are notably less than the magnetism-induced Voigt and Sch\"{a}fer-Hubert effects. In the strained 3Q state, the total Voigt and Sch\"{a}fer-Hubert angles were contributed from the topological origin and natural linear birefringence because the strain brings scalar spin chirality and crystal anisotropy simultaneously. The topological Voigt and Sch\"{a}fer-Hubert effects, originating from scalar spin chirality, were successfully identified. A unique fingerprint for experimentally probing the topological Voigt and Sch\"{a}fer-Hubert effects was also proposed. Our work not only deepens the understanding of second-order magneto-optical effects but also facilitates the applications of antiferromagnets in magneto-optical devices.
\begin{acknowledgments}
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11874085, 11734003, and 12061131002), the Sino-German Mobility Programme (Grant No. M-0142), the National Key R\&D Program of China (Grant No. 2020YFA0308800), and the Science \& Technology Innovation Program of Beijing Institute of Technology (Grant No. 2021CX01020).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,104 |
\section{Introduction}
The human civilisation still lacks a conclusive knowledge about the physics at galaxies and beyond, in the sense that either dark matter exists or the dynamics of gravity is more involved than that predicted by the Einstein's gravitional theory. In the first approach, we have not yet found any dark matter. The later approach started by the Milgrom's theory of the Modified Newtonian Dynamics (MOND)\cite{MOND}, changed to the a-quadratic Lagrangian model for gravity (AQUAL)\cite{AQUAL}. A generally covariant realisation of the AQUAL model is the TeVeS theory \cite{Bekenstein:2004ne}. TeVeS is in agreement with the Gravitomagnetism and the near horizon geometry of super massive black hole \cite{Exirifard:2011vb}. It has been shown that changing EH gravity to TeVeS can not reproduce the CMB \cite{Skordis:2005xk}. Though adding a $4^{\text{th}}$ sterile neutrino for $11-15~eV$ to matter content of TeVeS theory fits the spectrum in a LCDM cosmology \cite{Angus:2008qz}, it wouldn't form the large structure of the universe \cite{Xu:2014doa}. TeVeS also appears not to be consistent with GW170817 \cite{Boran:2017rdn}. There is also Moffat's theory of gravity \cite{Moffat:2005si} that passes many tests and observations when TeVeS fails; particularly it produces the acoustic peaks in the cosmic microwave background radiation and the matter power spectrum \cite{Moffat:2007ju}, and it is consistent with gravitational wave's data \cite{Green:2017qcv, Rahvar:2018nhx}.
We would like to study the gravitomagnetism in Moffat's theory. In the first section we review Moffat's theory of gravity around the Earth space-time geometry\cite{Moffat:2014aja}. In the second section we present the gravitomagnetism approximation to Moffat's theory around the space-time of the Earth. We report how Gravity Probe B \cite{Everitt: }, LAGEOS and LAGEOS 2\cite{LAGEOS}, and with a number of GRACE and Laser Lunar ranging measurements \cite{Murphy:2007nt} constrain the $\alpha$ parameter to $|\alpha|<0.0013$. In the last section we provide a discussion.
\section{The action of MOG theory around the Earth}
The action of MOG theory \cite{Moffat:2005si} is given by:
\begin{eqnarray}
S = \int d^4x (L_G + L_\Phi + L_S + L_M)\,,
\end{eqnarray}
where $L_M$ is the Lagrangian density of the matter and
\begin{eqnarray}
L_S &=& \sqrt{-g}
\left\{
\frac{1}{G^3} [\frac{1}{2} g^{\mu\nu} \nabla_\mu G \nabla_\nu G - V(G)]\right.\nonumber
\\
& &~~~~\left.+ \mu^2 G [\frac{1}{2}g^{\mu\nu} \nabla_\mu \mu \nabla_\nu \mu - V(\mu)]
\right\} \,,\\
L_\Phi &=& \frac{1}{4\pi} \sqrt{-g}
\left[ \frac{1}{4} B^{\mu\nu} B_{\mu\nu} -\frac{1}{2} \mu^2 \Phi^\mu \Phi_\mu + V({\Phi})
\right]\,,\\
L_G &=& \frac{1}{16\pi G} \sqrt{-g} (R+2\Lambda)\,,
\end{eqnarray}
are the Lagrangian densities of the scalar, the vector and the tensor, respectively. $G(x)$ is related to the Newton's constant. $\mu(x)$ represents the mass of the vector field, $B_{\mu\nu}= \partial_\mu \Phi_\nu - \partial_\nu \Phi_\mu$, and $V(G), V(\mu), V(\Phi)$ are potential.
Following \cite{Moffat:2014aja}, ``we neglect the mass of the $\Phi_\mu$ field, for in the determination of galaxy rotation curves and galactic cluster dynamics $\mu=0.042\,(\rm kpc)^{-1}$, which corresponds to the vector field $\Phi_\mu$ mass $m_\Phi=2.6\times 10^{-28}$ eV~\cite{MoffatRahvar1,TothMoffat,MoffatRahvar2}. The smallness of the $\Phi_\mu$ field mass in the present universe justifies our ignoring it when solving the field equations for compact objects such as the vicinity of Earth, neutron stars and black holes. However, the scale $\mu=0.042\,{\rm kpc}^{-1}$ does play an important role in fitting galaxy rotation curves and cluster dynamics". Ref. \cite{Green:2019cqm} provides the latest most detailed fitting of MOG to galaxy dynamics. Around the Earth space-time geometry $G_N$ is constant. So we set $\partial_\mu G=0$. Since $\mu$ is very small we ignore $\partial_\mu \mu$ at the vicinity of the Earth. We ignore the cosmological constant as well. Since the vector field around the space-time geometry of the Earth is small, and assuming that $V(\Phi)$ has a Taylor expansion around $\Phi_\mu=0$, we can ignore $V(\Phi)$ as well. These simplify Moffat's action at the vicinity of the Earth to \cite{Moffat:2014aja}:
\begin{equation}\label{MoGLB}
S \,=\, \frac{1}{16\pi G} \int d^4 x \sqrt{-g} (R + G B^{\mu \nu} B_{\mu\nu} )\,.
\end{equation}
The matter current density is defined in terms of the matter action $S_M$:
\begin{equation}
\frac{1}{\sqrt{-g}} \frac{\delta S_M}{\delta \Phi_\mu} \,=\, - J^\mu\,,
\end{equation}
where $J^\mu$ is proportional to the mass density. A test particle action is given by
\begin{equation}\label{STP}
S_{TP} = - m \int d\tau\sqrt{g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu} - k m \int d\tau \Phi_\mu \dot{x}^\mu \,,
\end{equation}
where dot is variation with respect to $\tau$ and $m$ denotes the test particle's mass. This means that geodesics in the Randers's geometry describes orbits of particles in the MOG \cite{Randers}. The coupling constant is assumed to be proportional to the mass of the test particle and the proportionality factor reads:
\begin{subequations}
\label{alphak}
\begin{eqnarray}
k &=& \pm \sqrt{\alpha G_N}\,,\\
\alpha & =& \frac{G -G_N }{G_N}\,,
\end{eqnarray}
\end{subequations}
where $G_N$ is the gravitational Newton's constant.
Let an auxiliary field $e(\tau)$ be defined on the world line of the test particle. Then consider the following action:
\begin{equation}\label{STPxe}
S_{TP}[x,e] = - m \int d\tau (\frac{1}{2 e} g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu + \frac{e}{2}) - k m \int d\tau \Phi_\mu \dot{x}^\mu\,,
\end{equation}
the variation of which with respect to $e(\tau)$ yields:
\begin{equation}
\frac{\delta S_{TP}[x,e] }{\delta e} = 0 ~\to~ e(\tau) = \sqrt{g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu}
\end{equation}
inserting which in \eqref{STPxe} reproduces \eqref{STP}. So \eqref{STPxe} is equivalent to \eqref{STP}. Notice that \eqref{STPxe} is invariant under the reparametrization of the world-line:
\begin{eqnarray}
\tau &\to & \tilde{\tau} = \tilde{\tau}(\tau)\,,\\
e(\tau) &\to & \tilde{e}(\tilde{\tau}) = \frac{d \tau}{d\tilde{\tau}} e(\tau)\,.
\end{eqnarray}
The reparametrization invariance allows to set
\begin{eqnarray}
e(\tau) &=& 1\,,\\
g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu &=&1
\end{eqnarray}
as the standard parametrization of the world-line. Doing so yields:
\begin{equation}\label{STPx}
S_{TP}[x] = - m \int d\tau (\frac{1}{2 } g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu + k m \Phi_\mu \dot{x}^\mu\,),
\end{equation}
We would like to study the theory around slow moving mass distribution. Consider that the we have a mass distribution composed of $N$ particles. The action of the these particles follow from \eqref{STPx}:
\begin{equation}
S_M \,=\, \sum_{i=1}^N S_{TP}[m_i,x_i] + \sum_{i\neq j} V(i,j) \,,
\end{equation}
where $S_{TP}[m_i,x_i]$ for each particle is given in \eqref{STPx} and $V(i,j)$ defines the interaction between particles. In the continuum limit, where we have a fluid at equilibrium, this suggests that
\begin{eqnarray}
S_M &=& - \int d^4 x \sqrt{-g} (\frac{1}{2}\rho g_{\mu\nu} u^\mu u^\nu + k \rho \Phi_\nu u^\nu) \\&&+ \text{Interaction of fluid with itself} \,,\nonumber
\end{eqnarray}
where $\rho$ is the local density of the fluid and $u^\mu$ is the local four velocity of the fluid. Notice that the fluid interaction with itself is not a functional of $\phi_\mu$. In so doing the source current for the vector/gauge field reads:
\begin{equation}
J^\mu = k \rho u^\mu.
\end{equation}
The equation of motion for the gauge field, in the gauge of $\nabla_\mu \Phi^\mu =0$, reads
\begin{equation}
\label{MOGPhi}
\Box \Phi^\mu \,=\,- 4 \pi k J^\mu\,.
\end{equation}
\section{Gravitomagnetism in MOG around the Earth}
In order to address the gravitomagnetism, we look at a small deviation from the flat space-time geometry:
\begin{eqnarray}
g_{\mu\nu} &=& \eta_{\mu\nu} + h_{\mu\nu}\,,\\
\Phi_\mu & =& 0 + \phi_\mu\,,
\end{eqnarray}
where in the natural unites it holds
\begin{eqnarray}
|h_{\mu\nu}|&\ll&1\,,\\
|\phi_\mu| &\ll&1\,.
\end{eqnarray}
We then consider a slow moving test particle:
\begin{eqnarray}
|\dot{t}|&\approx&1\,,\\
|\dot{x}^i| &\ll&c\,.
\end{eqnarray}
These simplify the action of the test particle \eqref{STPx} to
\begin{equation}
\label{TPEM}
S_{TP} \,=\,m \int dt (\frac{1}{2} |\dot{x}^i|^2 + \frac{1}{2} h_{00} + k \phi_0 + (h_{0i} + k \phi^i)\dot{x}^i)\,,
\end{equation}
in analogy with electromagnetism, the gravitoelectric and gravitomagnetic potentials are then identified to:
\begin{subequations}
\label{EBgravity}
\begin{eqnarray}
\phi_{\text{Phys}} & = & \frac{1}{2} h_{00} + k \phi_0 \,,\\
A^{\text{Phys}}_i & = & h_{0i} + k \phi_i\,.
\end{eqnarray}
\end{subequations}
The equation of motion derived from \eqref{TPEM} can be rewritten as follows
\begin{equation}
\label{appendinxneeds}
\ddot{x} = -\nabla \Phi_{\text{Phys}} + \frac{v}{c} \times (\nabla \times A_{\text{Phys}})\,.
\end{equation}
This allows interpreting $\nabla \times A$ as a gravitomagnetic field. $\nabla \times A$ causes precessions of the orbits of a test particle. This precession is referred to as the Lense-Thirring precession \cite{Lense}. Ref. \cite{Iorio:2010rk} provides a decent recent review on Lense-Thirring precession for planets and satellites in the Solar system.
The similarity between the gravitomagnetic field and magnetic field beside the spin precession formula in electrodynamics ($\dot{S}= \mu \times B , \mu = \frac{e}{2m} S$) dictates that the spin of a gyroscope precesses by \cite{padi}
\begin{equation}
\Omega_{LT} = - \frac{1}{2}\nabla \times A_{\text{Phys}}\,.
\end{equation}
This precession is called the Pugh-Schiff frame-dragging precession \cite{Pugh, Schiff}. The Pugh-Schiff frame-dragging precession due to the rotation of the earth recently has been measured by the gravity probe B with the precision of 19\% \cite{Everitt: }. The gravitomagnetic field due to geodesic effects are measured at an accuracy
of 0.28\%\cite{Everitt: }. GINGER, aiming to improve the sensitivity of the ring resonators, plans to measure the gravitomagnetic effect with a precision at least one order better than that of the gravity probe B \cite{Tartaglia:2012fd}. Also LAGEOS and LAGEOS 2, and with a number of GRACE (Gravity Recovery and Climate Experiment) have confirmed the prediction of General Relativity for the Earth's gravitomagnetic field with with an accuracy of approximately 10\% \cite{LAGEOS}. Ref. \cite{Murphy:2007nt} shows that the gravitomagnetic field of the Earth is in agreement with the Einstein theory's prediction with approximately 0.1\% accuracy via lunar laser ranging (LLR). All the results have been consistent with the Einstein prediction.
We notice that the Einstein's gravitational theory in the Gravitomagnetism approximation holds
\begin{subequations}
\label{EB-EH}
\begin{eqnarray}
\Box \phi_{EH}&=& \frac{1}{2}\Box h_{00} = 4\pi G \rho \,,\\
\Box A^i_{EH} &=&\Box h_{0i} = 16 \pi G \rho u^i \,.
\end{eqnarray}
\end{subequations}
While in Moffat's theory of gravity due to \eqref{EBgravity}, \eqref{EB-EH} and \eqref{MOGPhi} we have:
\begin{subequations}
\label{EB2}
\begin{eqnarray}
\Box \phi_{\text{Phys}}
& = & 4\pi G_N \rho \,,\\
\Box A^{\text{Phys}}_i
& = & 16\pi G_N (1 + \frac{3}{4}\alpha) \rho u^i\,.
\end{eqnarray}
\end{subequations}
where \eqref{alphak} are utilised. The theoretical consistency between Moffat's theory and Einstein theory at the level of gravitomagnetism demands $\alpha=0$. But this converts Moffat's theory to the Einstein theory and renders it useless. The consistency between Moffat's theory and the results of the Gravity Probe B demands $|\alpha| < 0.25$ and $|\alpha|<0.0037$ respectively for the measured frame-dragging and the geodetic effects \cite{Everitt: }. Its consistency with LAGEOS and LAGEOS 2, and GRACE (Gravity Recovery and Climate Experiment) demands $|\alpha|<0.13$. Its consistency with the LLR demands $|\alpha|< 0.0013$ \cite{Murphy:2007nt}.
\section{Conclusion and discussion}
We have reported that the consistency between the MOG gravitomagnetic field and that predicted by the Einstein's gravitional theory and measured by Gravity Probe B, LAGEOS and LAGEOS 2, and with a number of GRACE and Laser Lunar ranging measurements requires $|\alpha|<0.0013$.
We notice that there are two estimations for the mass of the supermassive central black hole in M87*: the stellar-dynamical model $(M=6.5 \times 10^9 M_\odot)$ and the gas-dynamical model ($M=3.5 \times 10^9\times M_\odot$). The former mass estimate is consistent with the measured size of the shadow and light emission region of M87* for GR, while the latter estimate is consistent with the MOG prediction with $\alpha=1.13^{0.30}_{-0.24}$ \cite{Moffat:2019uxp}. We observe that comparing the value of $\alpha$ from the near horizon geometry of M87*'s black hole to the value of $\alpha$ in the solar system is not as simple as it looks. One should first fix $V(G), V(\mu)$ and $V(\Phi)$ in such a way to allow the existence of an interpolating solution from $|\alpha|<0.0013$ in the solar system to its boundary in the Milky Way where $\alpha=O(10)$. The interpolating solution should not contradict any other data at the Solar system as well. Next one should show that the found potentials of the theory allow the existence of an interpolating solution from the boundary of M87* where $\alpha=O(10)$ to the event horizon of its central super massive black hole where $\alpha=1.13^{+0.30}_{-0.24}$. Taking these steps are outside the scope of the current work and are remained to be addressed.
\section*{Acknowledgements}
I would like to thank Atish Dabholkar for the invitation to ICTP, ICTP for its nice hospitality, Iva Kordic for converting a large wall in my apartment into a beautiful blackboard. I thank John Moffat for his very valuable feedback and discussion on the paper, Viktor T. Toth, Niayesh Afshordi, Constantinos Skordis, Stacy McGaugh, Takeshi Kobayashi, Paolo Creminelli, Loriano Bonora and Goran Senjanovic for discussions and email correspondences.
\providecommand{\href}[2]{#2}\begingroup\raggedright | {
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} | 9,610 |
{"url":"https:\/\/en.academic.ru\/dic.nsf\/enwiki\/388340","text":"# Rotary variable differential transformer\n\n\ufeff\nRotary variable differential transformer\n\nA rotary variable differential transformer (RVDT) is a type of electrical transformer used for measuring angular displacement.\n\nMore precisely, a Rotary Variable Differential Transformer (RVDT) is an electromechanical transducer that provides a variable alternating current (AC) output voltage that is linearly proportional to the angular displacement of its input shaft. When energized with a fixed AC source, the output signal is linear within a specified range over the angular displacement.\n\nRVDT\u2019s utilize brushless, non-contacting technology to ensure long-life and reliable, repeatable position sensing with infinite resolution. Such reliable and repeatable performance assures accurate position sensing under the most extreme operating conditions.\n\nMost RVDT are composed of a wound, laminated stator and a salient two-pole rotor. The stator, containing four slots, contains both the primary winding and the two secondary windings. Some secondary windings may also be connected together.\n\nOperation of RVDT's\n\nThe two induced voltages of the secondary windings, $V_1$ and $V_2$, varies lineary to the mechanical angle of the rotor, \u03b8:\n\n:$heta = G cdot left\\left( frac\\left\\{V_1 - V_2\\right\\}\\left\\{V_1 + V_2\\right\\} ight\\right)$\n\nwhere $G$ is the gain or sensitivity. The second voltage can be reverse determined by:\n\n:$V_2 = V_1 pm G cdot heta$\n\nThe difference $V_1 - V_2$ gives a proportional voltage:\n\n:$Delta V = 2 cdot G cdot heta$\n\nand the sum of the voltages is a constant:\n\n:$C= sum V = 2 cdot V_0$\n\nThis constant gives the RVDT great stability of the angular information, independence of the input voltage or frequency, or temperature, and enables it to also detect a malfunction.\n\nPutting the above mathematical equations in some theorotical form, the working of RVDT can be explanied as below :-\n\nBasic RVDT construction and operation is provided by rotating an iron-core bearing supported within a housed stator assembly. The housing is passivated stainless steel. The stator consists of a primary excitation coil and a pair of secondary output coils.A fixed alternating current excitation is applied to the primary stator coil that is electromagnetically coupled to the secondary coils. This coupling is proportional to the angle of the input shaft. The output pair is structured so that one coil is in-phase with the excitation coil, and the second is 180 degrees out-of-phase with the excitation coil.When the rotor is in a position that directs the available flux equally in both the in-phase and out-of-phase coils, the output voltages cancel and result in a zero value signal. This is referred to as the electrical zero position or E.Z. When the rotor shaft is displaced from E.Z., the resulting output signals have a magnitude and phase relationship proportional to the direction of rotation.Because RVDT\u2019s perform essentially like a transformer, excitation voltages changes will cause directly proportional changes to the output (transformation ratio). However, the voltage out to excitation voltage ratio will remain constant. Since most RVDT signal conditioning systems measure signal as a function of the transformation ratio (TR), excitation voltage drift beyond 7.5% typically has no effect on sensor accuracy and strict voltage regulation is not typically necessary. Excitation frequency should be controlled within +\/- 1% to maintain accuracy\n\nAlthough the RVDT can theoretically operate between \u00b145\u00b0, accuracy decreases quickly after \u00b135\u00b0. Thus, it operational limits lies mostly within \u00b130\u00b0, but some up to \u00b140\u00b0. Certain types can operate up to \u00b160\u00b0.\n\nThe advantages of the RVDT are :\n* low sensitivity to temperature, primary voltage & frequency variations\n* sturdiness\n* low cost\n* simple control electronics\n* small size\n\nRVDT varieties\n\nAn RVDT can also be designed with two laminations, one containing the primary and the other, the secondaries. These types can operate on larger rotations.\n\nA similar transformer is called the Rotary Variable Transformer and contains only one secondary winding giving only one voltage:\n\n:$V = G cdot heta$\n\n* Rotary encoder\n* Synchro\n* Resolver\n* LVDT, the RVDT function in the longitudinal position.\n\nWikimedia Foundation. 2010.\n\n### Look at other dictionaries:\n\n\u2022 Rotary Variable Differential Transformer \u2014 Un RVDT (de l anglais Rotary Variable Differential Transformer) est un capteur \u00e9lectrique actif (inductif) de d\u00e9placements de rotation. Le d\u00e9battement est en g\u00e9n\u00e9ral limit\u00e9 jusqu \u00e0 +\/ 40\u00b0. Principe Un RVDT est compos\u00e9 le plus souvent d un paquet\u2026 \u2026 \u00a0 Wikip\u00e9dia en Fran\u00e7ais\n\n\u2022 Rotary encoder \u2014 A rotary encoder, also called a shaft encoder, is an electro mechanical device used to convert the angular position of a shaft or axle to an analog or digital code, making it an angle transducer. These devices are used in industrial controls,\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Variable capacitor \u2014 Rotary variable capacitor A variable capacitor (also known as a variable air condenser ) is a capacitor whose capacitance may be intentionally and repeatedly changed mechanically or electronically. Variable capacitors are often used in L\/C\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 RVDT \u2014 Rotary Variable Differential Transformer Un RVDT (de l anglais Rotary Variable Differential Transformer) est un capteur \u00e9lectrique actif (inductif) de d\u00e9placements de rotation. Le d\u00e9battement est en g\u00e9n\u00e9ral limit\u00e9 jusqu \u00e0 +\/ 40\u00b0. Principe Un RVDT \u2026 \u00a0 Wikip\u00e9dia en Fran\u00e7ais\n\n\u2022 RVDT \u2014 Rotary Variable Differential Transformer Contributor:\u00a0GSFC \u2026 \u00a0 NASA Acronyms\n\n\u2022 Electronic component \u2014 Various components An electronic component is a basic electronic element and may be available in a discrete form having two or more electrical terminals (or leads). These are intended to be connected together, usually by soldering to a printed\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Abk\u00fcrzungen\/Luftfahrt\/L\u2013R \u2014 Dies ist der vierte Teil der Liste Abk\u00fcrzungen\/Luftfahrt. Liste der Abk\u00fcrzungen Teil 1 A A Teil 2 B\u2013D B; C; D Teil 3 E\u2013K \u2026 \u00a0 Deutsch Wikipedia\n\n\u2022 List of sensors \u2014 * Accelerometer * Touch sensor * Active pixel sensor * Air flow meter * Alarm sensor * Bedwetting alarm * Bhangmeter * Biochip * Biosensor * Breathalyzer * Capacitance probe * Carbon paste electrode * Carbon monoxide detector * Catadioptric\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Position sensor \u2014 A position sensor is any device that enables position measurement. It can either be an absolute position sensor or a relative one (displacement sensor). Position sensors can be either linear or angular.Some position sensors available today: *\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 List of Tesla patents \u2014 Below is a list of Tesla patents. Dr. Nikola Tesla was an inventor who obtained around 300 patents [Snezana Sarbo, [http:\/\/www.tesla symp06.org\/papers\/Tesla Symp06 Sarboh.pdfNikola Tesla s Patents] , Sixth International Symposium Nikola Tesla,\u2026 \u2026 \u00a0 Wikipedia","date":"2020-09-27 20:22:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 9, \"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\": 0.4484080672264099, \"perplexity\": 9019.384023206898}, \"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-2020-40\/segments\/1600401578485.67\/warc\/CC-MAIN-20200927183616-20200927213616-00090.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.beatthegmat.com\/remainder-t297233.html","text":"\u2022 NEW! FREE Beat The GMAT Quizzes\nHundreds of Questions Highly Detailed Reporting Expert Explanations\n\u2022 7 CATs FREE!\nIf you earn 100 Forum Points\n\nEngage in the Beat The GMAT forums to earn\n100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Remainder tagged by: rolandprowess ##### This topic has 3 expert replies and 0 member replies ## Remainder How many integers from 0 to 1000 inclusive, have a remainder of 3 when divided by 16? A. 61 B. 62 C. 63 D. 64 E. 65 QA is c Can some experts help me with this? How to come up with the correct answer? Thanks ### GMAT\/MBA Expert Legendary Member Joined 14 Jan 2015 Posted: 2666 messages Followed by: 125 members Upvotes: 1153 GMAT Score: 770 Roland2rule wrote: How many integers from 0 to 1000 inclusive, have a remainder of 3 when divided by 16? A. 61 B. 62 C. 63 D. 64 E. 65 QA is c Can some experts help me with this? How to come up with the correct answer? Thanks Call the integers that give a remainder of 3 when divided by 16: 16x + 3 Smallest number in the range that gives a remainder of 3 when divided by 16: 3; (when x = 0) Largest number in the range that gives a remainder of 3 when divided by 16: 995; (when x = 62) note: it's easy to see that 16*60 + 3 = 963. From there, it's straightforward to extrapolate the largest possible value. 16*61 + 3 = 979; 16*62 + 3 = 995. Number of integers between 0 and62 inclusive: 63. The answer is C _________________ Veritas Prep | GMAT Instructor Veritas Prep Reviews Save$100 off any live Veritas Prep GMAT Course\n\nEnroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now!\n\n### GMAT\/MBA Expert\n\nGMAT Instructor\nJoined\n04 Oct 2017\nPosted:\n551 messages\nFollowed by:\n11 members\n180\nHello Roland2rule.\n\nThe idea here is to find a number N such that $$16\\cdot N+3\\le1000\\ and\\ N\\ge0.$$\n\nLet's try some test values:\n\n$$N=50=>16\\cdot50+3=803\\ \\le\\ 1000.$$\n\nThis tell us that all there are 51 integers that satisfy the condition above. Now, let's try with a greater N.\n\n$$N=60\\ =>\\ 16\\cdot60+3=963\\le1000.$$\n\nIf we try with N=62 and N=63 we will get\n\n$$N=62\\ =>\\ 16\\cdot62+3=995\\le1000$$ $$N=63\\ =>\\ 16\\cdot63+3=1011>1000$$\n\nSo, the 63 doesn't satisfy the condition.\n\nIn conclusion, all the integers from 0 to 62 satisfy the condition, that is to say, there are 63 integers that have a remainder of 3 when divided by 16.\n\nRegards.\n\n_________________\nGMAT Prep From The Economist\nWe offer 70+ point score improvement money back guarantee.\nOur average student improves 98 points.\n\nFree 7-Day Test Prep with Economist GMAT Tutor - Receive free access to the top-rated GMAT prep course including a 1-on-1 strategy session, 2 full-length tests, and 5 ask-a-tutor messages. Get started now.\n\n### GMAT\/MBA Expert\n\nGMAT Instructor\nJoined\n09 Apr 2015\nPosted:\n1465 messages\nFollowed by:\n19 members\n39\nRoland2rule wrote:\nHow many integers from 0 to 1000 inclusive, have a remainder of 3 when divided by 16?\n\nA. 61\nB. 62\nC. 63\nD. 64\nE. 65\nThe first number from 0 to 1000 inclusive that has a remainder of 3 when divided by 16 is 3, and the last number is 995.\n\nThus, there are (995 - 3)\/16 + 1 = 63 integers from 0 to 1000 inclusive that have a remainder of 3 when divided by 16.\n\n_________________\n\nJeffrey Miller\njeff@targettestprep.com\n\nSee why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews\n\n\u2022 Magoosh\nStudy with Magoosh GMAT prep\n\nAvailable with Beat the GMAT members only code\n\n\u2022 5 Day FREE Trial\nStudy Smarter, Not Harder\n\nAvailable with Beat the GMAT members only code\n\n\u2022 5-Day Free Trial\n5-day free, full-access trial TTP Quant\n\nAvailable with Beat the GMAT members only code\n\n\u2022 Free Practice Test & Review\nHow would you score if you took the GMAT\n\nAvailable with Beat the GMAT members only code\n\n\u2022 Award-winning private GMAT tutoring\nRegister now and save up to $200 Available with Beat the GMAT members only code \u2022 Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code \u2022 Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code \u2022 Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code \u2022 FREE GMAT Exam Know how you'd score today for$0\n\nAvailable with Beat the GMAT members only code\n\n\u2022 1 Hour Free\nBEAT THE GMAT EXCLUSIVE\n\nAvailable with Beat the GMAT members only code\n\n### Top First Responders*\n\n1 Ian Stewart 43 first replies\n2 Brent@GMATPrepNow 35 first replies\n3 Scott@TargetTestPrep 33 first replies\n4 Jay@ManhattanReview 28 first replies\n5 GMATGuruNY 19 first replies\n* Only counts replies to topics started in last 30 days\nSee More Top Beat The GMAT Members\n\n### Most Active Experts\n\n1 Scott@TargetTestPrep\n\nTarget Test Prep\n\n149 posts\n2 Max@Math Revolution\n\nMath Revolution\n\n93 posts\n3 Brent@GMATPrepNow\n\nGMAT Prep Now Teacher\n\n53 posts\n4 Ian Stewart\n\nGMATiX Teacher\n\n52 posts\n5 GMATGuruNY\n\nThe Princeton Review Teacher\n\n32 posts\nSee More Top Beat The GMAT Experts","date":"2019-07-19 17:24:09","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\": 0.18898436427116394, \"perplexity\": 9185.886264043173}, \"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-2019-30\/segments\/1563195526324.57\/warc\/CC-MAIN-20190719161034-20190719183034-00426.warc.gz\"}"} | null | null |
Home News Emergency Response Injured woman airlifted to hospital from Spanish Point
Injured woman airlifted to hospital from Spanish Point
Rescue 115 on the helipad at University Hospital Limerick – File Photo: © Pat Flynn 2017
Emergency services responding to an incident in Co Clare this afternoon were provided with Garda escorts as part of a traffic management plan put in place for the Dubai Duty Free Irish Open golf being held at Lahinch.
Coast Guard and National Ambulance Service crews had to travel through Lahinch where tens of thousands of people have gathered for the European Tour golfing event.
A woman had to be airlifted to hospital after she was injured in a fall this afternoon. It's understood she suffered leg injuries after she fell on rocks at around 4.30pm.
Coast Guard volunteers from Doolin, who had to pass through Lahinch to get to the scene, received a Garda escort involving patrol cars and motorcycle outriders. An ambulance from Ennistymon was also provided with a Garda escort to ensure they also reached the scene without delay.
The Shannon based Irish Coast Guard helicopter was also tasked to the incident.
The woman, understood to be in her 60s and from Limerick, was treated at the scene by ambulance paramedics and the helicopter winchman/paramedic before she was taken on board the aircraft and flown to University Hospital Limerick for treatment.
Members of Garda Road Policing Unit provided an escort for the emergency services – Photo: © Pat Flynn 2018
A Garda spokesman confirmed: "We have a very comprehensive traffic management plan in place for the Irish Open and this includes providing for all contingencies including facilitating emergency services who have to pass through Lahinch to reach incidents. This involves ensuring emergency services can safely and efficiently pass through areas where crowds have gathered and has been very effective."
National Ambulance Service
Two men injured in Sixmilebridge shooting
Man arrested after car crashes into house near Kilkee
Woman killed in Kilrush collision
Flight diverts to Shannon with 'unruly passenger' | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,813 |
set -euo pipefail
ROOT_DIR=$(cd "$(dirname $0)/.." && pwd)
# Check the number of parameters
test $# -eq 1 || { echo "Usage: $(basename $0) PROJECT_ID"; exit 1; }
PROJECT_ID=$1
# Authenticate to GCP
TMP_DIR="$ROOT_DIR/build/tmp"
mkdir -p ${TMP_DIR}
GCLOUD_SERVICE_KEY_JSON="$TMP_DIR/gcloud-service-key.json"
echo ${GCLOUD_SERVICE_KEY} | base64 -di > ${GCLOUD_SERVICE_KEY_JSON}
gcloud auth activate-service-account --key-file ${GCLOUD_SERVICE_KEY_JSON}
rm ${GCLOUD_SERVICE_KEY_JSON}
# Set project
gcloud config set project ${PROJECT_ID}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,877 |
In regions with large tracer gradients and/or velocities the advection scheme that is used in many ocean models leads to significant under- and overshoot of the tracer values. In the case of the U.K. Fine Resolution Antarctic Model (FRAM), this lead to unphysical negative surface temperatures in some regions and overheating in others. In this paper, a new advection scheme is proposed and tested in the ocean model context using a limited-area model centered on a region where problems occurred in FRAM. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,147 |
namespace v8 {
namespace internal {
namespace compiler {
// Converts InstructionOperands from a given instruction to
// architecture-specific
// registers and operands after they have been assigned by the register
// allocator.
class InstructionOperandConverter {
public:
InstructionOperandConverter(CodeGenerator* gen, Instruction* instr)
: gen_(gen), instr_(instr) {}
// -- Instruction operand accesses with conversions --------------------------
Register InputRegister(size_t index) {
return ToRegister(instr_->InputAt(index));
}
FloatRegister InputFloatRegister(size_t index) {
return ToFloatRegister(instr_->InputAt(index));
}
DoubleRegister InputDoubleRegister(size_t index) {
return ToDoubleRegister(instr_->InputAt(index));
}
Simd128Register InputSimd128Register(size_t index) {
return ToSimd128Register(instr_->InputAt(index));
}
double InputDouble(size_t index) { return ToDouble(instr_->InputAt(index)); }
float InputFloat32(size_t index) { return ToFloat32(instr_->InputAt(index)); }
int32_t InputInt32(size_t index) {
return ToConstant(instr_->InputAt(index)).ToInt32();
}
uint32_t InputUint32(size_t index) {
return bit_cast<uint32_t>(InputInt32(index));
}
int64_t InputInt64(size_t index) {
return ToConstant(instr_->InputAt(index)).ToInt64();
}
int8_t InputInt8(size_t index) {
return static_cast<int8_t>(InputInt32(index));
}
int16_t InputInt16(size_t index) {
return static_cast<int16_t>(InputInt32(index));
}
uint8_t InputInt3(size_t index) {
return static_cast<uint8_t>(InputInt32(index) & 0x7);
}
uint8_t InputInt4(size_t index) {
return static_cast<uint8_t>(InputInt32(index) & 0xF);
}
uint8_t InputInt5(size_t index) {
return static_cast<uint8_t>(InputInt32(index) & 0x1F);
}
uint8_t InputInt6(size_t index) {
return static_cast<uint8_t>(InputInt32(index) & 0x3F);
}
ExternalReference InputExternalReference(size_t index) {
return ToExternalReference(instr_->InputAt(index));
}
Handle<Code> InputCode(size_t index) {
return ToCode(instr_->InputAt(index));
}
Label* InputLabel(size_t index) { return ToLabel(instr_->InputAt(index)); }
RpoNumber InputRpo(size_t index) {
return ToRpoNumber(instr_->InputAt(index));
}
Register OutputRegister(size_t index = 0) {
return ToRegister(instr_->OutputAt(index));
}
Register TempRegister(size_t index) {
return ToRegister(instr_->TempAt(index));
}
FloatRegister OutputFloatRegister() {
return ToFloatRegister(instr_->Output());
}
DoubleRegister OutputDoubleRegister() {
return ToDoubleRegister(instr_->Output());
}
Simd128Register OutputSimd128Register() {
return ToSimd128Register(instr_->Output());
}
// -- Conversions for operands -----------------------------------------------
Label* ToLabel(InstructionOperand* op) {
return gen_->GetLabel(ToRpoNumber(op));
}
RpoNumber ToRpoNumber(InstructionOperand* op) {
return ToConstant(op).ToRpoNumber();
}
Register ToRegister(InstructionOperand* op) {
return LocationOperand::cast(op)->GetRegister();
}
FloatRegister ToFloatRegister(InstructionOperand* op) {
return LocationOperand::cast(op)->GetFloatRegister();
}
DoubleRegister ToDoubleRegister(InstructionOperand* op) {
return LocationOperand::cast(op)->GetDoubleRegister();
}
Simd128Register ToSimd128Register(InstructionOperand* op) {
return LocationOperand::cast(op)->GetSimd128Register();
}
Constant ToConstant(InstructionOperand* op) {
if (op->IsImmediate()) {
return gen_->instructions()->GetImmediate(ImmediateOperand::cast(op));
}
return gen_->instructions()->GetConstant(
ConstantOperand::cast(op)->virtual_register());
}
double ToDouble(InstructionOperand* op) {
return ToConstant(op).ToFloat64().value();
}
float ToFloat32(InstructionOperand* op) { return ToConstant(op).ToFloat32(); }
ExternalReference ToExternalReference(InstructionOperand* op) {
return ToConstant(op).ToExternalReference();
}
Handle<Code> ToCode(InstructionOperand* op) {
return ToConstant(op).ToCode();
}
const Frame* frame() const { return gen_->frame(); }
FrameAccessState* frame_access_state() const {
return gen_->frame_access_state();
}
Isolate* isolate() const { return gen_->isolate(); }
Linkage* linkage() const { return gen_->linkage(); }
protected:
CodeGenerator* gen_;
Instruction* instr_;
};
// Eager deoptimization exit.
class DeoptimizationExit : public ZoneObject {
public:
explicit DeoptimizationExit(int deoptimization_id, SourcePosition pos)
: deoptimization_id_(deoptimization_id), pos_(pos) {}
int deoptimization_id() const { return deoptimization_id_; }
Label* label() { return &label_; }
SourcePosition pos() const { return pos_; }
private:
int const deoptimization_id_;
Label label_;
SourcePosition const pos_;
};
// Generator for out-of-line code that is emitted after the main code is done.
class OutOfLineCode : public ZoneObject {
public:
explicit OutOfLineCode(CodeGenerator* gen);
virtual ~OutOfLineCode();
virtual void Generate() = 0;
Label* entry() { return &entry_; }
Label* exit() { return &exit_; }
const Frame* frame() const { return frame_; }
TurboAssembler* tasm() { return tasm_; }
OutOfLineCode* next() const { return next_; }
private:
Label entry_;
Label exit_;
const Frame* const frame_;
TurboAssembler* const tasm_;
OutOfLineCode* const next_;
};
inline bool HasCallDescriptorFlag(Instruction* instr,
CallDescriptor::Flag flag) {
return MiscField::decode(instr->opcode()) & flag;
}
} // namespace compiler
} // namespace internal
} // namespace v8
#endif // V8_COMPILER_BACKEND_CODE_GENERATOR_IMPL_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,128 |
Q: numpy sum result and transpose on it give same answer I'm having some troubles to understand how to do this thing in the right way.
print np.sum(X,axis=1)
and
print np.sum(X,axis=1).T
gives me the same result.
What is the best way to fix it?
Why such a thing should not be considered as a bug in numpy?
For example: X=[[1,2],[3,4]].
For first result I wish to get array([[3,7]]), and array([[3],[7]]) for second. can be in the opposite way too..doesnt really matter.
A: I presume you want to transpose first:
print np.sum(X.T,axis=1)
You are getting a flat array after summing so obviously transposing a 1d array is going to give you the same output as the original array when transposed.
In [14]: X=np.array([[1,2,3],[4,5,6]])
In [15]: np.sum(X, axis=1)
Out[15]: array([ 6, 15])
In [16]: np.sum(X, axis=1).T
Out[16]: array([ 6, 15])
In [17]: np.sum(X.T, axis=1)
Out[17]: array([5, 7, 9])
A: If it's summing across the wrong axis, why not change it?
>> np.sum(np.array([[1, 2, 3], [2, 3, 4]]), axis=1)
array([6, 9])
>> np.sum(np.array([[1, 2, 3], [2, 3, 4]]), axis=0)
array([3, 5, 7])
Edit
You can reshape the resulting array, like this:
>> np.sum(np.array([[1, 2, 3], [2, 3, 4]]), axis=1).reshape((2, 1))
array([[6],
[9]])
A: If you want to keep the extra singleton dimension after computing the sum, you can pass keepdims=True to sum:
X = np.array([[1, 2, 3], [2, 3, 4]])
print(np.sum(X, axis=0).shape)
# (3,)
print(np.sum(X, axis=0, keepdims=1).shape)
# (1, 3)
print(np.sum(X, axis=1, keepdims=1).shape)
# (2, 1)
Apart from using keepdims, you could also reshape the output to insert a new axis to replace the one that was lost in the reduction, e.g.:
# the '-1' here means that numpy will infer the size in the first dimension to
# match the number of elements in the result array
print(np.sum(X, axis=1).reshape(-1, 1).shape)
# (2, 1)
print(np.sum(X, axis=1)[:, np.newaxis].shape)
# (2, 1)
# indexing with 'None' is equivalent to 'np.newaxis'
print(np.sum(X, axis=1)[:, None].shape)
# (2, 1)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,337 |
from __future__ import absolute_import, print_function, unicode_literals
import os
import io
import unittest
import subprocess
import tempfile
import shutil
import re
from vcstools.svn import SvnClient, canonical_svn_url_split, get_remote_contents
class SvnClientUtilTest(unittest.TestCase):
def test_canonical_svn_url_split(self):
self.assertEqual({'root': 'foo',
'type': None,
'name': None, 'subfolder': None,
'query': None, 'fragment': None},
canonical_svn_url_split('foo'))
self.assertEqual({'root': None,
'type': None,
'name': None, 'subfolder': None,
'query': None, 'fragment': None},
canonical_svn_url_split(None))
self.assertEqual({'root': 'svn://gcc.gnu.org/svn/gcc',
'type': 'branches',
'name': 'foo', 'subfolder': None,
'query': None, 'fragment': None},
canonical_svn_url_split('svn://gcc.gnu.org/svn/gcc/branches/foo'))
self.assertEqual({'root': 'svn://gcc.gnu.org/svn/gcc',
'type': 'branches',
'name': 'foo', 'subfolder': None,
'query': None, 'fragment': None},
canonical_svn_url_split('svn://gcc.gnu.org/svn/gcc/branches/foo/'))
self.assertEqual({'root': 'svn://gcc.gnu.org/svn/gcc',
'type': 'branches',
'name': 'foo', 'subfolder': 'sub/bar',
'query': None, 'fragment': None},
canonical_svn_url_split('svn://gcc.gnu.org/svn/gcc/branches/foo/sub/bar'))
self.assertEqual({'root': 'svn://gcc.gnu.org/svn/gcc',
'type': 'trunk',
'name': None, 'subfolder': None,
'query': None, 'fragment': None},
canonical_svn_url_split('svn://gcc.gnu.org/svn/gcc/trunk'))
self.assertEqual({'root': 'svn://gcc.gnu.org/svn/gcc',
'type': 'trunk',
'name': None, 'subfolder': 'sub',
'query': None, 'fragment': None},
canonical_svn_url_split('svn://gcc.gnu.org/svn/gcc/trunk/sub'))
self.assertEqual({'root': 'svn://gcc.gnu.org/svn/gcc',
'type': 'trunk',
'name': None, 'subfolder': 'sub/foo',
'query': None, 'fragment': None},
canonical_svn_url_split('svn://gcc.gnu.org/svn/gcc/trunk/sub/foo'))
self.assertEqual({'root': 'svn://gcc.gnu.org/svn/gcc',
'type': 'tags',
'name': '1.2.3', 'subfolder': None,
'query': None, 'fragment': None},
canonical_svn_url_split('svn://gcc.gnu.org/svn/gcc/tags/1.2.3'))
self.assertEqual({'root': 'svn://gcc.gnu.org/svn/gcc',
'type': 'tags',
'name': '1.2.3', 'subfolder': 'sub/foo',
'query': None, 'fragment': None},
canonical_svn_url_split('svn://gcc.gnu.org/svn/gcc/tags/1.2.3/sub/foo'))
self.assertEqual({'root': 'file://localhost/svn/gcc',
'type': 'tags',
'name': '1.2.3', 'subfolder': 'sub/foo',
'query': None, 'fragment': None},
canonical_svn_url_split('file://localhost/svn/gcc/tags/1.2.3/sub/foo'))
self.assertEqual({'root': 'https://frodo@gcc.gnu.org/svn/gcc',
'type': 'tags',
'name': '1.2.3', 'subfolder': 'sub/foo',
'query': 'pw=guest', 'fragment': 'today'},
canonical_svn_url_split('https://frodo@gcc.gnu.org/svn/gcc/tags/1.2.3/sub/foo?pw=guest#today'))
class SvnClientTestSetups(unittest.TestCase):
@classmethod
def setUpClass(self):
self.root_directory = tempfile.mkdtemp()
self.directories = dict(setUp=self.root_directory)
self.remote_path = os.path.join(self.root_directory, "remote")
self.init_path = os.path.join(self.root_directory, "init")
# create a "remote" repo
subprocess.check_call("svnadmin create %s" % self.remote_path, shell=True, cwd=self.root_directory)
self.local_root_url = "file://localhost" + self.remote_path
self.local_url = self.local_root_url + "/trunk"
# create an "init" repo to populate remote repo
subprocess.check_call("svn checkout %s %s" % (self.local_root_url, self.init_path), shell=True, cwd=self.root_directory)
for cmd in [
"mkdir trunk",
"mkdir branches",
"mkdir tags",
"svn add trunk branches tags",
"touch trunk/fixed.txt",
"svn add trunk/fixed.txt",
"svn commit -m initial"]:
subprocess.check_call(cmd, shell=True, cwd=self.init_path)
self.local_version_init = "-r1"
# files to be modified in "local" repo
for cmd in [
"touch trunk/modified.txt",
"touch trunk/modified-fs.txt",
"svn add trunk/modified.txt trunk/modified-fs.txt",
"svn commit -m initial"]:
subprocess.check_call(cmd, shell=True, cwd=self.init_path)
self.local_version_second = "-r2"
for cmd in [
"touch trunk/deleted.txt",
"touch trunk/deleted-fs.txt",
"svn add trunk/deleted.txt trunk/deleted-fs.txt",
"svn commit -m modified"]:
subprocess.check_call(cmd, shell=True, cwd=self.init_path)
self.local_version_master = "-r3"
# files to be modified in "local" repo
for cmd in [
"mkdir branches/foo",
"touch branches/foo/modified.txt",
"svn add branches/foo",
"svn commit -m 'foo branch'"]:
subprocess.check_call(cmd, shell=True, cwd=self.init_path)
self.branch_url = self.local_root_url + "/branches/foo"
self.local_version_foo_branch = "-r4"
self.local_path = os.path.join(self.root_directory, "local")
@classmethod
def tearDownClass(self):
for d in self.directories:
shutil.rmtree(self.directories[d])
def tearDown(self):
if os.path.exists(self.local_path):
shutil.rmtree(self.local_path)
class SvnClientTest(SvnClientTestSetups):
def test_get_url_by_reading(self):
client = SvnClient(self.local_path)
client.checkout(self.local_url)
self.assertTrue(client.path_exists())
self.assertTrue(client.detect_presence())
self.assertEqual(self.local_url, client.get_url())
self.assertEqual(client.get_version(), self.local_version_master)
self.assertEqual(client.get_version("PREV"), "-r2")
self.assertEqual(client.get_version("2"), "-r2")
self.assertEqual(client.get_version("-r2"), "-r2")
# test invalid cient and repo without url
client = SvnClient(os.path.join(self.remote_path, 'foo'))
self.assertEqual(None, client.get_url())
def test_get_type_name(self):
local_path = "/tmp/dummy"
client = SvnClient(local_path)
self.assertEqual(client.get_vcs_type_name(), 'svn')
def test_get_url_nonexistant(self):
local_path = "/tmp/dummy"
client = SvnClient(local_path)
self.assertEqual(client.get_url(), None)
def test_checkout(self):
url = self.local_url
client = SvnClient(self.local_path)
self.assertFalse(client.path_exists())
self.assertFalse(client.detect_presence())
self.assertFalse(client.detect_presence())
self.assertTrue(client.checkout(url))
self.assertTrue(client.path_exists())
self.assertTrue(client.detect_presence())
self.assertEqual(client.get_path(), self.local_path)
self.assertEqual(client.get_url(), url)
def test_checkout_dir_exists(self):
url = self.local_url
client = SvnClient(self.local_path)
self.assertFalse(client.path_exists())
os.makedirs(self.local_path)
self.assertTrue(client.checkout(url))
# non-empty
self.assertFalse(client.checkout(url))
def test_checkout_emptyversion(self):
url = self.local_url
client = SvnClient(self.local_path)
self.assertFalse(client.path_exists())
self.assertFalse(client.detect_presence())
self.assertFalse(client.detect_presence())
self.assertTrue(client.checkout(url, version=''))
self.assertTrue(client.path_exists())
self.assertTrue(client.detect_presence())
self.assertEqual(client.get_path(), self.local_path)
self.assertEqual(client.get_url(), url)
self.assertTrue(client.update(None))
self.assertTrue(client.update(""))
def test_checkout_specific_version_and_update_short(self):
"using just a number as version"
url = self.local_url
version = "3"
client = SvnClient(self.local_path)
self.assertFalse(client.path_exists())
self.assertFalse(client.detect_presence())
self.assertTrue(client.checkout(url, version))
self.assertTrue(client.path_exists())
self.assertTrue(client.detect_presence())
self.assertEqual(client.get_version(), "-r3")
new_version = '2'
self.assertTrue(client.update(new_version))
self.assertEqual(client.get_version(), "-r2")
def test_get_remote_version(self):
url = self.local_url
client = SvnClient(self.local_path)
client.checkout(url)
self.assertEqual(client.get_remote_version(fetch=True),
self.local_version_master)
self.assertEqual(client.get_remote_version(fetch=False),
None)
def test_get_remote_branch_version(self):
url = self.branch_url
client = SvnClient(self.local_path)
client.checkout(url)
self.assertEqual(client.get_remote_version(fetch=True),
self.local_version_foo_branch)
self.assertEqual(client.get_remote_version(fetch=False),
None)
def testDiffClean(self):
client = SvnClient(self.remote_path)
self.assertEquals('', client.get_diff())
def testStatusClean(self):
client = SvnClient(self.remote_path)
self.assertEquals('', client.get_status())
def test_get_environment_metadata(self):
# Verify that metadata is generated
directory = tempfile.mkdtemp()
self.directories['local'] = directory
local_path = os.path.join(directory, "local")
client = SvnClient(local_path)
self.assertTrue('version' in client.get_environment_metadata())
class SvnClientLogTest(SvnClientTestSetups):
@classmethod
def setUpClass(self):
SvnClientTestSetups.setUpClass()
client = SvnClient(self.local_path)
client.checkout(self.local_url)
def test_get_log_defaults(self):
client = SvnClient(self.local_path)
client.checkout(self.local_url)
log = client.get_log()
self.assertEquals(3, len(log))
self.assertEquals('modified', log[0]['message'])
for key in ['id', 'author', 'date', 'message']:
self.assertTrue(log[0][key] is not None, key)
# svn logs don't have email, but key should be in dict
self.assertTrue(log[0]['email'] is None)
def test_get_log_limit(self):
client = SvnClient(self.local_path)
client.checkout(self.local_url)
log = client.get_log(limit=1)
self.assertEquals(1, len(log))
self.assertEquals('modified', log[0]['message'])
def test_get_log_path(self):
client = SvnClient(self.local_path)
client.checkout(self.local_url)
log = client.get_log(relpath='fixed.txt')
self.assertEquals('initial', log[0]['message'])
class SvnDiffStatClientTest(SvnClientTestSetups):
@classmethod
def setUpClass(self):
SvnClientTestSetups.setUpClass()
client = SvnClient(self.local_path)
client.checkout(self.local_url)
# after setting up "local" repo, change files and make some changes
subprocess.check_call("rm deleted-fs.txt", shell=True, cwd=self.local_path)
subprocess.check_call("svn rm deleted.txt", shell=True, cwd=self.local_path)
f = io.open(os.path.join(self.local_path, "modified.txt"), 'a')
f.write('0123456789abcdef')
f.close()
f = io.open(os.path.join(self.local_path, "modified-fs.txt"), 'a')
f.write('0123456789abcdef')
f.close()
f = io.open(os.path.join(self.local_path, "added-fs.txt"), 'w')
f.write('0123456789abcdef')
f.close()
f = io.open(os.path.join(self.local_path, "added.txt"), 'w')
f.write('0123456789abcdef')
f.close()
subprocess.check_call("svn add added.txt", shell=True, cwd=self.local_path)
def tearDown(self):
pass
def assertStatusListEqual(self, listexpect, listactual):
"""helper fun to check scm status output while discarding file ordering differences"""
lines_expect = listexpect.splitlines()
lines_actual = listactual.splitlines()
for line in lines_expect:
self.assertTrue(line in lines_actual, 'Missing entry %s in output %s' % (line, listactual))
for line in lines_actual:
self.assertTrue(line in lines_expect, 'Superflous entry %s in output %s' % (line, listactual))
def assertEqualDiffs(self, expected, actual):
"True if actual is similar enough to expected, minus svn properties"
def filter_block(block):
"""removes property information that varies between systems, not relevant fo runit test"""
newblock = []
for line in block.splitlines():
if re.search("[=+-\\@ ].*", line) == None:
break
else:
# new svn versions use different labels for added
# files (working copy) vs (revision x)
fixedline = re.sub('\(revision [0-9]+\)', '(working copy)', line)
newblock.append(fixedline)
return "\n".join(newblock)
filtered_actual_blocks = []
# A block starts with \nIndex, and the actual diff goes up to the first line starting with [a-zA-Z], e.g. "Properties changed:"
for block in actual.split("\nIndex: "):
if filtered_actual_blocks != []:
# restore "Index: " removed by split()
block = "Index: " + block
block = filter_block(block)
filtered_actual_blocks.append(block)
expected_blocks = []
for block in expected.split("\nIndex: "):
if expected_blocks != []:
block = "Index: " + block
block = filter_block(block)
expected_blocks.append(block)
filtered = "\n".join(filtered_actual_blocks)
self.assertEquals(set(expected_blocks), set(filtered_actual_blocks))
def test_diff(self):
client = SvnClient(self.local_path)
self.assertTrue(client.path_exists())
self.assertTrue(client.detect_presence())
self.assertEqualDiffs('Index: added.txt\n===================================================================\n--- added.txt\t(revision 0)\n+++ added.txt\t(revision 0)\n@@ -0,0 +1 @@\n+0123456789abcdef\n\\ No newline at end of file\nIndex: modified-fs.txt\n===================================================================\n--- modified-fs.txt\t(revision 3)\n+++ modified-fs.txt\t(working copy)\n@@ -0,0 +1 @@\n+0123456789abcdef\n\\ No newline at end of file\nIndex: modified.txt\n===================================================================\n--- modified.txt\t(revision 3)\n+++ modified.txt\t(working copy)\n@@ -0,0 +1 @@\n+0123456789abcdef\n\\ No newline at end of file',
client.get_diff().rstrip())
def test_diff_relpath(self):
client = SvnClient(self.local_path)
self.assertTrue(client.path_exists())
self.assertTrue(client.detect_presence())
self.assertEqualDiffs('Index: local/added.txt\n===================================================================\n--- local/added.txt\t(revision 0)\n+++ local/added.txt\t(revision 0)\n@@ -0,0 +1 @@\n+0123456789abcdef\n\\ No newline at end of file\nIndex: local/modified-fs.txt\n===================================================================\n--- local/modified-fs.txt\t(revision 3)\n+++ local/modified-fs.txt\t(working copy)\n@@ -0,0 +1 @@\n+0123456789abcdef\n\\ No newline at end of file\nIndex: local/modified.txt\n===================================================================\n--- local/modified.txt\t(revision 3)\n+++ local/modified.txt\t(working copy)\n@@ -0,0 +1 @@\n+0123456789abcdef\n\\ No newline at end of file', client.get_diff(basepath=os.path.dirname(self.local_path)).rstrip())
def test_status(self):
client = SvnClient(self.local_path)
self.assertTrue(client.path_exists())
self.assertTrue(client.detect_presence())
self.assertStatusListEqual('A added.txt\nD deleted.txt\nM modified-fs.txt\n! deleted-fs.txt\nM modified.txt\n', client.get_status())
def test_status_relpath(self):
client = SvnClient(self.local_path)
self.assertTrue(client.path_exists())
self.assertTrue(client.detect_presence())
self.assertStatusListEqual('A local/added.txt\nD local/deleted.txt\nM local/modified-fs.txt\n! local/deleted-fs.txt\nM local/modified.txt\n', client.get_status(basepath=os.path.dirname(self.local_path)))
def test_status_untracked(self):
client = SvnClient(self.local_path)
self.assertTrue(client.path_exists())
self.assertTrue(client.detect_presence())
self.assertStatusListEqual('? added-fs.txt\nA added.txt\nD deleted.txt\nM modified-fs.txt\n! deleted-fs.txt\nM modified.txt\n', client.get_status(untracked=True))
class SvnExportRepositoryClientTest(SvnClientTestSetups):
@classmethod
def setUpClass(self):
SvnClientTestSetups.setUpClass()
client = SvnClient(self.local_path)
client.checkout(self.local_url)
self.basepath_export = os.path.join(self.root_directory, 'export')
def tearDown(self):
pass
def test_export_repository(self):
client = SvnClient(self.local_path)
self.assertTrue(
client.export_repository('',
self.basepath_export)
)
self.assertTrue(os.path.exists(self.basepath_export + '.tar.gz'))
self.assertFalse(os.path.exists(self.basepath_export + '.tar'))
self.assertFalse(os.path.exists(self.basepath_export))
class SvnGetBranchesClientTest(SvnClientTestSetups):
@classmethod
def setUpClass(self):
SvnClientTestSetups.setUpClass()
client = SvnClient(self.local_path)
client.checkout(self.local_url)
# def tearDown(self):
# pass
def test_get_remote_contents(self):
self.assertEqual(['branches', 'tags', 'trunk'], get_remote_contents(self.local_root_url))
def test_get_branches_non_canonical(self):
remote_path = os.path.join(self.root_directory, "remote_nc")
init_path = os.path.join(self.root_directory, "init_nc")
local_path = os.path.join(self.root_directory, "local_nc")
# create a "remote" repo
subprocess.check_call("svnadmin create %s" % remote_path, shell=True, cwd=self.root_directory)
local_root_url = "file://localhost/" + remote_path
local_url = local_root_url + "/footest"
# create an "init" repo to populate remote repo
subprocess.check_call("svn checkout %s %s" % (local_root_url, init_path), shell=True, cwd=self.root_directory)
for cmd in [
"mkdir footest",
"mkdir footest/foosub",
"touch footest/foosub/fixed.txt",
"svn add footest",
"svn commit -m initial"]:
subprocess.check_call(cmd, shell=True, cwd=init_path)
client = SvnClient(local_path)
client.checkout(local_url)
self.assertEqual([], client.get_branches())
def test_get_branches(self):
client = SvnClient(self.local_path)
self.assertEqual(['foo'], client.get_branches())
# slyly create some empty branches
subprocess.check_call("mkdir -p branches/foo2", shell=True, cwd=self.init_path)
subprocess.check_call("mkdir -p branches/bar", shell=True, cwd=self.init_path)
subprocess.check_call("svn add branches/foo2", shell=True, cwd=self.init_path)
subprocess.check_call("svn add branches/bar", shell=True, cwd=self.init_path)
subprocess.check_call("svn commit -m newbranches", shell=True, cwd=self.init_path)
self.assertEqual([], client.get_branches(local_only=True))
self.assertEqual(['bar', 'foo', 'foo2'], client.get_branches())
# checkout branch foo
local_path2 = os.path.join(self.root_directory, "local_foo")
client = SvnClient(local_path2)
client.checkout(self.local_root_url + '/branches/foo')
self.assertEqual(['foo'], client.get_branches(local_only=True))
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,299 |
{"url":"https:\/\/rufuspollock.com\/2007\/06\/22\/tell-no-one\/","text":"# Tell No One\n\nJUNE 22, 2007\n\n8\/10. A well above average thriller slightly let down by its ending (as is frequently the way with these things).","date":"2022-08-07 15:56:13","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\": 0.9063571691513062, \"perplexity\": 9037.975651376493}, \"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-2022-33\/segments\/1659882570651.49\/warc\/CC-MAIN-20220807150925-20220807180925-00564.warc.gz\"}"} | null | null |
The Girl and the Count () is a 1966 Danish family film directed by Finn Henriksen and starring Dirch Passer.
Cast
Dirch Passer - Andreas Lillelys
Josephine Passer - Lulu Hansen
Lene Tiemroth - Susanne 'Sus' Hansen
Peter Steen - Grev Ditmar Gyldenstjerne
Karin Nellemose - Grevinde Constance Gyldenstjerne
Malene Schwartz - Irene Gyldenstjerne
Cleo Jensen - Doris
Karl Stegger - Søren
Ove Sprogøe - Nielsen
Sigrid Horne-Rasmussen - Kokkepigen 'Putte'
Carl Ottosen - Godsforvalter Lauritsen
Paul Hagen - Bastian Gyldenstjerne
Preben Mahrt - Theodor Gyldenstjerne
Preben Kaas - Skuespiller
Poul Bundgaard - Skuespiller
Daimi - Skuespiller
Bjørn Spiro - Sælger af mejetærsker
References
External links
1966 films
Danish children's films
1960s Danish-language films
Films directed by Finn Henriksen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,226 |
using System;
using System.Runtime.InteropServices;
internal static partial class Interop
{
internal static partial class Crypt32
{
[StructLayout(LayoutKind.Sequential)]
internal struct CMSG_RECIPIENT_ENCODE_INFO
{
internal CMsgCmsRecipientChoice dwRecipientChoice;
//union
//{
// CMSG_KEY_TRANS_RECIPIENT
// PCMSG_KEY_TRANS_RECIPIENT_ENCODE_INFO pKeyTrans;
// CMSG_KEY_AGREE_RECIPIENT
// PCMSG_KEY_AGREE_RECIPIENT_ENCODE_INFO pKeyAgree;
// CMSG_MAIL_LIST_RECIPIENT
// PCMSG_MAIL_LIST_RECIPIENT_ENCODE_INFO pMailList;
//}
internal IntPtr pCmsRecipientEncodeInfo;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,204 |
{"url":"https:\/\/ask.sagemath.org\/question\/23895\/elliptic-curve-complex-numbers\/","text":"# elliptic curve complex numbers\n\nHi, I want to look at the curve E=EllipticCurve(CC,[-35\/4,-49\/4]) over the complex numbers. I want to find the 3-Torsion Points on the curve, so I tried to use the function E.division_polynomial(3, two_torsion_multiplicity=0) which gave me the 3-Division-Polynomial g=3x^4 - 105\/2x^2 - 147x - 1225\/16 which is an univariate Polynomial. The zeros of this Polynomial should be the x-coordinates of the 3-Torsion-Points. One of the zeros is a=5.26556730825188 Then I tried to compute the y-coordinates via the curve-equation y^2 = x^3 + (-8.75000000000000)x + (-12.2500000000000) The point I got was P=(5.26556730825188 , 9.36325015678742) which is clearly lying on the curve, because it fulfills the equation of the curve E, what I have tested.\n\nSo I wanted to use the function P = E(5.26556730825188 , 9.36325015678742)\n\nHere I got an error, telling me \"TypeError: Coordinates [5.26556730825188, 9.36325015678742, 1.00000000000000] do not define a point on Elliptic Curve defined by y^2 = x^3 + (-8.75000000000000)*x + (-12.2500000000000) over Complex Field with 53 bits of precision\" Why does that happen?\n\nNext problem is the following: If I use the function Q = E(0); Q.division_points(3)\n\nthis should give me the 3-torsion-points, but the x-coordinates of the points I get by this metod are different from the method with the 3-divison-polynomial! actually the function does not find any 3-torsion points! How can that happen? Sorry, I'm a sage-beginner from germany and my english is terrible! But this is really really important for me, so I would be very very thankful for any help!!!\n\ngreetings pittersen!!\n\nedit retag close merge delete\n\ncould you please accept my answer if you think it is good ?\n\n( 2014-08-23 01:24:47 -0500 )edit\n\nHi Frederic, I'm really happy with your answer. It helped me a lot! Thank you. I just klicked the green button on the left; is that what you mean by \"accept\" ?\n\n( 2014-08-25 10:47:15 -0500 )edit\n\nyes. Thanks\n\n( 2014-08-25 11:52:13 -0500 )edit\n\nSort by \u00bb oldest newest most voted\n\nMaybe something like that:\n\nsage: E = EllipticCurve(QQ,[-35\/4,-49\/4])\nsage: E2 = E.change_ring(CC)\nsage: p = E.torsion_polynomial(3)\nsage: p.complex_roots()\n[-0.682991613296036,\n5.26556730825188,\n-2.29128784747792 - 1.35880032042306*I,\n-2.29128784747792 + 1.35880032042306*I]\nsage: x = p.complex_roots()[1]\nsage: E2.lift_x(x)\n(5.26556730825188 : 9.36325015678740 : 1.00000000000000)\n\n\nSometimes it is better (but maybe slower) to work over QQbar for exact results.\n\nsage: E3 = E.change_ring(QQbar)\nsage: p = E3.torsion_polynomial(3)\nsage: p.roots()\n[(-0.6829916132960358?, 1),\n(5.265567308251876?, 1),\n(-2.291287847477920? - 1.358800320423061?*I, 1),\n(-2.291287847477920? + 1.358800320423061?*I, 1)]\nsage: x = p.roots()[1][0]\nsage: g = E3.lift_x(x); g\n(5.265567308251876? : 9.363250156787399? : 1)\nsage: g+g\n(5.265567308251876? : -9.363250156787399? : 1)\nsage: g+g+g\n(0 : 1 : 0)\n\n\nYou could also work over the splitting field of the division polynomial.\n\nmore","date":"2018-03-17 12:49:05","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\": 0.30264952778816223, \"perplexity\": 2138.1791005889913}, \"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-2018-13\/segments\/1521257645069.15\/warc\/CC-MAIN-20180317120247-20180317140247-00550.warc.gz\"}"} | null | null |
Q: Running script in host server from my computer and shutting down my computer I'm very new to ssh and working with servers. I know how to make a connection to a server, and how to run a local script in a host server.
However I always have to have my terminal open while the script is running. The thing is that I'm about to run a script which is expected to finish in a couple of weeks. I cannot have my computer on for two weeks. How can I "send" the instruction to run my script and be able to shut down my computer while the server is running?
If this is not possible, can I copy the script to the server's hard drive and "send" the instruction to run that script, and be able to shut down my computer?
Thanks for your comments.
A: Take a look at the screen program.
You can ssh to your server, then create a new login session by typing:
screen
In that session, you can start off your script that will take a couple of weeks.
You can then detach from that session by typing:
Ctrl-a Ctrl-d
You can then log off your ssh session, and your script is still running in the detached screen session.
Later, you can ssh in to your server again and type:
screen -r
This will reattach you to the detached session from before, and you can see how your script is going. If not finished, just detach again.
Also note that if your screen session is accidentally detached (e.g. ssh stops working, or a network outage) your screen session will still be there and you can ssh again and screen -r to reattach!
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\section{Introduction}\label{sec:intro}
Robotic Surgical Assistants (RSAs)
such as Intuitive Surgical's da Vinci~\cite{dvrk2014} are regularly used in hospitals and surgical procedures through teleoperation.
The introduction of RSAs gave surgeons greater ability to perform complex tasks through improved dexterity and visualization. This increased the number of surgeons able to offer minimally invasive surgery to patients, reducing their post operative pain and length of stay in the hospital compared to open surgery~\cite{danyal_fer_suggestion2,danyal_fer_suggestion}. RSAs also provide a platform for the automation of some surgical tasks, which have potential to aid surgeons by reducing fatigue or tedium~\cite{yip2017robot}.
We consider the well-known peg transfer task from the Fundamentals of Laparoscopic Surgery (FLS)~\cite{fls_1998} and how it could be automated at a speed and reliability level on par with a professional surgeon. This is an extremely high bar, as human surgeons perform this task with great dexterity~\cite{fls_2004}. In addition, commands to cable-driven RSAs yield errors in motion due to backlash, cable tension, and hysteresis~\cite{pastor2013,miyasaka2015,kalman_filter_2016,mahler2014case,Kehoe2014}.
In this task, the surgeon transfers six hollow triangular blocks from pegs on one half of a board to the pegs on the other half (see Fig.~\ref{fig:money-shot} inset).
To the best of our knowledge, the only prior work that focuses on automating a version of the peg transfer task is by Rosen and Ma~\cite{auto_peg_transfer_2015}, who used a single Raven II~\cite{raven2013} arm to perform handover-free peg transfer with three blocks. The present paper revisits this pioneering work and applies depth sensing to a monochrome variant of the peg transfer task using six blocks and the da Vinci Research Kit (dVRK)~\cite{Ballantyne2003,dvrk2014}.
\begin{figure}[t]
\center
\includegraphics[width=0.45\textwidth]{teaser_v04.png}
\caption{
\small
\textbf{Automated peg-transfer task setup.} The da Vinci surgical robot utilizes either one or both of its arms to transfer blocks between pegs. The workspace (zoomed-in inset to the bottom right) consists of six hollow triangular blocks on a peg board for a procedure in progress. The task starts with all six blocks on the left side of the board. One must transfer the blocks to the right side of the board, and then bring them back to the left side. The blocks, pegs, and peg board are monochrome (painted red) to simulate a surgical setting. We use an overhead Zivid RGBD camera.
}
\vspace*{-10pt}
\label{fig:money-shot}
\end{figure}
To sense the blocks and pegs, we use a Zivid One Plus RGB+depth (RGBD) camera which can provide 1920x1200 pixel images at 13 frames per second with depth resolution \SI{0.5}{\milli\meter}. RGBD technology is advancing\footnote{\url{https://bair.berkeley.edu/blog/2018/10/23/depth-sensing/}} and is widely used in industrial automation. Additionally, the size of RGBD sensors is not a severe restriction in open surgical environments, and robots such as the Taurus from SRI International are being developed for this purpose~\cite{madapana2019desk}. While depth sensing is invaluable for robotics applications such as data-driven grasping~\cite{mahler2017dexnet}, we are unaware of prior applications in RSAs, as depth sensing is not yet available for minimally-invasive surgery but should be considered for open-body tele-surgical systems operated remotely via intermittent supervision, where limited autonomy may be necessary.
This paper contributes: 1) the first application of depth-sensing to an RSA, 2) a robust depth-sensing perception algorithm for peg transfer, and 3) results on 20 and 5 episodes of automated peg transfer with single and bilateral arms, respectively, suggesting reliability of \SI{86.9}{\percent} and \SI{78.0}{\percent}, with transfer speeds of 10.02 and 5.72 seconds per block. Code and videos are available at \url{https://sites.google.com/view/peg-transfer}.
\section{Related Work}\label{sec:rw}
Although surgical robotics has a long history, summarized in surveys~\cite{Moustris2011,Beasley2012,surgical_robotics_2016}, no procedures are fully autonomous due to uncertainty in perception and control.
\subsection{Autonomous Robot Surgery}
In non-clinical research settings, several groups have explored autonomous robot surgery~\cite{yip2017robot}. Key tasks of interest, some of which are part of the FLS curriculum, include pattern cutting~\cite{murali2015learning,thananjeyan2017multilateral,intermittent_synch_2018}, suturing~\cite{sen2016automating,rosen_icra_suturing_2017,saeidi_suturing_icra_2019,Schulman2013, thananjeyan2019safety}, debridement for rigid and soft objects in static~\cite{Kehoe2014,mahler2014case,seita_icra_2018} and dynamic~\cite{surgical_camera_2018,stewart_platform} cases, needle extraction and insertion~\cite{extraction_needles_2019,needle_insertion_deformation_2019,automated_needle_pickup_2018}, knot-tying~\cite{vandenBerg2010,knot_tying_2002,improved_knots_case_2013}, and tumor localization~\cite{mckinley2015,garg2016gpas,mckinley2016}.
Automation in robot surgery has catalyzed the development of novel techniques in trajectory optimization and planning~\cite{Osa-RSS-14}, compliance manipulation~\cite{compliance_2018}, 3D reconstructions of deformable objects~\cite{super_yip_2019}, simulators or demonstrator data for imitation learning~\cite{seita_fabrics_2019}, reinforcement learning~\cite{thananjeyan2017multilateral,rosen_icra_tissues_2019,rosen_tissues_qlearn_2019}, or task segmentation~\cite{tsc-dl-2016,krishnan2017ddco,madapana2019desk,rahmantransferring}.
One challenge with autonomous minimally invasive surgery is that commercial robot systems, such as the Raven II~\cite{raven2013} and the da Vinci~\cite{dvrk2014}, are cable-driven and thus can have inaccurate motion and actuation~\cite{pastor2013,miyasaka2015,kalman_filter_2016,mahler2014case,Kehoe2014,seita_icra_2018} due to their susceptibility to cable stretch, backlash, hysteresis, and decalibration. We present a calibration procedure in Section~\ref{ssec:calibration} which was able to reliably achieve accuracy within 1--2~mm of error in the workspace.
\subsection{Peg Transfer Task}
The peg transfer task is one of the five tasks in the Fundamentals of Laparoscopic Surgery (FLS)~\cite{fls_2004}. The goal is to transfer six rubber triangular blocks from one half of a pegboard to the other, and then back. Transferring a single block requires grasping the block, lifting it off of a peg, handing it to the other arm, moving it over another peg, and then placing the block around the targeted peg. In this paper, we do not consider the handoff step because it requires large wrist motions --- the gripper that picks up the block also places it on the target peg.
Fried~et~al.~\cite{fls_2004} performed a study in which surgical residents performed the peg transfer task, which aims to assess and develop the surgeons' depth and visual-spatial perception. Task performance was measured based on completion time, with a score penalty whenever a block fell outside the surgeon's view. The study found that superior performance in peg transfer correlates with performance during real laparoscopic surgical settings, validating the rationale for having surgeons practice peg transfer.
Prior work looks into improving human performance of the peg-transfer task.
Abiri~et~al.~\cite{haptic-feedback-surgery-2019} focus on providing better haptic feedback to compensate for the loss of force feedback when teleoperating surgical robots, and test on the peg transfer task. Brown~et~al.~\cite{rating_peg_transfer_2017} provide ways to automatically evaluate a surgeon's skill at the peg transfer task by using data from contact forces and robot arm accelerations. Rivas-Blanco~et~al.~\cite{ur5-peg-transfer-2019} apply learning from demonstrations to autonomously control camera motion during teleoperated peg transfer. Other work~\cite{laparoscopic_transfer_2014} provides additional teleoperated peg transfer benchmarks or proposes novel methods to identify stages of the peg transfer task, called surgemes, across robot platforms~\cite{madapana2019desk, rahmantransferring}. None of these prior works focus on automating peg transfer.
Rosen and Ma~\cite{auto_peg_transfer_2015} attempted to automate a variant of the peg transfer task using a Raven II~\cite{raven2013}. They used one robot arm and three blocks per episode, and transferred in one direction. They compared performance with a human using Omni VR masters to control the arms, with each doing 20 episodes, for 60 total block transfer attempts. Results showed that the autonomous robot was able to achieve nearly the same block transfer success rate (56/60) as the human (60/60), and was twice as fast ($25\pm0.0$ vs $49\pm5.7$ seconds for each episode). We use the da Vinci with depth sensing, and transfer six blocks in both directions.
\section{Problem Statement}\label{sec:PS}
\begin{figure}[t]
\center
\includegraphics[width=0.48\textwidth]{different_height_v03.png}
\caption{
\small
\textbf{Block properties and sizes}. The left image shows all six blocks on a peg board, demonstrating that the height of the top layer of the blocks is not uniform. We overlay a scale of \SI{10}{\milli\meter}. The right image shows a top-down view of one of the blocks with a shadow, showing the hollow interior.
}
\vspace*{-10pt}
\label{fig:different-heights}
\end{figure}
The setup for the peg transfer task is shown in Fig.~\ref{fig:money-shot}.
The task uses six triangular rubber blocks shown in Fig.~\ref{fig:different-heights}. Each block is roughly \SI{15}{\milli\meter} in height and has triangular edges of length \SI{18}{\milli\meter} and a hollow center spanning \SIrange{5}{10}{\milli\meter}. Fig.~\ref{fig:different-heights} demonstrates that these are approximations, as the block edges are not at uniform heights.
We define one \emph{episode} of the task to be the full procedure where the surgical robot attempts to move the six blocks from the left to the right, and then moves all of them back. We test two variants: one with a single dVRK arm moving the blocks, and a bilateral one with both arms moving blocks simultaneously.
The visual cues in a surgical environment are rarely distinct and surgical decision making relies heavily on perception of minor differences in color, depth and texture. For this reason we paint the board, pegs, and blocks red to mimic real surgical settings in which tissue may be a more uniform red hue, so color may not provide a sufficient signal to automate sensing the state of the environment.
\subsection{Failure Modes}\label{ssec:failure-cases}
\begin{figure}[t]
\center
\includegraphics[width=0.48\textwidth]{dvrk_failure_grasp_v01.png}
\caption{
\small
Time lapse of an example \emph{pick} failure from the dVRK. The pick operation initially opens the gripper and lowers it (first image), but the gripper only barely touches the block edge (second image). When the gripper closes, it is unable to get the block within its grip as it rises (third image).
}
\vspace*{-6pt}
\label{fig:failure-grasp}
\end{figure}
\begin{figure}[t]
\center
\includegraphics[width=0.48\textwidth]{dvrk_failure_v03_stuck.png}
\caption{
\small
Time lapse of an example \emph{place-stuck} failure from the dVRK. The placing operation results in the bottom of the block making contact with a block underneath (overlaid circle in third image), preventing it from being fully inserted.
}
\vspace*{-10pt}
\label{fig:failure-place-stuck}
\end{figure}
\begin{figure}[t]
\center
\includegraphics[width=0.48\textwidth]{dvrk_failure_drop_v03.png}
\caption{
\small
Time lapse of an example \emph{place-fall} failure from the dVRK. The placing operation results in the bottom of the block making contact with the top of the peg. When the gripper releases the block and moves up (first images) the block eventually falls off the peg (last two images).
}
\vspace*{-10pt}
\label{fig:failure-place-fall}
\end{figure}
To better understand the performance of either the surgical robot or a human operator at the peg transfer task, we consider a set of failure modes. For consistency, we use and expand on failure definitions from prior work~\cite{auto_peg_transfer_2015}. We calculate failures based on each individual attempt at moving a block within an episode. The failure cases are:
\begin{enumerate}
\item \textbf{Pick failure}: an error in grasping the block from its starting peg, so that the block is not lifted free from the peg. As described in Section~\ref{ssec:error_recovery}, after this type of failure, we allow the robot one more attempt at transferring the block.
\item \textbf{Place failure}: when a block is not fully inserted onto its target peg and does not make contact with the bottom of the workspace. We sub-divide placing failures into two categories: \textbf{place-stuck failures} for when placing results in blocks stuck on top of a peg or another block, and \textbf{place-fall failures} for when placing results in blocks that fall on the surface.
\end{enumerate}
Fig.~\ref{fig:failure-grasp} shows an example grasping failure, and Figs.~\ref{fig:failure-place-stuck} and~\ref{fig:failure-place-fall} show examples of the two placing failures, all from the automated dVRK system we use for experiments. We sub-divide placing failures because place-stuck and place-fall failures have different effects in practice. Typically it is easier to recover from place-stuck failures because the blocks still lie on top of their target peg and a gentle nudge can usually slide the block into place. In contrast, a place-fall failure requires an entirely new grasp to pick a fallen block.
\section{Robot Calibration}\label{sec:setup}
\label{ssec:calibration}
\begin{figure*}[t]
\center
\includegraphics[width=1.00\textwidth]{calibration_overview_v02.png}
\caption{
\small
\textbf{Calibration.} Left: when calibrating the dVRK, we use a checkerboard with each edge \SI{16}{\milli\meter} long, and move the robot's end-effector towards the corners, which are at a known height offset of the peg tips. The right arm moves to all corners included in the overlaid blue grid above, while the arm to the left would be calibrated independently and go to all points in the overlaid green grid. At a given position, however, rotating the robot's arm will result in the tip being at a different location. For example, the second image shows the tip at a checkerboard corner, and the third image shows the result when rotating the roll angle by \SI{90}{\degree}, which means the actual tip is at a spot \SIrange{2}{4}{\milli\meter} away. For this reason, we discretize the roll angle and calibrate once per roll angle discretization.
}
\vspace*{-6pt}
\label{fig:calibration}
\end{figure*}
The calibration technique we use is based on the procedure from Seita~et~al.~\cite{seita_fabrics_2019}. We calibrate the positions by using a checkerboard located at a plane that roughly mirrors the height of the blocks when they are inserted into pegs. We servo each end effector with the gripper facing down to each corner of the checkerboard and record positions. During deployment, when given an image, we map the image pixels to a 2D coordinate on the checkerboard. For a given coordinate frame, we perform bilinear interpolation to estimate the robot position from the four surrounding known points. Fig.~\ref{fig:calibration} provides an overview.
We observe that the roll angle of the end effector's pose affects the positioning of the robot (see Fig.~\ref{fig:calibration}), so we follow the method from Seita~et~al.~\cite{seita_icra_2018} and discretize the roll. We choose two discretizations of the roll value, \SI{0}{\degree} and \SI{90}{\degree}, as these are approximately close to the roll values the robot would use in practice. We conduct the calibration procedure outlined above independently for each arm at each angle, giving us two calibration tables for each arm. Thus, for a given arm and roll angle, we interpolate between these tables to generate a new table, which can then be used to conduct the bilinear interpolation method described above for any point on the board. After calibration, both arms of the robot reached positions on the checkerboard's plane with \SIrange{1}{2}{\milli\meter} of error for the two roll values tested.
\newcommand{\pegblocksubfloat}[2]{%
\subfloat[#1]
\adjustimage{width=228pt,origin=c,angle=-1.25,Clip=36pt}{#2}}}
\begin{figure*}[t]
\center
\begin{tabular}{c@{\quad}c@{\quad}c}
\pegblocksubfloat{RGB image}{img_color.png} &
\pegblocksubfloat{Depth image}{img_depth_processed_for_humans.png} &
\pegblocksubfloat{Output of depth-sensing algorithm}{img_blocks_05.png}
\end{tabular}
\caption{
\small
Images as seen through the overhead RGBD camera (i.e., the depth sensor is not attached to the robot arm), and subsequently processed for later usage. Images (a) and (b) show sample RGB and depth images, respectively, that we might get at the beginning of an episode, where all pegs are informally dropped in the left half of the peg board. In (a), several blocks exhibit specular reflections that one might find in a surgical setting and could hinder other sensing methods. The board, pegs, and blocks are painted red. Image (c) shows the 12 detected pegs, circled in blue, along 6 grasp points (left half) and 6 place points (right half), all 12 of which are circled white. To the left, image (c) shows the location of the blocks that are within a pre-determined depth interval for block heights. Finally, 12 contours of the blocks are overlaid. To perform an episode, we follow a pre-determined order of grasp and place movements based on the white circles, and repeat the process going from right to left.
}
\vspace*{-6pt}
\label{fig:block-grasping}
\end{figure*}
\section{Depth-Sensing Algorithm}\label{sec:sensing}
\begin{algorithm}[htb!]
\caption{Depth Sensing}
\label{alg:sensing}
\begin{algorithmic}[1]
\Statex{\textbf{Require:} Depth Image $I$, number of blocks $n$, block masks $\{M_{\phi_i}\}_{i=1}^k$, depth target $d$, depth tolerance $\epsilon$}
\State $\mathtt{I}_\mathrm{thresh} = \mathtt{clip}(I, d-\epsilon, d+\epsilon) > 0$ // threshold depth
\State // Get activation maps to find objects matching the masks
\State $\{A_{\phi_i}\}_{i=1}^k = \{\mathtt{cross\_correlate(I_\mathrm{thresh}, M_{\phi_i})}\}_{i=1}^k$
\For{$j \in \{1,\dots,n\}$}
\State $\theta_j \leftarrow$ orientation $\phi_i$ of map with highest activation
\State $p_j \leftarrow \arg\max_p A_{\theta_j}(p)$
\State Zero out all activations in area of size of $M_{\theta_j}$ at $p_j$
\EndFor
\State Return $\{(p_j, \theta_j)\}_{j=1}^n$
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[htb!]
\caption{Grasp Planner}
\label{alg:grasp_plan}
\begin{algorithmic}[1]
\Statex{\textbf{Require:} Block pose $(p_\mathrm{block}, \theta)$, Arm $A \in \{\mathrm{left}, \mathrm{right}\}$, Peg location $p_\mathrm{peg}$}
\State $(s_1, s_2) \leftarrow$ closest two (out of three) sides to $A$
\State $\mathcal{G} \leftarrow 2$ grasp candidates along $s_1$ and $s_2$
\State Return $\arg\max_{g\in \mathcal{G}} \|g - p_\mathrm{peg}\|_2$
\end{algorithmic}
\end{algorithm}
As shown in Fig.~\ref{fig:block-grasping} and Algorithm \ref{alg:sensing}, the block-detection algorithm uses a depth image from the overhead RGBD camera and thresholds it to only include the top surfaces of $n$ blocks by using a target depth value $d$ and a tolerance parameter $\epsilon$. In this work, $n=6$ but the algorithm can scale to larger values of $n$ as long as all blocks are reachable. The algorithm then cross-correlates the thresholded image with pre-computed masks $\{M_{\phi_i}\}_{i=1}^k$ of the block rotated in 30 different orientations in the plane of the image. The algorithm then proceeds iteratively to find the blocks. The best match at iteration $j$ is saved as a pixel coordinate $p_j = (u_j, v_j)$ and orientation $\theta_j$. The procedure zeroes out the region of the thresholded image occupied by the best match's mask then proceeds to the next iteration to find the next best match. We find the peg locations in a similar way. By computing cross-correlations with masks of the target objects (blocks, pegs), we are searching the image for objects that match the geometry of the target. Algorithm~\ref{alg:sensing} is implemented using SciPy~\cite{bressert2012scipy} signal processing code.
After detecting the blocks, we compute grasps (Algorithm~\ref{alg:grasp_plan}) by first sub-dividing each block edge into two potential grasp points, for a total of six potential grasp points per block. We select the two sides that are closest to the end effector's current position; this is to prevent having the robot reach ``behind'' a peg. Of the grasp points on those edges, we select the grasp point furthest from the peg to get the most clearance for an open gripper and thus decreases the chances of collisions with pegs. Fig.~\ref{fig:block-grasping} shows an example setup of RGB and depth images and the corresponding proposed pick and place points circled in white, for the single-arm case.
\section{Trajectory Motion for Grasping}\label{sec:trajectory-motion}
\begin{figure*}[t]
\center
\includegraphics[width=1.00\textwidth]{trajectory_v01.png}
\caption{
\small
A visualization of the dVRK's gripper as it picks up a block. In the first two images, an arm that descends to grasp a block with an open gripper can risk colliding with the top of the peg (circled above) and damage the hardware. To avoid this, as shown in the sequence in the last three images, the gripper is kept closed until the tip surpasses the height of the peg (dashed line) and then it opens, allowing for safer grasping of a block.
}
\vspace*{-10pt}
\label{fig:trajectory-motion}
\end{figure*}
After the block detection and grasp analysis, the system moves blocks from one set of pegs to another in a predefined order, iterating through motions from block grasp points to placement points until all blocks are moved.
For a given grasp point, the system commands the gripper to go slightly above the grasp point with a closed gripper. Then, the system opens the gripper, lowers it, closes the gripper, and raises it (ideally with the block). The system then moves the block over its placement point and opens the gripper, dropping the block on to a peg. This full motion sequence is executed in an open loop without feedback or visual servoing, and thus depends on accurate calibration.
Extending to the bilateral-arm case, we select an ordering of the pegs/blocks for both arms such that they do not collide with each other during the trajectory; the left and right arms go to two neighboring pegs, such that each arm grasps the block closest to it. By simultaneously commanding both arms motions, the robot never makes motions that cross over each other, thus avoiding arm-to-arm collisions.
The gripper remains closed during most motions, reducing the chances that motion, block detection, or calibration errors will result in the gripper colliding with a block or peg.
Initial peg transfer trials showed a danger in that an open gripper could hit the top of a peg, and the resulting force applied from the dVRK's arm to its gripper could damage a cable (Fig.~\ref{fig:trajectory-motion}).
The safety measure of keeping the gripper closed was sufficient to avoid gripper-on-peg collisions.
All motions in the peg transfer system require adding a bit of extra clearance to add a safety margin to overcome a limitation we encountered when commanding the robot's motions in terms of the gripper pose. The robot internal control software translates poses into joint angles through the calculation of the inverse kinematics. We have observed that, while this software achieves poses accurately, pose-to-pose motions are linear interpolations in joint angles, and not in the gripper's pose. Due to the non-linear translation between pose and joint angles, this linear-in-joint-space interpolation results in non-linear motions of the end effector. By adding extra clearance to every pose, we add a safety margin and reduce failures. However, this may result in motions that are less efficient than possible were we to instead command the robot using more advanced motion planning and trajectory optimization techniques, as we plan to do in future work.
\subsection{Error Recovery Stage}\label{ssec:error_recovery}
When a pick failure occurs, the block often still remains in its original peg. After the first set of block transfer attempts, we perform a scan through the six known starting block positions, and detect if any still remain, and allow the dVRK a second attempt to grasp each block if needed. (In principle, this process could repeat ad infinitum, but we found that the robot would often make similar errors in subsequent actions, so we limit to two attempts.)
\section{Experiments and Results}\label{sec:results}
We initialize a peg transfer task episode by randomly dropping six blocks on the six pegs on the left side of the peg board, producing variation in the pose of each block. We evaluate failures as described in Section~\ref{ssec:failure-cases}, abbreviating failure modes as Pick (pick failures), Stuck (place-stuck failures) and Fall (place-fall failures).
\subsection{Single-Arm Results}
\begin{table}[t]
\caption{
\small
Results from experiments for both the single and bilateral arm cases, with 20 and 5 episodes, respectively. We report the success rate over the number of total block transfer attempts (236 and 59, respectively), along with the average time for each of those attempts. In addition, we report the fraction of the three failure modes on each block transfer attempt.
}
\centering
\begin{tabular}{l | l r l l l}
Task & Success Rate & Time (s) & Pick & Stuck & Fall \\ \hline
Single & 0.869 (205/236) & 10.02 & 0.013 & 0.072 & 0.046 \\
Bilateral & 0.780 (46/59) & 5.72 & 0.034 & 0.068 & 0.119 \\
\end{tabular}
\vspace*{-10pt}
\label{tab:dvrk-results}
\end{table}
We perform 20 episodes of the single arm case with the dVRK, and report results in Table~\ref{tab:dvrk-results}.
Across all 20 episodes, the robot performed 236 total attempts at moving a block from one half of the board to another, each of which took 10.02 seconds on average. A ``perfect'' episode with no failures involves 12 total attempts (6 per direction). The number of attempts in a given episode may be higher or lower depending on if repeated grasp attempts are needed or if placing failures occur during the first set of six block transfer attempts. For those 236 block attempts, we recorded 31 failure cases, of which 3 were pick failures, 17 were place-stuck failures, and 11 were place-fall failures. The success rate for a single block attempt is thus $205 / 236$ = \SI{86.9}{\percent}. For 4 of the 20 episodes, the dVRK executed the entire task without failures. Each full episode lasted $118.2 \pm 9.4$ seconds, which is nearly twice as long compared to the human operator (Section~\ref{ssec:danyal}), and is in part due to the safety checks that are built into the motion planning. The project website contains videos of full episodes.
\subsection{Bilateral Results}
We also study a bilateral case, where the second arm is the same instrument type as the arm used in the single experiment, and run for five episodes. As the two arms can both move their blocks simultaneously, the average length of each full episode is shorter, and was timed at $67.6 \pm 7.3$ seconds. The bilateral case raises the possibility of having a failure case with collisions among the arms, but we did not experience any due to carefully chosen block orderings as described in Section~\ref{sec:trajectory-motion}.
Over 5 episodes, the success rate of block transfer attempts was $46 / 59$ = \SI{78.0}{\percent} (Table~\ref{tab:dvrk-results}). We ran these episodes after the single-arm case, and there may have been extra wear and tear. Furthermore, the bilateral results require slight adjustments of the placing angle for each block to avoid arms and blocks colliding with each other, which may increase the chances of placing errors. Across the 59 attempts, there were 2 pick, 4 place-stuck, and 7 place-fall failures.
\subsection{Human Surgeon Teleop}\label{ssec:danyal}
\begin{table}[t]
\caption{
\small
Physical experiments from Dr. Danyal Fer for the single and bilateral arm cases, with two episodes each. Results are presented in a similar manner as in Table~\ref{tab:dvrk-results}.
}
\centering
\begin{tabular}{l | l r l l l }
Task & Success Rate & Time (s) & Pick & Stuck & Fall \\ \hline
Single & 0.958 (23/24) & 5.08 & 0.000 & 0.042 & 0.000 \\
Bilateral & 0.833 (20/24) & 3.45 & 0.046 & 0.125 & 0.000 \\
\end{tabular}
\vspace*{-10pt}
\label{tab:danyal-1}
\end{table}
Dr. Danyal Fer, a surgical resident, performed two episodes of the single- and double-arm peg transfer tasks following the same experiment protocol and setup as the automated system. Table~\ref{tab:danyal-1} summarizes peg transfer results from Dr. Fer. In addition, Appendix~\ref{sec:danyal-nonpainted} has results on Dr. Fer's corresponding episodes for the task with standard, off-the-shelf FLS peg transfer materials that were not painted red, and thus were less sticky and allowed for more color cues.
Dr. Fer did not experience any place-fall failures in his episodes, but had one failure (place-stuck) in the single-arm case and four failures (one grasp, three place-stuck) in the bilateral case. Some of Dr. Fer's placing attempts resulted in the block hitting part of the target peg, but he recovered by raising the block and repeating the placing motion, thus avoiding failures. Dr. Fer completed the single- and bilateral-arm trials in $61.0 \pm 3.0$ and $41.5 \pm 4.9$ seconds, which is significantly faster than the automated system.
\subsection{Failure Cases}
The vast majority of the dVRK's failures were placing failures (28 for single-arm, 11 for double-arm), such as those shown in Figs.~\ref{fig:failure-place-stuck} and~\ref{fig:failure-place-fall}). Placing is challenging because, even if we command the robot to drop at a fixed target for each peg, the orientation of the gripped block varies at the time of release. Dr. Fer was able to more reliably avoid placing failures because he could react in real time in case his initial placing did not fully insert the block into a peg.
A place-stuck failure is not as severe as a fallen block, because a gentle nudge can slide the block in place. In the dVRK experiments, we observed that several place-stuck cases were unintentionally ``corrected'' by a subsequent action, which either knocked the block into the peg or removed an underlying block that prevented full insertion. If we count those ``corrected'' blocks as successes, the dVRK's success rate for the single- and bilateral-arm setups would be $210/236$ = \SI{89.0}{\percent} and $49/59$ = \SI{83.1}{\percent}.
\section{Discussion and Future Work}\label{sec:conclusions}
In this paper, we explore the potential for depth sensing in automating the FLS peg transfer task. We demonstrate a proof of concept of the procedure and show how using a high-quality depth camera and calibration can allow a da Vinci surgical robot to autonomously perform the task with \SI{86.9}{\percent} and \SI{78.0}{\percent} success rates (single- and bilateral-arms, respectively).
Results suggest depth-sensing can be effective for automated peg transfer but there remains a significant gap between automated and expert human performance.
In future work, we will address placing failures. We will use more sophisticated calibration~\cite{seita_icra_2018}, motion-planning~\cite{Latombe:1991:RMP:532147}, visual servoing~\cite{kragic_servoing_2002}, and error recovery techniques using the depth-sensor and tactile feedback from joint motor currents to enhance error detection precision and speed. We will also explore the use of a surgical robot simulator to run reinforcement learning~\cite{gym,opensource_rl_surgical_2019,Sutton_2018} to train closed-loop controllers for the task, and will additionally use ideas from deep reinforcement learning for placing and insertion tasks~\cite{deepmind_insertion_2018,nair_insertion_2019}.
\section*{Acknowledgments}
{\small
This research was performed at the AUTOLAB at UC Berkeley in affiliation with the Berkeley AI Research (BAIR) Lab, Berkeley Deep Drive (BDD), the Real-Time Intelligent Secure Execution (RISE) Lab, the CITRIS ``People and Robots'' (CPAR) Initiative, and with UC Berkeley's Center for Automation and Learning for Medical Robotics (Cal-MR). The authors were supported in part by donations from SRI International, Siemens, Google, Toyota Research Institute, Honda, Intel, and Intuitive Surgical. The da Vinci Research Kit was supported by the National Science Foundation, via the National Robotics Initiative (NRI), as part of the collaborative research project ``Software Framework for Research in Semi-Autonomous Teleoperation'' between The Johns Hopkins University (IIS 1637789), Worcester Polytechnic Institute (IIS 1637759), and the University of Washington (IIS 1637444). Daniel Seita is supported by a National Physical Science Consortium Fellowship.
}
\appendices
\section{Human Operator on Non-Painted Case}\label{sec:danyal-nonpainted}
\begin{table}[h]
\caption{
\small
Physical experiments from Dr. Danyal Fer on the peg transfer task setup without the red paint. Results are presented in a similar manner as in Table~\ref{tab:dvrk-results}.
}
\centering
\begin{tabular}{l | l r l l l l}
Task & Success Rate & Time (s) & Pick & Stuck & Fall \\ \hline
Single & 1.000 (120/120) & 4.61 & 0.000 & 0.000 & 0.000 \\
Bilateral & 0.959 (117/122) & 2.71 & 0.016 & 0.025 & 0.000 \\
\end{tabular}
\vspace*{-10pt}
\label{tab:danyal-raw}
\end{table}
As an extra comparison, Table~\ref{tab:danyal-raw} reports results from Dr. Fer using the non-painted setup. This version allows for stronger color cues, avoids sticky red paint, and is closer to the standard FLS peg transfer task. Results indicate that Dr. Fer's performance in this case is superior to that of the painted setup, with success rates of \SI{100.0}{\percent} and \SI{95.9}{\percent} for the single and bilateral arm setups (versus \SI{95.8}{\percent} and \SI{83.3}{\percent} with red paint), along with faster block attempts of \SI{4.61}{\second} and \SI{2.71}{\second} (versus \SI{5.08}{\second} and \SI{3.45}{\second} with red paint). He completed episodes in $55.3 \pm 4.5$ and $33.0 \pm 3.4$ seconds, respectively.
\bibliographystyle{IEEEtranS}
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El Partido Fascista Mexicano (PFM) fue un partido político que se formó en México en 1922, que se basaba oficialmente en el fascismo italiano. El partido fue fundado por Gustavo Sáenz de Sicilia. Fue formado en gran parte por grupos opositores a las políticas de la Revolución mexicana de la clase media urbana y rural que se oponían al socialismo y a la reforma agraria que vieron el fascismo como una alternativa. La base del partido de los seguidores era en gran parte conservadora y antirrevolucionaria.
Historia
El partido fue visto con consternación por los fascistas italianos, con el embajador de Italia en 1923, afirmando que "Este partido no era otra cosa que una mala imitación de la nuestra, y no poseen las causas de origen y las finalidades de la misma. Es, de hecho, asumió el aspecto de un movimiento político que tiende a reunir en el conjunto del viejo país las fuerzas conservadoras y católicas dispersas por la revolución, y formar, de este modo, un partido claramente opuestos al gobierno actual".
Disolución
El PFM se disolvió en 1924, sus militantes formaron la Confederación de la Clase Media, un grupo contrarrevolucionario de corte católico y conservador contraria al cardenismo, donde apoyaron de manera activa y directa la rebelión de Saturnino Cedillo. Años después algunos de sus militantes pasaron a formar parte de la Unión Nacional Sinarquista, una organización ultranacionalista de ultraderecha con una base ideológica más sólida y propia, con principios de doctrina mejor definidos.
Referencias
Partidos políticos desaparecidos de México
Partidos políticos fundados en 1922
Partidos políticos disueltos en 1924
Partidos fascistas
Fascismo en México | {
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Minority lawmakers upset by audit of agencies
Black and Hispanic legislators said Monday they were outraged by a state auditor's report that showed many agencies had not complied with a Texas law designed to help minority- and women-owned businesses win state contracts.
The report released Friday by State Auditor Lawrence Alwin said nine of the 10 agencies reviewed are not fully complying with the state's Historically Underutilized Business program. Three of those, the Department of Protective and Regulatory Services, the State Board for Educator Certification and Southwest Texas State University, did not even make a "good-faith" effort to comply, the audit said.
"We're very disappointed," said Sen. Leticia Van de Putte, D-San Antonio. "It's unacceptable."
Rep. Joe Deshotel, D-Port Arthur, said the issue cuts across racial lines.
"This is an economic development issue, not an African-American or Hispanic issue," he said.
The program is designed to increase contracting opportunities through planning and strategy, outreach programs, reporting statistics and looking for qualified subcontractors.
According to the lawmakers, contracting with HUBs reached its peak at 16.4 percent of total state purchases in the 1996-97 fiscal year. The program was overhauled in 1999 and HUB contracting has fallen off to 9.7 percent in 2001-02.
The audit was the second in two years to show that agencies were not complying. Two agencies in the previous report, the Texas Education Agency and the Health and Human Services Commission _ still aren't fully complying, the new report said.
"The Texas Education Agency respectfully disagrees with the state auditor," agency spokeswoman Debbie Graves Ratcliffe told the Austin-American Statesman when the report was issued.
"For the last three fiscal years, the TEA has had a greater percentage of expenditures made through HUBs than the state," she said.
The report also cited Prairie View A&M University, a historically-black school, as not fully complying with three of the four prongs of the program, although it was cited for making a good faith effort.
Sen. Royce West, D-Dallas, said lawmakers should hold the agencies accountable. He said the Legislature could ask the Legislative Budget Board to take over agency purchasing if the problem isn't fixed.
"I think it's time to take the hammer out of our pocket and start waving it," West said.
State Auditor's Office: http//:www.sao.state.tx.us.com | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/2363375\/the-maximum-principle-for-subharmonic-functions","text":"# The maximum principle for subharmonic functions\n\nLet $V$ be a bounded open set in $\\mathbb{R}^{n}$ with $n\\geq2$, and $u$ a subharmonic function on an open set containing the closure $\\overline{V}$ of $V$. Suppose we have $u<M$ on $V$ ($M$ is a constant). Can we conclude that $u\\leq M$ on $\\overline{V}$?\n\n\u2022 Do you allow your subharmonic functions to attain the value $-\\infty$? \u2013\u00a0Daniel Fischer Jul 21 '17 at 15:07\n\u2022 Yes, but cannot be identically $-\\infty$. \u2013\u00a0M. Rahmat Jul 22 '17 at 0:52\n\nI have a partial answer to this question: if $u$ is continuous at each point of the boundary $\\partial V$ of $V$, the answer is yes! In fact, we have by continuity of $u$ and the maximum principle, $$u(\\zeta)=\\limsup_{\\substack{x\\rightarrow\\zeta\\\\(x\\in V)}}u(x)\\leq M$$ for all point $\\zeta\\in\\partial V$, which proves the claim!\n\u2022 This actually works for all functions that are continuous on the boundary, as $u(x)<M$ for all $x\\in V$ by assumption. No need for the maximum principle. \u2013\u00a0Giuseppe Negro Jul 27 '17 at 11:29","date":"2021-01-22 05:54:35","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\": 0.9321331977844238, \"perplexity\": 82.88182780542641}, \"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-2021-04\/segments\/1610703529128.47\/warc\/CC-MAIN-20210122051338-20210122081338-00024.warc.gz\"}"} | null | null |
\section{Introduction and motivations}
A quite intriguing aspect of cosmological models based on the low
energy string theory effective action is the occurrence of a phase of
growing dilaton coupling that might trigger a number of interesting
implications which were specifically addressed in the framework of the
pre-big-bang inflationary models \cite{ven1}.
One of the distinctive aspects of this class of models is the
amplification, via gravitational instability,
of the vacuum inhomogeneities associated both with the metric
and with the Abelian gauge fields
\cite{m2}. If this picture is correct, we should live, at later times
in an Universe filled with stochastic backgrounds of gravitational
waves and of of Abelian gauge fields which, thanks to the magnetic
flux conservation in a hot plasma, might also have interesting effects.
A lot of efforts have certainly been devoted to the analysis of
the cosmological solutions of the low energy string theory effective
action \cite{LTCS} in order to understand if the physical properties
of those solutions were at all relevant for the explanation of some general
features of our Universe. In all these studies one of the key
assumptions was the complete homogeneity
of the classical evolution of
the geometry. It has been recently realized that
it would be quite interesting to
relax this very stringent assumption and, therefore,
inhomogeneous string cosmological models were discussed
from different points of view \cite{barrow,ven2} but always in the
framework of the low energy string theory effective action.
There are of course different motivations for these investigations.
In general, it is always quite dangerous to make
strict assumptions about the form of the cosmological solutions. We
deal here with solutions of a completely non-linear system of
evolution equations and we cannot exclude, in principle, physical
situations were the Universe was very inhomogeneous at early times
evolving, ultimately, towards a homogeneous (and possibly isotropic)
epoch at later times. In order to address these questions one
should start, from the very beginning, with a completely inhomogeneous
metric.
Completely inhomogeneous configurations are also required in
the analysis of the initial conditions of cosmological (inflationary)
models. To relax the homogeneity assumption turned out to be relevant
for understanding if (and when) inflation could take place
starting from more generic initial conditions.
An example in this direction are the investigations
concerning the role played by initial (spherical) inhomogeneities
in chaotic inflationary models \cite{piran}.
With the same motivations (but in the context of string inspired
cosmological models) quasi-homogenous metrics were used
for the discussion of the initial value problem
\cite{ven2} in the pre-big-bang scenario \cite{ven1,gasven}.
In this last class of models,
the discussion of the initial conditions in a
completely homogeneous metric can lead to quite different results
\cite{turner}. We argue, therefore, that there is an urgent need
of a better understanding of completely inhomogeneous configurations
of the massless string modes.
We cannot a priori exclude that the features of
inhomogeneous string cosmological models might be significantly
different from the ones of their completely homogeneous counterpart.
An open question of (homogeneous) string cosmological models
is the occurrence of
curvature singularities
\cite{gasven} which turned out to be closely connected with the
problem of a graceful exit from a phase of growing dilaton coupling
\cite{brusven}, towards a phase of declining dilaton coupling. In
a completely homogeneous metrics quite general (numerical)
studies suggested that a phase of growing coupling is compatible
with the regularity of the curvature invariants (in the String frame)
provided higher order (curvature) corrections to the low energy
(tree-level) effective action are included in the game
\cite{brusven}. This idea received recently new attention \cite{mm}.
Moreover, the idea of a graceful exit transition from the initial
phase of growing coupling to the subsequent standard
radiation-dominated evolution has been the subject of many
investigations \cite{prr}.
In this paper we will not address the problem of the graceful exit but
we will be concerned with the regularity of the curvature
invariants.
In the case of a completely inhomogeneous metric is it still
mandatory to hit a singularity after a phase of growing coupling?
Is it possible to regularize the curvature invariants if the Universe
was sufficiently inhomogeneous at early times?
These are the main question we will try to address and, in this sense, our
approach is qualitatively different from the ones followed in Ref.
\cite{mm,prr} (where homogeneity was assumed)
and could be related to some pioneering
attempts discussed in Ref. \cite{GMV}.
Regular (inhomogeneous) solutions of the low energy string
theory effective action were constructed \cite{GMV} by ``boosting away'' the
curvature singularities through the $O(d,d)$
symmetries which were extensively studied in the past \cite{odd,meiss} and
which represent a natural generalization of the scale factor duality
\cite{ven1,sfd}. The key observation was that starting with a generic singular
cosmology in $1+1$ dimensions or with a $1+1$ dimensional black hole,
the singularity gets ``boosted away'' by $O(2,2)$ transformation involving
a third (originally flat) dimension. Moreover, by considering the
inhomogeneous cosmological model of Nappi and Witten \cite{NW} it was
also shown that the singularities of the model can be ``boosted away'' by
a $O(3,3)$ transformations involving a fifth dimension.
The main conclusion of Ref. \cite{GMV} was that, provided the coupling
oscillates and provided an antisymmetric tensor background is
included, the curvature invariants are regular. Therefore, there are
no doubts that it is certainly physically relevant to relax the
homogeneity assumption usually made in string inspired cosmological
models. Following this logical line and applying the Occam razor
we want to study inhomogeneous string cosmological models
with the minimal field content
(i.e. the gravi-dilaton action). In other words,
we want to understand if it is possible to have
simultaneously satisfied these requirements: growing coupling,
regularity of the curvature invariants, absence of antisymmetric
tensor field, absence of any dilaton potential.
In this sense the aim of our investigation is different from (but
complementary to) the one of Ref. \cite{GMV}.
To tackle this problem many different strategies can be certainly
employed. The most general one would consist in trying to solve the
low energy beta functions for a generic metric $g_{\mu\nu}(\vec{x},t)$
depending simultaneously upon {\em space and time}. A simplification
adopted in previous studies \cite{ven2,barrow} was also to stick to a
{\em diagonal} inhomogeneous metric.
Also the form of the inhomogeneous metric is, in a way, ``minimal''.
In fact the simplest non-trivial case of fully inhomogeneous
metrics is the one
where the inhomogeneities are distributed along
a line. In this case the whole spatial dependence is reduced to only
one coordinate (which we took, in our examples, to run along the the $x$ axis).
Needless to say that this is not the most general situation. At the
same time there are no reasons to forbid this choice.
A posteriori we finally found useful to impose an extra symmetry on
the metric which will turn out to possess, in our discussion, two
killing vector fields which are hypersurface orthogonal and
orthogonal to each other. This form of the
metric seems to be widely used in the study of inhomogeneous
cosmological models in general relativity \cite{kra}.
It is interesting to notice (as we will explicitly
show) that this ansatz for the
metric is compatible with a completely {\em homogenous} dilaton
coupling which turns out to be, ultimately, only dependent upon time.
We solved
exactly the low energy beta functions in the String
frame by using all the assumptions listed above. We computed
the curvature invariants associated with the obtained solutions and we
found a class of regular (and parity-invariant) solutions
where the curvature invariants are
all bounded. These solutions describe the physical situation where a
growing dilaton coupling smoothly evolves between two asymptotically
constant regimes. The physical parameter describing the solutions is
essentially the maximal energy density of the dilaton background in
string units.
From a technical point
of view, we have to mention that the most difficult part of our present
exercise is to solve the off-diagonal component of the beta functions
which do not contain second order derivatives with respect to time and
which is then a constraint mixing first order {\em time} derivatives
with first order {\em spatial} derivatives.
Therefore, the first step will be the solution of
the constraint. If the obtained solution will be compatible with the
dilaton equation and with the remaining components of the beta
functions, then, we will have a particular solution whose singularity
properties can be investigated by the direct calculation of the
curvature invariants.
Before closing this introduction we want to recall that the work
presented in this paper is certainly overlapping with previous work
performed in the context of general relativity. In particular in the
limit of constant dilaton field our solutions match the well known
inhomogeneous vacuum solutions of general relativity. The
simplest inhomogeneous vacuum models are those with two space-like
commuting killing vectors, known as orthogonally transitive $G_2$
cosmologies \cite{v1,v2}. Moreover, useful background material for the
present investigation can be also found in \cite{verd} where soliton
solutions were discussed in space-times with two space-like Killing
fields.
The plan of our paper is then the following. In Section 2 we will
derive the explicit form of the evolution equations of the dilaton and
of the metric in the String frame picture.
In Section 3 we will briefly derive the dilaton vacuum
solutions in the String frame and
we will generalise them to
the case of homogeneous dilaton background with inhomogeneous metric.
In Section 4 we will focus our attention on the case of growing
dilaton solutions and we will provide two physical examples whose
main physical aspect is the parity invariance of the background geometry.
Motivated the results of Section 4 we generalized our examples to a
class of parity even solutions (Section 5) and we computed the
curvature invariants in this situation. We found that they are regular
as in the case of the examples of Section 4.
Section 6 contains our concluding remarks.
Because of the excessive
length of the formulas contained in this paper we made the choice of
reporting few (indeed relevant) technical results in the Appendix which then
collects useful calculations for the interested reader.
\renewcommand{\theequation}{2.\arabic{equation}}
\setcounter{equation}{0}
\section{Basic Equations}
The low energy string theory effective action can be written in the
String frame and in the absence of antisymmetric tensor as \cite{LTCS}:
\begin{equation}
S= - \frac{1}{\lambda^2_{s}}
\int d^4 x \sqrt{-g} e^{-\phi} \biggl[ R + g^{\alpha\beta}
\partial_{\alpha}\phi\partial_{\beta} \phi \biggr],
\label{action}
\end{equation}
where $\lambda_{s}$ is the string scale. We assume that the dilaton potential
is completely negligible. We shall consider only the particular
case of critical superstring theory with vanishing cosmological
constant and six (frozen) internal dimensions.
The evolution equations for the massless modes
can be easily derived by varying the action
with respect to the dilaton field and the metric:
\begin{eqnarray}
& & R - g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi + 2
\Box \phi=0,
\label{dilaton}\\
& & R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \frac{1}{2}g_{\mu\nu}
g^{\alpha\beta} \partial_{\alpha}\phi\partial_{\beta}\phi +
\nabla_{\mu}\nabla_{\nu}\phi - g_{\mu\nu} \Box\phi =0
\label{equation}
\end{eqnarray}
($\nabla_{\alpha}$ is the covariant derivative with respect to
the metric $g_{\mu\nu}$;
$\nabla_{\alpha}\nabla_{\beta}\phi =
\partial_{\alpha}\partial_{\beta}\phi - \Gamma_{\alpha\beta}^{\sigma}
\partial_{\sigma}\phi$; $\Gamma_{\alpha\beta}^{\sigma}$ are the
Christoffel symbols).
Using Eq. (\ref{dilaton}) into Eq. (\ref{equation}) we find the
familiar form of the tree-level beta functions
\begin{eqnarray}
& & R - g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi + 2
\Box \phi=0,
\label{beta1}\\
& & R_{\mu}^{\nu} + \nabla_{\mu}\nabla^{\nu} \phi=0.
\label{beta2}
\end{eqnarray}
We now specialize to the case of a diagonal metric which admits two
commuting space-like killing vector fields, both of which are
hypersurface orthogonal \cite{kra}:
\begin{equation}
~g_{00}= A(x,t),~~~ g_{x x} = -A(x, t), ~~~g_{yy} = - B(x, t) C(x, t), ~~~
g_{zz}= - \frac{B(x,t)}{C(x,t)}.
\label{metric}
\end{equation}
The corresponding line element is then
\begin{equation}
ds^2 = A(x,t)[ dt^2 - dx^2] - B(x,t)\biggl[ C(x,t) dy^2 +
\frac{dz^2}{C(x,t)}\biggr].
\label{line}
\end{equation}
From Eqs. (\ref{metric}) and (\ref{line})
the Christoffel symbols, the Riemann tensors and the
Weyl tensors can be easily computed. In Appendix A we report the
Christoffel symbols, the Ricci tensors and the curvature scalar which
are required in order to express directly
Eqs. (\ref{beta1}) and (\ref{beta2}) in the metric (\ref{metric}).
In the metric (\ref{metric}), Eqs (\ref{beta1}) and (\ref{beta2}) will
produce a non-linear system of (partial) differential equations
which we are going to solve. First of all suppose that the
space-time dependence of the
metric functions can be factorised:
\begin{equation}
A(x,t) = a(t)~\alpha(x),~~~~B(x,t) = b(t)~\beta(x),~~~~
C(x,t) = c(t)~\kappa(x),
\label{ansatz}
\end{equation}
(notice that the latin letters will denote the temporal part whereas the Greek
ones will denote the spatial part of the metric functions).
This decomposition has the great advantage, once inserted in
Eqs. (\ref{beta1}) and (\ref{beta2}), of transforming the original
system into a system of
non-linear {\em ordinary} differential equations in the two variables
$x$ and $t$.
By inserting Eq (\ref{ansatz}) into the components of the Ricci
tensors and into the curvature scalar given in Appendix A
(see Eqs. (\ref{rxx})-(\ref{r})) we can
write down explicitly Eq. (\ref{beta1}) and all the components of
Eq. (\ref{beta2}) in the metric given in Eq. (\ref{metric}).
The $(xx)$, $(yy)$, $(zz)$ and $(00)$ components of Eq. (\ref{beta2}) will be
respectively:
\begin{eqnarray}
&&\frac{\ddot{a}}{a} + \frac{\dot{a}}{a} \frac{\dot{b}}{b}
-\left(\frac{\dot{a}}{a}\right)^2 - \frac{\dot{a}}{a} \dot{\phi} = 2
\frac{\beta''}{\beta}+\left(\frac{\alpha'}{\alpha}\right)' +
\left(\frac{\kappa'}{\kappa}\right)^2 - \left(\frac{\beta'}{\beta}\right)^2 -
\frac{\alpha'}{\alpha}~\frac{\beta'}{\beta} ~~~~~~~~~~(xx),
\label{xx}\\
&&\frac{\ddot{c}}{c} + \frac{\ddot{b}}{b} + \frac{\dot{b}}{b}
\frac{\dot{c}}{c} - \left(\frac{\dot{c}}{c}\right)^2 - \left(\frac{\dot{b}}{b}
+ \frac{\dot{c}}{c}\right)\dot{\phi} = \frac{\kappa''}{\kappa}
+\frac{\beta''}{\beta}
- \left(\frac{\kappa'}{\kappa}\right)^2
+ \frac{\beta'}{\beta}~\frac{\kappa'}{\kappa}~~~~~~~~~~~~~~(yy),
\label{yy}\\
&&\frac{\ddot{b}}{b} + (\frac{\dot{c}}{c})^2
-\frac{\dot{b}}{b}~\frac{\dot{c}}{c} - \frac{\ddot{c}}{c} -
\left(\frac{\dot{b}}{b} -\frac{\dot{c}}{c}\right)\dot{\phi}=
\frac{\beta''}{\beta} - \frac{\kappa''}{\kappa} + (\frac{\kappa'}{\kappa})^2
-\frac{\beta'}{\beta}\frac{\kappa'}{\kappa}~~~~~~~~~~~~~~~~~(zz),
\label{zz}\\
&&2 \ddot{\phi} - \frac{\dot{a}}{a} \dot{\phi} +
\left(\frac{\dot{a}}{a}\right)^2 +\frac{\dot{a}}{a}~\frac{\dot{b}}{b}+
\left(\frac{\dot{b}}{b}\right)^2 - \left(\frac{\dot{c}}{c}\right)^2 -
\frac{\ddot{a}}{a}
- 2 \frac{\ddot{b}}{b} =
- \frac{\alpha'}{\alpha}~\frac{\beta'}{\beta}
- \left(\frac{\alpha'}{\alpha}\right)' ~~~~(00),
\label{00}
\end{eqnarray}
whereas for Eq. (\ref{beta1}) and for the mixed
component (i.e. $(x0)$) of Eq. (\ref{beta2}) we will have respectively:
\begin{eqnarray}
&&2 \ddot{\phi} + 2 \frac{\dot{b}}{b}\dot{\phi} - {\dot{\phi}}^2
+ \frac{1}{2} \left(\frac{\dot{b}}{b}\right)^2
- \frac{1}{2} \left(\frac{\dot{c}}{c}\right)^2
- \frac{d}{dt}\left(\frac{\dot{a}}{a}\right) - 2 \frac{\ddot{b}}{b} =
\frac{1}{2}(\frac{\beta'}{\beta})^2
- \frac{1}{2}\left(\frac{\kappa'}{\kappa}\right)^2
\nonumber\\
&&- \left(\frac{\alpha'}{\alpha}\right)' - 2
\frac{\beta''}{\beta}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(\phi)
\label{phi}\\
&&\frac{\dot{b}}{b} ~\frac{\alpha'}{\alpha} + \frac{\dot{a}}{a}~
\frac{\beta'}{\beta} - \frac{\beta'}{\beta}~ \frac{\dot{b}}{b} -
\frac{\dot{c}}{c}~ \frac{\kappa'}{\kappa}-
\frac{\alpha'}{\alpha}\dot{\phi} = 0
\label{x0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(x0)
\end{eqnarray}
(in these equations and in all the paper we adopted the compact
notation : $'= \frac{\partial}{\partial x}$ and $\cdot=
\frac{\partial}{\partial t}$).
In the system of Eqs. (\ref{xx})--(\ref{phi}) at the left hand side
of each equation we only have {\em time}-dependent quantities whereas
at the right hand side we only have {\em space}-dependent quantities.
The exception is represented by Eq. (\ref{x0}) which is
{\em not} a dynamical equation but a
constraint mixing spatial and temporal derivatives.
The coupled system of non-linear differential equations is written in
the case where the dilaton coupling is completely {\em homogeneous},
namely $\phi' =0,~~~~\phi(x,t) \equiv \phi(t)$.
The solution of the system of coupled, non-linear differential
equations derived in the previous equations will be the main goal of
the next Section.
\renewcommand{\theequation}{3.\arabic{equation}}
\setcounter{equation}{0}
\section{Homogeneous dilaton solutions}
The first step in the derivation of our solutions with homogeneous
dilaton will be to consider the constant dilaton case where $\dot{\phi}(t)=0$.
In order to find a general expression able to solve the constraint,
we sum and subtract
Eqs. (\ref{yy}) and (\ref{zz}), obtaining, respectively,
\begin{equation}
\frac{\ddot{b}}{b} = \frac{\beta''}{\beta},~~~~
\Biggl[\frac{d}{d t}\left(\frac{\dot{c}}{c}\right)
+ \frac{\dot{b}}{b}\frac{ \dot{c}}{c}\Biggr] = \Biggl[\frac{d}{dx} \left(
\frac{\kappa'}{\kappa}\right) +
\frac{\beta'}{\beta}\frac{\kappa'}{\kappa} \Biggr].
\label{condition}
\end{equation}
A particular solution of
Eq. (\ref{condition}) which might solve the constraint is
\begin{equation}
b(t)= \cosh{\mu t},~~~~~\beta(x) = \sinh{\mu x}.
\label{ans}
\end{equation}
It is true that Eq. (\ref{condition}) can be solved also for another
choices of trial functions: not all the possible choices are allowed
by the off-diagonal constraint given in Eq. (\ref{x0}), which can now
be re-written as
\begin{equation}
\Biggl[\mu \tanh{\mu t} \left(\frac{\alpha'}{\alpha}\right) +
\left(\frac{\dot{a}}{a}\right)\frac{\mu}{\tanh{\mu x}} - \mu^2 \frac{\tanh{\mu
t}}{\tanh{\mu x}} - \left(\frac{\dot{c}}{c}\right)
\left(\frac{\kappa'}{\kappa}\right)\Biggr]=0.
\label{constr}
\end{equation}
Provided
\begin{equation}
\frac{\dot{a}}{a} = c_{1}\tanh{\mu t},~{\rm and}~~~\frac{\dot{c}}{c}= c_{2}
\tanh{\mu t},
\label{ansatz1}
\end{equation}
the time dependence appearing in Eq.
(\ref{constr}) factorizes
leading, ultimately, to a first-order (ordinary) differential relation
involving only functions depending upon the spatial
coordinates. Therefore direct (trivial) integration of
Eqs. (\ref{condition}) and (\ref{constr}), together with the conditions
(\ref{ans}) and (\ref{ansatz1}), gives us the following first
integrals
\begin{eqnarray}
&&\left(\frac{\kappa'}{\kappa}\right) = \frac{c_{2} +
c_{3}}{2}\biggl(\tanh{\frac{\mu x}{2}}\biggr)^{-1} + \frac{c_{2}-c_{3}}{2}
\biggl(\tanh{\frac{\mu x}{2}}\biggr),
\label{kappa}\\
&&\left(\frac{\alpha'}{\alpha}\right) = \Biggl[
\biggl(\frac{c^2_{2}+c_{2} c_{3} - \mu
c_{1} + \mu^2}{2 \mu}\biggr)\biggl(\tanh{\frac{\mu x}{2}}\biggr)^{-1}
+ \biggl(\frac{c^2_{2} - c_{2} c_{3} - \mu
c_{1} +\mu^2 }{2\mu}\biggr) \tanh{\frac{\mu x}{2}}\Biggr],
\label{alpha}
\end{eqnarray}
where $c_{3}$ is now an integration constant and
$c_{1}$ and $c_{2}$ are nothing but arbitrary constants which have to be
fixed by requiring the compatibility of Eqs. (\ref{ans}) and
(\ref{ansatz1}) with the complete system given by by Eqs.
(\ref{xx})--(\ref{phi}).
Let us therefore find the compatibility relations which have to be
satisfied by $c_{1}$, $c_{2}$ and $c_{3}$.
By direct substitution of Eqs. (\ref{ans}), (\ref{ansatz1}), (\ref{kappa}) and
(\ref{alpha}) into Eqs. (\ref{xx}) and (\ref{00}) we get, respectively
\begin{equation}
c^2_{3} = 3 \mu^2 + c^2_{2} - 2 \mu c_{1},~~~c^2_{2} =2 \mu c_{1} + \mu^2.
\label{c3}
\end{equation}
Surprisingly enough the same two conditions also solve
Eq. (\ref{phi}). Moreover Eqs. (\ref{yy}), (\ref{zz}) and (\ref{x0})
are automatically satisfied for every $c_{1}$, $c_{2}$ and $c_{3}$
not leading (as expected) to any further relation among the
constants appearing in our trial solution.
The present solutions are the String frame version of
analogous inhomogeneous vacuum solutions previously discussed in the
context of general relativity \cite{kra,v1}.
After having solved the case where the dilaton was constant we can now
discuss the case where the dilaton is time dependent.
The ansatz given in Eqs. (\ref{ans}) and (\ref{ansatz1}) can be
slightly modified in order to
match the time dependence of the dilaton field, which now appears, in
the equations of motion (\ref{00})--(\ref{phi})
with its first and second derivatives.
The idea is then to assume that in the presence of the dilaton field
the solutions of our system of equations can be written as
\begin{equation}
\left(\frac{\dot{a}}{a}\right) = c_{1} \tanh{\mu t} +
\Gamma(t),~~~\left(\frac{\dot{b}}{b}\right)= \mu \tanh{\mu t} + \Delta(t), ~~~
\left(\frac{\dot{c}}{c}\right) = c_{2} \tanh{\mu t} + \Sigma(t)~~,
\label{trial}
\end{equation}
where, at the moment $\Gamma(t)$, $\Delta(t)$ and $\Sigma(t)$, are
three unknown functions which should be hopefully determined by
solving explicitely the evolution equations of the metric and of the
dilaton field.
By comparing this ansatz with one made in the previous Section (see
Eq. (\ref{ansatz1})) it is evident which kind of logic we
followed since for $\Gamma=\Delta=\Sigma=0$ we exactly recover our
Eq. (\ref{ansatz1}) . Notice that $\Gamma$, $\Delta$ and $\Sigma$ are only
dependent upon time since we want to accommodate a {\em homogenous}
dilaton .
With the ansatz (\ref{trial}) we then repeat step by step the
procedure outlined in Section 1. First of all, we insert
Eq. (\ref{trial}) into Eq. (\ref{x0}) and we get the condition which
needs to be satisfied in order to solve the constraint:
\begin{eqnarray}
\Biggl[\left(\frac{\alpha'}{\alpha}\right) + \frac{ \mu - c_{1}}{\tanh{\mu x}} \Biggr]= \frac{c_{2}}{\mu}~
\left(\frac{\kappa'}{\kappa}\right)
- \frac{1}{\mu~\tanh{\mu t}}\Biggl\{ \frac{\mu~(\Gamma
- \Delta)}{\tanh{\mu x}} -
\left(\frac{\kappa'}{\kappa}\right)~ \Sigma -
\left(\frac{\alpha'}{\alpha}\right)~ \dot{\phi}\Biggr\}.
\label{first}
\end{eqnarray}
By summing and subtracting Eqs. (\ref{yy}) and (\ref{zz}), we obtain
\begin{equation}
\frac{\ddot{b}}{b} -\frac{\dot{b}}{b}\dot{\phi} =
\frac{\beta''}{\beta},~~~~\Biggl[\frac{d}{d t}\left(\frac{\dot{c}}{c}\right)
- \frac{\dot{b}}{b}\frac{ \dot{c}}{c} +\frac{\dot{c}}{c} \dot{\phi}\Biggr]
= \Biggl[\frac{d}{dx} \left(
\frac{\kappa'}{\kappa}\right) +
\frac{\beta'}{\beta}\frac{\kappa'}{\kappa}\Biggr].
\label{I}
\end{equation}
Inserting Eq. (\ref{trial}) into Eqs. (\ref{I}), we
find, respectively:
\begin{eqnarray}
&&\dot{\Delta} + [ \Delta - \dot{\phi}]~\Delta + \mu~\tanh{\mu t}~ [ 2~
\Delta - \dot{\phi}] =0,
\label{second}\\
&&\Biggl[\frac{d}{d x}\left( \frac{\kappa'}{\kappa}\right) +
\frac{\beta'}{\beta} \frac{\kappa'}{\kappa} -
c_{2}\mu\Biggr] = \dot{\Sigma} + \tanh{\mu t}~ [ \mu~ \Sigma + c_{2}~ ( \Delta
- \dot{\phi})] + \Sigma~(\Delta - \dot{\phi}).
\label{third}
\end{eqnarray}
Eqs. (\ref{first}), (\ref{second}) and (\ref{third}) can be
consistently solved if $\dot{\phi}(t) = \Delta(t) =
\Gamma(t),~~~\Sigma=0$ leading to the solution
\begin{eqnarray}
\left(\frac{k'}{k}\right) &=& \Biggl[\frac{c_{3} +c_{2}}{2}\left(
\tanh{\frac{\mu x}{2}}\right)^{-1}
+ \frac{c_{2} - c_{3}}{2}\tanh{\frac{\mu x}{2}}\Biggr],
\nonumber\\
\left(\frac{\alpha'}{\alpha}\right)&=&\Biggl[ \frac{c_{2}^2
+ c_{2}c_{3} - \mu c_{1} +
\mu^2}{2 \mu} \left( \tanh{\frac{\mu x}{2}}\right)^{-1} +
\frac{c_{2}^2 + c_{2}c_{3} - \mu c_{1} + \mu^2}{2
\mu}\tanh{\frac{\mu x}{2}}\Biggr],
\nonumber\\
\Gamma(t) &=& \frac{c_{4}}{\cosh{\mu t}},
\label{system2}
\end{eqnarray}
where $c_{3}$ and $c_{4}$ are integration constants.
Substituting Eqs. (\ref{trial}) and (\ref{system2})
into Eq. (\ref{xx}) and Eq. (\ref{00}) we
obtain, respectively, the following compatibility conditions
\begin{equation}
c^2_{3} = 3 \mu^2 + c^2_{2} - 2 \mu c_{1} \geq 0,~~~
c^2_{4} = c^2_{2} - \mu^2 - 2 \mu c_{1} \geq 0.
\label{rel2}
\end{equation}
Finally going into Eq. (\ref{phi}) we deduce that
\begin{equation}
2c_{4}^2 - \mu^2 + 6 \mu c_{1} = 3 c_{2}^2 - c_{3}^2,
\label{rel3}
\end{equation}
which turns out to be noting but a trivial linear combination of the
two conditions given in Eq. (\ref{rel2}).
As expected, Eqs. (\ref{yy}) and (\ref{zz}) are then satisfied without
imposing any further condition on $c_{1}$, $c_{2}$, $c_{3}$ and
$c_{4}$.
We are now ready to write down the explicit form of our solutions, namely:
\begin{eqnarray}
A(x,t) &=& e^{c_{4} gd(\mu t)}~ \Biggl[\cosh{\mu
t}\Biggr]^{\frac{c_1}{\mu}}
~\Biggl[\sinh{\left(\frac{\mu x}{2}\right)}
\Biggr]^{\frac{c_2^2 + c_2 c_3 - \mu c_1 + \mu^2}{\mu^2}}~
\Biggl[\cosh{\left(\frac{\mu x}{2}\right)}
\Biggr]^{ \frac{c_2^2 - c_2 c_3 - \mu c_1
+ \mu^2}{\mu^2} },
\nonumber\\
B(x,t) &=& e^{c_{4} gd(\mu t)}~\cosh{\mu t}~ \sinh{\mu x},
\nonumber\\
C(x,t) &=&\Biggl[\cosh{\mu t}\Biggr]^{\frac{c_2}{\mu}}~
\Biggl[\sinh{\left(\frac{\mu x}{2}\right)}\Biggr]^{\frac{c_{2} + c_{3}}{\mu}}~
\Biggl[\cosh{\left(\frac{\mu x}{2}\right)}\Biggr]^{\frac{c_{2} - c_{3}}{\mu}},
\nonumber\\
\phi(t) &=& \frac{c_{4}}{\mu}~gd(\mu t) - c_{5},
\label{thesolution}
\end{eqnarray}
where $gd(\mu t)= \arctan{[\sinh{\mu t}]}$
is the ``Gudermannian'' \cite{ryz} or hyperbolic amplitude. Notice
that the limit $c_{4}\rightarrow 0$ is slightly ill defined since, from
Eq. (\ref{rel2}), $c_2$ might take imaginary values and the
inequalites are, in this limit, strict equalities. Therefore, the
general expression of the solution (\ref{thesolution}) also changes,
in this limit.
Since the hyperbolic amplitude always grows, the dilaton
will either increase or decrease depending upon the sign of $c_{4}$.
Therefore, the physical meaning of $c_{5}$ is connected with the
initially small (perturbative) value of the coupling constant, whereas
$c_4$ is connected with its ``kinetic'' energy. The
physical situation we are describing with this solutions is really a
regime of very small dilaton coupling and in this sense we will
fine-tune $c_{5}$ to a quite small value. In this way higher loop
(genus) corrections will be automatically subleading. This procedure
is very reminiscent of what happens in the pre-big-bang scenario
\cite{ven1} where a (quite long) small coupling regime emerges
naturally from the tree-level (Kasner-like) solutions. In the pre-big-bang
case, the Kasner-like solutions have two ``branches'' separated by a
(curvature) singularity, and then, for each growing coupling solution
there is also a decreasing coupling solution.
We notice that, also in this sense, our solutions have close
analogies to the pre-big-bang case since, depending upon the value of
$c_{4}$ there are two branches: one of increasing coupling
($c_{4}>0$) and the other of decreasing coupling ($c_{4}<0$).
In the pre-big-bang case the singularity cannot be removed
(at tree-level). We will show, in the next
two sections, that in our solutions the two branches are not
analytically connected but, at the same time, the curvature
invariants, the dilaton kinetic energy are always well defined and
regular for any
$x,t$ at least for a family of solutions contained in
Eq. (\ref{thesolution}) (see Sections 4 and
5 for a discussion of this point).
\renewcommand{\theequation}{4.\arabic{equation}}
\setcounter{equation}{0}
\section{Two physical examples}
From Eq. (\ref{thesolution}) each particular solution of the system of
Eqs. (\ref{beta1}) and (\ref{beta2}) can be specified by fixing
$c_{1}$ and $c_{2}$ and by computing $c_{3}$ and $c_{4}$ from
Eqs. (\ref{rel2}). We tried different possible sets
of parameters and we found that not {\em all} the choices
of $c_{1}$ and $c_{2}$ will necessarily lead to the regularity of the
curvature invariants. The class of
regular solutions seems to be connected to the behaviour of the metric
tensor under (discrete) parity transformations (see the following
Section). At the same
time we point out that {\em there are} choices of $c_{1}$ and $c_{2}$
which lead to completely regular curvature invariants.
In this Section we discuss then particular examples of solutions
contained in Eq. (\ref{thesolution}) that share this quite interesting
property. In order to do that we will be inspired by some physical
requirement, namely we would like to have regularity of the curvature
invariants and, simultaneously, growing dilaton coupling which
corresponds to $\dot{\phi} > 0$.
For the set of parameters given by
\begin{equation}
c_{1} = 6 \mu,~~~c_{2}= 5 \mu,~~~c_{3} = - 4 \mu, ~~~c_{4}= 2\sqrt{3} \mu,
\label{ex1a}
\end{equation}
Eqs. (\ref{rel2}) are clearly solved.
The explicit form of the solution of Eq. (\ref{thesolution}) can then
be written, using Eqs. (\ref{ex1a}), as:
\begin{eqnarray}
&&A(x,t) = e^{2~\sqrt{3}~ gd(\mu t)}~ \Biggl[\cosh{\mu t}\Biggr]^{6}~
\Biggl[\cosh{\left(\frac{\mu x}{2}\right)}\Biggr]^{40},~~~~
B(x,t) = e^{2~\sqrt{3}~ gd(\mu t)}~\cosh{\mu t}~ \sinh{\mu x},
\nonumber\\
&&C(x,t) =\Biggl[\cosh{\mu t}\Biggr]^{5}~
\Biggl[\sinh{\left(\frac{\mu x}{2}\right)}\Biggr]~
\Biggl[\cosh{\left(\frac{\mu x}{2}\right)}\Biggr]^{9},
~~~~\phi(t) = 2\sqrt{3}~gd(\mu t) - c_{5}.
\label{thesolution1}
\end{eqnarray}
The dilaton is an increasing function
whose maximal value is given by $\phi(+\infty)= \sqrt{3}\pi - c_{5}$
(recall that $\lim_{t\rightarrow - \infty} gd(\mu t) = - (3/2) \pi$
and that $\lim_{t\rightarrow +\infty} gd(\mu t) = (\pi/2)$).
Now, our discussion is based on the (tree-level) action given by
Eq. (\ref{action}) which is valid provided
$g(\phi) \equiv \exp{[\phi/2]} = \exp{[\frac{c_{4}}{2} gd(\mu t) -
c_{5}]}\ll 1$.
If this is not the case, higher genus corrections should be
included. Consequently, for the compatibility of the solutions
discussed in the present Section with the effective description
adopted in Section 2 we have to require that the dilaton starts its
time evolution deep in its perturbative regime and this can be
achieved by fixing $c_{5}$. In the example given by the solution
(\ref{thesolution1}) we have that $g(+\infty) = \exp{( \sqrt{3} \pi
-c_{5})}\ll 1$, which implies that $c_{5} \gg \sqrt{3} \pi$. Therefore
choosing, for instance $c_{5} \sim 10 \sqrt{3} \pi$ we get that
$g(-\infty) = \exp{(- 13\sqrt{3} \pi)}$ and
$g(+\infty) = \exp{(-9\sqrt{3}\pi)}$.
Consider now a further example, namely
\begin{equation}
c_{1} = 5 \mu,~~~c_{2}= 4\mu,~c_{3}= - 3\mu,~~{\rm},~~~c_{4}=\sqrt{5}\mu.
\label{ex2a}
\end{equation}
Using Eqs. (\ref{ex2a}) into Eq. (\ref{thesolution})
we get
\begin{eqnarray}
&&A(x,t) = e^{\sqrt{5}~ gd(\mu t)}~ \Biggl[ \cosh{\mu
t}\Biggr]^{5}
\Biggl[\cosh{\left(\frac{\mu x}{2}\right)}\Biggr]^{24},
~~~~B(x,t) = e^{\sqrt{5}~ gd(\mu t)}~\cosh{\mu t}~ \sinh{\mu x},
\nonumber\\
&&C(x,t) =\Biggl[\cosh{\mu t}\Biggr]^{4}~
\Biggl[\sinh{\left(\frac{\mu x}{2}\right)}\Biggr]~
\Biggl[\cosh{\left(\frac{\mu x}{2}\right)}\Biggr]^{7},
~~~~\phi(t) = \sqrt{5}~gd(\mu t) - c_{5}.
\label{thesolution2}
\end{eqnarray}
The calculation of the curvature invariants for the solutions
(\ref{thesolution1}) and (\ref{thesolution2}) can be now performed.
In order to summarise our results we can say that each of the
curvature invariants
can be written (in a generalised notation) as:
\begin{equation}
{\cal I }= \mu^4~ f(x,t) [\cosh{\mu t}]^{-a}~
\Biggl[\cosh{\frac{\mu x}{2}}\Biggr]^{-b}
\label{inv}
\end{equation}
(${\cal I}$ labels a generic curvature invariant;
$f(x,t)$ is a regular expression containing combinations of
hyperbolic functions and it changes for each specific invariant
[see Appendix B];
$a$ and $b$ also change depending
upon the invariant under consideration, [see also Table 1]).
From
Eq. (\ref{inv}) and from the various forms of $f(x,t)$ which can be
deduced from Appendix B, we notice that ${\cal I}$ does not have any
pole for any finite value od $x$ and $t$. Moreover, we point out that for
large $x$ and $t$ the curvature invariants are suppressed. The
magnitude of the suppression depends upon $a$ and $b$ which are
reported in Table 1 but can also be read-off directly from
Eqs. (\ref{riemann})--(\ref{scalar}).
The only nasty thing which could then occur is that for some (large)
value of $x$ and/or $t$ some of the invariants explode because
$f(x,t)$ grows much faster than the suppression factor, parameterised,
for each invariant, by $a$ and $b$.
Keeping $x=x_c$ (where $x_{c}$ is some finite value of $x$)
and leaving $t$ to run we find that $f(x_{c},t)\sim [\cosh{\mu t}]^4$
for large $t$. Vice-versa keeping $t$ frozen at $t_{c}$ (where $t_{c}$
is some finite value of $t$) we find that (in the worse case)
$f(x,t_{c}) \sim [\cosh{\frac{\mu x}{2}}]^{4}$. Finally, by varying
$x$ and $t$ simultaneously we see that $f(x,t)$ can grow, at most
like $[\cosh{\frac{\mu x}{2}}]^{2}~ [\cosh{\mu t}]^4$ or like
$[\cosh{\frac{\mu x}{2}}]^{4}~ [\cosh{\mu t}]^2$. Therefore we see
that the growth of $f(x,t)$ is always suppressed by the high
(negative) powers (i.e. $a$ and $b$)
of the hyperbolic cosinus appearing in Eq. (\ref{inv}).
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$c_{4} = 2~\sqrt{3}~\mu$&
$R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$&
$C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$&
$R_{\alpha\beta}R^{\alpha\beta}$&
$R^2$\\
\hline
$a$ & $16$ & $16$ & $16$ & $16$\\
\hline
$b$ & $84$ & $84$ & $82$ & $80$\\
\hline
\end{tabular}
\end{center}
\nonumber
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$c_{4} = ~~\sqrt{5}~\mu $&
$R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$&
$C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$&
$R_{\alpha\beta}R^{\alpha\beta}$&
$R^2$ \\
\hline
$a$ & $14$ & $14$ & $14$ & $14$\\
\hline
$b$ & $52$ & $52$& $50$ & $48$\\
\hline
\end{tabular}
\end{center}
\label{ab}
\caption{We summarize the exponents of the suppression factor appearing
in Eq. (\ref{inv}). The upper part of the table refers to the case of
Eq. (\ref{thesolution1}), whereas the lower part of the table refers
to the case of Eq. (\ref{thesolution2}). The explicit analytical
results leading to this table can be found in the Appendix B, by
setting, respectively, $\alpha\equiv c_{4}/\mu = 2\sqrt{3}$ and
$\alpha=c_{4}/\mu \sqrt{5}$ in Eqs. (\ref{riemann}-(\ref{scalar}).}
\end{table}
We want now to make few more physical comments concerning the validity
of the present solutions.
The only dimension-full parameter appearing in the solution
(\ref{thesolution1}) and (\ref{thesolution2}) is $\mu = 1/L $ where
$L$ is, for the moment, a length scale which completely specifies each
solution which describes the ``spreading'' of the curvature
invariants around the origin (i.e. $x=0$, $t=0$).
The only fundamental scale appearing in Eq. (\ref{action})
is the string scale $\lambda_{s}$.
Correspondingly there is also a typical
energy connected with $\lambda_{s}$ which is the string energy, namely
$E_{s}= 1/\lambda_{s}$. The effective action (\ref{action}) which was
the starting point of our considerations holds for energies $E\ll
E_{s}$. In fact for $E\sim E_{s}$ higher order corrections in the
string tension $\alpha'= \lambda_{s}^{-2}$ should be included.
In order to be consistent with our approach we have to assume that the
typical scale $L$ has to be large compared to the string scale and
this means that $L> \lambda_{s}$. In the opposite case
(i.e. $L<\lambda_{s}$), $\lambda_{s}$ would not turn out as the
minimal length scale. Therefore if $L >\lambda_{s}$, we will also have
${\cal I}~<~ \lambda_{s}^{-4}$.
In Fig. \ref{rsquare} the scalar curvature (squared) is reported
for the two solutions given
in Eqs. (\ref{thesolution1}) and (\ref{thesolution2}) when $L=
10~\lambda_{s}$.
The corresponding analytical results are reported
in Eq. (\ref{scalar}) choosing, respectively $\alpha = 2 \sqrt{3}$ and
$\alpha = \sqrt{5}$. From these analytical
expressions we can clearly see that by tuning $L= 1/\mu$ for length
scales larger than $\lambda_{s}$, $R^2$ decreases (in string units) and
its spreading around the origin increases.
\begin{figure}
\begin{center}
\begin{tabular}{|c|c|}
\hline
\hbox{\epsfxsize = 7.5 cm \epsffile{r1curv.eps}} &
\hbox{\epsfxsize = 7.5 cm \epsffile{r2curv.eps}}\\
\hline
\end{tabular}
\end{center}
\caption{We plot the $R^2$ invariants corresponding
to the solution (\ref{thesolution1}) (left) and to the
solution (\ref{thesolution2})(right). The analytic form of these
invariants is given in Appendix B (Eq. (\ref{scalar})) by choosing
$\alpha = 2\sqrt{3}$ (for the plot at the left) and $\alpha= \sqrt{5}$ (for
the plot at the right). Here we also took $L= 10~\lambda_{s}$}
\label{rsquare}
\end{figure}
We can also look at the behaviour of
the time derivative of the dilaton which is illustrated ( Fig.
\ref{dilatonfig}) and of the related coupling constant
(Fig. \ref{couplingfig}).
\begin{figure}
\begin{center}
\begin{tabular}{|c|c|}
\hline
\hbox{\epsfxsize = 7.5 cm \epsffile{dilfig2.eps}}&
\hbox{\epsfxsize = 7.5 cm \epsffile{dilfig4.eps}}\\
\hline
\end{tabular}
\end{center}
\caption{We plot the dilaton kinetic term for the two exact
solutions discussed in this Section. The left figure corresponds
to the case given in Eq. (\ref{thesolution1}), whereas the right one
corresponds to Eq. (\ref{thesolution2}). We notice the absence of
any singularity.}
\label{dilatonfig}
\end{figure}
\begin{figure}
\begin{center}
\begin{tabular}{|c|c|}
\hline
\hbox{\epsfxsize = 7.5 cm \epsffile{dilfig1.eps}}&
\hbox{\epsfxsize = 7.5 cm \epsffile{dilfig3.eps}}\\
\hline
\end{tabular}
\end{center}
\caption{We report the evolution of the dilaton coupling in the
solutions given by Eq. (\ref{thesolution1}) (left) and by Eq.
(\ref{thesolution2}) (right). We took $c_{5} = 10~\sqrt{3}~\pi$ (left)
and $c_{5} =10~\sqrt{5}~\pi$ (right). Provided the
dilaton coupling starts its evolution deep in the perturbative regime,
$g(\phi)\ll 1$ for every time.}
\label{couplingfig}
\end{figure}
The solutions derived in Section 3 and the examples reported in the
present Section can be
described either in the String frame or in the Einstein frame
\cite{MC}.
In the String frame the fundamental
parameter of the theory is the String length $\lambda_{s}$. Notice
that in this frame the String scale is truly a {\em constant}, whereas
the Planck scale evolves according to a relation which can be
parameterised (at low energy and low coupling) as
$\lambda_{P}(t) = e^{\frac{\phi}{2}} \lambda_{s}$.
The Einstein frame is defined by the conformal transformation which
diagonalizes the gravitational action by decoupling the dilaton from
the Einstein-Hilbert Lagrangian.In the Einstein frame
$\lambda_{P} \sim 10^{-33} {\rm cm}$ is {\em
constant} whereas the String scale evolves according to
$\lambda_{s}(t) = e^{-\frac{\Phi}{2}}\lambda_{P}$
(where $\Phi$ is the Einstein frame dilaton).
The String frame and Einstein frame dilatons are simply related
and, moreover, in four dimensions they are indeed equal, whereas the
String frame ($g_{\mu\nu}$) and Einstein frame ($G_{\mu\nu}$) metrics
are connected by a conformal rescaling involving the dilaton coupling
($g_{\mu\nu} = e^{\Phi} G_{\mu\nu}$).
We transformed our solutions in the Einstein frame and we investigated
the emerging picture which can be summarized by saying that the
curvature invariants are also regular and everywhere defined.
At this point, we would be
tempted to call the solutions we just discussed
``singularity-free''. In order to name a generic solution of the low energy
String effective action ``singularity-free'' we should discuss the
geodesic completeness of the metric both in the Einstein and in the String
frame pictures. From the point of view of the interpretation of our solutions,
moreover, the geodesic completeness in the String frame seems even
more crucial, since strings moving in curved space follow geodesics in
the String frame but not in the Einstein frame \cite{sanchez}.
To show that the geodesics do not hit a boundary in both frames is
anyway a quite long computational task. We prefer to postpone it to
further investigations mainly because the calculations reported here
are already quite heavy. We do not have, at the moment, any argument
either favouring or disfavouring the geodesic completeness and we
believe that this can only come out from a precise calculation.
There is a second very important issue. The Einstein frame picture
of our solutions involves an energy momentum tensor which {\em does
not} violate any of the energy conditions which are part of the
hypotheses of the powerful Hawking-Penrose (HP) theorems \cite{haw,wald}.
Therefore the
question which naturally arises is which other hypothesis of the
HP theorems is violated. There are indeed examples of
truly non-singular (inhomogeneous) solutions in general relativity
which have an energy-momentum tensor of a perfect fluid \cite{sing}.
In that context it was rigourously proven that the so-called initial
or boundary condition of the HP theorems was the one failing to
hold. This would mean, in our context, that we have to fine-tune
(as we had!) the
initial conditions quite heavily in order not to bump into a
singularity. This point is indeed
closely connected with the behaviour of the geodesics and with their
completeness and therefore it certainly requires further studies.
The same considerations should be performed in the String frame
which is perhaps the most useful for the physical intuition (as it was
noticed in the case of the pre-big-bang scenario \cite{ven1}). We are
not aware, at the moment, of any general singularity theorem in this
case and therefore we are not in condition of making any kind of
general statement going beyond the solutions we just presented.
\renewcommand{\theequation}{5.\arabic{equation}}
\setcounter{equation}{0}
\section{Parity invariant solutions with growing coupling}
The two physical examples presented in the previous Section share a
quite amusing physical property, namely the invariance of the
line element corresponding to each one of the solutions
given in Eqs. (\ref{thesolution1}) and (\ref{thesolution2}) under
discrete parity transformations. In fact
by computing, from Eq. (\ref{line}), the line element associated to the
examples reported in Eq. (\ref{thesolution1}) and (\ref{thesolution2})
we find that it is invariant under the transformation $x\rightarrow -
x$. Surprisingly enough, we will show that parity invariant
metrics lead to regular curvature invariants. This will allow us to
define a class of regular, inhomogeneous models with growing
coupling.
We start by noting that, from the structure of the metric written in
Eqs. (\ref{thesolution}) the diagonal elements of the metric tensor
given in Eq. (\ref{metric}) are even under parity transformations
provided the spatial ($x$) dependence appears in functions which are
themselves even under parity. Given the form of the metric
(\ref{thesolution}) we see that, in order to preserve parity, the
spatial dependence has to occur either through powers of the
hyperbolic cosinus (which is parity even) or through even powers of
the hyperbolic sinus. We then impose
the following two extra conditions to the parameters
$c_1,~c_2,~,c_3,~c_4$ specifying the solutions (\ref{thesolution}):
\begin{equation}
c^2_2 + c_2 c_3 - \mu c_1 + \mu^2 =0,~~~~~c_2 + c_3 = \mu
\label{parity}
\end{equation}
In Eqs. (\ref{thesolution}) the first condition of Eq. (\ref{parity})
sets to zero the power of the hyperbolic sinus
appearing in $A(x,t)$ whereas the second condition sets to one the
power of the hyperbolic cosinus appearing in $C(x,t)$. Notice that
with this choices the components of the metric are all even under
$x\rightarrow - x$.
Therefore, the full system of conditions obeyed by the solution given
in Eq. (\ref{thesolution}) is
\begin{eqnarray}
&&\biggl(\frac{c_{3}}{\mu}\biggr)^2 = 3 +
\biggl(\frac{c_2}{\mu}\biggr)^2 - 2 \biggl(\frac{c_1}{\mu}\biggr),
\label{a}\\
&&\biggl(\frac{c_{4}}{\mu}\biggr)^2 =
\biggl(\frac{c_2}{\mu}\biggr)^2 - 2 \biggl(\frac{c_1}{\mu}\biggr) -1,
\label{b}\\
&&\biggl(\frac{c_{2}}{\mu}\biggr)^2 = \biggl(\frac{c_1}{\mu}\biggr) -
\biggl(\frac{c_2}{\mu}\biggr) \biggl(\frac{c_3}{\mu}\biggr),
\label{c}\\
&&\biggl(\frac{c_3}{\mu}\biggr) + \biggl(\frac{c_2}{\mu}\biggr) =1.
\label{d}
\end{eqnarray}
Now calling, for simplicity $\alpha = c_4/\mu$, $\gamma = c_2/\mu$,
$\delta= c_1/\mu$ and and eliminating $c_{3}/\mu$ through Eq. (\ref{a})
from the previous equations we obtain the following system:
\begin{eqnarray}
&&\gamma +1 = \delta,
\label{a1}\\
&&\alpha^2 = \gamma^2 -1 - 2 \delta,
\label{b1}\\
&&2\gamma + 2 = 2 \delta.
\label{c1}
\end{eqnarray}
Eqs. (\ref{a1}) and (\ref{c1}) are clearly not independent
and, therefore, the solution given in Eq. (\ref{thesolution}) can be
expressed only in terms of one (physical) parameter which we choose to
be $\alpha$. This turns out to be a physically interesting
parametrization since $\mu^2 \alpha^2 = \dot{\phi}^2(0)$. In this sense
$|\alpha|$ estimates the (maximal) energy density of the dilaton
background in string units.
The sign of $\alpha$ automatically selects solutions
either with growing coupling or with decreasing coupling.
Thus, using Eqs. (\ref{d}), (\ref{a1}) and (\ref{b1}), we have that
the metric of Eq. (\ref{thesolution}) can be re-written as
\begin{eqnarray}
A(x,t) &=& e^{\pm\alpha~ gd(\mu t)}~ \Biggl[\cosh{\mu
t}\Biggr]^{2 \pm \sqrt{\alpha^2 + 4}}
\Biggl[\cosh{\left(\frac{\mu x}{2}\right)}
\Biggr]^{ 2\sqrt{\alpha^2 + 4}[\sqrt{\alpha^2 + 4} \pm 1] },
\nonumber\\
B(x,t) &=& e^{\pm\alpha~gd(\mu t)}~\cosh{\mu t}~ \sinh{\mu x},
\nonumber\\
C(x,t) &=&\Biggl[\cosh{\mu t}\Biggr]^{1 \pm \sqrt{\alpha^2 + 4}}~
\Biggl[\sinh{\left(\frac{\mu x}{2}\right)}\Biggr]
\Biggl[\cosh{\left(\frac{\mu x}{2}\right)}\Biggr]^{1 \pm
2\sqrt{\alpha^2 + 4}},
\nonumber\\
\phi(t) &=& \pm\alpha ~gd(\mu t) - \beta.
\label{theclass}
\end{eqnarray}
(the plus and minus signs correspond, respectively, to solutions with
growing and decreasing dilaton coupling). In Appendix B we computed
the curvature invariants in the case of growing coupling
solutions. We indeed found that the requirement of parity invariance of
the solutions seem to be winning since the curvature invariants are
regular. Notice that the curvature invariants of the
two previous examples (reported in Section 4)
can be derived from the general result of this Section by setting,
respectively, $\alpha = 2 \sqrt{3}$ and $\alpha = \sqrt{5}$.
Also in the case of the class of solutions discussed in the present
Section the curvature invariants vanish asymptotically as can be
argued from the results reported in Appendix B.
The curvature invariants can be also computed in the case of
decreasing dilaton solution. We did this exercice and we found, again
regular solutions. From the more mathematical results reported in this
Section (and from the two examples of Section 4 illustrating
the physical properties of our solutions) we see that something quite
unexpected happens, namely the fact that the regularity of the
curvature invariants can already be achieved at tree level (without
any curvature correction). Moreover, as shown in Section 4 for two
particular cases, we can set initial conditions in such a way that the
coupling constant is always much smaller than one. We notice that
these results are not in contrast with previous discussions
\cite{brusven} suggesting that the only way of regularizing the
curvature invariants was to invoke higher curvature corrections. In
fact those studies were always postulating a completely homogeneous
(and often isotropic) four dimensional geometry. The physical picture
emerging from our detailed analytical calculations is that the
inhomogeneities are quite an essential ingredient in order to
regularize the curvature invariants. On a purely physical ground,
moreover, it does not seem unreasonable to have strong inhomogeneities
at very small scales.
As we noted in the introduction
our results confirm and extend previous ideas concerning possible
avoidance of the singularities in completely inhomogenous string
cosmological models \cite{GMV}. In agreement with \cite{GMV} we showed
that regular solutions are possible at tree-level. At the same time we
showed that this conclusion holds also in the presence of a growing
coupling solution and in the absence of any extra field. We believe
that our new results should be taken as a serious motivation in order to
perform a fully consistent numerical study of inhomogeneous string
cosmological models. These studies might extend (and perhaps change)
some of the conclusions drawn in the completely homogeneous case
\cite{brusven}. In particular, numerical studies might clarify if it
is at all possible to find inhomogeneous examples evolving at late
times towards a completely homogeneous state. In fact a quite
unphysical feature shared by our models and by the ones of \cite{GMV}
is their eternal inhomogeneity. It is certainly tue \cite{kra} that the
homogeneous limit of a inhomogeneous model is already quite delicate
in general relativity. In spite of the possible subtleties we can say
that a good measure of how inhomogeneous and anisotropic is a solution
at late times can be the ratio of the Weyl invariant
($C^{\mu\nu\alpha\beta}C_{\mu\nu\alpha\beta}$) over the Riemann
invariant ($R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$). In our case,
from the results of Appendix B, this quantity goes to a constant at
for $t\rightarrow +\infty$. A similar behaviour can be found in \cite{GMV}.
Numerical studies could also give us some clue concerning the role
played by the compactification radii which we took frozen but which
should be included in the game of inhomogeneous models. Up to now this
possibility was never considered.
\renewcommand{\theequation}{6.\arabic{equation}}
\setcounter{equation}{0}
\section{Concluding remarks}
In this paper we gave some examples of regular dilaton
driven solutions of the low energy beta functions without including
any dilaton potential, any higher $\alpha'$ (curvature)
corrections, any higher
genus corrections, any antisymmetric tensor field any dilaton
potential. We took
the six compactification radii to be constant and we looked at some
special class of inhomogeneous metric.
We showed that if the metric is sufficiently inhomogeneous the
curvature invariants are bounded for every $x$ and $t$.
The only definite conclusion that we feel like stating
is that regular (but inhomogeneous)
solutions of the evolution equations of the dilaton and of the
geometry are more common (at tree-level) than in the completely
homogenous case where no-go theorems probably apply.
Since in our examples the coupling is always very
small and the curvature invariants always smaller than one (in string
units) the tree-level treatment seems, a posteriori, not
meaningless. The
curvature invariants (including the Weyl invariants) vanish,
asymptotically for large times. Moreover the metric is invariant under
parity transformations.
We leave to future studies many unanswered questions.
Can we promote the regular solutions we spotted to true singularity
free solutions? If this is the case are the geodesics complete in the
Einstein frame? Which hypotheses of the singularity theorems
turn out to be violated? Can we formulate the singularity theorems (in
a consistent way) directly in the String frame or we have always to go to the
Einstein frame? Should we consider regular only those solutions which
are bounded in both frames?
\section*{Acknowledgments}
The author would like to thank M. Gasperini and
G. Veneziano for a careful reading of
the draft and for very useful comments. The author wishes also to thank
former discussions with H. Soleng.
\begin{appendix}
\section*{APPENDIX}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
\section{Christoffel symbols and Ricci tensors}
In this Appendix we will give the general formulas for the Christoffel
symbols and Ricci tensors computed in the case of the diagonal metric
given in Eq. (\ref{metric})
with two commuting space-like killing vectors which are hypersurface
orthogonal.
The Christoffel symbols can be easily derived from the metric
(\ref{metric}) and they are reported in Table {\ref{tab1}},
whereas the Ricci tensors and the curvature scalar
can be obtained, after some algebra, once the Christoffel symbols are
known. Our conventions for the Ricci tensor will be
$R_{\mu\nu}= R_{\alpha\mu\nu}^{~~~~\alpha}$ (where
$R_{\mu\nu\alpha}^{~~~~~\beta}=
\partial_{\mu}\Gamma_{\nu\alpha}^{~~~\beta}-...$ is the Riemann tensor).
The $(xx)$, $(yy)$, $(zz)$ and $(00)$ components of the Ricci tensors
are, respectively
\begin{eqnarray}
&&R_{x}^{x} = - \frac{1}{2~A}\Biggl\{ -
(\frac{\dot{A}}{A})^2 + \frac{\dot{A}}{A}\frac{\dot{B}}{B} +
\frac{\ddot{A}}{A} + (\frac{A'}{A})^2 + \frac{A'}{A}\frac{B'}{B}
+(\frac{B'}{B})^2 - (\frac{C'}{C})^2 - \frac{A''}{A} - 2 \frac{B''}{B}\Biggr\},
\label{rxx}\\
&&R_{y}^{y} = - \frac{1}{2~A}\Biggl\{ \frac{\dot{B}}{B}\frac{\dot{C}}{C} -
(\frac{\dot{C}}{C})^2 +
\frac{\ddot{B}}{B} + \frac{\ddot{C}}{C} - \frac{B'}{B} \frac{C'}{C} +
(\frac{C'}{C})^2 - \frac{B''}{B}
- \frac{C''}{C}\Biggr\},
\label{ryy}\\
&&R_{z}^{z} = -\frac{1}{2~A} \Biggl\{ (\frac{\dot{C}}{C})^2
- \frac{\dot{B}}{B}\frac{\dot{C}}{C}
+ \frac{\ddot{B}}{B} - \frac{\ddot{C}}{C}
+ \frac{B'}{B}\frac{C'}{C} - (\frac{C'}{C})^2 - \frac{ B''}{B}
+ \frac{C''}{C}\Biggr\},
\label{rzz}\\
&&R_{0}^{0} = \frac{1}{2~A} \Biggl\{(\frac{\dot{A}}{A})^2 +
\frac{\dot{A}}{A}\frac{\dot{B}}{B} + (\frac{\dot{B}}{B})^2 - (
\frac{\dot{C}}{C})^2
-\frac{\ddot{A}}{A} -2 \frac{\ddot{B}}{B} - (\frac{A'}{A})^2 +
\frac{A'}{A}\frac{B'}{B} + \frac{A''}{A}\Biggr\},
\label{r00}
\end{eqnarray}
The $(0x)$ component and the curvature scalar will be instead:
\begin{equation}
R_{x}^{0} = \frac{1}{2~A} \Biggl\{ \frac{\dot{B}}{B}\frac{A'}{A} +
\frac{\dot{A}}{A}\frac{B'}{B} + \frac{B'}{B} \frac{\dot{B}}{B} -
\frac{\dot{C}}{C}\frac{ C'}{C} - 2 \frac{{\dot{B}}'}{B} \Biggr\},
\label{r0x}
\end{equation}
\begin{equation}
R \equiv R_{\alpha}^{\alpha} = \frac{1}{2~A} \Biggl\{2 (\frac{\dot{A}}{A})^2
+(\frac{\dot{B}}{B})^2 - (\frac{\dot{C}}{C})^2
- 2\frac{\ddot{A}}{A} - 4 \frac{\ddot{B}}{B}
- 2(\frac{A'}{A})^2 -(\frac{B'}{B})^2 +
(\frac{C'}{C})^2 + 2 \frac{A''}{A} + 4 \frac{B''}{B}\Biggr\}.
\label{r}
\end{equation}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline
$\Gamma_{x x}^{x} = \frac{1}{2}\frac{\partial \log{A}}{\partial x}$ &
$\Gamma_{yy}^{x} =
-\frac{C~B}{2~A}\frac{\partial \log{(B~C)}}{\partial x} $ \\
\hline
$ \Gamma_{zz}^{x} = - \frac{B}{2~ A ~C} \frac{\partial\log{(B~C)}}{\partial x}
$ & $\Gamma_{x 0}^{x} = \frac{1}{2}\frac{\partial\log{A}}{\partial~t} $\\
\hline
$\Gamma_{00}^{x} =\frac{1}{2}\frac{\partial \log{A}}{\partial x}$ &
$\Gamma_{xy}^{y}=
\frac{1}{2}\frac{\log{(B~C)}}{\partial x}$ \\
\hline
$\Gamma_{y0}^{y} = \frac{1}{2}\frac{\log{(B~C)}}{\partial t} $ &
$\Gamma_{xz}^{z}=
\frac{1}{2}\frac{\partial\log{(B/C)}}{\partial x} $ \\
\hline
$ \Gamma_{z0}^{z}=\frac{1}{2}\frac{\partial\log{(B/C)}}{\partial t} $
& $\Gamma_{xx}^{0}=\frac{1}{2} \frac{\partial\log{A}}{\partial t}$ \\
\hline
$ \Gamma_{yy}^{0}= \frac{C~B}{2~A}\frac{\partial\log{(C~B)}}{\partial t}$ &
$ \Gamma_{zz}^{0}=\frac{B}{2~A~C} \frac{\partial\log{(B/C)}}{\partial t}$\\
\hline
$\Gamma_{x0}^{0} = \frac{1}{2} \frac{\partial\log{A}}{\partial x}$& $
\Gamma_{00}^{0}
=\frac{1}{2}\frac{\partial\log{A}}{\partial t}$\\
\hline
\end{tabular}
\end{center}
\caption{We report the Christoffel symbols computed from the metric
given in Eq. (\ref{metric}).}
\label{tab1}
\end{table}
\renewcommand{\theequation}{B.\arabic{equation}}
\setcounter{equation}{0}
\section{Curvature invariants for growing dilaton solutions}
From the (parity invariant) class of solutions given in
Eq. (\ref{theclass}) we can compute the curvature invariants, namely
curvature scalar together with
the squares of the Riemann, Weyl and Ricci tensors. The result is the
following
\begin{eqnarray}
R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta} &=& \frac{\mu^4}{32}
~e^{- 2\alpha gd(\mu t)} [\cosh{\mu t}]^{-8 - 2 \sqrt{\alpha^2 +4}}
\Biggl[\cosh{\frac{\mu x}{2}}\Biggr]^{-4 - 4\sqrt{\alpha^2 + 4}(1 +
\sqrt{\alpha^2 +1})}
\Biggl\{ s_1(\alpha)
\nonumber\\
&+& s_2(\alpha) \cosh{ 2 \mu t} +s_{3}(\alpha)
\cosh{4 \mu t} + s_4(\alpha) \cosh{\mu x} + s_5(\alpha) \cosh{2 \mu x}
\nonumber\\
&+& s_6(\alpha) \cosh{2 \mu t} \cosh{2\mu x} + s_7(\alpha) \cosh{2 \mu t} \cosh{\mu
x}
\nonumber\\
&+& s_8(\alpha) \cosh{4 \mu t} \cosh{\mu x}+ s_9(\alpha) \sinh{\mu t}
+ s_{10}(\alpha) \sinh{ \mu t}\cosh{2 \mu x}
\nonumber\\
&+& s_{11}(\alpha) \sinh{\mu t} \cosh{ \mu x} \Biggr\},
\label{riemann}\\
C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta} &=& \frac{\mu^4}{96} ~e^{- 2\alpha
gd(\mu t)}~[\cosh{\mu t}]^{-8 - 2 \sqrt{\alpha^2 + 4}}~\Biggl[\cosh{\frac{\mu
x}{2}}\Biggr]^{ -4 - 4 \sqrt{\alpha^2 + 4}(1 + \sqrt{\alpha^2 + 4})}
\Biggl\{w_{1} (\alpha)
\nonumber\\
&+& w_{2}(\alpha) \cosh{4 \mu t} +
w_{4}(\alpha) \cosh{\mu x} + w_{5}(\alpha) \cosh{2 \mu x}
\nonumber\\
&+& w_{6}(\alpha) \cosh{2\mu t} \cosh{2\mu x}
+ w_{7}(\alpha) \cosh{2\mu t} \cosh{\mu x}
\nonumber\\
&+& w_{8}(\alpha) \cosh{4 \mu t} \cosh{\mu x}\Biggr\},
\label{weyl}\\
R_{\alpha\beta}R^{\alpha\beta} &=&\frac{\alpha^2 \mu^4}{16}~ e^{-2\alpha gd(\mu t)}
~[\cosh{\mu t}]^{-8-2\sqrt{\alpha^2 +4}} ~\Biggl[\cosh{\frac{\mu x}{2}}
\Biggr]^{-2 - 4 \sqrt{\alpha^2 + 4} (1 + \sqrt{\alpha^2 + 4})}
~\Biggl\{ r_1(\alpha)
\nonumber\\
&+&r_{2} (\alpha) \cosh{2\mu t} + r_{3}(\alpha) \cosh{\mu x} + r_{4}(\alpha)
\cosh{2\mu t} \cosh{\mu x}
\nonumber\\
&+& r_5(\alpha) \sinh{\mu t} + r_{6}(\alpha) \sinh{\mu t}
\cosh{\mu x} \Biggr\},
\label{ricci}\\
R^2 &=& \alpha^4 ~\mu^4 ~e^{- 2\alpha gd(\mu t)} ~[\cosh{\mu t}]^{-8
-2\sqrt{\alpha^2 + 4}}
\Biggl[\cosh{\frac{\mu x}{2}}\Biggr]^{-4 \sqrt{\alpha^2 + 4} (1 +
\sqrt{\alpha^2 + 4})}.
\label{scalar}
\end{eqnarray}
The $\alpha$-dependent coefficients appearing in our expressions
are reported in Table
\ref{tab3}.
Notice that the two examples specifically discussed in Section 4 arise
by setting, respectively, $\alpha = 2\sqrt{3}$ and $\alpha =
\sqrt{5}$. Plugging these values in
Eqs. (\ref{riemann})--(\ref{scalar}) we get the corresponding
curvature invariants. We want to mention that we did the calculations
of the curvature invariants also in the Einstein frame after having
transformed the solutions according to the general conventions
(discussed in Section 4) which relate the two frames.
The curvature invariants turned out to be regular also in the
Einstein frame.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline
\hline
$s_1(\alpha)$ &
$ 1152 + 416 \alpha^2 + 824 \alpha^4 + 6 \alpha^6 + \sqrt{ \alpha^2 + 4}( 504 + 98
\alpha^2 + 14 \alpha^4) $ \\
\hline
$s_2(\alpha)$ &
$ 384 + 536 \alpha^2 + 124 \alpha^4 + 8 \alpha^6 + \sqrt{\alpha^2 + 4} ( 96 + 164 \alpha^2
+ 20 \alpha^4)$\\
\hline
$s_3(\alpha)$ & $ 384 + 216 \alpha^2 + 38 \alpha^4 + 2 \alpha^6 + \sqrt{\alpha^2 + 4} (
168 + 66\alpha^2 + 6\alpha^4)$\\
\hline
$s_4(\alpha)$ & $ 48 - 108 \alpha^2 - 78\alpha^4 - 6 \alpha^6 - \sqrt{\alpha^2 +4}( 24 +
190\alpha^2 + 18\alpha^4)$\\
\hline
$s_{5}(\alpha)$ &$ 192 + 104 \alpha^2 + 16\alpha^4 + \sqrt{\alpha^2 + 4}(192 + 36 \alpha^2
+ 4 \alpha^4)$\\
\hline
$s_{6}(\alpha)$ & $ 72 \alpha^2 + 12\alpha^4 + \sqrt{\alpha^2 + 4}( 36 \alpha^2 +
4\alpha^4)$\\
\hline
$s_{7}(\alpha)$ & $ -1728 - 688 \alpha^2 -120 \alpha^4 - 8\alpha^6 - ( 864 + 232 \alpha^2
+ 24 \alpha^4 ) \sqrt{\alpha^2 + 4})$\\
\hline
$s_{8}(\alpha) $ & $ -144 - 108 \alpha^2 - 18\alpha^4 - (72 + 42 \alpha^2 +
6\alpha^4)\sqrt{\alpha^2 + 4}$\\
\hline
$s_{9}(\alpha) $ & $ 96 \alpha^3 + 24\alpha^3 \sqrt{\alpha^2 + 4}$\\
\hline
$s_{10}(\alpha) $ & $ 32\alpha^3 + 8\alpha^3 \sqrt{\alpha^2 + 4}$\\
\hline
$s_{11}(\alpha) $ & $ 128 \alpha^3 + 32 \alpha^3 \sqrt{\alpha^2 + 4}$\\
\hline
\hline
$w_{1}(\alpha) $ & $ 3456 + 1488 \alpha^2 + 216 \alpha^4 + 12 \alpha^6 + \sqrt{\alpha^2 +
4} ( 1512 + 390 \alpha^2 + 30 \alpha^4)$\\
\hline
$w_{2}(\alpha) $ & $ 1152 + 1128 \alpha^2 + 282 \alpha^4 + 18 \alpha^6 + \sqrt{\alpha^2 +
4} ( 288 + 300 \alpha^2 + 48 \alpha^4 ) $\\
\hline
$w_{3}(\alpha) $ & $ 1152 + 648 \alpha^2 + 114 \alpha^4 + 6 \alpha^6 + \sqrt{\alpha^2 + 4}
(504 + 198 \alpha^2 + 18 \alpha^4)$\\
\hline
$w_4(\alpha) $ & $ -144 - 492 \alpha^2 - 266 \alpha^4 - 18 \alpha^6 - ( 72 + 378 \alpha^2 +
54 \alpha^4)\sqrt{ \alpha^2 + 4}$\\
\hline
$w_{5}(\alpha) $ & $ 1152 + 552 \alpha^2 + 94 \alpha^4 + 6 \alpha^6 + \sqrt{\alpha^2 + 4}
(576 + 204 \alpha^4 + 24 \alpha^4)$\\
\hline
$w_{6}(\alpha) $ & $ 216 \alpha^2 + 74 \alpha^4 + 6 \alpha^6 + \sqrt{\alpha^2 + 4} ( 108
\alpha^2 + 24 \alpha^4 ) $ \\
\hline
$w_{7}(\alpha) $ & $ - 5184 -1824 \alpha^2 -1128 \alpha^4 -24 \alpha^6 - \sqrt{\alpha^2 +
4} ( 2592 + 888 \alpha^2 + 72 \alpha^4)$\\
\hline
$w_{8}(\alpha) $ & $ -( 216 + 126 \alpha^2 + 18\alpha^4 ) \sqrt{\alpha^2 + 4}$ \\
\hline
\hline
$r_{1}(\alpha)$ & $ 22 \alpha^2 + 2\alpha^4 + 4\alpha^2 \sqrt{\alpha^2 + 4}$\\
\hline
$r_{2}(\alpha)$ & $ 80 + 22\alpha^2 + 2\alpha^4 + (32 + 4\alpha^2)\sqrt{\alpha^2 + 4}$\\
\hline
$r_{3}(\alpha)$ & $ -80 -14 \alpha^2 - 2\alpha^4 - (32 + 4\alpha^2) \sqrt{\alpha^2 + 4}$\\
\hline
$r_{4}(\alpha)$ & $ -14 \alpha^2 -2 \alpha^4 - 4\alpha^2 \sqrt{\alpha^2 +4}$\\
\hline
$r_{5}(\alpha)$ & $ 32 \alpha + 8\alpha \sqrt{\alpha^2 + 4}$\\
\hline
$r_{6}(\alpha)$ & $ 32 \alpha + 8\alpha \sqrt{\alpha^2 +4}$\\
\hline
\hline
\end{tabular}
\end{center}
\caption{We report the coefficients appearing in the expression of
the Riemann, Weyl and Ricci invariants given, respectively, in Eqs.
(\ref{riemann}), (\ref{weyl}) and (\ref{ricci}).}
\label{tab3}
\end{table}
\end{appendix}
\newpage
| {
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<li><h2>Interfaces</h2>
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<li class="api apilevel-1"><a href="../../../../../reference/org/xml/sax/ext/Attributes2.html">Attributes2</a></li>
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<div class="col-12" id="doc-col">
<div id="api-info-block">
<div class="sum-details-links">
Summary:
<a href="#pubctors">Ctors</a>
| <a href="#pubmethods">Methods</a>
| <a href="#inhmethods">Inherited Methods</a>
| <a href="#" onclick="return toggleAllClassInherited()" id="toggleAllClassInherited">[Expand All]</a>
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Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a>
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<!-- ======== START OF CLASS DATA ======== -->
<div id="jd-header">
public
class
<h1 itemprop="name">Attributes2Impl</h1>
extends <a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html">AttributesImpl</a><br/>
implements
<a href="../../../../../reference/org/xml/sax/ext/Attributes2.html">Attributes2</a>
</div><!-- end header -->
<div id="naMessage"></div>
<div id="jd-content" class="api apilevel-1">
<table class="jd-inheritance-table">
<tr>
<td colspan="3" class="jd-inheritance-class-cell"><a href="../../../../../reference/java/lang/Object.html">java.lang.Object</a></td>
</tr>
<tr>
<td class="jd-inheritance-space"> ↳</td>
<td colspan="2" class="jd-inheritance-class-cell"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html">org.xml.sax.helpers.AttributesImpl</a></td>
</tr>
<tr>
<td class="jd-inheritance-space"> </td>
<td class="jd-inheritance-space"> ↳</td>
<td colspan="1" class="jd-inheritance-class-cell">org.xml.sax.ext.Attributes2Impl</td>
</tr>
</table>
<div class="jd-descr">
<h2>Class Overview</h2>
<p itemprop="articleBody">SAX2 extension helper for additional Attributes information,
implementing the <code><a href="../../../../../reference/org/xml/sax/ext/Attributes2.html">Attributes2</a></code> interface.
<blockquote>
<em>This module, both source code and documentation, is in the
Public Domain, and comes with <strong>NO WARRANTY</strong>.</em>
</blockquote>
<p>This is not part of core-only SAX2 distributions.</p>
<p>The <em>specified</em> flag for each attribute will always
be true, unless it has been set to false in the copy constructor
or using <code><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#setSpecified(int, boolean)">setSpecified(int, boolean)</a></code>.
Similarly, the <em>declared</em> flag for each attribute will
always be false, except for defaulted attributes (<em>specified</em>
is false), non-CDATA attributes, or when it is set to true using
<code><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#setDeclared(int, boolean)">setDeclared(int, boolean)</a></code>.
If you change an attribute's type by hand, you may need to modify
its <em>declared</em> flag to match.
</p></p>
</div><!-- jd-descr -->
<div class="jd-descr">
<h2>Summary</h2>
<!-- ======== CONSTRUCTOR SUMMARY ======== -->
<table id="pubctors" class="jd-sumtable"><tr><th colspan="12">Public Constructors</th></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#Attributes2Impl()">Attributes2Impl</a></span>()</nobr>
<div class="jd-descrdiv">
Construct a new, empty Attributes2Impl object.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#Attributes2Impl(org.xml.sax.Attributes)">Attributes2Impl</a></span>(<a href="../../../../../reference/org/xml/sax/Attributes.html">Attributes</a> atts)</nobr>
<div class="jd-descrdiv">
Copy an existing Attributes or Attributes2 object.
</div>
</td></tr>
</table>
<!-- ========== METHOD SUMMARY =========== -->
<table id="pubmethods" class="jd-sumtable"><tr><th colspan="12">Public Methods</th></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#addAttribute(java.lang.String, java.lang.String, java.lang.String, java.lang.String, java.lang.String)">addAttribute</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName, <a href="../../../../../reference/java/lang/String.html">String</a> qName, <a href="../../../../../reference/java/lang/String.html">String</a> type, <a href="../../../../../reference/java/lang/String.html">String</a> value)</nobr>
<div class="jd-descrdiv">
Add an attribute to the end of the list, setting its
"specified" flag to true.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#isDeclared(java.lang.String, java.lang.String)">isDeclared</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Returns false unless the attribute was declared in the DTD.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#isDeclared(java.lang.String)">isDeclared</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Returns false unless the attribute was declared in the DTD.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#isDeclared(int)">isDeclared</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Returns false unless the attribute was declared in the DTD.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#isSpecified(int)">isSpecified</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Returns the current value of an attribute's "specified" flag.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#isSpecified(java.lang.String)">isSpecified</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Returns the current value of an attribute's "specified" flag.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#isSpecified(java.lang.String, java.lang.String)">isSpecified</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Returns the current value of an attribute's "specified" flag.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#removeAttribute(int)">removeAttribute</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Remove an attribute from the list.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#setAttributes(org.xml.sax.Attributes)">setAttributes</a></span>(<a href="../../../../../reference/org/xml/sax/Attributes.html">Attributes</a> atts)</nobr>
<div class="jd-descrdiv">
Copy an entire Attributes object.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#setDeclared(int, boolean)">setDeclared</a></span>(int index, boolean value)</nobr>
<div class="jd-descrdiv">
Assign a value to the "declared" flag of a specific attribute.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#setSpecified(int, boolean)">setSpecified</a></span>(int index, boolean value)</nobr>
<div class="jd-descrdiv">
Assign a value to the "specified" flag of a specific attribute.
</div>
</td></tr>
</table>
<!-- ========== METHOD SUMMARY =========== -->
<table id="inhmethods" class="jd-sumtable"><tr><th>
<a href="#" class="toggle-all" onclick="return toggleAllInherited(this, null)">[Expand]</a>
<div style="clear:left;">Inherited Methods</div></th></tr>
<tr class="api apilevel-" >
<td colspan="12">
<a href="#" onclick="return toggleInherited(this, null)" id="inherited-methods-org.xml.sax.helpers.AttributesImpl" class="jd-expando-trigger closed"
><img id="inherited-methods-org.xml.sax.helpers.AttributesImpl-trigger"
src="../../../../../assets/images/triangle-closed.png"
class="jd-expando-trigger-img" /></a>
From class
<a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html">org.xml.sax.helpers.AttributesImpl</a>
<div id="inherited-methods-org.xml.sax.helpers.AttributesImpl">
<div id="inherited-methods-org.xml.sax.helpers.AttributesImpl-list"
class="jd-inheritedlinks">
</div>
<div id="inherited-methods-org.xml.sax.helpers.AttributesImpl-summary" style="display: none;">
<table class="jd-sumtable-expando">
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#addAttribute(java.lang.String, java.lang.String, java.lang.String, java.lang.String, java.lang.String)">addAttribute</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName, <a href="../../../../../reference/java/lang/String.html">String</a> qName, <a href="../../../../../reference/java/lang/String.html">String</a> type, <a href="../../../../../reference/java/lang/String.html">String</a> value)</nobr>
<div class="jd-descrdiv">
Add an attribute to the end of the list.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#clear()">clear</a></span>()</nobr>
<div class="jd-descrdiv">
Clear the attribute list for reuse.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
int</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getIndex(java.lang.String)">getIndex</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Look up an attribute's index by qualified (prefixed) name.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
int</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getIndex(java.lang.String, java.lang.String)">getIndex</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Look up an attribute's index by Namespace name.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
int</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getLength()">getLength</a></span>()</nobr>
<div class="jd-descrdiv">
Return the number of attributes in the list.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getLocalName(int)">getLocalName</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Return an attribute's local name.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getQName(int)">getQName</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Return an attribute's qualified (prefixed) name.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getType(java.lang.String)">getType</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Look up an attribute's type by qualified (prefixed) name.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getType(java.lang.String, java.lang.String)">getType</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Look up an attribute's type by Namespace-qualified name.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getType(int)">getType</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Return an attribute's type by index.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getURI(int)">getURI</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Return an attribute's Namespace URI.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getValue(java.lang.String)">getValue</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Look up an attribute's value by qualified (prefixed) name.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getValue(int)">getValue</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Return an attribute's value by index.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#getValue(java.lang.String, java.lang.String)">getValue</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Look up an attribute's value by Namespace-qualified name.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#removeAttribute(int)">removeAttribute</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Remove an attribute from the list.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#setAttribute(int, java.lang.String, java.lang.String, java.lang.String, java.lang.String, java.lang.String)">setAttribute</a></span>(int index, <a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName, <a href="../../../../../reference/java/lang/String.html">String</a> qName, <a href="../../../../../reference/java/lang/String.html">String</a> type, <a href="../../../../../reference/java/lang/String.html">String</a> value)</nobr>
<div class="jd-descrdiv">
Set an attribute in the list.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#setAttributes(org.xml.sax.Attributes)">setAttributes</a></span>(<a href="../../../../../reference/org/xml/sax/Attributes.html">Attributes</a> atts)</nobr>
<div class="jd-descrdiv">
Copy an entire Attributes object.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#setLocalName(int, java.lang.String)">setLocalName</a></span>(int index, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Set the local name of a specific attribute.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#setQName(int, java.lang.String)">setQName</a></span>(int index, <a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Set the qualified name of a specific attribute.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#setType(int, java.lang.String)">setType</a></span>(int index, <a href="../../../../../reference/java/lang/String.html">String</a> type)</nobr>
<div class="jd-descrdiv">
Set the type of a specific attribute.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#setURI(int, java.lang.String)">setURI</a></span>(int index, <a href="../../../../../reference/java/lang/String.html">String</a> uri)</nobr>
<div class="jd-descrdiv">
Set the Namespace URI of a specific attribute.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#setValue(int, java.lang.String)">setValue</a></span>(int index, <a href="../../../../../reference/java/lang/String.html">String</a> value)</nobr>
<div class="jd-descrdiv">
Set the value of a specific attribute.
</div>
</td></tr>
</table>
</div>
</div>
</td></tr>
<tr class="api apilevel-" >
<td colspan="12">
<a href="#" onclick="return toggleInherited(this, null)" id="inherited-methods-java.lang.Object" class="jd-expando-trigger closed"
><img id="inherited-methods-java.lang.Object-trigger"
src="../../../../../assets/images/triangle-closed.png"
class="jd-expando-trigger-img" /></a>
From class
<a href="../../../../../reference/java/lang/Object.html">java.lang.Object</a>
<div id="inherited-methods-java.lang.Object">
<div id="inherited-methods-java.lang.Object-list"
class="jd-inheritedlinks">
</div>
<div id="inherited-methods-java.lang.Object-summary" style="display: none;">
<table class="jd-sumtable-expando">
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/Object.html">Object</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#clone()">clone</a></span>()</nobr>
<div class="jd-descrdiv">
Creates and returns a copy of this <code>Object</code>.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#equals(java.lang.Object)">equals</a></span>(<a href="../../../../../reference/java/lang/Object.html">Object</a> o)</nobr>
<div class="jd-descrdiv">
Compares this instance with the specified object and indicates if they
are equal.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#finalize()">finalize</a></span>()</nobr>
<div class="jd-descrdiv">
Invoked when the garbage collector has detected that this instance is no longer reachable.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
final
<a href="../../../../../reference/java/lang/Class.html">Class</a><?></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#getClass()">getClass</a></span>()</nobr>
<div class="jd-descrdiv">
Returns the unique instance of <code><a href="../../../../../reference/java/lang/Class.html">Class</a></code> that represents this
object's class.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
int</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#hashCode()">hashCode</a></span>()</nobr>
<div class="jd-descrdiv">
Returns an integer hash code for this object.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
final
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#notify()">notify</a></span>()</nobr>
<div class="jd-descrdiv">
Causes a thread which is waiting on this object's monitor (by means of
calling one of the <code>wait()</code> methods) to be woken up.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
final
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#notifyAll()">notifyAll</a></span>()</nobr>
<div class="jd-descrdiv">
Causes all threads which are waiting on this object's monitor (by means
of calling one of the <code>wait()</code> methods) to be woken up.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#toString()">toString</a></span>()</nobr>
<div class="jd-descrdiv">
Returns a string containing a concise, human-readable description of this
object.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
final
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#wait()">wait</a></span>()</nobr>
<div class="jd-descrdiv">
Causes the calling thread to wait until another thread calls the <code>notify()</code> or <code>notifyAll()</code> method of this object.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
final
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#wait(long, int)">wait</a></span>(long millis, int nanos)</nobr>
<div class="jd-descrdiv">
Causes the calling thread to wait until another thread calls the <code>notify()</code> or <code>notifyAll()</code> method of this object or until the
specified timeout expires.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
final
void</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/java/lang/Object.html#wait(long)">wait</a></span>(long millis)</nobr>
<div class="jd-descrdiv">
Causes the calling thread to wait until another thread calls the <code>notify()</code> or <code>notifyAll()</code> method of this object or until the
specified timeout expires.
</div>
</td></tr>
</table>
</div>
</div>
</td></tr>
<tr class="api apilevel-" >
<td colspan="12">
<a href="#" onclick="return toggleInherited(this, null)" id="inherited-methods-org.xml.sax.Attributes" class="jd-expando-trigger closed"
><img id="inherited-methods-org.xml.sax.Attributes-trigger"
src="../../../../../assets/images/triangle-closed.png"
class="jd-expando-trigger-img" /></a>
From interface
<a href="../../../../../reference/org/xml/sax/Attributes.html">org.xml.sax.Attributes</a>
<div id="inherited-methods-org.xml.sax.Attributes">
<div id="inherited-methods-org.xml.sax.Attributes-list"
class="jd-inheritedlinks">
</div>
<div id="inherited-methods-org.xml.sax.Attributes-summary" style="display: none;">
<table class="jd-sumtable-expando">
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
int</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getIndex(java.lang.String, java.lang.String)">getIndex</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Look up the index of an attribute by Namespace name.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
int</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getIndex(java.lang.String)">getIndex</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Look up the index of an attribute by XML qualified (prefixed) name.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
int</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getLength()">getLength</a></span>()</nobr>
<div class="jd-descrdiv">
Return the number of attributes in the list.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getLocalName(int)">getLocalName</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Look up an attribute's local name by index.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getQName(int)">getQName</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Look up an attribute's XML qualified (prefixed) name by index.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getType(java.lang.String)">getType</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Look up an attribute's type by XML qualified (prefixed) name.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getType(java.lang.String, java.lang.String)">getType</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Look up an attribute's type by Namespace name.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getType(int)">getType</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Look up an attribute's type by index.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getURI(int)">getURI</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Look up an attribute's Namespace URI by index.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getValue(java.lang.String)">getValue</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Look up an attribute's value by XML qualified (prefixed) name.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getValue(java.lang.String, java.lang.String)">getValue</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Look up an attribute's value by Namespace name.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
<a href="../../../../../reference/java/lang/String.html">String</a></nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/Attributes.html#getValue(int)">getValue</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Look up an attribute's value by index.
</div>
</td></tr>
</table>
</div>
</div>
</td></tr>
<tr class="api apilevel-" >
<td colspan="12">
<a href="#" onclick="return toggleInherited(this, null)" id="inherited-methods-org.xml.sax.ext.Attributes2" class="jd-expando-trigger closed"
><img id="inherited-methods-org.xml.sax.ext.Attributes2-trigger"
src="../../../../../assets/images/triangle-closed.png"
class="jd-expando-trigger-img" /></a>
From interface
<a href="../../../../../reference/org/xml/sax/ext/Attributes2.html">org.xml.sax.ext.Attributes2</a>
<div id="inherited-methods-org.xml.sax.ext.Attributes2">
<div id="inherited-methods-org.xml.sax.ext.Attributes2-list"
class="jd-inheritedlinks">
</div>
<div id="inherited-methods-org.xml.sax.ext.Attributes2-summary" style="display: none;">
<table class="jd-sumtable-expando">
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2.html#isDeclared(java.lang.String, java.lang.String)">isDeclared</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Returns false unless the attribute was declared in the DTD.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2.html#isDeclared(java.lang.String)">isDeclared</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Returns false unless the attribute was declared in the DTD.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2.html#isDeclared(int)">isDeclared</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Returns false unless the attribute was declared in the DTD.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2.html#isSpecified(int)">isSpecified</a></span>(int index)</nobr>
<div class="jd-descrdiv">
Returns true unless the attribute value was provided
by DTD defaulting.
</div>
</td></tr>
<tr class="alt-color api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2.html#isSpecified(java.lang.String)">isSpecified</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</nobr>
<div class="jd-descrdiv">
Returns true unless the attribute value was provided
by DTD defaulting.
</div>
</td></tr>
<tr class=" api apilevel-1" >
<td class="jd-typecol"><nobr>
abstract
boolean</nobr>
</td>
<td class="jd-linkcol" width="100%"><nobr>
<span class="sympad"><a href="../../../../../reference/org/xml/sax/ext/Attributes2.html#isSpecified(java.lang.String, java.lang.String)">isSpecified</a></span>(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</nobr>
<div class="jd-descrdiv">
Returns true unless the attribute value was provided
by DTD defaulting.
</div>
</td></tr>
</table>
</div>
</div>
</td></tr>
</table>
</div><!-- jd-descr (summary) -->
<!-- Details -->
<!-- XML Attributes -->
<!-- Enum Values -->
<!-- Constants -->
<!-- Fields -->
<!-- Public ctors -->
<!-- ========= CONSTRUCTOR DETAIL ======== -->
<h2>Public Constructors</h2>
<A NAME="Attributes2Impl()"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
</span>
<span class="sympad">Attributes2Impl</span>
<span class="normal">()</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Construct a new, empty Attributes2Impl object.
</p></div>
</div>
</div>
<A NAME="Attributes2Impl(org.xml.sax.Attributes)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
</span>
<span class="sympad">Attributes2Impl</span>
<span class="normal">(<a href="../../../../../reference/org/xml/sax/Attributes.html">Attributes</a> atts)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Copy an existing Attributes or Attributes2 object.
If the object implements Attributes2, values of the
<em>specified</em> and <em>declared</em> flags for each
attribute are copied.
Otherwise the flag values are defaulted to assume no DTD was used,
unless there is evidence to the contrary (such as attributes with
type other than CDATA, which must have been <em>declared</em>).
<p>This constructor is especially useful inside a
<code><a href="../../../../../reference/org/xml/sax/ContentHandler.html#startElement(java.lang.String, java.lang.String, java.lang.String, org.xml.sax.Attributes)">startElement</a></code> event.</p></p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>atts</td>
<td>The existing Attributes object.
</td>
</tr>
</table>
</div>
</div>
</div>
<!-- ========= CONSTRUCTOR DETAIL ======== -->
<!-- Protected ctors -->
<!-- ========= METHOD DETAIL ======== -->
<!-- Public methdos -->
<h2>Public Methods</h2>
<A NAME="addAttribute(java.lang.String, java.lang.String, java.lang.String, java.lang.String, java.lang.String)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
void
</span>
<span class="sympad">addAttribute</span>
<span class="normal">(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName, <a href="../../../../../reference/java/lang/String.html">String</a> qName, <a href="../../../../../reference/java/lang/String.html">String</a> type, <a href="../../../../../reference/java/lang/String.html">String</a> value)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Add an attribute to the end of the list, setting its
"specified" flag to true. To set that flag's value
to false, use <code><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#setSpecified(int, boolean)">setSpecified(int, boolean)</a></code>.
<p>Unless the attribute <em>type</em> is CDATA, this attribute
is marked as being declared in the DTD. To set that flag's value
to true for CDATA attributes, use <code><a href="../../../../../reference/org/xml/sax/ext/Attributes2Impl.html#setDeclared(int, boolean)">setDeclared(int, boolean)</a></code>.</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>uri</td>
<td>The Namespace URI, or the empty string if
none is available or Namespace processing is not
being performed.</td>
</tr>
<tr>
<th>localName</td>
<td>The local name, or the empty string if
Namespace processing is not being performed.</td>
</tr>
<tr>
<th>qName</td>
<td>The qualified (prefixed) name, or the empty string
if qualified names are not available.</td>
</tr>
<tr>
<th>type</td>
<td>The attribute type as a string.</td>
</tr>
<tr>
<th>value</td>
<td>The attribute value.</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">See Also</h5>
<ul class="nolist"><li><code><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#addAttribute(java.lang.String, java.lang.String, java.lang.String, java.lang.String, java.lang.String)">addAttribute(String, String, String, String, String)</a></code></li>
</ul>
</div>
</div>
</div>
<A NAME="isDeclared(java.lang.String, java.lang.String)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
boolean
</span>
<span class="sympad">isDeclared</span>
<span class="normal">(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Returns false unless the attribute was declared in the DTD.
This helps distinguish two kinds of attributes that SAX reports
as CDATA: ones that were declared (and hence are usually valid),
and those that were not (and which are never valid).
<p>Remember that since DTDs do not "understand" namespaces, the
namespace URI associated with an attribute may not have come from
the DTD. The declaration will have applied to the attribute's
<em>qName</em>.</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>uri</td>
<td>The Namespace URI, or the empty string if
the name has no Namespace URI.</td>
</tr>
<tr>
<th>localName</td>
<td>The attribute's local name.</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Returns</h5>
<ul class="nolist"><li>true if the attribute was declared in the DTD,
false otherwise.</li></ul>
</div>
</div>
</div>
<A NAME="isDeclared(java.lang.String)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
boolean
</span>
<span class="sympad">isDeclared</span>
<span class="normal">(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Returns false unless the attribute was declared in the DTD.
This helps distinguish two kinds of attributes that SAX reports
as CDATA: ones that were declared (and hence are usually valid),
and those that were not (and which are never valid).</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>qName</td>
<td>The XML qualified (prefixed) name.</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Returns</h5>
<ul class="nolist"><li>true if the attribute was declared in the DTD,
false otherwise.</li></ul>
</div>
</div>
</div>
<A NAME="isDeclared(int)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
boolean
</span>
<span class="sympad">isDeclared</span>
<span class="normal">(int index)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Returns false unless the attribute was declared in the DTD.
This helps distinguish two kinds of attributes that SAX reports
as CDATA: ones that were declared (and hence are usually valid),
and those that were not (and which are never valid).</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>index</td>
<td>The attribute index (zero-based).</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Returns</h5>
<ul class="nolist"><li>true if the attribute was declared in the DTD,
false otherwise.</li></ul>
</div>
</div>
</div>
<A NAME="isSpecified(int)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
boolean
</span>
<span class="sympad">isSpecified</span>
<span class="normal">(int index)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Returns the current value of an attribute's "specified" flag.</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>index</td>
<td>The attribute index (zero-based).</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Returns</h5>
<ul class="nolist"><li>current flag value</li></ul>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Throws</h5>
<table class="jd-tagtable">
<tr>
<th><a href="../../../../../reference/java/lang/ArrayIndexOutOfBoundsException.html">ArrayIndexOutOfBoundsException</a></td>
<td>When the
supplied index does not identify an attribute.
</td>
</tr>
</table>
</div>
</div>
</div>
<A NAME="isSpecified(java.lang.String)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
boolean
</span>
<span class="sympad">isSpecified</span>
<span class="normal">(<a href="../../../../../reference/java/lang/String.html">String</a> qName)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Returns the current value of an attribute's "specified" flag.</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>qName</td>
<td>The XML qualified (prefixed) name.</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Returns</h5>
<ul class="nolist"><li>current flag value</li></ul>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Throws</h5>
<table class="jd-tagtable">
<tr>
<th><a href="../../../../../reference/java/lang/IllegalArgumentException.html">IllegalArgumentException</a></td>
<td>When the
supplied name does not identify an attribute.
</td>
</tr>
</table>
</div>
</div>
</div>
<A NAME="isSpecified(java.lang.String, java.lang.String)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
boolean
</span>
<span class="sympad">isSpecified</span>
<span class="normal">(<a href="../../../../../reference/java/lang/String.html">String</a> uri, <a href="../../../../../reference/java/lang/String.html">String</a> localName)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Returns the current value of an attribute's "specified" flag.</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>uri</td>
<td>The Namespace URI, or the empty string if
the name has no Namespace URI.</td>
</tr>
<tr>
<th>localName</td>
<td>The attribute's local name.</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Returns</h5>
<ul class="nolist"><li>current flag value</li></ul>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Throws</h5>
<table class="jd-tagtable">
<tr>
<th><a href="../../../../../reference/java/lang/IllegalArgumentException.html">IllegalArgumentException</a></td>
<td>When the
supplied names do not identify an attribute.
</td>
</tr>
</table>
</div>
</div>
</div>
<A NAME="removeAttribute(int)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
void
</span>
<span class="sympad">removeAttribute</span>
<span class="normal">(int index)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Remove an attribute from the list.</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>index</td>
<td>The index of the attribute (zero-based).</td>
</tr>
</table>
</div>
</div>
</div>
<A NAME="setAttributes(org.xml.sax.Attributes)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
void
</span>
<span class="sympad">setAttributes</span>
<span class="normal">(<a href="../../../../../reference/org/xml/sax/Attributes.html">Attributes</a> atts)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Copy an entire Attributes object. The "specified" flags are
assigned as true, and "declared" flags as false (except when
an attribute's type is not CDATA),
unless the object is an Attributes2 object.
In that case those flag values are all copied.</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>atts</td>
<td>The attributes to copy.</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">See Also</h5>
<ul class="nolist"><li><code><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#setAttributes(org.xml.sax.Attributes)">setAttributes(Attributes)</a></code></li>
</ul>
</div>
</div>
</div>
<A NAME="setDeclared(int, boolean)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
void
</span>
<span class="sympad">setDeclared</span>
<span class="normal">(int index, boolean value)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Assign a value to the "declared" flag of a specific attribute.
This is normally needed only for attributes of type CDATA,
including attributes whose type is changed to or from CDATA.</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>index</td>
<td>The index of the attribute (zero-based).</td>
</tr>
<tr>
<th>value</td>
<td>The desired flag value.</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Throws</h5>
<table class="jd-tagtable">
<tr>
<th><a href="../../../../../reference/java/lang/ArrayIndexOutOfBoundsException.html">ArrayIndexOutOfBoundsException</a></td>
<td>When the
supplied index does not identify an attribute.</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">See Also</h5>
<ul class="nolist"><li><code><a href="../../../../../reference/org/xml/sax/helpers/AttributesImpl.html#setType(int, java.lang.String)">setType(int, String)</a></code></li>
</ul>
</div>
</div>
</div>
<A NAME="setSpecified(int, boolean)"></A>
<div class="jd-details api apilevel-1">
<h4 class="jd-details-title">
<span class="normal">
public
void
</span>
<span class="sympad">setSpecified</span>
<span class="normal">(int index, boolean value)</span>
</h4>
<div class="api-level">
<div>
Added in <a href="../../../../../guide/topics/manifest/uses-sdk-element.html#ApiLevels">API level 1</a></div>
</div>
<div class="jd-details-descr">
<div class="jd-tagdata jd-tagdescr"><p>Assign a value to the "specified" flag of a specific attribute.
This is the only way this flag can be cleared, except clearing
by initialization with the copy constructor.</p></div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Parameters</h5>
<table class="jd-tagtable">
<tr>
<th>index</td>
<td>The index of the attribute (zero-based).</td>
</tr>
<tr>
<th>value</td>
<td>The desired flag value.</td>
</tr>
</table>
</div>
<div class="jd-tagdata">
<h5 class="jd-tagtitle">Throws</h5>
<table class="jd-tagtable">
<tr>
<th><a href="../../../../../reference/java/lang/ArrayIndexOutOfBoundsException.html">ArrayIndexOutOfBoundsException</a></td>
<td>When the
supplied index does not identify an attribute.
</td>
</tr>
</table>
</div>
</div>
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| {
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The HOT DOG CARTS, are Street Food Solutions with excellent profitability, compared with a reasonable cost and a High Return on Investment.
These can be very useful for those who already have a Bar or a Pub and wants to offer an outside service to customers using it as Exterior Workbench and simultaneously to intrigue people, luring them.
A Hot Dog Cart is one of the Simple and Profitable Solution as Street Food carts. Our series of Hot Dog Bike Cart with Speedy Workbench is based on simplicity and lightness. We realized a Small Version Tricycle , an XL version a little bigger, and a bike trailer, which is a real Hot Dog Cooking Cart, towable with any Velocipede.
These Hot Dog Speedy Carts, don't have heavy and complicated equipment, but a simple Steamer and a Fridge that runs with Gas Cartridges (like camping) or 220V (if you have a 220v plug nearby).
To simplify the way in your work, it would be useful to cook WURSTEL purchased already pre-cooked (keeping them in Bain Marie to finish cooking).
You can also roast on a plate or on a pan (with the Camping Stove supplied, or on a little Oven with Grill but, at that point, you will need a gas Plant with LPG cylinder and Cook machinery , which will be optional, to make even more tasty hotdog baked in Water Bain Marie. You can add by yourself these Machinery in the Cart. The Perfume .... you know .... attracts customers Hungry !!!
However, if you decide to add the extra equipment, it will be important to choose models workin g withLPG gas. The electric ones consume too much energy and you can not feed them with batteries. So either alcohol or gas.
If you can find the right location to base yourself with Your cart, profitability and return on investment will be really interesting. A sale of 1 hotdog sandwich and 1 Drink can make an earning about 3,5 € / 4 € on average.
For example, by placing you near a Large School or University, at the moment of greatest flow of students (interval or lunch break), you could sell a 80 sandwiches in 1 or 2 hours. So make You the calculation about How Much can produce a Little Hot Dog Cart (€ 3.5 x 80 = 280 €). with this simplistic calculation, we can already say that, if you will find the right situation / Location (and this depends only by you), In about 1 month you could amortize the investment.
Taking into account that you could move it in the afternoon in front of another school or park, and in the evening in an area frequented by young people (for example a disco or a Pub, for example making an agreement with the managers) could also double the income of the day.
Multiply that Daily Cash Value for about 15 business days in one month, taking into account that not always Weather conditions allow you to work outdoors. You must also keep in mind that hotdogs are sold well both in Summer and in Winter, and this type of Cart, can be easily converted into Itinerant Crepes Shop. Then you can quickly change from Hot Dog toCrepes depending on the situations.
So, of course it all depends on how much you are willing to commit and how much you'll be good at to figure out which location or situation may be the most suitable in your town. Among other things after a few days you will find more profitable locations, and so on. The important thing is to select good stations where there is a high concentration of people.
Consider that these carts Hot Dog, are easy to move with pedal over short distances. Using a small Van or Trailer Cart you can move in other cities, which hosts events or fairs. There are Web sites that publish calendars of many events or fairs and, if you follow carefully these appointments, you can be present where there are high concentrations of people.
Even creating your Web site, or your Facebook account, it is very likely that you will come of requests to provide catering services to Parties or Events. The Barrow Hot Dog is something that stoking lot and generates sympathy.
We have also made a little cart Hot Dog that you can tow with an ordinary bicycle, then using a means of sustainable transport, with the possibility to enter ZTL zones or otherwise off-limits to motorized vehicles. | {
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} | 9,677 |
Q: Arranging letters with two letters not next to each other just wanna know what's the answer to this one.
How many ways can the word "ARRANGED" be arranged so that N and D aren't next to each other?
Thanks. :)
A: Notice, in the given word $ARRANGED$ there are total $8$ letters out of which $A$ & $R$ are repetitive letters hence, we have
Total number of linear arrangements of $ARRANGED$ $$=\frac{8!}{2!2!}$$
Number of linear arrangements when $N$ & $D$ are kept together $$=\frac{7!}{2!2!}\cdot 2!=\frac{7!}{2!}$$
Hence, the number of linear arrangements when $N$ & $D$ are not kept together
$$=\text{(total number of linear arrangements)}-\text{(number of linear arrangements when N & D are kept together)}$$
$$=\frac{8!}{2!2!}-\frac{7!}{2!}=\frac{3\cdot 7!}{2!}=\color{red}{7560}$$
A: The total number of words is $\binom82\cdot\binom62\cdot\binom41\cdot\binom31\cdot\binom21\cdot\binom11=10080$.
The number of words with ND is $\binom72\cdot\binom52\cdot\binom31\cdot\binom21\cdot\binom11=1260$.
The number of words with DN is $\binom72\cdot\binom52\cdot\binom31\cdot\binom21\cdot\binom11=1260$.
Hence the number of words with no ND and no DN is $10080-1260-1260=7560$.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,180 |
{"url":"https:\/\/www.primidi.com\/realization_systems\/lti_system\/canonical_realizations","text":"# Realization (systems) - LTI System - Canonical Realizations\n\nCanonical Realizations\n\nAny given transfer function which is strictly proper can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system)):\n\nGiven a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:\n\n.\n\nThe coefficients can now be inserted directly into the state-space model by the following approach:\n\n$dot{textbf{x}}(t) = begin{bmatrix} -d_{1}& -d_{2}& -d_{3}& -d_{4}\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0 end{bmatrix}textbf{x}(t) + begin{bmatrix} 1\\ 0\\ 0\\ 0\\ end{bmatrix}textbf{u}(t)$\n.\n\nThis state-space realization is called controllable canonical form (also known as phase variable canonical form) because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).\n\nThe transfer function coefficients can also be used to construct another type of canonical form\n\n$dot{textbf{x}}(t) = begin{bmatrix} -d_{1}& 1& 0& 0\\ -d_{2}& 0& 1& 0\\ -d_{3}& 0& 0& 1\\ -d_{4}& 0& 0& 0 end{bmatrix}textbf{x}(t) + begin{bmatrix} n_{1}\\ n_{2}\\ n_{3}\\ n_{4} end{bmatrix}textbf{u}(t)$\n.\n\nThis state-space realization is called observable canonical form because the resulting model is guaranteed to be observable (i.e., because the output exits from a chain of integrators, every state has an effect on the output).","date":"2020-12-02 06:24:19","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\": 2, \"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\": 0.9434248805046082, \"perplexity\": 522.172810941209}, \"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-2020-50\/segments\/1606141692985.63\/warc\/CC-MAIN-20201202052413-20201202082413-00539.warc.gz\"}"} | null | null |
{"url":"https:\/\/codeselfstudy.com\/blog\/problems-with-copying-and-pasting-code-in-vim\/","text":"# Problems with Copying and Pasting Code in Vim?\n\nI've written a tutorial on how to copy\/paste from Vim registers. I wanted to add a few more tips:\n\n## Problems with Pasting and Code Indentaion\n\nIf you're having trouble with indentation when pasting code into Vim (especially with indentation-sensitive languages like Python), there are a couple of possible solutions:\n\n### 1. Paste using Vim registers\n\nTip: read my other tutorial on Vim registers or the vim help using :help registers.\n\nThere are ten types of registers in Vim. You can think of them like separate clipboards. Each clipboard has a name. The important one for copying\/pasting between Vim and the system is the register called + (#8, below).\n\n1. The unnamed register \"\"\n2. 10 numbered registers \"0 to \"9\n3. The small delete register \"-\n4. 26 named registers \"a to \"z or \"A to \"Z\n5. three read-only registers \":, \"., \"%\n6. alternate buffer register \"#\n7. the expression register \"=\n8. The selection and drop registers \"*, \"+ and \"~\n9. The black hole register \"_\n10. Last search pattern register \"\/\n...\n8. Selection and drop registers \"*, \"+ and \"~\nUse these registers for storing and retrieving the selected text for the GUI.\nSee |quotestar| and |quoteplus|. When the clipboard is not available or not\nworking, the unnamed register is used instead. For Unix systems the clipboard\nis only available when the |+xterm_clipboard| feature is present. {not in Vi}\n\nNote that there is only a distinction between \"* and \"+ for X11 systems. For\nan explanation of the difference, see |x11-selection|. Under MS-Windows, use\nof \"* and \"+ is actually synonymous and refers to the |gui-clipboard|.\n\n### 2. Use :set paste\n\nIf you're using a Mac and trying to paste with command-v, the indentation might get distorted, depending on the contents of your .vimrc file.\n\nYou can fix that by typing :set paste before you paste and then doing :set nopaste when done. Those could be assigned to keybindings in your .vimrc file.\n\nHere are the relevant docs on paste, which you can read by typing :help paste in Vim:\n\n'paste' boolean (default off)\nglobal\n{not in Vi}\nPut Vim in Paste mode. This is useful if you want to cut or copy\nsome text from one window and paste it in Vim. This will avoid\nunexpected effects.\nSetting this option is useful when using Vim in a terminal, where Vim\ncannot distinguish between typed text and pasted text. In the GUI, Vim\nknows about pasting and will mostly do the right thing without 'paste'\nbeing set. The same is true for a terminal where Vim handles the\nmouse clicks itself.\nThis option is reset when starting the GUI. Thus if you set it in\nyour .vimrc it will work in a terminal, but not in the GUI. Setting\n'paste' in the GUI has side effects: e.g., the Paste toolbar button\nwill no longer work in Insert mode, because it uses a mapping.\nWhen the 'paste' option is switched on (also when it was already on):\n- mapping in Insert mode and Command-line mode is disabled\n- abbreviations are disabled\n- 'autoindent' is reset\n- 'expandtab' is reset\n- 'formatoptions' is used like it is empty\n- 'revins' is reset\n- 'ruler' is reset\n- 'showmatch' is reset\n- 'smartindent' is reset\n- 'smarttab' is reset\n- 'softtabstop' is set to 0\n- 'textwidth' is set to 0\n- 'wrapmargin' is set to 0\nThese options keep their value, but their effect is disabled:\n- 'cindent'\n- 'indentexpr'\n- 'lisp'\nNOTE: When you start editing another file while the 'paste' option is\non, settings from the modelines or autocommands may change the\nsettings again, causing trouble when pasting text. You might want to\nset the 'paste' option again.\nWhen the 'paste' option is reset the mentioned options are restored to\nthe value before the moment 'paste' was switched from off to on.\nResetting 'paste' before ever setting it does not have any effect.\nSince mapping doesn't work while 'paste' is active, you need to use\nthe 'pastetoggle' option to toggle the 'paste' option with some key.\n\n## Conclusion\n\nI would use \"+p and \"+y, because once one understands registers, they can be used for many things. Learn more about them by opening up Vim and typing: :help registers.","date":"2018-07-17 08:05:41","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\": 0.2620193660259247, \"perplexity\": 10352.249066674887}, \"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-2018-30\/segments\/1531676589618.52\/warc\/CC-MAIN-20180717070721-20180717090721-00175.warc.gz\"}"} | null | null |
Acanthascus hamatus är en svampdjursart som beskrevs av Lendenfeld 1915. Acanthascus hamatus ingår i släktet Acanthascus och familjen Rossellidae. Inga underarter finns listade i Catalogue of Life.
Källor
Glassvampar
hamatus | {
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} | 9,258 |
package com.google.gwt.core.client.js;
import java.lang.annotation.Documented;
import java.lang.annotation.ElementType;
import java.lang.annotation.Retention;
import java.lang.annotation.RetentionPolicy;
import java.lang.annotation.Target;
/**
* JsProperty marks a method in a {@link com.google.gwt.core.client.js.JsType} as a property accessor and recognizes
* JavaBean style naming convention. Instead of translating method calls to JsProperty methods
* as method calls in JS, they will be replaced with dotted property lookups.
*<p> Examples:
* <ul>
* <li> {@code @JsProperty getX()} translates as <tt>this.x</tt>
* <li> {@code @JsProperty x()} translates as <tt>this.x</tt>
* <li> {@code @JsProperty setX(int y)} translates as <tt>this.x=y</tt>
* <li> {@code @JsProperty x(int x)} translates as <tt>this.x=y</tt>
* <li> {@code @JsProperty hasX(int x)} translates as <tt>x in this</tt>
*</ul>
* <p>
* In addition, fluent style <tt>return this</tt> syntax is supported for setters, so
* {@code @JsProperty T setX(int x)} translates as <tt>this.x=x, return this</tt>.
*/
@Retention(RetentionPolicy.RUNTIME)
@Target(ElementType.METHOD)
@Documented
public @interface JsProperty {
String value() default "";
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,181 |
Gliwice-Silesia 2019
Eurovision Events
ALL JUNIOR EUROVISION SONGS FROM BELARUS! 🇧🇾
Posted 17 December 2016 at 09:30 CET
There are two countries who participated in every edition of the Junior Eurovision Song Contest: Belarus and The Netherlands. Time to take a trip down memory lane and look at the entries from Belarus
tags Belarus 2016 Junior history Eurovision Song all entries contest all editions
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Subscribe to the official Junior Eurovision Song Contest YouTube channel for music videos, live show footage, historic moments, exclusive behind-the-scenes material and our regular updates.
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The Junior Eurovision Song Contest is organized by the European Broadcasting Union, the world's foremost alliance of public service media, representing 117 member organizations in 56 countries and an additional 34 Associates in Asia, Africa, Australasia and the Americas.
© EBU 2002-2019. All rights reserved.
Website by Scrn.
This website uses cookies to provide you with a better user experience. Further information on cookies and how we use them is contained in the Cookies section of the Privacy & Cookie Policy. By continuing to actively use this website, you agree to our use of cookies. | {
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the cost best https://fr.upscalerolex.to/ review is considered to be corresponded with show casing really price tag. swiss www.iqosvape.com owns a high factor within throughout the world watch business sector. best swiss https://www.replicasalvatoreferragamo.ru passed down that chic neoclassical trend, but more straight into the cutting-edge substances. we supply the high quality https://tomtops.ru with cheap price. https://www.vapesstores.es/ vape for sale is good and chic. swiss https://www.watchesbuy.pl/ the big ten started fees are raised yet unfortunately customers continues to be feel valuable.
Globalisation on human terms
Citizen Correspondent
50 years after Allende at the UN: a corporate triumph named multistakeholderism
Harris Gleckman
Interviews, UN, Dialogue
*Originally published on December 30, 2022 **Produced by Lynn Fries / GPEnewsdocs TRANSCRIPT LYNN FRIES: Hello and welcome. I'm Lynn Fries producer of Global Political Economy or GPEnewsdocs with guest Harris Gleckman. A recent symposium marking the 50th anniversary of Salvador Allende's speech at the UNGA in 1972 delved into the topic of "Corporate Power: Then…
"Trust is the philosopher's stone of political systems" – A conversation with Raquel Chanto on the state of democracy in Costa Rica and in the World
Yoriko Yasukawa
Costa Rica is an exceptional country in a number of ways. It is a small nation in a region, Central America, torn by civil wars well into the 1980s, that abolished its army over 70 years ago and has lived in peace ever since. The country has achieved exceptionally high levels of human development despite…
The unacknowledged present: An interview with the artist Stine Marie Jacobsen
Richard Willmsen
Especially in times of populism and growing right-wing movements, artists hold a mirror up to societies and sense tendencies long before these become conscious.
"Neoliberalism had some good points": An interview with Nick Currie aka Momus about Europe, politics, identity and Japan
Momus is currently doing a series of appearances around Europe, travelling mostly by train.
Jónasson: The Icelandic Minister who refused cooperation with the FBI
Marta Pacheco
Famous for standing up to the FBI, Ögmundur Jónasson spoke to Katoikos about whistleblower protection, countering the rise of populism and Iceland's unique approach to the financial crisis.
Elia Suleiman: Hope and Action Despite Pessimism
Daniel Tkatch
Film, Interviews, Arts
On the role of engaged art in an increasingly globalized world and old and new Jerusalems. An interview with the Nazareth-born filmmaker Elia Suleiman (1960), President of the Jury at this year's, 22nd Sarajevo Film Festival.
'European Politics Isn't Kittens': An Interview with the Founders of Politix EU
Would the British vote to leave the EU had there been a pan-European public sphere? Would nationalist and populist arguments have the same appeal in Britain and other European countries? An interview with the realists behind Politix EU
Profession: Parliamentary Assistant. An interview with Georgios Oikonomides
e(u)-generation
It´s Our Voice, Interviews, e(u)-Generation
In his late 20s, Georgios Oikonomides has a lot to be proud of. His political activism, European studies and determination to succeed have taken him from small and distant Cyprus to the centre of European democracy, the European Parliament. He is not an MEP himself, at this point at least, but is "the right hand" of another Cypriot, MEP, Demetris Papadakis. We approached Georgios as a possible role model for young people who want to work for Europe and the European Institutions.
A conversation with Shlomo Ben-Ami; on Israel, Palestine and beyond
Georgios Kostakos
Interviews, Op-ed
Shlomo Ben-Ami is an old hand in Israeli and international politics. He has been Minister of Internal Security and Minister of Foreign Affairs of Israel, and now serves as Vice President of the Toledo International Center for Peace in Madrid. This is a summary of a 30-minute discussion with him following the 17 March 2015 Israeli election. He talks about Mr. Netanyahu's reelection, the state of the Israeli-Palestinian peace process, and possible steps by the international community to move things forward under the current circumstances.
Interview with Yves Pascouau
Raluca Raduta
Interviews, Dialogue, Op-ed
"If you give people more opportunities to move, then you decrease the pressure on irregular migration".
Yves Pascouau is Director of Migration and Mobility Policies at the European Policy Centre. He holds a PhD in Law from the University of Pau in France and he worked extensively on migration management. Beside his current position at EPC, he is the editor of the online legal website European Migration Law www.europeanmigrationlaw.eu.
Send your article
Episode 9 – Dissecting the Alt-Right Movement
26 January, 2023 | Katoikos
Today's guest is Ipsita Chatterjee, a human geographer interested in the economic, cultural, and geopolitical impacts of globalization. Ipsita is also a prolific author, and her latest book "The Alt-Right Movement: Dissecting Racism, Patriarchy and Anti-immigrant Xenophobia" served as the basis of our discussion. To help me better understand this phenomenon, she took me through…
Russia is less than its myths and the truth will set its peoples free
26 January, 2023 | Francis M. O'Donnell
Lately, I came across a very interesting article "The Roots of Russia", written exactly sixty years ago by Dr. Lev Dobriansky (1918-2008), a renowned Professor of Economics at Georgetown University and former US Ambassador to The Bahamas[1]. Dobriansky's article was re-published in January 1964, in An Cosantóir (The Defender), The Irish Defence Journal, courtesy of…
Katoikos is a publication that wants to bring out the common elements that unite humanity and, at the same time, to celebrate its diversity. It is inspired by the shared interest for peace, human dignity and well-being for all.
© 2023 Katoikos, all rights are reserved. Developed by eMutation | New Media
We use cookies to ensure optimum user experience on our website. Your continued use of this site constitutes your acceptance.I agreeRead more | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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Published May 20, 2015 by Pearson IT Certification. Part of the LiveLessons series.
The 300+ videos contained in this product provide you more than 32 hours of instruction. Modules are divided into easy to digest lessons and conclude with summaries and interactive module and glossary quizzes to help assess your knowledge. In addition to the review quizzes, the product also contains interactive exercises to help you truly learn the topics in each module. Each product concludes with a series of lessons that give you valuable advice to help prepare for the actual exam.
· The troubleshooting toolkit, including change control procedures, managing configuration files, limiting show command output with modifiers, hardware diagnostic commands, and SPAN, NetFlow, and EEM configuration. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,671 |
{"url":"http:\/\/mathhelpforum.com\/algebra\/199084-algeb.html","text":"1. ## Algeb\n\nIf $x^2+abx+c=0 \\mbox\\ {and} \\ x^2+acx+b=0$ quadratic equations have a common root.prove b=c or the other roots are the roots of $a(b+c)x^2+(b+c)-abc=0$\n\n2. ## Re: Algeb\n\nOriginally Posted by srirahulan\nIf $x^2+abx+c=0 \\mbox\\ {and} \\ x^2+acx+b=0$ quadratic equations have a common root.prove b=c or the other roots are the roots of $a(b+c)x^2+(b+c)-abc=0$\nThe roots of the first equation are\n\n\\displaystyle \\begin{align*} x &= \\frac{-ab \\pm \\sqrt{(ab)^2 - 4(1)(c)}}{2(1)} \\\\ &= \\frac{-ab \\pm \\sqrt{a^2b^2 - 4c}}{2} \\end{align*}\n\nand the roots of the second equation are\n\n\\displaystyle \\begin{align*} x &= \\frac{-ac \\pm \\sqrt{(ac)^2 - 4(1)(b)}}{2(1)} \\\\ &= \\frac{-ac \\pm \\sqrt{a^2c^2 - 4b}}{2} \\end{align*}\n\nSince they share a root, there is some value such that\n\n\\displaystyle \\begin{align*} \\frac{-ab \\pm \\sqrt{a^2b^2 - 4c}}{2} &= \\frac{-ac \\pm \\sqrt{a^2c^2 - 4b}}{2} \\\\ -ab \\pm \\sqrt{a^2b^2 - 4c} &= -ac \\pm \\sqrt{a^2c^2 - 4b} \\end{align*}\n\nIf \\displaystyle \\begin{align*} b = c \\end{align*}, we have\n\n\\displaystyle \\begin{align*} LHS &= -ab \\pm \\sqrt{a^2b^2 - 4c} \\\\ &= -ab \\pm \\sqrt{a^2b^2 - 4b} \\\\ \\\\ RHS &= -ac \\pm \\sqrt{a^2c^2 - 4b} \\\\ &= -ab \\pm \\sqrt{a^2b^2 - 4b} \\\\ &= LHS \\end{align*}\n\nSo that means for the two equations to share a root, it's possible that \\displaystyle \\begin{align*} b = c \\end{align*}.\n\n3. ## Re: Algeb\n\nI can't understand you statements.pls reply,\n\n4. ## Re: Algeb\n\nOriginally Posted by srirahulan\n....are the roots of $a(b+c)x^2+(b+c)-abc=0$\nk = [abc - (b+c)] \/ [a(b+c)]\nx = +- SQRT(k)\n\n5. ## Re: Algeb\n\nOriginally Posted by srirahulan\nIf $x^2+abx+c=0 \\mbox\\ {and} \\ x^2+acx+b=0$ quadratic equations have a common root.prove b=c or the other roots are the roots of $a(b+c)x^2+(b+c)-abc=0$\nx^2 + abx + c = 0 [1]\nx^2 + acx + b = 0 [2]\n\nRoots of [1] = u,v\nRoots of [2] = u,w\nThen:\nab = u + v\nc = uv : v = c\/u\nab = u + c\/u : a = (u + c\/u) \/ b [3]\n\nSimilarly: a = (u + b\/u) \/ c [4]\n\n[3][4]: (u + c\/u) \/ b = (u + b\/u) \/ c\nSimplify:\nbu^2 + b^2 = cu^2 + c^2\nb = c","date":"2017-09-23 16:56:11","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\": 13, \"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\": 1.0000100135803223, \"perplexity\": 8496.478857686692}, \"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-2017-39\/segments\/1505818689752.21\/warc\/CC-MAIN-20170923160736-20170923180736-00454.warc.gz\"}"} | null | null |
Kugel ( , pronounced ) is a baked casserole, most commonly made from lokshen or Jewish egg noodles ( ) or potato. It is a traditional Ashkenazi Jewish dish, often served on Shabbat and Jewish holidays. American Jews also serve it for Thanksgiving dinner.
Etymology
The name of the dish comes from the Middle High German meaning 'sphere, globe, ball'; thus the Yiddish name likely originated as a reference to the round, puffed-up shape of the original dishes (compare to German —a type of ring-shaped cake). However, nowadays kugel is often baked in square pans.
Litvaks (Jews from Lithuania, northeastern Poland and northern Russia) call the pudding , Galitzianers (Jews from southeastern Poland and western Ukraine) call it .
History
The first kugels were made from bread and flour and were savory rather than sweet. About 800 years ago, Jewish cooks in Germany replaced bread mixtures with lokshen noodles or farfel. Eventually eggs were incorporated. The addition of cottage cheese and milk created a custard-like consistency common in today's dessert dishes. In Poland, Jewish homemakers added raisins, cinnamon and sweet curd cheese to noodle kugel recipes. In the late 19th century, Jerusalemites combined caramelized sugar and black pepper in a noodle kugel known as the Jerusalem kugel (), which is a commonly served at Shabbat kiddushes and is a popular side dish served with cholent during Shabbat lunch.
In Romania, this dish is called ("macaroni pudding") or . It is made with or without cheese, but almost always includes raisins. In Transylvania, especially in the Hungarian-speaking regions, a very similar dish is called .
Savory kugel may be based on potatoes, matzah, cabbage, carrots, zucchini, spinach, or cheese.
Romani people call it pirogo. The Romani version is made with raisins, cream cheese, and butter.
Varieties
Jerusalem Kugel
Kugel Yerushalmi, ( kugl yerushalmi in Hebrew), also known as Jerusalem kugel, or Galilean kugel, is an Israeli kugel dish originating from the local Jewish community of Jerusalem in the 1700s.
Noodle kugel
Noodle kugel, also known as lokshen kugel, is an Ashkenazi Jewish casserole, side dish and popular variety of kugel made with lokshen noodles and either a variety of dairy or pareve ingredients.
Potato kugel
Potato kugel is a potato-based kugel of Ashkenazi Jewish origin, made with grated or pureed potatoes, onions, eggs, flour or matzo meal, oil, salt and pepper.
Jewish festivals
Kugels are a mainstay of festive meals in Ashkenazi Jewish homes, particularly on the Jewish Sabbath and other Jewish holidays or at a tish. Some Hasidic Jews believe that eating kugel on the Jewish Sabbath brings special spiritual blessings, particularly if that kugel was served on the table of a Hasidic Rebbe.
While noodle kugel, potato kugel, and other variations are dishes served on Jewish holiday meals, matzo kugel is a common alternative served at Passover seders which is adjusted to meet Passover kosher requirements.
South African slang usage
Among South African Jews, the word kugel was used by the elder generation as a term for a young Jewish woman who forsook traditional Jewish dress values for those of the ostentatiously wealthy and became overly materialistic and overgroomed, mirroring how the kugel is a plain pudding garnished as a delicacy. The women thus described made light of the term, and it has since become an amusing rather than derogatory slang in South African English for a materialistic young woman.
Similar dishes
Hotdish
Potatonik
Zucchini slice
See also
Cuisine of Israel
List of casserole dishes
References
External links
Ashkenazi Jewish cuisine
Desserts
Jewish cuisine
Israeli cuisine
Pasta dishes
Potato dishes
Yiddish words and phrases
Casserole dishes
Shabbat
Savory puddings
Jewish American cuisine
Jewish noodle dishes
Romani cuisine
Romanian cuisine
Thanksgiving food | {
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Q: I don t understand how value of variable is changing or initializing: I don't understand what is new value of variable. I don't get it how it works..
I cant find the answer nowhere..
i run it and x = 10 , n = 2 and y = 0? but i don t see the logic behind it.
int x = 5, y = 3, n;
x *= (2 & 2);
n = 2 | 2;
y *=(5 & 8); // y = 0 ??
Console.WriteLine(x); // 10
Console.WriteLine(y); // 0
Console.WriteLine(n); // 2
x = 10 , but y = 0 ? I just don t get the logic...
I don't remember this form class but all of the sudden there it is.
How it works?
A: 5 is like 0101(2) which means 5 contains 4 (0100) and 1(0001)
8 is like 1000(2) which means 8 only contains 8 (1000)
& and | operator checks if values contains same elements which is true at same points or not.
so 5|8 will be contains 8 4 1 => 1101 and it converts to decimal like 13.
and 5&8 will be zero because 5 doesn't contains 8 and 8 also doesn't contains 4 and 1.
I'd like to introduce this webpage to learn about bitwise operators.
https://code.tutsplus.com/articles/understanding-bitwise-operators--active-11301
| {
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{"url":"https:\/\/mathematica.stackexchange.com\/questions\/204796\/output-of-reduce-simplify","text":"# Output of Reduce + Simplify\n\nI'm using Reduce to solve some equations and then simplify to eliminate some undesired results. The problem is that, when I apply Simplify, it messes with my output from reduce. As an example:\n\n eq = Cos[x] + Sin[x] == y\nReduce[eq, x]\nSimplify[Reduce[eq, x]]\n\n\nMy outputs are\n\n (C[1] \\[Element] Integers && y == -1 && (x == -(\\[Pi]\/2) + 2 \\[Pi] C[1] || x == \\[Pi] + 2 \\[Pi] C[1])) || (C[1] \\[Element] Integers && 1 + y != 0 && (x == 2 ArcTan[(1 - Sqrt[2 - y^2])\/(1 + y)] + 2 \\[Pi] C[1] ||x == 2 ArcTan[(1 + Sqrt[2 - y^2])\/(1 + y)] + 2 \\[Pi] C[1]))\n\n\nAnd with Simplify:\n\n (C[1] \\[Element] Integers && y == -1 && (\\[Pi] + 2 x == 4 \\[Pi] C[1] || \\[Pi] + 2 \\[Pi] C[1] ==x)) || (C[1] \\[Element] Integers && 1 + y != 0 && (x == 2 (ArcTan[(1 - Sqrt[2 - y^2])\/(1 + y)] + \\[Pi] C[1]) || x == 2 (ArcTan[(1 + Sqrt[2 - y^2])\/(1 + y)] + \\[Pi] C[1])))\n\n\nOf course, Simplify here is not necessary, but my real equations are rather long to put here. My point is, when I use simplify, I get x to the lhs, so I can't use ToRules to make a list of rules from my output. I sounds like a silly question, but I've tried a lot of different things and I cant figure it out how to do it...\n\nThank you so much\n\n\u2022 I can make it work for some cases: ToRules[x - a == b + c + d] \/. ((x + lhs_) -> rhs_) :> (x -> (-lhs + rhs)) will produce the desired result, but if my lhs is (-x+a) it won't work... \u2013\u00a0F\u00e1bio Sep 5 at 1:22\n\u2022 Sometimes you get x==complicated and you would like x==simple but Simplify thinks the result is simpler if it pushes things across the ==. For those cases try x==complicated\/.Equal[left_,right_]:>Equal[Simplify[left],Simplify[right]] For x+complicated==morecomplicated try x+complicated==morecomplicated \/.Equal[left_,right_]:>Solve[Equal[left,right],x] or even replace that Solve with Reduce` and see if that works for you. If you can show more specific examples that still don't work then perhaps someone can show a general method that works for you. \u2013\u00a0Bill Sep 5 at 4:43\n\u2022 Well, it's working so far. I'll try to come with a more general method if someone comes with the same issue. Thank you! \u2013\u00a0F\u00e1bio Sep 6 at 19:43","date":"2019-10-15 19:59:33","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\": 0.5326432585716248, \"perplexity\": 1785.249265092521}, \"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\/1570986660231.30\/warc\/CC-MAIN-20191015182235-20191015205735-00530.warc.gz\"}"} | null | null |
I'm glad you are here! You can contact me at mail2homeis@gmail.com or by twitter @henjenca or through this contact form. I do not engage guest post content or article content without an invitation from me, if you email me asking to do a guest post, I likely will not answer the email because of the volume of such inquiries. | {
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Белиз принимал участие в Летних Олимпийских играх 2008 года в Пекине (Китай) в десятый раз за свою историю, но не завоевал ни одной медали. Сборная страны состояла из 4 спортсменов (3 мужчины, 1 женщина).
Состав олимпийской сборной Белиза
Лёгкая атлетика
Спортсменов — 1
Мужчины
Ссылки
База МОК
Официальные олимпийские отчёты
www.sports-reference.com
Страны на летних Олимпийских играх 2008 года
2000-е годы в Белизе | {
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ФК «Болонья» в сезоні 1931—1932 — сезон італійського футбольного клубу «Болонья».
Склад команди
Чемпіонат Італії
Підсумкова таблиця
Матчі
Статистика в чемпіонаті
Товариські матчі
30-08-1931, Форлі — Болонья — 1-2
06-09-1931, Болонья — Лаціо — 1-0
13-09-1931, Болонья — Про Кальчо (Модена) — 4-2
17-09-1931, Болонья — Шабарія (Угорщина) — 2-0
10-04-1932, Болонья — Фаєнца — 4-0
21-04-1932, Болонья — Падова — 5-0
18-05-1932, Болонья — Олімпік (Антіб, Франція) — 3-0
Посилання
1931–1932
Болонья | {
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} | 9,598 |
{"url":"http:\/\/stackoverflow.com\/questions\/6313507\/how-to-regex-search-replace-with-filemap-in-largish-text-file-with-to-avoid-o","text":"# How to regex search\/replace with File::Map in largish text file with to avoid \u201cOut of Memory\u201d-Error?\n\nUPDATE 2: Solved. See below.\n\nI am in the process of converting a big txt-file from an old DOS-based library program into a more usable format. I just started out in Perl and managed to put together a script such as this one:\n\nBEGIN {undef $\/; }; open$in, '<', \"orig.txt\" or die \"Can't read old file: $!\"; open$out, '>', \"mod.txt\" or die \"Can't write new file: $!\"; while( <$in> )\n{\n$C=s\/foo\/bar\/gm; print \"$C matches replaced.\\n\"\netc...\nprint $out$_;\n}\nclose $out; It is quite fast but after some time I always get an \"Out of Memory\"-Error due lack of RAM\/Swap-Space (I'm on Win XP with 2GB of Ram and a 1.5GB Swap-File). After having looked around a bit on how to deal with big files, File::Map seemed to me as a good way to avoid this problem. I'm having trouble implementing it, though. This is what I have for now: #!perl -w use strict; use warnings; use File::Map qw(map_file); my$out = 'output.txt';\nmap_file my $map, 'input.txt', '<';$map =~ s\/foo\/bar\/gm;\n\nprint $out$map;\n\n\nHowever I get the following error: Modification of a read-only value attempted at gott.pl line 8.\n\nAlso, I read on the File::Map help page, that on non-Unix systems I need to use binmode. How do I do that?\n\nBasically, what I want to do is to \"load\" the file via File::Map and then run code like the following:\n\n$C=s\/foo\/bar\/gm; print \"$C matches found and replaced.\\n\"\n\n$C=s\/goo\/far\/gm; print \"$C matches found and replaced.\\n\"\nwhile(m\/complex_condition\/gm)\n{\n$C=s\/complex\/regex\/gm;$run_counter++;\n}\nprint \"$C matches replaced. Script looped$run_counter times.\\n\";\netc...\n\n\nI hope that I didn't overlook something too obvious but the example given on the File::Map help page only shows how to read from a mapped file, correct?\n\nEDIT:\n\nIn order to better illustrate what I currently can't accomplish due to running out of memory I'll give you an example:\n\nOn http:\/\/pastebin.com\/6Ehnx6xA is a sample of one of our exported library records (txt-format). I'm interested in the +Deskriptoren: part starting on line 46. These are thematic classifiers which are organised in a tree hierarchy.\n\nWhat I want is to expand each classifier with its complete chain of parent nodes, but only if none of the parent nodes are not already present before or after the child node in question. This means turning\n\n+Deskriptoren\n-foo\n-Cultural Revolution\n-bar\n\n\ninto\n\n+Deskriptoren\n-foo\n-History\n-Modern History\n-PRC\n-Cultural Revolution\n-bar\n\n\nThe currently used Regex makes use of Lookbehind and Lookahead in order to avoid duplicates duplicates and is thus slightly more complicated than s\/foo\/bar\/g;:\n\ns\/(?<=\\+Deskriptoren:\\n)((?:-(?!\\QParent-Node\\E).+\\n)*)-(Child-Node_1|Child-Node_2|...|Child-Node_11)\\n((?:-(?!Parent-Node).+\\n)*)\/${1}-Parent-Node\\n-${2}\\n${3}\/g; But it works! Until Perl runs out of memory that is... :\/ So in essence I need a way to do manipulations on a large file (80MB) over several lines. Processing time is not an issue. This is why I thought of File::Map. Another option could be to process the file in several steps with linked perl-scripts calling each other and then terminating, but I'd like to keep it as much in one place as possible. UPDATE 2: I managed to get it working with Schwelm's code below. My script now calls the following subroutine which calls two nested subroutines. Example code is at: http:\/\/pastebin.com\/SQd2f8ZZ Still not quite satisfied that I couldn't get File::Map to work. Oh well... I guess that the line-approach is more efficient anyway. Thanks everyone! - Why do you think you need to slurp the entire file? Why won't processing it a line-at-a-time work? \u2013 tadmc Jun 11 '11 at 3:00 just guessing, but maybe the complex regexp needs to span several lines, in which case line-based processing wouldn't work. \u2013 mirod Jun 11 '11 at 3:23 yes exactly, as my updated post shows. \u2013 screen12345 Jun 11 '11 at 15:50 ## 3 Answers Some simple parsing can break the file down into manageable chunks. The algorithm is: 1. Read until you see +Deskriptoren: 2. Read everything after that until the next +Foo: line 3. Munge that bit. 4. Goto 1. Here's the sketch of the code: use strict; use warnings; use autodie; open my$in, $input_file; open my$out, $output_file; while(my$line = <$in>) { # Print out everything you don't modify # this includes the +Deskriptoren line. print$out $line; # When the start of a description block is seen, slurp in up to # the next section. if($line =~ m{^ \\Q Deskriptoren: }x ) {\nmy($section,$next_line) = _read_to_next_section($in); # Print the modified description print$out _munge_description($section); # And the following header line. print$out $next_line; } } sub _read_to_next_section { my$in = shift;\n\nmy $section = ''; my$line;\nwhile( $line = <$in> ) {\nlast if $line =~ \/^ \\+ \/x;$section .= $line; } # When reading the last section, there might not be a next line # resulting in$line begin undefined.\n$line = '' if !defined$line;\nreturn($section,$line);\n}\n\n# Note, the +Deskriptoren line is not on $description sub _munge_description { my$description = shift;\n\n...whatever you want to do to the description...\n\nreturn $description; } I haven't tested it, but something like that should do you. It has the advantage over dealing with the whole file as a string (File::Map or otherwise) that you can deal with each section individually rather than trying to cover every base in one regex. It also will let you develop a more sophisticated parser to deal with things like comments and strings that might mess up the simple parsing above and would be a huge pain to adapt a massive regex to. - Thank you. I ended up using your code for the problematic bit. \u2013 screen12345 Jun 28 '11 at 8:34 One last problem is however that Perl complains about Use of uninitialized value$next_line in print at D:\\path\\to\\script.pl line 98, <$in> line 5416772. Both seem to be in the while-loop. The script finishes fine despite the warning but I'd like to know how to correct the script... I tried to initialize $next_line via our $next_line = 0 at the beginning of the script, but it still shows the same. \u2013 screen12345 Jun 28 '11 at 11:20 @screen12345 Trying to initialize it with our$next_line won't work. That creates the global $next_line which my$next_line masks by creating a lexical $next_line (which only exists for the block its in). The problem is in _read_to_next_section(). It initializes $line (which will be returned to $next_line) but at the last line of the file $line = <$in> will be undefined. This is normal and something I didn't account for. Setting $line if its undefined after the while loop fixes that. I'll edit it now. \u2013\u00a0Schwern Jun 28 '11 at 18:24\nthat did it. updated the code on pastebin. thank you again. I still have a lot to learn. \u2013\u00a0screen12345 Jun 28 '11 at 20:53\n\nWhen you set $\/ (the input record separator) to undefined, you are \"slurping\" the file -- reading the entire content of the file at once (this is discussed in perlvar, for example). Hence the out-of-memory problem. Instead, process it one line at a time, if you can: while (my$line = <$in>){ # Do stuff. } In situations where the file is small enough and you do slurp the file, there is no need for the while loop. The first read gets everything: { local$\/ = undef;\nmy $file_content = <>; # Do stuff with the complete file. } Update After seeing your massive regex I would urge you reconsider your strategy. Tackle this as a parsing problem: process the file one line at a time, storing information about the parser's state as needed. This approach allows you to work with the information using simple, easily understood (even testable) steps. Your current strategy -- one might call it the slurp and whack with massive regex strategy -- is difficult to understand and maintain (in 3 months will your regex makes immediate sense to you?), difficult to test and debug, and difficult to adjust if you discover unanticipated deviations from your initial understanding of the data. In addition, as you've discovered, the strategy is vulnerable to memory limitations (because of the need to slurp the file). There are many questions on StackOverflow illustrating how one can parse text when the meaningful units span multiple lines. Also see this question, where I delivered similar advice to another questioner. - Thanks, I'll try that. \u2013 screen12345 Jun 11 '11 at 15:49 @screen12345 Answer updated. \u2013 FMc Jun 11 '11 at 19:10 You are using mode <, which is read-only. If you want to modify the contents, you need read-write access, so you should be using +<. If you are on windows, and need binary mode, then you should open the file separately, set binary mode on the file handle, then map from that handle. I also noticed that you have an input file and an output file. If you use File::Map, you are changing the file in-place... that is, you can't open the file for reading and change the contents of a different file. You would need to copy the file, then modify the copy. I've done so below. use strict; use warnings; use File::Map qw(map_file); use File::Copy; copy(\"input.txt\", \"output.txt\") or die \"Cannot copy input.txt to output.txt:$!\\n\";\n\nopen my $fh, '+<', \"output.txt\" or die \"Cannot open output.txt in r\/w mode:$!\\n\";\n\nbinmode($fh); map_handle my$contents, $fh, '+<'; my$n_changes = ( $contents =~ s\/from\/to\/gm ); unmap($contents);\nclose($fh); The documentation for File::Map isn't very good on how errors are signaled, but from the source, it looks as if $contents being undefined would be a good guess.\n\n-\nOn error, it will throw an exception. I'll add that to the documentation. Also, map_file will binmode automatically, unless you pass it a layer. And you don't really need that unmap, it's taken care of automatically when \\$contents falls out of scope. \u2013\u00a0Leon Timmermans Jan 14 '13 at 11:44","date":"2016-06-29 16:57:12","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\": 0.31607523560523987, \"perplexity\": 4788.252338208541}, \"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-2016-26\/segments\/1466783397749.89\/warc\/CC-MAIN-20160624154957-00162-ip-10-164-35-72.ec2.internal.warc.gz\"}"} | null | null |
This is one little bow tie for one little man! Our solid Crimson Red Infant Pre-Tied Bow Tie is perfect for your little baby boy who needs to be dressed for that we... read more This is one little bow tie for one little man! Our solid Crimson Red Infant Pre-Tied Bow Tie is perfect for your little baby boy who needs to be dressed for that wedding, or wants to make one heck of an impression in baby play group! Already tied, the bow tie comes with a clip that is easy to attach to a collared shirt. Made of 100% silk, and our Crimson Red color matches our Elite Solid Silks. | {
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{"url":"https:\/\/people.cs.georgetown.edu\/~maloof\/cosc570.f22\/p2.html","text":"### COSC-570: Artificial Intelligence\n\nProject 2\nFall 2022\n\nDue: F 10\/7 @ 5 PM ET\n9 points\n\nFor this project, you will implement a resolution theorem prover for propositional logic.\n\nI have divided the project into three phases:\n\n1. For the first phase, write the classes and methods necessary to read, parse, store, and output propositional well-formed formulas (wffs). Formulas must be stored internally as expression trees.\n\nThe following grammar specifies the syntax for wffs.\n\nformula ::= '(' proposition ')'\n| '(' 'not' formula-or-proposition ')'\n| '(' 'or' formula-or-proposition formula-or-proposition ')'\n| '(' 'and' formula-or-proposition formula-or-proposition ')'\n| '(' 'cond' formula-or-proposition formula-or-proposition ')'\n\nformula-or-proposition ::= formula | proposition\n\nproposition ::= letter letters-and-digits\n\nletters-and-digits ::= letter-or-digit letters-and-digits | \\epsilon\n\nletter-or-digit ::= letter | digit\n\nletter ::= 'a'..'z' | 'A'..'Z'\n\ndigit ::= '0'..'9'\n\n\nExamples of wffs written in this syntax are:\n\n\u2022 Wishes are horses provided that horses cannot fly.\n\u2022 $$\\neg\\mbox{HF} \\rightarrow \\mbox{W}$$\n\u2022 (cond (not hf) w)\n\u2022 If it is not the case that both beggars ride and wishes are nonequine, then horses can fly.\n\u2022 $$\\neg (\\mbox{BRD} \\; \\& \\; \\neg\\mbox{W}) \\rightarrow \\mbox{HF}$$\n\u2022 (cond (not (and brd (not w))) hf)\n\nTo support development, I put some starter code on cs-class. To get started, log on to cs-class and copy over the following zip file:\n\ncs-class-1% cd\ncs-class-1% cp ~maloofm\/cosc570\/p2.zip .\/\ncs-class-1% unzip p2.zip\n\nThere is also a copy on Canvas. In the p2 directory, you will find a class implementation for Tokenizer and a partial implementation for Formula. The Tokenizer class demonstrates how to configure Scanner to tokenize propositional logic formulae. The directory also contains files for the two proofs we discussed in lecture, Example 1 and the proof that beggars do not ride horses. As you can see in the files, comments begin with a forward slash, formulas begin with a left parenthesis, and both are confined to a single physical line. By convention, the last formula in the file is the conclusion.\n\nIn addition to the these proofs, you must find two additional proofs from reputable sources consisting of at least three formulas and three propositions. Include these proof files with your submission to Autolab. Naturally, I will test your program on my own set of proofs.\n\nFor this first phase, you must implement the class Formula with the methods\n\n\u2022 public Formula()\n\u2022 public void set( String s ), which constructs an expression tree from the specified string s\n\u2022 public String toString(), which returns a string representation of the expression tree\n\nWhen you submit to Autolab, the autograder will call these three methods. Naturally, you will need to implement additional methods.\n\n2. For the second phase, implement the routines to convert wffs to clausal form. Clauses must be stored internally as expression trees or more precisely a linked list of literals. The following grammar specifies the syntax for clauses.\nclause ::= '{' literals '}'\n\nliterals ::= literal literals-comma-separated | \\epsilon\n\nliterals-comma-separated ::= ',' literal literals-comma-separated | \\epsilon\n\nliteral ::= proposition\n| '(' 'not' proposition ')'\n\n\nYou must implement\n\n\u2022 public void Formula.cnf(), which converts a formula to conjunctive normal form\n\u2022 public class Clauses, which must extend ArrayList<Clause>\n\u2022 public Clauses Formula.getClauses(), which converts a formula to clausal form and returns the resulting clauses\n\u2022 public String Clause.toString(), which returns the string representation of a clause\n\nThe transformations must preserve the order of literals. For example, (cond p q) should be written in CNF as ((not p) q), not as (q (not p)); (cond p q) should be written as the clause {(not p), q}, not as {q, (not p)}.\n\n3. Finally, implement the routines required to conduct proofs using linear resolution. For this final phase, you must implement Main.java so it takes a file name as a command-line argument (e.g., java Main lecture.txt), reads the formulas in the file, negates the conclusion, converts the formulas to clauses, and conducts a proof using linear resolution. Main.main should print the formulas, the converted clauses, and a message indicating whether the conclusion follows from the premises.\n\nYou must implement\n\n\u2022 public Clause()\n\u2022 public void Clause.set( String ), which sets the Clause to the clause that the specified String represents\n\u2022 public boolean Clause.empty(), which returns true if the Clause is empty and returns false otherwise\n\u2022 public Clause Clause.resolve( Clause clause ), which resolves this Clause with the specified Clause clause and returns the resolvent. If the two clauses do not resolve, then the method sets resolvent's internal state so Clause.failure() returns true\n\u2022 public Clause Clause.resolve( Clause clause ), which resolves this Clause with the specified Clause clause and returns the resolvent. If the two clauses do not resolve, then the method sets resolvent's internal state so Clause.failure() returns true\n\u2022 public Clauses()\n\u2022 public Clauses Clauses.resolve( Clause clause ), which resolves all of the clauses in Clauses with the Clause clause and returns them. If there are no resolvents, then resolve returns an empty clause.\n\u2022 public static boolean LinearResolution.prove( String filename ), which returns true if the conclusion follows from the premises and returns false otherwise\n\nInclude with your submission the proof files and a transcript of your program's execution for the four proofs. The transcript should be a plain ASCII file named README. In a file named HONOR, include the following statement:\n\nIn accordance with the class policies and Georgetown's Honor Code,\nI certify that, with the exceptions of the class resources and those\nitems noted below, I have neither given nor received any assistance\non this project.\n\nName\nNetID\n\n\nWhen you are ready to submit your project for grading, put your source files, Makefile, proof files, transcript, and honor statement in a zip file named submit.zip. Upload the zip file to Autolab using the assignment p2. 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Natula averni är en insektsart som först beskrevs av Costa, O.G. 1855. Natula averni ingår i släktet Natula och familjen syrsor. Inga underarter finns listade i Catalogue of Life.
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averni | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,294 |
{"url":"https:\/\/mundialcardans.com.br\/germany-trade-eebf\/chlorophyll-d-colour-e231f6","text":"# chlorophyll d colour\n\nHome \/ Sem categoria \/ chlorophyll d colour\n\n## chlorophyll d colour\n\n\u2014 Edited in Bilbao. It yields far better colour when heated and cooled multiple times so I took it into the studio already soaked and heated over several days to maximize the depth of dye we could achieve. Molecular formula . Subscribe to our daily newsletter to recieve articles and another updates. Now imagine that Chl a is treated with weak $\\ce{HCl}$. Donostia International Physics Center (DIPC) is a singular research center born in 2000 devoted to research at the cutting edge in the fields of Condensed Matter Physics and Materials Science. In some, the porphyrin is modified, for example by replacing the chlorine atoms with hydrogen atoms. Commercial pigments with structures similar to chlorophyll have been produced in a range of colors. It is a greenish colour pigment, which is able to capture energy from sunlight and produce foods in photoautotrophs. (2012). Discover how the pigments chlorophyll, anthocyanins, anthoxanthins, and carotenoids determine a plant's colour. The two main chlorophylls are chlorophyll a and chlorophyll b. Chlorophyll a absorbs purple and orange light the most. Is the colour of leaves due to d-d transitions in chlorophyll? Can you escape a grapple during a time stop (without teleporting or similar effects)? Range: 0-100 ppm Path Length: Variable (used in calculation) Chlorophyll is a green pigment found in almost all plants. You might think that it is as simple as preparing a solution of chlorophyll and use a spectrometer to get the answer. Author: C\u00e9sar Tom\u00e9 L\u00f3pez is a science writer and the editor of Mapping Ignorance. These two types are efficient in absorbing the light, and are effective photoreceptors. Chlorophyll b absorbs mostly blue and yellow. It is more soluble than chlorophyll a in polar solvents because of its carbonyl group. Chlorophylls are the pigments primarily responsible for photosynthesis. Chlorophyll d Found in red algae and some microorganisms ( cyanobacteria ), chlorophyll d is a minor pigment that is involved in the capture of the red spectrum of light (far end spectrum of red light). Since its conception DIPC has stood for the promotion of excellence in research, which demands a flexible space where creativity is stimulated by diversity of perspectives. Chlorophyll d is found in a type of cyanobacterium that lives in areas lacking visible light, but containing infrared radiation (700 nm to \u2026 Your email address will not be published. Therefore, in this study, the effect of roasting temperature, velocity of hot air, and exposure time on the colour, chlorophyll, and xanthophyll degradation and their correlation was studied. using natural colorants. The problem with this method is that these solvents have an effect on the electronic structure of the chlorophyll molecule, namely on the electron cloud at the porphyrin ring, thus modifying its optical behaviour. Chlorophyll in plants is a pigment molecule that imparts a green colour to the leaf and stems by absorbing a red and blue spectrum of light. There are two types of photosystems that involve with the light reaction. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Origin of \u201cGood books are the warehouses of ideas\u201d, attributed to H. G. Wells on commemorative \u00a32 coin? Plants use chlorophyll along with sunlight to get their nutrients. C7 group -CH 3-CHO -CH 3-CH 3-CH 3. D. Chlorophyll content extractio n . Chlorophyll or leaf green is a porphyrin derivative with magnesium as the central atom and is hence a metal complex dye. The colour of green leaves results from the absorption of the polycyclic pheophytin ligand and its extended $\\pi$ system. Donate Login Sign up. Let's simplify the Pheophytin a molecule: Removal of the E ring and replacing all the fancy substituents with ethyl groups leads to octaethylporphyrin as the core structure. Chlorophyll C has a blue-green color and is mainly found in brown algae. Chlorophyll gives leaves their green color and absorbs light that is used in photosynthesis. Therefore, they are seen the least. ABSORPTION SPECTRUM FOR CHLOROPHYLL d 1-9. Many photosynthetic organisms have a \u2026 Da dies nicht nur von PS II selbst, sondern auch von Kompnenten des Elektronentransports davor und danach, und z.B. So, what is the true colour of chlorophylls? While reading a chapter on transition metals, I came to know that d-Block elements have colour compounds due to d-d transitions. Si ta b\u00ebjm\u00eb klorofilin - Si ekstrakt klorofili - Ngjyrosje natyrale e gjelb\u00ebr e ushqimitThank you for watching my video on: How to make Chlorophyll. Chlorophyll d is a form of chlorophyll, identified by Harold Strain and Winston Manning in 1943. Use MathJax to format equations. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. The spectra obtained by the researchers using this method demonstrate negligible dependence on the nature of the tag. Chlorophyll as a green dye has been used commercially in processed foods, toothpaste, soaps and cosmetics. They absorb red and blue light, and reflect green light, which is what gives leaves their green colour. We cannot really draw a conclusion from chlorophyll A vs. B, as both these elements, give nature its beautiful green color. In the case of chlorophyll a the maximal absorption in the red region is at 642 nm and in the blue region at 372 nm; for chlorophyll b the values are 626 nm and 392 nm, respectively. Encyclop\u00e6dia Britannica, Inc.See all videos for this article. chlorophyll a and chlorophyll b. A chelate consists of a central metal ion bonded to a large organic molecule, composed of carbon, hydrogen, and other elements such as oxygen and nitro\u2026 In the UV-VIS absorption spectrum, however, we would still observe the two bands. Chlorophyll A reflects blue-green color, which is responsible for the green color of most of the land plants. 53,55 Chlorophyll a has a methyl group (CH 3 ) attached to R 1 position (Fig.1) thus its chemical formula is C 55 H 72 O 5 N 4 Mg Class 10 Science MCQs [\u2026] Chlorophyll molecules have a ring shape at one end, called a porphyrin ring, with a magnesium ion in the center (see figures 2 and 3; the magnesium ion is represented in green). Search. Does the green color represents existence of Copper in it. (A) bluish green, yellowish green (B) yellowish green, bluish C3 group -CH=CH 2-CH=CH 2-CH=CH 2-CH=CH 2-CHO . Chlorophyll is found in the chloroplasts of plants. These two types of chlorophyll differ only slightly, in the composition of a single side chain. I know the green colours of leaves is due to presence of chlorophyll and also there are coloured pigments in plants. That particular light wavelength is reflected from the plant, so it appears green. What is the origin of the colour of azo dyes? I may be wrong in my thought, but I really want to clarify my doubt. Chlorophyll is the bedrock of all plant life, and the similarities to the molecules of our red blood cells is uncanny. The name chlorophyll comes from the Greek words chloros (green) and phyllon (leaf). a natural colorant. Effect of pH on the chlorophyll degradation and visual green colour loss in blanched green peas were studied at 70, 80, 90 and 100 \u00b0C in buffered solutions of pH 5.5, 6.5 and 7.5. Chlorophyll concentration in the fruit peel can be estimated by extraction and spectrophotometry or in terms of the reflectance of the apple surface at 675 nm. The method consists in tagging chlorophyll (a and b only) molecules with three different ammonium cations with no mobile protons, so that in each case the distance between the chlorophyll and the electric charge is known. Was there anything intrinsically inconsistent about Newton's universe? If chlorophyll were red instead of green making plants red in appearance which color of light would you expect to produce the LOWEST rate of photosynthesis A. white light B. blue light C. red D. green 1 Chlorophyll demand continuously . Chlorophyll b is a form of chlorophyll.Chlorophyll b helps in photosynthesis by absorbing light energy. Function & characteristics: Green food colour. This structure is the one that is found in the photosynthetic reaction center. Pea puree of pH 6.95 and pH 8.45 was heat processed in thermal death\u2010time tubes at temperatures between 115.6\u00b0C and 148.9\u00b0C to a process value of F 0 = 6.0. Chlorophyll-Fluoreszenz: Misst die Quantenausbeute der verschiedenen Wege, auf denen ein von PS II absorbiertes Lichtquant seine Energie abgeben kann. Spirulina color extract is available as a powder or liquid. If you're seeing this message, it means we're having trouble loading external resources on our website. In any case neither one absorbs green, so the leaf looks green because that light is reflected to our eyes instead of being absorbed by the leaf. Fig. C 55 H 72 O 5 N 4 Mg . Spirulina contains green chlorophyll and phycobilins (a pigment\u2013protein complex). The coloring properties of spirulina are the phycobilins, these are blue phycocyanins that on extraction create a range of blue hues. To learn more, see our tips on writing great answers. rev\u00a02021.1.7.38271, The best answers are voted up and rise to the top, Chemistry 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. The reflected colors are what give pigments their color. Chlorophyll A is the most important pigment in photosynthesis, which serves as the primary electron donor in the electron transport chain of photosynthesis. DOI: Your email address will not be published. Animals that eat plants or other animals are called heterotrophs. Chlorophyll is a pigment that causes a green colour. Green leaves contain two types of chlorophyll, Chl a and Chl b, both have $\\ce{Mg}$ as the central atom and show two distinct bands in the UV-VIS absorption spectrum. 1: Chlorophyll, the green colour of plants . The intense green colour of chlorophyll is due to its strong absorbencies in the red and blue regions of the spectrum, shown in fig. Purified chlorophyll is used as a food colour with E-number E140, the more stable copper complexes of chlorophyll are number E141. In the resulting product, known as Pheophytin a, the central atom ($\\ce{Mg^{2+}}$) is replaced with two protons. Chlorophyll d: C 54 H 70 O 6 N 4 Mg Biosynthesis of chlorophy ll: Succinyl-coA and glycine react to form an intermediate Product \u03b4-Aminolevulinic acid 2 molecules of \u03b4-Aminolevulinic acid react with each other to give porphobilinogen 4 molecules of porphobilinogen fuse to give protochlorophyll.Then protochlorophyll react with 2H to synthesis chlorophyll in the presence of sunlight. chlorophylls a, b, c1, c2, d. All plants, \"plant-like\" protists, cyanobacteria and prochlorophytes that photosynthesize have chlorophyll a. It is this interaction with the surrounding microenvironment what finetunes chlorophylls to cover as much of the visible spectrum as possible. Chlorophyll A is the primary photosynthetic pigment present in plants and algae. Chlorophyll c is found in eukaryotic microbes, like marine and freshwater algae, and absorbs red light (between 450 and 640). There are different types of chlorophyll (chlorophyll-a, chlorophyll-b, chlorophyll-c1, chlorophyll-c2, chlorophyll-d, divinyl chlorophyll-a). The differences occur at the level of division or class. So if you eat green chips and the ingredient lists E141, know that it was colored using chlorophyll. Chlorophyll d absorbs far-red wavelengths beyond the optical range and some wavelengths of the blue-green region. It is the most widely distributing type of chlorophylls. The Chlorophyll Molecule Chemical and Physical Properties. Zero correlation of all functions of random variables implying independence, Reflection - Method::getGenericReturnType no generic - visbility, Macbook in Bed: M1 Air vs M1 Pro with Fans Disabled, Why is the in \"posthumous\" pronounced as (\/t\u0283\/). Chlorophyll pigments, Hunterlab colour indices and pH were determined before and after high temperature\u2010short time (H.T.S.T.) Chlorophyll molecules have a ring shape at one end, called a porphyrin ring, with a magnesium ion in the center (see figures 2 and 3; the magnesium ion is represented in green). Chlorophyll E140 is a chlorin pigment found in cyanobacteria and the chloroplasts of algae and plants. This is not an easy question to answer, though. green color of chlorophyll has been long time used as . They include other forms of chlorophyll, such as chlorophyll b in green algal and higher plant antennae, while other algae may contain chlorophyll c or d. In addition, there are many non-chlorophyll accessory pigments, such as carotenoids or phycobiliproteins which also absorb light and transfer that light energy to the photosystem chlorophylls. Chlorophyll is the green photosynthetic pigment present in chloroplasts which provides the energy necessary for photosynthesis. In the UV-VIS spectrum, we would still be able to see the strong Soret band, which corresponds to a $\\pi,\\pi^*$ transition from the ground state to the second excited singlet state ($S_0 \\rightarrow S_2$). Use this link to get alternative options to subscribe. Chlorophyll is the green photosynthetic pigment present in chloroplasts which provides the energy necessary for photosynthesis. Thanks for contributing an answer to Chemistry Stack Exchange! Its dynamic research community integrates local host scientists and a constant flow of international visiting researchers. C 54 H 70 O 6 N 4 Mg . This is due to the chlorophylls. We may conclude that our initial hypothesis is wrong. Just a slight change and the optical behaviour of the molecule changes. The different pigments, chlorophyll a, chlorophyll b, and beta carotene have different polarities, due to which the separation of these pigments is possible with chromatography paper. Therefore both these types are important for effective photosynthesis. Milne B.F., Angel Rubio & Steen Br\u00f8ndsted Nielsen (2014). This is where photosynthesis takes place. MCQ Questions for Class 10 Science with Answers was Prepared Based on Latest Exam Pattern. There are six types of chlorophylls in plants. \u2014 ISSN 2529-8992 It is vital for photosynthesis, which allows plants to obtain energy from light. They are slightly shifted and show somewhat different extinction coefficients, but the pattern is fundamentally the same. Let's rephrase your hypothesis and test it: The green colour of leaves results from a d-d transition of the central atom of metal complexes known as chlorophylls. MathJax reference. How do you take into account order in linear programming? C 35 H 28 O 5 N 4 Mg . Chlorophyll d absorbs far-red light, at 710 nm wavelength, just outside the optical range. Beta carotene is non-polar, chlorophyll b is the most polar, chlorophyll a is more polar than beta carotene, but less polar than chlorophyll \u2026 By clicking \u201cPost Your Answer\u201d, you agree to our terms of service, privacy policy and cookie policy. The degradation of chlorophylls a and b followed a first-order reaction and the temperature-dependence of these reactions was modelled by the Arrhenius equation. Chlorophyll is an important molecule that plays a critical role in photosynthesis. Imm e diat ely a ft er SPA D and imag e me asu rem ent, th e . This has already been done with different solvents. As you can see on the photo, we failed to put chlorophyll back into plants. C 55 H 70 O 6 N 4 Mg . How chlorophylls and other pigments absorb light. Chlorophyll a appears in colour and chlorophyll b appears in colour in the chromatogram. It is present in the chloroplasts in all green parts of plants as a mixture of blue green chlorophyll a and yellow green chlorophyll b, and constitutes the catalyst for photosynthesis.Chlorophyll c occurs in lacustrine algae and chlorophyll d in red algae. The LCC developed for rice by the University of California Cooperative Extension has eight plastic panels ranging from yellow\u2010green to a dark green ().Mutters and Eckert (2004) collected leaves of different rice cultivars, with different amounts of leaf N, and measured the spectral reflectances. It only takes a minute to sign up. Then dissolved mixtures of chlorophyll and tags are electro sprayed in a vacuum within a spectrometer. Three natural pigments \u2013 red beet powder, chlorophyll powder (in the form of a copper complex) and aqueous cochineal \u2013 were examined in order to understand better their behaviour or colouring ability in different model systems. Extract the value in the line after matching pattern, Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology. The excitation provoked by light produces the dissociation of chlorophylls and tags allowing the measurement of the spectrum of chlorophylls for different cations. If there are no other pigments present, that is. 1. Properties of light. The current implementation for the default chlorophyll algorithm (chlor_a) employs the standard OC3\/OC4 (OCx) band ratio algorithm merged with the color index (CI) of Hu et al. This means that the environment red-shifts the absorption spectra of chlorophyll in plant cells or, in other words, chlorophyll pigments are bluer than we think. There are various types of chlorophyll structures, but plants contain chlorophyll a and b. increases inline with increasing awareness for. Chlorophyll gives plants their green color because it does not absorb the green wavelengths of white light. Chlorophyll E140 natural green color pigment. In others, different metal \u2026 When I first started looking for an answer I was surprised to discover that chlorophyll is classified as an effective coloring agent.. Yep, it even has a registered E number E141.. This work by Mapping Ignorance is licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0, \u00a9 2021 Mapping Ignorance The change in colour from green to yellow during ripening of apples depends essentially upon disappearance of chlorophyll. To know how much this microenvironment modulates the colour we need to know first which is the true colour of chlorophylls. CHLOROPHYLL D, A GREEN PIGMENT OF RED ALGAE. E. aureum and F. benjamina), cut into pieces measuring approximately 2 cm x 2 cm Photosynthetic chlorophylls are not alone in the leaf cells, they are usually in a protein pocket. This possibility was confirmed using first-principle calculations: the results showed that the tag has minimum influence on the excitation energy and therefore on the wavelengths the chlorophyll molecules absorb; actually less than 10 nm. C 35 H 30 O 5 N 4 Mg . And this is very important if we are ever going to understand how photosynthesis work and use this knowledge to build truly efficient photovoltaic devices. Chlorophyll pigments, Hunterlab colour indices and pH were determined before and after high temperature\u2010short time (H.T.S.T.) They publish their results in Angewandte Chemie International Edition. The green colour of leaves results from a d-d transition of the central atom of metal complexes known as chlorophylls. Fig. Chlorophyll can be further be divided in two main groups: chlorophyll A and chlorophyll B. The most important part of chlorophyll is to absorb and transfer the light to the dedicated reaction centers in each photosystem. What is the explanation of the energy gap law in radiationless transitions? A question came to my mind as to whether there is any d-d kind of transitions in plants as they also show characteristics colours. Show more Chlorophyll a and b, which are identical except for the part indicated in the red box, are responsible for the green color of leaves. Chlorophyll d . Chlorophyll d is greenish yellow and is found in red algae. Purified chlorophyll is used as a food colour with E-number E140, the ... Chlorophyll c2 . Making statements based on opinion; back them up with references or personal experience. Its color is green, and it primarily absorbs blue light. ). Summary. processing and during storage for 18 months at 20\u00b0C, 2.8\u00b0C, and \u221223.3\u00b0C. Degradation of chlorophyll in broccoli juice occurred at temperatures exceeding 60 \u00b0C. Chlorophyll is located in a plant\u2019s chloroplasts, which are tiny structures in a plant\u2019s cells. The other forms of chlorophyll are found in different taxa as accessory photosynthetic pigments. However, standard algorithms have been parameterized using global data sets that severely underrepresent the AO. But, as you can see in figure 1, they also absorb light with other wavelengths with less intensity. Structure Chemically chlorophyll is a mixture of several highly complex molecules, which consist of a ring structure (the porphyrin structure) with a central magnesium ion, and a long hydrophobic side chain (Fig 2. Some of these more delicate colours (from molecules such as carotene and quercetin) are revealed when the chlorophyll molecule decays in the autumn, and the woodlands turn red, orange, and golden brown. Synthetic copper complex of chlorophyll (E140), a natural green colour, which is present in all plants and algae. If you boil a leaf in water, this magnesium ion gets replaced by a hydrogen ion (a proton), and the color changes from bright green to the dull color of overcooked broccoli. Why? But chlorophyll absorbs so strongly that it can mask other less intense colours. Carotenoids, on the other hand, reflect yellow, orange and red \u2013 the colour of leaves during autumn. Refer to the preceding graph in arriving at your answers. chlorophyll (kl\u00f4r`\u0259f\u012dl'), green pigment that gives most plants their color and enables them to carry on the process of photosynthesis photosynthesis, process in which green plants, algae, and cyanobacteria utilize the energy of sunlight to manufacture carbohydrates from carbon dioxide and water in the presence of chlorophyll. bcmwl-kernel-source broken on kernel: 5.8.0-34-generic. 1. site design \/ logo \u00a9 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Chemical analysis revealed that degradation of chlorophyll a and b to pheophytin a and b, respectively, followed first-order kinetics and that chlorophyll a was more heat sensitive than chlorophyll b. Due to the green colour of chlorophyll, it has many uses as dyes and pigments. 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The instability of chlorophyll in harvested and processed plant products has promoted extensive research into methods for its preservation as well as to methods for its isolation, analysis, and concentration from cheaper sources, with a view to its use for natural color reinforcement. Most leaves are various shades of green. In the case of chlorophyll a the maximum absorption in the red region is at 642 nm and in the blue one at 372 nm; for chorophyll b the values are 626 nm and 392 nm respectively. E141 is commercially extracted from nettles, grass and alfalfa. Each pigment has (d) a unique absorbance spectrum. Because of these absorbencies the light it reflects and transmits appears green. What causes dough made from coconut flour to not stick together? 1 - The uv\/visible adsorption spectrum for chlorophyll. Required fields are marked *. The highest peaks represent colors that chlorophyll absorbs the most. If it is not the absorption of the central atom in our complex, the ligand must be responsible. Structurally, chlorophyll d is similar to chlorophyll b but differs from chlorophyll a in the presence of a formyl group in ring A of the structure. Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and students in the field of chemistry. This means that the environment red-shifts the spectra of chlorophylls or, in other words, chlorophylls are bluer than we think. The stronger ($\\epsilon \\approx 100000\\, \\mathrm{cm}^{-1}\\cdot \\mathrm{M}^{-1}$) band in the 400 nm (blue) region is known as the Soret band, the weaker $\\epsilon \\approx 30000\\, \\mathrm{cm}^{-1}\\cdot \\mathrm{M}^{-1}$) in the 660 nm (red) region is known as the Q band. Green leaves contain two types of chlorophyll, Chl a and Chl b, both have $\\ce{Mg}$ as the central atom and show two distinct bands in the UV-VIS absorption spectrum. Why? Underwater prison for cyborg\/enhanced prisoners? Courses. ABSTRACT: Chlorophyll is the most widely distributed natural pigment and occurs in the leaves and other parts of almost all plants. Due to chemical de-esterification of chlorophyll, phaeophytins are formed. \u2663 Chlorophyll b (C 55 H 70 O 6 N 4 Mg) \u2663 Chlorophyll c1 (C 35 H 30 O 5 N 4 Mg) \u2663 Chlorophyll c2 (C 35 H 2 5 N 4 Mg) \u2663 Chlorophyll d (C 54 H 70 O 6 N 4 Mg) This was some information related to the green pigment in plants. Chlorophyll d is greenish yellow and is found in red algae. 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. The molecular formula of chlorophyll d \u2026 Chlorophyll is vital for photosynthesis, which allows plants to absorb energy from light. Chlorophyll absorbs mostly in the blue and, to a lesser extent, red portions of the electromagnetic spectrum, hence its intense green color. Chlorophyll A will not be able to function to the fullest without chlorophyll B. Chlorophyll in plants is a pigment molecule that imparts a green colour to the leaf and stems by absorbing a red and blue spectrum of light. This would mean that the cations are far enough from the porphyrin ring that confers its optical properties to chlorophylls. Now, a group of researchers, including Angel Rubio from the Max Planck Institute for the Structure and Dynamics of Matter and DIPC, have developed a method1 to measure the true colour of chlorophyll in the absence of perturbations from its surroundings. Dog likes walks, but is terrified of walk preparation, What do this numbers on my guitar music sheet mean, Seeking a study claiming that a successful coup d\u2019etat only requires a small percentage of the population. Students can solve NCERT Class 10 Science Life Process Multiple Choice Questions with Answers to know their preparation level. Author links open overlay panel Winston M. Manning Harold H. Strain. So if you eat green chips and the ingredient lists E141, know that it was colored using chlorophyll. What authority does the Vice President have to mobilize the National Guard? How chlorophylls and other pigments absorb light. The intense green color of chlorophyll is due to its strong absorbencies in the red and blue regions of the electromagnetic spectrum, and because of these absorbencies the light it reflects and transmits appears green. Chlorophyll is a pigment that gives plants their green color. When you drink really well sourced Chlorophyll it\u2019s like breathing fresh air straight into our bloodstream. So when you say go green, you know what lies beneath the green mystery! When I first started looking for an answer I was surprised to discover that chlorophyll is classified as an effective coloring agent.. Yep, it even has a registered E number E141.. \u03b2-carotene is responsible for the orange color in carrots. How true is this observation concerning battle? Leaf samples (e.g. Unraveling the Intrinsic Color of Chlorophyll, Angewandte Chemie, 127 (7) 2198-2201. Chlorophyll C has a blue-green color and is mainly found in brown algae. processing and during storage for 18 months at 20\u00b0C, 2.8\u00b0C, and \u221223.3\u00b0C. Plants that use photosynthesis to make their own food are called autotrophs. How to stop writing from deteriorating mid-writing? Chlorophyll pigments are green because they reflect green light. Furthermore, as you can see in figures 2 and 3, chlorophyll a and b only differ in a substituent of the porphyrin ring, for chlorophyll a it is a -CH3 and a -CHO for chlorophyll b, but it is sufficient to alter the spectrum of the molecule. These different types of chlorophyll are the same basic molecule with very slight differences in their chemical structures. Below is a graph showing the percent of light energy reflected for the absorption spectrum for chlorophyll. Phytoplankton, the microscopic floating plants that form the basis of the entire marine food web, contain chlorophyll, which is why high phytoplankton concentrations can make water look green. As you can see on the photo, we failed to put chlorophyll back into plants. It is present in cyanobacteria which use energy captured from sunlight for photosynthesis. Free PDF Download of CBSE Class 10 Science Chapter 6 Life Process Multiple Choice Questions with Answers. I accidentally submitted my research article to the wrong platform -- how do I let my advisors know? If you boil a leaf in water, this magnesium ion gets replaced by a hydrogen ion (a proton), and the color changes from bright green to the dull color of overcooked broccoli. 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\section{Introduction }
Two of the oldest problems in classical mechanics involve the study of (1) celestial bodies interacting with each other gravitationally and (2) pendulums. These two types of problems may be derived from simple and foundational laws of nature, making them introductory examples in math and physics curricula.
In the case of celestial dynamics, all one requires is Newton's second law, $F = ma$, and the law of universal gravitation to produce the equations of motion. These laws produce simple and surprisingly accurate models of how bodies interact in our solar system, which can be used to design fuel efficient satellite and shuttle transport~\cite{NBody1,RossBook,Surfing,MarsdenAMS,JJM1,Caillau} and provide evidence for the existence of extreme trans-Neptunian objects~\cite{PlanetNine,PlanetNine2,PlanetNine3}.
Similar physical laws can be used to describe the motion of pendulums, which are bodies connected by rods swinging under the influence of gravity. Studies of a single pendulum, consisting of a single mass suspended by a rod, go back at least to Galileo, and their periodic motion has long been used in time keeping, and even to estimate the gravitational constant~\cite{SinglePendulum}.
As depicted in Figure~\ref{fig:SaddleComparison}, one can add another mass and rod, suspended from the mass of the first pendulum, to produce the {\em double pendulum}. The study of the double pendulum goes back at least to the work of Bernoulli~\cite{Bernoulli}, and now provides a prototypical example of a chaotic dynamical system~\cite{DPChaos}. Despite the many superficial differences between the motion of celestial bodies and pendulums, this manuscript will demonstrate that the phase space of the double pendulum bears a striking resemblance to that of the three-body problem in astrophysics, thus uniting the study of these centuries old problems.
Thus, the double pendulum provides a simple experimental analog~\cite{kaheman2022experimental} with which to explore various learning and control algorithms for efficient space mission design.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{Figures/SaddleComparison_Revised.pdf}
\caption{Despite representing dynamics on vastly different scales, the planar restricted 3-body problem (PCR3BP) and the double pendulum bear a striking resemblance. The Lagrange points $L_1$ and $L_2$ in the PCR3BP are analogous to the saddle Down-Up and Up-Down equilibria since they all have one-dimensional stable and unstable directions and a two-dimensional center direction.}
\label{fig:SaddleComparison}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth]{Figures/HillsRevised.pdf}
\caption{Hill's region comparison of PCR3BP and double pendulum. The energy values $E_i$ are meant to denote the transitions in which the Hill's region is extended to include another critical point of the energy surface.}
\label{fig:HillsResionComparison}
\end{figure}
\subsection{Multi-Body Problems}
The study of celestial bodies in the presence of Newtonian gravity traditionally begins by examining only two bodies. This constitutes the so-called Kepler problem and it successfully describes the (approximate) elliptical motion of planets in the solar system about the sun, a satellite orbiting a planet, and binary stars orbiting each other. From a mathematical perspective, the Kepler problem can be reduced to a planar ordinary differential equation (ODE) for which all solutions lie along periodic orbits~\cite{Arnold}, thus giving simple and predictable motion.
Adding even one more body to the system significantly complicates the dynamics, giving way to chaos~\cite{TBPreview}. This was first articulated by Poincar\'e in the late nineteenth century.
In an effort to understand the motion of three celestial bodies -- the three-body problem -- he pioneered many of the methods that led to the development of modern chaos theory.
Due to the complication of analyzing the three-body problem in general, a number of simplifications of the problem have been studied~\cite{TBPreview}.
A prominent simplified model is the restricted three-body problem (R3BP), which consists of two massive bodies that generate a gravitational field and a third relatively massless body that moves in this field~\cite{Koon,RossBook}.
In the planar circular restricted three-body problem (PCR3BP), the two massive bodies are assumed to form their own Kepler problem, maintaining a constant distance from each other and so generating a circular orbit, while the motion of the third body is confined to the orbital plane of the massive bodies. The result is a first-order four-dimensional ODE representing the planar position and velocity of the massless body, whose equations of motion are left to the appendix. Specific applications often take the massive bodies to be either the Earth and Moon~\cite{Caillau,NBody1} or the Sun and Jupiter~\cite{Koon,RossBook}, while the third body may represent a satellite, a shuttle, or a comet~\cite{TBPreview}. In Figure~\ref{fig:SaddleComparison} we present an illustration of the motion of a satellite governed by the PCR3BP in physical space, along with its potential energy landscape.
Figure~\ref{fig:SaddleComparison} shows the five steady-state equilibrium solutions of the PCR3BP, which are referred to as the Lagrange points $L_1$ through $L_5$. As the energy of the system increases, more of the potential energy surface, referred to as the {\em Hill's region}, becomes accessible to the third body as illustrated in Figure~\ref{fig:HillsResionComparison}.
The Lagrange points $L_1$ and $L_2$ are saddle points, and for a continuum of energies slightly above the energy of these points, there exist unstable periodic orbits (UPOs)~\cite{Moser,Wiggins}, known as Lyapunov orbits, that resemble halos about $L_1$ and $L_2$.
These Lyapunov orbits have two-dimensional stable and unstable manifolds, which we refer to throughout as {\em tubes} since they are diffeomorphic to a circle crossed with the real line.
Moreover, these invariant manifold tubes have codimension-1 in the ambient phase space, acting as separatrices that define an interior and exterior region.
It has been proven that these tubes can connect to form homoclinic orbits to these UPOs~\cite{Hom3BP,McGehee,Conley}, resulting in trajectories of the system that can travel significant distances in phase space before returning to where they started. Further numerical studies~\cite{Koon} have demonstrated the presence of heteroclinic connections between Lyapunov orbits. These homoclinic and heteroclinic orbits were used to construct itineraries through phase space that allow one to jump from following the homoclinics to the heteroclinics and back.
Thus, it is possible to construct trajectories that allow the exploration of phase space with no additional energy expenditure. Comparing observations of the orbit of the comet {\em Oterma} with the paths of the homoclinic and heteroclinic orbits in a Sun-Jupiter system reveals that its trajectory is guided by these global tube structures. Similarly, the trajectory for the {\em Genesis Discovery Mission} transited from the $L_1$ to the $L_2$ Lagrange point through nearly heteroclinic motion in the Earth-Moon system~\cite{LoGenesis,KoonGenesis}.
Japan's {\em Hiten} lunar mission used this transport network for a low energy transfer~\cite{Hiten}, as did the ISEE-3 spacecraft~\cite{ISEE3}.
Therefore, in the case of the PCR3BP we find that these invariant manifold tubes associated with the UPOs are crucial to not only understanding the dynamics of the phase space, but also for optimally designing itineraries for exploratory missions around planet and/or moon systems~\cite{RossL1}.
The discovery of the tube structure in the PCR3BP marks a major landmark in our understanding of celestial dynamics and has prompted the study of more complex multi-body systems~\cite{JJM2,JJM3,NBody1}, again aimed at optimal space mission design~\cite{NBody1,RossBook,Surfing,MarsdenAMS,JJM1,MasdemontReview}. Due to the abundance of these tubes in multi-body systems, they have been termed the {\em interplanetary transport network}~\cite{Interplanetary}. These tubes provide gravitationally determined pathways that require little energy to follow and can be used to design itineraries that explore major bodies of our solar system. Further, recent studies have employed computer-assisted proofs to obtain the existence of transverse homoclinic and heteroclinic connections which form the backbone of this network~\cite{CompAst1,CompAst2,CompAst3,CompAst4}.
\subsection{The Double Pendulum}
Much like multi-body systems, the base case of a single pendulum is deceptively simple compared to more general pendulum models. The single pendulum gives rise to purely periodic motion, and it resembles the Kepler problem at low energies with trajectories given by ellipsis in phase space. Both the Kepler problem and the single pendulum have integrable dynamics, meaning that their trajectories are fully described by the one-dimensional level sets of their Hamiltonian functions~\cite{Integrable}.
Furthermore, according to the Poincar\'e--Bendixson theorem, neither system can be chaotic because of their two-dimensional phase spaces.
The Kepler problem is to the single pendulum as the three-body problem is to the double pendulum. That is, like the three-body problem, the double pendulum has become a prototypical example of a chaotic system. In this work, we will develop this analogy, showing that the phase space dynamics of the double pendulum bears a striking resemblance to that of the PCR3BP. The double pendulum has two saddle steady-state solutions given by one arm hanging down with the other balanced in the upright position, referred to as the ``Down-Up" and ``Up-Down" equilibria.
There exist a neighborhood about these points where the dynamics are topologically conjugate to those of the Lagrange points $L_1$ and $L_2$ in the PCR3BP. We illustrate this correspondence in Figure~\ref{fig:SaddleComparison} with a comparison of the potential energy landscapes of the PCR3BP and the double pendulum, a visualization of the Down-Up and Up-Down equilibria, and an illustration of the dynamics near the saddle equilibria. Like the PCR3BP, increasing the energy of the double pendulum opens up more of the potential energy landscape, increasing the size of its associated ``Hill's region'', as illustrated in Figure~\ref{fig:HillsResionComparison}. Using similar techniques to those developed for the PCR3BP~\cite{Koon}, we demonstrate numerically the existence of homoclinic and heteroclinic orbits between the UPOs at energies slightly above those of the Down-Up and Up-Down steady-states. In particular, we demonstrate that the phase space of the double pendulum is similarly organized by global tube structures that enable macroscopic transport in the system with no additional energy expenditure.
Our work herein establishes the tube structure of the double pendulum, making it a low-stakes and relatively low-cost testbed to explore saddle mediated transport, with direct relevance to the PCR3BP~\cite{Koon}. For a fraction of the cost of building a satellite or space shuttle, a number of researchers have built double pendulums~\cite{spong1995swing,fantoni2000energy,driver2004design,timmermann2011discrete,hesse2018reinforcement,DPChaos,christini1996experimental,myers2020low,rubi2002swing,rafat2009dynamics,freidovich2008periodic,kaheman2019learning}, including the authors of this manuscript~\cite{kaheman2022experimental}.
Thus, the theoretical work put forth here can and will be tested on these physical realizations of the system in a follow-up investigation.
Our work shows that it is possible to design an itinerary of complex acrobatic motions of the pendulum arms by combining a sequence of homoclinic and heteroclinic connections; homoclinic orbits correspond to full rotations of the unstable pendulum arm, while heteroclinic orbits connect the Up-Down and Down-Up orbits.
These orbits provide a surprisingly tight analogy with orbits in the PCR3BP.
\subsection{Contributions}
In this manuscript we present a detailed study of the double pendulum while drawing comparisons to previous work on the PCR3BP. In particular, we follow a similar numerical procedure to~\cite{RossBook,Koon,Barcelona} to determine the global tube structure emanating from the Down-Up and Up-Down saddle points of the double pendulum. We demonstrate that there exist homoclinic and heteroclinic connections that allow for macroscopic transport in both physical space and phase space. We then use these numerical findings to formulate a general set of hypotheses that give way to our major theoretical contribution of this manuscript. Specifically, we show that the connecting trajectories found numerically can be used to prove the existence of long homoclinic, heteroclinic, and periodic trajectories of the double pendulum that can spend arbitrarily long times near each saddle point before continuing on and transiting to the neighborhood of another saddle point. This results in a collection of trajectories that can be used to fully explore the phase space.
Our main result is sufficiently general to also apply to existing work on the PCR3BP, and so it therefore comes as a generalization of the itineraries result proven in~\cite{Koon}. Much like the work on the PCR3BP, we show that there is not only one trajectory that follows a given itinerary, but infinitely many. However, our proofs are constructive in that they explicitly demonstrate that the long trajectories are built up by shadowing the `base' homoclinic and heteroclinic orbits found to exist by the numerical methods herein. This comes from the fact that our proofs are similar to the work in~\cite{BramLDS,BramIsola}, whereby we move to local Poincar\'e sections near the UPOs and use the existence of heteroclinic connections to transit between these local sections. The result is a general description of saddle mediated transport in two degree of freedom Hamiltonian systems that goes beyond the existing theory.
Outside of multi-body systems and the double pendulum, there are a number of other systems for which transport over vast regions of phase space can be attributed to the tubes of stable and unstable manifolds of UPOs near an index-1 saddle. A notable example is that of chemical reactions where tubes are shown to mediate the reaction and can be used to compute reaction rates and scattering phenomenon~\cite{Chemical,Chemical2,Chemical3}. Saddle mediated transport is also particularly useful in describing transitions in systems with two or more potential wells. For example, the works~\cite{Ship1,Ship2} provide a nonlinear model for ship motion with three potential wells: one for the ship being upright in the middle and wells on either side representing capsizing. Here the tubes associated to the index-1 saddles separating the potential wells explain how one transitions from the upright well into either of the capsizing wells~\cite{Naik}. Similar ideas have been used to explain snap-through buckling of a shallow arch~\cite{Arch} and the motion of a ball rolling on a saddle surface~\cite{Ball}. Our work is relevant to these studies since our analytical results are general enough to apply to all of these situations, thus extending the itineraries of the PCR3BP to a variety of Hamiltonian systems.
This paper is organized as follows. We begin in Section~\ref{sec:Background} with an introduction to index-1 saddles, of which $L_1$ and $L_2$ in the PCR3BP and the Down-Up and Up-Down equilibria in the double pendulum are examples. We describe the linear and nonlinear dynamics near these saddles and also review the importance of the global tube structure associated with them. Much of this is illustrated with recreations of known results for the PCR3BP. In Section~\ref{sec:EOM} we present a derivation of the equations of motion for the {\em experimental double pendulum}. This model differs from the simple theoretical double pendulum as it includes the mass of the pendulum arms. A comparison between these two pendulum models is presented in Figure~\ref{Fig:DP_Illustrate} below and our motivation for studying the experimental double pendulum comes from our physical model~\cite{kaheman2022experimental} and the goal of implementing our theory in practice in subsequent work. Section~\ref{sec:TubesDP} presents our numerical findings on the global tube structure of the double pendulum. These results include a description of the symmetries of our model in~\S\ref{sec:Symmetry}, linear analysis of the various steady-states which identifies the Down-Up and Up-Down equilibria as index-1 saddles in~\S\ref{sec:linearanalysis}, and a discussion of the numerically identified homoclinic and heteroclinic orbits in~\S\ref{sec:L1Hom}-\ref{sec:Heteroclinic}. Our main theoretical result is presented in Section~\ref{sec:Theory}, in particular in~\S\ref{sec:Theorem}. We then discuss the implication of our theoretical results on the double pendulum in~\S\ref{subsec:DPapplication} and discuss how these results extend the known results of the PCR3BP in~\S\ref{subsec:TBPapplication}. The proof of our main result is left to Section~\ref{sec:Proofs} and then we conclude in Section~\ref{sec:Conclusion} with a discussion of our findings and possible extensions.
\section{Index-1 Saddles}\label{sec:Background}
Throughout this section we will seek to demonstrate the importance of saddle steady-states to the global dynamics of a Hamiltonian system. In particular, the global dynamics of both the PCR3BP~\cite{RossBook,Koon} and the double pendulum are mediated by {\em index-1} saddle points.
Index-1 saddle points are fixed points characterized by a single unstable saddle degree of freedom, with corresponding positive and negative eigenvalue, and the remainder of the degrees of freedom being linearly stable center dynamics, each with corresponding pairs of purely imaginary, complex conjugate eigenvalues.
The terminology index-1 refers to the fact these orbits have one-dimensional stable and unstable manifolds associated to them.
Index-1 saddles are sometimes referred to as \emph{rank-1} saddles.
Throughout we denote the Hamiltonian function by $\mathcal{H}$.
Much of the theory presented for index-1 saddles generalizes to arbitrary numbers of degrees of freedom; however, for simplicity, we consider Hamiltonian systems with two degrees of freedom in this work.
The resulting dynamical system has a four-dimensional phase space. Our interest will be in understanding the local dynamics near these index-1 saddles.
\subsection{Linear Dynamics Near an Index-1 Saddle}
Let us begin by assuming that we have an equilibrium of the resulting 4-dimensional ODE and that linearizing about this equilibrium results in four eigenvalues $\pm \lambda$ and $\pm \mathrm{i}\omega$, for some real $\lambda,\omega > 0$. Through an invertible transformation we may translate this equilibrium to the origin and bring the resulting linearized dynamics into the Jordan normal form
\begin{equation}\label{Linear}
\begin{bmatrix}
\dot p_1 \\ \dot q_1 \\ \dot p_2 \\ \dot q_2
\end{bmatrix} = \begin{bmatrix}
-\lambda & 0 & 0 & 0 \\
0 & \lambda & 0 & 0 \\
0 & 0 & 0 & -\omega \\
0 & 0 & \omega & 0 \\
\end{bmatrix}\cdot\begin{bmatrix}
p_1 \\ q_1 \\ p_2 \\ q_2
\end{bmatrix}.
\end{equation}
The linearized system~\eqref{Linear} is itself a conservative system with quadratic Hamiltonian given by
\begin{equation}\label{LinearHam}
\mathcal{H}_\mathrm{loc}(p_1,q_1,p_2,q_2) = \lambda p_1q_1 + \frac{\omega}{2}(p_2^2 + q_2^2).
\end{equation}
Solutions of~\eqref{Linear} are given by
\begin{equation}\label{LinearSol}
p_1(t) = p_1^0\mathrm{e}^{-\lambda t}, \quad q_1(t) = q_1^0\mathrm{e}^{\lambda t}, \quad p_2(t) + \mathrm{i}q_2(t) = (p_2^0 + \mathrm{i}q_2^0)\mathrm{e}^{-\mathrm{i}\omega t},
\end{equation}
for any constants $p_1^0,q_1^0,p_2^0,q_2^0\in\mathbb{R}$. Importantly,~\eqref{Linear} has three constants of motion
\begin{equation}
C_1 = p_1q_1, \quad C_2 = p_2^2 + q_2^2, \quad C_3 = \mathcal{H}_\mathrm{loc}
\end{equation}
which we will use to understand the flow near the saddle point.
From the above, we can see that projecting the dynamics into the $(p_1,q_1)$-plane results in a standard planar saddle system. Similarly, projecting into the $(p_2,q_2)$-plane results in a linear center consisting of harmonic oscillator motion. See Figure~\ref{fig:SaddleComparison} for an illustration. For some fixed $h \in \mathbb{R}^+$ we can then consider the dynamics of~\eqref{Linear} inside the level set $\mathcal{H}_\mathrm{loc} = h$. First, rearranging~\eqref{LinearHam} gives
\begin{equation}
p_2^2 + q_2^2 = \frac{2}{\omega}(h - \lambda p_1q_2),
\end{equation}
so that in the bounding scenario that $C_1 = h/\lambda$ we have that the dynamics in the $(p_2,q_2)$-plane collapse to the point $(0,0)$ and trajectories lie along the hyperbolas $p_1q_1 = h/\lambda$. For $0 < C_1 < h/\lambda$ the dynamics in the $(p_2,q_2)$-plane lie along a circle and the trajectories are diffeomorphic to a circle crossed with a line.
When $p_1 = q_1 = 0$ the resulting dynamics are confined to the circles
\begin{equation}
S^1_h := \bigg\{(p_2,q_2):\ p_2^2 + q_2^2 = \frac{2h}{\omega}\bigg\}
\end{equation}
for each $h > 0$. These invariant circles constitute periodic orbits of the system, referred to as {\em Lyapunov orbits} in the context of celestial dynamics governed by many body equations. The above circles are only one of three nontrivial examples of trajectories with the constant of motion $C_1 = 0$. Simply restricting to the invariant manifold $q_1 = 0$ again gives $C_1 = 0$, but now the dynamics of $p_1$ need not be trivial. Using the solution~\eqref{LinearSol} we find that $p_1(t) \to 0$ as $t \to \infty$ while the dynamics in the $(p_2,q_2)$-plane are still restricted to $S^1_h$. Therefore, the set $q_1 = 0$ is the stable manifold of the periodic orbit $S^1_h$, denoted $W^s(S^1_h)$. Similarly, taking $p_1 = 0$ and using the fact that $q_1(t) \to 0$ as $t \to -\infty$, we have that the set $p_1 = 0$ is the unstable manifold of the period orbit $S^1_h$, denoted $W^u(S^1_h)$. Both the stable and unstable manifolds of $S^1_h$ are homeomorphic to circles crossed with lines, which we henceforth refer to as tubes. Furthermore, the presence of both stable and unstable manifolds for the periodic orbits implies they are all saddles like the equilibrium they surround, and therefore are unstable.
\subsection{Nonlinear Dynamics Near an Index-1 Saddle}
In the previous subsection we described the linearized dynamics near an index-1 saddle point in a Hamiltonian system. We now extend this discussion to the nonlinear dynamics near the index-1 saddle. The same invertible change of variable that brings the linear dynamics to the system~\eqref{Linear} transforms the Hamiltonian of the nonlinear system into the form
\begin{equation}\label{LinearHam2}
\mathcal{H}(p_1,q_1,p_2,q_2) = \mathcal{H}_\mathrm{loc}(p_1,q_1,p_2,q_2) + h.o.t.,
\end{equation}
where {\em h.o.t.} denotes the `higher order terms' which in this case are at least cubic and $\mathcal{H}_\mathrm{loc}$ is given in~\eqref{LinearHam}. Similarly, the dynamics in a neighborhood of the equilibrium (shifted to the origin in the $p_1,q_1,p_2,q_2$ variables) takes the form
\begin{equation}\label{Nonlinear}
\begin{bmatrix}
\dot p_1 \\ \dot q_1 \\ \dot p_2 \\ \dot q_2
\end{bmatrix} = \begin{bmatrix}
-\lambda & 0 & 0 & 0 \\
0 & \lambda & 0 & 0 \\
0 & 0 & 0 & -\omega \\
0 & 0 & \omega & 0 \\
\end{bmatrix}\cdot\begin{bmatrix}
p_1 \\ q_1 \\ p_2 \\ q_2
\end{bmatrix} + h.o.t.
\end{equation}
where now the higher order terms are at least quadratic. It is apparent from the fact that the linearization includes a complex conjugate pair of purely imaginary eigenvalues that one cannot apply results such as the Hartman--Grobman theorem to conclude that the dynamics~\eqref{Nonlinear} are locally conjugate to the dynamics of~\eqref{Linear}. Fortunately, the works~\cite{Moser,Wiggins} exploit the Hamiltonian structure of the underlying dynamical system to provide the necessary results that demonstrate this. Precisely, there exists a small ball around $(p_1,q_1,p_2,q_2) = (0,0,0,0)$ for which the dynamics of~\eqref{Nonlinear} are equivalent to the dynamics of~\eqref{Linear}.
The consequence of the above is that in a neighborhood of any index-1 saddle point comprises a continuum of UPOs. Furthermore, these UPOs have two-dimensional stable and unstable manifolds resembling tubes that eventually leave the region of validity for the conjugacy with the linear system~\eqref{Linear}. Using~\eqref{LinearHam2}, it follows that these UPOs can only be guaranteed to exist for $\mathcal{H} = h$ with $0 < h \ll 1$ since the higher order terms become more important as $h$ increases. By abuse of notation we will refer to these UPOs again as $S^1_h$ since the UPOs of~\eqref{Linear} and~\eqref{Nonlinear} lie in one-to-one correspondence when $h$ is small. Moreover, the local components of the stable and unstable manifolds, $W^s_\mathrm{loc}(S^1_h)$ and $W^u_\mathrm{loc}(S^1_h)$, respectively, are approximated by the sets $\{q_1 = 0\}$ and $\{p_1 = 0\}$, respectively. These points become important in the following sections for numerically following the global stable and unstable manifolds of different UPOs.
\subsection{Global Invariant Manifold Tube Dynamics}
The previous subsections focus on local aspects of the Hamiltonian phase space in neighborhoods of index-1 saddles. Per the previous discussion, the saddles and their nearby UPOs all have stable and unstable manifolds that extend beyond these neighborhoods. Importantly, intersections of these stable and unstable manifolds give the existence of global structures such as homoclinic and heteroclinic orbits in the phase space. These homoclinic and heteroclinic orbits organize the global geometry of phase space and enable transport over vast regions of space with no extra energy expenditure.
Note that with a 2 degree of freedom Hamiltonian system, we are restricted to the (generically) three-dimensional level sets of the Hamiltonian function. In one of these three-dimensional level sets, the index-1 saddles have one-dimensional stable and unstable manifolds, meaning that the existence of a homoclinic orbit is unlikely since these stable and unstable manifolds would have to coincide. Such a phenomenon is not robust with respect to generic parameter manipulation, and so this situation should not be expected without additional assumptions such as symmetries. Alternatively, for a connected continuum of values of the Hamiltonian slightly above the value at the index-1 saddle we have the UPOs, each of which have two-dimensional stable and unstable manifolds inside the three-dimensional level sets. In this case an intersection of these manifolds is robust and expected to be one-dimensional, leading to the existence of a homoclinic orbit. The robustness of these intersections implies that the existence of a homoclinic orbit for one value of the Hamiltonian is expected to imply the existence for values of the Hamiltonian in an open neighborhood of said value. Heteroclinic connections between index-1 saddles are similarly robust and are made up of intersections of one UPO's stable manifold intersecting along a generically one-dimensional curve with another UPO's unstable manifold. We again expect the existence of such a heteroclinic connection to exist for a range of values of the Hamiltonian, if it exists at all.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\textwidth]{Figures/TBP_HomoclinicOrbit.pdf}
\caption{Numerically confirming the existence of homoclinic trajectories to the $L_1$ and $L_2$ Lyapunov orbits in the PCR3BP is done by finding intersections between their stable and unstable manifolds in the Poincar\'e section $y = 0$ with $x < 0$. The manifolds are shown in (a) and (d), while their intersections with the Poincar\'e section are illustrated in (b) and (e). Intersections of the stable and unstable manifolds of each Lyapunov orbit results in a homoclinic orbit, some of which are depicted in (c) and (f). In (c), both symmetric (solid) and asymmetric (dashed) $L_1$ homoclinic orbits are shown. }
\label{fig:TBP_HomoclinicOrbit}
\end{figure}
The existence of homoclinic orbits to the UPOs near index-1 saddles has notably been proven for the Lyapunov orbits around the $L_1$ Lagrange point in the PCR3BP~\cite{Conley,McGehee}. These results were improved by~\cite{LMS} to show that under appropriate conditions one could conclude that these intersections were transverse, meaning that they are indeed robust. More in line with our work herein, the work~\cite{Koon} followed the ideas put forth by~\cite{Barcelona} to explore the system numerically and detect the existence of these homoclinic orbits to not only Lyapunov orbits of the $L_1$ Lagrange point, but also the $L_2$ point. In a process that will be described in more detail below, one simulates the unstable manifold of the Lyapunov orbit forward in time and the stable manifold backward in time until they meet at an appropriately defined Poinacar\'e section. The intersection of these manifolds with the Poincar\'e section will be diffeomorphic to a circle and the intersection of these circles represent the desired homoclinic trajectories. This process and the found homoclinic orbits are illustrated in Figure~\ref{fig:TBP_HomoclinicOrbit}. We refer to the appendix for details on the PCR3BP including the equations of motion, the location of the Lagrange points $L_1$ and $L_2$, and the parameter values used in our computations.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{Figures/TBP_HeteroclinicOrbit_Revised.pdf}
\caption{Heteroclinic orbits between the $L_1$ and $L_2$ Lyapunov orbits of the PCR3BP are found numerically in the same way as the homoclinic orbits in Figure~\ref{fig:TBP_HomoclinicOrbit}, as shown in (a) and (b). A heteroclinic orbit from an $L_1$ Lyapunov orbit to an $L_2$ Lyapunov orbit is shown in (c) and is identified from manifold intersections in the Poincar\'e section in (d). Similarly, (e) and (f) show manifold intersections in the Poincar\'e section resulting in heteroclinic orbits from an $L_2$ to an $L_1$ Lyapunov orbit.}
\label{fig:TBP_HeteroclinicOrbit}
\end{figure}
Beyond just the homoclinic orbits,~\cite{Koon} also numerically demonstrates the existence of heteroclinic orbits between the $L_1$ and $L_2$ Lyapunov orbits. These are obtained in the same way as the homoclinic orbits in that stable and unstable manifolds are simulated backward and forward in time, respectively, to meet an appropriate Poincar\'e section to find intersections of these manifolds. This process and the results are summarized in Figure~\ref{fig:TBP_HeteroclinicOrbit}. The effect this has is that for values of the Hamiltonian slightly above that of the index-1 saddles there exist level sets containing Lyapunov orbits to both $L_1$ and $L_2$ which each have homoclinic orbits and are connected by a heteroclinic orbit. These orbits are used in~\cite{Koon} to construct itineraries through phase space that allow one to jump from following the homoclinics to the heteroclinics and back, thus providing trajectories that allow the lightweight object to fully explore the phase space with no additional energy expenditure. In what follows we will perform a similar investigation for the double pendulum by identifying homoclinic and heteroclinic trajectories to index-1 saddles. We further prove a general result that uses these orbits that we identify numerically to construct arbitrarily long trajectories in phase space that are analogous to the itineraries identified.
\section{Equations of Motion for the Experimental Double Pendulum} \label{sec:EOM}
In this section we provide the relevant equations of motion for the physical double pendulum. We note that many mathematical investigations of the double pendulum use the {\em point-mass double pendulum}, which ignores the mass of the pendulum rod. To make our investigation applicable to future laboratory experiments, we will consider the {\em experimental double pendulum} which includes the mass of the pendulum arms. In Figure~\ref{Fig:DP_Illustrate} we provide a comparison between the theoretical and experimental double pendulum models and we direct the reader to the appendix for the equations of motion for the theoretical double pendulum.
To begin, let us denote the mass of the first pendulum arm by $m_1$ and the mass of second pendulum arm by $m_2$.
Then, let $\ell_1$ and $\ell_2$ be the lengths of the first and second arms, respectively. Next, define $a_1$ and $a_2$ as the positions of the centers of mass of the first and second pendulum arm, respectively. Let $J_1$ and $J_2$ denote the moment of inertia of the first and second pendulum arm. The constant of gravitational acceleration is $g$. Finally, we take $\theta_1$ and $\theta_2$ to be the time-dependent rotational angles of pendulum arms, as illustrated in Figure~\ref{Fig:DP_Illustrate}.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{Figures/IllustrateDP_Revised.pdf}
\caption{This figure illustrates the difference between the point-mass model of the double pendulum (left) and the more realistic experimental double pendulum (right) studied herein. The experimental model includes both the inertial ($J_i$) and mass $(m_i)$ of the pendulum arms.}
\label{Fig:DP_Illustrate}
\end{figure}
The kinetic energy of the experimental double pendulum is given by
\begin{equation}\label{eq:edp_kinetic_energy}
T=\frac{1}{2}\left(m_1a_1\dot{\theta}_1^2 +m_2\left(\ell_1^2\dot{\theta}_1^2+a_2^2\dot{\theta}_2^2+2\ell_1a_2\dot{\theta}_1\dot{\theta}_2\cos(\theta_1-\theta_2)\right)+J_1\dot{\theta}_1^2+J_2\dot{\theta}_2^2\right)
\end{equation}
and the potential energy is
\begin{equation}\label{eq:edp_potential_energy}
\begin{split}
V&=-g\left(m_1a_1\cos(\theta_1)+m_2(\ell_1\cos(\theta_1)+a_2\cos(\theta_2))\right).
\end{split}
\end{equation}
Using~\eqref{eq:edp_kinetic_energy} and~\eqref{eq:edp_potential_energy}, the Lagrangian of the experimental double pendulum can be calculated as
\begin{equation}\label{eq:edp_lagrangian}
L=T-V,
\end{equation}
and the Hamiltonian of the system is given by
\begin{equation}\label{eq:edp_hamiltonian}
\mathcal{H} =T+V.
\end{equation}
Throughout our investigation in this manuscript we will neglect the effect of friction, although this is important for real experimental control.
Using the Lagrangian~\eqref{eq:edp_lagrangian}, the equations of motion of the double pendulum are given by
\begin{subequations}
\label{eq:edp_eom}
\begin{align}
\frac{\partial}{\partial t}\frac{\partial L}{\partial \dot{\theta}_1}-\frac{\partial L}{\partial \theta_1}&=0, \label{eq:edp_eom1}\\
\frac{\partial}{\partial t}\frac{\partial L}{\partial \dot{\theta}_2}-\frac{\partial L}{\partial \theta_2}&=0. \label{eq:edp_eom2}
\end{align}
\end{subequations}
Equation~\eqref{eq:edp_eom1} yields
\begin{equation}\label{eq:edp_eom_cal1}
\begin{split}
0 &= J_1\ddot{\theta}_1 + \ell_1^2\ddot{\theta}_1m_2
+ a_1^2\ddot{\theta}_1m_1 +\ell_1gm_2\sin(\theta_1) + a_1gm_1\sin(\theta_1) \\ &\qquad + \ell_1a_2\dot{\theta}_2^2m_2\sin(\theta_1 - \theta_2)
+ \ell_1a_2\ddot{\theta}_2m_2\cos(\theta_1 - \theta_2),\\
\end{split}
\end{equation}
while equation~\eqref{eq:edp_eom2} yields
\begin{align}\label{eq:edp_eom_cal2}
0 &= J_2\ddot{\theta}_2 - \ell_1a_2\dot{\theta}_1^2m_2\sin(\theta_1 - \theta_2) + \ell_1a_2\ddot{\theta}_1m_2\cos(\theta_1 - \theta_2).
\end{align}
From~\eqref{eq:edp_eom_cal1} and~\eqref{eq:edp_eom_cal2} we can solve for $\ddot\theta_1$ and $\ddot\theta_2$, giving
\begin{subequations}\label{eq:edp_ddtheta}
\begin{align}
\ddot{\theta}_1&=\frac{A_{10}\sin(\theta_1)+A_{11}\sin(\theta_1-\theta_2)+A_{12}\sin(\theta_1-2\theta_2)+A_{22}\sin(2\theta_1-2\theta_2)}{D(\theta_1 - \theta_2)}\\
\ddot{\theta}_2&=\frac{B_{01}\sin(\theta_2)+B_{11}\sin(\theta_1-\theta_2)+B_{21}\sin(2\theta_1-\theta_2)+B_{22}\sin(2\theta_1-2\theta_2)}{D(\theta_1 - \theta_2)}
\end{align}
\end{subequations}
where
\begin{equation}
\begin{split}
A_{10} &= - \ell_1ga_2^2m_2^2 - 2a_1gm_1a_2^2m_2 - 2J_2l_1gm_2 - 2J_2a_1gm_1 \\
A_{11} &= - 2\ell_1a_2^3\dot{\theta}_2^2m_2^2 - 2J_2\ell_1a_2\dot{\theta}_2^2m_2 \\
A_{12} &= -\ell_1a_2^2gm_2^2 \\
A_{22} &= -\ell_1^2a_2^2\dot{\theta}_1^2m_2^2 \\
B_{01} &= -a_2m_2(gm_2\ell_1^2 - gm_1\ell_1a_1 + 2gm_1a_1^2 + 2J_1g)\\
B_{11} &= a_2m_2(2m_2\ell_1^3\dot{\theta}_1^2 + 2m_1\ell_1a_1^2\dot{\theta}_1^2 + 2J_1\ell_1\dot{\theta}_1^2)\\
B_{21} &= a_2m_2(gm_2\ell_1^2 + a_1gm_1\ell_1)\\
B_{22} &= 2m_2(gm_2\ell_1^2 + a_1gm_1\ell_1)\\
D(\theta_1 - \theta_2) &=-\ell_1^2a_2^2m_2^2cos(\theta_1-\theta_2)^2 + \ell_1^2a_2^2m_2^2 + J_2\ell_1^2m_2 + m_1a_1^2a_2^2m_2 \\&\qquad + J_2m_1a_1^2 + J_1a_2^2m_2 + J_1J_2.
\end{split}
\end{equation}
Here we use the convention that $A_{ij},B_{ij}$ are the coefficients of the energy preserved Hamiltonian system for the $\ddot\theta_1$ and $\ddot\theta_2$ equations, respectively, while the subscripts denote the coefficients on the $\theta_1$ and $\theta_2$ terms inside the sine function they are multiplied against. The notation $D(\theta_1 - \theta_2)$ represents the denominator of both equations. Writing~\eqref{eq:edp_ddtheta} as a first-order ODE gives
\begin{subequations}\label{eq:edp_ode}
\begin{align}
\dot{\theta}_1&=\omega_1,\\
\dot{\theta}_2&=\omega_2,\\
\dot{\omega}_1&=\frac{A_{10}\sin(\theta_1)+A_{11}\sin(\theta_1-\theta_2)+A_{12}\sin(\theta_1-2\theta_2)+A_{22}\sin(2\theta_1-2\theta_2)}{D(\theta_1 - \theta_2)},\\
\dot{\omega}_2&=\frac{B_{01}\sin(\theta_2)+B_{11}\sin(\theta_1-\theta_2)+B_{21}\sin(2\theta_1-\theta_2)+B_{22}\sin(2\theta_1-2\theta_2)}{D(\theta_1 - \theta_2)}.
\end{align}
\end{subequations}
System~\eqref{eq:edp_ode} therefore represents the full equations of motion for the double pendulum on a cart that we investigate for the remainder of this manuscript.
Throughout this manuscript our numerical experiments will use the parameter values
\begin{equation}\label{ParamVals}
\begin{split}
&m_1 = 0.0938\mathrm{kg}, \quad m_2 = 0.1376\mathrm{kg}, \quad a_1 = 0.1086\mathrm{m}, \quad a_2 = 0.1168\mathrm{m}, \\
&\ell_1= 0.1727\mathrm{m}, \quad \quad J_1 = 10^{-4}\mathrm{kgm}^2, \quad J_2 = 10^{-4}\mathrm{kgm}^2, \quad g = 9.808\mathrm{m}/\mathrm{s}^2.
\end{split}
\end{equation}
All of the above values were obtained via parameter estimation from the physical model constructed in~\cite{kaheman2022experimental} and chosen to demonstrate that these results are physically realizable. We have also found that other similar choices of parameter values lead to nearly identical results and so we do not expect that the restriction to these parameter values is a limitation of this work. Finally, notice that the length of the second arm, $\ell_2$, is notably absent from the equations of motion~\eqref{eq:edp_ode} and so its value is not necessary for our investigation of the experimental double pendulum.
\section{Tube Dynamics of the Double Pendulum}\label{sec:TubesDP}
In this section we will consider the tube structure of transport between index-1 saddles of system~\eqref{eq:edp_ode}. Over the following subsections we seek to identify its index-1 saddles and their global tube dynamics. This will help us to identify the macroscopic transport mechanisms of the experimental double pendulum modelled by equations~\eqref{eq:edp_ode} in a similar manner to what has been done for a variety of other Hamiltonian systems, including the PCR3BP.
We begin in~\S\ref{sec:Symmetry} by describing the reversible symmetry of~\eqref{eq:edp_ode}. This reversible symmetry will help us to classify the trajectories between neighborhoods of index-1 saddles as either symmetric or asymmetric, with those that are asymmetric coming in pairs related by applying the symmetry transformation. We then proceed to identify the index-1 saddles of~\eqref{eq:edp_ode} in~\S\ref{sec:linearanalysis} by analyzing the linearization of the system about each of its steady-states. We identify two index-1 saddles given by one of the pendulum arms standing straight up (unstable) and the other hanging down (stable). We provide a numerical justification for the existence of homoclinic orbits to UPOs in the neighborhoods of these saddles in~\S\ref{sec:L1Hom} and~\S\ref{sec:L2Hom}. We then conclude this section in~\S\ref{sec:Heteroclinic} by numerically demonstrating the existence of heteroclinic orbits between UPOs in the neighborhoods of each of the index-1 saddles.
\subsection{Reversible Symmetry}\label{sec:Symmetry}
Beyond the Hamiltonian structure of~\eqref{eq:edp_ode}, we note that the system is also reversible in the sense that if $(\theta_1(t),\theta_2(t),\omega_1(t),\omega_2(t))$ is a solution, then so is $(\theta_1(-t),\theta_2(-t),-\omega_1(-t),-\omega_2(-t))$. Precisely, we define the reverser
\begin{equation}\label{Reverser}
\mathcal{R} = \begin{bmatrix}
1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\
\end{bmatrix}
\end{equation}
and say that $\Theta(t) := (\theta_1(t),\theta_2(t),\omega_1(t),\omega_2(t))$ and $\mathcal{R}\Theta(-t)$ are solutions. Using this property, we will refer to a solution as {\em symmetric} if $\Theta(t) = \mathcal{R}\Theta(-t)$ for all $t \in \mathbb{R}$. Importantly, if $\Theta_*$ is a symmetric equilibrium solution of~\eqref{eq:edp_ode}, i.e. $\mathcal{R}\Theta_* = \Theta_*$, then the associated stable manifold $W^s(\Theta_*)$ and unstable manifold $W^u(\Theta_*)$ are such that $W^s(\Theta_*) = \mathcal{R}W^u(\Theta_*)$. Therefore, homoclinic orbits of symmetric equilibria are either symmetric or asymmetric, for which the latter case implies that they come in pairs, related by applying $\mathcal{R}$ and reversing the flow of time. The preceding discussion also holds for the symmetric UPOs near a symmetric index-1 saddle which are the focus of much of our discussion going forward. We will therefore emphasize the symmetric and asymmetric homoclinic orbits that can be observed in our numerical findings.
\subsection{Linear Analysis}\label{sec:linearanalysis}
From the equations of motion~\eqref{eq:edp_ode} we can see that equilibria come in the form $(\theta_1,\theta_2,\omega_1,\omega_2) = (\pi k_1,\pi k_2,0,0)$ for every pair of integers $(k_1,k_2) \in \mathbb{Z}^2$. All of these equilibria are symmetric with respect to the action of the reverser $\mathcal{R}$, defined in~\eqref{Reverser}. Although we have obtained a lattice of equilibria, periodicity of the components $\theta_{1,2}$ implies that we may restrict our analysis to the four distinct steady-states
\begin{subequations}\label{DPsteady}
\begin{align}
\mathrm{Down-Down}: \quad &(\theta_1,\theta_2,\omega_1,\omega_2) = (0,0,0,0) \\
\mathrm{Down-Up}: \quad &(\theta_1,\theta_2,\omega_1,\omega_2) = (0,\pi,0,0) \\
\mathrm{Up-Down}: \quad &(\theta_1,\theta_2,\omega_1,\omega_2) = (\pi,0,0,0) \\
\mathrm{Up-Up}: \quad &(\theta_1,\theta_2,\omega_1,\omega_2) = (\pi,\pi,0,0).
\end{align}
\end{subequations}
The naming of the states represents the vertical position of the arms of the double pendulum. Precisely, Down-Down has both arms hanging straight down, Down-Up has the first arm with parameters $(\ell_1,m_1)$ hanging down and the other standing straight up, Up-Down has the first arm standing straight up and the other hanging down, and the Up-Up state has both arms standing straight up. We note that we have listed the steady-states in~\eqref{DPsteady} in order of ascending energy values, according to the Hamiltonian~\eqref{eq:edp_hamiltonian}. Our results that follow apply for all choices of parameters, not just those provided in~\eqref{ParamVals}.
Linearizing~\eqref{eq:edp_ode} about an equilibrium $(\theta_1,\theta_2,\omega_1,\omega_2) = (\pi k_1,\pi k_2,0,0)$ with $k_1,k_2\in\mathbb{Z}$ results in a Jacobian matrix of the form
\begin{equation}\label{DPLinMat}
\begin{bmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\frac{\sigma_2\sigma_3}{\sigma_1^2}-\frac{\sigma_4}{2\sigma_1} & \frac{\sigma_5}{\sigma_1}-\frac{\sigma_2\sigma_3}{\sigma_1^2} & 0 & 0 \\
\frac{\sigma_6}{\sigma_1}-\frac{\sigma_7\sigma_8}{\sigma_1^2} & -\frac{\sigma_{9}}{\sigma_1}-\frac{\sigma_7\sigma_8}{\sigma_1^2} & 0 & 0
\end{bmatrix}
\end{equation}
with
\begin{equation}
\begin{split}
\sigma_1&=- \ell_1^2a_2^2m_2^2\cos(k_1\pi - k_2\pi)^2 + \ell_1^2a_2^2m_2^2 \\
& \quad + J_2\ell_1^2m_2 + m_1a_1^2a_2^2m_2 + J_2m_1a_1^2 + J_1a_2^2m_2 + J_1J_2, \\
\sigma_2 &=\ell_1^2 a_2^2 g m_2^2 \cos(k_1\pi - k_2\pi) \sin(k_1\pi - k_2\pi), \\
\sigma_3 &=(\ell_1 a_2^2 m_2^2 \sin(k_1\pi) + 2 J_2 \ell_1 m_2 \sin(k_1\pi) \\
&\quad + \ell_1 a_2^2 m_2^2 \sin(k_1\pi - 2 k_2\pi) + 2 J_2 a_1 m_1 \sin(k_1\pi) + 2 a_1 a_2^2 m_1 m_2 \sin(k_1\pi)), \\
\sigma_4 &=(\ell_1 a_2^2 m_2^2 \cos(k_1\pi) + 2 J_2 \ell_1 m_2 \cos(k_1\pi)\\
&\quad + \ell_1 a_2^2 m_2^2 \cos(k_1\pi - 2 k_2\pi) + 2 J_2 a_1 m_1 \cos(k_1\pi) + 2 a_1 a_2^2 m_1 m_2 \cos(k_1\pi)),\\
\sigma_5 &=\ell_1 a_2^2 g m_2^2 \cos(k_1\pi - 2 k_2\pi), \\
\sigma_6 &=\ell_1 a_2 g m_2 \cos(2 k_1\pi - k_2\pi) (\ell_1 m_2 + a_1 m_1), \\
\sigma_7 &=2 \ell_1^2 a_2^3 g m_2^3 \cos(k_1\pi - k_2\pi) \sin(k_1\pi - k_2\pi), \\
\sigma_8 &=(J_1 \sin(k_2\pi) + \ell_1^2 m_2 \sin(k_2\pi) + a_1^2 m_1 \sin(k_2\pi)\\
&\quad - \ell_1^2 m_2 \cos(k_1\pi - k_2\pi) \sin(k_1\pi) - \ell_1 a_1 m_1 \cos(k_1\pi - k_2\pi) \sin(k_1\pi)), \\
\sigma_9 &=a_2 g m_2 (J_1 \cos(k_2\pi) + \ell_1^2 m_2 \cos(k_2\pi) + a_1^2 m_1 \cos(k_2\pi) \\
&\quad - \ell_1^2 m_2 \sin(k_1\pi - k_2\pi) \sin(k_1\pi) - \ell_1 a_1 m_1 \sin(k_1\pi - k_2\pi) \sin(k_1\pi)).
\end{split}
\end{equation}
Due to the block structure of~\eqref{DPLinMat}, the square of its eigenvalues are equal to those of the lower left $2\times 2$ matrix
\begin{equation}\label{MiniMatEDP}
\begin{bmatrix}
\frac{\sigma_2\sigma_3}{\sigma_1^2}-\frac{\sigma_4}{2\sigma_1} & \frac{\sigma_5}{\sigma_1}-\frac{\sigma_2\sigma_3}{\sigma_1^2} \\
\frac{\sigma_6}{\sigma_1}-\frac{\sigma_7\sigma_8}{\sigma_1^2} & -\frac{\sigma_{9}}{\sigma_1}-\frac{\sigma_7\sigma_8}{\sigma_1^2}
\end{bmatrix}.
\end{equation}
We will now proceed to classify the stability of the four equilibria in~\eqref{DPsteady} using the matrix~\eqref{MiniMatEDP}.\\
\noindent{\bf Down-Down.} The Down-Down state is obtained by setting $k_1 = k_2 = 0$. In this case~\eqref{MiniMatEDP} becomes
\begin{equation}
\begin{bmatrix}
-\frac{g(m_2a_2^2+J_2)(\ell_1m_2+a_1m_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} & \frac{\ell_1a_2^2gm_2^2}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} \\
\frac{\ell_1a_2gm_2(\ell_1m_2+a_1m_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} & -\frac{a_2gm_2(m_2\ell_1^2+m_1a_1^2+J_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2}
\end{bmatrix}.
\end{equation}
Since the trace of the above matrix is negative and the determinant is positive for all the physical parameters $m_1$,$m_2$,$a_1$,$a_2$,$J_1$,$J_2$,$g$,$\ell_1$,$\ell_2$, it follows that both eigenvalues are distinct and strictly negative. Therefore, the linearization~\eqref{DPLinMat} about the Down-Down state has two pairs of purely complex eigenvalues, making it a linear center. \\
\noindent{\bf Down-Up.} The Down-Up equilibrium is obtained by setting $k_1 = 0$ and $k_2 = 1$. The matrix~\eqref{MiniMatEDP} becomes
\begin{equation}
\begin{bmatrix}
-\frac{g(m_2a_2^2+J_2)(\ell_1m_2+a_1m_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} & \frac{\ell_1a_2^2gm_2^2}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} \\
-\frac{\ell_1a_2gm_2(\ell_1m_2+a_1m_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} & \frac{a_2gm_2(m_2\ell_1^2+m_1a_1^2+J_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2}
\end{bmatrix}.
\end{equation}
which has a negative determinant for all relevant parameter values. Therefore, the above matrix has one positive and one negative real eigenvalue along with a pair of purely imaginary eigenvalues. Hence, the Down-Up equilibrium is an index-1 saddle. \\
\noindent{\bf Up-Down.} We obtain the Up-Down equilibrium by setting $k_1 = 1$ and $k_2 = 0$. The matrix~\eqref{MiniMatEDP} is then given by
\begin{equation}
\begin{bmatrix}
\frac{g(m_2a_2^2+J_2)(\ell_1m_2+a_1m_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} & -\frac{\ell_1a_2^2gm_2^2}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} \\
\frac{\ell_1a_2gm_2(\ell_1m_2+a_1m_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} & -\frac{a_2gm_2(m_2\ell_1^2+m_1a_1^2+J_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2}
\end{bmatrix}.
\end{equation}
and following as in the Down-Up equilibrium, we find that the Up-Down equilibrium is also an index-1 saddle. \\
\noindent{\bf Up-Up.} The Up-Up state has $k_1 = k_2 = 1$, resulting in~\eqref{MiniMatEDP} taking the form
\begin{equation}
\begin{bmatrix}
\frac{g(m_2a_2^2+J_2)(\ell_1m_2+a_1m_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} & -\frac{\ell_1a_2^2gm_2^2}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} \\
-\frac{\ell_1a_2gm_2(\ell_1m_2+a_1m_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2} & \frac{a_2gm_2(m_2\ell_1^2+m_1a_1^2+J_1)}{J_2m_2\ell_1^2+m_1m_2a_1^2a_2^2+J_2m_1a_1^2+J_1m_2a_2^2+J_1J_2}
\end{bmatrix}.
\end{equation}
The above matrix is the result of negating all of the entries of the Down-Down analysis. It follows that the linearization~\eqref{DPLinMat} about the Up-Up state has two positive eigenvalues and two negative eigenvalues. From the nomenclature above, the Up-Up state is an {\em index-2 saddle}. \\
\begin{figure}
\centering
\includegraphics[height=0.55\textwidth]{Figures/EPeriodicOrbitComparison_Revised.pdf}
\caption{Near the index-1 saddles of the double pendulum there are infinitely many UPOs. These UPOs are projected into (a) the $(\theta_1,\dot\theta_1)$-plane and the $(\theta_2,\dot\theta_2)$-plane with a cartoon of their physical motion in the double pendulum shown in (b).}
\label{fig:DP_UPO}
\end{figure}
From the above we find that the Down-Up and Up-Down equilibria are index-1 saddles. Importantly, our work in Section~\ref{sec:Background} gives that in a neighborhood of these index-1 saddles there exist infinitely many UPOs. These UPOs are analogous to the Lyapunov orbits of the $L_1$ and $L_2$ Lagrange points of the PCR3BP. However, in the case of the double pendulum these UPOs may be easier to visualize. Precisely, the `Down' arm of these index-1 saddles should be understood as being stable, while the `Up' arm is unstable, both due to the force of gravity. Therefore, the UPOs near these Down-Up and Up-Down saddles have the `Up' arm undergoing a slight wobble, while the `Down' arm exhibits little variation since it is stable. This is demonstrated in Figure~\ref{fig:DP_UPO} where one can see a collection of these UPOs projected into both the $(\theta_1,\dot\theta_1)$ and $(\theta_2,\dot\theta_2)$ planes. In the following subsections we will work to numerically identify homoclinic and heteroclinic trajectories associated to these UPOs near the index-1 saddles of the double pendulum.
\subsection{Tube Dynamics Near the Down-Up State}\label{sec:L1Hom}
We begin by describing our process of identifying homoclinic trajectories to the UPOs near the Down-Up equilibrium of the double pendulum. We emphasize that although we refer to these orbits throughout as 'homoclinic' they are actually heteroclinic orbits in phase space. This potential source of confusion comes from the periodicity of the $\theta_1$ and $\theta_2$ components in~\eqref{eq:edp_ode}. Precisely, a trajectory that connections the UPOs near the Down-Up equilibrium $(\theta_1,\theta_2,\omega_1,\omega_2) = (0,\pi,0,0)$ to those near another Down-Up equilibrium $(\theta_1,\theta_2,\omega_1,\omega_2) = (0,\pi\pm 2\pi,0,0)$ comes as a heteroclinic orbit in phase space if one does not quotient by the periodicity of the $\theta_1$ and $\theta_2$ components, but in physical space such a connection appears to return to where it started while having the second arm undergoing a full clockwise or counterclockwise rotation. As the motion of the double pendulum is best understood in physical space, we will hereby refer to trajectories that connect UPOs near any of the Down-Up equilibriums, $(0,\pi \pm 2\pi k,0,0)$, $k \in \mathbb{Z}$, as homoclinic. Finally, we comment that our numerical investigations have revealed that true homoclinic trajectories that asymptotically approach one of the UPOs near $(0,\pi,0,0)$ do not exist, meaning that only 'physical' homoclinic trajectories described above can exist.
We can numerically obtain the UPOs near the Down-Up equilibrium by searching for symmetric periodic solutions with the value of the Hamiltonian~\eqref{eq:edp_hamiltonian} slightly above that of the Down-Up equilibrium. The search for these periodic solutions can be posed as a root-finding problem constrained by the fact that we must remain within a Hamiltonian level set everywhere on the trajectory. This process is identical to that used in~\cite{RossBook} to identify homoclinic trajectories in the PCR3BP and to implement this process numerically we use the Julia package $Zygote$~\cite{liao2019differentiable}. A demonstration of these techniques is included in the repository associated with this manuscript. With these UPOs, as displayed in Figure~\ref{fig:DP_UPO}, we then use their Floquet spectrum to locally approximate the stable and unstable directions associated to the UPO. By simulating these approximate stable and unstable directions backward and forward in time, respectively, according to the ODE~\eqref{eq:edp_ode} we obtain accurate representations of the stable and unstable manifolds, i.e. the tubes, associated to each UPO.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Figures/EDP_TubeStructure_L1.pdf}
\caption{(a) The clockwise rotating portion of the tube structure of the stable and unstable manifolds associated to a UPO near the Down-Up equilibrium with Hamiltonian value $\mathcal{H} = 0.2$. In (b) we show the counterclockwise portion of the tube structure, related to (a) by applying the reverser~\eqref{Reverser}. In (c) and (d) we present the intersection of the tubes with the Poincar\'e section $\theta_2 = 2\pi k$, $k \in \mathbb{Z}$. Intersections of the stable and unstable manifolds in the Poincar\'e section represent 'physical' homoclinic trajectories that connect the UPOs near the Down-Up equilibrium.}
\label{fig:EDP_TubeStructure_L1}
\end{figure}
To identify homoclinic trajectories we flow the unstable manifold forward in time until $\theta_2 = 2\pi$. Similarly, we flow the stable manifold backward in time until $\theta_2 = 0$. Notice that upon quotienting for the periodicity in $\theta_2$ both the stable and unstable manifolds have been simulated to the same region in physical space, given by the second arm hanging straight down. We use the hyperplane $\theta_2 = 2\pi k$ with $k \in \mathbb{Z}$ as our Poincar\'e section to identify homoclinic trajectories. We further note that the reversible symmetry of~\eqref{eq:edp_ode} guarantees that we could equivalently flow the unstable manifold forward in the direction that decreases the value of $\theta_2$ until it reaches $\theta_2 = 0$ to meet the Poincar\'e section, and similarly for the stable manifold meeting $\theta_2 = 2\pi$. The tube structure of these stable and unstable manifolds are given in Figure~\ref{fig:EDP_TubeStructure_L1}(a) and their reversible counterparts are shown in Figure~\ref{fig:EDP_TubeStructure_L1}(b) for $\mathcal{H} = 0.2$, above the value of the Down-Up equilibrium at $\mathcal{H} = -0.1754$. Having the unstable manifold meet $\theta_2 = 2\pi$ represents the second arm moving clockwise, while having the unstable manifold meet $\theta_2 = 0$ represents the second arm moving counterclockwise.
We can identify `physical' homoclinic trajectories near the Down-Up equilibrium of the double pendulum using the Poincar\'e section $\theta_2 = 2\pi k$, $k \in \mathbb{Z}$. Indeed, Figure~\ref{fig:EDP_TubeStructure_L1}(c) and (d) plot the $(\theta_1,\dot\theta_1)$ plane corresponding to the tube structures presented in (a) and (b), respectively, of the same figure. Intersections of the stable and unstable manifolds in the Poincar\'e section represent the homoclinic orbits that are the focus of this section. Such intersections are homoclinic trajectories since $\theta_1,\theta_2$, and $\dot\theta_1$ are equal at these points, and it can be verified numerically that the Hamiltonian structure of system~\eqref{eq:edp_ode} confines the value of $\dot\theta_2$ to be the same here as well. As one can see, for this value of the Hamiltonian there are numerous intersections. Those with $\theta_1 = 0$ represent symmetric, or reversible, homoclinic orbits while those intersections with $\theta_1 \neq 0$ represent the asymmetric homoclinic orbits. As discussed in \S~\ref{sec:Symmetry}, the asymmetric orbits come in pairs, represented in the Poincar\'e section by the symmetry over the $\theta_1 = 0$ line. From our choice of the Poincar\'e section, all such homoclinic trajectories represent the double pendulum starting near the Down-Up equilibrium and having the second arm completing a full rotation before returning to a neighborhood of the Down-Up equilibrium.\footnote{To visualize the behavior of those homoclinic orbits presented in Fig.~\ref{fig:EDP_TubeStructure_L1} in physical space see \href{https://github.com/dynamicslab/Saddle-Mediated-Transport-of-Double-Pendulum}{https://github.com/dynamicslab/Saddle-Mediated-Transport-of-Double-Pendulum}}.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{Figures/EDP_PoincareCut_L1_VariesEng.pdf}
\caption{Increasing the energy of the double pendulum up from the energy of the Down-Up steady-state results in an increasing number of homoclinic trajectories of the UPOs near the Down-Up equilibrium. Energy values are (a) $\mathcal{H} = -0.147$, (b) $\mathcal{H} = -0.07$, (c) $\mathcal{H} = -0.034$, (d) $\mathcal{H} = 0.06$, (e) $\mathcal{H} = 0.102$, and (f) $\mathcal{H} = 0.157$. The Down-Up equilibrium has energy $\mathcal{H} = -0.1754$.}
\label{fig:EDP_PoincareCut_L1_VariesEng}
\end{figure}
Figure~\ref{fig:EDP_PoincareCut_L1_VariesEng} shows that increasing the Hamiltonian energy~\eqref{eq:edp_hamiltonian}, from that of the Down-Up equilibrium, results in a steadily increasing number of homoclinic trajectories. These homoclinic orbits arise via saddle-node and symmetry-breaking pitchfork bifurcations as the energy is increased, coming from further manifold intersections. For relatively large values of the energy, as in panels (e) and (f), flowing some parts of stable and unstable manifolds towards the Poincar\'e section takes a long time. The result is that some of our intersections of the stable and unstable manifolds with the Poincar\'e section do not present themselves as closed loops. We expect that simulating the tubes for a significantly longer time will complete their intersections with the Poincar\'e section, but this would require significant computational time and is not necessary for our exploration here.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{Figures/EDP_L1_ColorWheel_LowEng.pdf}
\caption{Trajectories that start inside of the unstable tube will remain inside of it for all forward time. This is demonstrated by flowing numerous points shown in (a) which lie inside both the stable and unstable tubes of a UPO near the Down-Up equilibrium with energy $\mathcal{H}=-0.06985$ forward until they intersect the Poincar\'e section again at (b) $\theta_2 = 4\pi$, (c) $\theta_2 = 6\pi$, and (d) $\theta_2 = 8\pi$. Flowing these same points backward in time produces results that are mirrored over $\theta_1 = 0$, likely resulting in a fractal set that remains inside both the stable and unstable tubes for all forward and backward time. The location of a symmetric periodic orbit that remains inside both tubes for all times is also plotted.}
\label{fig:EDP_L1_ColorWheel_LowEng}
\end{figure}
The fact that these stable and unstable tubes intersect means that the system exhibits macroscopic transport mechanisms that allow one to traverse large distances in phase space, eventually returning to the region where it started. Precisely, trajectories that are initialized inside of an unstable tube will remain inside it for all forward time, thus repeatedly coming back to the Poincar\'e section $\theta_2 = 2\pi k$ and intersecting within the closed curve created by the intersection with the unstable tube. This is demonstrated in Figure~\ref{fig:EDP_L1_ColorWheel_LowEng} where we simulate numerous trajectories with initial conditions from the Poincar\'e section that lie in the intersection of the stable and unstable tubes for energy $\mathcal{H}=-0.06985$. After each pass from the Poincar\'e section up to the UPO near the Down-Up equilibrium and back, one sees that these trajectories intersect the Poincar\'e section inside the closed curve created by the unstable tube, while spreading out over the interior of the tube. One can achieve a similar phenomenon by simulating these initial conditions backward in time, forcing them to remain inside of the stable tube forever. Visualizing this can be achieved by simply mirroring the forward iterates in Figure~\ref{fig:EDP_L1_ColorWheel_LowEng} over $\theta_1 = 0$. Thus, it appears that those trajectories that remain inside both tubes for all forward and backward time become a fractal set, which we suspect forms a horseshoe set~\cite{Smale}. In fact, our theoretical results below in Theorem~\ref{thm:Main} demonstrate that this is indeed the case in a neighborhood of each homoclinic orbit, while Figure~\ref{fig:EDP_L1_ColorWheel_LowEng} provides evidence for a more global horseshoe structure.
This apparent horseshoe structure means that there exist a plethora of bounded solutions that return infinitely often to a neighborhood of the UPO near the Down-Up equilibrium. As an example, we have identified a simple symmetric periodic orbit that in physical space has the second arm of the double pendulum completing a full $2\pi$ rotation over each period.
The intersection of this periodic orbit with the Poincar\'e section is plotted in Figure~\ref{fig:EDP_L1_ColorWheel_LowEng}. Interestingly, it lies almost perfectly between the location of the two symmetric homoclinic orbits that form the boundary of the tubes restricted to $\theta_1 = 0$. Although we do not have an explanation for this, we do not believe it to be a coincidence and expect it to be related to the precise horseshoe structure generated by the intersection of the interiors of the stable and unstable tubes.
\subsection{Tube Dynamics Near the Up-Down State}\label{sec:L2Hom}
The process of identifying homoclinic trajectories to the UPOs near the Up-Down equilibrium is similar to what was done in the previous subsection. Again, we were not able to identify homoclinic trajectories in phase space but have sought to identify the `physical' homoclinic trajectories that connect UPOs near the Up-Down equilibrium after accounting for the periodicity of the $\theta_1$ and $\theta_2$ variables. For the UPOs near the Up-Down equilibrium we find that these physical homoclinic trajectories have the first arm making a full clockwise or counterclockwise rotation before returning to where they started. In phase space this means that $\theta_1$ increases or decreases by exactly $2\pi$ over the course of the homoclinic trajectory.
\begin{figure}
\centering
\includegraphics[width=0.85\textwidth]{Figures/EDP_TubeStructure_L2.pdf}
\caption{Panel (a) presents the stable and unstable tubes of a UPO near the Up-Down steady-state in the variables $(\theta_1,\theta_2,\dot\theta_1)$ with energy $\mathcal{H} = 0.2$. In (b) and (c) we present the intersection of the tubes with the Poincar\'e section $\theta_1 = 2\pi k$, $k \in \mathbb{Z}$. Intersections of the stable and unstable manifolds in the Poincar\'e section represent 'physical' homoclinic trajectories that connect the UPOs near the Up-Down state.}
\label{fig:EDP_TubeStructure_L2}
\end{figure}
We can identify the UPOs near the Up-Down equilibrium and use their Floquet spectrum to follow the stable and unstable tubes backward and forward in time, respectively. We flow these stable and unstable tubes associated to these UPOs according to~\eqref{eq:edp_ode} until they meet the Poincar\'e section $\theta_1 = 2\pi k$, $k \in \mathbb{Z}$. In Figure~\ref{fig:EDP_TubeStructure_L2}(a) we present the stable and unstable manifolds of a UPO near the Up-Down equilibrium with energy $\mathcal{H} = 0.2$. The resulting intersections with the Poincar\'e section are further plotted in panels (b) and (c), for which the former represents clockwise rotations of the first arm, while the latter represents counterclockwise rotations. Again we are able to identify multiple symmetric and asymmetric homoclinic orbits that enable macroscopic transport throughout the phase space of the double pendulum\footnote{To visualize the behavior of those homoclinic orbits presented in Fig.~\ref{fig:EDP_TubeStructure_L2} in physical space see \href{https://github.com/dynamicslab/Saddle-Mediated-Transport-of-Double-Pendulum}{https://github.com/dynamicslab/Saddle-Mediated-Transport-of-Double-Pendulum}.}. As with the UPOs near the Down-Up equilibrium, we find that increasing the energy above that of the Up-Down equilibrium results in sequences of saddle-node and symmetry-breaking pitchfork bifurcations to more and more homoclinic orbits. We do not present a figure with these results for brevity.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Figures/EDP_L2_ColorWheel.pdf}
\caption{The apparent fractal structure generated by those trajectories that remain inside of the unstable tube of a UPO near the Up-Down equilibrium with energy $\mathcal{H} = 0.2$. As in Figure~\ref{fig:EDP_L1_ColorWheel_LowEng} we flow numerous initial conditions shown in (a) forward in time until they intersect the Poincar\'e section for a (b) first, (c) second, and (d) third time. The location of a symmetric periodic orbit that remains inside both tubes for all times is also plotted.}
\label{fig:EDP_L2_ColorWheel}
\end{figure}
As with the tubes associated to UPOs near the Down-Up equilibrium, we find a fractal/horseshoe structure of trajectories that remain inside the tubes associated to UPOs near the Up-Down equilibrium for all forward and backward time. This structure is illustrated in Figure~\ref{fig:EDP_L2_ColorWheel} for energy $\mathcal{H} = 0.2$, which is analogous to Figure~\ref{fig:EDP_L1_ColorWheel_LowEng} presented previously. We again exploit this complex structure to identify an exemplary symmetric periodic orbit that represents the first arm making a full $2\pi$ rotation in physical space. These symmetric orbits intersect the Poincar\'e section at $\theta_2 = 0$ and again we find this point of intersection lies almost perfectly in the center of the intersections of two symmetric homoclinic orbits.
\subsection{Heteroclinic Transitions} \label{sec:Heteroclinic}
The approach to identifying heteroclinic connections between neighborhoods of the Down-Up and Up-Down equilibria is similar to that outlined above for identifying homoclinic trajectories. We fix the energy and flow the unstable manifold of the a UPO near the Down-Up equilibrium forward in time until it reaches the Poincar\'e section $\theta_1 = \theta_2$ in phase space. We then flow the stable manifold of a UPO with the same energy near the Up-Down equilibrium backwards in time until it also reaches the Poincar\'e section $\theta_1 = \theta_2$. Intersections of these stable and unstable manifolds in the Poincar\'e section therefore represent the desired heteroclinic orbits that transport one through phase space between neighborhoods of index-1 saddles. The process we have just described allows one to obtain orbits from near the Down-Up equilibrium to near the Up-Down state, although applying the reverser~\eqref{Reverser} reverses the direction of these orbits and therefore gives orbits that originate near the Up-Down equilibrium and move to a neighborhood of the Down-Up equilibrium as well.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\textwidth]{Figures/EDP_HeteroclinicPoints_Revised.pdf}
\caption{Heteroclinic connections between UPOs in the neighborhoods of the Down-Up and Up-Down equilibria are found by inspecting where their tubes meet the Poincar\'e section $\theta_1 = \theta_2$. (a) and (c) show the tubes flowing towards the section in forward and backward time for $\mathcal{H}=0.2$. Panels (b) and (d) demonstrate the intersection of the tubes with the Poincar\'e section and panels (A) and (B) provide a zoom of the intersection of these tubes, representing heteroclinic trajectories.}
\label{fig:EDP_HeteroclinicPoints}
\end{figure}
Our results are illustrated in Figure~\ref{fig:EDP_HeteroclinicPoints}. Panel (a) demonstrates the unstable manifold of a UPO near the Down-Up equilibrium moving toward the Poincar\'e section, while panel (c) demonstrates the time reversed flow of the stable manifold of the same UPO moving backward in time toward the Poincar\'e section. Panels (b) and (d) show the intersection of the UPO tubes with the Poincar\'e section for panels (a) and (c), respectively. We further present a zoom-in of the Poincar\'e section near the intersections, illustrating two distinct heteroclinic connections. These heteroclinic orbits are plotted in $(\theta_1,\theta_2,\dot\theta_1)$ space in Figure~\ref{fig:EDP_HeteroclinicOrbits3D} with their asymptotic UPOs shown in solid black. Our numerical experiments have shown that such heteroclinic connections exist for $\mathcal{H} \geq 0.1754$. Like the homoclinic orbits of the previous subsections, the existence of these heteroclinic orbits demonstrates a transport mechanism that allows one to move between neighbourhoods of the index-1 saddles in phase space.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\textwidth]{Figures/EDP_HeteroclinicOrbits3D.pdf}
\caption{A visualization in $(\theta_1,\theta_2,\dot\theta_1)$ space of the two distinct heteroclinic orbits (red and blue) found as intersections in the Poincar\'e section $\theta_1 = \theta_2$ in Figure~\ref{fig:EDP_HeteroclinicPoints}. The black closed loops represent the asymptotic UPOs near each of the index-1 saddles.}
\label{fig:EDP_HeteroclinicOrbits3D}
\end{figure}
\section{Existence of Long Periodic and Connecting Orbits}\label{sec:Theory}
Here we seek to leverage the numerical findings of the previous section to establish the existence of periodic and homo-/heteroclinic orbits of the double pendulum that transit over vast regions of phase space. Our main result, Theorem~\ref{thm:Main}, in~\S\ref{sec:Theorem} provides this existence for completely general four dimensional Hamilton systems. In~\S\ref{subsec:DPapplication} we will apply our result to the double pendulum, thus extending the numerical results of the previous section. The generality of this result means that it is applicable to a wide variety of Hamiltonian systems with two degrees of freedom. In fact, it provides similar results to the itinerary construction for the PCR3BP in~\cite{Koon} and we briefly comment on how our results extend the known structure of phase space for the PCR3BP in~\S\ref{subsec:TBPapplication}. Further, our result is general enough to apply to similar studies where homoclinic and heteroclinic orbits near index-1 saddles are known to exist without needing to know the specific structure of the system. Therefore, these results may be applied to chemical reaction models~\cite{Chemical,Chemical2,Chemical3}, models for ship motion~\cite{Ship1,Ship2,Naik}, snap-through buckling of shallow arches~\cite{Arch}, and the motion of a ball rolling on a saddle surface~\cite{Ball}.
\subsection{Abstract Setting and Results}\label{sec:Theorem}
Let us consider a general ODE
\begin{equation}\label{ODE}
\dot{u} = F(u),
\end{equation}
with $u \in \mathbb{R}^4$ and a smooth function $F:\mathbb{R}^4 \to \mathbb{R}^4$. To mimic the setting of the double pendulum, we first assume the existence of a conserved quantity, corresponding to a Hamiltonian of the system~\eqref{ODE}.
\begin{hyp}\label{hyp:Ham}
There exists a smooth function $\mathcal{H}:\mathbb{R}^4\to\mathbb{R}$ such that $\langle F(u),\nabla \mathcal{H}(u) \rangle = 0$ for all $u \in \mathbb{R}^4$.
\end{hyp}
The existence of a conserved quantity allows us to reduce ourselves to its (generically) three-dimensional level sets. Next we assume that there exists a collection of UPOs that reside in the same level set of $\mathcal{H}$.
\begin{hyp}\label{hyp:Periodic}
There exists $p \geq 1$ such that~\eqref{ODE} has periodic orbits $\gamma_1(t), \gamma_2(t), \dots, \gamma_p(t)$ satisfying
\begin{equation}
\mathcal{H}(\gamma_i(t)) = \mathcal{H}(\gamma_j(t))
\end{equation}
for all $1 \leq i,j \leq p$ and all $t \in \mathbb{R}$. Moreover, each $\gamma_j$ is hyperbolic and so has precisely two Floquet multipliers at one and no others on the unit circle.
\end{hyp}
Note that each periodic orbits has exactly two Floquet multipliers at one under Hypothesis~\ref{hyp:Ham}. Indeed, one multiplier comes from the Floquet exponent at zero related to translations around the periodic orbit, while the other comes from the fact that the conserved quantity reduces the flow of system~\eqref{ODE} to its level sets. Furthermore, the conserved quantity forces the symmetry relationship that if $\lambda \in \mathbb{C}$ is a Floquet multiplier of some periodic orbit, then so is $\lambda^{-1},\bar \lambda,$ and $\bar \lambda^{-1}$. Therefore, we find that the two remaining Floquet multipliers that are not on the unit circle are real, taking the form $\lambda,\lambda^{-1} \in \mathbb{R}\setminus\{0\}$.
From the assumption that each $\gamma_i(t)$ is hyperbolic, each periodic orbit has a stable and unstable manifold associated to it. As was discussed in Section~\ref{sec:Background}, these stable and unstable manifolds are diffeomorphic to cylinders inside of the level set $\mathcal{H}(\gamma_i(t))$. We denote these stable and unstable manifolds for each periodic orbit $\gamma_i$ by $W^s(\gamma_i)$ and $W^u(\gamma_i)$, respectively. Then we define
\begin{equation}
\mathcal{W} := \bigcup_{1 \leq i,j\leq p} W^s(\gamma_i) \cap W^u(\gamma_j)
\end{equation}
to be the set of all homoclinic and heteroclinic trajectories of~\eqref{ODE}. Of course, for a given system it is nearly impossible to know the full structure of the set $\mathcal{W}$, but our main result shows that we can use a finite collection of elements in $\mathcal{W}$ to construct infinitely many more. The following hypothesis makes our assumptions precise.
\begin{hyp}\label{hyp:HomHet}
The set $\mathcal{W}$ is nonempty and the subset $\mathcal{W}_0 \subset \mathcal{W}$ is comprised of finitely many transverse homoclinic and heteroclinic orbits of~\eqref{ODE} belonging to $\mathcal{W}$.
\end{hyp}
The set $\mathcal{W}_0$ comprises the set of `base' homoclinic and heteroclinic orbits of~\eqref{ODE}. The assumption that these trajectories are transverse means that they lie along a transverse intersection of stable and unstable manifolds and is necessary to providing the results of this manuscript.
Using Hypotheses~\ref{hyp:Periodic} and \ref{hyp:HomHet} we will define a directed graph, written $\mathcal{G} = (\mathcal{V},\mathcal{E})$ as a collection of vertices $\mathcal{V}$ and edges $\mathcal{E}$. The vertices, $\mathcal{V}$, will be labelled as $\{1,\dots,p\}$ meant to represent each of the periodic orbits $\gamma_1(t),\dots,\gamma_p(t)$. The edges of the graph lie in one-to-one correspondence with the finitely many unique elements of $\mathcal{W}_0$. That is, the element $h(t) \in \mathcal{W}_0$ defines a directed edge on the graph $\mathcal{G}$ from vertex $i \in\mathcal{V}$ to $j\in\mathcal{V}$ if, and only if, $h(t) \in W^u(\gamma_i)$ and $h(t) \in W^s(\gamma_j)$ for all $t \in \mathbb{R}$. Informally, an edge connection from vertex $i$ to vertex $j$ exists if, and only if, there exists an orbit in $\mathcal{W}_0$ that goes asymptotically from $\gamma_i(t)$ to $\gamma_j(t)$.
We note that our definition of the graph not only contains directed edges, but also allows for the possibility of loops (an edge that goes from one vertex back to itself) and multiple edges between two vertices. The loops come from the potential presence of homoclinic orbits in $\mathcal{W}_0$, while we also allow for the presence of multiple homoclinic and heteroclinic orbits to and from the same pair of periodic orbits, thus giving multiple edges between the same two vertices. The reader is referred to Figure~\ref{fig:Graphs} below for examples of these graphs coming from our numerical results in Section~\ref{sec:TubesDP}. We will denote a walk of length $k \geq 1$ on the directed graph $\mathcal{G}$ by the tuple of edges $(E_1,E_2,\dots,E_k) \in \mathcal{E}^k$. Such a walk represents starting at vertex $V_1 \in \mathcal{V}$ and moving along edge $E_1$ to vertex $V_2$, then moving along edge $E_2$ to vertex $V_3$, and continuing until finally moving along edge $E_k$ to finish the walk on $V_{k+1}$. This leads to our main result whose proof is left to Section~\ref{sec:Proofs}.
\begin{theorem}\label{thm:Main}
Assume Hypotheses~\ref{hyp:Ham}-\ref{hyp:HomHet}. For any integer $k \geq 1$ and a walk $(E_1,E_2,\dots,E_k)$ on $\mathcal{G}$ moving through the sequence of vertices $(V_1,V_2,\dots,V_{k+1})$, there exists $M_k \geq 1$ such that the following is true:
\begin{enumerate}
\item For each set of integers $N_2,\dots,N_k \geq M_k$ there exists a trajectory of~\eqref{ODE} belonging to $W^u(\gamma_{V_1})\cap W^s(\gamma_{V_{k+1}})$.
\item If $V_{k+1} = V_1$, for each set of integers $N_1,N_2,\dots,N_{k-1},N_k \geq M_k$ there exists a periodic trajectory of~\eqref{ODE}.
\end{enumerate}
In both cases above the trajectory follows the walk on the graph, in that it leaves a neighborhood of $\gamma_{V_1}(t)$ following closely along the element of $\mathcal{W}_0$ corresponding to $E_1$, enters a neighborhood of $\gamma_{V_2}(t)$ and leaves, following closely to the element of $\mathcal{W}_0$ corresponding to $E_2$, repeating this process until finally entering a neighborhood of $\gamma_{V_{k+1}}(t)$ along the element of $\mathcal{W}_0$ corresponding to $E_k$.
\end{theorem}
In the statement of Theorem~\ref{thm:Main} the integers $N_i$ roughly correspond to how many times the trajectory wraps around the periodic orbit $\gamma_{V_j}(t)$ while in its neighborhood. Since there is no upper bound on the number of wraps, the above theorem gives infinitely many trajectories that follow the prescribed walk on the graph. Furthermore, the lack of upper bound on the integers $N_i$ means that the trajectories described in Theorem~\ref{thm:Main} can be arbitrarily long. In this way, a trajectory can reside near one of the periodic orbits $\gamma_i(t)$ for an arbitrarily long time before jumping to another one. The reason we only have $(k-1)$ integers $N_i$ in the first case of homoclinic and heteroclinic trajectories is that going backwards or forwards in $t$, the trajectory asymptotically enters the neighbourhoods of $\gamma_{V_1}(t)$ and $\gamma_{V_{k+1}}(t)$, respectively, not to ever leave it again. In the second point of a periodic trajectory we have $k$ integers $N_i$ since the trajectory initially wraps around $\gamma_{V_1}(t)$, but since $V_{k+1} = V_1$, it follows that once the trajectory re-enters the neighborhood $\gamma_{V_1}(t)$ along the edge $E_k$, it completes one full period of the periodic orbit.
Another useful application of Theorem~\ref{thm:Main} is that it can be applied iteratively. That is, one may use one of the homoclinic or heteroclinic connections from point (1) and add it into the set $\mathcal{W}_0$ to add another edge to the graph $\mathcal{G}$. The result is a new graph with more edges, to which we can then apply Theorem~\ref{thm:Main} again. This process can be continued in order to construct longer and more complex trajectories of the system~\eqref{ODE}. However, we emphasize that the most important aspect when applying these results is the original set $\mathcal{W}_0$, since all trajectories constructed using Theorem~\ref{thm:Main} can only `shadow' the base homoclinic and heteroclinic orbits in $\mathcal{W}_0$.
\begin{rmk}
We remark that the results of Theorem~\ref{thm:Main} also hold for non-Hamiltonian systems since Hypothesis~\ref{hyp:Ham} only restricts the dimension of the phase space to the Hamiltonian level sets of~\eqref{ODE}. For non-Hamiltonian ODEs we can alternatively take~\eqref{ODE} to have a three-dimensional phase space and simply remove the assumption that the periodic orbits belong to the same energy level set in Hypothesis~\ref{hyp:Periodic}. Although such a relaxation is easily achieved, we have elected to include the Hamiltonian assumption on~\eqref{ODE} to emphasize the applicability of our results to the double pendulum, as well as other notable Hamiltonian systems such as the PCR3BP.
\end{rmk}
\subsection{Application to the Double Pendulum}\label{subsec:DPapplication}
In this subsection we provide a number of applications of Theorem~\ref{thm:Main} to the double pendulum model~\eqref{eq:edp_ode}. From our work in Section~\ref{sec:EOM} we have that Hypothesis~\ref{hyp:Ham} is satisfied with Hamiltonian function~\eqref{eq:edp_hamiltonian}. In all examples we will take the periodic orbits in Hypothesis~\ref{hyp:Periodic} to be the UPOs near the index-1 saddles, thus satisfying the hyperbolicity assumptions as well. Unlike the work~\cite{Conley,McGehee} on the PCR3BP that proves the existence of homoclinic orbits to the Lyapunov orbits about the index-1 saddles, the double pendulum lacks such proofs and so we rely on our numerical work in Section~\ref{sec:TubesDP}. Thus, the set $\mathcal{W}_0$ is formed from collections of the numerically observed homoclinic and heteroclinic orbits detailed in Figures~\ref{fig:EDP_PoincareCut_L1_VariesEng}, \ref{fig:EDP_TubeStructure_L2}, and \ref{fig:EDP_HeteroclinicPoints}, with examples of the associated directed graphs given in Figure~\ref{fig:Graphs}. We will work exclusively with the periodic phase space that identifies $\theta_{1,2}$ with $\theta_{1,2} \pm 2\pi$ to ensure that our orbits are truly homoclinic. Beyond this numerical existence, we also lack a proof of transversality of the orbits, but again the figures appear to confirm these properties, as one can see they lie along transverse intersections between stable and unstable manifolds in the Poincar\'e section.
In what follows we will detail three separate applications of Theorem~\ref{thm:Main}. We begin with only the homoclinic orbits to the UPOs near the Down-Up saddle state. In this case we can use the base homoclinic orbits found in Section~\ref{sec:L1Hom} to create longer multi-homoclinic trajectories and periodic orbits. Although not explored here, these same results can be extended to the UPOs near the Up-Down saddle using the homoclinic orbits found in Section~\ref{sec:L2Hom}. Then, we use the heteroclinic orbits from Section~\ref{sec:Heteroclinic} to create new homoclinic and periodic orbits that bounce between neighborhoods of the index-1 saddles. Finally, we use all of our numerical work together to show that when the energy is large enough we can transit across vast regions of phase space by following all of the homoclinic and heteroclinic orbits found in Section~\ref{sec:TubesDP}.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Figures/Graphs.pdf}
\vspace{-.1in}
\caption{To apply Theorem~\ref{thm:Main} one uses the homoclinic and heteroclinic connections between UPOs to create a directed graph. (a) The graph whose single vertex corresponds to the UPO near the Down-Up equilibrium with $\mathcal{H} = -0.07$ for which Figure~\ref{fig:EDP_PoincareCut_L1_VariesEng} demonstrates the existence of four homoclinic orbits, resulting in four edges originating and terminating at the single vertex. (b) The heteroclinic orbits from Figure~\ref{fig:EDP_HeteroclinicPoints} and their reversed counterparts lead to a graph with two vertices corresponding to the UPOs near each of the index-1 saddles with $\mathcal{H} = 0.2$ and two pairs of directed edges from one edge to the other. (c) The resulting directed graph obtained by putting together the information from Figures~\ref{fig:EDP_TubeStructure_L1}, \ref{fig:EDP_TubeStructure_L2}, and \ref{fig:EDP_HeteroclinicPoints} with $\mathcal{H} = 0.2$.}
\label{fig:Graphs}
\end{figure}
\subsubsection{Long Trajectories Near the Down-Up State}
An immediate corollary of Theorem~\ref{thm:Main} is that a single transverse homoclinic orbit to one of the periodic orbits $\gamma_i(t)$ gives rise to infinitely many more homoclinic and periodic orbits that `shadow' the original homoclinic orbit. In the language of the graph $\mathcal{G}$, we would have for any integer $k \geq 1$ a walk given by $E_1 = E_2 = \dots = E_k$ and $V_1 = V_2 = \dots = V_{k+1}$, where the edge $E_1$ is a loop on the vertex $V_1$. This result is a well-known consequence of having a transverse homoclinic orbit in three or more dimensional ODEs~\cite{Champneys,Haller,Homburg,Homburg2,Jens,2Pulse}. Thus, Theorem~\ref{thm:Main} comes as a generalization of some of these results. In Figure~\ref{fig:MultiHom}(a) we present an illustration of a `base' homoclinic orbit that is asymptotic to a UPO near an index-1 saddle, along with cartoons of (b) a longer homoclinic orbit to the same UPO that shadows the base orbit twice ($k = 2$) and (c) a periodic orbit that shadows the base homoclinic once ($k = 1$), both of which are guaranteed by Theorem~\ref{thm:Main}.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{Figures/MultiHom.pdf}
\caption{Theorem~\ref{thm:Main} gives the existence of multi-homoclinic and periodic solutions that shadow a single homoclinic orbit. (a) The `base' homoclinic orbit that is asymptotic to a single UPO near an index-1 saddle. (b) A double homoclinic orbit which shadows the base homoclinic twice with an intermediary transition wrapping around the UPO. (c) A periodic orbit that shadows the base homoclinic.}
\label{fig:MultiHom}
\end{figure}
Beyond just a single homoclinic orbit emanating from a UPO near the Down-Up saddle, one may turn to our numerical findings in Figure~\ref{fig:EDP_PoincareCut_L1_VariesEng}, which demonstrate multiple homoclinic orbits. For example, inside the energy level set $\mathcal{H} = -0.07$ as in panel (b) of Figure~\ref{fig:EDP_PoincareCut_L1_VariesEng} we have four distinct homoclinic orbits. Thus, we can apply Theorem~\ref{thm:Main} with $\mathcal{W}_0$ comprised of these four homoclinic orbits. An illustration of the resulting directed graph is presented in Figure~\ref{fig:Graphs}(a). The result is an abundance of homoclinic and periodic trajectories formed by shadowing the elements of $\mathcal{W}_0$ for which each shadow causes the second pendulum arm to undergo a full rotation and the intermediary transition that switches between elements given by a long wobble of the second arm that shadows the UPO near the index-1 Down-Up equilibrium, as illustrated in Figure~\ref{fig:DP_UPO}. Finding these orbits is outlined in~\cite{Koon}, but we can also use similar numerical techniques from Section~\ref{sec:TubesDP} to identify longer homoclinic orbits. Figure~\ref{fig:L1DoubleHom}~(a) shows the result of continuing to flow the unstable manifold of the UPO near the Down-Up equilibrium forward to $\theta_2 = 4\pi$. Intersections in the Poincar\'e section now represent homoclinic orbits that have the second arm making two full rotations over the course of the trajectory. The curves are incomplete since many homoclinic orbits take a long time to reach the Poincar\'e section again, corresponding to large values of $N_i$ in Theorem~\ref{thm:Main}. One can see that the curve wraps around itself, appearing to give infinitely many intersections with the stable manifold near the location of the original homoclinic orbits from Figure~\ref{fig:EDP_PoincareCut_L1_VariesEng}(b). Figure~\ref{fig:L1DoubleHom}~(b) also includes the flow of the unstable manifold forward to $\theta_2 = 6\pi$, for which the resulting homoclinic orbits now have the second arm making three full rotations. Increasing the value of $\mathcal{H}$ results in more `base' trajectories with which to build $\mathcal{W}_0$, thus increasing the orbits to shadow for producing periodic and homoclinic trajectories.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\textwidth]{Figures/EDP_L1_DoubleCut.pdf}
\vspace{-.15in}
\caption{Flowing the unstable manifold of the UPO near the Down-Up equilibrium at $\mathcal{H} = -0.07$ forward to $\theta = 2\pi k$, $k \geq 2$, can numerically demonstrate the existence of longer homoclinic orbits to it. (a) Intersections in the Poincar\'e section of the unstable manifold flowed forward in time to $\theta_2 = 4\pi$ and the stable manifold flowed backward to $\theta_2 = 0$ gives the existence of doubly-homoclinic orbits. (b) Similarly, flowing the unstable manifold forward in time to $\theta_2 = 6\pi$ demonstrates the existence of triply-homoclinic orbits.}
\label{fig:L1DoubleHom}
\end{figure}
\subsubsection{Transitions Between the Down-Up and Up-Down State}
Section~\ref{sec:Heteroclinic} numerically demonstrates heteroclinic orbits that transfer one from a UPO near the Down-Up equilibrium to a UPO near the Up-Down equilibrium. Furthermore, applying the reversible symmetry of the double pendulum to these orbits also gives the existence of heteroclinic orbits that transfer from a UPO near the Up-Down equilibrium to a UPO near the Down-Up equilibrium. Using this information we can take $\mathcal{W}_0$ to be these four heteroclinic orbits that go back and forth between UPOs near the Down-Up and Up-Down equilibria at $\mathcal{H} = 0.2$, resulting in the directed graph presented in Figure~\ref{fig:Graphs}(b). Applying Theorem~\ref{thm:Main} leads to the existence of homoclinic, heteroclinic, and periodic orbits that bounce between the neighborhoods of the index-1 saddles and can be understood by traversing the edges of the corresponding graph. Such orbits begin by wobbling around one of the index-1 saddles, mimicking the motion of the nearby UPO, and then shadow one of the heteroclinic orbits to move to a wobbling motion about the other index-1 saddle, according to its nearby UPO. All orbits given by Theorem~\ref{thm:Main} continue this process of wobbling, transferring, wobbling, transferring, and so on to either generate a homoclinic, heteroclinic, or periodic orbit of the system~\eqref{eq:edp_ode}. What differentiates these orbits is where they start and end, with homoclinics starting and ending with wobbles about the same index-1 saddle, heteroclinics starting and ending with wobbles around different saddles, and periodic orbits repeating the wobble, transfer, wobble process ad infinitum.
\begin{figure}[t]
\centering
\includegraphics[width=.95\textwidth]{Figures/EDP_L1_HeteroDoubleCut.pdf}
\vspace{-.15in}
\caption{Continuing to flow the unstable manifold of the UPO near the Down-Up equilibrium (black) forward in time reveals the existence of new homoclinic orbits to it. (a) The manifolds in $(\theta_1,\theta_2,\dot\theta_1)$-space. (b) The intersection of the stable and unstable manifolds in the Poincar\'e section $\theta_1 = \theta_2$, but now with the unstable manifold crossing in the other direction. Intersections of the curves in the Poincar\'e section provide the existence of homoclinic orbits.}
\label{fig:HetApplication}
\end{figure}
Similar to the previous section, we can flow the manifolds of the UPO for a longer time to identify more complex homoclinic and heteroclinic orbits guaranteed by Theorem~\ref{thm:Main}. For example, we have flowed the unstable manifold of the UPO near the Down-Up equilibrium forward in time until we meet the Poincar\'e section $\theta_1 = \theta_2$ again, but this time crossing in the negative direction. In Figure~\ref{fig:HetApplication} we provide this new intersection with the Poincar\'e section, along with the intersection of the stable manifold from~\S\ref{sec:Heteroclinic}. Intersections of these curves represent homoclinic orbits to the UPO near the Down-Up equilibrium that are entirely distinct from those found in~\S\ref{sec:L1Hom}. The curves are not connected in the Poincar\'e section, partially attributed to the extremely long time to flow parts of the manifold back to the Poincar\'e section and partially due to the fact that a portion of the unstable manifold does not turn around and come back to the Poincar\'e section from the other side. We have performed the same calculation for the UPO near the Up-Down equilibrium but the results are not included for brevity.
\subsubsection{Full State-Space Dynamics}
Our final application of Theorem~\ref{thm:Main} unites all of our numerical findings from Section~\ref{sec:TubesDP}. Notice that Figures~\ref{fig:EDP_PoincareCut_L1_VariesEng}, \ref{fig:EDP_TubeStructure_L2}, and \ref{fig:EDP_HeteroclinicPoints} all provide the existence of homoclinic and heteroclinic orbits for $\mathcal{H} = 0.2$. Thus, when the energy $\mathcal{H}$ is sufficiently large, it is possible to transit around the homoclinic orbits to the UPOs near the index-1 saddles and between these UPOs along the heteroclinic orbits. Taking $\mathcal{W}_0$ to be the set of all such homoclinic and heteroclinic orbits found numerically for $\mathcal{H} = 0.2$, we can generate the directed graph presented in Figure~\ref{fig:Graphs}(c) and apply Theorem~\ref{thm:Main}. This gives the existence of long trajectories in phase space that temporarily shadow the base orbits in $\mathcal{W}_0$ to transit back and forth between neighborhoods of the index-1 saddles.
Precisely, one is able to determine the existence of orbits that perform the acrobatic motion of one arm making a full rotation by following one of the homoclinic orbits in Sections~\ref{sec:L1Hom} and \ref{sec:L2Hom} and transferring between these motions by shadowing the transitory heteroclinic orbits of Section~\ref{sec:Heteroclinic}. Much like the previous applications, Theorem~\ref{thm:Main} allows one to stitch these motions together to form a long, complex orbit of the double pendulum that follows such an itinerary. In fact, Theorem~\ref{thm:Main} gives infinitely many such orbits that follow a given itinerary since one can choose the integers $N_i$ that approximately prescribe how long the intermediary wobbling motion will take place before performing an arm swing or a transition between saddle neighborhoods. Therefore, much like the Interplanetary Transport Network of the solar system, coming from similar results in the PCR3BP and related models, the double pendulum has a similar transport network given by similar tube dynamics.
\subsection{Application to the PCR3BP}\label{subsec:TBPapplication}
Let us briefly comment on the application of our results to the PCR3BP. In~\cite{Koon} the authors undertook a similar numerical investigation of the Lyapunov orbits about the $L_1$ and $L_2$ Lagrange points of the PCR3BP. Their numerical findings were summarized in Figures~\ref{fig:TBP_HomoclinicOrbit} and \ref{fig:TBP_HeteroclinicOrbit}. These findings were then leveraged to provide analytical results that detail the existence of long trajectories that follow these homoclinic and heteroclinic orbits according to a prescribed itinerary. The results of Theorem~\ref{thm:Main} can similarly be applied to the PCR3BP in the same way we did for the double pendulum in the previous subsection, and in many instances our results herein provide more details on the itineraries than those in~\cite{Koon}. We summarize our extensions as follows:
\begin{enumerate}
\item Theorem~\ref{thm:Main} demonstrates exactly how trajectories follow a given itinerary by shadowing the homoclinic and heteroclinic orbits in $\mathcal{W}_0$.
\item The presence of multiple homoclinic and heteroclinic orbits provides different tracks for the itineraries to follow as they leave neighborhoods of the Lagrange points.
\item The periodic orbits $\gamma_i(t)$ need not be Lyapunov orbits, and so Theorem~\ref{thm:Main} can be applied to establish transit between any hyperbolic periodic orbits in the PCR3BP for which a transverse heteroclinic connection is known to exist.
\end{enumerate}
In the context of the PCR3BP we have that the values of the integers $N_i$ in Theorem~\ref{thm:Main} approximately correspond to how long the trajectory will orbit around a Lagrange point in the PCR3BP by shadowing its Lyapunov orbit. The larger the value of $N_i$, the longer the trajectory orbits a Lagrange point. Thus, one may use Theorem~\ref{thm:Main} to not only follow a given itinerary, but also to rest for arbitrarily long periods of time near the Lagrange points. This was similarly established in \cite[Theorem~4.2]{Koon} with the values $r_i$, although by different methods to the proofs in the following section.
\section{Proof of Theorem~\ref{thm:Main}}\label{sec:Proofs}
Throughout this section we will consider $k \geq 1$ and the sequences
\begin{equation}
(E_1,E_2,\dots,E_k) \in \mathcal{E}^k, \quad (V_1,V_2,\dots,V_{k+1})\in\mathcal{V}^{k+1}
\end{equation}
as they are given in the statement of Theorem~\ref{thm:Main}. In an effort to simplify the notation of this section we will define
\begin{equation}\label{tildegamma}
\tilde{\gamma}_i(t) = \gamma_{V_i}(t)
\end{equation}
for all $i = 1,2,\dots,k+1$. This gives a new sequence of periodic orbits $\{\tilde\gamma_1(t),\tilde\gamma_2(t),\dots,\tilde\gamma_{k+1}(t)\}$ which corresponds to the sequence of vertices traversed in the walk on the graph $\mathcal{G}$. We note that depending on the choices of $E_i$ and $V_i$, it may be the case that $\tilde{\gamma}_i(t) = \tilde{\gamma}_j(t)$ for some $i \neq j$, which will necessarily be true when $k > p$. The effect is that we may now define a graph $\mathcal{G}' = (\mathcal{V}',\mathcal{E}')$ for which the vertex set $\mathcal{V}' = \{1,2,\dots,k+1\}$ is meant to represent each of the periodic orbits $\tilde\gamma_1(t),\tilde\gamma_2(t),\dots,\tilde\gamma_{k+1}(t)$. The edge set $\mathcal{E}'$ is solely comprised of the edges that make up the walk, thus giving that we have a directed edge from vertex $i$ to vertex $i+1$ for all $1 \leq i \leq k$. Therefore, $\mathcal{G}'$ is a directed chain meant to represent transits from $\gamma_{V_i}(t)$ to $\gamma_{V_{i+1}}(t)$ dictated by the walk $(E_1,E_2,\dots,E_k)$ on $\mathcal{G}$. In the following subsections we will work exclusively with the $\tilde\gamma_i(t)$ periodic orbits and the graph $\mathcal{G}'$ to simplify notation.
The proof of Theorem~\ref{thm:Main} is broken down over the following subsections. We begin in~\S\ref{subsec:Local} with an understanding of the dynamics of the differential equation~\eqref{ODE} in a neighborhood of each $\tilde\gamma_i(t)$. To achieve this understanding we define an appropriate Poincar\'e section and consider iterates through this section governed by a Poincar\'e mapping. When we restrict the domain of the Poincar\'e mappings close enough to the intersection with the periodic orbits we have that each iterate of the map roughly represents a trajectory of the continuous-time system~\eqref{ODE} that wraps around the periodic orbit once. Upon establishing these local results, we then move to~\S\ref{subsec:Transfer} where we define maps that transport iterates through phase space from one local description to another. Then~\S\ref{subsec:HomHet} and~\S\ref{subsec:Periodic} set up appropriate matching conditions which, when satisfied, construct the trajectories outlined in point (1) and point (2), respectively, of Theorem~\ref{thm:Main}.
\subsection{Dynamics Near the Periodic Orbits}\label{subsec:Local}
In this subsection we will work to understand the dynamics of system~\eqref{ODE} in neighbourhoods of the periodic orbits $\tilde\gamma_1(t), \dots, \tilde\gamma_{k+1}(t)$. The results apply to all $\tilde\gamma_i(t)$ with $1 \leq i \leq k+1$, and so whenever possible we will define global constants that apply in all neighbourhoods of the periodic orbits. Such global constants can always be found since they can be defined as either the maximum or the minimum, depending on the context, of the individual constants needed for each $\tilde\gamma_i(t)$. We also emphasize that all results moving forward can be obtained independently of the value of $k$. This comes from the definition of the $\tilde\gamma_i(t)$ in~\eqref{tildegamma} and the fact that there are only $p \geq 1$ periodic orbits, meaning that we need only understand the dynamics near each of the $p\geq 1$ periodic orbits and carry them over to the redundant $\tilde\gamma_i(t) = \tilde\gamma_j(t)$ with $i \neq j$. As discussed at the beginning of this section, we will continue to use the $\tilde\gamma_i(t)$ notation for convenience, but the reader should keep in mind that there are only finitely many unique choices for our periodic orbits.
To begin, Hypothesis~\ref{hyp:Ham} implies that we may reduce the full four-dimensional phase space of~\eqref{ODE} to the three-dimensional level set of $\mathcal{H}$ that contains the periodic orbits $\tilde\gamma_i(t)$. Inside of this three-dimensional phase space we can define $\Sigma_i$ to be a Poincar\'e section comprised of the plane orthogonal to $\gamma_i(0)$ restricted to a small neighborhood of this point. Let us define the local Poincar\'e mapping, denoted $P_i$, by
\begin{equation}\label{Psec}
P_i: \Sigma'_i \to \Sigma_i,
\end{equation}
where the domain $\Sigma'_i \subset \Sigma_i$ is such that $P(\Sigma'_i) \subset \Sigma_i$, making the mapping well-defined. Note that $P_i(\tilde\gamma_i(0)) = \tilde\gamma_i(0)$, and from Hypothesis~\ref{hyp:Periodic} the linearization $DP(\tilde\gamma_i(0))$ has two nonzero eigenvalues $\lambda^{-1}_i,\lambda_i \in \mathbb{R}$, corresponding to the nontrivial Floquet multipliers of $\tilde\gamma_i(t)$. Upon potentially relabelling $\lambda_i$ and $\lambda_i^{-1}$, we will assume $0 < |\lambda_i^{-1}| < 1 < |\lambda_i|$, making $\lambda_i$ the unstable eigenvalue and $\lambda_i^{-1}$ the stable eigenvalue. This leads to our first result which details a change of variables to locally straighten the stable and unstable manifolds of the fixed point $\tilde\gamma_i(0)$ in a small neighborhood.
\begin{lem}\label{lem:Shil}
Assume Hypotheses~\ref{hyp:Ham} and \ref{hyp:Periodic} are met. Then there exists $\delta > 0$ such that for each ${i \in \{1,\dots,k+1\}}$ there is a smooth change of coordinates mapping $u\in\Sigma'_i$ to $v_i = (v^s_i,v^u_i)$ near the fixed point $u = \tilde\gamma_i(0)$, and smooth functions $f^s_{i,j},f^u_{i,j}:\mathcal{I}\times\mathcal{I}\to \mathbb{R}$, $j = 1,2$, so that (\ref{Psec}) is of the form
\begin{equation}\label{Shil}
\begin{split}
v^s_{i,n+1} &= [\lambda_i^{-1} + f_{i,1}^s(v^s_{i,n},v^u_{i,n})v^s_{i,n} + f_{i,2}^s(v^s_{i,n},v^u_{i,n})v^u_{i,n}]v^s_{i,n}, \\
v^u_{i,n+1} &= [\lambda_i + f_{i,1}^u(v^s_{i,n},v^u_{i,n})v^s_{i,n} + f_{i,2}^u(v^s_{i,n},v^u_{i,n})v^u_{i,n}]v^u_{i,n}, \\
\end{split}
\end{equation}
where $v^s_{i,n},v^u_{i,n} \in \mathcal{I} := [-\delta,\delta]$.
\end{lem}
\begin{proof}
Let $\xi^s_i$ be the eigenvector of $DP_i(\tilde\gamma_i(0))$ associated to $\lambda_i^{-1}$ and $\xi^u_i$ be the eigenvector associated to $\lambda_i$. It follows from the stable manifold theorem for maps that there exists a $\delta_i > 0$ and smooth functions $w^s_i,w^u_i:[-\delta_i,\delta_i] \to \mathbb{R}$ with $w^s_i(0) = (w^s_i)'(0) = w^u_i(0) = (w^u_i)'(0) = 0$ such that the local stable and unstable manifolds of $\tilde\gamma_i(0)$ can be written
\begin{equation}
\begin{split}
W^s_\mathrm{loc}(\tilde\gamma_i(0)) &= \{\tilde\gamma_i(0) + v^s_i\xi^s_i + w^s_i(v^s_i)\xi^u_i:\ v^s_i \in [-\delta_i,\delta_i]\}, \\
W^u_\mathrm{loc}(\tilde\gamma_i(0)) &= \{\tilde\gamma_i(0) + v^u_i\xi^u_i + w^u_i(v^u_i)\xi^s_i:\ v^u_i \in [-\delta_i,\delta_i]\}.
\end{split}
\end{equation}
Then, for $u\in\Sigma_i'$ in a small neighborhood of $\tilde\gamma_i(0)$ we may introduce the change of variable
\begin{equation}
(v^s_i,v^u_i) \mapsto u = \tilde\gamma_i(0) + (v^s_i\xi^s_i + w^s_i(v^s_i)\xi^u_i) + (v^u_i\xi^u_i + w^u_i(v^u_i)\xi^s_i).
\end{equation}
From the tangency properties of the $w^s_i,w^u_i$ functions at $(v^s_i,v^u_i) = (0,0)$, it follows that the above change of variable is a local diffeomorphism. Putting this change of variable into the Poincar\'e map $P_i$ and expanding in a neighborhood of $(v^s_i,v^u_i) = (0,0)$ gives the expansion~\eqref{Shil}. Moreover, this change of variable gives that $W^s_\mathrm{loc}(\tilde\gamma_i(0)) = \{v^u_i = 0\}$ and $W^u_\mathrm{loc}(\tilde\gamma_i(0)) = \{v^s_i = 0\}$. Finally, we may take $\delta := \min \{\delta_1,\dots,\delta_{k+1}\}$ to arrive at the $\delta > 0$ in the statement of the lemma. This completes the proof.
\end{proof}
With $\delta > 0$ taken sufficiently small, each iterate of~\eqref{Shil} represents a solution of~\eqref{ODE} that completes a full revolution of $\tilde\gamma_i(t)$. In this way, we can use~\eqref{Shil} to quantify orbits of~\eqref{ODE} that pass through a neighborhood of the periodic orbits $\tilde\gamma_i(t)$. Precisely, the saddle structure gives that orbits enter the neighborhood closely following the stable manifold and exit the neighborhood closely following the unstable manifold, all while wrapping around the periodic orbit. This is made precise with the following lemma which originally appeared in~\cite{BramLDS} and comes as the discrete-time analogue of the main result of~\cite{Schecter}.
\begin{lem}[\cite{BramLDS} Lemma~3.2] \label{lem:ShilSol}
There exists constants $\eta \in (0,1)$ and $M > 0$ such that the following is true: for each $1 \leq i \leq k+1$, $N > 0$, and $a^s,a^u \in \mathcal{I}$ there exists a unique solution to~\eqref{Shil}, written $v_{i,n} = (v^s_{i,n},v^u_{i,n}) \in \mathcal{I}\times\mathcal{I}$ with $n \in \{0,\dots,N\}$, such that
\begin{equation}
v_{i,0}^s = a^s, \quad v_{i,N}^u = a^u.
\end{equation}
Furthermore, this solution satisfies
\begin{equation}\label{LocalBnds}
|v^s_{i,n}| \leq M\eta^n, \quad |v^u_{i,n}| \leq M\eta^{N-n},
\end{equation}
for all $n \in \{0,\dots,N\}$, $v_{i,n} = v_{i,n}(a^s,a^u)$ depends smoothly on $(a^s,a^u)$, and the bounds~\eqref{LocalBnds} also hold for the derivatives of $v$ with respect to $(a^s,a^u)$.
\end{lem}
\subsection{Transferring Between Periodic Orbits}\label{subsec:Transfer}
Recall that $\mathcal{W}$ comprises the set of all homoclinic and heteroclinic orbits of~\eqref{ODE} between the $p \geq 1$ periodic orbits $\{\gamma_i(t)\}_{i = 1}^p$. From Hypothesis~\ref{hyp:HomHet} we have that $\mathcal{W} \neq \emptyset$, and we have assumed the existence of a set $\mathcal{W}_0 \subset \mathcal{W}$ which is a nonempty finite collection of transverse homoclinic and heteroclinic orbits between the periodic orbits. Moreover, the walk along the edges $(E_1,E_2,\dots,E_k)$ can equivalently be written as an ordered list of elements in $\mathcal{W}_0$. Let us denote the element $h_i(t) \in \mathcal{W}_0$ to be the orbit used to form the edge $E_i$ on the graph $\mathcal{G}$. Then, from Hypothesis~\ref{hyp:HomHet} and the definition of the graph $\mathcal{G}'$, each $h_i(t) \in \mathcal{W}_0$ lies on a transverse intersection of $W^u(\tilde\gamma_i(t))$ and $W^s(\tilde\gamma_{i+1}(t))$. Using the local coordinates of Lemma~\ref{lem:Shil}, there exists $h^s_{i+1}$ so that
\begin{equation}
(v^s_{i+1},v^u_{i+1}) = (h^s_{i+1},0) \in \{h_i(t) \cap (-\delta,\delta)\times \{0\}\} \subset W^s(\tilde\gamma_{i+1}(0))\cap W^u(\tilde\gamma_i(0))
\end{equation}
near the fixed point $\tilde\gamma_{i+1}(0)$ of the Poincar\'e mapping $P_{i+1}$. That is, the point $(h^s_{i+1},0)\in\mathcal{I}\times\mathcal{I}$ represents a point of intersection of the orbit $h_i(t)$ with the local Poincar\'e section $\Sigma_{i+1}$, lying along the stable manifold of $\tilde\gamma_{i+1}(t)$. Similarly, there exists $h_i^u$ so that
\begin{equation}
(v^s_i,v^u_i) = (0,h_i^u) \in \{h_i(t) \cap \{0\}\times(-\delta,\delta)\} \subset W^s(\tilde\gamma_{i+1}(0))\cap W^u(\tilde\gamma_i(0)),
\end{equation}
representing a point of intersection of the orbit $h_i(t)$ with the local Poincar\'e section $\Sigma_i$, lying along the unstable manifold of $\tilde\gamma_i(t)$. We note that there are infinitely many choices of $h_{i+1}^s$ and $h_i^u$ and it does not matter which we choose except that they lie entirely in the interior of $\mathcal{I}\times\mathcal{I}$. Furthermore, the assumption of transversality of the orbit $h_i(t)$ implies that $(h^s_{i+1},0)$ represent a transverse point of intersection between the stable manifold of $\tilde\gamma_{i+1}(t)$ inside the Poincar\'e sections $\Sigma_{i+1}$. The analogous statement also holds in that the point $(0,h_i^u)$ lies along a transverse point of intersection inside the Poincar\'e section $\Sigma_i$.
The following lemma makes explicit use of the assumption that the intersections of $W^s(\tilde\gamma_{i+1}(0))\cap W^u(\tilde\gamma_i(0))$ are transverse. We will adopt the notation $B_r(x)$ to denote the ball of radius $r > 0$ about the point $x$.
\begin{lem}\label{lem:GFn}
There exists $\varepsilon > 0$ such that the following is true for all $1 \leq i \leq k$:
\begin{enumerate}
\item There exists a smooth function $G^u_{i+1}:B_\varepsilon(h^s_{i+1},0)\to\mathbb{R}$ such that $G^u_{i+1}(v^s_{i+1},v^u_{i+1}) = 0$ if and only if $(v^s_{i+1},v^u_{i+1})\in W^u(\tilde\gamma_i(0))\cap B_\varepsilon(h^s_{i+1},0)$. Furthermore, $\partial_{v^s_{i+1}}G^u_{i+1}(h^s_{i+1},0) \neq 0$.
\item There exists a smooth function $G^s_i:B_\varepsilon(0,h^u_i)\to\mathbb{R}$ such that $G^s_i(v^s_i,v^u_i) = 0$ if and only if $(v^s_i,v^u_i)\in W^s(\tilde\gamma_{i+1}(0))\cap B_\varepsilon(0,h^u_i)$. Furthermore, $\partial_{v^u_i}G^u_{i,j}(0,h^u_i) \neq 0$.
\end{enumerate}
\end{lem}
\begin{proof}
We will only prove the first statement since the second is handled identically. By assumption we have that $(h^s_{i+1},0) \in \mathcal{I}\times\mathcal{I}$ represents a transverse point of intersection between $W^s(\tilde\gamma_{i+1}(0))$ and $W^u(\tilde\gamma_i(0))$. Therefore, we can locally parameterize $W^u(\tilde\gamma_i(0))$ near $(h^s_{i+1},0)$ by the function $v^s_{i+1} = g(v^u_{i+1})$ so that $h^s_{i+1} = g(0)$. Then, defining
\begin{equation}
G^u_{i+1}(v^s_{i+1},v^u_{i+1}) = v^s_{i+1} - g(v^u_{i+1})
\end{equation}
gives a function that locally vanishes on $W^u(\tilde\gamma_i(0))$ and satisfies $\partial_{v^s_{i+1}}G^u_{i+1}(h^s_{i+1},0) \neq 0$. We can therefore take $\varepsilon > 0$ small enough to ensure that $B_\varepsilon(h^s_{i+1},0)$ lies inside of $\mathcal{I}\times\mathcal{I}$ and that $G^u_{i+1}$ only vanishes when $(v^s_{i+1},v^u_{i+1}) \in W^u(\tilde\gamma_{i+1}(0))$. This completes the proof of the first point.
\end{proof}
The final result of this subsection describes a push-forward map that transfers one between the local Poincar\'e sections $\Sigma_i$. This transfer is done by shadowing the homoclinic/heteroclinic orbits $h_i(t)$ which make up the edges $E_i$ in the walk on $\mathcal{G}$.
\begin{lem}\label{lem:PushForward}
There exists an $\varepsilon > 0$ such that for each $1 \leq i \leq k$ there exists a smooth map $\Pi_i: B_\varepsilon(0,h^u_i) \to \mathcal{I}\times\mathcal{I}$ such that $\Pi_i(0,h^u_i) = (h^s_{i+1},0)$ and $\Pi_i$ is diffeomorphism in a neighborhood of $(0,h^u_i)$.
\end{lem}
\begin{proof}
Smooth dependence on initial conditions for the continuous-time flow~\eqref{ODE} guarantees that for some $\varepsilon > 0$ chosen sufficiently small, each point in $B_\varepsilon(0,h^u_i)$ defines a trajectory that lies close to the orbit $h_i(t)$ that eventually will intersect the local component of the Poincar\'e section $\Sigma_{i+1}$ in a small neighborhood of $(h^s_{i+1},0)$. We define $\Pi_i$ to be the map that transports the initial conditions in $B_\varepsilon(0,h^u_i)$ to their image in the small neighborhood of $(h^s_{i+1},0)$ in $\Sigma_{i+1}$, with $\varepsilon$ taken sufficiently small that the range lies entirely inside of $\mathcal{I}\times\mathcal{I}$ when converted to the coordinates $(v^s_{i+1},v^u_{i+1})$ defined in Lemma~\ref{lem:Shil}. By definition, $\Pi_i(0,h^u_i) = (h^s_{i+1},0)$, smooth dependence on initial conditions guarantees that $\Pi_i$ is a smooth map, and uniqueness of solutions to~\eqref{ODE} guarantees that $\Pi_i$ is invertible on its range. This completes the proof.
\end{proof}
\subsection{Constructing Homoclinic and Heteroclinic Trajectories}\label{subsec:HomHet}
With the previous subsections we were able to understand the local dynamics near each of the periodic orbits $\tilde\gamma_i(t)$ in the Poincar\'e sections $\Sigma_i$ and how we can transfer between these local dynamical systems. Here we will now prove Theorem~\ref{thm:Main}, starting with the homoclinic and heteroclinic orbits described in the point (1). It should be noted that we need only consider $k \geq 2$ since the case $k = 1$ is handled by the elements $h_i(t) \in \mathcal{W}_0$.
We start by employing Lemma~\ref{lem:ShilSol} to obtain solutions of~\eqref{Shil} in a neighborhood of $\tilde\gamma_i(0)$, $2 \leq i \leq k$, given by
\begin{equation}
\begin{split}
\{(v^s_{2,n},v^u_{2,n})\}_{n = 0}^{N_2},& \quad v^s_{2,0} = h^s_2 +a^s_2, \quad v^u_{2,N_2}= h^u_2 +a^u_2 \\
\{(v^s_{3,n},v^u_{3,n})\}_{n = 0}^{N_3},& \quad v^s_{3,0} = h^s_3 +a^s_3, \quad v^u_{3,N_3}= h^u_3 +a^u_3 \\
&\vdots \\
\{(v^s_{k,n},v^u_{k,n})\}_{n = 0}^{N_{k}},& \quad v^s_{k,0} = h^s_k +a^s_{k}, \quad v^u_{k,N_{k}}= h^u_k +a^u_{k},
\end{split}
\end{equation}
respectively. Here the $a^s_i$ and $a^u_i$ are taken to be small and the $N_i \geq 1$ sufficiently large to guarantee that
\begin{equation}\label{Expansions}
\begin{split}
(v^s_{i,0},v^u_{i,0}) &= (h^s_i+a_i^s,\mathcal{O}(\eta^{N_i})) \in B_\varepsilon(h^s_i,0) \\
(v^s_{i,N_j},v^u_{i,N_j}) &= (\mathcal{O}(\eta^{N_i}),h^u+a_i^u) \in B_\varepsilon(0,h^u_i) \\
\end{split}
\end{equation}
for each $2 \leq j \leq k$, where $\varepsilon > 0$ is chosen small enough to satisfy the requirement for Lemmas~\ref{lem:GFn} and \ref{lem:PushForward}. Each of the above solutions represents a trajectory of the continuous-time system~\eqref{ODE} that wraps at least $N_i \geq 1$ times around $\tilde\gamma_i(t)$. We may stitch these solutions together to create a homoclinic or heteroclinic orbit by attempting to satisfy the following matching conditions:
\begin{equation}\label{HomMatch}
\begin{split}
G^u_2(v^s_{2,0},v^u_{2,0}) &= 0 \\
\Pi_2(v^s_{2,N_2},v^u_{2,N_2}) - (v^s_{3,0},v^u_{3,0}) &= 0 \\
\Pi_3(v^s_{3,N_3},v^u_{3,N_3}) - (v^s_{4,0},v^u_{4,0}) &= 0 \\
&\vdots \\
\Pi_{k-1}(v^s_{k-1,N_{k-1}},v^u_{k-1,N_{k-1}}) - (v^s_{k,0},v^u_{k,0}) &= 0 \\
G^s_k(v^s_{k,N_{k}},v^u_{k,N_{k}}) &= 0.
\end{split}
\end{equation}
Indeed, from Lemma~\ref{lem:GFn} we have that the first condition guarantees that the trajectory belongs to $W^u(\tilde\gamma_1(0))$, and the final condition guarantees that the trajectory belongs to $W^s(\tilde\gamma_{k+1}(0))$. The intermediary conditions use the push-forward maps $\Pi_i$ to guarantee that each time the trajectory leaves the neighborhood $\tilde\gamma_i(0)$ it moves to the neighborhood of $\tilde\gamma_{i+1}(0)$ to complete $N_i \geq 1$ iterations before leaving for the next neighborhood. We are now in position to prove Theorem~\ref{thm:Main}(1). Prior to doing so, we present the following lemma coming from~\cite{RadialPulse} that will be used throughout the proof.
\begin{lem}[\cite{RadialPulse} Lemma~7.2]\label{lem:Roots}
If $H:\mathbb{R}^n \to \mathbb{R}^n$ is smooth and there are constants $0 < \kappa < 1$ and $\rho > 0$, a vector $w_0 \in \mathbb{R}^n$, and an invertible matrix $A \in \mathbb{R}^{n\times n}$ so that
\begin{enumerate}
\item $\|1 - A^{-1}DH(w)\| \leq \kappa$ for all $w \in B_\rho(w_0)$, and
\item $\|A^{-1}H(w_0)\| \leq (1 - \kappa)\rho$
\end{enumerate}
then $H$ has a unique root $w_*$ in $B_\rho(w_0)$, and this root satisfies $|w_* - w| \leq \frac{1}{1-\kappa}\|A^{-1}F(w_0)\|$.
\end{lem}
\begin{proof}[Proof of Theorem~\ref{thm:Main}(1)]
Let us begin with the case $k = 2$ for illustration. In this case~\eqref{HomMatch} becomes
\begin{equation}\label{2HomMatch}
\begin{split}
G^u_2(v^s_{2,0},v^u_{2,0}) &= 0 \\
G^s_2(v^s_{2,N_{2}},v^u_{2,N_{2}}) &= 0 \\
\end{split}
\end{equation}
meaning that the orbit asymptotically connects $\tilde\gamma_1(0)$ and $\tilde\gamma_3(0)$ while iterating $N_2$ times through the neighborhood of $\tilde\gamma_2(0)$. Let us define the function $H_2:\mathbb{R}^2 \to \mathbb{R}^2$ by
\begin{equation}
H_2(a^s_2,a^u_2) := \begin{bmatrix}
G^u_2(v^s_{2,0}(a^s_2,a^u_2),v^u_{2,0}(a^s_2,a^u_2)) \\ G^s_2(v^s_{2,N_2}(a^s_2,a^u_2),v^u_{2,N_2}(a^s_2,a^u_2))
\end{bmatrix}
\end{equation}
so that for $(a^s_2,a^u_2)$ small the roots of $H_2$ are exactly the values of $(a^s_2,a^u_2)$ that satisfy the matching conditions~\eqref{2HomMatch}. Using the expansions~\eqref{Expansions} we have
\begin{equation}\label{2HomExpansion}
\begin{split}
G^s_2(v^s_{2,0},v^u_{2,0}) &= G^s_2(h^s_2+a^s_2,\mathcal{O}(\eta^{N_2})) = G^s(h^s_2+a^s_2,0) + \mathcal{O}(\eta^{N_2}) \\
G^u_2(v^s_{2,N_2},v^u_{2,N_2}) &= G^u_2(\mathcal{O}(\eta^{N_2}),h^u_2+a^u_2) = G^u(0,h^u_2+a^u_2) + \mathcal{O}(\eta^{N_2})
\end{split}
\end{equation}
when $N_2$ is taken sufficiently large.
Let us define the matrix
\begin{equation}
A_2 = \begin{bmatrix}
\partial_{v^s_2}G^s_2(h^s_2,0) & 0 \\ 0 & \partial_{v^u_2}G^u_2(0,h^u_2)
\end{bmatrix},
\end{equation}
which from Lemma~\ref{lem:GFn} is invertible since the partial derivatives are assumed to be nonzero. We then have $H_2(0,0) = \mathcal{O}(\eta^{N_2})$, and so $\|A_2^{-1}H_2(0,0)\| = \mathcal{O}(\eta^{N_2})$. Furthermore,
\begin{equation}
\|1 - A_2^{-1}DH_2(a^s_2,a^u_2)\| = \mathcal{O}(|a^s_2| + |a^u_2| + \eta^{N_2}).
\end{equation}
Then, with $\rho = \frac{1}{4}$, $\kappa = \frac{1}{2}$, and $N_2 \gg 1$ sufficiently large, Lemma~\ref{lem:Roots} gives that there exists a unique solution $(a^s_2,a^u_2) = (\bar a^s_2,\bar a^u_2)$ satisfying $H_2(\bar a^s_2,\bar a^u_2) = 0$ and
\begin{equation}
\|(\bar a^s_2,\bar a^u_2)\| = \mathcal{O}(\eta^{N_2}).
\end{equation}
Hence, we have satisfied the matching conditions~\eqref{2HomMatch} and therefore proven Theorem~\ref{thm:Main}(1) in the case $k = 2$.
The cases $k \geq 3$ are similar except that they include more matching conditions coming from the push-forward functions $\Pi_i$. Let us illustrate with $k = 3$. Then,~\eqref{HomMatch} becomes
\begin{equation}\label{3HomMatch}
\begin{split}
G^u_2(v^s_{2,0},v^u_{2,0}) &= 0 \\
\Pi_2(v^s_{2,N_2},v^u_{2,N_2}) - (v^s_{3,0},v^u_{3,0}) &= 0 \\
G^s_3(v^s_{3,N_{3}},v^u_{3,N_{3}}) &= 0 \\
\end{split}
\end{equation}
and as in the previous case, we will define the function $H_3:\mathbb{R}^4 \to \mathbb{R}^4$ by
\begin{equation}
H_3(a^s_2,a^u_2,a^s_3,a^u_3) := \begin{bmatrix}
G^u_2(v^s_{2,0}(a^s_2,a^u_2),v^u_{2,0}(a^s_2,a^u_2)) \\ \Pi_2(v^s_{2,N_2}(a^s_2,a^u_2),v^u_{2,N_2}(a^s_2,a^u_2)) - (v^s_{3,0}(a^s_3,a^u_3),v^u_{3,0}(a^s_3,a^u_3)) \\ G^s_3(v^s_{3,N_3}(a^s_3,a^u_3),v^u_{3,N_3}(a^s_3,a^u_3))
\end{bmatrix}
\end{equation}
so that when $(a^s_2,a^u_2,a^s_3,a^u_3)$ are sufficiently small the roots of $H_4$ are exactly the values of $(a^s_2,a^u_2,a^s_3,a^u_3)$ that satisfy the matching conditions~\eqref{3HomMatch}.
The expansions~\eqref{2HomExpansion} are still valid here with the appropriate changes of indices to accommodate the presence of $(v^s_{3,N_3},v^u_{3,N_3})$, and therefore we focus on the second condition. From~\eqref{Expansions} we have
\begin{equation}
\begin{split}
\Pi_2(v^s_{2,N_2},v^u_{2,N_2}) - (v^s_{3,0},v^u_{3,0}) &= \Pi_2(\mathcal{O}(\eta^{N_2}),h^u_2 + a^u_2) - (h^s_3 + a^s_3,\mathcal{O}(\eta^{N_3})) \\
&= \Pi_2(0,h^u_2 + a^u_2 ) - (h^s_3 + a^s_3,0) + \mathcal{O}(\eta^{\min\{N_2,N_3\}}).
\end{split}
\end{equation}
Let us define the matrix
\begin{equation}\label{A3}
A_3 := \begin{bmatrix}
\partial_{v^s_2}G^u_2(h^s_2,0) & 0 & 0 & 0 \\ 0 & \zeta^s_2 & -1 & 0 \\ 0 & \zeta^u_2 & 0 & 0 \\ 0 & 0 & 0 & \partial_{v^u_3}G^s(0,h^u_3)
\end{bmatrix} + \mathcal{O}(\eta^{\min\{N_2,N_3\}})
\end{equation}
where
\begin{equation}\label{xiEqn}
[\zeta^s_2, \zeta^u_2]^T = \partial_{a^u_2}\Pi_2(0,h^u_2).
\end{equation}
The matrix $A_3$ is invertible because Lemma~\ref{lem:GFn} gives that $\partial_{v^s_2}G^u_2(h^s_2,0),\partial_{v^u_3}G^s(0,h^u_3) \neq 0$ and $\zeta^u_2 \neq 0$ since vary $a^u_2$ in a neighborhood of zero causes $\Pi_2(0,h^u_2+a^u_2)$ to locally parametrize a connected component of $W^u(\tilde\gamma_2(0))$, which by assumption transversely intersects $W^s_\mathrm{loc}(\tilde\gamma_3(0)) = \{v^u_3 = 0\}$ at $(h^s_3,0) \in \mathcal{I}\times\mathcal{I}$ in a neighborhood of $\tilde\gamma_3(0)$. Then, $H_3(0,0,0,0) = \mathcal{O}(\eta^{\min\{N_2,N_3\}})$, thus giving that $\|A_3^{-1}H_3(0,0,0,0)\| = \mathcal{O}(\eta^{\min\{N_2,N_3\}})$. Furthermore,
\begin{equation}
\|1 - A_3^{-1}DH_3(a^s_2,a^u_2,a^s_3,a^u_3)\| = \mathcal{O}(|a^s_2| + |a^u_2| + |a^s_3| + |a^u_3| + \eta^{\min\{N_2,N_3\}}),
\end{equation}
and therefore with $\rho = \frac{1}{4}$, $\kappa = \frac{1}{2}$, and $N_2,N_3 \gg 1$, Lemma~\ref{lem:Roots} gives that there exists a unique solution $(a^s_2,a^u_2,a^s_3,a^u_3) = (\bar a^s_2,\bar a^u_2,\bar a^s_3,\bar a^u_3)$ satisfying $H_3(\bar a^s_2,\bar a^u_2,\bar a^s_3,\bar a^u_3) = 0$ and
\begin{equation}
\|(\bar a^s_2,\bar a^u_2,\bar a^s_3,\bar a^u_3)\| = \mathcal{O}(\eta^{\min\{N_2,N_3\}}).
\end{equation}
This solution satisfies the matching conditions~\eqref{3HomMatch} and therefore proves Theorem~\ref{thm:Main}(1) for the case $k = 3$.
The general setting of $k \geq 4$ is almost identical to the case of $k = 3$. Indeed, we can define a function $H_k:\mathbb{R}^{2k-2}\to\mathbb{R}^{2k-2}$ so that when $(a^s_2,a^u_2,\dots,a^s_k,a^u_k)$ are sufficiently small the roots of $H_k$ are exactly the values of $(a^s_2,a^u_2,\dots,a^s_k,a^u_k)$ that satisfy the matching conditions~\eqref{HomMatch}. We then define the matrix
\begin{equation}
A_k:= \begin{bmatrix}
\partial_{v^s_2}G^u_2(h^s_2,0) & 0 & 0 & \cdots & 0 & 0& 0 \\ 0 & \zeta^s_2 & -1 & \cdots & 0 & 0 & 0 \\ 0 & \zeta^u_2 & 0 & \cdots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & \zeta^s_{k-1} & -1 & 0\\
0 & 0 & 0 & \cdots &\zeta^u_{k-1} & 0 & 0\\
0 & 0 & 0 & \cdots & 0 & 0 & \partial_{v^u_k}G^s_k(0,h^u_k)
\end{bmatrix}
\end{equation}
where
\begin{equation}\label{Zeta}
[\zeta^s_i, \zeta^u_i]^T = \partial_{a^u_i}\Pi_i(0,h^u_i).
\end{equation}
for all $2\leq i \leq k-1$. The same arguments that were used to show that $A_3$ defined in~\eqref{A3} can be used to show that $A_k$ is invertible for every $k \geq 4$. Then, the application of Lemma~\ref{lem:Roots} to obtain a root of $H_k$ is as in the cases $k = 2,3$ above, thus satisfying the matching conditions~\eqref{HomMatch} and proving Theorem~\ref{thm:Main}(1) for the remain cases with $k \geq 4$.
\end{proof}
\subsection{Constructing Periodic Trajectories}\label{subsec:Periodic}
In this subsection we will prove Theorem~\ref{thm:Main}(2), concerning the existence of periodic trajectories. Much of the proofs will be the same as in the previous section and therefore we seek to only highlight the differences. Recall that the initial vertex $V_1$ of the walk is the same as the terminal vertex $V_{k+1}$, thus giving that $\tilde\gamma_{k+1}(t) = \tilde\gamma_1(t)$. This means that the desired periodic orbit enters and leaves the neighbourhoods of the orbits $\gamma_i(t)$ exactly $k$-times, while the final entry into the neighborhood of $\tilde\gamma_{k+1}(t)$ marks a completion of one full period of the trajectory. In these neighbourhoods we may employ Lemma~\ref{lem:ShilSol} to obtain $k \geq 1$ solutions of the form
\begin{equation}
\begin{split}
\{(v^s_{1,n},v^u_{1,n})\}_{n = 0}^{N_1},& \quad v^s_{1,0} = h^s_1 +a^s_1, \quad v^u_{1,N_1}= h^u_1 +a^u_1 \\
\{(v^s_{2,n},v^u_{2,n})\}_{n = 0}^{N_2},& \quad v^s_{2,0} = h^s_2 +a^s_2, \quad v^u_{2,N_2}= h^u_2 +a^u_2 \\
\{(v^s_{3,n},v^u_{3,n})\}_{n = 0}^{N_3},& \quad v^s_{3,0} = h^s_3 +a^s_3, \quad v^u_{3,N_3}= h^u_3 +a^u_3 \\
&\vdots \\
\{(v^s_{k,n},v^u_{k,n})\}_{n = 0}^{N_{k}},& \quad v^s_{k,0} = h^s_k +a^s_{k}, \quad v^u_{k,N_{k}}= h^u_k +a^u_{k},
\end{split}
\end{equation}
with the $a^s_i$ and $a^u_i$ taken to be small and the $N_i \geq 1$ sufficiently large to again satisfy~\eqref{Expansions}. The required matching conditions to prove Theorem~\ref{thm:Main}(2) then become
\begin{equation}\label{PerMatch}
\begin{split}
\Pi_1(v^s_{1,N_1},v^u_{1,N_1}) - (v^s_{2,0},v^u_{2,0}) &= 0 \\
\Pi_2(v^s_{2,N_2},v^u_{2,N_2}) - (v^s_{3,0},v^u_{3,0}) &= 0 \\
&\vdots \\
\Pi_{k-1}(v^s_{k-1,N_{k-1}},v^u_{k-1,N_{k-1}}) - (v^s_{k,0},v^u_{k,0}) &= 0 \\
\Pi_k(v^s_{k,N_k},v^u_{k,N_k}) - (v^s_{1,0},v^u_{1,0}) &= 0 .
\end{split}
\end{equation}
Notice the similarity between~\eqref{PerMatch} and~\eqref{HomMatch}, with the only difference coming from the first and last matching conditions. The first matching condition in~\eqref{PerMatch} simply describes iterates in the Poincar\'e section near $\tilde\gamma_1(0)$, while the final condition is the periodicity constraint that guarantees that after the trajectory transfers between $k$ local Poincar\'e sections it returns to where it began.
The process of satisfying the matching conditions~\eqref{PerMatch} is nearly identical to that of satisfying~\eqref{HomMatch} and will therefore be omitted. Briefly, one defines a function whose roots are exactly the choices of $(a^s_1,a^u_1,\dots,a^s_k,a^u_k)$ near the origin that satisfy the matching equations~\eqref{PerMatch}. The invertible matrix required to apply Lemma~\ref{lem:Roots} is given by
\begin{equation}
\begin{bmatrix}
0 & \zeta^s_2 & -1 & \cdots & 0 & 0 & 0 \\ 0 & \zeta^u_2 & 0 & \cdots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & \zeta^s_{k-1} & -1 & 0\\
0 & 0 & 0 & \cdots &\zeta^u_{k-1} & 0 & 0\\
-1 & 0 & 0 & \cdots & 0 & 0 & \zeta^s_k \\
0 & 0 & 0 & \cdots & 0 & 0 & \zeta^u_k
\end{bmatrix} \in \mathbb{R}^{2k\times 2k},
\end{equation}
where the $\zeta^s_i$ and $\zeta^u_i$ are as they are defined in~\eqref{Zeta}. The argument that the above matrix is invertible follows from the previous arguments that the $\zeta^u_i \neq 0$ for each $1 \leq i \leq k$, and so the application of Lemma~\ref{lem:Roots} is entirely analogous to the proof of Theorem~\ref{thm:Main}(1) in the previous subsection.
\clearpage
\section{Discussion}\label{sec:Conclusion}
This manuscript has provided a comprehensive exploration of saddle mediated transport in the double pendulum. Our work describes the state space organization of a benchmark chaotic system with a focus on the implications of our findings in physical space. Our methods combined both numerical and analytical techniques, guided by previous investigations into saddle mediated transport, while also providing a general result that describes macroscopic transport in general Hamiltonian dynamical systems. We began by reviewing index-1 saddles of a Hamiltonian system, while also demonstrating the importance of these equilibria by recreating known results from the PCR3BP. We then showed that the state space organization of the experimental double pendulum is similarly organized to the PCR3BP, primarily attributed to the presence of two index-1 saddles. The end result of this manuscript is twofold: we have extended the literature on saddle mediated transport to the double pendulum and provided a powerful theorem that can be used to produce itineraries throughout phase space for a large class of Hamiltonian systems.
The results of this manuscript have extended the concept of the Interplanetary Transport Network to the double pendulum, similarly showing that there are physically determined pathways that allow one to move throughout phase space at no extra energy expenditure. The relative ease of building a double pendulum means that these transport mechanisms can be tested and followed physically as a table-top analogue of interplanetary space travel. Unfortunately, unlike the PCR3BP, a physical double pendulum model has frictional forces which destroy the tube structures as global invariant manifolds. However, for small dissipation in the system, these tubes will approximately remain on long timescales for which a control algorithm can be implemented to correct any discrepancies due to friction. Many researchers have sought to control the chaotic dynamics of the double pendulum by putting its anchored arm on a movable cart. These previous investigations are primarily concerned with stabilizing the unstable steady-states of the pendulum~\cite{driver2004design,zhong2001energy,ControlPend,hesse2018reinforcement}, such as the index-1 saddles, while the work herein opens up the possibility to stabilize a variety of acrobatic movements. With a double pendulum on a cart, methods of controlling chaos~\cite{OGY,ControlChaos1,ControlChaos2} could be implemented to introduce small movements of the cart to re-orient the pendulum arms to correct for any divergence of the tubes due to friction. Implementing these control methods on a physical system may require the aid of data-driven control methods~\cite{BramUPO} and/or discrepancy modelling~\cite{kaheman2019learning}, while also being guided by recent theoretic investigations into the effect of small amounts of dissipation or periodic forcing that break the Hamiltonian structure~\cite{DissRoss,ZhongGeometry,PerForced}. This is an avenue of ongoing work that builds on the theory in this manuscript.
In a follow-up theoretical investigation, one may wish to understand the dynamics generated by the index-2 Up-Up saddle point in the double pendulum. This equilibrium has a two-dimensional stable and unstable manifold, and so does not have the nearby UPOs which give the tube structure near the index-1 saddles. Nonetheless, the two-dimensional manifolds of the Up-Up state could intersect along a one-dimensional curve, resulting in homoclinic orbits that similarly produce macroscopic transport mechanisms at higher energy values in the double pendulum. One may similarly seek to identify the transport pathways in the triple pendulum, formed by appending a third arm to the double pendulum. The triple pendulum has three index-1 saddles: Down-Down-Up, Down-Up-Down, and Up-Down-Down, which are named analogously to the states in the double pendulum. The `tube' structures near these index-1 saddles are now three-dimensional, corresponding to a sphere crossed with a line, and so intersections should be expected to be two-dimensional~\cite{Chemical2}. Thus, the homoclinic and heteroclinic orbits in the double pendulum become two-dimensional invariant manifolds whose dynamics asymptotically approach neighborhoods of the index-1 saddles of the triple pendulum. Both situations require a delicate numerical treatment that will be left to a follow-up investigation.
\section*{Acknowledgments}
The authors acknowledge funding support from the Army Research Office (ARO W911NF-19-1-0045) and National Science Foundation AI Institute in Dynamic Systems (grant number 2112085).
The authors also would like to thank Shane Ross for valuable discussions.
\begin{spacing}{.77}
\small{
\setlength{\bibsep}{.8pt}
\addcontentsline{toc}{section}{References}
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,715 |
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