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The Trenton Psychiatric Hospital is a state run mental hospital located in Trenton and Ewing, New Jersey. It previously operated under the name New Jersey State Hospital at Trenton and originally as the New Jersey State Lunatic Asylum.
Founded by Dorothea Lynde Dix on May 15, 1848, it was the first public mental hospital in the state of New Jersey, and the first mental hospital designed on the principle of the Kirkbride Plan. The architect was the Scottish-American John Notman.
Under the hospital's first superintendent, Dr. Horace A. Buttolph, the hospital admitted and treated 86 patients. In 1907, Dr. Henry Cotton became the medical director. Believing that infections were the key to mental illness, he had his staff remove teeth and various other body parts that might become infected from the hospital patients. Cotton's legacy of hundreds of fatalities and thousands of maimed and mutilated patients did not end with his leaving Trenton in 1930 or his death in 1933; in fact, removal of patients' teeth at the Trenton asylum was still the norm until 1960.
See also
John Forbes Nash (1928–2015), patient
Nathan Trupp (born 1947), patient
Howard Unruh (1921–2009), patient
Human experimentation in the United States
Willowbrook State School
Greystone Park Psychiatric Hospital, the second "lunatic asylum" opened in New Jersey (1876).
Madhouse: A Tragic Tale of Megalomania and Modern Medicine
References
External links
Trenton Psychiatric Hospital, New Jersey Department of Human Services, Division of Mental Health Services
http://www.rootsweb.com/~asylums/trenton_nj/
http://ajp.psychiatryonline.org/cgi/content/full/156/12/1982
http://www.forgottenphotography.com
Psychiatric hospitals in New Jersey
Hospitals established in 1848
Buildings and structures in Mercer County, New Jersey
Kirkbride Plan hospitals
Psychiatry controversies
Buildings and structures in Trenton, New Jersey | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,147 |
\section{Introduction}
Large-scale structure (LSS) surveys constrain cosmology with ever higher
precision (e.g.~\cite{Anderson1312} and references therein). However,
the unkown bias between invisible dark matter and observable tracers
introduces degeneracies which weaken constraints on cosmological
parameters. In particular, the normalization of the matter power
spectrum $\sigma_8$ is degenerate with the unknown linear bias $b_1$
if only the $2$-point correlation function or the power spectrum in
Fourier space are measured, because they only depend on the
combination $b_1^2\sigma_8^2$ (on large scales). This degeneracy is
reduced if anisotropic redshift-space distortions are
included. Further improvements can be obtained by probing statistics
beyond the $2$-point level such as the $3$-point correlation function
or the bispectrum in Fourier space, exploiting the fact that different
combinations of cosmological and bias parameters are associated with
different functional dependencies of the bispectrum on triangle shape.
This has been demonstrated on galaxy survey data in
\cite{2001ApJ...546..652S,2001PhRvL..86.1434F,2002MNRAS.335..432V,2004ApJ...607..140J,2004MNRAS.353..287W,2013MNRAS.432.2654M,2011ApJ...739...85M,2011ApJ...726...13M}. Recently,
Gil-Marin et al.~\cite{Hector1407Data1,Hector1408Data2} measured the
bispectrum of SDSS DR11 BOSS galaxies with unprecedented accuracy
obtaining constraints on the growth rate $f$ and the clustering
amplitude $\sigma_8$ from galaxy clustering data alone. An additional
important motivation to measure large-scale structure bispectra is to
constrain primordial non-Gaussianity that can help to distinguish
inflation models.
All analyses of real data to date have estimated the bispectrum or
$3$-point correlation function in a brute-force approach, going
manually over triangle configurations and orientations. The number of
these possible triangles is very large because it is cubic in the
number of grid points per dimension (e.g.~choosing ten $k$-bins in
every direction gives $\mathcal{O}(10^3)$ triangles). This leads to
several problems. First, the estimation of the bispectrum itself is
computationally expensive, which is typically overcome by considering
only a subset of possible triangles which comes however at the expense
of losing a potentially significant fraction of the signal in the
data. Second, the covariance between thousands or more triangles is
hard to estimate from simulations because it would require a large
number of independent realizations.\footnote{Analytical covariances can only
partially overcome this problem because it is hard to include
real-world effects from e.g.~the survey window function. A possible
work-around is to neglect non-trivial bispectrum covariances in the
estimation procedure and only estimate the covariance between final
cosmological and bias parameters from simulations
\cite{hector1407theory,Hector1407Data1,Hector1408Data2}. However, this
leads to a sub-optimal estimator that is only unbiased in the limit of
infinitely many observations.} Third, the task of testing theoretical
models of the bispectrum against simulations is rather complex because
it is not obvious which triangle configurations are relevant for final
parameter constraints, making it nontrivial to find a suitable
metric for comparing models against simulations. Lastly, modeling
anisotropic effects due to redshift-space distortions or survey
geometry at the level of triangles can be cumbersome.
To avoid some of these issues, the goal of this paper is to propose
alternative
simpler statistics that do not require direct
bispectrum estimation for individual triangles but still include the full information of the
bispectrum on bias and cosmological parameters in an optimal way. The
basic idea to achieve this is to square the density and cross-correlate it
with the density itself. This cross-spectrum then depends on the
bispectrum.
In fact, the separability of dark matter (DM) and halo bispectra can be exploited to
show that optimal bispectrum estimation is equivalent to
cross-correlating three fields that are quadratic in the density with
the density itself. Due to the particular form of DM and halo
bispectra, these three quadratic fields are the
squared density in configuration space $\delta^2(\vec{x})$, a shift term that contracts the density
gradient with the displacement field,
$\Psi^i(\vec{x})\partial_i\delta(\vec{x})$, and a tidal term, $s^2(\vec{x})$.
The cross-spectra of these quadratic fields
with the density encode the full bispectrum information but require
only the computational cost of typical power spectrum analyses, which
is much faster than brute-force bispectrum estimation. Additionally,
since the cross-spectra only depend on a single scale $k$, covariances
are much simpler to estimate from simulations than covariances between
all triangle configurations.
The quadratic fields themselves can be computed efficiently as the
product of fields in configuration space at the same location $\vec{x}$. The
factors are given by the density itself or related quantities like
e.g.~the density gradient $\partial_i\delta$ or the displacement field
$\Psi_i=-(\partial_i\partial^{-2})\delta$.
The displacement field is already conventionally
computed from real data when employing reconstruction to sharpen the
BAO peak, so it should be straightforward to compute the other fields
we propose in a similar fashion.
One of the three proposed statistics, the cross-spectrum between the
squared density and the density, has been studied previously in the
literature \cite{Bernardeau:1996,Bel:2012,Hoffmann1403}, but mostly in
configuration space rather than Fourier space and in the large separation
limit. Some of these studies speculated that this cross-spectrum contains
information not present in the full three point function. The same
statistic was also analysed in Fourier space in \cite{Pollack1309}.
We will point out that the inverse variance weighted integral over
this statistic in Fourier space is in fact an optimal estimator for the
amplitude of the angle-independent part of the bispectrum, which is
sensitive to a combination of $b_1$ and $b_2$, and we calculate this
statistic consistently using one loop perturbation theory.
Furthermore, we are extending these studies in considering a more
realistic bias model and will show in detail that considering the
other two cross-spectra involving other transformations than squaring
are instrumental in constraining the bias parameters.
Our proposed procedure is also
related to (but different from) a method recently proposed by Chiang et
al.~\cite{chiang1403}, which similarly aims at compressing bispectrum
information to simpler observable quantities. Their work considers the
correlation between locally measured power spectra with the local mean
density, which is determined by the squeezed limit of the bispectrum.
However, the squeezed limit of bispectra generated by nonlinear
gravity is suppressed in absence of primordial non-Gaussianity. In
contrast, we systematically derive the types of cross-spectra that
explicitly probe bispectrum shapes that are optimal to estimate bias
parameters, arriving at different statistics than the one proposed in
\cite{chiang1403}.
To some extent
our method can be regarded as a simplification of the
separable expansion method to estimate general bispectra
\cite{shellard1008,shellard1108,marcel1207} by tailoring it to estimate
bias parameters from large-scale structure bispectra, keeping the
scale-dependence more obvious by not integrating over
scales $k$. The method proposed here could be combined with separable
expansion estimators to probe bispectrum contributions beyond those
included in our modeling, but we leave this for future work.
In the
context of the CMB, the fact that separable
bispectra and trispectra can be estimated more efficiently than
non-separable bispectra has been exploited for a long time. For
example, the KSW estimator \cite{ksw} is often used
to estimate separable primordial bispectra directly
(e.g.~$f_\mathrm{NL}^\mathrm{local}$), primordial bispectra are often
approximated by separable templates to facilitate their estimation,
the ISW-lensing bispectrum can be measured by cross-correlating the
lensing reconstruction field (which is quadratic in the CMB
temperature) with the CMB temperature \cite{lewis1101,Planck1303ISW},
and primordial and non-primordial general bispectra can be estimated
by expanding in separable basis functions
\cite{shellard0912,shellard1006,Planck1303NG}. Similarly, the
separability of the lensing-induced trispectrum of the CMB temperature
is exploited when computing the auto-power spectrum of the CMB
lensing reconstruction field, which is a quadratic function in the CMB
temperature with derivative operations dictated by the form of the CMB
trispectrum
\cite{ZaldarriagaSeljak98LensingRec,okamotoHu0301,duncan1008}.
However, the explicit separability of bispectra or trispectra induced
by nonlinear gravity and bias relations has so far not been exploited
in the field of large-scale structure. The aim of this paper is to
provide a theoretical framework for this and test it with simulations.
We restrict ourselves to real space in this paper, but note
that redshift space distortions (RSDs) should be included before
applying the proposed technique to real data. Since RSDs
change the expected bispectrum signal, in particular rendering it
non-isotropic, the specific estimators we derive in real space
should be modified and extended in order to be optimal in redshift
space. While this will likely increase the number of cross-spectra,
making the analysis somewhat more complicated,
we do not anticipate any new conceptual challenges because the
leading-order RSDs and certain Fingers-of-God models are still
product-separable. Since venturing into redshift
space is beyond the scope of this paper we leave it for future
work.
The paper is organized as follows. We
start with general definitions of quadratic fields and
bispectra in Section \ref{se:QuadraticFields}. Section \ref{se:MaxLikeliBispEsti} describes the
relationship between optimal bispectrum estimation and cross-spectra
of quadratic fields. Theoretical predictions for the cross-spectra are
computed in Section \ref{se:TheoryCrossSpectra}. Section
\ref{se:Simulations} describes simulation results in comparison with
theoretical predictions. An extension to primordial
non-Gaussianity is briefly discussed in Section
\ref{se:Extensions}. Finally we conclude in Section
\ref{se:Conclusions}. Two appendices provide technical details
of large-scale limits and analytical covariances.
\section{Quadratic fields and bispectrum decomposition}
\label{se:QuadraticFields}
\subsection{Quadratic fields}
As will be shown in Section \ref{se:MaxLikeliBispEsti}, maximum-likelihood bispectrum
estimators for bias parameters can be cast in form of cross-spectra of
the density field with three fields that are quadratic in the
configuration-space density, with different dependencies on the cosine $\mu$
between the Fourier space wavevectors $\vec{q}$ and $\vec{k}-\vec{q}$,
\begin{equation}
\label{eq:mu_def}
\mu \equiv \frac{\vec{q}\cdot(\vec{k}-\vec{q})}{q|\vec{k}-\vec{q}|}.
\end{equation}
Explicitly, these three quadratic fields are:
\begin{itemize}
\item
The \emph{squared
density} $\delta^2(\vec{x})$, which can be written as a convolution in
Fourier space,
\begin{equation}
\label{eq:delta2_x}
\delta^2(\vec{x}) = \int\frac{\d^3 k}{(2\pi)^3}e^{i\vec{k}\vec{x}}
\int\frac{\d^3 q}{(2\pi)^3} \mathsf{P}_0(\mu)\delta(\vec{q})\delta(\vec{k}-\vec{q}),
\end{equation}
where $\mathsf{P}_0(\mu)=1$ is the Legendre polynomial for
$l=0$.\footnote{$\mathsf{P}_l(\mu)$ with $l\in\{0,1,2\}$ always denotes Legendre
polynomials in this paper and should not be confused with power spectra $P(k)$.}
\item
The \emph{shift-term}
\begin{equation}
\label{eq:shift_x}
-\Psi^i(\vec{x})\partial_i\delta(\vec{x})=
-{\mathbf{\Psi}}(\vec{x})\cdot\nabla\delta(\vec{x})
= -\int\frac{\d^3 k}{(2\pi)^3}e^{i\vec{k}\vec{x}}\int\frac{\d^3 q}{(2\pi)^3}
F_2^1(q,|\vec{k}-\vec{q}|)\mathsf{P}_1(\mu) \delta(\vec{q})\delta(\vec{k}-\vec{q}),
\end{equation}
which depends on the $l=1$ Legendre polynomial $\mathsf{P}_1(\mu)=\mu$
and is obtained by contracting the density gradient $\nabla\delta$
with the displacement field
\begin{equation}
\label{eq:vPsi_def}
{\mathbf{\Psi}}(\vec{k}) = -\frac{i\vec{k}}{k^2}\delta(\vec{k}).
\end{equation}
The symmetric kernel $F_2^1$ in \eqq{shift_x} is defined as
\begin{equation}
\label{eq:40}
F_2^1(k_1,k_2) = \frac{1}{2}\left( \frac{k_1}{k_2} + \frac{k_2}{k_1} \right).
\end{equation}
\item
The \emph{tidal term}\footnote{An overall factor of $3/2$ is absorbed
compared to e.g.~\cite{tobias1201},
i.e.~$s^2_\mathrm{here}(\vec{x})=\frac{3}{2}s^2_\mathrm{there}(\vec{x})$ and
$\mathsf{P}_{2}^\mathrm{here}(\mu)=\frac{3}{2}S_2^\mathrm{there}(\vec{q},\vec{k}-\vec{q})$.}
\begin{equation}
\label{eq:s2_x}
s^2(\vec{x})\equiv \frac{3}{2} s_{ij}(\vec{x})s_{ij}(\vec{x}) =
\int\frac{\d^3 k}{(2\pi)^3}e^{i\vec{k}\vec{x}}
\int \frac{\d^3 q}{(2\pi)^3}\mathsf{P}_2(\mu)
\delta(\vec{q})\delta(\vec{k}-\vec{q}),
\end{equation}
which is defined by contracting
the tidal tensor
\begin{equation}
\label{eq:sij_vk}
s_{ij}(\vec{k}) = \left(
\frac{k_ik_j}{k^2} - \frac{1}{3}\delta^\mathrm{(K)}_{ij}
\right)\delta(\vec{k})
\end{equation}
with itself. $\delta^\mathrm{(K)}_{ij}$ denotes the Kronecker delta.
The corresponding convolution kernel in \eqq{s2_x} is given by the $l=2$ Legendre polynomial
\begin{equation}
\label{eq:46}
\mathsf{P}_2(\mu) = \frac{3}{2}\left(\mu^2-\frac{1}{3}\right).
\end{equation}
\end{itemize}
While these three quadratic fields will be derived more rigorously
below, some intuition for their appearance can be gained as
follows. In standard perturbation theory (see \cite{BernardeauReview}
for a review) the dark matter density is expanded in powers of the
linear perturbation $\delta_0$.
In configuration space, truncating at second order, this can be written as
(e.g.~\cite{Bouchet1992ApJ...394L...5B,tobias1201,SherwinZaldarriaga1202})
\begin{equation}
\label{eq:delta_2ndorder_x}
\delta_\mathrm{m}(\vec{x}) = \delta_0(\vec{x}) + \frac{17}{21} \delta^2_0(\vec{x}) + {\mathbf{\Psi}}_0(\vec{x})\cdot
\nabla \delta_0(\vec{x}) + \frac{4}{21}s_0^2(\vec{x}).
\end{equation}
The biased halo density can be modeled by
\cite{McDonaldRoy09,tobias1201,KwanScoccimarro1201}
\begin{equation}
\label{eq:bias_relation}
\delta_\mathrm{h}(\vec{x}) = b_1\delta_\mathrm{m}(\vec{x}) + b_2\left[
\delta_\mathrm{m}^2(\vec{x})-\langle \delta_\mathrm{m}^2(\vec{x})\rangle\right]
+\frac{2}{3}b_{s^2}\left[s_\mathrm{m}^2(\vec{x})-\langle s_\mathrm{m}^2(\vec{x})\rangle\right],
\end{equation}
where the normalization of the bias parameters $b_1$, $b_2$ and
$b_{s^2}$ is the same as in \cite{tobias1201}. Since the linear
perturbation $\delta_0$ is assumed to be Gaussian, the leading order
$3$-point function is due to expectation values of the form
$\langle\delta^{(2)}\delta_0\delta_0\rangle$, where $\delta^{(2)}$ can
be any of the terms in Eqs.~\eq{delta_2ndorder_x} and
\eq{bias_relation} that are quadratic in $\delta_0$. There are only
three types of such terms; the squared density \eq{delta2_x}, the
shift term \eq{shift_x} and the tidal term \eq{s2_x}. The
corresponding $k$-dependencies that these fields imprint on the
bispectra will later be used to reduce optimal bispectrum estimation
to cross-spectra of these quadratic fields with the density.
To simplify calculations in the rest of the paper we define a general
quadratic field $D[\delta](\vec{k})$ obtained from a density realization
$\delta(\vec{k})$ and some kernel $D(\vec{q}, \vec{k}-\vec{q})$, always assumed to be
symmetric in its arguments,
by\footnote{$D[\delta]$ with square brackets denotes the functional
that turns some density realization $\delta(\vec{k})$ into the r.h.s.~of
\eqq{D_functional}. $D(\vec{q},\vec{k}-\vec{q})$ denotes the corresponding
kernel.}
\begin{equation}
\label{eq:D_functional}
D[\delta](\vec{k}) \equiv \int\frac{\d^3 q}{(2\pi)^3}D(\vec{q},\vec{k}-\vec{q})\delta(\vec{q})\delta({\vec{k}-\vec{q}}).
\end{equation}
The squared density \eq{delta2_x} corresponds to the identity kernel
$D(\vec{q},\vec{k}-\vec{q})=\mathsf{P}_0(\mu)=1$, the shift term \eq{shift_x} to
$D(\vec{q},\vec{k}-\vec{q})=-F_2^1(q,|\vec{k}-\vec{q}|)\mathsf{P}_1(\mu)$ and the tidal term \eq{s2_x} to
$D(\vec{q},\vec{k}-\vec{q})=\mathsf{P}_2(\mu)$. We will usually suppress the arguments and
just write $D\in\{\mathsf{P}_0, -F_2^1\mathsf{P}_1, \mathsf{P}_2\}$.
\subsection{Cross-spectra}
The cross-spectrum of a quadratic field $D[\delta]$ with the density is
\begin{equation}
\label{eq:32}
\langle
D[\delta](\vec{k})\delta(\vec{k}')
\rangle \equiv (2\pi)^3 \delta_D(\vec{k}+\vec{k}')P_{D[\delta],\delta}(k),
\end{equation}
where $\delta_D$ always denotes the Dirac delta function. Given a
realization of the density, an unbiased estimator for the
cross-spectrum is given by
\begin{equation}
\label{eq:hat_cross_power}
\hat P_{D[\delta],\delta}(k) = \frac{1}{4\pi L^3}\int
\d\Omega_{\hat\vec{k}}
D[\delta](\vec{k})\delta(-\vec{k}),
\end{equation}
where $L$ is the box size.
The normalization is fixed by noting that
$(2\pi)^3\delta_D(\mathbf{0})=L^3$ and requiring
\begin{equation}
\label{eq:39}
\langle \hat P_{D[\delta],\delta}(k)\rangle = P_{D[\delta],\delta}(k).
\end{equation}
On a discrete grid the angular integral is replaced by a sum over
discrete $\vec{k}$ vectors with wavenumber $|\vec{k}|$ belonging to the bin
centered at $k$,
\begin{equation}
\label{eq:discrete_sum_over_modes}
\frac{1}{4\pi} \int \d\Omega_{\hat\vec{k}} \quad
\rightarrow
\quad
\frac{1}{N_\mathrm{modes}(k)}\sum_{\vec{k}, [k-\Delta k/2\le |\vec{k}|\le k+\Delta
k/2]},
\end{equation}
where $N_\mathrm{modes}(k)= 4\pi (k/\Delta k)^2$ at high $k$ if the binning
width is $\Delta k$. In practice, we count the number of
modes manually in the code because this is more accurate at low $k$.
\subsection{Legendre decomposition of gravitational bispectra}
At leading order in perturbation theory the DM bispectrum is given
by\footnote{The $\vec{k}_i$ must be such that they form a closed triangle,
$\vec{k}_1+\vec{k}_2+\vec{k}_3=0$. Up to the overall orientation, the triangle
can be specified e.g.~by three sidelengths $k_1$, $k_2$ and $k_3$,
which fixes the angle between any two sides,
e.g.~$\vec{k}_1\cdot\vec{k}_2=(k_3^2-k_1^2-k_2^2)/2$. We will parametrize
triangles by sidelengths and/or angles, using whichever is most
convenient in the context.}
\begin{equation}
\label{eq:BDM}
B_\mathrm{mmm}(k_1, k_2, k_3) = 2P_\mathrm{mm}^\mathrm{lin}(k_1)P_\mathrm{mm}^\mathrm{lin}(k_2)F_2(\vec{k}_1,\vec{k}_2)+\mbox{ 2 perms in }\, k_1,k_2,k_3,
\end{equation}
where $P^\mathrm{lin}_\mathrm{mm}$ is the
linear DM power spectrum and $F_2$ denotes the symmetrized kernel for
the second order density perturbation,
\begin{equation}
\label{eq:F2}
F_2(\vec{k}_1,\vec{k}_2) = \frac{17}{21} + \frac{1}{2}\left(
\frac{k_1}{k_2}+\frac{k_2}{k_1}
\right) \hat\vec{k}_1\cdot\hat\vec{k}_2
+
\frac{4}{21}\,\frac{3}{2}\left((\hat\vec{k}_1\cdot\hat\vec{k}_2)^2 -\frac{1}{3}\right),
\end{equation}
where $\hat \vec{k}_i=\vec{k}_i/k_i$.
The decomposition of this kernel in Legendre polynomials in the cosine
$\hat\vec{k}_1\cdot\hat\vec{k}_2$ is
\begin{equation}
\label{eq:F2LegendreDecomp}
F_2(\vec{k}_1,\vec{k}_2) = \sum_{l=0}^2 F_2^{l}(k_1,k_2)\mathsf{P}_l(\hat\vec{k}_1\cdot\hat\vec{k}_2)
\end{equation}
with coefficients $F_2^l$ given by
\begin{eqnarray}
F_{2}^{0}(k_1,k_2) &=& \frac{17}{21},\\
F_{2}^{1}(k_1,k_2) &=&
\frac{1}{2}\left(\frac{k_1}{k_2}+\frac{k_2}{k_1}\right),
\\
F_{2}^{2}(k_1,k_2) &=&
\frac{4}{21}.
\end{eqnarray}
The bispectrum \eq{BDM} can thus be split
in parts that depend on different Legendre
polynomials in the angle $\hat\vec{k}_1\cdot\hat\vec{k}_2$ (and permutations),
\begin{eqnarray}
\label{eq:B_split_in_Bmu_parts}
B_\mathrm{mmm}(k_1,k_2,k_3)
&=&\sum_{l=0}^2 B^{(l)}_\mathrm{mmm}(k_1,k_2,k_3),
\end{eqnarray}
where
\begin{equation}
\label{eq:BDM_l}
B_\mathrm{mmm}^{(l)}(k_1,k_2,k_3) =
2P_\mathrm{mm}^\mathrm{lin}(k_1)P_\mathrm{mm}^\mathrm{lin}(k_2)F_2^{l}(k_1,k_2)\mathsf{P}_l(\hat\vec{k}_1\cdot\hat\vec{k}_2) + \mbox{2 perms}.
\end{equation}
The leading-order unsymmetrized matter-matter-halo bispectrum can be
obtained from the bias relation \eq{bias_relation},
\begin{align}
\label{eq:Bmmh_unsymm}
B_\mathrm{mmh}^\mathrm{unsym}(k_1,k_2,k_3) =
&\;b_1B_\mathrm{mmm}(k_1,k_2,k_3)
+ 2b_2P_\mathrm{mm}^\mathrm{lin}(k_1)P_\mathrm{mm}^\mathrm{lin}(k_2)
+
\frac{4}{3}b_{s^2}P_\mathrm{mm}^\mathrm{lin}(k_1)P_\mathrm{mm}^\mathrm{lin}(k_2)\mathsf{P}_2(\hat\vec{k}_1\cdot\hat\vec{k}_2),
\end{align}
where $k_3$
is for the halo density.
This can similarly be decomposed into $l=0,1,2$
Legendre polynomials:
\begin{eqnarray}
\label{eq:Bmmh_l0}
B_\mathrm{mmh}^{\mathrm{unsym},(0)}(k_1,k_2,k_3) &=& b_1 B_\mathrm{mmm}^{(0)}(k_1,k_2,k_3) +
2b_2P_\mathrm{mm}^\mathrm{lin}(k_1)P_\mathrm{mm}^\mathrm{lin}(k_2),\\
\label{eq:Bmmh_l1}
B_\mathrm{mmh}^{\mathrm{unsym},(1)}(k_1,k_2,k_3) &=& b_1
B_\mathrm{mmm}^{(1)}(k_1,k_2,k_3),\\
\label{eq:Bmmh_l2}
B_\mathrm{mmh}^{\mathrm{unsym},(2)}(k_1,k_2,k_3) &=& b_1 B_\mathrm{mmm}^{(2)}(k_1,k_2,k_3) +
\frac{4}{3}b_{s^2}P_\mathrm{mm}^\mathrm{lin}(k_1)P_\mathrm{mm}^\mathrm{lin}(k_2)\mathsf{P}_2(\hat\vec{k}_1\cdot\hat\vec{k}_2).
\end{eqnarray}
Similarly, the halo-halo-halo bispectrum
\begin{equation}
\label{eq:Bhhh}
B_\mathrm{hhh}(k_1,k_2,k_3)
=
2P_\mathrm{mm}^\mathrm{lin}(k_1)P_\mathrm{mm}^\mathrm{lin}(k_2)\left[
b_1^3F_2(\vec{k}_1,\vec{k}_2)+b_1^2b_2+\frac{2}{3}b_1^2b_{s^2}\mathsf{P}_2(\hat\vec{k}_1\cdot\hat\vec{k}_2)
\right]+\mbox{2 perms}
\end{equation}
can be decomposed into
\begin{eqnarray}
\label{eq:Bhhh_l0}
B_\mathrm{hhh}^{(0)}(k_1,k_2,k_3) &=& b_1^3 B_\mathrm{mmm}^{(0)}(k_1,k_2,k_3) +
2b_1^2b_2\left[P_\mathrm{mm}^\mathrm{lin}(k_1)P_\mathrm{mm}^\mathrm{lin}(k_2)+\mbox{2
perms}\right],\\
\label{eq:Bhhh_l1}
B_\mathrm{hhh}^{(1)}(k_1,k_2,k_3) &=& b_1^3
B_\mathrm{mmm}^{(1)}(k_1,k_2,k_3),\\
\label{eq:Bhhh_l2}
B_\mathrm{hhh}^{(2)}(k_1,k_2,k_3) &=& b_1^3 B_\mathrm{mmm}^{(2)}(k_1,k_2,k_3) +
\frac{4}{3}b_1^2b_{s^2}\left[P_\mathrm{mm}^\mathrm{lin}(k_1)P_\mathrm{mm}^\mathrm{lin}(k_2)\mathsf{P}_2(\hat\vec{k}_1\cdot\hat\vec{k}_2)
+\mbox{2 perms}\right].
\end{eqnarray}
Since the $l=1$ part depends only on $b_1$ and is not
contaminated by
the nonlinear bias parameters $b_2$ and $b_{s^2}$ we expect it to be
most powerful for determining $b_1$. The $l=0$ contribution depends on a
mixture of $b_1$ and $b_2$, so one could constrain $b_2$ if $b_1$ was
known. Similarly, the $l=2$ part depends on $b_1$ and $b_{s^2}$, so
$b_{s^2}$ can be constrained once $b_1$ is known.
Note that all bispectrum contributions are product-separable and only
depend on $l=0,1,2$ Legendre polynomials. In the following section we
will discuss how this can be used to simplify estimators for the
amplitude of bispectrum contributions, which can in turn be used to
estimate bias parameters.
\section{Maximum likelihood bispectrum estimation}
\label{se:MaxLikeliBispEsti}
\subsection{General bispectra}
Assuming a fiducial theoretical power spectrum $P_\delta(k)$ and
bispectrum $f_\mathrm{NL}^{B_\mathrm{theo}}{B_\delta^{\textrm{theo}}}$ for the density
perturbation $\delta$, the maximum likelihood estimator (in the limit
of weak non-Gaussianity) for the amplitude\footnote{In our
notation $f_\mathrm{NL}^B$ denotes the nonlinearity amplitude
of an arbitrary bispectrum $B$, no matter if it is generated by
primordial non-Gaussianity or nonlinear gravity.}
$f_\mathrm{NL}^{B_\mathrm{theo}}$ of the bispectrum is given by
\cite{shellard1008,shellard1108}
\begin{align}
\label{eq:fnl_esti}
\hat{f}_\mathrm{NL}^{B^\mathrm{theo}} =&
\frac{(2\pi)^3}{N_\mathrm{theo}} \int
\frac{\d^3 k}{(2\pi)^3}\int\frac{\d^3q}{(2\pi)^3}
\frac{
{B_\delta^{\textrm{theo}}}(\vec{q},\vec{k}-\vec{q},-\vec{k})[\delta(\vec{q})\delta(\vec{k}-\vec{q})\delta(-\vec{k})-3\langle
\delta(\vec{q})\delta(\vec{k}-\vec{q})\rangle \delta(-\vec{k})]}{P_\delta(q)P_\delta(|\vec{k}-\vec{q}|)P_\delta(k)}
\end{align}
where $P_\delta$ factors in the denominator represent
inverse-variance weighting of the observed density perturbation
$\delta$, and $N_\mathrm{theo}$ is a normalization factor depending
on the theoretical bispectrum whose amplitude we aim to measure (see
\cite{marcel1207} for the explicit definition). The term
linear in the observed density will be
omitted in the following because it is only relevant if the field is
statistically inhomogeneous (e.g.~due to inhomogenous noise).
The estimator of \eqq{fnl_esti} is unbiased and includes information
from the full bispectrum, i.e.~all triangle configurations and
orientations. It is optimal in the limit of weak non-Gaussianity
since it is derived by Edgeworth-expanding the likelihood around a
Gaussian pdf truncating higher orders of $f_\mathrm{NL}$ and connected
$n$-point functions beyond the bispectrum. The
inverse-variance weighting of the density in
\eqq{fnl_esti} is optimal if higher-order contributions to the
bispectrum covariance are neglected. The estimator can be improved by
relaxing these assumptions, but we leave this to future
work.\footnote{Alternatively, constraints can be tightened by
pushing the analysis to smaller scales, which could be achieved by improving
the theory modeling of the bispectrum (see
e.g.~\cite{scocci_fitting_formula,hector1111,hector1407theory,tobias1406EFT,angulo1406EFT}),
or by applying clipping or logarithmic density transforms
\cite{Neyrinck0903,SimpsonClipping1107}.}
\subsection{Estimating separable bispectra by cross-correlating linear and
quadratic fields}
The bispectra discussed above are given by sums
of terms that have a product-separable form.\footnote{At leading
order this is the case because the second order density
\eq{delta_2ndorder_x} and the halo density \eq{bias_relation}
depend on products of fields evaluated at the same location $\vec{x}$
which can be written as convolutions with separable kernels in
Fourier space.} For simplicity, let us start with a
bispectrum given by just one such product-separable term
\begin{equation}
\label{eq:generic_separable_bisp_fgh}
B_\delta^\mathrm{theo}(\vec{k}_1,\vec{k}_2,\vec{k}_3) =
f(\vec{k}_1)g(\vec{k}_2)h(\vec{k}_3)
\end{equation}
for some functions $f, g$ and $h$. Then the estimator \eq{fnl_esti} becomes
\begin{equation}
\label{eq:fnl_split_integrals}
\hat{f}_\mathrm{NL}^{B^\mathrm{theo}}
=
\frac{(2\pi)^3}{N_\mathrm{theo}} \int
\frac{\d^3 k}{(2\pi)^3}
\left[
\int \frac{\d^3 q}{(2\pi)^3}
\frac{f(\vec{q})\delta(\vec{q})}{P_\delta(q)}
\frac{g(\vec{k}-\vec{q})\delta(\vec{k}-\vec{q})}{P_\delta(|\vec{k}-\vec{q}|)}
\right]
\frac{h(-\vec{k})\delta(-\vec{k})}{P_\delta(k)}.
\end{equation}
The integral over $\vec{q}$ is a convolution of two filtered densities
$f\delta/P_\delta$ and $g\delta/P_\delta$, which we denote as
\begin{equation}
\label{eq:fg_quadratic_field}
\left[\frac{f\delta}{P_\delta} * \frac{g\delta}{P_\delta}\right](\vec{k})
\equiv \int\frac{\d^3q}{(2\pi)^3}
\frac{
f(\vec{q})\delta(\vec{q})}{P_\delta(q)}
\frac{g(\vec{k}-\vec{q})\delta(\vec{k}-\vec{q})}{P_\delta(|\vec{k}-\vec{q}|)}.
\end{equation}
Then, the estimator \eq{fnl_split_integrals} simplifies to
\begin{eqnarray}
\nonumber
\hat{f}_\mathrm{NL}^{B^\mathrm{theo}}
&=&
\frac{1}{N_\mathrm{theo}} \int
\d k\frac{ k^2}{P_\delta(k)}\int\d\Omega_{\hat\vec{k}}
\left[\frac{f\delta}{P_\delta} * \frac{g\delta}{P_\delta}\right]\!(\vec{k})\,
[h\delta](-\vec{k})
\\
\label{eq:fnl_separable_fgh}
&=&
\frac{4\pi L^3}{N_\mathrm{theo}} \int
\d k \frac{k^2}{P_\delta(k)}
\hat P_{\frac{f\delta}{P_\delta} *
\frac{g\delta}{P_\delta},\,h\delta}(k),
\end{eqnarray}
where we used \eqq{hat_cross_power}.
This is a (weighted) integral over the estimated cross-spectrum
between the
quadratic field \eq{fg_quadratic_field} and the
filtered density
\begin{equation}
\label{eq:54}
[h\delta](\vec{k})
\equiv h(\vec{k})\delta(\vec{k}).
\end{equation}
Rather than performing the
integration of the cross-spectrum over wavenumbers $k$ in
\eqq{fnl_separable_fgh}, it is useful to consider the cross-spectrum
as the fundamental observable. This can simplify comparisons of theory,
simulations and observations as a function of scale $k$, and offers
the possibility to incorporate covariances between cross-spectra.
The convolution in \eqq{fg_quadratic_field} can be calculated efficiently by filtering the
density in $k$-space with two filters $f/P$ and $g/P$, Fourier
transforming to configuration space, multiplying the fields with each other in configuration space, and
Fourier transforming back to $k$-space
(e.g.~for $f=g=P_\delta$, \eqq{fg_quadratic_field} is the Fourier transform of
$\delta^2(\vec{x})$).
If the theoretical bispectrum consists of a sum of separable terms, each
term gives rise to a cross-spectrum of accordingly filtered
densities. Each cross-spectrum can be measured to analyse
individual contributions to the bispectrum, or they can be combined to
constrain the overall bispectrum amplitude.
\subsection{Gravitational bispectrum and bias estimators}
Eqs.~\eq{BDM_l}, \eq{Bmmh_l0}-\eq{Bmmh_l2} and \eq{Bhhh_l0}-\eq{Bhhh_l2} show that
matter-matter-matter, matter-matter-halo and halo-halo-halo bispectra
can be constructed from contributions of the form\footnote{In all
equations $l$ is fixed to $l=0,1,2$, unless we explicitly write
$\sum_l$ to sum over $l$. We never use Einstein summation notation
for $l$.}
\begin{equation}
\label{eq:Blunsym}
B^{(l)}_\mathrm{unsym}(\vec{k}_1,\vec{k}_2,\vec{k}_3) \equiv
2P^\mathrm{lin}_\mathrm{mm}(k_1)P^\mathrm{lin}_\mathrm{mm}(k_2)F_2^{l}(k_1,k_2)\mathsf{P}_l(\hat\vec{k}_1\cdot\hat\vec{k}_2),
\end{equation}
which are not symmetric in the $k_i$. Note that every $F_2^{l}\mathsf{P}_l$
kernel is separable (or a sum of separable terms), because squaring
the triangle condition $\vec{k}_1+\vec{k}_2=-\vec{k}_3$ implies $\vec{k}_1\cdot\vec{k}_2 =
\tfrac{1}{2} (k_3^2-k_1^2-k_2^2)$. Consequently,
the maximum likelihood estimator for the amplitude of a bispectrum
contribution \eq{Blunsym} can be expressed as an integral over a
cross-spectrum between a quadratic field and the density. Explicitly, the
maximum likelihood estimator \eq{fnl_esti} for the amplitude of a
bispectrum \eq{Blunsym} involving the densities $\delta_a$, $\delta_a$ and $\delta_b$
(for $a,b\in\{\mathrm{m},\mathrm{h}\}$ labeling DM or halo densities)
gives\footnote{Note that the estimators for the $l=0,1,2$ contributions to the full
symmetric gravitational bispectrum $B^{(l)}$ contain an additional factor of
$3$ because \eqq{Blunsym} picked one out of three permutations in the
$k_i$.}
\begin{equation}
\label{eq:fnl_Bgrav_unsym_general}
\hat{f}^{B^{(l)}_\mathrm{unsym}}_\mathrm{NL}
=
\frac{8\pi L^3}{N_{B^{(l)}_\mathrm{unsym}}}
\int\d k \frac{ k^2}{P_{bb}(k)}
\hat P_{\tilde F_2^{l}\mathsf{P}_l[\delta_a], \,\delta_b}(k)
\end{equation}
where we defined
\begin{equation}
\label{eq:general_f2tildeell_applied_to_delta_a}
\tilde F_2^{l}\mathsf{P}_l[\delta_a](\vec{k}) \equiv
\int\frac{\d^3q}{(2\pi)^3}
\frac{P^\mathrm{lin}_\mathrm{mm}(q)P^\mathrm{lin}_\mathrm{mm}(|\vec{k}-\vec{q}|)}{P_{aa}(q)P_{aa}(|\vec{k}-\vec{q}|)}
F_2^{l}(q,|\vec{k}-\vec{q}|)\mathsf{P}_l(\mu)
\delta_a(\vec{q})\delta_a(\vec{k}-\vec{q}),
\end{equation}
where $\mu$ is the cosine \eq{mu_def} between $\vec{q}$ and $\vec{k}-\vec{q}$.
The power spectrum ratio in the integrand
serves as a weight which becomes unity on large scales for dark
matter. To gain intuition, we choose a
slightly less optimal weight by setting the power spectrum ratio to
unity on all scales\footnote{If $\delta_a$ is the halo
density this implies a sub-optimal weighting at high $k$, but
results are not biased because we treat simulations and theory
consistently. Also, high $k$ modes are suppressed by smoothing,
which we apply before squaring any fields to suppress nonlinear mode
coupling, similarly to cutting off bispectrum analyses at some
maximum wavenumber $k_\mathrm{max}$. We do this because we do not have a
reliable way to model these high $k$ modes. In our simulations, the power
ratio $P^\mathrm{lin}_\mathrm{mm}/P_\mathrm{hh}$ drops by $30\%$ or
less between $k\rightarrow 0$ and $k=0.3h/\mathrm{Mpc}$. All
smoothing kernels we use asymptote to $0$ much faster for increasing
$k$, so that corrections due to the power spectrum ratio weight are
likely small.} (dropping the tilde on $F_2$),
\begin{equation}
\label{eq:general_f2ell_applied_to_delta_a}
F_2^{l}\mathsf{P}_l[\delta_a](\vec{k}) \equiv
\int\frac{\d^3q}{(2\pi)^3}F_2^{l}(q,|\vec{k}-\vec{q}|)\mathsf{P}_l(\mu)
\delta_a(\vec{q})\delta_a(\vec{k}-\vec{q}).
\end{equation}
In configuration space, the quadratic fields
\eq{general_f2ell_applied_to_delta_a} are proportional to
\begin{eqnarray}
\label{eq:55}
\mathsf{P}_0[\delta](\vec{x}) &=&
\delta^2(\vec{x})\\
-F_2^{1}\mathsf{P}_1[\delta](\vec{x}) &=&
-\Psi^i(\vec{x})\partial_i\delta(\vec{x})\\
\mathsf{P}_2[\delta](\vec{x}) &=&
s^2(\vec{x}).
\end{eqnarray}
According to \eqq{fnl_Bgrav_unsym_general} the (integrated)
cross-spectra of these three quadratic fields with the density are
optimal estimators for the amplitude of the $l=0,1,2$ contributions
to the gravitational bispectrum,
\begin{eqnarray}
\label{eq:fnl_B0_k}
\hat{f}^{B^{(0)}_\mathrm{unsym}}_\mathrm{NL}
&=&
\frac{17}{21}\frac{8\pi L^3}{N_{B^{(0)}_\mathrm{unsym}}}
\int\d k \frac{ k^2}{P_{bb}(k)}
\hat P_{\delta_a^2, \,\delta_b}(k)
\\
\label{eq:fnl_B1_k}
\hat{f}^{B^{(1)}_\mathrm{unsym}}_\mathrm{NL}
&=&
-\frac{8\pi L^3}{N_{B^{(1)}_\mathrm{unsym}}}
\int\d k \frac{ k^2}{P_{bb}(k)}
\hat P_{-\Psi^i_a\partial_i\delta_a, \,\delta_b}(k)
\\
\label{eq:fnl_B2_k}
\hat{f}^{B^{(2)}_\mathrm{unsym}}_\mathrm{NL}
&=&
\frac{4}{21}
\frac{8\pi L^3}{N_{B^{(2)}_\mathrm{unsym}}}
\int\d k \frac{ k^2}{P_{bb}(k)}
\hat P_{s_a^2, \,\delta_b}(k).
\end{eqnarray}
These nonlinearity amplitudes $\hat f_\mathrm{NL}$ depend on bias
parameters, e.g.~on $b_1^3$, $b_1^2b_2$ and $b_1^2b_{s^2}$ for
halo-halo-halo cross-spectra. Bias parameters can therefore be
estimated by combining the measured nonlinearity amplitudes
appropriately. Similarly, the bias parameters can be obtained by
jointly fitting them to the three measured cross-spectra $\hat
P_{D[\delta_a]\delta_b}(k)$ for $D\in\{\mathsf{P}_0, -F_2^1\mathsf{P}_1, \mathsf{P}_2\}$.
These cross-spectra contain the entire information that a full
bispectrum analysis would yield, if we are only interested in the
amplitudes of fixed shape contributions to the bispectrum, which is
often the case, e.g.~for estimating bias parameters.
As will be discussed later, the $k^2/P_{bb}(k)$ weighting of the
cross-spectra corresponds to inverse-variance weighting in the limit
of $k\rightarrow 0$ assuming Gaussian bispectrum covariance. This
weighting is improved if cross-spectrum variances and covariances
obtained from $N$-body simulations or mock catalogues are used to fit
cross-spectrum models to measurements. Estimating covariances from
simulations is computationally much less expensive for cross-spectra
than for bispectra because the three cross-spectra depend only on a single
argument $k$ whereas bispectra depend on triangle configurations
$(k_1,k_2,k_3)$, of which there can be many thousands, especially if
fine $k$-binning is used to distinguish different bispectrum shapes.
\subsection{Configuration space estimators}
We mainly worked in Fourier space so far which has the
advantage that different modes are uncorrelated at leading
order. Sometimes it is more convenient to work in configuration
space because it may be easier to include effects that are localized in
configuration space (e.g.~the survey selection function).
It turns out that the optimal
bispectrum estimator \eq{fnl_separable_fgh} for the amplitude of a
generic separable bispectrum \eq{generic_separable_bisp_fgh} can be
rewritten in configuration space instead of Fourier space as\footnote{To see
this, express the quadratic field in \eq{fnl_separable_fgh} as $\int
\d^3 x e^{-i\vec{k}\vec{x}} \frac{f\delta}{P}(\vec{x})\frac{g\delta}{P}(\vec{x})$
and the linear field as $\int \d^3 y
e^{i\vec{k}\vec{y}}\frac{h\delta}{P}(\vec{y})$. Then, integrating over $\vec{k}$
gives $\vec{y}=\vec{x}$ and the result \eq{fnlBtheoRealSpace}
follows. Alternatively, \eqq{fnlBtheoRealSpace} can be derived by
introducing $\int \d^3 q' \delta_D(\vec{q}'-\vec{k}+\vec{q})$ in
\eqq{fnl_split_integrals} and using $(2\pi)^3\delta_D(\vec{k})=\int \d^3
x e^{i\vec{k}\vec{x}}$.}
\begin{equation}
\label{eq:fnlBtheoRealSpace}
\hat{f}_\mathrm{NL}^{B^\mathrm{theo}}
=
\frac{(2\pi)^3}{N_\mathrm{theo}}
\int \d^3 x\,
\frac{f\delta}{P_\delta}(\vec{x})
\frac{g\delta}{P_\delta}(\vec{x})
\frac{h\delta}{P_\delta}(\vec{x}),
\end{equation}
where the filtered densities are defined as
\begin{equation}
\label{eq:53}
\frac{f\delta}{P_\delta}(\vec{x}) =
\int \frac{\d ^3k}{(2\pi)^3}e^{i\vec{k}\vec{x}}\frac{f(\vec{k})\delta(\vec{k})}{P_\delta(k)},
\end{equation}
and similarly if $f$ is replaced by $g$ and $h$. If one wants to avoid
Fourier space entirely, the filtering can be performed with
convolutions in configuration space.
For the gravitational bispectrum contributions
$B^{(l)}_\mathrm{unsym}$ we get\footnote{Again, we assume
$P_\delta=P^\mathrm{lin}_\mathrm{mm}$ to get a simple weighting, but
it would be straightforward to include the optimal
$P_\mathrm{mm}^\mathrm{lin}/P_\delta$ weighting. }
\begin{eqnarray}
\label{eq:fnl_B0_x}
\hat{f}^{B^{(0)}_\mathrm{unsym}}_\mathrm{NL}
&=&
\frac{34}{21}\frac{(2\pi)^3}{N_{B^{(0)}_\mathrm{unsym}}}\int\d^3 x\,
\delta_a^2(\vec{x})\frac{\delta_b}{P_{bb}}(\vec{x}),
\\
\label{eq:fnl_B1_x}
\hat{f}^{B^{(1)}_\mathrm{unsym}}_\mathrm{NL}
&=&
2\frac{(2\pi)^3}{N_{B^{(1)}_\mathrm{unsym}}}\int\d^3 x\,
\Psi^i_a(\vec{x})[\partial_i\delta_a(\vec{x})]\frac{\delta_b}{P_{bb}}(\vec{x}),
\\
\label{eq:fnl_B2_x}
\hat{f}^{B^{(2)}_\mathrm{unsym}}_\mathrm{NL}
&=&
\frac{8}{21} \frac{(2\pi)^3}{N_{B^{(2)}_\mathrm{unsym}}}\int\d^3 x\,
s_a^2(\vec{x})\frac{\delta_b}{P_{bb}}(\vec{x}).
\end{eqnarray}
\section{Theory cross-spectra}
\label{se:TheoryCrossSpectra}
\subsection{Smoothing}
To suppress small-scale modes, we apply a smoothing filter $W_R(k)$ to
the nonlinear (DM or halo)
density,
\begin{equation}
\label{eq:smoothing}
\delta^R(\vec{k}) \equiv W_R(k)\delta(\vec{k}),
\end{equation}
where $R$ is the smoothing radius.
For Gaussian smoothing,
\begin{equation}
\label{eq:W_R_Gauss}
W_R^\mathrm{Gauss}(k) = e^{-\frac{1}{2}k^2R^2}.
\end{equation}
The power spectrum and bispectrum of the smoothed field $\delta^R$ are
\begin{eqnarray}
\label{eq:30}
P_{\delta^R\delta^R}(k) &=& P^R_{\delta\delta}(k)= W^2_R(k)P_{\delta\delta}(k),
\\
B_{\delta^R\delta^R\delta^R}(k_1,k_2,k_3) &=& W_R(k_1)W_R(k_2)W_R(k_3)B_{\delta\delta\delta}(k_1,k_2,k_3).
\end{eqnarray}
More small-scale modes can be included by choosing smaller smoothing
scale $R$. This will generally increase the signal to noise in the
observables at the expense of worse agreement with models, so in
practice some trade-off smoothing scale $R$ should be chosen. Note
that the window function is applied to every density field, so an
isotropic survey window function could be included in the model
predictions below by simply modifying the smoothing kernel $W_R$ appropriately.
Haloes are finite size objects and thus the correlators of halo centers should not have structure on scales below the halo scale. This is equivalent to modelling the halo density field in terms of the density field smoothed on the halo scale. There is indeed evidence for a cutoff of the protohalo power spectrum in Lagrangian space \cite{Tobias14}. Gravitational evolution, in particular the collapse, shrinks this scale and generates non-linear contributions to the clustering statistics. While a self consistent theoretical understanding of the combined effects of Lagrangian smoothing and non-linear gravitational evolution does not exist, there are hints from simulation for the existence of a finite smoothing scale in the halo density field in Eulerian space \cite{Pollack1309}.
Such a smoothing scale would also affect our perturbation theory calculations of the correlators of the squared field. In particular, the integral over the smoothed power spectrum would decrease the amplitude of the low-$k$ limit of the $I_{DE}^R$ terms (Eq.~\eq{I_DE_R} below) by a few percent for realistic smoothing scales and thus increase the inferred bias parameters by this amount. At higher wavenumbers, the smoothing scale enters more explicitly, also in the $I_{DE}^{\text{bare},R}$ (Eq.~\eq{I_DE_R_bare} below) and can thus lead to larger changes. We have calculated these effects and found that the additional fitting parameter did not improve our fits of halo-halo-halo statistics. With no apparent improvements and the lack of a theoretical model, we decide to neglect the smoothing corrections in this study but remark that they should be understood and included in the future.
\subsection{General bispectrum}
To compute general theoretical predictions for cross-spectra
we work with two smoothed fields
$\delta^R_a$ and $\delta^R_b$ that can be dark matter or halo densities,
$a,b\in\{\mathrm{m}, \mathrm{h}\}$.
The expectation value of the cross-spectrum of a quadratic field $D[\delta^R_a]$
for $D\in\{\mathsf{P}_0,-F_2^1\mathsf{P}_1, \mathsf{P}_2\}$ with the density $\delta^R_b$
is given by an integral over the
bispectrum $B$,
\begin{eqnarray}
\label{eq:48}
\langle D[\delta_a^R](\vec{k})\delta_b^R(\vec{k}')\rangle
&=&
\int\frac{\d^3q}{(2\pi)^3} D(\vec{q},\vec{k}-\vec{q}) \langle
\delta_a^R(\vec{q})\delta_a^R(\vec{k}-\vec{q})
\delta^R_b(\vec{k}')\rangle
\\
&=&
(2\pi)^3\delta_D(\vec{k}+\vec{k}')
\int\frac{\d^3q}{(2\pi)^3} D(\vec{q},\vec{k}-\vec{q})
B_{\delta_a^R\delta_a^R\delta^R_b}(\vec{q},\vec{k}-\vec{q},-\vec{k}).
\end{eqnarray}
For $a\ne b$, the bispectrum $B$ is not symmetric in its arguments and
the last argument $-\vec{k}$ is associated with $\delta_b$.
Writing the smoothing kernels explicitly, we have
\begin{equation}
\label{eq:2}
P_{D[\delta^R_a],\delta^R_b}(k) = W_R(k)\int\frac{\d^3q}{(2\pi)^3}W_R(q)W_R(|\vec{k}-\vec{q}|) D(\vec{q},\vec{k}-\vec{q})
B_{\delta_a\delta_a\delta_b}(\vec{q},\vec{k}-\vec{q},-\vec{k}).
\end{equation}
\subsection{Matter-matter-matter cross-spectra}
For cross-spectra of smoothed dark matter fields ($a=b=\mathrm{m}$),
the DM bispectrum \eq{BDM} gives
\begin{eqnarray}
\label{eq:Pcross_mmm_D_in_terms_of_B}
P_{D[\delta^R_\mathrm{m}],\delta^R_\mathrm{m}}(k) &=& \int\frac{\d^3 q}{(2\pi)^3}
D(\vec{q},\vec{k}-\vec{q})
B_\mathrm{\delta^R_\mathrm{m}\delta^R_\mathrm{m}\delta^R_\mathrm{m}}(\vec{q},
\vec{k}-\vec{q}, -\vec{k})\\
\label{eq:Pcross_mmm_D}
&=&2I^R_{DF_2}(k)
+4 I_{DF_2}^{\mathrm{bare},R}(k),
\end{eqnarray}
where we used that the kernel $D(\vec{q}, \vec{k}-\vec{q})$ is assumed to be symmetric in its
arguments and we defined for kernels $D$ and $E$
\begin{equation}
\label{eq:I_DE_R}
I_{DE}^R(k)\equiv
W_R(k)
\int\frac{\d^3 q}{(2\pi)^3}W_R(q)W_R(|\vec{k}-\vec{q}|)P_\mathrm{mm}^\mathrm{lin}(q)P_\mathrm{mm}^\mathrm{lin}(|\vec{k}-\vec{q}|)
D(\vec{q}, \vec{k}-\vec{q})E(\vec{q}, \vec{k}-\vec{q}),
\end{equation}
which is symmetric under $D\leftrightarrow E$,
and
\begin{equation}
\label{eq:I_DE_R_bare}
I_{DE}^{\mathrm{bare},R}(k)\equiv W_R(k) P_\mathrm{mm}^\mathrm{lin}(k) \int\frac{\d^3 q}{(2\pi)^3}
W_R(q) W_R(|\vec{k}-\vec{q}|) P_\mathrm{mm}^\mathrm{lin}(q) D(\vec{q},\vec{k}-\vec{q})E(\vec{q},-\vec{k}),
\end{equation}
which is not symmetric under $D\leftrightarrow E$.
Explicit predictions for the cross-spectra $P_{\delta_\mathrm{m}^2, \delta_\mathrm{m}}$,
$P_{-\Psi_\mathrm{m}^i\partial_i\delta_\mathrm{m},\delta_\mathrm{m}}$ and $P_{s_\mathrm{m}^2, \delta_\mathrm{m}}$
can be obtained from \eqq{Pcross_mmm_D} by plugging in $D=\mathsf{P}_0$,
$D=-F_2^{1}\mathsf{P}_1$ and $D=\mathsf{P}_2$, respectively, and setting $E=F_2$.
Note that there are three powers
of the smoothing kernel because we smooth the nonlinear rather than
the linear field.
The integrals in Eqs.~\eq{I_DE_R} and \eq{I_DE_R_bare} are similar to
typical 1-loop expressions and can be reduced to two-dimensional
integrals over scale $q$ and cosine $\hat \vec{q}\cdot\hat \vec{k}$, which can
be evaluated numerically with little computational cost (see
e.g.~\cite{MPTBREEZE,taruyaRegPT} for public codes that compute similar
integrals). The factors
$W_R(|\vec{k}-\vec{q}|)$ and $P_\mathrm{mm}^\mathrm{lin}(|\vec{k}-\vec{q}|)$ introduce
a non-trivial angle dependence so that the angular integration
generally needs to be performed numerically.
The only ingredient for the theory prediction of \eqq{Pcross_mmm_D} is
the model for the DM bispectrum. Improved bispectrum models that have
the same form as \eqq{BDM} could easily be included, e.g.~by replacing
the perturbation theory $F_2$ kernel by an effective $F_2$ kernel
fitted to $N$-body simulations
\cite{hector1111,scocci_fitting_formula,hector1407theory}.
\subsection{Matter-matter-halo cross-spectra}
From the unsymmetric unsmoothed
matter-matter-halo bispectrum of \eqq{Bmmh_unsymm}
we find for the cross-spectrum of a quadratic matter field with the halo density
\begin{eqnarray}
\label{eq:Pcross_mmh_D}
P_{D[\delta_\mathrm{m}^R],\delta_\mathrm{h}^R}(k) &=& \int\frac{\d^3 q}{(2\pi)^3}
D(\vec{q},\vec{k}-\vec{q}) B^\mathrm{unsym}_\mathrm{\delta_\mathrm{m}^R\delta_\mathrm{m}^R\delta_\mathrm{h}^R}(\vec{q}, \vec{k}-\vec{q}, -\vec{k})\\
&=&
2b_1I^R_{DF_2}(k) + 4b_1I_{DF_2}^{\mathrm{bare},R}(k)
+2b_2I^R_{D\mathsf{P}_0}(k)+\frac{4}{3}b_{s^2}I^R_{D\mathsf{P}_2}(k).
\end{eqnarray}
The part depending on $b_1$ can be expressed in terms of the
matter-matter-matter cross-spectrum so that
\begin{equation}
\label{eq:Pcross_mmh_D_in_terms_of_mmm}
P_{D[\delta_\mathrm{m}^R],\delta_\mathrm{h}^R}(k) - b_1P_{D[\delta^R_\mathrm{m}],\delta^R_\mathrm{m}}(k)
=
2b_2I^R_{D\mathsf{P}_0}(k)+\frac{4}{3}b_{s^2}I^R_{D\mathsf{P}_2}(k).
\end{equation}
\subsection{Halo-halo-halo cross-spectra}
The halo-halo-halo bispectrum \eq{Bhhh} gives for the halo-halo-halo
cross-spectra
\begin{equation}
\label{eq:hhh_theory_all_cross_spectra}
P_{D[\delta_\mathrm{h}^R],\delta^R_\mathrm{h}}(k)
=2b_1^3\left[I^R_{DF_2}(k)
+2 I_{DF_2}^{\mathrm{bare},R}(k)\right] +
2b_1^2b_2 \left[I^R_{D\mathsf{P}_0}(k) +
2 I_{D\mathsf{P}_0}^{\mathrm{bare},R}(k)\right]+
\frac{4}{3}b_1^2b_{s^2} \left[I^R_{D\mathsf{P}_2}(k) +
2 I_{D\mathsf{P}_2}^{\mathrm{bare},R}(k)\right].\quad
\end{equation}
Contributions depending on $b_1^3$ also appear in matter-matter-halo
cross spectra, so that
\begin{equation}
\label{eq:27}
P_{D[\delta_\mathrm{h}^R],\delta^R_\mathrm{h}}(k)
- b_1^2 P_{D[\delta_\mathrm{m}^R],\delta^R_\mathrm{h}}(k)
=
4b_1^2\left[
b_2 I_{D\mathsf{P}_0}^{\mathrm{bare},R}(k)
+\frac{2}{3}b_{s^2} I_{D\mathsf{P}_2}^{\mathrm{bare},R}(k)
\right].
\end{equation}
Decomposing the
$F_2$ kernel in Legendre polynomials as in \eqq{F2LegendreDecomp},
\eqq{hhh_theory_all_cross_spectra} can be rewritten as
\begin{align}
\nonumber
P_{D[\delta_\mathrm{h}^R],\delta^R_\mathrm{h}}(k)
=\,&
\left(\frac{34}{21}b_1^3 + 2b_1^2b_2\right)\left[
I^R_{D\mathsf{P}_0}(k) + 2I^{\mathrm{bare},R}_{D\mathsf{P}_0}(k)
\right]\\
\nonumber
&\,+
2b_1^3\left[I^R_{D,F_2^1\mathsf{P}_1}(k) +2I^{\mathrm{bare},R}_{D,F_2^1\mathsf{P}_1}(k) \right]\\
&\,+
\left(\frac{8}{21}b_1^3 + \frac{4}{3}b_1^2b_{s^2}\right)
\left[I^{R}_{D\mathsf{P}_2}(k) + 2I^{\mathrm{bare},R}_{D\mathsf{P}_2}(k) \right].
\end{align}
\begin{figure}[tb]
\centerline{
\includegraphics[width=0.8\textwidth]{figs_bias_estimators/mmm_model_only_RGauss20_nk200_quad2_splitintegrals_nn.pdf}
}
\caption{Theory contributions \eq{hhh_theory_all_cross_spectra} to
halo-halo-halo cross-spectra scaling like $b_1^3$ (dashed),
$b_1^2b_2$ (dash-dotted) and $b_1^2 b_{s^2}$ (dotted) for squared
density $\delta^2_\mathrm{h}(\vec{x})$ (blue), shift term
$-\Psi^i_\mathrm{h}(\vec{x})\partial_i \delta_\mathrm{h}(\vec{x})$
(red) and tidal term $s_\mathrm{h}^2(\vec{x})$ (green), evaluated
for fixed bias
parameters $b_1=1$, $b_2=0.5$ and $b_{s^2}=2$,
Gaussian smoothing with $R_G=20h^{-1}\mathrm{Mpc}$,
at $z=0.55$,
with linear matter power spectra in integrands. Thin gray lines show the large-scale
(low $k$) limit given by \eqq{lowk_hhh_theory}. The cross-spectra are
divided by the partially smoothed FrankenEmu emulator matter power spectrum
$W_R^{3/2}P_{\mathrm{mm}}^\mathrm{emu}$ \cite{FrankenEmuExt,Emu1,Emu2,Emu3} for plotting convenience.
}
\label{fig:hhhtheory_RGauss20}
\end{figure}
The contributions to the theory expression of \eqq{hhh_theory_all_cross_spectra} are shown in
Fig.~\ref{fig:hhhtheory_RGauss20}
for Gaussian smoothing with $R=20h^{-1}\mathrm{Mpc}$ (see
Fig.~\ref{fig:hhhtheory_RGauss10} in the appendix for
$R=10h^{-1}\mathrm{Mpc}$). Different colors describe
different cross-spectra, $D\in\{\mathsf{P}_0,-F_2^1\mathsf{P}_1,\mathsf{P}_2\}$, while different
line styles correspond to the contributions with different
dependencies on bias parameters, scaling like $b_1^3$, $b_1^2b_2$ or
$b_1^2b_{s^2}$. The characteristic $k$-dependencies of the different
contributions can be exploited to fit $b_1$, $b_2$ and $b_{s^2}$ to
the three cross-spectra at the same time. In practice, the fitted bias
parameters can still be degenerate because sampling variance at low
$k$ and modeling uncertainty at high $k$ limit the usable $k$ range.
In particular, in the range $0.01h/\mathrm{Mpc}\lesssim k\lesssim
0.1h/\mathrm{Mpc}$, every cross-spectrum depends rather similarly
on $b_1^3$ and $b_1^2b_2$ leading to a degeneracy where larger $b_1$
can be compensated by a smaller $b_2$, which is also present when
considering individual bispectrum triangles rather than
cross-spectra. Consequently, models that extend leading-order PT to
higher $k$ are expected to improve bias constraints
significantly.
In the large-scale limit, $k\ll q$, the three halo-halo-halo
cross-spectra of \eqq{hhh_theory_all_cross_spectra} equal each other
(see Appendix \ref{se:lowk_integral_limits} for
details),
\begin{align}
\nonumber
\lim_{k\rightarrow 0} P_{D[\delta^R_\mathrm{h}],\delta^R_\mathrm{h}}(k)
= W_R(k)\bigg[
b_1^3 P_\mathrm{mm}^\mathrm{lin}(k)\left(
\frac{68}{21}\sigma_{R}^2 - \frac{1}{3}
\sigma_{R,P'}^2
\right)
+
2 b_1^2 b_2 \left(\tau^4_R
+
2 P_\mathrm{mm}^\mathrm{lin}(k)\sigma_R^2 \right)
+
&
\frac{4}{3}b_1^2b_{s^2}\tau^4_R\bigg],&\\
\label{eq:lowk_hhh_theory}
& D\in\big\{\mathsf{P}_0, -F_2^{1}\mathsf{P}_1,
\mathsf{P}_2\big\},
\end{align}
where $W_R(k)\rightarrow 1$ for $k\rightarrow 0$, and
$\sigma^2_{R,P'}$ and $\tau^4_R$ are defined in
Eqs.~\eq{sigma2_R_Pprime} and \eq{tau_def}, respectively. In this
limit, the $b_1^3$ term is proportional to the linear matter power
spectrum, whereas the term involving $b_2$ scales like the linear
matter power spectrum plus a $k$-independent correction and the term
involving $b_{s^2}$ is entirely $k$-independent (see thin gray lines
in Figs.~\ref{fig:hhhtheory_RGauss20} and \ref{fig:hhhtheory_RGauss10}
for the the ratio of these limits over the matter power spectrum).
Due to large sampling variance at low $k$, this scale-dependence is
expected to be less powerful in distinguishing bias parameters than
constraints obtained from the different scale-dependencies at high
$k$.
\subsection{Shot noise}
The bispectrum of the smoothed halo density has an additional stochasticity contribution,
\begin{equation}
\label{eq:43}
\langle \hat B_\mathrm{hhh}^R(k_1,k_2,k_3)\rangle =
B_\mathrm{hhh}^R(k_1,k_2,k_3)+
B_\mathrm{hhh}^{R,\mathrm{shot}}(k_1,k_2,k_3),
\end{equation}
whose Poissonian prediction is (see e.g.~\cite{jeongThesis})
\begin{equation}
\label{eq:Bhhhshot}
B_\mathrm{hhh}^{R,\mathrm{shot}}(k_1,k_2,k_3)
=
W_R(k_1)W_R(k_2)W_R(k_3)\left\{
\frac{1}{\bar n_\mathrm{h}}\left[
P_\mathrm{hh}(k_1)
+ 2\mbox{ perms}
\right]
+ \frac{1}{\bar{n}^2_\mathrm{h}}
\right\}.
\end{equation}
Here, $\bar n_\mathrm{h}$ is the mean halo number density, and
$P_\mathrm{hh}$ is the power spectrum of the unsmoothed continuous
halo density field, which we approximate by the ensemble-averaged,
CIC- and shot-noise-corrected power spectrum of the
unsmoothed halo density measured in the simulations. The stochasticity
bispectrum contributes to halo cross-spectra as
\begin{equation}
\label{eq:hhh_cross_spectra_shotnoise}
P^\mathrm{shot}_{D[\delta^R_\mathrm{h}], \delta^R_\mathrm{h}}(k)
=
\left[\frac{1}{\bar n^2_\mathrm{h}} + \frac{P_\mathrm{hh}(k)}{\bar
n_\mathrm{h}}\right]J^R_D(k)
+ \frac{2}{\bar n_\mathrm{h}}\tilde J^R_D(k),
\end{equation}
where we defined
\begin{equation}
\label{eq:50}
J^R_D(k) \equiv W_R(k)\int\frac{\d^3
q}{(2\pi)^3}W_R(q)W_R(|\vec{k}-\vec{q}|)D(\vec{q}, \vec{k}-\vec{q})
\end{equation}
and
\begin{equation}
\label{eq:51}
\tilde J^R_D(k) \equiv W_R(k)\int\frac{\d^3
q}{(2\pi)^3}W_R(q)W_R(|\vec{k}-\vec{q}|)D(\vec{q}, \vec{k}-\vec{q}) P_\mathrm{hh}(q),
\end{equation}
which depends on the mass bin through $P_\mathrm{hh}$.
The full model is
\begin{equation}
\label{eq:Pcross_shotnoisecorrection}
\langle \hat P_{D[\delta^R_\mathrm{h}],\delta^R_\mathrm{h}}(k)\rangle =
P_{D[\delta^R_\mathrm{h}],\delta^R_\mathrm{h}}(k) + P^\mathrm{shot}_{D[\delta^R_\mathrm{h}],\delta^R_\mathrm{h}}(k),
\end{equation}
where the first term on the r.h.s.~depends on bias parameters to be
fitted from data, while the stochasticity (shot noise) term does not explicitly
depend on bias parameters because we use the measured
ensemble-averaged halo power spectrum there.
The Poisson stochasticity should be corrected for exclusion and
clustering effects, similarly to the power spectrum results of
\cite{tobias1305}. Phenomenologically, we can model these shot noise
corrections with two scale-independent parameters, $\Delta_1$ and
$\Delta_2$, by adding $\Delta_1[P_\mathrm{hh}(k_1)+2\mbox{perms}]$ to
$\bar{n}_\mathrm{h}^{-1}[P_\mathrm{hh}(k_1)+2\mbox{perms}]$ and
$\Delta_2$ to $\bar{n}_\mathrm{h}^{-2}$ in \eqq{Bhhhshot}, so that
\begin{equation}
\label{eq:Pcross_shotnoisecorrection_D1_D2}
P^{\mathrm{shot}}_{D[\delta^R_\mathrm{h}],\delta^R_\mathrm{h}}(k) =
\left[\left(\bar n^{-2}_\mathrm{h}+\Delta_2\right) +
\left(\bar
n^{-1}_\mathrm{h} + \Delta_1\right)P_\mathrm{hh}(k)\right]J^R_D(k)
+ 2\left(\bar n^{-1}_\mathrm{h}+\Delta_1\right)\tilde J^R_D(k).
\end{equation}
For $\Delta_2=\Delta_1/\bar{n}_\mathrm{h}$, this is
equivalent to rescaling the Poisson shot noise by an overall
scale-independent amplitude
as done in e.g.~\cite{hector1407theory}. The Poisson prediction is
recovered for $\Delta_1=\Delta_2=0$.
\subsection{Covariances}
\label{se:TheoryCovariances}
To estimate bias and cosmological parameters from cross-spectra we
need to know their noise and covariance properties. Leading-order
perturbation theory predicts for the covariance between two
cross-spectra at the same wavenumber (see Appendix
\ref{se:TheoryCovariancesAppendix})
\begin{equation}
\label{eq:cov_cross_spectra_diagonal}
\mathrm{cov}(\hat P_{D[\delta^R_a],\delta^R_b}(k),
\hat P_{E[\delta^R_a],\delta^R_b}(k)) =
\frac{2}{N_\mathrm{modes}(k)}P^R_{bb}(k)I^{P^R_\mathit{aa}P^R_\mathit{aa}}_{DE}(k),
\end{equation}
where $I^{P^R_\mathit{aa}P^R_\mathit{aa}}_{DE}$ is defined in \eqq{I_DE_PR_PR_cov}.
The correlation between two cross-spectra is therefore
\begin{equation}
\label{eq:correl_cross_spectra_diagonal}
\mathrm{correl}(\hat P_{D[\delta^R_a],\delta^R_b}(k),
\hat P_{E[\delta^R_a],\delta^R_b}(k)) =
\frac{I^{P^R_\mathit{aa}P^R_\mathit{aa}}_{DE}(k)}
{\sqrt{I^{P^R_\mathit{aa}P^R_\mathit{aa}}_{DD}(k) I^{P^R_\mathit{aa}P^R_\mathit{aa}}_{EE}(k)}},
\end{equation}
which does not depend on the type $b$ of the linear field.
Note that the perturbative
calculation in Appendix \ref{se:TheoryCovariancesAppendix} predicts
additional covariances between cross-spectra at different wavenumbers
$k\ne k'$, but we neglect them here for simplicity.
Eqs.~\eq{cov_cross_spectra_diagonal} and
\eq{correl_cross_spectra_diagonal} will be compared against
simulations in Section \ref{se:EstimatedCovs} and
Fig.~\ref{fig:correl_mmm_RGauss20}.
For sufficiently large smoothing scale $R$ and low $k$, we can approximate
$P^R_\mathrm{hh}\approx b_1^2P^R_\mathrm{mm}$ in the integrand of
\eqq{I_DE_PR_PR_cov}, so that the halo correlation
($a=\mathrm{h}$ in \eqq{correl_cross_spectra_diagonal}) approaches the dark matter correlation
($a=\mathrm{m}$ in \eqq{correl_cross_spectra_diagonal}).
In the large scale limit $k\rightarrow 0$ the kernels $D,
E\in\{\mathsf{P}_0,-F_2^1\mathsf{P}_1,\mathsf{P}_2\}$ become unity (see Appendix
\ref{se:lowk_integral_limits}), so that
\begin{equation}
\label{eq:42}
\lim_{k\rightarrow 0} \mathrm{correl}(\hat P_{D[\delta^R_a],\delta^R_b}(k),
\hat P_{E[\delta^R_a],\delta^R_b}(k)) =1, \qquad\qquad D, E\in\{\mathsf{P}_0,
-F_2^{1}\mathsf{P}_1, \mathsf{P}_2\}
\end{equation}
i.e.~the three cross-spectra are perfectly correlated on large scales.
Therefore all information is already contained in any one of the three
cross-spectra. On intermediate scales (higher $k$) the cross-spectra
are less correlated (and have different expectation values), so that
constraints are expected to improve if more than just one
cross-spectrum is considered. On
small scales, smoothing destroys clustering information and the
cross-spectra become perfectly correlated or anti-correlated (see
Fig.~\ref{fig:correl_mmm_RGauss20} below; this happens at higher $k$ if smaller
smoothing scale $R$ is chosen).
On large scales, $k\rightarrow 0$, the variance of the cross-spectra
scales like $P^R_{bb}(k)/k^2$, because the large-scale limit of
\eqq{I_DE_PR_PR_cov} is independent of $k$ and
$N_\mathrm{modes}\propto k^2$. This confirms that the $k^2/P$
weighting in the optimal bispectrum estimators in
Eqs.~\eq{fnl_separable_fgh}, \eq{fnl_Bgrav_unsym_general},
\eq{fnl_B0_k}, \eq{fnl_B1_k} and \eq{fnl_B2_k} corresponds to
inverse-variance weighting on large scales.
\section{Simulations}
\label{se:Simulations}
\subsection{Setup}
We use ten realizations of $N$-body simulations that were also used in
\cite{beth1404,mwhite1408}. The simulations were run with the TreePM
code of \cite{mwhiteTreePM2002}. Each realization has $2048^3$ DM
particles in a box of side length $L=1380h^{-1}\mathrm{Mpc}$. The
cosmology is flat $\Lambda\mathrm{CDM}$ with $\Omega_bh^2 = 0.022,
\Omega_mh^2 = 0.139, n_s = 0.965, h = 0.69$ and $\sigma_8 = 0.82$, and
we only use the snapshot at $z=0.55$. Details of the simulations can
be found in \cite{beth1404,mwhite1408}.
To obtain the DM density, for each realization the full set of
$2048^3$ DM particles is interpolated to a $N_g^3=512^3$ grid using
the cloud-in-cell (CIC) scheme. Halos are identified using the FoF
algorithm with linking length $b=0.168$. The halo sample is split into
four mass bins, each spanning a factor of three in mass, and
interpolated to halo density grids using CIC. The CIC window is
deconvolved from DM and halo densities. The inverse number density
$1/\bar{n}$ in units of $h^{-3}\mathrm{Mpc}^3$ is $0.306$ for dark
matter and $351.5$, $746.6$, $2026.2$ and $6561.3$ for halo bins ordered by
increasing mass.
Before squaring any fields, we apply a Gaussian smoothing filter \eq{W_R_Gauss} to
the density.
The squared density $\delta^2(\vec{x})$ in \eqq{delta2_x} is obtained by
squaring this smoothed density in configuration space. To obtain the shift term
$-\Psi^i(\vec{x})\partial_i\delta(\vec{x})$ of \eqq{shift_x}, the density is
first Fourier-transformed to $k$ space, where it is multiplied by
$\vec{k}/k^2$ or $\vec{k}$ to get the displacement or density gradient fields in
$k$ space. Then both fields are Fourier-transformed back to configuration
space, where they are multiplied and contracted as in \eqq{shift_x}. A
similar procedure is used to obtain $s_{ij}(\vec{k})$ and $s^2(\vec{x})$ as
defined in \eqq{s2_x}. Finally, the three quadratic fields
$\delta^2(\vec{x})$, $-\Psi^i(\vec{x})\partial_i\delta(\vec{x})$ and $s^2(\vec{x})$
are Fourier-transformed back to $k$-space, where their cross-spectra with
the density $\delta(\vec{k})$ are estimated using
Eqs.~\eq{hat_cross_power} and \eq{discrete_sum_over_modes}.
The computational cost is dominated by Fourier transforms which can be
evaluated efficiently as FFTs, requiring only $\mathcal{O}(N_g^3\log
N^3_g)$ operations. Therefore the cross-spectrum analysis with
quadratic fields has the same complexity as a usual power spectrum
analysis in $k$-space, but it is sensitive to the full bispectrum
information. Note that brute-force estimation of the bispectrum triangle
by triangle is computationally more expensive by several orders of
magnitude because it requires $\mathcal{O}(N_g^6)$ operations.
Theoretical expressions for expectation values and covariances use the
linear matter-matter power spectrum computed by CAMB \cite{camb} at $z=0.55$
for our fiducial cosmology. If theoretical expressions involve
halo-halo power spectra, we use the estimated ensemble-averaged halo power
spectrum corrected for shot noise and CIC. For plotting convenience,
the cross-spectrum expectation values are typically divided by the
partially smoothed nonlinear matter power spectrum
$W_R^{3/2}P_\mathrm{mm}^\mathrm{emu}$ which is calculated with the FrankenEmu emulator
\cite{FrankenEmuExt,Emu1,Emu2,Emu3}.
Error bars in all plots of this
paper show the standard error of the mean of the ten realizations,
which is estimated as the sample standard deviation divided by
$\sqrt{10}$. This corresponds to $1\sigma$ errors in the total volume
$26.3h^{-3}\mathrm{Gpc}^3$ of the ten realizations.\footnote{Due to
the small number of realizations the estimated error bars are rather
uncertain, see Fig.~\ref{fig:correl_mmm_RGauss20} below for a
comparison with theoretical error bars.}
\begin{figure}[tp]
\centerline{
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/mmm_model_vs_data_Ashot1_RGauss20_nk200_quad2_splitintegrals_nn.pdf}
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/mmm_model_vs_data_Ashot1_RGauss10_nk200_quad2_splitintegrals_nn.pdf}}
\caption{Matter-matter-matter cross-spectra measured from $10$
realizations at $z=0.55$ (crosses with error bars), compared with leading order theory
prediction of \eqq{Pcross_mmm_D} (solid lines), neglecting shot noise. Upper panels show
cross-spectra divided by the partially smoothed emulator matter
power spectrum $W_R^{3/2}P^\mathrm{emu}_\mathrm{mm}$, lower panels
show the ratio of measured cross-spectra over their theory
expectation \eq{Pcross_mmm_D}. Gaussian smoothing is applied with
$R_G=20h^{-1}\mathrm{Mpc}$ (left) and $R_G=10h^{-1}\mathrm{Mpc}$
(right). Different colors represent different cross-spectra (squared
density in blue, shift term in red and tidal term in green). }
\label{fig:mmm_model_vs_data_RGauss10}
\end{figure}
\begin{figure}[tp]
\centerline{
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/mmh_minus_b1_mmm_RGauss20_mass0_nk200_quad2_splitintegrals_Ashot1_fullErrors_b_nn.pdf}
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/mmh_minus_b1_mmm_RGauss20_mass1_nk200_quad2_splitintegrals_Ashot1_fullErrors_b_nn.pdf}}
\centerline{
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/mmh_minus_b1_mmm_RGauss20_mass2_nk200_quad2_splitintegrals_Ashot1_fullErrors_b_nn.pdf}
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/mmh_minus_b1_mmm_RGauss20_mass3_nk200_quad2_splitintegrals_Ashot1_fullErrors_b_nn.pdf}}
\caption{Test of $b_2$ and $b_{s^2}$ contributions to
matter-matter-halo cross-spectra. First, $\hat b_1$ is obtained
from $\hat P_\mathrm{hm}/\hat P_\mathrm{mm}$ at
$k<0.04h/\mathrm{Mpc}$. Then $b_2$ and $b_{s^2}$ are obtained by
fitting the model \eq{Pcross_mmh_D_in_terms_of_mmm} to the measured
excess cross-spectra $\hat
P_{D[\delta^R_\mathrm{m}]\delta^R_\mathrm{h}} - \hat b_1\hat
P_{D[\delta^R_\mathrm{m}]\delta^R_\mathrm{m}}$ (crosses). The
best-fit total model (solid lines) consists of the $b_2$
contribution (dash-dotted) and the $b_{s^2}$ contribution (dotted),
while shot noise is neglected. Different plots show different mass
bins (increasing from upper left to lower right). The upper
sub-panels show excess cross-spectra divided by the partially
smoothed emulator matter power
$W_R^{3/2}P^\mathrm{emu}_\mathrm{mm}$, the lower sub-panels show
measured excess cross-spectra divided by their theory
expectation. The fit is obtained from the grey shaded region,
assuming estimated standard errors of the mean without any
covariances. Best-fit bias parameters, reduced $\chi^2$ and halo
mass range are reported
at the top of each plot. Gaussian smoothing with
$R_G=20h^{-1}\mathrm{Mpc}$ is applied to matter and halo densities.
}
\label{fig:mmh_model_vs_data_RGauss20}
\end{figure}
\subsection{Cross-spectrum expectation values}
We test the consistency of the model by comparing
theory expressions for matter-matter-matter, matter-matter-halo and
halo-halo-halo cross-spectra with simulations. Since the goal of this
section is to test the model, we consider statistics involving the
dark matter field although they cannot directly be observed.
Fig.~\ref{fig:mmm_model_vs_data_RGauss10} compares
matter-matter-matter cross-spectra measured in simulations against
the theory expression of \eqq{Pcross_mmm_D}, finding agreement at the
$5\%$ level for $k\lesssim 0.09h/\mathrm{Mpc}$ for
$R=20h^{-1}\mathrm{Mpc}$ and $R=10h^{-1}\mathrm{Mpc}$. This demonstrates that
the model for matter-matter-matter cross-spectra works well on large scales.
Figs.~\ref{fig:mmh_model_vs_data_RGauss20} and
\ref{fig:mmh_model_vs_data_RGauss10} test the model for
matter-matter-halo cross-spectra by comparing the excess cross-spectra
\begin{equation}
\label{eq:69}
\hat P_{D[\delta^R_\mathrm{m}]\delta^R_\mathrm{h}}(k) - \hat b_1\hat
P_{D[\delta^R_\mathrm{m}]\delta^R_\mathrm{m}}(k)
\end{equation}
to the theory expectation of \eqq{Pcross_mmh_D_in_terms_of_mmm}, where
$\hat b_1$ is obtained from large-scale $\hat P_\mathrm{hm}/\hat
P_\mathrm{mm}$, while $b_2$ and $b_{s^2}$ are jointly fitted to the
three excess cross-spectra at $k\le
0.09h/\mathrm{Mpc}$.\footnote{The fits in
Figs.~\ref{fig:mmh_model_vs_data_RGauss20},
\ref{fig:mmh_model_vs_data_RGauss10} and
\ref{fig:hhh_model_vs_data_RGauss20_fitAshot} use estimated
variances of the cross-spectra in the likelihood and neglect
covariances for simplicity. In contrast,
Fig.~\ref{fig:contours_hhh_RGauss20_and_Rgauss10_mass3} uses
theoretical covariances \eq{cov_cross_spectra_diagonal} between
different cross-spectra with kernels $D\ne E$ at $k=k'$. Generally,
error bars of bias parameters are consistent at the few percent
level if theoretical instead of estimated variances are
used. Including theoretical covariances
\eq{cov_cross_spectra_diagonal} between different cross-spectra with
kernels $D\ne E$ at $k=k'$ typically leads to fractional changes of
bias parameter error bars by $\sim 10\%$ or less.
} The plots show that for all mass bins there is a
combination of $b_2$ and $b_{s^2}$ that describes the simulations
within statistical uncertainties for $k \lesssim
0.09h/\mathrm{Mpc}$.\footnote{In fact, the model seems to overfit the
data in Figs.~\ref{fig:mmh_model_vs_data_RGauss20} and
\ref{fig:mmh_model_vs_data_RGauss10} because the reduced $\chi^2$ is
less than $1$. This might be attributed to the fact that $b_2$ and
$b_{s^2}$ are degenerate in the excess matter-matter-halo
cross-spectra, so that the fitting procedure can pick a parameter
combination along the degeneracy that overfits the data. We do not
address this issue further because the goal of this section is only
to show that there are bias parameters for which the model of
\eqq{Pcross_mmh_D_in_terms_of_mmm} agrees with simulations. }
The fits are somewhat better for $R_G=20h^{-1}\mathrm{Mpc}$ than for
$R_G=10h^{-1}\mathrm{Mpc}$ because the former excludes nonlinear mode
coupling more efficiently.
Fig.~\ref{fig:hhh_model_vs_data_RGauss20_fixAshot0} tests the
theory prediction of \eqq{hhh_theory_all_cross_spectra} for
halo-halo-halo cross-spectra for $R_G=20h^{-1}\mathrm{Mpc}$. The full measured halo-halo-halo
cross-spectra are compared with their theory prediction
\eq{hhh_theory_all_cross_spectra} for bias parameters fixed to the
values obtained from matter-halo statistics (see caption for details)
and for halo-halo-halo shot noise (stochasticity) fixed to be Poissonian,
corresponding to $\Delta_1=\Delta_2=0$ in
\eqq{Pcross_shotnoisecorrection_D1_D2}. While theory and simulations do not
differ strongly for the lowest and highest mass bins, the simulations
show a clear excess over theory for the two intermediate mass
bins. Possible reasons for this could be that the perturbative
treatment breaks down or shot noise is not Poissonian.
To test the latter, Fig.~\ref{fig:hhh_model_vs_data_RGauss20_fitAshot}
shows the same cross-spectra if the shot noise correction $\Delta_1$
in \eqq{Pcross_shotnoisecorrection_D1_D2} is varied as a free
parameter and fitted to the measured halo-halo-halo cross-spectra,
imposing $\Delta_2=\Delta_1/\bar{n}_\mathrm{h}$ and keeping bias
parameters fixed to their values from matter-halo statistics. This
clearly improves the agreement between simulations and theory,
especially for the two intermediate mass bins.\footnote{The worst
reduced $\chi^2$ is $2.16$, which would improve further if the
lowest $k$-bin was removed (the theory of this bin is rather noisy
because the estimated halo-halo power is used to compute the shot
noise in \eqq{Pcross_shotnoisecorrection_D1_D2}).} For our fiducial
four mass bins the reduced $\chi^2$ does not improve significantly if
$\Delta_2$ is treated as a free parameter,
i.e.~$\Delta_2=\Delta_1/\bar{n}_\mathrm{h}$ seems to be an acceptable
approximation within the error bars of the simulations.
To further test the shot noise corrections, we consider two additional
mass bins above the fiducial four mass bins used in the rest of the
paper, with linear bias $b_1=2.8$ and $4.3$. Fitting $\Delta_1$ and
imposing $\Delta_2=\Delta_1/\bar{n}_\mathrm{h}$ gives
$\Delta_1=-4.7h^{-3}\mathrm{Gpc}^3$ with
$\chi^2/\mathrm{d.o.f.}=1.8$ for the
$b_1=2.8$ mass bin, and $\Delta_1=-16.6h^{-3}\mathrm{Gpc}^3$ with
$\chi^2/\mathrm{d.o.f.}=3.4$ for the $b_1=4.3$ mass bin. For these
two high-mass bins, the reduced $\chi^2$ of the fits improve to $1.4$ and
$0.67$ if $\Delta_2$ is treated as a free parameter; see
Fig.~\ref{fig:hhh_model_vs_data_RGauss20_fitAshot_mass45}.
The shot noise correction $\Delta_1$ is positive for the lowest four
mass bins, but becomes negative for the two very high mass bins shown
in Fig.~\ref{fig:hhh_model_vs_data_RGauss20_fitAshot_mass45}. This
mass dependence is qualitatively consistent with Fig.~11 of
\cite{tobias1305}, where the shot noise correction to the power
spectrum turns negative at around $3\times 10^{13}h^{-1}M_{\odot}$,
because the exclusion effect dominates at high mass. While
alternative modifications of the model might be able to describe the
simulations similarly well, the qualitative agreement of the mass
dependence of the shot noise correction and the fact that the shot
noise corrections are capable to capture the cross-spectrum
measurements for all mass bins indicate that deviations from Poisson
shot noise caused by exclusion and nonlinear biasing
are indeed responsible for the disagreement in
Fig.~\ref{fig:hhh_model_vs_data_RGauss20_fixAshot0}.
If so one should be able to model these effects rather than treat them as a
free parameter (see Fig.~11 of
\cite{tobias1305} for a theoretical model prediction that qualitatively
agrees with our measurements). We note that a more detailed modeling
in \cite{tobias1305} predicts this stochasticity term to be constant
(i.e.~shot noise like) only for low $k$, and is expected to vanish at high $k$ (with
the transition given by the halo radius scale). A more detailed analysis is needed
to investigate what the appropriate form is for the bispectrum analysis, and we
expect that the phenomenological approach adopted here can be improved considerably
with a more detailed modeling.
\begin{figure}[tp]
\centerline{
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/hhh_fix_b1_b2_bs2_Ashot0_RGauss20_mass0_nk200_quad2_splitintegrals_nn_Delta1b.pdf}
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/hhh_fix_b1_b2_bs2_Ashot0_RGauss20_mass1_nk200_quad2_splitintegrals_nn_Delta1b.pdf}}
\centerline{
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/hhh_fix_b1_b2_bs2_Ashot0_RGauss20_mass2_nk200_quad2_splitintegrals_nn_Delta1b.pdf}
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/hhh_fix_b1_b2_bs2_Ashot0_RGauss20_mass3_nk200_quad2_splitintegrals_nn_Delta1b.pdf}}
\caption{Measured halo-halo-halo cross-spectra (crosses) compared
against theory (thick solid, \eqq{hhh_theory_all_cross_spectra})
with bias parameters $\hat b_1$ from $\hat P_\mathrm{hm}/\hat
P_\mathrm{mm}$ and $b_2$ and $b_{s^2}$ from $\hat
P_{D[\delta^R_\mathrm{h}]\delta^R_\mathrm{m}}-\hat b_1\hat
P_{D[\delta^R_\mathrm{m}]\delta^R_\mathrm{m}}$ for
$R_G=20h^{-1}\mathrm{Mpc}$ smoothing. Upper panels also show theory
contributions scaling like $b_1^3$ (dashed), $b_1^2b_2$
(dash-dotted) and $b_1^2b_{s^2}$ (dotted), as well as the
halo-halo-halo shot noise contribution (thin solid), which is
assumed to be Poissonian (i.e.~$\Delta_1=\Delta_2=0$ in
\eqq{Pcross_shotnoisecorrection_D1_D2}). The reduced $\chi^2$ on top of
the plots quantifies the (dis-)agreement between halo-halo-halo
cross-spectra measurements and model for the fixed bias
parameters. It is computed over the gray region, neglecting
covariances. Note that the shot
noise contribution fluctuates on very large scales because it is
computed using the ensemble-averaged estimated halo-halo power spectrum. }
\label{fig:hhh_model_vs_data_RGauss20_fixAshot0}
\end{figure}
\begin{figure}[tp]
\centerline{
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/hhh_fix_b1_b2_bs2_fit_Ashot_RGauss20_mass0_nk200_quad2_splitintegrals_nn_Delta1b.pdf}
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/hhh_fix_b1_b2_bs2_fit_Ashot_RGauss20_mass1_nk200_quad2_splitintegrals_nn_Delta1b.pdf}}
\centerline{
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/hhh_fix_b1_b2_bs2_fit_Ashot_RGauss20_mass2_nk200_quad2_splitintegrals_nn_Delta1b.pdf}
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/hhh_fix_b1_b2_bs2_fit_Ashot_RGauss20_mass3_nk200_quad2_splitintegrals_nn_Delta1b.pdf}}
\caption{Same as Fig.~\ref{fig:hhh_model_vs_data_RGauss20_fixAshot0}
if the shot noise correction $\Delta_1$ in
\eqq{Pcross_shotnoisecorrection_D1_D2} is fitted to measured
halo-halo-halo cross-spectra, fixing $\Delta_2=\Delta_1/\bar{n}_\mathrm{h}$ (still keeping $b_1$, $b_2$ and
$b_{s^2}$ fixed to the values obtained from matter-halo and
matter-matter-halo measurements). }
\label{fig:hhh_model_vs_data_RGauss20_fitAshot}
\end{figure}
\begin{figure}[tp]
\centerline{
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/hhh_fix_b1_b2_bs2_fit_Ashot_Povern_Ashot_1overn2_RGauss20_mass4_nk200_quad2_splitintegrals_nn_Delta12e.pdf}
\includegraphics[width=0.5\textwidth]{figs_bias_estimators/hhh_fix_b1_b2_bs2_fit_Ashot_Povern_Ashot_1overn2_RGauss20_mass5_nk200_quad2_splitintegrals_nn_Delta12e.pdf}}
\caption{Same as Fig.~\ref{fig:hhh_model_vs_data_RGauss20_fitAshot} for two higher mass bins and treating both
shot noise corrections $\Delta_1$ and $\Delta_2$ in
\eqq{Pcross_shotnoisecorrection_D1_D2} as independent parameters. }
\label{fig:hhh_model_vs_data_RGauss20_fitAshot_mass45}
\end{figure}
\begin{figure}[tp]
\centerline{
\includegraphics[width=0.45\textwidth]{figs_bias_estimators/std_mmm_model_vs_data_mass3_RGauss20_nk200_kmax0pt4_rtol1em5_quad2_splitintegrals_nn_withShotNoise.pdf}
\includegraphics[width=0.45\textwidth]{figs_bias_estimators/correl_mmm_model_vs_data_mass3_RGauss20_nk200_kmax0pt4_rtol1em5_quad2_splitintegrals_nn_withShotNoise.pdf}
}
\caption{\emph{Left:}
Ratio of estimated over theoretical standard deviation of
matter-matter-matter cross-spectra (using $D=E$ and $a=b=\mathrm{m}$
in \eqq{cov_cross_spectra_diagonal}).
\emph{Right:} Correlations between matter-matter-matter cross-spectra at
the same scale $k=k'$ predicted by theory
(\eqq{correl_cross_spectra_diagonal}, dashed) and estimated from simulations
(solid).
Both panels assume Gaussian smoothing with
$R_G=20h^{-1}\mathrm{Mpc}$ and use the linear matter power spectrum in
theory expressions. For smaller smoothing scale $R$, the zero
crossing would move to higher $k$.}
\label{fig:correl_mmm_RGauss20}
\end{figure}
\begin{figure}[tp]
\centerline{
\includegraphics[width=0.45\textwidth]{figs_bias_estimators/std_mmh_model_vs_data_mass3_RGauss20_nk200_kmax0pt4_rtol1em5_quad2_splitintegrals_nn_withShotNoise.pdf}
\includegraphics[width=0.45\textwidth]{figs_bias_estimators/correl_mmh_model_vs_data_mass3_RGauss20_nk200_kmax0pt4_rtol1em5_quad2_splitintegrals_nn_withShotNoise.pdf}
}
\centerline{
\includegraphics[width=0.45\textwidth]{figs_bias_estimators/std_hhh_model_vs_data_mass3_RGauss20_nk200_kmax0pt4_rtol1em5_quad2_splitintegrals_nn_withShotNoise.pdf}
\includegraphics[width=0.45\textwidth]{figs_bias_estimators/correl_hhh_model_vs_data_mass3_RGauss20_nk200_kmax0pt4_rtol1em5_quad2_splitintegrals_nn_withShotNoise.pdf}
}
\caption{Same as Fig.~\ref{fig:correl_mmm_RGauss20} but for
matter-matter-halo cross-spectra (top) and halo-halo-halo
cross-spectra (bottom), using the halo mass
bin with $b_1=1.98$. Theoretical standard deviations \eq{cov_cross_spectra_diagonal}
are evaluated using the estimated ensemble-averaged halo-halo power
spectrum including shot noise for $P_\mathrm{hh}$ and the linear
matter power spectrum for $P_\mathrm{mm}$.
Theory correlations of \eqq{correl_cross_spectra_diagonal} are
the same as for the matter-matter-matter case. Results for other
mass bins are similar (not shown). }
\label{fig:correl_mmh_and_hhh_RGauss20_mass3}
\end{figure}
\subsection{Cross-spectrum covariances}
\label{se:EstimatedCovs}
For the case of matter-matter-matter cross-spectra and $R_G=20h^{-1}\mathrm{Mpc}$,
Fig.~\ref{fig:correl_mmm_RGauss20} tests the predicted cross-spectrum
covariances of \eqq{cov_cross_spectra_diagonal} (counting the number
of modes manually in the code for the given $k$ binning) against
estimates obtained from ten realizations. Due to the small number of
realizations, the covariance estimates are rather noisy. Within this
large uncertainty, the standard deviations of the three cross-spectra
(Fig.~\ref{fig:correl_mmm_RGauss20} left) as well as the
cross-correlations between the cross-spectra at the same scale $k=k'$
(Fig.~\ref{fig:correl_mmm_RGauss20} right) are consistent between
simulations and theory at $k\lesssim 0.2h/\mathrm{Mpc}$.
For matter-matter-halo and halo-halo-halo cross-spectra, we find similar agreement; see
Fig.~\ref{fig:correl_mmh_and_hhh_RGauss20_mass3} for the $b_1=1.98$ halo mass
bin. Similar results are obtained for lower mass bins (not shown for
brevity). In particular, the ratio of measured over theoretical
standard deviations fluctuates between $0.5$ and $2$ for all mass
bins for $k\lesssim 0.2h/\mathrm{Mpc}$.
More work is needed to test the covariances at higher precision,
e.g.~by running more realizations or by dividing simulation boxes into
sub-boxes. We also leave it for future work to test the theoretical
covariances \eq{cov_DE_final} between cross-spectra at different
scales $k\ne k'$.
\subsection{Bias estimation from halo-halo-halo cross-spectra}
\label{se:biasContours}
While the goal of this section thus far has been to test the theory
predictions against simulations, we now aim to get a rough
sense of how well bias parameters could be measured from observable
halo-halo-halo cross spectra alone. This should be regarded mainly as
a motivation to study these observables in more detail in the future
rather than a realistic forecast, because we make a number of
idealistic assumptions that are not valid in practice: the
covariance between cross-spectra is assumed to be given by the
leading-order theoretical expression of \eqq{correl_cross_spectra_diagonal}, neglecting any
covariance between different wavenumbers $k'\ne k$; observations
are assumed to be in periodic boxes in real space, neglecting
redshift-space distortions; the shot noise correction
is treated as a free parameter; and the cosmology is
fixed to the fiducial cosmology of the $N$-body runs. Most of these
assumptions are likely to impact the constraining
power of the cross-spectra and should be addressed before considering
real data.
\begin{figure}[tp]
\centerline{
\includegraphics[width=0.8\textwidth]{figs_bias_estimators/contours_marcel_44_kmax0pt09_R20.pdf}
}
\caption{Results of fitting $b_1$, $b_2$, $b_{s^2}$ and
$A_\mathrm{shot}\equiv -\bar{n}_\mathrm{h}\Delta_1$ (imposing
$\Delta_2=\Delta_1/\bar{n}_\mathrm{h}$) to halo-halo-halo
cross-spectra involving the squared density (blue), shift term (red)
or tidal term (green), or all three combined (black), choosing Gaussian
$R=20h^{-1}\mathrm{Mpc}$ smoothing and
$k_\mathrm{max}=0.09h/\mathrm{Mpc}$ for the halo mass bin with
$b_1=1.98$.
The 2d contours of the posterior show $68\%$ and $95\%$
confidence regions corresponding to the full volume of
$V=26.3h^{-3}\mathrm{Gpc}^3$ (i.e.~errors in a single realization
would be larger by a factor of $\sqrt{10}$).
Thin black lines show the
maximum-likelihood points corresponding to the black contours.
The joint likelihood for the
three halo-halo-halo cross-spectra is assumed to be Gaussian in the
cross-spectra, with non-zero covariance between cross-spectra at
$k=k'$ given by the theory expression of
\eqq{cov_cross_spectra_diagonal}. The green contours are somewhat
uncertain because it is not clear how well the MCMC chains sampled
the elongated degeneracies. }
\label{fig:contours_hhh_RGauss20_and_Rgauss10_mass3}
\end{figure}
Under the above idealistic assumptions, we use the
Monte-Carlo sampler emcee \cite{emcee} to fit the bias parameters
$b_1$, $b_2$ and $b_{s^2}$ as well as the dimensionless shot noise
amplitude $A_\mathrm{shot}\equiv -\bar{n}_\mathrm{h}\Delta_1$
(imposing $\Delta_2=\Delta_1/\bar{n}_\mathrm{h}$) to the three
ensemble-averaged halo-halo-halo cross-spectra
$P_{D[\delta^R_\mathrm{h}],\delta^R_\mathrm{h}}(k)$,
$D\in\{\mathsf{P}_0, -F_2^1\mathsf{P}_1, \mathsf{P}_2\}$ for the
halo mass bin with $b_1=1.98$. Colored contours in
Fig.~\ref{fig:contours_hhh_RGauss20_and_Rgauss10_mass3} show results
if each cross-spectrum is fitted individually, while the black
contours correspond to the joint fit to all three cross-spectra.
Focusing on the leftmost column of
Fig.~\ref{fig:contours_hhh_RGauss20_and_Rgauss10_mass3}, we see that
the linear bias $b_1$ is best determined by the shift
cross-spectrum. Adding the other two cross-spectra does not tighten
the $b_1$ constraint much, i.e.~the shift cross-spectrum contains almost the
entire bispectrum information on $b_1$. Once $b_1$ is known, the
constraint on $b_2$ is mostly tightened by the squared density
cross-spectrum (blue degeneracy regions are thinner than green ones in the
$b_1$ vs $b_2$ panel of
Fig.~\ref{fig:contours_hhh_RGauss20_and_Rgauss10_mass3}), while the
constraint on $b_{s^2}$ is mostly
tightened by the tidal cross-spectrum (green degeneracy regions are
thinner than blue ones in the $b_1$ vs $b_{s^2}$ panel). Intuitively, this can be understood
from the Legendre decomposition of the halo-halo-halo bispectrum in
Eqs.~\eq{Bhhh_l0}-\eq{Bhhh_l2}: The shift term corresponding to $l=1$ is the only
term that picks up $b_1$ without any contribution from nonlinear
bias $b_2$ or $b_{s^2}$ (in absence of velocity bias), while the $l=0$ squared
density term is the only one that gets contributions from $b_2$ and
the $l=2$ tidal term is the only term depending on
$b_{s^2}$.\footnote{This intuitive picture is not exact though: In
practice, finite $k_\mathrm{max}$ and smoothing
imply that each cross-spectrum actually picks up dependencies on all
bias parameters, see
Fig.~\ref{fig:contours_hhh_RGauss20_and_Rgauss10_mass3}.
}
Overall, the best-fit parameters
from the combined fit, $(b_1,b_2,b_{s^2},A_\mathrm{shot})=(1.96,
-0.02, -0.44, -0.007)$, agree with the values obtained from matter-halo
statistics, $(1.98, -0.08, -0.52,
-5\times 10^{-5})$, within the uncertainties shown in
Fig.~\ref{fig:contours_hhh_RGauss20_and_Rgauss10_mass3},
but the $1\sigma$ error of the mean of $b_1$ is relatively large
($\sim 5\%$, using the full $26.3h^{-3}\mathrm{Gpc}^3$ volume). The
uncertainties decrease significantly for higher $k_\mathrm{max}$ and
smaller smoothing scales $R$, e.g.~$b_1$ could be measured with $\sim
0.4\%$ precision for $k_\mathrm{max}=0.24h/\mathrm{Mpc}$ and
$R=5h^{-1}\mathrm{Mpc}$, but
best-fit parameters become inconsistent with the true values, demonstrating that
modeling needs to be improved or appropriate clipping or
logarithmic density transforms \cite{Neyrinck0903,SimpsonClipping1107} need to be applied
before it is possible to push to such
small scales. This would also make the full advantage of the
efficient cross-spectrum method over conventional brute-force methods more
apparent because the latter become computationally unfeasible at high $k$.
The degeneracies of the combined contours in
Fig.~\ref{fig:contours_hhh_RGauss20_and_Rgauss10_mass3} correspond to the
typical degeneracies between bias parameters estimated from the
halo-halo-halo bispectrum if redshift-space distortions are neglected,
e.g.~higher $b_1$ can be compensated by lower $b_2$ or
$b_{s^2}$. Since the shot noise correction parameterised by $A_\mathrm{shot}$ is also rather degenerate
with the bias parameters, it would be interesting to model this
analytically instead of treating it as a free parameter.
As a consistency check of the assumed likelihood, jack-knife error
bars are obtained by fitting to each of the ten realizations
individually and calculating the scatter of the best-fit values among
realizations. These jack-knife error estimates are larger than the
average uncertainty predicted by the likelihood width for a single
realization by a factor of $\sim 1.8$ for the conservative $k_\mathrm{max}=0.09h/\mathrm{Mpc}$
and by a factor of $\sim 1.25$ for the more ambitious $k_\mathrm{max}=0.24h/\mathrm{Mpc}$. This could
be attributed to additional contributions to the true covariance that
are neglected in our theoretical covariance (e.g.~at $k'\ne k$), or
uncertainty in the determination of the jack-knife errors due to the
small number of realizations, or a departure of the true likelihood
from a Gaussian pdf.
Summarizing the above, it seems possible to
reach percent-level estimates of the linear bias $b_1$ from
halo-halo-halo cross-spectra in real surveys, but we re-emphasize that
we made a number of unrealistic assumptions, which could easily change
this conclusion (to the better or worse).
\section{Extension to primordial non-Gaussianity}
\label{se:Extensions}
So far we have assumed Gaussian initial conditions. Multiple-field
inflation models can generate local primordial non-Gaussianity that
induces an additional contribution to the matter-matter-matter
bispectrum of the local form
\begin{equation}
\label{eq:Bmmmlocal}
B_\mathrm{mmm}^\mathrm{loc}(k_1,k_2,k_3) =
2 f_\mathrm{NL}^\mathrm{loc} \left[
\frac{M(k_3)}{M(k_1)M(k_2)}P_\mathrm{mm}(k_1)P_\mathrm{mm}(k_2)+2\mbox{
perms}\right],
\end{equation}
where $M(k)=M(k,z)$ is the linear Poisson conversion factor between the primordial
potential $\Phi$ and the late-time matter density at redshift $z$,
\begin{equation}
\label{eq:8}
M(k,z) \equiv \frac{2}{3}\,\frac{k^2T(k)D(z)}{\Omega_\mathrm{m}H_0^2},
\end{equation}
so that $\delta_\mathrm{m}^\mathrm{lin}(\vec{k},z)=M(k,z)\Phi(\vec{k})$.
Here, $T(k)$ is the linear transfer function normalized to $T(k)=1$ on
large scales, and the linear growth factor $D(z)$ for
$\Omega_\mathrm{rad}=0$ is normalized to $D(z)=1/(1+z)$ during matter
domination. Note that $M(k)\propto k^2$ for $k\ll k_\mathrm{eq}$ and
$M(k)\propto k^0$ for $k\gg k_\mathrm{eq}$. The bispectrum
\eq{Bmmmlocal} is maximal in the squeezed limit (e.g.~$k_1\ll
k_2\approx k_3$).
Plugging the bispectrum \eq{Bmmmlocal} into
\eqq{fnl_separable_fgh}, we get
\begin{equation}
\label{eq:fnl_local_esti}
\hat f^\mathrm{loc}_\mathrm{NL} = \frac{24\pi
L^3}{N_\mathrm{loc}}\int \d k\frac{k^2 M^2(k)}{P_\mathrm{mm}(k)}
\hat P_{[\frac{\delta_\mathrm{m}}{M}]^2, \frac{\delta_\mathrm{m}}{M}}(k),
\end{equation}
where we defined the quadratic field
\begin{equation}
\label{eq:64}
\left[\frac{\delta_\mathrm{m}}{M}\right]^2(\vec{k}) \equiv \int \frac{\d^3
q}{(2\pi)^3}\,\frac{\delta_\mathrm{m}(\vec{q})}{M(q)} \frac{\delta_\mathrm{m}(\vec{k}-\vec{q})}{M(|\vec{k}-\vec{q}|)}
\end{equation}
and the filtered density
\begin{equation}
\label{eq:67}
\frac{\delta_\mathrm{m}}{M}(\vec{k}) \equiv \frac{\delta_\mathrm{m}(\vec{k})}{M(k)}.
\end{equation}
At leading order, this equals the primordial potential $\Phi$
reconstructed from the DM density $\delta_\mathrm{m}$. The
cross-spectrum in \eqq{fnl_local_esti} then probes the cross-spectrum
of this reconstructed $\Phi$ with $\Phi^2(\vec{x})$, which corresponds to
the mechanism that generates primordial non-Gaussianity of the local
kind (adding $f_\mathrm{NL}\Phi^2(\vec{x})$ to $\Phi(\vec{x})$).
The cross-spectrum appearing in \eqq{fnl_local_esti} could be used to
estimate $f_\mathrm{NL}^\mathrm{loc}$ if the dark matter density was
directly observable. The extension to observable halo densities is
left for future work. It would also be
straightforward to extend the cross-spectra to other separable types
of primordial non-Gaussianity generated by other inflation models (e.g.~equilateral or
orthogonal).
\section{Conclusions}
\label{se:Conclusions}
In this paper we explore methods to probe large-scale
structure bispectrum parameters in a nearly optimal way.
The tree level bispectrum receives contributions from
gravity at second order, which can be Legendre decomposed into
the squared density $\delta^2(\vec{x})$, the shift term
$-\Psi^i(\vec{x})\partial_i \delta(\vec{x})$ and the tidal term
$s^2(\vec{x})=\tfrac{3}{2}s_{ij}(\vec{x})s_{ij}(\vec{x})$.
When applied to galaxies or halos the gravity term is multiplied by the
appropriate linear bias $b_1$ factor (e.g.~$b_1^3$ when investigating
the halo bispectrum). In addition, nonlinear biasing can introduce two
additional terms that contribute at second order, $b_2\delta^2(\vec{x})$ and $b_{s^2}s^2(\vec{x})$.
There is no nonlinear bias associated with the shift term in the absence of velocity bias.
Since any velocity bias must vanish in the $k \rightarrow 0$ limit as a consequence of
Galilean invariance, we do not include any such term.\footnote{At a $k^2$ level there could be a velocity bias,
but we ignore this here since we work at the lowest order in $k$. Indeed, all the biasing terms can
receive $k^2$ type corrections \cite{Tobias14}. }
These terms correspond to individual components of the bispectrum in a separable form. In this case, in the
limit where tree level theory is valid, one can write an optimal bispectrum estimator using these terms. Specifically,
given a density
$\delta(\vec{x})$, smoothed on the smallest scale where we
still trust the theory predictions, the procedure we propose is as
follows:
\begin{enumerate}
\item Compute the density gradient $\partial_i\delta(\vec{x})$, the
displacement field $\Psi_i(\vec{x})=-\partial_i\partial^{-2} \delta(\vec{x})$, and
the tidal tensor $s_{ij}(\vec{x})=\big[\partial_i\partial_j\partial^{-2}
- \tfrac{1}{3}\delta_{ij}^{(K)}\big]\delta(\vec{x})$.
\item Compute the squared density $\delta^2(\vec{x})$, the shift term
$-\Psi^i(\vec{x})\partial_i \delta(\vec{x})$ and the tidal term
$s^2(\vec{x})=\tfrac{3}{2}s_{ij}(\vec{x})s_{ij}(\vec{x})$.
\item Fourier transform
the three quadratic fields to get $[\delta^2](\vec{k})$,
$[-\Psi^i\partial_i\delta](\vec{k})$ and $[s^2](\vec{k})$.
\item Compute the cross-spectra between the quadratic fields and the
density, i.e.~(suppressing division by the number of modes)
\begin{eqnarray}
\label{eq:delta2_conclusion}
\hat P_{\delta^2,\delta}(k)& \sim &
\sum_{\vec{k},|\vec{k}|=k}
[\delta^2](\vec{k})\delta(-\vec{k}),
\\
\label{eq:shift_conclusion}
\hat P_{-\Psi^i\partial_i\delta,\delta}(k)&\sim &
\sum_{\vec{k},|\vec{k}|=k}
[-\Psi^i\partial_i\delta](\vec{k})\delta(-\vec{k})
\\
\label{eq:s2_conclusion}
\hat P_{s^2,\delta}(k)&\sim &
\sum_{\vec{k},|\vec{k}|=k} [s^2](\vec{k})\delta(-\vec{k}).
\end{eqnarray}
\end{enumerate}
As expected from the Legendre decomposition of the
halo bispectrum, the $l=1$ shift
cross-spectrum \eq{shift_conclusion} contains almost the entire
bispectrum information on the linear bias $b_1$, while the $l=0$ squared density
cross-spectrum \eq{delta2_conclusion} and the $l=2$ tidal
cross-spectrum \eq{s2_conclusion} mostly improve constraints on $b_2$
and $b_{s^2}$ once $b_1$ is known.
Measuring all three cross-spectra and comparing them to their theory
predictions is equivalent to an optimal maximum-likelihood estimation
of the amplitudes of contributions to dark matter or halo bispectra
(under certain regularity conditions; see Section
\ref{se:MaxLikeliBispEsti}). Therefore, these cross-spectra contain the same
constraining power on bias parameters and $\sigma_8$ as a full optimal
bispectrum analysis.
Measuring cross-spectra is both
simpler and computationally cheaper than performing direct bispectrum
measurements for individual triangle configurations. Since they only depend on a single rather than three
wavenumbers, modeling the covariance is also simpler.
We have derived leading-order perturbation theory predictions for the
expectation values and covariances of the three cross-spectra, where
both the quadratic and the single field can be dark matter or halo
fields, and second order bias $b_2$ and tidal tensor bias $b_{s^2}$
are included. The results are given by integrals over
matter-matter-matter, matter-matter-halo or halo-halo-halo bispectra.
The proposed cross-spectra were measured on a set of ten large $N$-body
simulations. The expectation values
are consistent with perturbation theory at the few percent level for
$k \lesssim 0.09h/\mathrm{Mpc}$ at $z=0.55$ for matter-matter-matter
and matter-matter-halo combinations, if all fields are smoothed by a
Gaussian with smoothing scale $R=20h^{-1}\mathrm{Mpc}$. For
halo-halo-halo cross-spectra,
one must include corrections to the Poisson stochasticity.
While these corrections are qualitatively similar to corrections to the halo-halo power spectrum due to exclusion and nonlinear biasing \cite{tobias1305}, future work should investigate and model them in more detail.
The predicted variance of the cross-spectra
and the covariance between any two cross-spectra at the same
wavenumber are found to be consistent with simulations (although the
numerical noise is somewhat large given the small number of
independent realizations).
The ultimate goal of this is to determine the three bias parameters and dark matter clustering power
spectrum by combining these three statistics with the measured galaxy power spectrum.
We have not performed this step in this paper: we plan to explore
potential improvements of the modeling by including higher order
perturbation theory terms as well as improved bias and stochasticity
models to measure the new observables in galaxy surveys in future
work. We also plan to include redshift space distortions
by accordingly modifying and extending the cross-spectra. Given the
simplicity of the method and the agreement with
leading-order perturbation theory on large scales, we hope it will become a useful
tool to break degeneracies of bias and cosmological parameters.
\section*{Acknowledgements}
We thank Florian Beutler and Blake Sherwin for many helpful
discussions, and Martin White and Beth Reid for providing the $N$-body
simulations used in this paper. We are also very grateful to Anatoly
Klypin for sharing a CIC and power spectrum code. We acknowledge useful
discussions with James Fergusson, Airam Marcos-Caballero, Paul
Shellard and Zvonimir Vlah. TB gratefully acknowledges support from
the Institute for Advanced Study through the Corning Glass Works
Foundation Fellowship. This research used resources of the National
Energy Research Scientific Computing Center, a DOE Office of Science
User Facility supported by the Office of Science of the
U.S. Department of Energy under Contract No. DE-AC02-05CH11231. It
also used the COSMOS Shared Memory system at DAMTP, University of
Cambridge operated on behalf of the STFC DiRAC HPC Facility and funded
by BIS National E-infrastructure capital grant ST/J005673/1 and STFC
grants ST/H008586/1, ST/K00333X/1.
| {
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"http:\/\/motls.blogspot.com\/2013\/10\/the-arxiv-aaronson-adopt-mathjax.html","text":"Thursday, October 17, 2013 ... \/\/\/\/\/\n\nThe arXiv, Aaronson adopt Mathjax\n\nArXiv authors expected to use dollar signs in abstracts\n\nThis blog, the world's most important personal physics blog, celebrated its 9th birthday last week. Because it has always been focusing on mathematically heavy topics, there has always been the question whether the dominant expert method to type mathematical formalism \u2013 $\\rm \\TeX$ and\/or $\\rm\\LaTeX$ \u2013 should be implemented.\n\nFor long years, there would be many solutions but all of them had some serious bugs that made me sure rather quickly that I wouldn't get used to them. Or you, the readers, wouldn't get used to them.\n\nTechexplorer required the readers to install a plugin, something that a sensible blogger simply shouldn't expect from most readers. LaTeXMathML.js produced a rather ugly outcome; it wasn't using fonts that look fine at any resolution; and there were probably problems with background colors. CodeCogs had also trouble with the background color and high-resolution outcomes. And I wasn't satisfied with Mimetex and its variations, either. There were probably a few other attempts I forgot.\n\nThings changed in Summer 2011, more than two years ago.\n\nMathjax headquartered at Mathjax.org is a Javascript that every modern Internet browser may understand. It looks for $\\rm\\TeX\/ \\LaTeX$-like sequences on the page and replaces them with the nice $\\rm\\TeX\/ \\LaTeX$ outcome using a few new fonts that may be automatically transmitted to the reader. They look great at any resolution. They may be printed. They don't require the viewer to install any plugins. It just works.\n\nIt seemed clear to me from Day 1 that I would keep it, despite the few seconds that your browser needs to reprocess the $\\rm \\TeX\/\\LaTeX$ formulae. If you've never noticed Mathjax, look at this equation:$J_\\alpha(x) = \\sum\\limits_{m=0}^\\infty \\frac{(-1)^m}{m! \\, \\Gamma(m + \\alpha + 1)}{\\left({\\frac{x}{2}}\\right)}^{2 m + \\alpha}$ This formula for Bessel's function is written in the \"displayed math\" mode. The HTML source looks like this:\n\n...your browser needs to reprocess the $\\rm \\TeX\/\\LaTeX$ formulae. If you've never noticed Mathjax, look at this equation:$J_\\alpha(x) = \\sum\\limits_{m=0}^\\infty \\frac{(-1)^m}{m! \\, \\Gamma(m + \\alpha + 1)}{\\left({\\frac{x}{2}}\\right)}^{2 m + \\alpha}$ This formula formula...\n\nOnce the javascript does its job, the esoteric expressions (featuring many commands that start with a backslash) written in between the backslash-parentheses are converted to nicely looking mathematical symbols. Search for \"$\\rm\\LaTeX$ help\" and \"Mathjax doc\" in the right sidebar of this blog to learn more about $\\rm\\LaTeX$ and Mathjax. Once you feel confident that you have learned some $\\rm\\TeX\/\\rm\\LaTeX$, you may test your skills here.\n\nSorry, the DISQUS comments don't allow Mathjax \u2013 or any website-injected script, for that matter.\n\nYou might say that your humble correspondent was an early adopter, probably the first personal blog with more than \"hundreds\" of visitors per day (4,000 in average for TRF) that installed this feature. It's easy and it just works. It was bound to spread. But it is amazing to see how slow the rest of the Internet is.\n\nNow, more than 2 years later, the most important server for high-energy physics (and some other) preprints, arXiv.org, finally installed Mathjax as well. Look e.g. at the following abstract of a (very interesting) paper that was released today:\nA realistic renormalizable supersymmetric $E_6$ model\nAmong other things, the abstract contains this piece of mathematics:$2\\times(27+\\bar{27})+351'+\\bar{351'}$ You may see that it is being typeset using slightly different fonts. I got the $\\rm\\TeX$ source simply by right-clicking the mathematical expression and choosing \"show math source as $\\rm\\TeX$ commands\"; you are invited to go through the right-click menus. You may \"disable Mathjax\" by clicking at this phrase at the bottom of the arXiv page (this button may then re-enable Mathjax, and so on). The mathematical expression changes to\n$2\\times(27+\\bar{27})+351'+\\bar{351'}$\nWell, the representations should have been written in boldface, the pluses should have been circled, and the bar's should have been overline's, i.e. the right form should have been $2\\times({\\bf 27}\\oplus\\overline{\\bf 27})\\oplus{\\bf 351'} \\oplus \\overline{\\bf 351'},$ but let me not be too picky because the paper is nice. ;-)\n\nNote that $...$ is the standard format to identify \"inline mathematics\" (in between the dollar signs) and arXiv.org allows that. I don't allow that because single dollar signs are reserved for financial applications. Instead, you need to type $...$ for inline mathematics. Similarly, $$...$$ is reserved for displayed maths. This double-dollar terminology is allowed on this blog, too, but I mostly use an equivalent notation with $...$ that you have already encountered.\n\nCongratulations to arXiv.org. It's not really at the forefront of the IT technologies ;-) but at least it reduced its lag to several years or decades. Authors of arXiv.org preprints are invited to try $...$ maths in their abstracts.\n\nScott Aaronson has installed Mathjax on his blog, too. You may check this comment of mine (and upvote it, to partially compensate the vandalism by a dozen of pro-global-warming imbeciles who downvoted it \u2013 and who apparently hope that quantum computing will \"solve\" global warming, a non-existent problem). ;-)\n\nWhat about others? Mr Adam Jester Falkowski and Mr Tommaso Dorigo, open this gate! Prof Matt Strassler, tear down this wall! ;-)\n\nLet me mention one more thing related to $\\rm\\LaTeX$, more precisely ShareLaTeX previously discussed on TRF. You may now become a $\\rm\\LaTeX$ adviser etc. The full (unedited) e-mail below that I received from the ShareLaTeX folks also contains links to some video lectures about $\\rm\\LaTeX$ and other things.\n\nHi Lubos, we've been working on some more cool stuff at ShareLaTeX recently! Here are some of the highlights that we're pleased to announce:\n\nYou can now join the ShareLaTeX team as an\u00a0official advisor. Selected advisors will\n\u2022 Teach introductory LaTeX sessions in your Institution,\n\u2022 Help us refine our LaTeX learning resources for use in your teaching sessions,\n\u2022 Give us feedback and new ideas,\n\u2022 Let us know the needs of your Institution, and\n\u2022 Get to know other advisors and the ShareLaTeX team.\nThere's also some other cool stuff in the works (including some freebies). If you would like to become a ShareLaTeX advisor then please\u00a0apply now!\n\nLaTeX Tutorial Videos\n\nA lot of people find the best way to learn is via video. So we now have four different free video series on youtube (all with a corresponding blog write up and example document) for your viewing pleasure:\nGroup plans\n\nWe have had a lot of requests from people who want to use ShareLaTeX with their entire lab or team so we've added official\u00a0group plans. With group plans you can manage your team members easily.\n\nSurvey\n\nFinally, we have\u00a0a new survey\u00a0which we would really appreciate you filling in. It should only take 60 seconds and the results help us build a better product for you.\u00a0Click here to help us out by filling it in.\n\nBest Regards\nHenry & James\nShareLaTeX co-founders\n\nsnail feedback (13) :\n\nreader Kimmo Rouvari said...\n\nI use it too. It produces very nice looking math stuff. Highly recommended!\n\nreader Lubo\u0161 Motl said...\n\nSorry if you were an early adopter and I forgot about you!\n\nreader Dimension10 (Abhimanyu PS) said...\n\nYay! Great!\n\nI noticed the Loading [MathJaX]\/... thing in LaTeX abstracts.\n\nDo you know why Math JaX LaTeX takes so much time to work on ArXiV, though? It is fine every where else.\n\nI wonder if this will make Physics.SE adopt a new policy of linking to the text-only version of the abstract page (Currently, we just have to link to the abstract and not the PDF) : )\n\nreader Lubo\u0161 Motl said...\n\nLOL, I don't know but this Mathjax in titles and abstracts is arguably the first visible improvement that Paul Ginsparg and his many employees have done since 1991, isn't it? So given their complete avoidance of all new trends in the information technologies, I am surprised that this Mathjax switch occurred at all.\n\nNo need to upvote your comment on Scott's blog---he removed the upvote\/downvote feature after a quick poll of readers wanted it removed.\n\nNice that they now have LaTex on the Arxiv :-)\n\nLaTex, together with loading the TP data dump is the most important thing that we want to work on the new physics site too ...\nDoes there exist som FM how to install the Java script on my Wordpress blog for exmple?\n\nMatt Strassler should not only enable LaTex, but he should make the comments up\/down votable too, such that heavily downvoted comments get hidden automatically. This would largely improve his comment sections, in particular below articles where keywords such as \"supersymmetry\" or \"string theory\" are featured too prominently ...\n\nWhy did Aron Scott do such a poll?\n\nMaking comments up\/down votable largely improves the comment sections.\n\nIt is clear that uninformed trolls do not like this feature, because their comments then have a high probability to get downvoted by the more numerous reasonable but often less vocal readers ...\n\nI kind of (but not strongly) disagree with you, Dilaton.\nThe upvote\/downvote feature allows sock puppets of the blog dictator ( :) ) to gang up on anyone posting a different viewpoint. Also, more reticent lurkers may be put off commenting because they haven't put their toes in the water enough to tolerate some flaming (what a mangled metaphor!) It does, however, allow identification of out-and-out crackpots.\n\nreader James Gallagher said...\n\nI agree Dilaton (Bizarrely I can't upvote or downvote on this site - but can on other disqus sites?) I wasn't sure it was suitable for Sean Carroll's preposterousuniverse blog at first but I think it has been pretty successful there.\n\nreader Dimension10 (Abhimanyu PS) said...\n\nIt is not possible to install MathJaX CDN onto a WordPress.Com blog.\n\nIt is only possible with a WordPress.Org or BloggerSpot blog.\n\nIn fact, no java script can be added to a WordPress.Com blog. They said, when I asked them, that only giyants like Google can take such risks : )\n\nHOWEVER, WordPress.Com has its own option for latex. Format it like this:\n\n$latex MATH EXPRESSION GOES HERE$\n\nThat's how I added the GS Formalism thing in the example questions.\n\nHm sure, it can have disadvantages and bad things can happen with votable comments as you say too ;-)\n\nBut my hope or expectation is that on physics blogs there is certain amount of reasonable people just silently lurking and reading who does not comment, but who would downvote low-level uninformed trolling comments if they have the possibility to do it.\n\nThat is what I have seen happen below this nice guest post of Lance Dixon\n\nhttp:\/\/www.preposterousuniverse.com\/blog\/2013\/10\/03\/guest-post-lance-dixon-on-calculating-amplitudes\/\n\nThe reasonable comments got upvoted and (the astonishingly few) trolls buried in downvotes as it should be ... :-)","date":"2016-10-21 18:19:41","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\": 3, \"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\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.39022281765937805, \"perplexity\": 2947.8680723438533}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2016-44\/segments\/1476988718296.19\/warc\/CC-MAIN-20161020183838-00466-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/thomas1111.wordpress.com\/2010\/09\/","text":"## Archive for September, 2010\n\n### MO birthday\n\nSeptember 28, 2010\n\nSo, MO is one year old today.\u00a0 Let\u2019s make some quick stats (following ones done 6 months ago).\n\nThere are about 2725 \u201cactive users\u201d (which I define as having a rep $\\geq 11$), that\u2019s about 1340 more users within 6 months. In particular, many more first rate contributors have joined since last time. Overall, the senior users (:=tenured folks) include 4 very active Fields Medalists\u00a0 \u2014 Terry Tao, Tim Gowers, Richard Borcherds, Bill Thurston \u2014 and actually the user with the most gold badges is Tim Gowers.\u00a0 Among all users, about 18 of them have reached the 10k reps required for becoming a moderator (the first to 10k was Greg Kuperberg, the first to 20k was Pete L. Clark, the current user with most reps is Joel David Hamkins with 27.2k).\n\nThere have been nearly 11,000 questions asked (some closed quickly, but surely 10,000 at least were legitimate ones).\u00a0 That figure is definitely stunning! I\u2019m not sure I have a favorite, not to mention very many superb answers.\n\nA fantastic website then, congratulations to them.\n\nRelatedly, MO\u2019s cousin for undergraduates, math.stackexange.com (still in beta as I write) is fast becoming very useful too, at another level.","date":"2018-01-19 13:15:10","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\": 1, \"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\": 0.569042980670929, \"perplexity\": 6515.158288413258}, \"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-2018-05\/segments\/1516084887981.42\/warc\/CC-MAIN-20180119125144-20180119145144-00625.warc.gz\"}"} | null | null |
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CELEBRATE LIFE'S STORIES DURING NATIONAL SKILLED NURSING CARE WEEK (MAY 13-19)
BETTY (MARY) DELANEY & MARIA DIRICO CELEBRATE THEIR 100TH BIRTHDAYS
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A healthy person needs 30 to 50 ounces of fluid per day.
Drinking fluids is crucial to staying healthy and maintaining the function of every system in your body, including your heart, brain, and muscles. Fluids carry nutrients to your cells, flush bacteria from your bladder, and prevent constipation.
Older adults often don't get enough fluids and risk becoming dehydrated, especially during summer when it's hotter and people perspire more.
"Older people don't sense thirst as much as they did when they were younger. And that could be a problem if they're on a medication that may cause fluid loss, such as a diuretic," says Dr. Julian Seifter, a kidney specialist and associate professor of medicine at Harvard Medical School. Warning signs of dehydration include weakness, low blood pressure, dizziness, confusion, or urine that's dark in color.
To ward off dehydration, Dr. Seifter says that healthy people should get 30 to 50 ounces of water per day (about 1 to 1.5 liters), but not all at once. "The kidneys lose some ability to eliminate water as we age. It's important to stay hydrated gradually, throughout the day," he says. He recommends drinking water or juices and eating water-rich foods such as salads, fruit, and applesauce. "An easy way to stay hydrated gradually is by getting fluids at meals, with medicine, and socially," says Dr. Seifter.
It's possible to take in too much water if you have certain health conditions, such as thyroid disease or kidney, liver, or heart problems, or if you're taking medications that make you retain water, such as nonsteroidal anti-inflammatory drugs (NSAIDs), opiate pain medications, and some antidepressants. Dr. Seifter says for that reason, you should check with your doctor to be sure you're getting the right amount. | {
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#pragma once
#ifdef __cplusplus
extern "C" {
#endif
enum attachsql_ret_t {
ATTACHSQL_RET_OK,
ATTACHSQL_RET_PARAMETER_ERROR,
ATTACHSQL_RET_CONNECTING,
ATTACHSQL_RET_DNS_ERROR,
ATTACHSQL_RET_CONNECT_ERROR,
ATTACHSQL_RET_OUT_OF_MEMORY_ERROR,
ATTACHSQL_RET_NO_SCRAMBLE,
ATTACHSQL_RET_USER_TOO_LONG,
ATTACHSQL_RET_SCHEMA_TOO_LONG,
ATTACHSQL_RET_NET_READ_ERROR,
ATTACHSQL_RET_NET_SSL_ERROR,
ATTACHSQL_RET_NET_WRITE_ERROR,
ATTACHSQL_RET_PACKET_OUT_OF_SEQUENCE,
ATTACHSQL_RET_BAD_PROTOCOL,
ATTACHSQL_RET_NO_OLD_AUTH,
ATTACHSQL_RET_BAD_SCRAMBLE,
ATTACHSQL_RET_COMPRESSION_FAILURE,
ATTACHSQL_RET_BAD_STMT_PARAMETER
};
#ifdef __cplusplus
}
#endif
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\section{Introduction}\label{ujmsec:intro}
{\leftskip\parindent\emph{I have this ``madly optimistic'' (Mermin called it) feeling that Bohrian--Paulian ideas will lead us to the next stage of physics. That is, that thinking about quantum foundations from their point of view will be the beginning of a new path, not the end of an old one.}\par\hfill--- \emph{Christopher A. Fuchs}\cite{FuchsPaulian}}\par\smallskip\noindent
The beginning of the 21st Century saw the launch of a new interpretation of quantum mechanics, by {Carlton Caves}, {Chris Fuchs}, and {Ruediger Schack}.\cite{CFS2002} Initially conceived as an extended personalist Bayesian theory of probability called {``Quantum Bayesianism},'' it has since been re-branded as ``QBism,'' the term David Mermin\cite{MerminQBnotCop} prefers, considering it ``as big a break with 20th century ways of thinking about science as Cubism was with 19th century ways of thinking about art.'' The big break lies not in the emphasis that the mathematical apparatus of quantum mechanics is a probability calculus---that ought to surprise no one---but in this \emph{plus} a radically subjective Bayesian interpretation of probability \emph{plus} a radically subjective interpretation of the events to which (and on the basis of which) probabilities are assigned.
Recently the referent of the ``B'' in QBism became moot. While Mermin\cite{MerminNow2013} at one time suggested that it should stand for Bruno de Finetti (``Quantum Brunoism''), he now endorses the term Bettabilitarianism\cite{MerminBetterSense} suggested by Chris Fuchs, which was coined by Oliver Wendell Holmes, Jr.:
\begin{quote}
I must not say necessary about the universe\dots. We don't know whether anything is necessary or not. I believe that we can \emph{bet} on the behavior of the universe in its contact with us. So I describe myself as a \emph{bet\,}tabilitarian.\cite{Holmes-Pollock}
\end{quote}
In the present paper, I shall argue that the ``B'' in QBism rightfully stands for Niels Bohr.
The paper is organized as follows. Section~\ref{ujmsec:obscure} explains why Bohr nowadays seems obscure to most physicists. Section~\ref{ujmsec:contextuality} identifies the contextuality of physical variables as Bohr's essential contribution to Kant's theory of science. (Affinities between Bohr's theory of science and Kant's have been noted by a number of scholars.\cite{Falkenburg2010, Brock2010, dEspagnat2010, Chevalley94, Folse94, Hooker94, Bitbol2010, Cuffaro2010, Honner1982, Kaiser1992, MacKinnon2012}) Section~\ref{ujmsec:decontexting} outlines Kant's theory of science, its own contextuality---human experience---and its decontextualization. Section~\ref{ujmsec:realism-gbu} distinguishes three kinds of realism: the ``good'' (internal) realism of Kant and Bohr, the ``bad'' naive realism recently defended by John Searle,\cite{Searle2004} and the ``ugly'' realism associated with the representative theory of perception.
For Kant, objectivity meant the possibility of thinking of experiences as experiences of objects. Realizing that in the new field of experience opened up by the quantum theory this possibility no longer exists, Bohr supplemented the object-oriented language of everyday discourse with the language of quantum phenomena. This is discussed in Sec.~\ref{ujmsec:Bohr-o2qp}. Section~\ref{ujmsec:Bohr-cl} explains why Bohr insisted on the need to use (i)~``ordinary language'' or ``plain language'' or ``the common human language'' and (ii)~ ``classical concepts'' or ``the terminology of classical physics'' yet never mentioned ``classical language'' nor ``the language of classical physics'' (i.e., not once in his \emph{Collected Works}). Section~\ref{ujmsec:amplification} addresses the apparent conflict in Bohr's writings between invocations of ``irreversible processes'' and ``irreversible amplification effects'' on the one hand and ``the essential irreversibility inherent in the very concept of observation'' on the other. Section~\ref{ujmsec:years} briefly surveys the philosophically rather barren period between the passing of Niels Bohr and the advent of QBism.
The discussion of QBism begins in Sec.~\ref{ujmsec:qbismWigner}, in which it is argued that by admitting incoherent superpositions of alternatives involving distinct cognitive states, the QBist solutions to ``Wigner's friend''\cite{Wigner61} and similar conundrums overshoot their marks. Section~\ref{ujmsec:qbismShifty} concerns the placement of Bell's shifty split.\cite{Bell90} QBists see only one alternative to placing the Heisenberg cut inside the objective world, to wit, between the objective world and the private experiences in which it originates. There is, however, another alternative, which is to place it between the objective world and the unspeakable domain beyond the reach of our concepts, which only becomes speakable by saying in ordinary language ``what we have done and what we have learned'' [BCW\,7:349].%
\footnote{In what follows, BCW (followed by volume and page number) refers to the \emph{Collected Works} of Niels Bohr.\cite{BCW}}
This, I content, is where it should be placed, and where to all intents and purposes it was placed by Bohr.
The concluding section deals with how QBism relates to Bohr, beginning with certain misreadings of Bohr that QBists share with the majority of current interpreters of quantum mechanics. My final verdict is that QBism, through its emphasis on the individual experiencing subject, brings home the intersubjective constitution of our common external world more forcefully than Bohr ever did. (The time wasn't ripe for this then. Perhaps it is now.) Bohr's insights, on the other hand, are exceedingly useful in clarifying the QBist position, attenuating its excesses, and enhancing its internal consistency.
\section{Why Bohr seems obscure}\label{ujmsec:obscure}
{\leftskip\parindent\emph{As a philosopher Niels Bohr was either one of the great visionary figures of all time, or merely the only person courageous enough to confront head on, whether or not successfully, the most imponderable mystery we have yet unearthed. --- N. David Mermin}\cite{MerminBoojums}\par}\smallskip\noindent
Today, Bohr seems obscure to most physicists. Catherine Chevalley\cite{Chevalley99} has identified three reasons for this deplorable situation. The first is that Bohr's mature views, which ``remained more or less stable at least over the latter thirty years of Bohr's life''\cite{Hooker1972} (i.e., since at least 1932), have come to be equated with one variant or another of the Copenhagen interpretation. The latter only emerged in the mid-1950's, in response to David Bohm's hidden-variables theory and the Marxist critique of Bohr's alleged idealism, which had inspired Bohm. The second reason is that Bohr's readers will usually not find in his writings what they expected to find, while they will find a number of things that they did not expect. What they expect is a take on problems arising in the context of $\Psi$-ontology---the spurious reification of a probability calculus---such as the problem of objectification or the quantum-to-classical transition. What they find instead is discussions of philosophical issues such as the meanings of ``objectivity,'' ``truth,'' and ``reality'' and the roles of language and communication. The third reason is that the task of making sense of quantum mechanics is seen today as one of grafting a metaphysical narrative onto a mathematical formalism, in a language that is sufficiently vague philosophically to be understood by all and sundry. For Bohr, as also for Werner Heisenberg and Wolfgang Pauli, the real issues lay deeper. They judged that the conceptual difficulties posed by quantum mechanics called in question the general framework of contemporary thought, its concepts, and its criteria of consistency.
\section{Contextuality}\label{ujmsec:contextuality}
It was Immanuel Kant, the most important philosopher of the modern era, who first demonstrated that it was possible to provide a scientific theory with much stronger justification than mere empirical adequacy. The kind of argument inaugurated by him to this end begins by assuming that a certain proposition \textbf{p} is true, and then shows that another proposition \textbf{q}, stating a precondition for the truth of \textbf{p}, must also be true: if \textbf{q} were not true, \textbf{p} could not be true. For his immediate purpose the relevant proposition \textbf{p} was that empirical knowledge is possible, and the corresponding proposition \textbf{q} was that certain universal laws must hold.
``Reason,'' Kant wrote, ``must approach nature with its principles in one hand \dots\ and, in the other hand, the experiments thought out in accordance with these principles.'' The concepts in terms of which reason's principles are formulated owe their meanings to our cognitive faculties of intuition%
\footnote{The German original, \emph{Anschauung}, covers both visual perception and visual imagination.}
and thought. They allow us to ask meaningful questions, and to make sense of the answers we obtain by experiment and observation: ``what reason would not be able to know of itself and has to learn from nature, it has to seek in the latter'' but it has to do this ``in accordance with what reason itself puts into nature''.\cite{KantCPR2} What Kant did not anticipate was that experiments would come to play the same constitutive role as our cognitive faculties do in defining the terms of our discourse with nature. The insight that certain questions are \emph{contextual}---that they have no answers unless their answers are elicited by actual experiments---is due to Bohr. To hitch the definition of physical quantities to the experimental conditions under which they are observed, is Bohr's ground-breaking contribution to Kant's theory of science.\cite{Bitbol98} It has, moreover, been spectacularly borne out by the no-go theorems of John Bell,\cite{Bell64} Simon Kochen and Ernst Specker,\cite{Kochen-Specker} and Alexander Klyachko and coworkers.\cite{Klyachko08}
\section{Decontextualization}\label{ujmsec:decontexting}
Bohr's contextuality, however, was not the first to play a role in natural philosophy. From the end of the 17th century onwards, it was widely accepted by philosophers that objects existed relative to a context, to wit, human experience. By placing the subject of empirical science squarely into the context of human experience, Kant dispelled many qualms that had been shared by thinkers at the end of the 18th century---qualms about the objective nature of geometry, about the purely mathematical nature of Newton's theory, about the unintelligibility of action at a distance, about Galileo's principle of relativity, to name a few.
Concerning the laws of geometry, which apply to objects constructed by us in the space of our imagination, the question was why they should also apply to the physical world. Kant's answer was that they apply to objects perceived as well as to objects imagined because visual perception and visual imagination share the same space.%
\footnote{\label{note:Kant1}It is noteworthy that Kant's argument applies, not to Euclidean geometry specifically, which was the only geometry known in Kant's time, but to geometry in general, and thus to whichever geometry is best suited to formulating the laws of physics. It has even been said that Kant's theory of science set in motion a series of re-conceptualizations of the relationship between geometry and physics that eventuated in Einstein's theories of relativity.\cite{Friedman2009}}
As to the mathematical nature of Newtonian mechanics, it was justified, not by the Neo-Platonic belief that the book of nature was written in mathematical language, but by its being a precondition of scientific knowledge. What made it possible to conceive of appearances as aspects of an objective world was the mathematical regularities that obtain between them. Newton's refusal to explain action at a distance was similarly justified, inasmuch as the only intelligible causality available to us consists in lawful mathematical relations between phenomena: for the Moon to be causally related to the Earth was for the Moon to stand in a regular mathematical relation to the Earth. As to the principle of relativity, ditto: lawful mathematical relations only exist between phenomena, and thus only between objects or objective events, but never between a particular phenomenon and space or time itself.%
\footnote{\label{note:Kant2}Here, too, it would be an anachronism to argue that Kant singled out Galilean relativity, which was the only relativity known in his time. His argument holds for every possible principle of relativity, including Einstein's.}
Kant's premise was that ``space and time are only forms of sensible intuition, and therefore only conditions of the existence of the things {as appearances}.'' It follows
\begin{quote}
that we have no concepts of the understanding and hence no elements for the cognition of things except insofar as an intuition can be given corresponding to these concepts, consequently that we can have cognition of no object as a thing in itself, but only insofar as it is an object of sensible intuition, i.e. as an appearance; from which follows the limitation of all even possible speculative cognition of reason to mere objects of \emph{experience}. Yet \dots\ even if we cannot \emph{cognize} these same objects as [i.e., \emph{know} them to be] things in themselves, we at least must be able to \emph{think} them as things in themselves. For otherwise there would follow the absurd proposition that there is an appearance without anything that appears.\cite{KantCPR3}
\end{quote}
Before Kant, there appears to have been no philosopher who did not have a correspondence theory of truth, and who did not think of the relation of sense impressions to the external world as a relation of similarity. Kant was the first to show that the predictive success of a scientific theory does not have to be attributed to some empirically inaccessible correspondence between the structure of the theory and the structure of the real world. Needless to say, this had to be done without calling into question the objectivity of the theory, i.e., in a way that allowed people to think of phenomena as appearances of things ``out there.'' We must be able to \emph{decontextualize} the objective world, to forget that it depends on us. And if there is only the single universal context of human experience, this is easily done. We are free to think of perceived objects as faithful representations of real objects (things in themselves), free to forget that the apparently mind-independent system of objects ``out there'' was a mental construct, and that the concepts that were used in its construction are meaningless outside the context of human experience.
\section{Realism good, bad, and ugly}\label{ujmsec:realism-gbu}
In an essay written during the last year of his life,\cite{SchrWhatIsReal} Erwin Schr\"odinger expressed his astonishment at the fact that despite ``the absolute hermetic separation of my sphere of consciousness'' from everyone else's, there was ``a far-reaching structural similarity between certain parts of our experiences, the parts which we call external; it can be expressed in the brief statement that we all live in the same world.'' This similarity, Schr\"odinger avowed, was ``not rationally comprehensible. In order to grasp it we are reduced to two irrational, mystical hypotheses,'' one of which%
\footnote{The alternative hypothesis, which he endorsed, was ``that we are all really only various aspects of the One''\cite{SchrWhatIsReal}: the multiplicity of minds ``is only apparent, in truth there is only one mind. This is the doctrine of the Upanishads. And not only of the Upanishads''.\cite{SchrLifeMindMatterAPOM} The One in question is the ultimate subject, from which we are separated by a veil of self-oblivion. The same veil (according to the Upanishads) also prevents us from perceiving the ultimate object, as well as its identity with the ultimate subject. If ``to Western thought this doctrine has little appeal,'' it is because our science ``is based on objectivation, whereby it has cut itself off from an adequate understanding of the Subject of Cognizance, of the mind.'' For Schr\"odinger,\cite{SchrLifeMindMatterPO} this was ``precisely the point where our present way of thinking does need to be amended, perhaps by a bit of blood-transfusion from Eastern thought.''}
is ``the so-called {hypothesis of the real external world}.'' Schr\"odinger left no room for uncertainty about what he thought of this hypothesis. To invoke ``the existence of a real world of bodies which are the causes of sense impressions and produce roughly the same impression on everybody \dots\ is not to give an explanation at all; it is simply to state the matter in different words. In fact, it means laying a completely useless burden on the understanding.'' It means uselessly translating the statement ``everybody agrees about something'' into the statement ``there exists a real world which causes everybody's agreement.'' Instead of explaining the fact expressed by the first statement, the second merely reinforces its incomprehensibility, for the relation between this real world and those aspects of our experiences about which there is agreement, is something we cannot know. The causal relations we know are internal to those of our experiences about which we agree.
In ancient and medieval philosophy, to \emph{be} was either to be a substance or to be a property of a substance. Substance was self-existent; everything else depended for its existence on a substance. With Descartes, the human conscious subject assumed the role of substance: to \emph{be} became either to be a subject or to exist as a representation for a subject. Thus was born the representative theory of perception. In the eyes of philosopher John Searle,\cite{Searle2004a} the move from the older view that ``we really perceive real objects'' to the view that we only perceive sense impressions was ``the greatest single disaster in the history of philosophy over the past four centuries.'' A disaster it was indeed, not least because it continues to muddy the scientific waters when it comes to sensory perception.%
\footnote{\label{note:sacop}The standard scientific account of perception begins by positing the mind-independent existence of a real world. Objects in this world are said to emit light or sound waves, which are said to stimulate peripheral nerve endings (retinas or ear drums). The stimulated nerves are said to send signals to the brain, where neural processes are said to give rise to perceptual experience. The trouble with this account is not simply that no one has any idea how to bridge the ``explanatory gap''\cite{Levine2001} between objective brain processes and subjective experiences. The trouble is that no one has any idea how we could have information about what happens or exists in that real world. While the standard scientific account of perception begins by invoking events in that world, it leads to the conclusion that we have access only to our experiences, and that there is no way we could know anything about what happens or exists in that world.}
The representative theory of perception poses this dilemma: either the gap between representations and the objects they are supposed to represent can never be bridged, or the world is reduced to representations. Either science deals with objects in the real world, in which case we have no justifiable idea of how we come to have representations, or it deals with representations, in which case we have no justifiable knowledge of the real world. Transcendental philosophy, inaugurated by Kant and continued in the 20th century by Edmund Husserl,\cite{HusserlEssential} emerged as a critique of the representative theory. In an attempt to defend the older, direct realism, Searle has invoked the fact that we are able communicate with other human beings, using publicly available meanings in a public language. For this to work, he argued,\cite{Searle2004b} we have to assume common, publicly available objects of reference:
\begin{quote}
So, for example, when I use the expression ``this table'' I have to assume that you understand the expression in the same way that I intend it. I have to assume we are both referring to the same table, and when you understand me in my utterance of ``this table'' you take it as referring to the same object you refer to in this context in your utterance of ``this table.''
\end{quote}
The implication then is that
\begin{quote}
you and I share a perceptual access to one and the same object. And that is just another way of saying that I have to presuppose that you and I are both seeing or otherwise perceiving the same public object. But that public availability of that public world is precisely the direct realism that I am here attempting to defend.
\end{quote}
Searle points out that his argument is transcendental in Kant's sense. Here \textbf{p} is the assumption that we are able to communicate with each other in a public language, and \textbf{q} is the conclusion that there must be publicly available objects in a public world about which we can communicate in a public language. The actual implication of his argument, however, is the agreement between our respective ``spheres of consciousness''---between what exists for me, in my experience, and what exists for you, in your experience---which so astonished Schr\"odinger. It allows us to communicate with each other \emph{as if} direct realism were true. What Searle succeeds in defending against the ``ugly'' representative realism is not the ``bad'' direct realism but the ``good'' \emph{internal realism} inaugurated by Immanuel Kant and defended (among others) by Hilary Putnam and Bernard d'Espagnat.%
\footnote{Putnam assumed the existence of a mind-independent real world but insisted that it does not dictate its own descriptions to us: ``talk of ordinary empirical objects is not talk of things-in-themselves but only talk of things-for-us''\cite{Putnam81}; ``we don't know what we are talking about when we talk about `things in themselves'\,''\cite{Putnam87}. D'Espagnat,\cite{d'Espagnat_VR} for his part, stressed the necessity of distinguishing between an empirically inaccessible veiled reality and an intersubjectively constructed objective reality.}
The key role that language plays in establishing the rationally incomprehensible correspondence between the ``external parts'' of our internal experiences, has also been emphasized by Schr\"odinger:
\begin{quote}
What does establish it is \emph{language}, including everything in the way of expression, gesture, taking hold of another person, pointing with one's finger and so forth, though none of this breaks through that inexorable, absolute division between spheres of consciousness.\cite{SchrWhatIsReal}
\end{quote}
\section{Bohr: from objects to quantum phenomena}\label{ujmsec:Bohr-o2qp}
The hallmark of empirical knowledge is objectivity. To Kant, objectivity meant the possibility of thinking of appearances as experiences of \emph{objects}. His inquiry into the preconditions of empirical science was therefore an inquiry into the conditions that make it possible to organize sense impressions into (identifiable) objects. Yet in the new field of experience opened up by the quantum theory, this possibility no longer seemed to exist. As Schr\"odinger\cite{SchrNGSH} wrote,
\begin{quote}
Atoms---our modern atoms, the ultimate particles---must no longer be regarded as identifiable individuals. This is a stronger deviation from the original idea of an atom than anybody had ever contemplated. We must be prepared for anything.
\end{quote}
For the present-day physicist, it is not easy to understand the bewilderment that the founders and their contemporaries experienced in the early days of the quantum theory:
\begin{quote}
All the verities of the preceding two centuries, held by physicists and ordinary people alike, simply fell apart---collapsed. We had to start all over again, and we came up with something that worked just beautifully but was so strange that nobody had any idea what it meant except Bohr, and practically nobody could understand him. So naturally we kept probing further, getting to smaller and smaller length scales, waiting for the next revolution to shed some light on the meaning of the old one.\cite{Mermin_epochs}
\end{quote}
That revolution never came. Quantum mechanics works as beautifully in the nucleus as it does in the atom; and it works as beautifully in the nucleon as it does in the nucleus, seven or eight orders of magnitude below the level for which it was designed. (It also works beautifully many orders of magnitude above that level, as for example in a superconductor.) It is therefore past time to try more seriously to understand what Bohr had been trying to drive home.
``Without sensibility no object would be given to us,'' Kant wrote,\cite{KantCPR5} ``and without understanding none would be thought.'' And again: ``we have no concepts of the understanding \dots\ except insofar as an intuition can be given corresponding to these concepts''.\cite{KantCPR3} Bohr could not have agreed more, insisting as he did that meaningful physical concepts have not only mathematical but also visualizable content. Such concepts are associated with pictures, like the picture of a particle following a trajectory or the picture of a wave propagating in space. In the classical theory, a single picture could accommodate all of the properties a system can have. When quantum theory came along, that all-encompassing picture fell apart. Unless certain experimental conditions obtained, it was impossible to picture the electron as following a trajectory (which was nevertheless a routine presupposition in setting up Stern--Gerlach experiments and in interpreting cloud-chamber photographs), and there was no way in which to apply the concept of position. And unless certain other, incompatible, experimental conditions obtained, it was impossible to picture the electron as a traveling wave (which was nevertheless a routine presupposition in interpreting the scattering of electrons by crystals), and there was no way in which to apply the concept of momentum.
Bohr settled on the nexus between pictures, physical concepts, and experimental arrangements as key to ``the task of bringing order into an entirely new field of experience''.\cite{BohrSchilpp} If the visualizable content of physical concepts cannot be described in terms of compatible pictures, it has to be described in terms of something that \emph{can} be so described, and what can be so described are the experimental conditions under which the incompatible physical concepts can be employed. What distinguishes these experimental conditions from the quantum systems under investigation is their accessibility to direct sensory experience.
What Bohr added to Kant's theory of science was his insight that empirical knowledge was not necessarily limited to what is \emph{directly} accessible to our senses, and that, therefore, it does not have to be \emph{solely} a knowledge of sense impressions organized into objects. It can also be a knowledge of objects that are \emph{not} objects of sensible intuition but instead are constituted by experimental conditions, which \emph{are} sense impressions organized into objects. This is why ``the objective character of the description in atomic physics depends on the detailed specification of the experimental conditions under which evidence is gained'' [BCW\,10:215]. Quantum mechanics does not do away with objects of sensible intuition but supplements them with \emph{quantum phenomena}:
\begin{quote}
all unambiguous interpretation of the quantum mechanical formalism involves the fixation of the external conditions, defining the initial state of the atomic system concerned and the character of the possible predictions as regards subsequent observable properties of that system. Any measurement in quantum theory can in fact only refer either to a fixation of the initial state or to the test of such predictions, and it is first the combination of measurements of both kinds which constitutes a well-defined phenomenon. [BCW\,7:312]
\end{quote}
\section{Bohr: concepts and language}\label{ujmsec:Bohr-cl}
The transition in Bohr's thinking from the familiar epistemology of objects to an epistemology supplemented with quantum phenomena resulted from his insight that ``the facts which are revealed to us by the quantum theory \dots\ lie outside the domain of our ordinary forms of perception'' [BCW\,6:217]. As early as 1922, Bohr opined that the difficulties physicists were facing at the time were ``of such a kind that they hardly allow us to hope, within the world of atoms, to implement a description in space and time of the kind corresponding to our usual sensory images'' [BCW\,10:513--514]. By 1926, the mature (non-relativistic) theory was in place, and by 1929 Bohr's thoughts had gelled into what to my mind remains the most astute understanding of quantum mechanics to date.
In his writings of that year, abundant reference is made to ``our (ordinary) forms of perception,'' time and space. As in: quantum theory has ``justified the old doubt as to the range of our ordinary forms of perception when applied to atomic phenomena'' [BCW\,6:209]; ``at the same time as every doubt regarding the reality of atoms has been removed, \dots we have been reminded in an instructive manner of the natural limitation of our forms of perception'' [BCW\,6:237]. This limitation was ``brought to light by a closer analysis of the applicability of the basic physical concepts in describing atomic phenomena'' [BCW\,6:242]. That is to say, the natural limitation of our forms of perception both implies and is implied by a natural limitation of the applicability of our basic physical concepts, which is a consequence of the uncertainty relations. Yet ``in spite of their limitation, we can by no means dispense with those forms of perception which colour our whole language and in terms of which all experience must ultimately be expressed'' [BCW\,6:283]. In other words, the conceptual framework of quantum physics is the same as that of classical physics, the difference being that in quantum physics its applicability is limited.
``When speaking of a conceptual framework,'' Bohr wrote, ``we merely refer to an unambiguous logical representation of relations between experiences'' [BCW\,10:84]. In Bohr's time and the cultural environment in which he lived, Kant's theory of science still exercised considerable influence. There can be little doubt that the unambiguous logical representation of relations between experiences that Bohr had in mind, was in all important respects the conceptual framework staked out by Kant, providing the general structure of an object-oriented language.
In Kant's theory of science, the relevant relations between experiences fall under the logical categories of substance, causality, and interaction. The logical relation between a (logical) subject and a predicate makes it possible for us to think of a particular nexus of perceptions as the properties of a \emph{substance}, connected to it as predicates are connected to a subject. It makes it possible for me to think of perceptions as connected not by me, in my experience, but in an object ``out there'' in the public world. The logical relation between antecedent and consequent (if \dots\ then\dots) makes it possible for us to think of what we perceive at different times as properties of substances connected by \emph{causality}. It makes it possible for me to think of asynchronous perceptions as connected not merely in my experience but also objectively, by a causal nexus ``out there.'' And the category of community or reciprocity, which Kant associated with the disjunctive relation (either\dots\ or\dots), makes it possible for us to think of what we perceive in different locations as properties of substances connected by a \emph{reciprocal action}. It makes it possible for me to think of simultaneous perceptions as connected not only in my experience but objectively. (Kant thought that by establishing a reciprocal relation, we establish not only an objective spatial relation but also an objective relation of simultaneity.)
But if we are to be able to think of perceptions as properties of substances, or as causally connected, or as affecting each other, the connections must be regular. For perceptions to be perceptions of a particular kind of thing (say, an elephant), they must be connected in an orderly way, according to a concept denoting a lawful concurrence of perceptions. For perceptions to be causally connected, like (say) lightning and thunder, they must fall under a causal law, according to which one perception necessitates the subsequent occurrence of another. (By establishing a causal relation falling under a causal law, we also establish an objective temporal relation.) And for perceptions to be reciprocally connected, like (say) the Earth and the Moon, they must affect each other according to a reciprocal law, such as Newton's law of gravity. It is through lawful connections in the ``manifold of appearances'' that we are able to think of appearances as perceptions of a self-existent system of causally evolving (and thus re-identifiable) objects, from which we, the experiencing subjects, can remove ourselves.
Even in a field of experience in which the concepts required to bundle sense impressions into objects cannot be applied, one has to rely on the common object-oriented language. Where one cannot speak of objects, one has to speak of quantum phenomena, i.e., of experimental arrangements and results indicated by measuring instruments:
\begin{quote}
The argument is simply that by the word ``experiment'' we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangement and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics. [BCW\,7:349]
\end{quote}
Two expressions are significant here: ``unambiguous language'' and ``the terminology of classical physics.'' Presently (August 2019) a combined Google search for ``Bohr'' and ``classical language'' (the latter term including the quotes) yields nearly 5,000 results, while a Google search for ``Bohr'' and ``language of classical physics'' yields more than 24,000 results. By contrast, searching the 13 volumes of the \emph{Complete Works} of Niels Bohr does not yield a \emph{single} occurrence of either ``classical language'' or ``language of classical physics.'' It is the ubiquity in the secondary literature of these latter expressions that is chiefly responsible for the widespread misconceptions about Bohr's thinking.
While Bohr insisted on the use of classical \emph{concepts} (or, the terminology of classical physics, or simply ``elementary physical concepts'' [BCW\,7:394]), the \emph{language} on the use of which he insisted was ``ordinary language'' [BCW\,7:355], ``plain language'' [BCW\,10:159], the ``common human language'' [BCW\,10:157--158], or the ``language common to all'' [10:xxxvii]. A distinction must therefore be drawn between the role that classical concepts played in Bohr's thinking and the role that was played by the common human language. The common human language is the object-oriented language of everyday discourse, while classical concepts are not proprietary to classical physics but denote attributes that owe their meanings to our forms of perception, such as position, orientation, and the ones that are defined in terms of invariances under spacetime transformations.
One day during tea at his institute, Bohr was sitting next to Edward Teller and Carl Friedrich von Weizs\"acker. Von Weizs\"acker\cite{vW_Structure} recalls that when Teller suggested that ``after a longer period of getting accustomed to quantum theory we might be able after all to replace the classical concepts by quantum theoretical ones,'' Bohr listened, apparently absent-mindedly, and said at last: ``Oh, I understand. We also might as well say that we are not sitting here and drinking tea but that all this is merely a dream.'' If we are dreaming, we are unable to tell others what we have done and what we have learned. Therefore
\begin{quote}
it would be a misconception to believe that the difficulties of the atomic theory may be evaded by eventually replacing the concepts of classical physics by new conceptual forms. \dots the recognition of the limitation of our forms of perception by no means implies that we can dispense with our customary ideas or their direct verbal expressions when reducing our sense impressions to order. [BCW\,6:294]
\end{quote}
Or, as Heisenberg put it,\cite{Heisenberg_PP56} ``[t]here is no use in discussing what could be done if we were other beings than we are.''%
\footnote{Heisenberg thought it possible that the forms of perception of other beings, and hence their concepts, could be different from ours: our concepts ``may belong to the species `man,' but not to the world as independent of men''.\cite{Heisenberg_PP91}}
Bohr's claim that the ``classical language'' (i.e., plain language supplemented with the terminology of classical physics) was indispensable, has also been vindicated by subsequent developments in particle physics:
\begin{quote}
This [claim] has remained valid up to the present day. At the \emph{individual} level of clicks in particle detectors and particle tracks on photographs, all measurement results have to be expressed in classical terms. Indeed, the use of the familiar physical quantities of length, time, mass, and momentum-energy at a subatomic scale is due to an extrapolation of the language of classical physics to the non-classical domain.\cite{Falkenburg2007-162}
\end{quote}
It is therefore an irony that Bohr, seeing Kant as arguing for the necessary validity and unlimited reach of classical concepts, regarded complementarity as an alternative to Kant's theory of science, thus drawing the battle lines in a way which put Kant and himself on opposing sides. Just as Kant did not argue for the universal validity of Euclidean geometry \emph{in particular} (see Note~\ref{note:Kant1}), nor for Galilean relativity \emph{in particular} (see Note~\ref{note:Kant2}), so his arguments did not, in effect, establish the \emph{unlimited reach} of classical concepts. As his arguments merely established the validity of whichever geometry, and whichever principle of relativity, was convenient, so they established the necessary validity of classical concepts but not their unlimited reach. What Kant did not anticipate was the possibility of empirical knowledge that did \emph{not} involve the organization of sense impressions into objects---an empirical knowledge that, while being obtained \emph{by means of} sense impressions organized into objects, was not a knowledge \emph{of} sense impressions organized into objects.
\section{Bohr and the irreversibility of measurements}\label{ujmsec:amplification}
If the terminology of quantum phenomena is used consistently, then nothing---at any rate, nothing we know how to think about---happens between ``the fixation of the external conditions, defining the initial state of the atomic system concerned'' and ``the subsequent observable properties of that system'' [BCW\,7:312].
Any story purporting to detail a course of events in the interval between a system preparation and a subsequent observation is inconsistent with ``the essential wholeness of a quantum phenomenon,'' which ``finds its logical expression in the circumstance that any attempt at its subdivision would demand a change in the experimental arrangement incompatible with its appearance'' [BCW\,10:278]. What, then, are we to make of the following passages [emphases added]?
\begin{quote}
[E]very well-defined atomic phenomenon is closed in itself, since its observation implies a permanent mark on a photographic plate \emph{left by the impact of an electron} or similar recordings \emph{obtained by suitable amplification devices of essentially irreversible functioning}. [BCW\,10:89]
\smallskip Information concerning atomic objects consists solely in the marks they make on these measuring instruments, as, for instance, a spot \emph{produced by the impact of an electron on a photographic plate} placed in the experimental arrangement. The circumstance that such marks are \emph{due to irreversible amplification effects} endows the phenomena with a peculiarly closed character pointing directly to the irreversibility in principle of the very notion of observation. [BCW\,10:120]
\smallskip In this connection, it is also essential to remember that all unambiguous information concerning atomic objects is derived from the permanent marks---such as a spot on a photographic plate, \emph{caused by the impact of an electron}---left on the bodies which define the experimental conditions. Far from involving any special intricacy, the\emph{ irreversible amplification effects on which the recording of the presence of atomic objects rests} rather remind us of the essential irreversibility inherent in the very concept of observation. [BCW\,7:390; BCW\,10:128]
\end{quote}
If a well-defined atomic phenomenon is closed, how can something happen between the fixation of the external conditions and a permanent mark on a photographic plate? Does not the interposition of the impact of an electron and/or of subsequent amplification effects amount to a subdivision of the phenomenon in question?
Ole Ulfbeck and Aage Bohr\cite{UlfbeckBohr} have shed light on this issue. For them, clicks in counters are ``events in spacetime, belonging to the world of experience.'' While clicks can be classified as electron clicks, neutron clicks, etc., ``there are no electrons and neutrons on the spacetime scene'' and ``there is no wave function for an electron or a neutron but [only] a wave function for electron clicks and neutron clicks.'' ``[T]here is no longer a particle passing through the apparatus and producing the click. Instead, the connection between source and counter is inherently non-local.'' The key to resolving the issue at hand is that each click has an ``onset''---``a beginning from which the click evolves as a signal in the counter.'' This onset
\begin{quote}
has no precursor in spacetime and, hence, does not belong to a chain of causal events. In other words, the onset of the click is not the effect of something, and it has no meaning to ask how the onset occurred\dots. [T]he occurrence of genuinely fortuitous clicks, coming by themselves, is recognized as the basic material that quantum mechanics deals with\dots. [T]he wave function enters the theory not as an independent element, but in the role of encoding the probability distributions for the clicks\dots. [T]he steps in the development of the click, envisaged in the usual picture, are not events that have taken place on the spacetime scene\dots. [T]he downward path from macroscopic events in spacetime, which in standard quantum mechanics continues into the regime of the particles, does not extend beyond the onsets of the clicks.
\end{quote}
If irreversible amplification effects---the steps in the development of the click---only occur ``in the usual picture,'' then they neither modify nor subdivide the quantum phenomenon in which---through an illegitimate extension of the object-oriented language of classical physics---they are said to occur.
For Niels Bohr, ``the physical content of quantum mechanics is exhausted by its power to formulate statistical laws governing observations obtained under conditions specified in plain language'' [BCW\,10:159]. A quantum phenomenon thus has a statistical component, which correlates events in the world of experience. The so-called irreversible amplification effects belong to this statistical component. The unmediated step from the source to the onset of the click, and the subsequent unmediated steps in the development of the click, are steps in a gazillion of alternative sequences of possible outcomes of \emph{unperformed} measurements, and unperformed measurements do not affect the essential wholeness of a quantum phenomenon.
Niels Bohr, moreover, strongly cautioned against the terminology of ``disturbing a phenomenon by observation'' and of ``creating physical attributes to objects by measuring processes'' [BCW\,7:316]. If there is nothing to be disturbed by observation, if even the dichotomy of objects and attributes created for them by measuring processes is unwarranted, then it is not just the measured property that is constituted by the experimental conditions under which it is observed; it is the quantum system itself that is so constituted. Recently this point was forcefully made by Brigitte Falkenburg in her monograph \emph{Particle Metaphysics}\cite{Falkenburg2007-205f}:
\begin{quote}
[O]nly the experimental context (and our ways of conceiving of it in classical terms) makes it possible to talk in a sloppy way of \emph{quantum objects}\dots. Bare quantum ``objects'' are just bundles of properties which underlie superselection rules and which exhibit non-local, acausal correlations\dots. They seem to be Lockean empirical substances, that is, collections of empirical properties which constantly go together. However, they are only individuated by the experimental apparatus in which they are measured or the concrete quantum phenomenon to which they belong\dots. They can only be individuated as context-dependent quantum \emph{phenomena}. Without a given experimental context, the reference of quantum concepts goes astray. In this point, Bohr is absolutely right up to the present day.
\end{quote}
In the following passages [emphases added], Bohr goes beyond invoking irreversible amplification effects, apparently arguing that the quantum features involved in the atomic constitution of a measurement apparatus (or the statistical element in its description) can be neglected because the relevant parts of a measurement apparatus are sufficiently large and heavy.
\begin{quote}
In actual experimentation this demand [that the experimental arrangement as well as the recording of observations be expressed in the common language] is met by the specification of the experimental conditions by means of bodies like diaphragms and photographic plates \emph{so large and heavy that the statistical element in their description can be neglected}. The observations consist in the recording of permanent marks on these instruments, and the fact that the amplification devices used in the production of such marks involves essentially irreversible processes presents no new observational problem, but merely stresses the element of irreversibility inherent in the definition of the very concept of observation. [BCW\,10:212]
\smallskip In actual physical experimentation this requirement [that we must employ common language to communicate what we have done and what we have learned by putting questions to nature in the form of experiments] is fulfilled by using as measuring instruments rigid bodies like diaphragms, lenses, and photographic plates \emph{sufficiently large and heavy to allow an account of their shape and relative positions and displacements without regard to any quantum features inherently involved in their atomic constitution}\dots. The circumstance that [recordings of observations like the spot produced on a photographic plate by the impact of an electron] involve essentially irreversible processes presents no special difficulty for the interpretation of the experiments, but rather stresses the irreversibility which is implied in principle in the very concept of observation. [BCW\,10:165]
\end{quote}
How can the size and weight of a measuring device justify
\begin{itemize}
\item[---] the irreversibility in principle of the very notion of observation [BCW\,10:120],
\item[---] the essential irreversibility inherent in the very concept of observation [BCW\,7:390; BCW\,10:128],
\item[---] the irreversibility which is implied in principle in the very concept of observation [BCW\,10:165], or
\item[---] the element of irreversibility inherent in the definition of the very concept of observation [BCW\,10:212]?
\end{itemize}
The only irreversibility that can justify the irreversibility of observations is the incontestable irreversibility of human sensory experience.
For Bohr, ``the emphasis on the subjective character of the idea of observation [was] essential'' [BCW\,10:496]. If, as he insisted, the description of atomic phenomena nevertheless has ``a perfectly objective character,'' it was ``in the sense that no explicit reference is made to any \emph{individual} observer and that therefore \dots\ no ambiguity is involved in the communication of information'' [BCW\,10:128, emphasis added]. It was never in the sense that no reference was made to the community of communicating observers or to the incontestable irreversibility of their experiences.
Like Kant, Bohr was concerned with the possibility of an objective knowledge of phenomena (or appearances, or experiences). Kant primarily addressed the possibility of organizing appearances into a world of objects. This involved concepts that allowed the individual not only to think of appearances as a world of external objects but also to communicate with others about a world of external objects. Bohr primarily addressed the requisite possibility of communication. This involved the use of concepts that allowed the individual not only to communicate with others about a world of external objects but also to think of appearances as a world of external objects. Bohr and Kant were on the same page.
What distinguishes the objectivity that could be achieved in the age of Kant from the objectivity that can be achieved in the age of quantum mechanics is that a complete elision of the subject is no longer feasible. Having asserted that ``we can have cognition of no object as a thing in itself, but only insofar as it is an object of sensible intuition, i.e. as an appearance,'' Kant (as we have seen) went on to affirm that
\begin{quote}
even if we cannot \emph{cognize} these same objects as things in themselves, we at least must be able to \emph{think} them as things in themselves. For otherwise there would follow the absurd proposition that there is an appearance without anything that appears.\cite{KantCPR3}
\end{quote}
If the only relevant context is human experience, or if the reach of human sensory experience is unlimited (as classical physics takes it to be), the elision of the subject can be achieved. It is possible to transmogrify calculational tools into objective physical processes with some measure of consistency.\cite{Mermin_badhabit} But if the experimental context is relevant as well, or if the reach of human sensory experience is limited, the elision of the subject is a lost cause, and so is the transmogrification of calculational tools into physical mechanism or natural processes. If one nevertheless wants to establish the irreversibility of measurements and the definiteness of outcomes without invoking the definiteness and irreversibility of human sensory experience, one has no choice but to invoke such ``macroscopic'' features as the size and weight of a measurement apparatus. This (to appropriate a well-known passage by John Bell),
\begin{quote}
is like a snake trying to swallow itself by the tail. It can be done---up to a point. But it becomes embarrassing for the spectators even before it becomes uncomfortable for the snake.\cite{Bell90}
\end{quote}
In the words of Bernard d'Espagnat\cite{dEspagnat2010}, quantum mechanics practically compels us
\begin{quote}
to adopt the idea that was, in fact, at the very core of Kantism and constitutes its truly original contribution to philosophical thinking, to wit, the view that things and events, far from being elements of a ``reality per se,'' are just phenomena, that is, elements of our experience.
\end{quote}
So what were Bohr's intentions in invoking, in lieu of the irreversibility of human experience, the size and weight of the measurement apparatus? Was it in order to appease the na\"{\i}ve realistic inclinations of lesser minds? Or was it in order to indicate the price one had to pay (but which he personally was not willing to pay) for exorcising every possible reference to human sensory experience? My suggested answer is ``yes'' to both questions.
\section{The intervening years}\label{ujmsec:years}
In 1932 John von Neumann\cite{vN1932} developed the mathematical part of the quantum theory into an autonomous formal language. In doing so he turned the theory into a mathematical formalism that was in need of a physical interpretation. In the 1950s, interpreting quantum mechanics began to turn into a growth industry. First David Bohm presented his hidden-variables interpretation,\cite{Bohm1952} then Hugh Everett put forward his relative-state interpretation,\cite{Everett} whereupon Heisenberg entered the fray, arguing that the Copenhagen interpretation was the only viable interpretation.\cite{Heisenberg_PPc3c8} He thereby transformed Bohr's views into just another interpretation of a mathematically formulated theory. Historically, Bohr's reply\cite{Bohr1935} to the argument by Einstein, Podolsky, and Rosen\cite{EPR1935} was taken as a definitive refutation by the physics community. During the ``shut up and calculate'' period of the post-war years, Bohr's perspective was lost. His paper, which only treated the mathematical formalism in a footnote, is now widely seen as missing the point.
By transmogrifying a probability algorithm---the wave function or the ``state'' vector---into a bona fide physical state, adopting the eigenvalue--eigenstate link,%
\footnote{Thus formulated by Dirac:\cite{DiracPQM} ``The expression that an observable `has a particular value' for a particular state is permissible in quantum mechanics in the special case when a measurement of the observable is certain to lead to the particular value, so that the state is an eigenstate of the observable.''} and modeling the ``process'' of measurement as a two-stage affair (``pre-measurement'' followed by ``objectification''), von Neumann created what is commonly known as the measurement problem but is more appropriately called ``the disaster of objectification''.\cite{vF1990} This is how quantum mechanics came to be labeled as ``the great scandal of physics'',\cite{Wallace2008} as a theory that ``makes absolutely no sense'',\cite{Penrose86} and as ``the silliest'' of all the theories proposed in the 20th century.\cite{Kaku95}
What is responsible for these mischaracterizations should not be hard to detect: think of a quantum state's dependence on time as the time-dependence of an evolving physical state, rather than as the dependence of probabilities on the time of the measurement to the possible outcomes of which they are assigned, and you have two modes of evolution whereas in reality there is not even one.
Nevertheless, today the reasons for these mischaracterizations \emph{are} hard to detect. One of these reasons is that whereas a junior-level classical mechanics course devotes a considerable amount of time to different formulations of classical mechanics, even graduate-level courses often emphasize one particular formulation of quantum mechanics almost to the exclusion of all variants, of which there are (at least) nine.\cite{Styeretal} It would seem reasonable to expect that an interpretation of quantum mechanics be based on features that are common to all formulations of the theory, not on the mathematical idiosyncrasies of a particular formulation, such as the wave-function formulation. What is common to all formulations is that they afford tools for calculating correlations between measurement outcomes.
Another reason is the axiomatic method by which quantum mechanics is now typically taught. First students are told that the state of a quantum-physical system is (or is represented by) a normalized element of a Hilbert space. Then they are told that observables are (or are represented by) self-adjoint operators, and that the possible outcomes of a measurement are the eigenvalues of such an operator. Then comes a couple of axioms concerning the time evolution of states---unitary \emph{between} measurements and as stipulated by the projection postulate \emph{at the time of} a measurement. A further axiom stipulates that the states of composite systems are (or are represented by) vectors in the tensor product of the Hilbert spaces of the component systems. And finally, almost as an afterthought, comes an axiom about probabilities, the Born rule. This is how the \emph{The Ashgate Companion to Contemporary Philosophy of Physics}\cite{Wallace2008} comes to distinguish between a ``bare quantum formalism,'' which it describes as ``an elegant piece of mathematics'' that is ``prior to any notion of probability, measurement etc.,'' and a ``quantum algorithm,'' which it describes as ``an ill-defined and unattractive mess,'' whose business is to extract ``empirical results'' from the former. In actual fact, there is no such thing as a bare quantum formalism. Every single axiom of the theory only makes sense as a feature of a probability calculus.\cite{Mohrhoff-QMexplained}
It is beyond doubt that significant progress was made during the roughly four decades between the passing of Niels Bohr and the advent of QBism. We now have a congeries of complex, sophisticated, and astonishingly accurate probability algorithms---the standard model%
\footnote{ ``Standard model is a grotesquely modest name for one of humankind's greatest achievements''.\cite{Wilczek2008}}%
---and we are witnessing rapid growth in the exciting fields of quantum information and quantum technology. By contrast, the contemporaneous progress in quantum theory's philosophical foundations mainly consisted in finding out what does \emph{not} work, such as the countless attempts to transmogrify statistical correlations between observations into physical processes that take place between and give rise to observations.
\section{QBism: Wigner's friend}\label{ujmsec:qbismWigner}
To make the centrality of human experience duly and truly stick, QBism emphasizes the \emph{individual} subject. To a QBist, all probabilities are of the subjective, personalist Bayesian kind. The so-called quantum state is something the individual user (of quantum mechanics) or agent (in a quantum world) assigns on the basis of her own experiences,%
\footnote{While Fuchs and Schack prefer the term ``agent,'' Mermin prefers the term ``user,'' in order to emphasize that QBists regard quantum mechanics as a ``user's manual''.\cite{MerminQBnotCop}}
and it is used by her to assign probabilities to a set of possible personal experiences, which are determined by the action she takes to elicit one these experiences. Such an action does not have to take place in a physics laboratory. It ``can be anything from running across the street at L'\'{E}toile in Paris (and gambling upon one's life) to a sophisticated quantum information experiment (and gambling on the violation of a Bell inequality)''.\cite{Fuchs_Notwithstanding} The only thing a QBist ``does not model with quantum mechanics is her own direct internal awareness of her own private {experience}''.\cite{FMS2014}
Two of the pseudo-problems that quantum-state realists have to contend with are thereby taken care of: the matter of Wigner's friend\cite{Wigner61} and the matter of Bell's shifty split.\cite{Bell90} In Wigner's scenario, Wigner's friend~$F$ performs a measurement on a system~$S$ using an apparatus~$A$. Treating $F$ as a quantum system, and treating quantum states as ontic states evolving unitarily between measurement-induced state reductions, Wigner concludes that a reduction of the combined system $S{+}A$ occurs for $F$ when she becomes aware of the outcome, while a reduction of the combined system $S{+}A{+}F$ occurs for him when he is informed of the outcome by~$F$. This scenario led Wigner to conclude that the theory of measurement was logically consistent only ``so long as I maintain my privileged position as ultimate observer.'' QBism, on the contrary, maintains that Wigner's state assignment, which is based on his actual past and possible future experiences, is as valid as his friend's, based as that is on a different set of actual past and possible future experiences. This point, however, can be made without envisioning Wigner's friend in a coherent superposition of two distinct cognitive states:
\begin{quote}
Wigner's quantum-state assignment and unitary evolution for the compound system are only about his \emph{own} expectations for his \emph{own} experiences should he take one or another action upon the system or any part of it. One such action might be his sounding the verbal question, ``Hey friend, what did you see?,'' which will lead to one of two possible experiences for him. Another such action could be to put the whole conceptual box into some amazing quantum interference experiment, which would lead to one of two completely different experiences for him. \cite{Fuchs_Notwithstanding}
\end{quote}
QBists distinguish between (subjective) agent-dependent realities and a common body of (objective) reality:
\begin{quote}
What is real for an agent rests entirely on what that agent experiences, and different agents have different experiences. An agent-dependent reality is constrained by the fact that different agents can communicate their experience to each other, limited only by the extent that personal experience can be expressed in ordinary language\dots. In this way a common body of reality can be constructed.\cite{FMS2014}
\end{quote}
What do we know about our common body of reality? Because we construct it from our experiences, and because our experiences are definite and irreversible, it is constructed from experiences that are definite and irreversible. I may be ignorant of your experiences and you may be ignorant of mine, but we cannot doubt the definiteness and irreversibility of our respective experiences. It is therefore inadmissible to assign to any (sane and healthy) subject a coherent superposition of distinct cognitive states. Wigner is not only perfectly justified but \emph{required} to assign to the system that includes his friend an incoherent mixture reflecting his ignorance of the outcome that his friend has obtained. To treat his own experiences as definite but not those of his friend---that would be the solipsism which Wigner feared and sought to avoid by proposing ``that the equations of motion of quantum mechanics cease to be linear, in fact that they are grossly non-linear if conscious beings enter the picture.''
QBists are united in rejecting ``the silly charges of solipsism''.\cite{MerminQBnotCop} In order to avoid these charges, however, they need to do more than acknowledge the fact that ``[m]y experience of you leads me to hypothesize that you are a being very much like myself, with your own private experience.'' They need to stop fantasizing about coherent superpositions involving distinct experiences.
\section{QBism: Bell's shifty split}\label{ujmsec:qbismShifty}
{\leftskip\parindent\emph{There is a straight ladder from the atom to the grain of sand, and the only real mystery in physics is the missing rung. Below it, particle physics; above it, classical physics; but in between, metaphysics.}\par\hfill--- \emph{Tom Stoppard, \emph{Hapgood}}}\par\smallskip\noindent
Bell's ``shifty split,'' a.k.a. the Heisenberg cut, is the mysterious boundary separating the system under investigation from the means of investigation. While for Heisenberg its location was more or less arbitrary, for Bohr it was determined by the measurement setup
\footnote{See Camilleri and Schlosshauer\cite{CamilleriSchlosshauer2015} for a discussion of Bohr's and Heisenberg's divergent views on this matter.}
In QBism, the experience of the individual user takes the place of the measurement setup. Accordingly there are as many splits as there are users, and there is nothing shifty about them. Mermin explains:
\begin{quote}
Each split is between an object (the world) and a subject (an agent's irreducible awareness of her or his own experience). Setting aside dreams or hallucinations, I, as agent, have no trouble making such a distinction, and I assume that you don't either. Vagueness and ambiguity only arise if one fails to acknowledge that the splits reside not in the objective world, but at the boundaries between that world and the experiences of the various agents who use quantum mechanics.\cite{Mermin_shifty}
\end{quote}
Let us disregard the ambiguity of ``awareness of one's own experience,'' which could mean either awareness of something one is experiencing or awareness of one's experiencing something. The question is: what is meant by ``objective world''? First guess: what is meant is ``the common external world we have all negotiated with each other'' or, equivalently, ``a model for what is common to all of our privately constructed external worlds''.\cite{MerminQBnotCop} In this case the split occurs between this world or model and the private experiences in which it originates.
Second guess: what is meant is something that induces experiences in conscious subjects. This interpretation is suggested by Mermin's statement that, according to QBism, ``my understanding of the world rests entirely on the experiences that the world has induced in me throughout the course of my life'',\cite{MerminQBnotCop} or by the equivalent statement that ``[t]he world acts on me, inducing the private experiences out of which I build my understanding of my own world''.\cite{MerminBetterSense} Judging by \emph{these} statements, the split occurs between my private experiences and a world that induces them, rather than between my private experiences and the world as I understand it on the basis of my private experiences. (On the relation between these two worlds see Note~\ref{note:sacop}.)
What we are faced with here is an attempt to throw the baby out with the bathwater. The bathwater is the shifty split; the baby is the measuring apparatus. And not only the measuring apparatus. If QBism, as Fuchs and Schack affirm, treats ``all physical systems in the same way, including atoms, beam splitters, Stern-Gerlach magnets, preparation devices, measurement apparatuses, all the way to living beings and other agents'',\cite{FS2015} then Bohr's crucial insight that the properties of quantum systems are \emph{contextual}---that they are defined by experimental arrangements---is lost.
For Bohr, the measurement apparatus was needed not only to indicate the possession of a property (by a system) or a value (by an observable) but also, and in the first place, to make a set of properties or values available for attribution to a system or an observable. The sensitive regions of an array of detectors \emph{define} the regions of space in which the system can be found. In the absence of an array of detectors, the regions of space in which the system can be found do not exist. The orientation of a Stern-Gerlach apparatus \emph{defines} the axis with respect to which a spin component is measured. In the absence of a Stern-Gerlach apparatus, the axis with respect to which a spin component can be up or down does not exist. What physical quantity is defined by running across the street at L'\'{E}toile in Paris?
From a different QBist point of view, espoused by Fuchs and Schack, the measurement apparatus should be understood as an extension of the agent, and quantum mechanics itself should be regarded as a theory of stimulation and response:
\begin{quote}
A quantum measurement device is like a prosthetic hand, and the outcome of a measurement is an unpredictable, undetermined ``experience'' shared between the agent and the external system.\cite{Fuchs_Notwithstanding}
\end{quote}\begin{quote}
The agent, through the process of quantum measurement stimulates the world external to himself. The world, in return, stimulates a response in the agent that is quantified by a change in his beliefs---i.e., by a change from a prior to a posterior quantum state. Somewhere in the structure of those belief changes lies quantum theory's most direct statement about what we believe of the world as it is without agents.\cite{FS2004}
\end{quote}
This invites two comments. The first is that the question where the apparatus ends and the rest of the world begins is once more open to dispute. It appears that one shifty split has been traded for another. Fuchs responds by pointing out that the physical extent of the agent is up to the agent:
\begin{quote}
The question is not where does the quantum world play out and the classical world kick in? But where does the agent expect his own autonomy to play out and the external world, with its autonomy and its capacity to surprise, kick in? The physical extent of the agent is a judgment he makes of himself. \cite{Fuchs2Wootters}
\end{quote}
By placing the the dividing line---wherever the agent chooses to place it---between the agent-cum-instrument and the rest of the physical world, Fuchs does precisely what Mermin objects to when he writes that ``[v]agueness and ambiguity only arise if one fails to acknowledge that the splits reside not in the objective world, but at the boundaries between that world and the experiences of the various agents who use quantum mechanics''.\cite{Mermin_shifty}
The second comment concerns ``the world as it is without agents.'' The phrase \emph{could} refer to the unspeakable domain beyond the reach of our concepts, which only becomes speakable through the manner in which it is stimulated (i.e., by saying in ordinary language what the agent has done) and through the manner in which it responds (i.e., by saying in ordinary language what the agent has learned). Mermin,\cite{Mermin2ujm} however, rejects this interpretation: ``QBists (at least this one) attach no meaning to `the world as it is without agents.' It only means `the common external world we have all negotiated with each other'.''
Regardless, there is another boundary at which the Heisenberg cut can be placed. As our common external world has a ``near'' boundary (between it and the private experiences in which it originates), so it has a ``far'' boundary (between it and the unspeakable domain beyond the reach of our concepts). My contention is that the cut ought to be placed there. I therefore agree with Mermin that the cut does not reside in the objective world---the world of sense impressions organized into objects, the world we have all negotiated with each other. But instead of placing it at its near boundary, I maintain that it should be placed at its far boundary, and I take it that, to all intents and purposes, Bohr did the same. What is definite in this case is not just the measurement apparatus but the entire objective world. Nothing indefinite is implied by the unpredictability in general of measurement outcomes.
\section{QBism and Bohr}\label{ujmsec:QBB}
As it is an irony that Bohr drew the battle lines in a way which put Kant and himself
on opposing sides, so it is an irony that QBists draw their battle lines in a way which puts Bohr and themselves on opposing sides, notwithstanding that ``QBism agrees with Bohr that the primitive concept of \emph{experience} is fundamental to an understanding of science''.\cite{FMS2014}
Thus Fuchs \emph{et al.}:
\begin{quote}
The Founders of quantum mechanics were already aware that there was a problem. Bohr and Heisenberg dealt with it by emphasizing the inseparability of the phenomena from the instruments we devised to investigate them. Instruments are the Copenhagen surrogate for experience\dots. [They are] objective and independent of the agent using them.\cite{FMS2014}
\end{quote}
And thus Mermin:
\begin{quote}
Those who reject QBism \dots\ reify the common external world we have all negotiated with each other, purging from the story any reference to the origins of our common world in the private experiences we try to share with each other through language. \dots by ``experience'' I believe [Bohr] meant the objective readings of large classical instruments\dots. Because outcomes of Copenhagen measurements are ``classical,'' they are \emph{ipso facto} real and objective.\cite{MerminQBnotCop}
\end{quote}
While QBists are generally aware of the important distinction between what Mermin calls ``reification'' and what Schr\"odinger\cite{SchrWhatIsReal} called ``objectivation,'' they share the now prevailing misappreciation of Bohr's thinking. Bohr was concerned with objectivation, the representation of a shared mental construct as an objective world, not with reification, which ignores or denies the origins of the objective world in our thoughts and perceptions. Objectivation means purging from the story any reference to these origins without ignoring or denying them, so that science may deal with the objective world as common-sense realism does---\emph{as if} it existed independently of our thoughts and perceptions. Reification is the assertion that the world we perceive does in fact exist independently of our perceptions, or that the world we mentally construct does in fact exist independently of our constructing minds, or that the world we describe does in fact exist---just as we describe it---independently of our descriptions.
To Bohr, measurement outcomes are ``classical'' (i.e., definite and irreversible) and instruments are objective not because (or in the sense that) they are reified but because (or in the sense that) they are situated in an intersubjectively constituted world---like everything else that is directly accessible to human sensory experience. Instead of being a ``surrogate of experience,'' instruments---like everything else in our common external world---are experiences that lend themselves to objectivation. They make it possible not only to apply classical concepts to quantum systems but also to extend their reach into the non-classical domain via principles of correspondence.%
\footnote{``[Q]uantum mechanics and quantum field theory only refer to individual systems due to the ways in which the quantum models of matter and subatomic interactions are linked by semi-classical models to the classical models of subatomic structure and scattering processes. All these links are based on tacit use of a generalized correspondence principle in Bohr's sense (plus other unifying principles of physics).'' This generalized correspondence principle serves as ``a semantic principle of continuity which guarantees that the predicates for physical properties such as `position', `momentum', `mass', `energy', etc., can also be defined in the domain of quantum mechanics, and that one may interpret them operationally in accordance with classical measurement methods. It provides a great many inter-theoretical relations, by means of which the formal concepts and models of quantum mechanics can be filled with physical meaning''.\cite{Falkenburg2007-XII-191}}
The statement that those who reject QBism reify the common external world we have all negotiated with each other, also rings false. One can certainly reject some of the (sometimes mutually inconsistent) claims made by QBists without reifying the objectivized world. What is true is the converse: those who reify the objectivized world will have to reject QBism.
Admittedly, Bohr obscured his original thinking by compounding the incontestable irreversibility of human sensory experience with amplification processes or apparatus features like being sufficiently large or heavy. To invoke such processes or features was for him the price that one had to pay for achieving complete objectivation (i.e., for banishing every reference to experience). He himself, however, was clearly not inclined to pay this price. It bears repetition: for him the description of atomic phenomena had ``a perfectly objective character, \emph{in the sense that} no explicit reference is made to any individual observer and that therefore \dots\ no ambiguity is involved in the communication of information'' [BCW\,7:390, emphasis added]---and thus \emph{not in the sense that} no reference was made to the community of communicating observers or to the incontestable irreversibility of their experiences.
To Bohr, objectivity meant ``a description by means of a language common to all \dots\ in which people may communicate with each other in the relevant field'' [BCW\,10:XXXVII]. What QBists mean by objectivity is less clear, though arguably they mean the same, to wit: ``language is the only means by which different users of quantum mechanics can attempt to compare their own private experiences,'' and it is only by communicating ``that we can arrive at a shared understanding of what is common to all our own experiences of our own external worlds''.\cite{MerminQBnotCop}
There is just one detail in Mermin's argument to which Bohr would probably have objected, namely the idea that each of us first constructs a private external world, and that language comes in only after this is done, as a means of figuring out what is common to all our privately constructed external worlds. One cannot construct a private external world before being in possession of a language providing the concepts that are needed for its construction. At any rate, Mermin's claim that ``[o]rdinary language comes into the QBist story in a more crucial way than it comes into the story told by Bohr,'' appears to me wholly unjustified.
The great merit of QBism is that it puts the spotlight back on the role that human experience plays in creating physical theories. One this is recognized, the mystery as to why measurements are irreversible and outcomes definite vanishes into this air: it is because our experiences are irreversible and definite. Bohr could have said the same, and arguably did, but in so many words that the core of his message has been lost or distorted beyond recognition. The fundamental difference between Bohr and QBism is that the former was writing before interpreting quantum mechanics became a growth industry, while the latter emerged in reaction to an ever-growing number of futile attempts at averting the disaster of objectification in the same realist framework in which it arose.
There are several ways in which QBism goes beyond Bohr, but this in no wise affects my claim that Bohr was a QBist---what with his contention that ``in our description of nature the purpose is not to disclose the real essence of the phenomena but only to track down, so far as it is possible, relations between the manifold aspects of our experience'' [BCW\,6:296]. And no, by ``our experience'' he did \emph{not} mean the objective readings of large classical instruments. What qualifies him as a QBist, if not as the unwitting \emph{founder} of QBism, is his insistence on \emph{individual} experience, which only through communication becomes \emph{objective} knowledge.
One of the ways in which contemporary QBism goes beyond Bohr consists in replacing the standard projector-valued probability measures by positive-operator-valued measures (POVMs). This makes it possible to formulate the Born rule entirely in terms of probabilities (as outlined in the Appendix) and to view quantum mechanics as nothing but a generalization of the (personalist) Bayesian theory of probability. In their justifiable enthusiasm, however, QBists also overshoot their mark, as when they permit incoherent superpositions to be assigned to possibilities involving distinct cognitive states, or when they claim that quantum mechanics is ``\emph{explicitly} local in the QBist interpretation''.\cite{FMS2014} According to Fuchs,\cite{Fuchs2010}
\begin{quote}
[QBism] gives each quantum state a home. Indeed, a home localized in space and time---namely, the physical site of the agent who assigns it! By this method, one expels once and for all the fear that quantum mechanics leads to ``spooky action at a distance.''
\end{quote}
A quantum state has its home in an agent's mind, not at any physical site---which would have to be a site in our common external world, since there are no agents in ``the world as it is without agents.'' Why inserting minds into our common external world is a bad idea has been explained by Schr\"odinger, who according to Fuchs \emph{et al.} took ``a QBist view'' of science.\cite{FMS2014} By the very fact that we treat some of our experiences as aspects of a shared external world, Schr\"odinger argued, we exclude ourselves from this world: ``We step with our own person back into the part of an onlooker who does not belong to the world, which by this very procedure becomes an objective world''.\cite{SchrLifeMindMatter} If we then reify the mind's creation, we are left with no choice but to insert the mind into its creation: ``I so to speak put my own sentient self (which had constructed this world as a mental product) back into it---with the {pandemonium} of disastrous logical consequences'' that flow from this error, such as ``our fruitless quest for the place where mind acts on matter or vice-versa.'' Locating experiences in our common external world therefore is not an option. Instead of asserting that QBism is explicitly local, QBists ought to assert that QBism is neither local nor nonlocal in any realist sense of these terms.
It is strange indeed to see a QBist look upon spooky action at a distance as something to be feared. To banish it by claiming that quantum mechanics is local is to concede way too much to those who see themselves as modeling a reality not of their own making. Something fearsome is implied only if one forgets that physics deals with our common external world. One then has to worry how, in the absence of a common cause, measurement outcomes in spacelike relation can be so spookily correlated. Keeping in mind that measurement outcomes are responses from a reality beyond the reach of human sensory experience, one realizes (as Bohr did) that the answer to this question is beyond the reach of our concepts. After all, diachronic correlations (between successive experiences of the same agent) are no less inexplicable than synchronic correlations (between simultaneous experiences of different agents).
All in all, QBism, through its emphasis on the individual experiencing subject, brings home the intersubjective constitution of our common external world more forcefully than Bohr ever did. (The time wasn't ripe for this then. Perhaps it is now.) Bohr's insights, on the other hand, are eminently useful in clarifying the QBist position, attenuating its excesses, and enhancing its internal consistency.
In concluding, I want to extend my gratitude to the QBists for making me finally come round to seeing that there is \emph{no} difference between observations qua experiences and observations qua measurement outcomes (which explains why measurements are irreversible and outcomes definite).
\section*{Appendix: The Born rule according to QBism}\label{ujmapx}
{\leftskip2\parindent\emph{If quantum theory is so closely allied with probability theory, why is it not written in a language that \emph{starts} with probability, rather than a language that ends with it? Why does quantum theory invoke the mathematical apparatus of Hilbert spaces and linear operators, rather than probabilities outright? --- Christopher A. Fuchs\cite{Fuchs2010}}\par}\smallskip\noindent
For QBists, quantum mechanics is a generalization of the Bayesian theory of probability. It is a calculus of consistency---a set of criteria for testing coherence between beliefs. As there are no {external} criteria for declaring a probability judgment right or wrong, so there are no {external} criteria for declaring a quantum state assignment right or wrong. The only criterion for the adequacy of a probability judgment or a state assignment is internal coherence between beliefs. The Born rule thus is not simply a rule for updating probabilities, for getting new ones from old. It is a rule for {relating} probability assignments and {constraining} them. As such, it can be expressed entirely in terms of probabilities.
The proof of the last claim requires the use of POVMs, which generalize the standard projector valued measures used by von Neumann\cite{vN1932} and Dirac.\cite{DiracPQM} It goes like this: While a density operator $\rho$ determines a potentially infinite number of probabilities, these cannot all be independent. On a $d$-dimensional Hilbert space, $\rho$~is completely determined by the $d^{\,2}$ probabilities it assigns to the outcomes (represented by linearly independent positive operators~$E_i$) of an \emph{informationally complete} measurement. Any density operator $\rho$ therefore corresponds to a vector whose $d^{\,2}$ components are the Born probabilities
\begin{equation}
p_i=\hbox{Tr}(\rho E_i)\,,
\end{equation}
and any POVM $\{F_j\}$ corresponds to a matrix whose elements are the conditional probabilities
\begin{equation}
R_{ji}=\hbox{Tr}(\Pi_i F_j)\,,
\end{equation}
where the $\Pi_i$ are 1-dimensional projectors proportional to~$E_i$. This makes it possible to write the Born rule in the generic form
\begin{equation}
q(F_j)= f \bigl(\{R_{ji}\},\{p_i\}\bigr)\,,
\label{eq:BornGeneric}
\end{equation}
where $f$ depends on the details of the informationally complete measurement $\{E_i\}$.
The function $f$ takes a particularly simple form if the positive operators $E_i$ constitute a \emph{symmetric} informationally complete (SIC) measurement. In this case one of the ways in which the Born rule can be written is\cite{Fuchs2010, Fuchs2004}
\begin{equation}
q(F_j)=\sum_{i=1}^{d^2}\left[(d+1)\,p_i-\frac1d\right]R_{ji}\,.
\label{eq:BornPOVM}
\end{equation}
If the positive operators $F_j$ are mutually orthogonal projectors representing the outcomes of a complete von Neumann measurement, the Born rule takes the even simpler form
\begin{equation}
q(F_j)=(d+1)\sum_{i=1}^{d^2}\,R_{ji}p_i-1\,.
\label{eq:BornPVM}
\end{equation}
While the probabilities (\ref{eq:BornPOVM}) and (\ref{eq:BornPVM}) are expressed in terms of (i)~the probabilities $p_i$ that an agent assigns to the possible outcomes of the SIC measurement and (ii)~the conditional probabilities $R_{ji}$ that the agent assigns to the possible outcomes of a subsequent measurement if the SIC measurement is actually made, they pertain to a situation in which the SIC measurement is \emph{not} made. If it \emph{is} made, the law of total probability applies, and we have
\begin{equation}
q(F_j)=\sum_{i=1}^{d^2}R_{ji}p_i \,.
\label{eq:BornAgain}
\end{equation}
Comparing Eqs. (\ref{eq:BornPOVM}) and (\ref{eq:BornPVM}) with Eq. (\ref{eq:BornAgain}), one can see that ``[t]he Born Rule is nothing but a kind of Quantum Law of Total Probability! No complex amplitudes, no operators---only probabilities in, and probabilities out''.\cite{Fuchs2010}
QBists hope to eventually be in a position to derive the standard Hilbert space formalism from the Born rule. And they hope so to distill the essence of quantum mechanics and the essential characteristic of the quantum world.%
\footnote{While the Born rule is normative---it guides an agent's behavior in a world that is fundamentally quantum---it is also an empirical rule. It is a statement about the quantum world, indirectly expressed as a calculus of consistency for bets placed on the outcomes of measurements.}
This a fascinating, highly ambitious, and seriously challenging project. Do SIC measurements even exist? Unfortunately, proofs of their existence are elusive. As of May 2017, such proofs have been found for all dimensions up to $d{=}151$, and for a few others up to 323.\cite{Fuchs_Notwithstanding} The mood of the QBist community nevertheless is that a SIC measurement should exist for every finite dimension. That said, it must be stressed that the general form of the Born rule, Eq.~(\ref{eq:BornGeneric}), does not depend on the existence of SIC measurements; it only presupposes informationally complete POVMs, and these are known to exist for all finite dimensions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 483 |
Audrey Marie Munson (Rochester, 8 de junho de 1891 - Ogdensburg, 20 de fevereiro de 1996) foi uma modelo e atriz de cinema americana, considerada a "primeira supermodelo da América". Em seu tempo, era conhecida como " Miss Manhattan ", a " Garota do Panamá-Pacífico ", a " Garota da Exposição " e " Vênus Americana ". Ela foi modelo ou inspiração para mais de doze estátuas na cidade de Nova York e muitas outras em outros lugares. Munson apareceu em quatro filmes mudos, incluindo uma aparição sem roupa em Inspiration (1915). Ela foi uma das primeiras atrizes americanas a aparecer nua em um filme não pornográfico.
Carreira
Modelo
Audrey Marie Munson nasceu em Rochester, Nova York, em 8 de junho de 1891, filha de Edgar Munson (1857-1945), condutor de bonde e especulador imobiliário descendente de puritanos ingleses, e Katherine "Kittie" Mahaney (1863-1958), filha de imigrantes irlandeses. Seu pai era de Mexico, Nova York, e mais tarde ela morou lá. Seus pais se divorciaram quando ela tinha oito anos, e Audrey e sua mãe se mudaram para Providence, Rhode Island .
Em 1909, as duas mudaram-se para Washington Heights, na cidade de Nova York, onde Audrey, de 17 anos, buscou uma carreira como atriz e corista. Seu primeiro papel na Broadway foi como "lacaio" em The Boy and The Girl no Aerial Gardens do New Amsterdam Theatre, que decorreu de 31 de maio a 19 de junho de 1909. Ela também apareceu em The Girl and the Wizard, Girlies e La Belle Paree.
Enquanto olhava as vitrines na Quinta Avenida com sua mãe, foi flagrada pelo fotógrafo Felix Benedict Herzog, que a pediu para posar para ele em seu estúdio no Lincoln Arcade Building na Broadway com a 65th Street. Herzog a apresentou a seus amigos do mundo da arte. Ela posou para o muralista William de Leftwich Dodge, que lhe deu uma carta de apresentação para Isidore Konti . Konti foi seu primeiro escultor e o primeiro artista para quem modelou nua. A partir desse ponto, Munson posaria para alguns artistas visuais conhecidos, incluindo o pintor Francis Coates Jones, os ilustradores Harrison Fisher, Archie Gunn e Charles Dana Gibson, e os fotógrafos Herzog e Arnold Genthe, mas ela era predominantemente um modelo de escultor.
O primeiro crédito reconhecido de Munson é a estatuária de mármore de Konti chamada Três Graças, revelada no novo Grand Ballroom no Hotel Astor em Times Square em setembro de 1909. Ela posou para todas as três Graças. Logo depois, e na década seguinte, Munson se tornou a modelo preferida para do primeiro escalão de escultores americanos, posando para uma longa lista de estátuas, monumentos e esculturas arquitetônicas alegóricas independentes em capitólios estaduais e outros edifícios públicos importantes. De acordo com o The Sun em 1913, "Mais de cem artistas concordam que se o nome de Miss Manhattan pertence a alguém em particular, é a esta jovem." Em 1915, ela estava tão bem estabelecida que se tornou a modelo preferida de Alexander Stirling Calder, quando se tornou Diretor de Escultura da Exposição Internacional Panamá-Pacífico realizada em San Francisco naquele ano. Sua figura foi "noventa vezes repetida contra o céu" em um único edifício, no topo das colunatas do Pátio do Universo, modelado aproximadamente na Praça de São Pedro no Vaticano. Na verdade, Munson posou para três quintos da escultura criada para o evento e ganhou fama como a "Garota Panamá-Pacífico".
Atriz de cinema
A recém-descoberta celebridade de Munson ajudou a lançar sua carreira na nascente indústria cinematográfica e ela estrelou quatro filmes mudos. No primeiro, Inspiration (1915), feito pela Thanhouser Film Corporation em New Rochelle, Nova York e dirigido por George Foster Platt, ela apareceu totalmente nua no papel de modelo de um escultor. Os censores relutaram em banir o filme, temendo que também tivessem que banir a arte renascentista . Os filmes de Munson foram um sucesso de bilheteria, embora os críticos estivessem divididos. Thanhouser contratou uma sósia chamada Jane Thomas para fazer as cenas de atuação de Munson, enquanto Munson fez as cenas em que posava nua. Embora a aparição de Munson em Inspiration às vezes seja considerada a primeira ocasião em que uma atriz americana apareceu nua em um filme não pornográfico, de acordo com a historiadora de cinema Karen Ward Mahr, na verdade foi Margaret Edwards a pioneira em Hypocrites, lançado no início de 1915.
O segundo filme de Munson, Purity (1916), feito pela American Film Company em Santa Bárbara, Califórnia e dirigido por Rae Berger, é o único de seus filmes a sobreviver, tendo sido redescoberto em 1993 em uma coleção de "pornografia" na França e adquirido pelo arquivo do cinema nacional francês. Seu terceiro filme, The Girl o' Dreams, também feito pela American em Santa Bárbara e provavelmente dirigido por Tom Ricketts a partir de uma história de William Pigott (o catálogo do American Film Institute lista Pigott como diretor, mas todos os seus outros créditos o listam como roteirista), foi concluído no outono de 1916, mas embora o filme seja mencionado nas listas de créditos de vários de seus atores no Motion Picture Studio Directory de 21 de outubro de 1916, ele não foi lançado naquela época e nem tinha direitos autorais até 31 de dezembro de 1918; não há menção subsequente ao filme e pode nunca ter sido lançado.
Munson voltou à Costa Leste de trem via Syracuse em dezembro de 1916, tendo se envolvido com a alta sociedade em Nova York e Newport, Rhode Island. Há relatos de que sua mãe insistiu para que ela se casasse com o filho de um herdeiro da prata "Comstock Lode", Hermann Oelrichs Jr., então o solteiro mais rico da América. Não há registro desse fato. Em 27 de janeiro de 1919, ela escreveu uma carta desconexa ao Departamento de Estado dos Estados Unidos denunciando Oelrichs como parte de uma rede pró-alemã que a havia afastado do ramo do cinema. Ela disse que planejava abandonar os Estados Unidos para reiniciar sua carreira no cinema na Inglaterra.
Notoriedade
Em 1919, Audrey Munson estava morando com sua mãe em uma pensão na 164 West 65th Street, Manhattan, de propriedade do Dr. Walter Wilkins. Wilkins se apaixonou por Munson e, em 27 de fevereiro, assassinou sua esposa, Julia, para que ele pudesse estar disponível para o casamento. Munson e sua mãe deixaram Nova York e a polícia as procurou para interrogatório. Depois de uma busca nacional, elas foram localizadas. Elas se recusaram a voltar para Nova York, mas foram questionadas por agentes da Burns Detective Agency em Toronto, Ontário, Canadá. O conteúdo das declarações que eles forneceram nunca foi revelado, mas Audrey Munson negou veementemente que tivesse qualquer relacionamento romântico com o Dr. Wilkins. Wilkins foi julgado, considerado culpado e condenado à cadeira elétrica . Ele se enforcou em sua cela antes que a sentença pudesse ser executada.
Como consequência direta ou não, o assassinato de Wilkins marcou o fim da carreira de modelo de Munson. Ela continuou a buscar cobertura jornalística regular. Em 1920 foi relatado que Munson, incapaz de encontrar trabalho em qualquer lugar, estaria morando em Syracuse, Nova York, sustentada por sua mãe, que vendia utensílios de cozinha de porta em porta. Em novembro de 1920, dizia-se que ela trabalhava como bilheteira em um museu barato.
De janeiro a maio de 1921, uma série de vinte artigos serializados foi publicada na Hearst's Sunday Magazine em dezenas de suplementos de jornais de domingo, sob o nome de Munson, toda a série intitulada By the 'Queen of the Artists' Studios'. Os vinte artigos relatam anedotas de sua carreira, com alertas sobre o destino de outras modelos. Em uma delas, ela pedia ao leitor que imaginasse seu futuro:
Em fevereiro daquele ano, o agente-produtor Allen Rock fez anúncios mostrando um cheque de $ 27.500 que ele disse ter pago a Munson para estrelar um quarto filme intitulado Heedless Moths, dirigido por Robert Z. Leonard a partir de seu próprio roteiro baseado nesses escritos. Mais tarde, ela disse que o cheque de $ 27.500 era apenas um "golpe publicitário" e entrou com uma ação contra Allen Rock. Esses procedimentos revelaram que os vinte artigos foram escritos pelo jornalista Henry Leyford Gates.
No verão de 1921, Munson conduziu uma busca nacional, realizada pela United Press, pelo homem perfeito para se casar. Ela encerrou a busca em agosto alegando que não queria se casar de qualquer maneira. Em 3 de outubro de 1921 ela foi presa no Royal <i id="mwzw">Theatre</i> (mais tarde Towne Theatre ) em St.Louis numa acusação moral relacionada à sua atuação pessoal no filme Innocence (o título de relançamento de Purity), no qual ela teve um papel de liderança . Ela e seu empresário, o produtor de cinema independente Ben Judell, foram ambos absolvidos. Semanas depois, ela ainda aparecia em St. Louis, junto com as exibições de Innocence, encenando "uma série de novas poses de pinturas famosas".
Em 27 de maio de 1922, Munson tentou suicídio engolindo uma solução de bicloreto de mercúrio.
Anos seguintes e morte
Em 8 de junho de 1931, sua mãe pediu a um juiz que a internasse em um asilo psiquiátrico. O juiz do condado de Oswego ordenou que Munson fosse internada em uma clínica psiquiátrica para tratamento em seu aniversário de 40 anos. Ela permaneceu no St. Lawrence State Hospital for the Insane em Ogdensburg, onde foi tratada de depressão e esquizofrenia, por 65 anos, até sua morte aos 104 anos. Durante sua permanência na instituição, muitas vezes ela cuidava de sua beleza com leite, iogurte e urina.
Em meados da década de 1950, Munson era famosa o suficiente para servir de tema de uma anedota em um livro de memórias que PG Wodehouse e Guy Bolton escreveram sobre seus anos na Broadway, Bring on the Girls! (1953), embora esse livro de memórias seja considerado mais ficção do que fato pelo biógrafo de Wodehouse.
Ela não recebeu visitas no asilo por mais de 25 anos depois da morte4 de sua mãe em 1958, mas foi redescoberta lá por uma meia-sobrinha, Darlene Bradley, em 1984, quando Munson tinha 93 anos Em meados da década de 1980, Munson, em meados dos anos 90, foi transferida para uma casa de repouso em Massena, Nova York, quando o hospital original fechou, no entanto, ela costumava fugir para um bar próximo, com funcionários da casa de repouso tendo que buscá-la. Como resultado, ela foi transferida de volta para a nova instituição mental. Quando completou 100 anos, tinha perdido todos os dentes e boa parte da audição, mas gozava de boa saúde. Pouco depois de seu 100º aniversário, Munson quebrou o quadril. Munson morreu em 20 de fevereiro de 1996, aos 104 anos. Foi enterrada no cemitério de New Haven em New Haven, Nova York, e recebeu uma lápide em seu túmulo em 8 de junho de 2016, 20 anos após sua morte e no que seria seu 125º aniversário.
Referências
Bibliografia
Bone, James (2016) The Curse of Beauty: The Scandalous & Tragic Life of Audrey Munson, America's First Supermodel. New York: Regan Arts. ISBN 978-1942872030
Donnelly, Elisabeth (Summer 2015) "Descending Night", The Believer, v.13 n.2.
Mullgardt, Louis Christian (1915) The Architecture and Landscape Gardening of the Exposition – A Pictorial Survey of the Most Beautiful of the Compositions of the Panama-Pacific International Exposition. San Francisco: Paul Elder and Company.
Neuhaus, Eugen (1915) The Art of the Exposition – Personal Impressions of the Architecture, Sculpture, Mural Decorations, Color Scheme & Other Aesthetic Aspects of the Panama-Pacific International Exposition. San Francisco: Paul Elder and Company.
Rozas, Diane & Gottehrer, Anita Bourne (1999) American Venus: The Extraordinary Life of Audrey Munson, Model and Muse. Los Angeles: Balcony Press. ISBN 1-890449-04-0
Ligações externas
Blog devoted to Munson in NYC
The Audrey Munson Project
Audrey Munson, J. Willis Sayre Photographs Collection, University of Washington
Portrait photo, 1922, The New York Times, December 9, 2007
"America's first supermodel", BBC News, May 31, 2016; video with images: photos, film, sculpture
"Miss Manhattan", 99% invisible, February 15, 2016, Podcast, video, images
Musas
Pessoas com esquizofrenia
Atrizes de teatro dos Estados Unidos
Centenários dos Estados Unidos
Norte-americanos de ascendência irlandesa
Mortos em 1996
Nascidos em 1891
Modelos dos Estados Unidos | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,502 |
It usually happens when you use filters on the homepage. In this case there is no any particular category id present in the URL and Virtuemart "decides" to show all available categories you have in the store.
Let's see how it's easily set up. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,145 |
package edu.cmu.lti.oaqa.bioasq.eval.calculator;
import static edu.cmu.lti.oaqa.baseqa.eval.EvalCalculatorUtil.sumMeasurementValues;
import static edu.cmu.lti.oaqa.bioasq.eval.calculator.AnswerEvalMeasure.*;
import static java.util.stream.Collectors.toList;
import static java.util.stream.Collectors.toSet;
import java.util.Collection;
import java.util.Comparator;
import java.util.List;
import java.util.Map;
import java.util.Set;
import java.util.function.Function;
import java.util.stream.IntStream;
import com.google.common.collect.Iterables;
import org.apache.uima.jcas.JCas;
import com.google.common.collect.ImmutableMap;
import com.google.common.collect.Sets;
import edu.cmu.lti.oaqa.baseqa.eval.EvalCalculator;
import edu.cmu.lti.oaqa.baseqa.eval.EvalCalculatorUtil;
import edu.cmu.lti.oaqa.baseqa.eval.Measure;
import edu.cmu.lti.oaqa.ecd.config.ConfigurableProvider;
import edu.cmu.lti.oaqa.type.answer.Answer;
import edu.cmu.lti.oaqa.util.TypeUtil;
/**
* Implementations of BioASQ Phase B Factoid, List, and YesNo QA evaluation metrics.
*
* @see AnswerEvalMeasure
*
* @author <a href="mailto:ziy@cs.cmu.edu">Zi Yang</a> created on 4/29/15
*/
public class AnswerEvalCalculator<T extends Answer> extends ConfigurableProvider
implements EvalCalculator<T> {
@Override
public Map<Measure, Double> calculate(JCas jcas, Collection<T> resultEvaluatees,
Collection<T> gsEvaluatees, Comparator<T> comparator,
Function<T, String> uniqueIdMapper) {
Set<String> gsVariants = gsEvaluatees.stream().map(TypeUtil::getCandidateAnswerVariantNames)
.flatMap(Collection::stream).map(String::toLowerCase).collect(toSet());
List<Answer> resultAnswers = resultEvaluatees.stream().sorted(comparator).collect(toList());
String questionType = TypeUtil.getQuestion(jcas).getQuestionType();
ImmutableMap.Builder<Measure, Double> builder = ImmutableMap.builder();
switch (questionType) {
case "FACTOID":
Set<String> strictResultVariants = resultAnswers.stream().limit(1)
.map(TypeUtil::getCandidateAnswerVariantNames).flatMap(Collection::stream)
.map(String::toLowerCase).collect(toSet());
Set<String> lenientResultVariants = resultAnswers.stream().limit(5)
.map(TypeUtil::getCandidateAnswerVariantNames).flatMap(Collection::stream)
.map(String::toLowerCase).collect(toSet());
int strictRetrieved = Sets.intersection(gsVariants, strictResultVariants).isEmpty() ? 0 : 1;
builder.put(FACTOID_STRICT_RETRIEVED, (double) strictRetrieved);
int lenientRetrieved = Sets.intersection(gsVariants, lenientResultVariants).isEmpty() ?
0 :
1;
builder.put(FACTOID_LENIENT_RETRIEVED, (double) lenientRetrieved);
double reciprocalRank = IntStream.range(0, resultAnswers.size())
.filter(i -> TypeUtil.getCandidateAnswerVariantNames(resultAnswers.get(i)).stream()
.map(String::toLowerCase).anyMatch(gsVariants::contains))
.mapToDouble(i -> 1.0 / (i + 1.0)).findFirst().orElse(0.0);
builder.put(FACTOID_RECIPROCAL_RANK, reciprocalRank);
builder.put(FACTOID_COUNT, 1.0);
break;
case "LIST":
int relevantRetrieved = (int) resultAnswers.stream()
.map(TypeUtil::getCandidateAnswerVariantNames).filter(names -> names.stream()
.map(String::toLowerCase).anyMatch(gsVariants::contains))
.count();
double precision = EvalCalculatorUtil.calculatePrecision(resultAnswers.size(),
relevantRetrieved);
builder.put(LIST_PRECISION, precision);
double recall = EvalCalculatorUtil.calculateRecall(gsVariants.size(), relevantRetrieved);
builder.put(LIST_RECALL, recall);
builder.put(LIST_F1, EvalCalculatorUtil.calculateF1(precision, recall));
builder.put(LIST_COUNT, 1.0);
break;
case "YES_NO":
String gs = Iterables.getOnlyElement(gsVariants);
String result = resultAnswers.stream().map(Answer::getText).findAny().orElse("");
int correctRetrieved = gs.equals(result) ? 1 : 0;
builder.put(YESNO_CORRECT, (double) correctRetrieved);
if (gs.equals("yes")) {
int truePositive = result.equals("yes") ? 1 : 0;
builder.put(YESNO_TRUE_POS, (double) truePositive);
} else {
int trueNegative = result.equals("no") ? 1 : 0;
builder.put(YESNO_TRUE_NEG, (double) trueNegative);
}
break;
}
return builder.build();
}
@Override
public Map<Measure, Double> accumulate(
Map<Measure, ? extends Collection<Double>> measure2values) {
ImmutableMap.Builder<Measure, Double> builder = ImmutableMap.builder();
if (measure2values.get(FACTOID_COUNT) != null) {
double factoidCount = sumMeasurementValues(measure2values.get(FACTOID_COUNT));
builder.put(FACTOID_COUNT, factoidCount);
builder.put(FACTOID_STRICT_ACCURACY,
sumMeasurementValues(measure2values.get(FACTOID_STRICT_RETRIEVED)) / factoidCount);
builder.put(FACTOID_LENIENT_ACCURACY,
sumMeasurementValues(measure2values.get(FACTOID_LENIENT_RETRIEVED)) / factoidCount);
builder.put(FACTOID_MRR,
sumMeasurementValues(measure2values.get(FACTOID_RECIPROCAL_RANK)) / factoidCount);
}
if (measure2values.get(LIST_COUNT) != null) {
double listCount = sumMeasurementValues(measure2values.get(LIST_COUNT));
builder.put(LIST_COUNT, listCount);
builder.put(LIST_MEAN_PRECISION,
sumMeasurementValues(measure2values.get(LIST_PRECISION)) / listCount);
builder.put(LIST_MEAN_RECALL,
sumMeasurementValues(measure2values.get(LIST_RECALL)) / listCount);
builder.put(LIST_MEAN_F1, sumMeasurementValues(measure2values.get(LIST_F1)) / listCount);
}
if (measure2values.get(YESNO_CORRECT) != null) {
Collection<Double> corrects = measure2values.get(YESNO_CORRECT);
double yesnoCount = corrects.size();
builder.put(YESNO_COUNT, yesnoCount);
builder.put(YESNO_MEAN_ACCURACY, sumMeasurementValues(corrects) / yesnoCount);
Collection<Double> truePositives = measure2values.get(YESNO_TRUE_POS);
builder.put(YESNO_MEAN_POS_ACCURACY,
sumMeasurementValues(truePositives) / truePositives.size());
Collection<Double> trueNegatives = measure2values.get(YESNO_TRUE_NEG);
builder.put(YESNO_MEAN_NEG_ACCURACY,
sumMeasurementValues(trueNegatives) / trueNegatives.size());
}
return builder.build();
}
@Override
public String getName() {
return "Answer";
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,226 |
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"redpajama_set_name": "RedPajamaC4"
} | 9,785 |
Q: ArcGIS API for Flex create customised widget I want to create customised widget using ArcGIS API for flex (in adobe flash builder). I surfed internet and all the samples I found of widgets use ArcGIS viewer for flex. Has anybody has the experience on creating widget using adobe flash builder? where should I start?
I am using Adobe Flex SDK 4.6 and ArcGIS API for Flex 3.6, thanks.
A: To start with your custom widget, you need
*
*ArcGIS for Flexviewer3.6 source code
*ArcGIS API 3.6 for Flex.
You can download
*
*Flexviewer3.6 source code from GitHub.
*AcrGIS API 3.6 for Flex from esri under ArcGIS API for Flex section.
Steps to import the source code to Flash Builder:
*
*Flash Builder -> File -> Import Flash Builder Project.
*Browse to downloaded source code zip file and Open the file.
*Give name to project. (optional)
*Click Finish.
Steps to add ArcGIS API to your flexveiwer:
*
*Extract the arcgis_api_for_flex_3_6.zip file.
*Copy the file ArcGIS_Flex\libs\agslib-3.6-2013-12-13.swc from extracted folder.
*Go to Flash Builder and expand the imported flexveiwer project.
*Paste the file in libs folder under flexveiwer project.
Now you are good to go.
you can add ArcGIS API to your flexveiwer from Flex Build Path too but above mentioned steps are simpler.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 47 |
So do your Valentine's Day celebrations mean a cosy night in with your partner or a hot date at the newest venue in town?
Whatever your evening looks like, take a little time to make sure you look and feel your best.
Dress codes are often more relaxed and we can have a tendency to dress down for most things.
In addition, we often stop making an effort with those we love most which, in time, can make us feel less attractive.
So, on the most romantic day of the year, take the opportunity to reverse the trend and make it different.
Kate Nicholson, Colour Analyst and Personal Stylist with House of Colour has some tips on how to do just that!
Red is the colour of love, so why not embrace it on Valentine's Day? We all suit different reds whether it be a cherry red, brick red, scarlet or geranium, but if you are unsure, go for a true red as it suits most skins tones.
Choose one feature to enhance, whether that's long legs, your cleavage or beautiful skin, but don't try and show everything off at once! Sizzle in your own style, whether that is classic, dramatic or, especially on this day, romantic!
Always balance your accessories - big earrings rarely look good paired with a flamboyantly big necklace! And choose accessories that really suit you rather than just following the latest fashion.
Without question, heels can lift an outfit and make you look more elegant, and 'Sitting down shoes' are great from kerb to restaurant, but you can't walk in them, your look will crumble! Always wear shoes that you can walk in, especially if your partner has planned a surprise and you have no idea how much walking there will be!
Finish your look with that all important slick of lipstick to make your eyes sparkle and you're good to go!
Impressing your partner doesn't mean you have to be suited and booted, but maybe something other than jeans would be nice for a change! Make sure everything fits well; there is nothing worse than trousers that are too long (or too short!), or shirts which are too baggy. Quality will show you care about yourself and therefore the person you are with.
And using colour doesn't just apply to women. Use it around your face to lift your look and if you know your best colours, use one in a (well-ironed) shirt, tie or jumper. If you are unsure, look at your face when you hold a colour next to it and make sure it's not making you look tired or pale. Paired with smart chinos and a neutral jacket a 'wow' colour will always ensure you'll look great!
Make sure that your hair is just as sharp as your outfit – get a good haircut in advance, make sure you are well-groomed and that your skin is moisturised.
For your finishing touches, detail is everything. Matching your belt with your shoe colour always brings an outfit together. Wear a nice watch – white metal if your colours are winter or summer seasons, or gold coloured if you are an autumn or spring colour person.
Wearing a smart, fitted coat will sharpen up any outfit and a scarf in one or more of your wow colours will again bring the whole ensemble together.
And remember, a little bit of thought will help you add a touch of romance to make Valentine's Day special, whilst bringing back the excitement and anticipation that getting dressed up to go out.
Kate Nicholson is one of a network of consultants with House of Colour.
To find out more or make a booking, contact her on 07885 541742, email kate.nicholson@houseofcolour.co.uk or visit the House of Colour website www.houseofcolour.co.uk/katenicholson. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,859 |
Q: How to return HTML file as Symfony response I'm new to Symfony and am trying to understand that controller response function. I just want to return a simple HTML file home.html, that at the moment just has a Hello World line in it.
How do I return the file as a response? As it stands with the code below it's just returning the string 'home.html'.
Many thanks.
<?php
namespace App\Controller;
use Symfony\Component\HttpFoundation\Response;
use Symfony\Component\Routing\Annotation\Route;
class MainController
{
/**
* @Route("/")
*/
public function homepage()
{
return new Response('home.html');
}
}
A: Because you need to extend your controller with AbstractController and use the render method to generate the view from html.
class MainController extends AbstractController
{
/**
* @Route("/")
*/
public function homepage()
{
return $this->render('home.html');
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,953 |
Menu Search My Evli
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Evli.com does not support your browser or browser version. To make sure the site works properly and to increase the security, please take note of the following technical requirements. Evli.com runs most reliably with Internet Explorer 11, and with the most recent versions of Firefox, Safari and Google Chrome.
Products Mutual funds Factor funds Evli Equity Factor USA
Fund NAVs
Corporate bond and fixed income
Evli Corporate Bond
Evli Emerging Markets Credit
Evli Euro Government Bond
Evli Euro Liquidity
Evli European High Yield
Evli European Investment Grade
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Evli Leveraged Loan
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Evli Target Maturity Nordic Bond 2023
Global equities
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Evli GEM
Evli Global
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Nordic equities
Evli Finland Select
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Evli Sweden Equity Index
Evli Swedish Small Cap
Factor funds
Evli Equity Factor Europe
Evli Equity Factor Global
Evli Equity Factor USA
Evli USA Growth
Evli Emerging Frontier
Asset allocation solutions
Evli Finland Mix
Evli Global Multi Manager 30
Kasvuyhtiörahasto
Evli Growth Partners
Equity fund that invests in American companies using a factor-based strategy
A (EUR)
B (EUR)
IA (EUR)
IB (EUR)
B (USD)
Return %
Return % p.a.
Since start
146.370 2.99 5.20 6.88 9.83 10.15
155.522 3.01 5.52 7.20 10.16 10.49
169.643 4.47 1.25 6.35 6.89 9.45
Recommended Investment Horizon
at least 7 years
Administrative fees
0.95 % p.a.
Suitable for investors
who want factors (certain characteristics of equities) to be focused on in the fund's holdings rather than traditional market value
who seek the additional return potential offered by factors
who seek high returns in the long term
who are prepared to tolerate greater price fluctuations in the short term
who want to invest responsibly and take into account not only economic analysis but also environmental, social and good governance (ESG) factors.
min. 1 000 € or 50 €/month
Evli Equity Factor USA Fund invests its assets in a diversified manner in the equities of major American companies. The fund focuses on four academically determined factors: value, low volatility, momentum and quality.
The fund's investment style is active. The fund is not subject to restrictions concerning business area or country within its geographical investment area.
The portfolio is managed by
Mattias Lagerspetz
Peter Lindahl
Antti Sivonen
Investment Objective and Risks
The aim is to earn a return which, in the long term, exceeds the return of the benchmark index.
As the fund's assets are invested in equities or equity-linked securities, the fund unit value can fluctuate significantly within a short period.
Evli Equity Factor USA focuses on four academically determined factors: value, low risk, momentum and quality. The fund complies with policies for responsible investment (ESG) and excludes from its investments companies that have substantial business in the following areas: weapons, alcohol, tobacco, mining of thermal coal, controversial weapons, gambling and adult entertainment. The fund also excludes companies with the lowest ESG rating and companies which have very severe confirmed ESG controversies.
Evli Equity Factor USA lost 7.1% in December, while the benchmark MSCI USA Daily Net TR was down by 9.4%. The fund lagged its benchmark by 17.2% points since inception as of the end of December.
American equity fund (UCITS)
Investment activity began
Benchmark index
MSCI USA TR Net (USD)
Profit distribution
Fund-units are divided into A and B units. Profit share of at least 4% is distributed on A units annually.
Financial product's sustainability information in accordance with EU Sustainable Finance Disclosure Regulation (SFDR) 2019/2088 (sustainability‐related disclosures in the financial services sector). This is a financial product in accordance with Article 8 of the SFDR.
Publication date: December 1, 2022
Legal Entity Identifier: 7437005N2I7MBZ97YL71
a) Summary
Read the summary in IT, NO, ES
This financial product promotes environmental or social characteristics, but does not have as its objective sustainable investment.
Environmental and social characteristics are promoted by observing Evli's Principles for Responsible Investment, Climate Change Principles and climate targets, and by requiring that target companies observe good governance practices. The fund excludes harmful industries on the basis of Evli's responsibility principles and Climate Change Principles. In addition, target companies are monitored regularly to ensure they have not violated the norms of specific international treaties and principles.
The fund promotes climate change mitigation in accordance with Evli's climate targets: the fund's carbon footprint and emission indicators are measured and monitored, and a regular scenario analysis is conducted to monitor the attainment of Evli's general climate targets. Evli's goal is to achieve carbon neutrality by 2050 at the latest, and it has set an interim target of a 50 percent reduction in indirect emissions from all investments by 2030, provided that this is possible in the investment environment. The comparison year is 2019. The fund-specific share of the emission reduction target may vary between funds.
Investment targets are monitored regularly, and efforts are made to engage with companies to influence their practices. Evli can engage with companies either independently or together with other investors. The themes of Evli's engagement are climate change mitigation, respect for human rights, anti-corruption measures, taking environmental issues into consideration, factors related to good governance and the reporting of responsibility factors. Evli's Responsible Investment Policy and Corporate Governance Principles set the framework for Evli's engagement and conduct in the event of perceived breaches of the Code.
The fund's target companies are analyzed before an investment decision is made and at regular intervals during the investment period with regard to environmental, social and corporate governance matters, or ESG factors. ESG factors are integrated into the analysis of target companies and their selection for investment by the fund. Evli has built an internal ESG database based on data produced by external service providers, which it uses to monitor ESG factors.
The achievement of the environmental or social characteristics promoted by the financial product is monitored through sustainability indicators, which are the target companies' carbon intensity and ESG scores and the number of target companies that have not committed serious norm violations.
Sustainability indicators are monitored through Evli's internal ESG database. The data is based on data provided by external service providers, which is not verified by a third party. The completeness of the data is reported in conjunction with the sustainability indicators. All active investments of the fund promote environmental and social characteristics by observing Evli's Principles for Responsible Investment and Climate Change Principles, and completeness of data has no impact on observance of the above-mentioned principles.
The fund's benchmark index is a money market-based index that does not consider sustainability factors. The benchmark index used by the fund can be found in the fund-specific key investor information document.
b) No sustainable investment objective
c) Environmental or social characteristics of the financial product
In addition to other characteristics, the fund promotes environmental and social characteristics by observing Evli's Principles for Responsible Investment, Climate Change Principles and climate targets, and requires that target companies observe good governance practices. The fund excludes harmful industries on the basis of Evli's responsibility principles and Climate Change Principles. In addition, target companies are regularly monitored for violations of norms.
Investment targets are monitored regularly, and efforts are made to engage with companies to influence their practices. Evli can engage with companies either independently or together with other investors. The themes of Evli's engagement are climate change mitigation, respect for human rights, anti-corruption measures, taking environmental issues into consideration, factors related to good governance and the reporting of responsibility factors.
d) Investment strategy
The attainment of the climate targets will be measured using data from external service providers to monitor the fund's carbon footprint and intensity, the degree of low-carbon transition, a scenario analysis in relation to the target of limiting global warming to 1.5 degrees Celsius and the warming ratio associated with the fund.
An assessment of the quality of corporate governance is an important part of the assessment of potential investments. Good governance refers in particular to effective management structures, employee relations, staff remuneration and tax compliance.
Evli's ownership control principles state that the companies it invests in must engage in good governance by complying with the Finnish Corporate Governance Code issued by the Securities Market Association, for example, or corresponding foreign guidelines, which often impose a partial framework on the remuneration models of the invested companies. In addition, Evli's Responsible Investment Team analyzes the fund's investments every three months for any breaches of norms (UN Global Compact and OECD's guidelines for multinational companies). The OECD's guidelines for multinational companies also cover disputes related to taxation. Consequently, such disputes may lead to the exclusion of an investment instrument.
e) Proportions of investments
All active investments of the fund promote environmental and social characteristics.
f) Monitoring of environmental or social characteristics
The achievement of the environmental or social characteristics promoted by the financial product is monitored through the target companies' carbon intensity and ESG scores and the number of target companies that have not committed serious norm violations. Evli has built an internal ESG database to monitor sustainability indicators. In addition, the Responsible Investment Team analyzes norm violation cases in accordance with the process set out in the Principles for Responsible Investment.
g) Methods concerning environmental or social characteristics
The environmental and social characteristics promoted by the financial product are monitored and reported using the sustainability indicators mentioned above.
h) Data sources and data processing
Evli has built an internal database based on data provided by external service providers, which is used for monitoring and reporting sustainability indicators and adverse impacts of investment decisions related to the promotion of the promoted environmental and social characteristics. The data from external providers is not verified by a third party and the completeness of the data is reported at the same time.
i) Limitations of methods and data
The achievement of the promoted environmental and social characteristics is reported annually through the sustainability indicators mentioned above, in conjunction with which the completeness of the data from the target companies is also reported. All active investments of the fund promote environmental and social characteristics by observing Evli's Principles for Responsible Investment and Climate Change Principles. The completeness of the data does not affect compliance with the above principles.
j) Due diligence
The fund's target companies are analyzed before an investment decision is made and at regular intervals during the investment period with regard to environmental, social and corporate governance matters, or ESG factors. ESG factors are integrated into the analysis of target companies and their selection for investment by the fund. Evli has built an internal ESG database based on data produced by external service providers, which it uses to monitor ESG factors. Evli regularly monitors its active investments and seeks to influence the companies' practices. If a company violates the principles of the UN Global Compact, the UN Guiding Principles on Business and Human Rights, the OECD Guidelines for Multinational Enterprises or Evli's Climate Change Principles, Evli will either seek to influence the company's actions through engaging with it or exclude it from its investments. The methods are based on data provided by an external service provider, which is not verified by a third party.
k) Engagement policies
The financial product can be used to engage with the target companies as part of the promotion of environmental and social characteristics. Evli's Responsible Investment Policy and Corporate Governance Principles set the framework for Evli's engagement and conduct in the event of perceived breaches of the Code.
l) Designated reference value
Read more about Evli's responsible investing
Information on environmental and social characteristics of the fund in accordance with article 8 of Sustainable Disclosure Regulation (in force from January 1, 2023)
Fund Rules
Avaintietoasiakirja
Nøkkelinformasjonsdokument (trer i kraft 1. januar 2023) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,044 |
For a very special, spiritual evening, join the congregation at St. John's Co-Cathedral in Valletta for candlelit carol singing. Other Baroque parish churches across the Islands are just as awe-inspiring during advent. Their interiors are decked out in papal crimson and altars are adorned with flowers. All churches have a calendar of events, ranging from the procession of Mary and Joseph, to streets with groups of children carol singing.
Cribs are positively everywhere, from private houses to small chapels. Visiting cribs is a popular activity at Christmastime and many locals spend time going round touring the various 'presepju' displays. Some are very artistic and elaborate, with figures moving around mechanically and a degree of detail that comes from hours of meticulous work by dedicated craftsmen.
One of the most spectacular cribs, is situated in Gozo, where a whole town brings the 'crib' to life. Bethlehem f' Għajnsielem is an animated nativity village spread on 20,000sqm of land. It offers naturalistic reproduction of the environ of Bethlehem and the timeless story of Nativity every December at 'Ta Passi' fields in Ghajnsielem. Between 6th December 2015 to 3rd January 2016, around 150 actors including three newborns bring the timeless Christmas story to life as never before. The atmosphere will simply be unique… horses turn mills, villagers go about their jobs, shepherds inhabit caves, animals roam in enclosed spaces and a poor unknown couple tend their newly-born in a grotto.
The Christmas Crib's origins can be traced directly back to the early 17th century, when Dominican friars in Rabat set up their own, local crib display. A tradition imported from neighbouring Naples and Sicily, the very first cribs were supposedly imported by noblemen, though they weren't immediately embraced by locals; rumour has it they many were burned as firewood. But despite that failing vote of confidence, it didn't take long for the tradition to take root, and for the cribs to adopt their own, uniquely local appearance.
Visually, Maltese presepju differ from generic nativity scenes in their depictions of the Maltese landscape. Mary, Joseph, and Jesus may be found within a manger, but here it's surrounded by rocky stones, porous caves, Maltese flour windmills, and ancient ruins – all signifiers of the traditional Maltese landscape, in other words. Aside from the setting, the figurines in the cribs, called pasturi, were also traditional, and produced by Maltese artisans out of sculpted and painted clay.
Today's cribs are, of course, more elaborate than their predecessors. Having grown in scale and level of detail, many now feature mechanical parts and moving figurines in addition to ornate local landscapes. And while most Maltese families have a crib on display at home, there are a number of viewing options for visitors looking to uncover this favourite of Maltese traditions. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,216 |
Q: Linear algebra homework problem involving basis and dual basis. Please help me get started on this problem:
Let $V = R^3$, and define $f_1, f_2, f_3 ∈ V^*$ as follows:
$f_1(x,y,z) = x - 2y$
$f_2(x,y,z) = x + y + z$
$f_3(x,y,z) = y-3z$
Prove that $\{f_1,f_2,f_3\}$ is a basis for $V^*$, and then find a basis for $V$ for which it is the dual basis.
A: $\bf Hint:$ Suppose that $a f_1(x,y,z)+b f_2(x,y,z)+c f_3(x,y,z)=0$. Try different values for $(x,y,z)$, for example, if you substitute $(2,1,-3)$. We obtain $f_1(2,1,-3)=0$, $f_2(2,1,-3)=0$ and $f_3(2,1,-3)=10$, hence $cf_3(2,1,-3)=10c$ which implies $c=0$.
A: Proving $\lbrace f_1,f_2,f_3\rbrace$ is a basis for $V^*$ can be done by row reducing the coefficients of $\lbrace f_1,f_2,f_3 \rbrace$ and showing that it has a rank of $3$.
The dual basis of $ \lbrace f_1,f_2,f_3 \rbrace $ is found by calculating the inverse of coefficients of $f_i$ which is:
$x_{1}=
\
\begin{bmatrix}
4/10 \\ -3/10 \\ -1/10
\end{bmatrix},
x_{2}=
\begin{bmatrix}
6/10 \\ 3/10 \\ 1/10
\end{bmatrix},
x_{3}=
\begin{bmatrix}
2/10 \\ 1/10 \\ -3/10
\end{bmatrix}
]\
$
Checking that $f_{i}(x_{j})=$ $1,$ if $i=j$ and $0$ if $i \ne j $
$f_{1}(x_1)=(4/10)-2(-3/10)=1\\
f_{1}(x_2)=(6/10)-2(3/10)=0\\
f_{1}(x_3)=(2/10)-2(1/10)=0$
$f_{2}(x_1)=(4/10)+(-3/10)+(-1/10)=0\\
f_{2}(x_2)=(6/10)+(3/10)+(1/10)=1\\
f_{2}(x_3)=(2/10)+(1/10)+(-3/10)=0$
$f_{3}(x_1)=(-3/10)-3(-1/10)=0\\
f_{3}(x_2)=(3/10)-3(1/10)=0\\
f_{3}(x_3)=(1/10)-3(-3/10)=1$
A: First, you should verify that the $f_i$ are elements of $V^*$; that is, that they are functions on $\Bbb R^3$ that satisfy:
$\ \ \ $1) $f({\bf x}+{\bf y})=f({\bf x})+f({\bf y})$ for all ${\bf x},{\bf y}\in\Bbb R^3$
and
$\ \ \ $2) $f(c{\bf x})=cf({\bf x})$ for all $c\in\Bbb R$ and all ${\bf x}\in \Bbb R^3$.
Of course, if you know this has been covered in your class already, you're probably safe just writing something like "as was demonstrated in lecture blah, these are elements of the dual of $V=\Bbb R^3$.
Another fact I assume you have use of is that the dimension of $V^*$ is three (as the dimension of the vector space $\Bbb R^3$ is three).
So, with three linear functionals on $\Bbb R^3$, towards showing that they are a basis of $V^*$, it suffices to show that they are independent. There are many ways towards achieving this end.
One way in particular is to show that the matrix $A$ formed by taking as its rows the coefficients of the $f_i$,
$$ A=\left[\matrix{1&-2&0\cr 1&1&1\cr 0&1&-3}\right], $$
has full rank (that this is so isn't hard to see: $c_1f_1+c_2f_2+c_3f_3={\bf 0}\iff
A\bigl[{{\scriptstyle c_1\atop\scriptstyle c_2}\atop c_3}\bigr]=\bf 0$; and the former equation has only the trivial solution if and only if $A$ has full rank).
So, let's row reduce $A$:
$$
A=
\left[\matrix{1&-2&0\cr 1&1&1\cr 0&1&-3}\right]
\buildrel{r_2-r_1\rightarrow r_2}\over{\longrightarrow}
\left[\matrix{1&-2&0\cr 0& 3& 1\cr 0&1&-3}\right]
\buildrel{r_2-3r_3\rightarrow r_3}\over{\longrightarrow}
\left[\matrix{1&-2&0\cr 0& 3& 1\cr 0&0&10}\right].
$$
At this point we can see that $A$ indeed has full rank. Thus the set $\{f_1,f_2,f_3\}$ is independent and consequently is a basis of $V^*$.
Towards finding the dual basis let's recall what this is: the dual basis of
$\{f_1,f_2,f_3\}$ by definition is the basis $\{ {\bf x}_1,{\bf x}_2,{\bf x}_3\}$ of $V$ for which
$$ f_i({\bf x}_j)=\cases{1,& if $i=j$\cr 0, & if $i\ne j$}. $$
So, in particular, the first basis element of dual basis, ${\bf x}_i$, would satisfy
$$
f_1({\bf x}_1)=1,\ f_2({\bf x}_1)=0, \text{and}, f_3({\bf x}_1)=0.
$$
In other words, we have the system:
$$
\eqalign{
x-2y&=1\cr
x+y+z&=0\cr
y-3z&=0
}
$$
whose matrix form is
$$
A {\bf x}_1=\left[\matrix{ 1\cr0\cr 0 }\right]
$$
and whose, necessarily unique, solution gives the coordinates of ${\bf x}_1$.
We would have two similar systems to solve in order to find ${\bf x}_2$ and ${\bf x}_3$.
That seems like a lot of work; but wait... suppose we wrote the dual basis as a matrix whose columns were the ${\bf x}_i$. Then we'd have:
$$
A[ {\bf x}_1\ {\bf x}_2\ {\bf x}_3]
=\left[\matrix{1&-2&0\cr 1&1&1\cr 0&1&-3}\right][ {\bf x}_1\ {\bf x}_2\ {\bf x}_3]=\left[\matrix{1&0&0\cr 0&1&0\cr 0&0&1}\right]
$$
So $[ {\bf x}_1\ {\bf x}_2\ {\bf x}_3]$ is the inverse of $A$. Rather than solving three systems of equations, we could instead find the inverse of $A$ and then the columns will be our dual basis.
This is what we'll do.
Towards that end, we may (and do) adjoin the identity matrix to $A$ and perform a full forward/back row reduction:
$$\eqalign{
[A \,|\,I\,]=
\left[\matrix{1&-2&0\cr 1&1&1\cr 0&1&-3}
\ \ \Biggl|\ \ \matrix{1& 0&0\cr 0&1&0\cr 0&0&1}\right]
&\buildrel{r_2-r_1\rightarrow r_2}\over{\longrightarrow}
\left[\matrix{1&-2&0\cr 0& 3& 1\cr 0&1&-3}
\ \ \Biggl|\ \ \matrix{1& 0&0\cr -1&1&0\cr 0&0&1}\right]\cr
&\buildrel{r_2-3r_3\rightarrow r_3}\over{\longrightarrow}
\left[\matrix{1&-2&0\cr 0& 3& 1\cr 0&0&10}
\ \ \Biggl|\ \ \matrix{1& 0&0\cr -1& 1&0\cr -1&1&-3}\right]\cr
&\buildrel{10r_2-r_3\rightarrow r_2}\over{\longrightarrow}
\left[\matrix{1&-2&0\cr 0& 30& 0\cr 0&0&10}
\ \ \Biggl|\ \ \matrix{1& 0&0\cr -9&9&3\cr -1&1&-3}\right]\cr
&\buildrel{15r_1+r_2\rightarrow r_1}\over{\longrightarrow}
\left[\matrix{15&0&0\cr 0& 30& 0\cr 0&0&10}
\ \ \Biggl|\ \ \matrix{ 6& 9&3\cr -9&9&3\cr -1&1&-3}\right]\cr
&\buildrel{ }\over{\longrightarrow}
\left[\matrix{1 &0&0\cr 0& 1& 0\cr 0&0&1 }
\ \ \Biggl|\ \ \matrix{ 6/15& 9/15&3/15\cr -9/30&9/30&3/30\cr -1/10&1/10&-3/10}\right].\cr
}
$$
So
$$
A^{-1}=\left[\matrix{2/5& 3/5&1/5\cr -3/10&3/10&1/10\cr -1/10&1/10&-3/10}\right],
$$
and the dual basis has as its elements the columns of $A^{-1}$:
$$
{\bf x}_1=\left[\matrix{2/5\cr-3/10\cr-1/10 }\right],\
{\bf x}_2=\left[\matrix{3/5\cr3/10\cr1/10 }\right],\
{\bf x}_3=\left[\matrix{1/5\cr1/10\cr-3/10 }\right].
$$
The basis $\{ {\bf x}_1,{\bf x}_2,{\bf x}_3 \}$ is the dual basis of $\{f_1,f_2,f_3\}$.
Note that the relative ordering is important. For instance, we must have $f_3({\bf x}_1)=f_3({\bf x}_2)=0$ and $f_3({\bf x}_3)=1$. As a spot check, let's verify this:
$$\eqalign{
f_3({\bf x}_1)&= (-3/10)-3(-1/10)=0\cr
f_3({\bf x}_2)&= (3/10)-3(1/10)=0\cr
f_3({\bf x}_3)&= (1/10)-3(-3/10)=1.\cr
}
$$
| {
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Collection of essays celebrates life of Dr. Luke Keefer, Jr.
In early October, two events were held in Ashland, Ohio to mark the release of the book, Celebrations and Convictions: Honoring the Life and Legacy of Luke L. Keefer, Jr., edited by J. Robert Douglass and Wyndy Corbin Reuschling, and published by the Historical Society.
Members of the Historical Society joined about 200 others at a conference at Ashland Theological Seminary in honor of the late Luke L. Keefer, Jr. Following the conference, the Society's annual meeting was held at the Ashland Brethren in Christ Church. The church provided dinner, including the traditional annual meeting fare of homemade pie.
At the annual meeting, Douglass and Reuschling, former colleagues of Luke's at Ashland Seminary, about their memories of Luke and their experience editing the book. After their remarks, a panel discussion was held featuring three of the Brethren in Christ contributors to the book: Grace Holland, Zach Spidel, and John Yeatts (pictured below right). Devin Manzullo-Thomas, another contributor, moderated the panel.
Attendance at annual meeting was not large, but those who were able to come appreciated the opportunity to celebrate their friend and colleague.
Copies of the book are now available for sale from the Historical Society at $15 each plus shipping of $2.75 for one copy. You won't want to miss essays that tell Luke's life story, describe his role as mentor and friend, offer Luke's own reflections on the "uneasy synthesis of heritage streams" in the Brethren in Christ Church, and address some of Luke's interests and core convictions: Brethren in Christ history, heresy (something Luke was researching when he became ill), spiritual formation, Christian history from a Wesleyan perspective, missional identity, Anabaptist ecclesiology, women in ministry, reading Scripture, pacifism and nonviolence, and Christian ethical commitments.
Contact the Society to order the book.
Living Simply, Giving Generously: A Biography of David and Jeannie Byer, was also published this fall. Written by Historical Society editor emeritus, E. Morris Sider, the biography was published in partnership with the Friends of Murray Library at Messiah College, one of many recipients of the Byers' generosity.
All members of the Historical Society received a free copy; additional copies are available for $10 each plus shipping of $2.75 for one copy. Contact the Society.
The Carland-Zion Brethren in Christ Church (located between Elsie and Owosso, Michigan) recently celebrated its 125th anniversary. "Tunker" families from Markham, Ontario Canada who emigrated to Michigan established the church in the mid-1880s. It was the first Brethren in Christ congregation in the state and played an important role in establishing other congregations in Michigan – many of which remain to this day.
Other pastors over the years included George Kiteley, Harry Brubaker, William H. Engle, Clinton Starr, Albert Brenaman, Harvey B. Stickley, Lloyd Melhorn, Jr., Roger Carr, Cyrus Lutz, Carl Lewis, Verle Brubaker, and Donald Bundenthal.
The original building remains in use today as the main worship sanctuary. A parsonage was built next door in 1948, and a fellowship hall was built in 1958. In the 1960s, the original church and fellowship hall were connected with a building that includes classrooms, a nursery, restrooms, and office space. At that time, the layout of the sanctuary was reversed, and the original entrance area became the pulpit/preaching area.
The congregation's oldest member, Leoda Kiteley Brady, and her daughter, Mary Brady Ford. Leola is a descendant of Bishop Henry Schneider. Photo by Laura Reppenhagan.
Current Pastor Eric Stanton, who has served the congregation for more than twenty-five years, led a hymn sing and a time of sharing memories and stories from the past. Bishop John Zuck gave a brief message and a time of fellowship followed. Church members are gathering memories and stories which will be placed in a time capsule to be buried this fall.
Who would follow Jesus in answer to His call?
Preaching the gospel everywhere…even in Michigan!
Because the Word that was preached to them was sharp as a two-edged sword.
To worship God and grow in grace and fellowship so sweet.
In which to carry out God's work, and thus send forth the light.
Its rays penetrating the darkness, the darkness of sin and night.
Salvation is through Christ alone, by works we can't attain.
Our sins all covered by the blood assures a home in heaven.
Instruction in holy living was among the things they taught.
Some chose to follow Christ – the only right choice to make.
How sad for them when called to account upon the Judgement Day!
Jonathan Stanton is the son of Eric Stanton, current pastor of the Carland-Zion congregation. Jon lives in Omaha, Nebraska with his wife and serves on the editorial committee for Shalom!, a Brethren in Christ publication on peace and justice issues. Much of this information is from a brief history of the church written by Anna Kiteley in 1959.
Recently, while conducting research in the Archives photo collection on another topic, we came across a folder labeled "California: Huron Cotton Camps." Markings on the photos led us to an article, "Migrant Work in California," in the missions insert of the January 28, 1957 issue of the Evangelical Visitor.
The article was a news items about a new Home Missions pastorate which began in October 1956. At that time, Carl and Marilyn Wolgemuth conducted their first service at a migrant labor camp in Fresno County, California. One of the larger cotton camps in the Huron community, "Camp 4," consisted of some 100 small quonset cabins and space for many trailers. The chapel was a double-size quonset formerly used as a cook-house.
In the article, the Wolgemuths reflect on the early endeavors in this mission project, the living conditions of the migrants, and the long-term prospects of the mission. Selections from the article appear below.
"Interest among teenagers in Sunday evening services led to a series of campfire services outside the chapel during November and early December. Singing choruses and gospel songs to the accompaniment of the guitar, the group are [sic] prepared to listen to testimonies and a short, pointed Gospel message at the close of the fireside service….
"Life is drab and tragically barren for the migrants. The privilege of sharing with the Mennonite Service Unit and others in a ministry to these people is much appreciated. The camp folks now have opportunity for activities under a Christian influence – after-school Bible Clubs with handcrafts for children, recreation for the teenagers, and sewing classes for mothers.
In recent months, several members of the Historical Society who are finding it necessary to "downsize" have offered to donate their complete set of Brethren in Christ History and Life to a young historian, church library, or anyone else interested in history.
For more information, contact Glen Pierce, director of the Brethren in Christ Archives by email ([email protected]), phone (717-691-6048), or postal mail (One College Ave., Suite 3002, Mechanicsburg, PA 17055).
Services were held in homes of members for the first ten years. The second photo shows Pastor and Mrs. Earl Sider standing in front of the church shortly after the congregation celebrated its twenty-fiftj anniversary of being in the building. | {
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After leaving several IAS officers disgruntled, Smriti Irani seems to have finally found a confidante in the bureaucracy. In her first year as HRD minister, at least seven IAS officers and her then PS had walked out amid rumours of her temper tantrums. The second PS lasted a tad longer, but decided to opt out. The person who seems to have survived the minister's mood swings is Imkongla Jamir, a 2002 batch IAS officer of the Karnataka cadre. The two ladies seemed to have hit it off so well that Irani had her transferred to the textile ministry.
Officials in the PMO are ducking for cover after an Income Tax raid revealed that a purported RSS member has been impersonating as a personal secretary to the PM. A cryptic SMS from the man citing Nagpur is all it took for top government officials to bend backwards. The imposter thus managed to get transfers and posting of government officials or dodgy business transactions done. What nailed the imposter finally was a tax raid on an erstwhile Delhi wheeler-dealer.
The ongoing cold war in Uttar Pradesh's ruling Yadav clan has worsened with the return of Amar Singh. Using his clout with Samajwadi Party supremo Mulayam Singh Yadav and his younger brother Shivpal Yadav, he is now nibbling his way into matters of governance. As it is, Akhilesh Yadav had moved heaven and earth to keep the wily Amar Singh out. Now the youthful chief minister has to bear with increasing interference from Amar Singh, who has managed to have a finger in every pie.
Ziona, the head of the world's largest family, with 162 members, has turned 72. The Mizoram resident is reputed to have 38 wives, 89 children and a large number of grandchildren.
The consumption of eggs in India is repor¬ted to have shot up 800 per cent between 1991 and 2013, making it one of the big winners of the economic liberalisation process.
Solar Impulse 2, the first sun-powered plane to encircle the globe without a drop of fuel, has completed 42,000 km across four continents, two oceans and three seas.
The manufacture of video cassette recorders, by one of the world's last known makers of the 1980s rage, the Funai Corporation of Japan. The VCR was launched 45 years ago. | {
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Томи Кархунен (фин. Tomi Karhunen; 29 октября 1989, Оулу, Финляндия) — хоккеист, вратарь.
Биография
Воспитанник хоккейной школы «Йокипоят», в составе которой прошёл всю вертикаль развития, вплоть до попадания в состав юниорской сборной, за которую отыграл на мемориале Глинки в 2006 году. В этом же году права на хоккеиста перешли в клуб «Кярпят». В 2007 году Кархунен был выбран на импорт-драфте клубом — «Сарния Стинг», в составе которого отыграл 39 матчей на уровне хоккейной лиги Онтарио, поссле чего вернулся на родину. После сезона в системе клуба «КалПа», в составе которого Кархунен выступал лишь за молодёжную команду, вратарь вернулся в «Кярпят».
В сезоне 2010/2011 Кархунен дебютировал на высшем, профессиональном уровне в SM-лиге. В сезоне 2011/2012 Томи Кархунен был отдан в аренду в словацкий куб из Братиславы «Слован», в составе которого стал чемпионом словацкой Экстралиги. В сезонах 2013/2014 и 2014/2015 Кархунен являлся основным вратарём «Кярпят» и дважды становился чемпионом Финляндии. Перед началом сезона 2014/2015 вратарь перешёл в другой финский клуб «Таппара», в составе которого вновь стал чемпионом Финляндии, тем самым заработав третий чемпионский титул подряд. Летом 2016 года Томи Кархунен перешёл в новообразованный, для участие в Континентальной хоккейной лиге, китайский клуб «Куньлунь Ред Стар». 5 сентября 2016 года, в домашнем матче против владивостокского «Адмирала», Томи дебютировал в КХЛ. Всего, в сезоне 2016/2017, вратарь принял участие в 31-ом матче (включая игры плей-офф), в которых одержал 13 побед. Новый сезон Томи начал также в составе «Куньлуня», однако, в конце октября 2017 года, вратарь был помещён в список-отказов. 6 ноября 2017 Кархунен перешёл в подмосковный «Витязь», в составе которого провёл на льду всего 3 матча. В своей третьей игре за «Витязь» Томи получил травму, после чего долгое время восстанавливался и в итоге, по обоюдному согласию, покинул команду. Зимой 2018 года Кархунен подписал соглашение, до конца сезона, с швейцарским клубом «Амбри-Пиотта». Сезон 2018/2019 Кархунен начал в составе шведского «Брюнес», однако, в декабре 2018 года принял решение вернуться в Китай, в состав уже знакомого ему «Куньлуня», который нуждался в усилении вратарской позиции. За оставшуюся часть сезона Томи провёл 12 матчей, в которых пропустил 40 шайб, при коэффициенте надёжности 88.3%.
Летом 2019 года Кархунен вернулся в родную Финляндию, и первую часть сезона провёл в составе клуба «Пеликанс». В конце ноября того же года першёл в швейцарский клуб «СК Берн». В сезоне 2020/2021 стал обладателем кубка Швейцарии. Летом 2021 года перешёл в немецкий клуб «Штраубинг Тайгерс». С февраля по май 2022 года выступал в составе австрийского клуба «Филлах».
Достижения
Чемпион словацкой Экстралиги в сезоне 2011/2012
Чемпион Финской Лииги в сезоне 2013/2014
Чемпион Финской Лииги в сезоне 2014/2015
Чемпион Финской Лииги в сезоне 2015/2016
Обладатель кубка Швейцарии в сезоне 2020/2021
Примечания
Ссылки
Профиль на сайте КХЛ
Хоккеисты Финляндии
Хоккеисты КХЛ
Игроки ХК «Йокипоят»
Игроки ХК «Кярпят»
Игроки «Сарния Стинг»
Игроки ХК КалПа
Игроки ХК РоКи
Игроки ХК «Слован» Братислава
Игроки ХК «Хокки»
Игроки ХК «Таппара»
Игроки ХК «Куньлунь Ред Стар»
Игроки ХК «Витязь»
Игроки ХК «Амбри-Пиотта»
Игроки ХК «Брюнес»
Игроки ХК «Пеликанз»
Игроки ХК «Берн»
Игроки ХК «Штраубинг Тайгерс»
Игроки ХК «Филлах» | {
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Аян () — мусульманин, який користується авторитетом у своєї громади. Є аналогом поняття деребей, що поширено було в Анатолії та Сирії.
Титул поширився на Балканському півострові у часи Османської імперії. У XVIII в Румелійському еялеті у провідними містами (насамперед на Дунаї) керували аяни, утворивши аянлики. На початку XIX століття Боснійський еялет був розділений на 39 районів, на чолі яких стояли аяни, титул передавався у спадок.
Примітки
Феодалізм
Суспільство Османської імперії | {
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{"url":"https:\/\/www.oxfordscholarship.com\/view\/10.1093\/0199251789.001.0001\/acprof-9780199251780-chapter-8?rskey=JuXxcr&result=17","text":"## Mario Diani and Doug McAdam\n\nPrint publication date: 2003\n\nPrint ISBN-13: 9780199251780\n\nPublished to Oxford Scholarship Online: November 2003\n\nDOI: 10.1093\/0199251789.001.0001\n\nShow Summary Details\nPage of\n\nPRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2019. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use.\u00a0 Subscriber: null; date: 23 October 2019\n\n# Networks, Diffusion, and Cycles of Collective Action\n\nChapter:\n(p.173) 8 Networks, Diffusion, and Cycles of Collective Action\nSource:\nSocial Movements and Networks\nPublisher:\nOxford University Press\nDOI:10.1093\/0199251789.003.0008\n\n# Abstract and Keywords\n\nUses simulation models to explore network mechanisms in diffusion processes and protest cycles. The network dimension is taken into account, focusing on three processes: information flows, influence flows, and the construction of joint action. The repeatable and reversible nature of protest requires models of diffusion that focus on the spread of actions and not the spread of ideas across actors. Moreover, while diffusion processes tend to generate waves or cycles of events, not all waves of events arise from diffusion processes. The effect of network structure varies greatly depending upon the nature of a particular network process.\n\nThis chapter shows how different \u2018network\u2019 arguments about how protest spreads imply quite different underlying mechanisms that in turn produce different diffusion processes. There is considerable ambiguity about the relationships among networks, diffusion, and action cycles and the way these can be identified in empirical data. We thus both seek to unpack the \u2018network\u2019 concept into different kinds of processes, and then show how these different network processes affect the diffusion processes we are studying. We sketch out some formal models to capture some of these distinctions.\n\nThis chapter extends recent work (Oliver and Myers forthcoming) that develops diffusion models of protest cycles, and focuses on discussing links between network concepts and diffusion concepts in understanding protest cycles. We conceive of social movements as diffuse action fields in which actions affect other actions and the action repertoires of the different actors coevolve through time and through interaction with each other. Movement activists and regimes engage in strategic interactions, each responding to the actions of the other. Different organizations within a movement respond to the actions of others, as successful tactical innovations and movement frames diffuse to new organizations. News media cover or fail to cover particular protests, and thus encourage or discourage future protests. Each of these processes affects the others, in a complex, multifaceted set of interactions. Over time, the action set of each actor evolves in response to the actions of the others and, thus, the whole field is one large coevolving environment in which the characteristics and actions of any actor is constrained and influenced by the characteristics and actions of all other actors in the environment.\n\nOne central concern about understanding diffusion and networks in protest waves is that we do not actually have straightforward data about the underlying social networks or mobilization processes. Protest event data usually just contain records of the timing and location of events along with some (often incomplete) information about the participants in the event, their forms of action, and their stated claims or other rhetoric (McAdam 1982; Olzak 1992; Kriesi et al. 1995; (p.174) Tilly 1995a\u2013d). Rarely, if ever, will the data contain information on the social relationships or communication processes that were involved in organizing and mobilizing that event. Lacking this kind of data, we want to know whether different patterns of social organization will give rise to different patterns in protest event data, and how what we already know about how protests get organized might influence our analysis of protest event data.\n\nAfter a brief review of the interplay between diffusion concepts and network effects, we develop some important distinctions among different processes often lumped together as \u2018network effects\u2019. We then develop preliminary models for three empirically important network processes in movements: the flow of information, the flow of influence, and the construction of joint action. All of these models are built on a core modelling \u2018engine\u2019 which we explain. Our models of information flow are most complex, as we stress on the importance of two kinds of networks: broadcast networks, and node\u2010to\u2010node networks. Finally, we show how the models we are constructing are capable of representing the strength of network ties, not just their presence or absence, and of permitting network ties themselves to evolve and be dependent on other processes.\n\n# Disaggregating Protest Waves to Get at Mechanisms of Diffusion\n\nThe ideas of cycles of protest, diffusion, and network effects are often discussed without making clear distinctions among them. Diffusion is the process whereby past events make future events more likely. In \u2018classic\u2019 diffusion models, there is a transmission of some innovation between people, and it is impossible to have any diffusion without some kind of contact or network tie between individuals. But this equation between networks and diffusion arises because of the assumption of permanent and irreversible \u2018adoption\u2019 in classic diffusion models, an assumption that is inappropriate for the diffusion of collective action (Myers and Oliver 2000; Oliver and Myers forthcoming). Individuals and groups or populations can and do protests or riot on multiple occasions, and the performance of an action by an individual or group often makes a repetition of that action more likely. One could insist on using the word \u2018diffusion\u2019 only when demonstrably different people are protesting or rioting, but this definition is problematic for at least two reasons. First, empirical data on protest events almost never contain sufficient detail to distinguish clearly between new actors and repeaters. If repeated events of the same type occur in the same geographic area (e.g. riots), the rioters are quite likely a mixture of previous and new participants. Available data generally provide only numerical counts of numbers of participants and perhaps the names of a few key leaders. They would never provide sufficient detail to track exactly how many new people are entering a form of action and where they came from. Data of that level of detail are only available in detailed case studies of well\u2010structured events, (p.175) not in data across a large number of events or more amorphous events. The second reason is theoretical. The reinforcement process, whereby an actor's own actions and its consequences influence that actor's future actions, is theoretically almost identical to a diffusion process, whereby one actor's actions and their consequences influence other actors' future actions. Most of the same processes and factors are involved in the repetition of actions by the same actors and the adoption of actions by new actors. Either way, the \u2018diffusion\u2019 effects of an action are mediated by whether the action is repressed, whether it gets media coverage, whether it affects policy, and so forth. The only difference is that actors presumably know about their own actions and its immediate consequences, while others cannot be affected by a group's actions unless they know about them. Only the \u2018network processes\u2019 themselves are different between self\u2010reinforcement and diffusion to other actors. Because protest is a repeatable, reversible action, diffusion models of protest must focus on the spread of actions, not the spread of actors (Myers 1996, 1997, 2001).\n\nAn additional distinction needs to be made between diffusion and cycles. Diffusion processes tend to generate waves or cycles of events, but not all waves of events arise from diffusion processes. Waves of protest can also arise from rhythms and from common responses to external events. A major event such as a disaster or an act of war may trigger independent responses in many locales. Rhythms are what the term \u2018cycle\u2019 most often means in other contexts, periodic rhythms of physical or social life that structure time. The ordinary rhythms of life structure protest just as they structure any other activity, so that protest generally occurs when people are awake and around the constraints of work, school, and political schedules. Beyond these quotidian rhythms are the rhythms of protest itself. There is a recovery or regrouping interval after most actions before a group is ready to act again. At a minimum, people must eat and sleep. Big events such as marches on Washington necessarily require relatively long intervals between them for organizing the logistics. Ritualized protests are often held at regular intervals. The presence of rhythms and external shocks does not, however, mean that diffusion processes are absent. Empirical research has often demonstrated diffusion processes in the spread of information about a major event (Shibutani 1966) and Myers (1996) found clear evidence of diffusion effects within the \u2018long hot summers\u2019 of the 1960s riots and after the assassination of Martin Luther King, Jr.\n\nFinally, we need to recognize the importance of diffusion processes nested within other diffusion processes. Long multi\u2010year protest waves are the accumulation of smaller protest waves arising from particular campaigns and the smaller\u2010scale diffusion processes that occur within them. McAdam (1983) showed that the bursts of activity in the civil rights movement followed tactical innovations. The diffusion of collective action across national boundaries also shows evidence of waves within waves, a general wave of mobilization that transcended national boundaries, and nation\u2010specific waves (Kriesi et al. 1995). Similarly, a broad social movement is always made up of smaller campaigns in particular localities or involving particular issues. These smaller campaigns usually arise either from (p.176) a burst of repeated actions by one group or in one locality, or the diffusion of a particular movement issue, frame, or tactic between groups or localities. The term \u2018network\u2019 is often used in both cases, but in the former, it tends to refer empirically to the existing social and political ties within a community that permit a set of people to act in concert, while in the latter, it refers empirically to communication channels through which information is spread between different local networks.\n\nSpecifying these nested diffusion processes is theoretically critical, as it is clear that big protest waves are built from smaller campaigns that have their own logics, while influencing each other in the larger wave. These campaigns implicate network processes. A wide variety of network forms are involved in campaigns. The most basic is a series of events around the same issue involving the same people in a single locale. If no new people are brought in, this is a simple case of repetitive action by the same actors, a pure \u2018reinforcement model\u2019 process, in which the consequences of earlier actions influence the rate of subsequent actions by these same people, but there is no interpersonal diffusion process involved. However, if these events become larger over time, then we would say that some kind of between\u2010person diffusion has occurred. Of course, even if the number of participants stays constant, there could well have been turnover in the participants. We have developed an approach that is capable of being modified to capture these waves within waves, but we will not be developing such modifications in the scope of this chapter.\n\n# Specifying Network Effects\n\nAs we dig into the mechanisms of diffusion, it is important to specify the very different kinds of \u2018network\u2019 relations that are involved in different kinds of diffusion. A very wide range of specific phenomena has been lumped together under the rubric of \u2018network effects\u2019 or \u2018social ties\u2019. If we are going to understand the role of network effects in diffusion, we need to unpack the concept. There are at least three distinct (although related) processes that occur through network ties: communication, influence, and joint action. The relation among these three processes is somewhat hierarchical. A communication tie provides a basis for disseminating information that something has occurred. An influence tie provides a basis for one actor to affect the opinions or actions of another actor; influence requires communication but involves additional social processes beyond mere communication. Joint action may be considered an extreme case of influence, in which initially separate actors come to make joint decisions and act in concert. Influence requires communication, but not all communication entails influence. Joint action requires both communication and influence. It is important to recognize the concept of joint action because empirically researchers may not be able to distinguish multiple acts from concerted joint actions. Many protest event series exhibit huge \u2018spikes\u2019 in which a very big action \u2018suddenly\u2019 occurs or many different actors (p.177) \u2018suddenly\u2019 engage in the same kind of action at the same time, and these spikes cannot possibly be modelled with standard diffusion models. However, we will show that a model of \u2018hidden organizing\u2019 outside the view of the data collectors can quite readily model such spikes. This chapter will provide detailed discussion of some of the issues involved in each of these three kinds of processes, and outline some approaches to formal modelling of each. In each case, we will give special attention to the question of how the process might be reflected in observed empirical data on protests. However, before moving to these three sections, it is important to consider some other distinctions and dimensions among network processes.\n\n## Dimensions of Proximity or Connection\n\nInformation and influence flow through social networks. But there are different ways in which actors can be \u2018connected\u2019. There are at least three dimensions to network proximity that are relevant to the study of social movements: spatial, organizational, and other social. These may be expected to play different roles in protest and social movements.\n\nSpatial\/social: Movement actions are space\u2010bound: people often congregate in the same place at the same time to act in concert. Riots and \u2018spontaneous\u2019 protests most often diffuse spatially: individuals become aware of the riot or protest because they are near it. However, there is no \u2018pure\u2019 space, and space itself is always socially organized. Neighbourhoods are usually segregated by class, ethnicity, or race, and are often segregated by political orientation, so that different \u2018kinds\u2019 of people are found in different kinds of public spaces. Social etiquette rules about class or ethnicity or gender, as well as language differences may create communication barriers that are the practical equivalent of great distances. A wide variety of routine social structures can create network ties. For example, Oberschall (1989) shows that early sit\u2010ins in North Carolina after the first Greensboro sit\u2010in diffused as black colleges played basketball games against each other. The mass media also have a decided spatial component. Mass media have clear geographic and linguistic catchments. Although there is \u2018national\u2019 news, which is usually broadly available, that \u2018national\u2019 news always has a bias toward events occurring near the site of publication or broadcast (Mueller 1997; Myers and Caniglia 2000). Myers (2000a) found for example that although large riots diffused nationally, presumably by way of national news coverage, smaller riots diffused within the boundaries of television broadcast ranges. Prior to electronic communication, collective disturbances diffused along transportation routes and took longer to diffuse (Rud\u00e9 1964; Hobsbawm and Rud\u00e9 1968; Charlesworth 1979; Myers 2000b).\n\nMovement\/organizational: Even within spaces, the participants in particular actions usually have additional ties to each other beyond mere proximity. Between spaces, actions may be coordinated through political\/movement ties between movement organizations. Local chapters of the same national organization would (p.178) be expected to have high political ties. Different organizations with similar political\/movement goals would tend to have positive ties, although they would also have some elements of competition between them. There obviously has to be some actual mechanism of communication between spatially dispersed elements of the same organization, such as organization newsletters, or telephone calls or e\u2010mail among members. But these actual mechanisms of communication are most often invisible to the protest events researcher, who merely notes that events were organized in five different cities by local chapters of the same organization.\n\nRelational\/social: Movement organizations may have ties to nonmembers through their members' \u2018other\u2019 social relationships and memberships. These other ties include kinship and friendship, attendance at the same school, membership in the same recreational club or religious congregation, employment at the same workplace, or membership in some secondary association that has no direct relation to the movement. In many cases, these \u2018other\u2019 ties become the basis for recruitment into a movement organization or its actions, as well as for increased support for the movement's opinions (Ohlemacher 1996). Movements whose members have social connections to the larger society through many different social ties are likely to be better able to mobilize support than those that lack such ties. However, as we consider influence models below, it will become apparent that these external ties can have both \u2018positive\u2019 and \u2018negative\u2019 effects on movement mobilization.\n\nIn the work that follows, we will not be able to explore the effects of these different kinds of proximity, but have set up general schemes that should be able to capture the structures that the different kinds of relations would imply.\n\n## Sizes of Networks and Numbers of Actors\n\nIf we are looking at the total numbers of participants in collective action, we often conceive of the network diffusion as reaching down to individual people. But it is well established that most people enter protest movements as parts of relatively cohesive groups, and that whole groups make decisions together about whether to participate in particular actions. This means that it is often most reasonable to think of the \u2018actors\u2019 as groups, not individuals. But when this is so, we will then also want to be able to consider the \u2018size\u2019 of each of these actors, which is the number of people it mobilizes. Although capturing this complexity in its totality is beyond the scope of this chapter, we will discuss how our models can be modified to deal with group size issues.\n\n## Network Structures and Collective Action\n\nNetwork theorists have devoted a fair amount of attention to measuring and categorizing qualitative differences in network structures, as well as quantifying the position of any one actor in a qualitatively\u2010defined network (Knoke and Kuklinski 1982; Wasserman and Faust 1995). The same number of ties in a network has (p.179) different effects depending upon their distribution, so that star\u2010like structures in which one central person has links to other actors who have no links to each other are, for example, quite different from circles in which each actor has exactly two ties to other actors and all actors are connected. Similarly, cliques can be defined within larger networks. Unless one wants to stay at the level of the case study, however, it is difficult to use these concepts in the study of the diffusion of collective action across a large and complex population. Instead we need to have summary measures of a movement group's network ties. In this chapter, we will give some simple examples of how structural effects can be incorporated, but will not pursue this dimension in any depth.\n\n## The Basic Model\n\nIn this model, each actor has a probability pk of acting at each time period. The number of people who actually act at each time period varies stochastically around the mean pk, where N is the number of actors. Each actor's pk may change across time as a function of the past actions of themselves or others. Elsewhere (Oliver and Myers 2001), we explore the question of the form of the underlying model for the diffusion of collective action. Plausible models for mobilization cycles that go up and down are not straightforward. Collective action always declines, and the question is whether this should be specified as arising from a natural tendency within actors that occurs regardless of outside influences, or whether it is a process of outside factors such as repression. Addressing these questions is beyond the scope of this chapter. Here, we will simplify the individual decision model and focus only on the upswing or accelerative phase (Oliver et al. 1985) of a protest cycle, where the feedback effect from others' actions is entirely positive. This underlying model does not produce event distributions that look like real protest cycles, which always come down again, but it will give us a basis for evaluating network effects.\n\nModels in this chapter are developed using the Stella simulation programme from High Performance Systems, Inc.1 The program has a graphical interface to represent differential equations. An appealing feature of Stella is that it generates a list of the equations implied by the graphical connections.2 The programme can handle one\u2010 or two\u2010dimensional arrays with sizes constrained only by the capacity of the computer. The acting probability and other characteristics of each actor are captured by one\u2010dimensional arrays, while network links and inter\u2010actor influences are captured by two\u2010dimensional arrays. The programme accepts hot links to inputs and outputs, so it is possible to set up a who\u2010to\u2010whom matrix of network linkages in a spreadsheet that can be read by the programme. All of the models in this chapter could readily be programmed in some other way, but we have found Stella to be a very useful development tool as it hugely reduces the ratio of programming to thinking in the process of model development.\n\n(p.180) For analysis, we have set up several fixed network configurations as well as a random network controlled by a random number generator and can choose between network configurations with a user\u2010controlled switch. For this chapter, the arrays are fixed at size 10, which is large enough to show some of the effects of random variations, but small enough to be manageable in a development phase. Substantively, an N this small could be understood as actions in different cities or by different groups in a movement. Representing a city of a million inhabitants as a matrix would tax our computer systems and be unlikely to be informative. The more reasonable way to proceed for representing large populations is to conceive them as subgroups with varying sizes, where the group's size is another variable in the model. Such an extension is beyond the scope of this chapter.\n\n## Baseline Model With No Communication\n\nFor baseline comparisons, we begin with a group of N actors who have no awareness of each other. Each group may randomly emit an action. We tally the plot of all actions. Initially, we have all actors with the same low probability. Because actors do not influence each other, this probability does not change. Because of its random component, each iteration of this model produces a slightly different outcome plot. Figure 8.1 shows plots of the baseline model for a system of 10 actors. Even though there is a constant probability of action, because it is a random model, there are varying numbers of actors at any given time, and the frequency plot exhibits a spiky sawtooth form with waves typical of protest event plots. The cumulative count, however, shows a different story: in a purely random model with a constant probability, the total rises essentially linearly with time. We will be using the total counts across five periods in subsequent models because they damp out some of the random variations of one\u2010period counts. These five\u2010period counts are roughly equivalent to the kinds of patterns you would get if you aggregate daily event counts to weeks, or weekly counts to months. This is shown in the bottom panel of Fig. 8.1. Note that this purely random process generates cycles and even small diffusion\u2010like S\u2010curves in the cumulative count.\n\nTo model information diffusion effects, we have to provide some specification of how one actor's probability of acting is affected by the actions of others. Here, we will assume that the tendency to repeat this action is a function of how many others are doing it. Although verbal theorists can relax into vague discussions of positive effects, and even quantitative empirical researchers can just specify a regression coefficient on the lag of prior action, when we write a mathematical model, we have to say exactly how we think people respond to others' actions, and this is not at all clear from empirical research. Shall we assume that others' actions always increase our own probabilities, no matter what? And, if so, in what functional form? Linearly in a power relationship? With rising and then falling marginal returns? Or should we assume that actors respond not to the absolute level of others' actions, but to whether it is increasing or not? The former assumption, that (p.181)\n\nFig. 8.1. Random processes produce apparent cycles. Top panel is number of events per time period, bottom panel is 5\u2010period moving average (used in other models)\n\nactors respond to the level of others' actions, would arise if there is an accelerating production function or if actors' behaviour is principally determined by influence or imitation processes. However, in the long run, such models produce unanimous action in which everyone is protesting with certainty forever, something that never happens. The latter assumption, that actors respond positively to the increases in others' actions, and negatively to decreases, would arise from an S\u2010shaped production function that first rises then falls, which seems consistent with an underlying process in which initial action obtains benefits, but there are declining marginal returns to action after it has been at a given level for some time. Our initial work with this second model indicates that, while interesting, it produces volatile results that are very sensitive to initial conditions, which makes (p.182) it unsuitable as a platform for investigating network effects.3 For this reason, in this chapter we use models employing the assumption that actors respond to the level of prior action.\n\nThe model we use assumes that actors respond to the total level of others' action in a diffusion\u2010like fashion. The basic elements of this model are: pt= probability of acting at time t; n = number of actors\u2014a random process determines whether each actor actually acts on a given trial; rk (t) = recent total number of actions across all actors within the past k trials, at time t; and k = the number of trials considered.\n\nThe algorithm for changing the probability of action as a function of past actions is\n\n$Display mathematics$\nwhere w 1 is a weighting coefficient on the feedback term. Actors simply respond to the total of others' actions, which means that \u2018full information\u2019 is assumed so that there are no network effects. This simple model produces an S\u2010shaped growth in the probability until a probability of 1.0 is reached, when it stabilizes with everyone acting. The weighting factor determines how quickly this happens; if the weighting factor is small enough relative to the time span of the model, the probabilities may remain essentially unchanged for the duration of the model. The distribution of current action exhibits random variation around an S\u2010shaped rise until unanimous action is reached; unanimous action is an absorbing state. The cumulative distribution is S\u2010shaped until unanimity is reached, and thenceforth rises linearly. In Fig. 8.2 we show examples of the effect of feedback from others' actions in this algorithm. The plot of cumulative protests clearly shows the S\u2010shaped growth pattern diagnostic of a diffusion process in the first phase, until unanimous action is achieved, and then it becomes a linear curve like any other constant\u2010probability model. We have parameterized the baseline model so that it has a low level of action if there is no feedback and a relatively rapid rise toward unanimity if there is 100 per cent feedback through all possible network ties. This will give us a backdrop against which to consider the effects of various network constructs. The upper panel shows the current action rate as well as the cumulative event count and the probability for a homogenous group in which everyone's initial probability is 5 per cent and the feedback weight is 0.005. The lower panel provides two variants of the initial probability of action. In the homogeneous case, all actors begin with a 5 per cent probability of acting; in the heterogeneous case, actor 1 has a 40 per cent chance of acting, while the other nine actors each have a 1 per cent chance. The average probability is about the same in the two cases. The lower panel compares the homogeneous and heterogeneous cases for the full feedback and zero feedback models. When there is no feedback, the heterogeneous group has slightly more action, due to the one high\u2010probability actor. When there is full feedback, the heterogeneous group reaches unanimous action a little more slowly than the homogeneous group. (p.183)\n\nFig. 8.2. Diffusion in networks with all ties. Top panel shows diffusion in a homogeneous group (all have p0=.05) with all ties present. Bottom pane contrasts all ties with no ties for homogeneous and heterogeneous groups (one actor has p0=.4, all others p0=.01). Feedback weight is .005.\n\n# Information Flows\n\nWhen ideas or actions are diffusing between actors, the \u2018thing\u2019 that is transmitted is information. Broadly speaking, there are two types of networks through which information may flow, node\u2010to\u2010node and broadcast. Node\u2010to\u2010node paths are the kind usually implied by the use of the term \u2018network\u2019. Actor A communicates with actor B, who communicates with actor C, and so forth. Many network analysts examine the efficiency of communication across node\u2010to\u2010node networks with different properties, such as overall density of ties, the tendency to cliquing, or the extent to which communication is channelled through a few key actors. By contrast, a broadcast network involves a single communication source that is directly (p.184) received by a very large number of people. In our era, this is the mass media. But previous eras also had broadcast communication on a smaller scale, in the form of town criers and travelling messengers.\n\nAlthough hard\u2010core network analysis focuses on the effects of network structure and chains of indirect ties (Knoke and Kuklinski 1982; Wasserman and Faust 1995), any \u2018network\u2019 analysis of communication in protest waves in the modern era is sterile if it does not treat the mass media. Large numbers of people who otherwise have no connection at all can be \u2018connected\u2019 by their responses to a common news or entertainment source. When the actions of one group are covered in the mass media, communication effects can spread as far as the media are broadcast, without prior connection between the actors. Myers (1996, 2000a) shows that large riots that received national television coverage increased riot propensities nationally, while smaller riots increased riot propensities within their local television catchment areas. Protest event data based on newspapers, especially if it is drawn from a single \u2018national\u2019 news source, is, by definition, data on the events that can be assumed to have been communicated to a broad population.\n\nBut, of course, the news media are not unbiased samplers of events. They are rather intentional actors who select news stories for reasonably well\u2010defined reasons, and it is well established that the size and disruptiveness of an event increase its probabilities of news coverage, as does the proximity of the event to the news organization (Snyder and Kelly 1977; McCarthy et al. 1996; Mueller 1997; Myers and Caniglia 2000). More recent research also suggests that news media cover some kinds of issues much more than others (Oliver and Myers 1999; Oliver and Maney 2000). The media themselves are subject to diffusion processes, both within one news organization, and between them. If a news organization has already published several stories about a particular issue, it is less likely to publish another because it is not \u2018news,\u2019 although there is some evidence that for at least some issues, the recent publication of one article about an issue will raise the probability of another article about the same issue, as the news organization follows the \u2018story\u2019. Between news organizations, once one outlet picks up a story, other outlets may pick it up. If enough outlets begin to cover the story, it becomes news, and the media will begin actively seeking more stories on the same theme. The result is the \u2018media attention cycle\u2019 which has been shown to under\u2010represent movements at the beginnings and ends of their cycles, and over\u2010represent them in the middle, when the issue is \u2018hot\u2019 (Downs 1972; Cancian and Ross 1981; McCarthy et al. 1996).\n\nEven though the mass media play a central role in our era, node\u2010to\u2010node networks are also important. Social ties between groups increase and deepen information flows beyond the information presented in the mass media, as posited in the classic \u2018two step\u2019 model for media influence on attitudes. Social influence appears to flow principally through social connections, not the mass media, so that we expect information coming only through news sources to be much less effective in changing opinions and orienting people toward action than information coming through social ties.\n\n(p.185) In the real world, patterns of diffusion and the ways diffusion uses different networks are messy, to say the least. In fact, the different kinds of networks patterns not only operate at the same time, but also are affected by one another. Recently, a number of scholars studying media coverage of protest and demonstrations have noted that larger events are more likely to get media coverage\u2014and more of it (Snyder and Kelly 1977; McCarthy et al. 1996; Mueller 1997; Oliver and Myers 1999; Myers and Caniglia 2000; Oliver and Maney 2000). This means that the larger a protest group's local network is and the stronger the ties in that network, the larger its events will be and the more press coverage it will receive.\n\nWhen the press covers a protest event, the protest issue and tactic are projected to other potential actors thereby invoking a completely different kind of network. In this way, recruitment through personal networks can piggyback on media coverage. Even if activists in one city have no direct communicative ties with activists in a second, they may be inspired to invoke their local network to produce an imitative event once they hear about the first event through the mass media. Thus the media operates directly through its distribution network to mobilize additional individuals to join existing protest groups and it can also invoke networks indirectly by mobilizing a node in a different activist network that will activate its local network.\n\nOther carriers of diffusing protest also interact with local networks and the media to reinforce and extend their influence. For example, some protest has been tied to travelling activists who give speeches or engage in direct attempts to organize. These activists do not just wander aimlessly, but select targets based partially on the likelihood that their efforts will be successful\u2014as indicated by some level of local organization which has the network ties to support the protest activity. Indeed, these activists may even be called upon by existing organization to come and help rally the troops. Furthermore, media coverage of the speeches and meetings helps to draw new recruits into the fold of potential activists and the ensuing actions give the media more to report.\n\nThe messages delivered to individuals by their personal contacts and by the media can also reinforce each other during the critical time when the individual is presented with an opportunity to decide whether or not to act (Oliver 1989). If, when approached by a friend or colleague and asked to act in support of civil rights, and the recruit has recently been watching the news about church burnings, that recruit may be more likely to respond to the personal network. The importance of the cycles of influence among distinct kinds of networks cannot be ignored.\n\n## Modelling Network Ties With No Media Coverage\n\nSuppose we have a taboo issue that the news media refuse to cover. Or, perhaps, instead of being \u2018taboo,\u2019 it is one of those positive and uplifting kinds of action which lack news value because it is not conflict\u2010oriented and not linked to institutional politics (Oliver and Myers 1999; Oliver and Maney 2000). To add network effects to the baseline model we create a who\u2010to\u2010whom network matrix with entries that are zeroes or ones. A matrix with all 1's is the \u2018everyone affects everyone\u2019 model and produces the same results as a model in which people's actions are affected by totals. Conversely, a matrix with all 0's produces the same result as the independent probabilities model. Because the underlying model is a growth model, where there is no decline, if actors are influenced by others' actions (or their own) there is a gradual increase in the probability of action and, thus, in the average level of total action, but the rate at which the action increases is a function of the density of communication. Between the \u2018full information\u2019 model and the \u2018no information\u2019 model lay the models in which there are some connections between actors. Theoretically, it is important to specify whether the diagonals are 1s or 0s, that is, whether people increase their action as a function of their own actions as well as of others' actions, but exploring these subtleties is beyond the scope of this chapter.4\n\nThis model can be used to assess the effects of varying network structures. Because it is stochastic, even for exactly the same determinate who\u2010to\u2010whom relationship matrix, there will be different results on each iteration of the model, depending on random fluctuations in exactly who acts when. We may use Stella's ability to use a seed for the random number generator to fix this process and compare network structures. Figure 8.3 compares one random and three fixed structures including a \u2018star\u2019 network in which all ties are through actor, a cliqued network in which all ties are within cliques (1, 2, 3 vs. 4, 5, 6 vs. 7, 8, 9, 10), and a bridged cliqued network in which there is an additional tie between 3 and 4, and (p.187)\n\nFig. 8.3. Comparison of different patterns of network ties in the diffusion of action for heterogeneous and homogeneous groups\n\nbetween 6 and 7. In this model, different random networks vary widely in their results, and the variability of results due to network ties is even greater when the initial probability distribution is heterogeneous. The particular random network in this figure is slightly more effective than the bridged clique network, which in turn is slightly more effective than the fully cliqued network. The \u2018star\u2019 network in this example fares little better than no feedback at all: this arises because the non\u2010stars only have information about one actor's actions and so the total level of action is too small to lead to much increase through feedback. In an influence model, shown later, a star can have a much bigger impact on action.\n\nThis approach can be readily generalized to much larger network matrices (e.g. 100 \u00d7 100), but these are quite difficult to analyse without prior theory of what (p.188) kinds of structures are relevant or interesting. Obviously, the approach of using a full matrix of who\u2010to\u2010whom ties becomes computationally impossible with very large groups such as the tens or hundreds of thousands in city populations, and seems most appropriate for modelling the relationships between groups.\n\n## Modelling Protest and the Media\n\nProtesters generally seek news coverage as the mechanism for having influence on a wider public and the authorities. Protests that receive no news coverage are often construed as failures. Protests that receive news coverage are likely to be invigorated, and activists are likely to prolong their activism and emit more total protests if they have received news coverage. But, of course, the news media do not cover all protests that occur, and their coverage is dependent on the amount of protest. There are \u2018media attention cycles,\u2019 which are diffusion cycles: news media tend to ignore a protest campaign in its small initial phases and then, when they do begin to cover it, there is a flurry of coverage for a while until it becomes \u2018old news,\u2019 and then coverage dies down again.\n\nAdding media effects into a model requires specifying how the media work. This is a complex problem, which will need to be the subject of a separate analysis. We need to consider both how the media affect protest, and how protest affects the media. In this chapter, we will assume that the media are simply a channel of communication, so news coverage of events affects protest by conveying to actors information about the protest rates of others. This means that we will assume that media coverage acts just like full feedback or network communication, in terms of the algorithm for the effect of others' actions on an actor's probability of acting. In terms of the relation between protest and the probability that the protest receives news coverage, there is some information from recent empirical work. We know that there are issue attention cycles that may be functions of factors exogenous protest, or may be set off by protest; an issue attention cycle raises the probability that an event will be covered. In addition, we know that the probability of an event being covered increases with its size, and recent large events may draw a higher rate of coverage to immediately subsequent events. There are also \u2018news hole\u2019 effects, so that there is a limit on the amount of action that can be reported on one day. Myers and Caniglia (2000) found, for example, that the New York Times under\u2010reported riots at the peak of a riot cycle: even though they reported that there was a lot of rioting going on, any particular riot was less likely to be mentioned when there were many riots happening.\n\nIn this chapter, we cannot provide a full analysis, but illustrate a possible approach to such a problem by showing the effects of several kinds of media factors separately. We begin by showing the effect of a flat percentage of news coverage on the rate of \u2018adoption\u2019 of action compared to full information. Figure 8.4 shows the rate of action diffusion with news coverage at a constant 50 and 20 per cent probability as compared with the full information model (equivalent to (p.189)\n\nFig. 8.4. Media coverage as flat percentage provides communication, which promotes event diffusion. There is little difference between heterogeneous and homogeneous groups.\n\n100 per cent probability of news coverage). In this initial model, the specification is that the news media has a single probability of news coverage. If it \u2018covers\u2019 action at all, it covers all the action that is occurring on that round. A more detailed specification would say that the media could be differentially sensitive to different actors, so that actors could have different probabilities of coverage or that different proportions of those acting on a round could be covered. That would yield different patterns of results.\n\nFigure 8.5 shows how the diffusion of action is affected when the probability of news coverage is not a flat percentage, but increases with the size of the action, for example, the number of actors. The \u2018functional\u2019 relation is parameterized so that actions involving all ten actors have a 50 per cent rate of coverage, while the probability for smaller actions is proportionately smaller. This dependence of news coverage on event size markedly slows the spread of action.\n\nIn most research, newspapers are the source of data and thus only news coverage of action is empirically observable. Figure 8.6 shows both action and news coverage of action when the probability of news coverage is a constant 50 per cent (upper panel), and when the probability of news coverage is 50 per cent for the largest actions (involving all ten actors) but is proportionately lower for smaller actions. Two patterns are clear is these figures. First, if the probability news coverage is proportional to the size of the events, diffusion is delayed relative to a constant probability of coverage, because the earlier smaller events (involving just one or two actors) are less likely to get news coverage. Additionally, the apparent level of protest from news coverage is even lower than the actual level, due to the lower probability of coverage. Second, note that the cycles of news stories differ (p.190)\n\nFig. 8.5. Comparison of action diffusion when news coverage rate is 100%, 50%, 20%, or a function of number of actors. Homogeneous groups.\n\nFig. 8.6. Comparison of actual event series with events reported in the news. Top panel shows a flat 50% coverage rate. Bottom panel shows coverage proportional to the number of actors. Homogeneous groups.\n\n(p.191)\n\nFig. 8.7. Independent issue attention cycle can distort apparent protest cycle. Dashed line shows actual events, solid line shows events reported in the news.\n\nmarkedly from the cycles of action. This is especially true when the probability of coverage is a function of event size. But even after action has reached unanimity, random fluctuations in news coverage give the appearance of protest cycles where there are none. However, in both these cases, news coverage does successfully track the difference between high\u2010action and low\u2010action periods.\n\nThere is substantial reason to believe that the news media's probability of covering protest is often determined not by the characteristics of the protest, but by external events or political cycles (Oliver and Maney 2000). In Fig. 8.7, the probability of news coverage is exogenously determined as a sine function, that is, a wave that goes up and down independently of protest levels. As before, past news coverage of protest raises future protesting. In this example, there is an early news cycle that helps to spark a diffusion process. Then the news coverage dies down while the protest is still rising. Coverage comes and goes again later when action is unanimous. Because very often the news coverage of protest is the only \u2018data\u2019 we have about protest, it is very important to recognize how easy it is for news cycles to be unrelated to protest cycles, and it is obviously important to do a more detailed study of how protest and news coverage relate to each other.\n\n# Influence\n\nThere are many network theorists working on influence models which assume that people's attitudes are shaped by those of the people to whom they have network ties, and in particular that the degree of influence will be affected by the homogeneity\/heterogeneity of the opinions in the networks to which one is tied. If virtually all of one's acquaintances share the same political perspective, one's mobilization level or attitude extremity will be greater than if one's acquaintances (p.192) vary in political perspectives (Pfaff 1996; Kim and Bearman 1997; Soule 1997; Van Dyke 1998; Chwe 1999; Sandell 1999). This suggests that there is an interesting dynamic in the way networks affect mobilization. The same factors that create higher influence (all one's acquaintances are similar) are likely also to reduce the extent to which a group has network ties into nonmovement organizations. Thus relatively closed, politicized networks tend to increase diffusion through self\u2010reinforcement processes, while relatively open networks have more potential to foster diffusion through mobilizing new participants, although the force of such a diffusion effect is likely to be weaker. Of particular concern is whether a group is relatively inbred, with ties only to itself or to other movement groups, or whether it has ties out into the general population of people who are not already mobilized. For example, Ohlemacher (1996) develops the concept of the social relay to distinguish the networks in two communities, one in which the protesters were relatively isolated, and the other in which protesters had substantial ties to non\u2010protest organizations in the community: the relatively isolated protesters were viewed as more radical and failed to generate a broad mobilization, while the protesters with substantial non\u2010protest ties built a broader, less marginalized, mobilization.\n\nWe may begin to model these processes by adapting Gould's (1993a,b) influence model, in which each person's probability of action is affected by the average of the action level of all the others to whom she\/he is tied. If there are zero network ties, each person's probability stays the same; if there are 100 per cent of all possible network ties, everyone's probability fairly rapidly converges to the same probability, with the initially higher probabilities dropping and the initially lower probabilities increasing. If we put a simple who\u2010to\u2010whom matrix in this system, network ties affect the speed with which these processes occur, but not the final outcomes. We can see how this works by setting up a two\u2010clique network with radically different initial values of opinions. If the cliques are completely unconnected, they will each reach their own equilibrium, as in the top panel of Fig. 8.8. Here, then, we have the gap between the isolated radical terrorist cell, for example, and the larger population. The radical cell can maintain its radicalism, but at the cost of having no influence on the larger population. If there are any bridges between the networks, however, influence will \u2018leak\u2019 across the system and the two cliques will move toward each other and will ultimately reach system\u2010wide equilibrium, as in the middle panel of the figure. However, the move toward equilibrium can take quite a while to happen and, in the mean time, there can be radical disjuncture between subnetworks. These two cliqued cases may be compared with the bottom panel, which shows how one random network fairly rapidly converges to a system\u2010wide equilibrium. In this particular case, it happens that one actor has no ties to other actors and so remains unchanged while everyone else converges toward equilibrium.\n\nNetwork analysts usually treat the structure of network ties as fixed and unchanging. But, of course, movement actors devote a great deal of effort toward (p.193)\n\nFig. 8.8. Influence processes in cliqued, bridged and random networks. Network ties fixed. (In the random network, one actor happens to have no ties to others.)\n\n(p.194) creating new ties, and even the less planned forms of social interaction create new ties. In a formal modelling approach, it is quite feasible to make the ties themselves change over time in response to prior interaction. We may demonstrate this with a modified influence model. Instead of fixed present\/absent ties, we begin with a who\u2010to\u2010whom matrix in which each entry is the probability that two actors will come into contact and influence each other. In this model, a matrix of 0,1 network ties is generated on each round probabilistically as a function of the given probabilities of influence. In addition, we add a feedback to these probabilities so that if a contact actually occurs (i.e. if there is a 1 in the matrix, even if it arises from a low probability of occurrence) that contact raises the probability of future contact by a given amount. To demonstrate how this model works, we set up an\n\nFig. 8.9. Network ties are probabilistic and can be increased by contact. Each clique of five actors initially has 50% probability within clique and 5% between cliques. Upper panel shows opinion convergence as actors come into contact. Lower panel shows that the average probability of contact gradually rises across many time periods\n\n(p.195) input matrix with two cliques, each of which has a 50 per cent probability of making contact within the clique and only a 5 per cent chance of making contact between cliques. As before, we give the two cliques widely different starting values on the opinion measure. As Fig. 8.9 demonstrates, this model also generates convergence toward an equilibrium value, although it happens more slowly and with random fluctuations around the trend. As the bottom panel of Fig. 8.9 indicates, the average overall density of ties within the network gradually increases as well, approaching saturation as a limit. The irregular shape of the plot exhibits the influence of the cliquing. There is an initial rapid increase in the average contact probability arising from increases within cliques. After this phase, there is a classic S\u2010shaped diffusion curve arising from the gradual increase in the probability of contact between cliques, which accelerates in the middle of the process, and then slows again as the network approaches saturation.\n\n# Joint Action\n\nAn important phenomenon in any sphere of social action is that individuals come together to form collective actors, and smaller collective actors come together to form larger collective actors. When people organize themselves into groups, they do not show the random patterns of individuals acting independently, but very different patterns that arise from coordinated action. In evaluating protest event data, it is important to recognize that the \u2018actors\u2019 producing the event plots can be of widely different sizes and, in addition, can often be shifting around, grouping and regrouping themselves into temporary coalitions and alliances. No existing models of the diffusion of action have addressed the ways in which these patterns affect the observable event distributions. We cannot provide a detailed analysis of this problem, but we present here one example of it in the empirical data, and show how that kind of phenomenon can be modelled.\n\n## Movement Networks and the Problem of Protest \u2018Spikes\u2019\n\nThe typical protest wave is more \u2018spiked\u2019 than standard diffusion models can possibly capture. That is, the empirical waves rise and fall much more quickly than can be accounted for by models of interactor transmission. One possible explanation for this pattern is that much of the protest event data is drawn from media sources and the attention cycle bias makes the peaks of action appear more extreme than they are. Another reason may be the failure to account for repeated actions by the same actor in network models. The density of connections drives diffusion between actors between them. If networks were conceived as operating across time, the network connections to self would increase the overall density of connections within the population and perhaps account for some of the steepness of the empirical curves for protest distribution.\n\n(p.196) In some cases the \u2018spike\u2019 is generated by a major external shock that has provoked a common response, without explicit coordination. When this occurs, however, the response will be something that requires relatively little coordination and has become a standard action form within a particular population. Identical actions involving complex coordination or novel tactics would not be expected to arise simultaneously in diverse locales simply from an external shock, without explicit coordination and communication through networks. The initial day of rioting after the assassination of Martin Luther King, Jr occurred in a context in which black urban populations were familiar with the \u2018riot\u2019 as an action form. The wave of protests at the beginning of the 1991 Gulf War bombings followed a build\u2010up of mobilization in which it was \u2018understood\u2019 that everyone would protest if the war started.\n\nPulling out diffusion effects in these cases of closely connected events requires thinking clearly about the nature of the event and the type of coordination involved. In the USA 1960s riots, there was clear evidence of diffusion of small riots to nearby communities within the next day or two. For major protests in Germany, where the demonstrations are generally held on weekends, particularly Saturdays, there would be a seven or more day lag for diffusion effects to occur. That is, there are good reasons to expect different time lags for different kinds of events.\n\nOther problems arise from using the news media as a data source when they are also one of the actors in the process. When the data source is one national news source, it is likely that there will be smaller regional diffusion effects that are not captured in the news source. What appears as a spike in the news accounts may simply be a failure to report the smaller events building up to and following a major event, and media attention cycles may exacerbate this spiking. (In sub\u2010sequent work with our media models, we can investigate these possibilities.) Myers' riot data is based on newspapers, but was compiled from a large collection of local newspapers by a clipping service and, as a consequence, had much more information about smaller and more localized riot waves. Nevertheless, even Myers' data shows greater peaking than would be predicted by most diffusion models, so there is clearly more work to do.\n\n## Joint Action as a Source of Spikes\n\nMany spikes in protest distributions arise from joint action that has clearly been organized. Sometimes this organization is overt and can actually be located in news sources, if it is looked for. Other times it is covert. We examined two data series available to us, Ruud Koopman's data on new social movements' protests in Germany and Kelley Strawn's data on Mexican protests, and identified a large number of cases in which similar events occurred almost simultaneously in multiple locales, often with no prior warning or build\u2010up of action. It is obvious in most of these cases that there has to have been prior communication and coordination, (p.197) whether or not it is visible in the data sources. There is clearly some sort of network diffusion process operating, but something else is diffusing other than the final action. Instead, it is an ideology or action plan that is diffusing and the simultaneous coordinated action that follows is an observable expression of a different diffusion process. From a diffusion modelling perspective, such \u2018multiple event days\u2019 create apparent discontinuous spikes in the flow of events.\n\nWe have modelled a simple process that generates a \u2018spike\u2019. Actors have a constant low probability of emitting protest actions. But in addition, actors are organizing. They are linked to other actors through their networks. Each actor has a probability of \u2018organizing\u2019 other actors (which is also assumed to be the probability of acting at the end). Actors \u2018organize\u2019 only those to whom they have a network tie. Each receipt of organizing raises an actor's probability of participating in the \u2018big event\u2019 at the appointed time, as well as of organizing other actors. (In this initial model, these two probabilities are treated as the same, but they could be readily differentiated.) But nothing \u2018happens\u2019 at the big event until the appointed day, when everyone acts at once. In this example, we assume that Actor 1 is the organizer and starts with a 100 per cent chance of organizing\/acting, while all the other actors begin with a zero per cent chance of organizing\/acting. Each time an actor receives an organizing contact, his\/her probability of acting rises 1 per cent. At the specified time period (time=100 in this example), each actor acts or not with the accumulated probability. This model produces a result that looks like Fig. 8.10. We have added random noise of a low probability of acting, to show how hidden organizing looks against a backdrop of random action. This discontinuous spike is the product of more gradually diffusing influence that is raising the probabilities of action. Figure 8.11 shows these probabilities rising for several different network configurations. The results in Fig. 8.11 differ dramatically from each other: the three random networks have widely different results, and the cliqued\n\nFig. 8.10. Hidden organizing, against a backdrop of random noise. Random networks\n\n(p.198)\n\nFig. 8.11. The average probability which is rising in hiding varies tremendously with network structure. Because of the skew in starting values, it is connection to Actor 1 that is especially important\n\nand bridged networks are different from each other. In this model of hidden organizing, the \u2018star\u2019 model is most effective. The size of the \u2018big event\u2019 differs markedly depending on the network organizing it.\n\nFigure 8.12 shows how the effects of network structure can be seen in this process by calculating and plotting the average probability within cliques. Only the bridged cliques show an \u2018interesting\u2019 plot where the spread of organizing through the bridges can be seen. Full cliques have zero probability outside the organizer's clique, while random networks rarely show much cliquing. In a star network, the average probabilities for all the non\u2010stars are about the same. A similar technique of examining different subgroups within a larger network could be used for the information and influence models, as well. It is important to note that the effects of different network structures vary greatly depending on how the network \u2018works\u2019. Information flows, influence flows, and hidden organizing appear to be impacted differently by different network structures.\n\nThis particular specification of hidden organizing assumed that actors were building up to an appointed day, which is the appropriate model for big demonstrations. An alternate specification would be that actors organize until they have mobilized a large enough critical mass, that is, until some size criterion is achieved; this alternate approach would seem more appropriate for the hidden organizing behind a coup or revolution. Hidden organizing mechanisms can be incorporated into an influence model or a communication model. In empirical cases, this behind\u2010the\u2010scenes organizing is occurring simultaneously with other actions. However, it would be expected that actors might have limited resources, which might lead to a decline in other forms of action as organizing increases. Modelling this would require some algorithm for how actors choose between organizing and acting, a complexity that is beyond the scope of this article. Another issue to explore is (p.199)\n\nFig. 8.12. Probability of organizing for a big event. Network structure can be seen in the average probability within cliques. Here the spread of organizing through the bridges can be seen. Full cliques have zero probability outside the organizer's clique. Random networks show wide variability in the degree of cliquing\n\nwhether these coordinated actions foster subsequent actions via a diffusion effect, or whether all possible actors act in concert, and action falls off afterwards.\n\n# Discussion and Conclusions\n\nThe term \u2018network\u2019 needs to be unpacked if it is to move beyond vague heuristic and actually structure research into social movements. We find that attempting to specify network effects in formal models forces us to grapple with the difficult questions of exactly what we think these effects are and how they work, and how they relate to concepts of diffusion. The models we are working with in this chapter are of a particular sort that is rarely attempted in sociology. We are not analysing empirical data and fitting regression coefficients. And we are not specifying elegant deductive models and deriving their formal properties. Both of us have done both of these in other works. But in this project, we are struggling with what empirical data patterns actually look like, and trying to model the underlying processes that could be giving rise to these patterns. This chapter has sketched an approach to this problem and has shown how the flows of information, influence, and joint action can be modelled and how these different processes can yield widely different results.\n\nAs we have worked on this problem, we have come to recognize that any empirically valid model needs to have a substantial random or stochastic element. Random fluctuations from constant probabilities produce the kind of spiky, jagged (p.200) plots of event counts over time that are characteristic of empirical data. These same random fluctuations frequently produce \u2018waves\u2019 of events, especially when they are aggregated across a few time periods. Once we made the shift to stochastic modelling, we have been forced to confront the huge effect which simple random variation produces in our models. Even with a fixed set of network ties, random fluctuations in who happens to act when can produce large effects on the pace with which action or influence diffuses. Random variations in which actors are tied to each other in a network can produce even larger differences in results. Substantively, this means that sheer chance appears to play a large role in affecting the trajectory of a protest cycle. It will take some time to absorb the theoretical and empirical implications of this result.\n\nAs we have unpacked different network processes and sought to pin them down so they could be modelled, we have found that the effect of \u2018network structure\u2019 varies greatly depending upon the nature of a particular network process. This can be seen most extremely with the \u2018star\u2019 networks in which all the network ties are with one central actor. This structure is a severe impediment to mobilization in a model which assumes that actors respond to their direct information about the number of others who have acted recently: because all the actors except the \u2018star\u2019 know about at most one other actor's actions, they do not increase their own probabilities of action to any significant degree. By contrast, the \u2018star\u2019 network is the most efficient in the \u2018hidden organizing\u2019 model, where it is in contact with an organizer that is assumed to increase the probability of behaviour, not the total of prior actions. It would be foolish to try to decide whether \u2018star\u2019 networks are \u2018good\u2019 or \u2018bad\u2019 for mobilization. Instead, it must be recognized that the impact of a network structure is intimately intertwined with exactly how actors affect each others' behaviour. Verbal theorists have talked vaguely for years about information flows and influence, but it is only when you actually try to pin these ideas down to formal representations that you realize how deeply the exact specification of what those relationships are influences not only the gross levels of outcomes, but the ways in which other factors affect outcomes.\n\nWe have shown how several different kinds of network effects can be modelled, and why they are important. Our model of information flow focused on the assumption that actors increase their probability of acting as a function of the number of others they know about who have previously acted, an assumption that leads to a gradual rise in everyone's rate of action. In this model, as information diffuses so does action, and we showed that different network structures affect the pace with which this occurs.\n\nConsistent with our other research, we also devoted attention to modelling the effects of news coverage. This is particularly important because most often the data we have about protest comes from newspapers. We first show that even if the newspapers are completely unbiased samplers of protests, simple random fluctuations in news coverage produce apparent cycles that are not present in the (p.201) underlying protest distribution. But, of course, newspapers are not unbiased samplers. We know that they respond to the size of protest and that they are subject to issue attention cycles that may be independent of protest. Both of these patterns produce additional distortions in the protest cycles in newspapers as compared to the underlying \u2018real\u2019 protest cycle. But, additionally, news coverage itself affects protest and changes the protest cycle. Methodologically, this helps protest researchers, because if news coverage increases protest, it brings the \u2018real\u2019 protest cycle more into line with news coverage of protest. However, if the causal effect of news coverage on protest is not recognized, researchers can draw quite erroneous conclusions about the effect of protest on policy debates. More detailed studies of the interplay between protest and news coverage must be the subjects of other analyses.\n\nInfluence models assume that people's opinions change in the direction of those with whom they are in frequent contact. This assumption generates a long\u2010term tendency for a population who has direct or indirect ties to each other to move toward one common opinion, while wholly distinct cliques move toward separate average opinions. We showed how network structures affect these processes. If networks are cliqued, these models provide some way of understanding the relationship between in\u2010group and out\u2010group ties in opinion formation. We also showed how this approach could be readily modified to make the network ties themselves fluid and changing, in response to contacts from others.\n\nThe approach we offered for studying influence immediately points to a large number of possible extensions. Our simple models employed only symmetric influence ties, and an obvious extension would be to see how asymmetric influence affects these results. Empirically, populations obviously do not seem to be tending toward a single common opinion, and empirically it is clear that contact between persons of different opinions can generate polarization of opinions rather than convergence. Thus, even though averaging rules like the ones we used are the most common in formal models of influence, they do not seem to generate results that fit empirical patterns. We suspect that the most promising avenue to pursue is a model that says actors will either polarize or converge when they encounter each other, with the probability of doing polarization versus convergence being a function of the distance of their opinions from each other.\n\nOur model of \u2018hidden organizing\u2019 is not necessarily very elegant, but it calls attention to an important empirical phenomenon that cannot be neglected in the analysis of empirically observable protest waves. Protest data are much more spiked than standard diffusion models can accommodate. These spikes violate all the assumptions that undergird standard statistical regression models, as well. Too many scholars have been willing to run models without confronting the implications of these spikes. Yet every social movements researcher knows that \u2018hidden\u2019 organizing (i.e. organizing that is not reported in observable data sources) occurs. This is one example of how important it is to think about what we already know about movement processes as we seek to develop formal theory that speaks to (p.202) empirical data, and as we seek to do quantitative analyses of empirical data that are soundly grounded in a theoretical understanding of the underlying processes that give rise to observable data.\n\nApart from providing an explanation for data spikes, our work on joint action points to the need for conceptual clarity about actors and units of analysis. Separate individuals come together to form groups, and once they are in groups, those groups act with a high level of unity. Thus protest cannot be modelled as if independent individuals are conducting it. But, of course, the groups themselves also may temporarily act together with some unity, and models of independent action of groups will not correctly describe observable data, either. We need to fit the model to the type of action. The black riots of the 1960s had relatively little coordination between communities and relatively little organization within communities, while new social movement protests have a great deal of preplanning and coordination associated with them. We should expect to see different kinds of empirical patterns arising from these different kinds of actions.\n\nWe need a middle ground between the statistical analysis of data and the development of pure formal theory. In this project, we are in dialogue with empirical data, seeking to determine the kinds of processes that could produce the patterns we can observe. As we have repeatedly stressed, the discipline of turning theory into equations reveals the ambiguity and imprecision of many past discussions of network effects, and forces us to think more seriously and deeply about just how we think things work.\n\n## Notes:\n\n$Display mathematics$","date":"2019-10-23 23:55:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 2, \"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\": 0.45446452498435974, \"perplexity\": 1258.2946671651762}, \"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\/1570987836368.96\/warc\/CC-MAIN-20191023225038-20191024012538-00169.warc.gz\"}"} | null | null |
Wazimap over USSD & SMS
=======================
Overview
--------
Code4SA is an awesome South African non-profit organisation that is active in
the open data, transparency and journalism space. During 2014 winter internship
Merada Richter created a USSD application that allows people to query Code4SA's
medicine price registry to allow people make better informed decisions around
which brand medicine to buy and at what price. It is available on all South
African mobile networks by dialling *120*8864*1399#. The code that powers is
this a Vumi Javascript sandbox application, the code is available here.
We'd like to do the same for Code4SA's Wazimap. Wazimap is an awesome Django
based web application that gives insight into the 2011 census data on a
provincial, municipal, and ward level.
* We would like to explore what information is available via Wazimap's APIs,
e.g.
* http://wazimap.co.za/place-search/json/?q=melkbosstrand
* http://wazimap.co.za/profiles/ward-19100023.json
* We would like to access this data from the Vumi's Javascript sandbox.
* We would like to design an application that allows people to query census data
via USSD and receive the results via SMS.
Things one would learn:
* Developing a USSD & SMS application using Vumi's Javascript sandbox
* UX design within the constraints of a USSD application
* Writing tests
People to poke: Simon Cross (hodgestar), Justin van der Merwe (justinvdm), Rudi
Giesler (rudigiesler)
Useful links
------------
* `Jsbox Toolkit documentation <http://vumi-jssandbox-toolkit.readthedocs.org/>`_
* `Jsbox Toolkit source <https://github.com/praekelt/vumi-jssandbox-toolkit/>`_
* `Jsbox application skeleton <https://github.com/praekelt/go-jsbox-skeleton>`_
* `Q promises <https://github.com/kriskowal/q>`_
* `Javascript <https://developer.mozilla.org/en-US/docs/Web/JavaScript>`_
* `Lodash <https://lodash.com/docs>`_
* `Mocha test framework <http://mochajs.org/>`_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,351 |
\section{Introduction}
Vortex matter in multiband superconductors demonstrates many unusual properties which are drastically different from that in single-band materials\cite{PhysRevB.81.020506,
PhysRevB.85.094511,PhysRevLett.102.117001,eskilsen2002,PhysRevLett.95.097004,
PhysRevB.81.214501,Fente2016}. The origin of non-trivial new effects comes from the greatly enhanced number of the available degrees of freedom in the system consisting of Cooper pairs and quasiparticles residing in several different bands. In this case the condensates in general tend to have different coherence lengths\cite{PhysRevLett.105.067003,
PhysRevB.72.180502, PhysRevLett.105.067003} sharing the same critical temperature and the single divergent scale near $T_c$ \cite{PhysRevB.84.094515} . With increasing the coupling between condensates their length scales become essentially the same\cite{PhysRevB.85.134514}. Although being quite important characteristics the coherence lengths are not directly measurable. For example the sizes of Abrikosov vortices in the two-band superconductor MgB$_2$ measured with the scanning tunnelling microscopy (STM) local probes appear to be significantly different from the coherence length inferred from the upper critical field\cite{eskilsen2002}. This physics is explained by the different localization scales of local density of states profiles in different bands determined by the disparity of diffusion coefficients\cite{PhysRevLett.90.177002}.
The high resolution of STM allows to explore individual vortex cores in details by measuring the quasiparticle local density of states (LDOS)
\cite{PhysRevLett.64.2711, PhysRevLett.62.214,GH,suderow2014, Fente2016}. The LDOS profiles $N(r)$ are essentially determined by the spatial order parameter distribution $\Delta(r)$ near the vortex core. However the vortex core size determined from the STM tunnelling conductance depends on the temperature and bias \cite{volodin1997} indicating the spatial and energy variation of the LDOS of localized quasiparticle states trapped close to the vortex center.
The relation between {zero-energy LDOS} $N(r)$ and $\Delta(r)$ is quite straightforward in diffusive superconductors when the magnetic field $B$ is close to the upper critical field $H_{c2}$ so that $H_{c2}-B \ll H_{c2}$.
In this regime as shown by de Gennes\cite{dG} the following relation holds
%
\begin{equation} \label{Eq:deGennes}
N(r)= 1 - 2|\Delta (r)|^2/\Delta_0^2
\end{equation}
%
where $N$ is normalized to the normal metal DOS and $\Delta_0$ is the gap function amplitude in the absence of magnetic field.
At lower fields the relation between $N(r)$ and $\Delta (r)$ has not been checked even in the simplest case of single-band superconductors.
In the present paper we demonstrate that in general the behaviour of these two profiles with decreasing magnetic field becomes quite different so that it is not possible to extract the information about coherence length from STM measurements by applying directly the Eq.(\ref{Eq:deGennes}).
The behaviour of gap and LDOS profiles can be even more intriguing in two-band superconductors. According to the recent experiments \cite{Fente2016} the vortex sizes measured by STM in 2H-NbSe$_2$ and
2H-NbS$_2$ compounds demonstrate much weaker magnetic field dependencies than in the single-band materials.
Interpreting these data using de Gennes relation \cite{dG} results in the conclusion about the mostly field-independent condensate length scales in the two-band superconductors.
Here we report the results of exact numerical calculations in the framework of the multiband Usadel theory.
We find that in the two-band superconductor the vortex core sizes $w_{\Delta_1}$, $w_{\Delta_2}$ determined by the gap function profiles in different bands $\Delta_{1,2}(r)$ in general have no distinct correlation with the widths $w_{\sigma_1}$, $w_{\sigma_2}$ of the corresponding LDOS distributions $N_{1,2}(r)$. We illustrate that
for the distinct disparity between diffusion coefficients in different bands, the vortex core sizes $w_{\sigma_1,\sigma_2}$ and $w_{\Delta_1,\Delta_2}$
can show qualitatively different behaviour as functions of the magnetic field.
For the large enough interband pairing the gap function distributions $\Delta_{1}(r)$
and $\Delta_{2}(r)$ are mostly identical so that $w_{\Delta_1}\approx w_{\Delta_2}$. However the profiles of $N_1(r)$
and $N_{2}(r)$ are strongly different except of the high field regime when the modified de-Gennes relation restores and all length scales coincide.
We demonstrate that in the superconducting band with the smallest diffusion coefficient the zero-energy LDOS length-scale shows quite weak magnetic field dependence in accordance with STM data in multiband superconductors\cite{Fente2016}.
At the same time the width of LDOS profile in the band with larger diffusion coefficient grows with decreasing magnetic field in a rate which is typically faster than the growth of healing lengths $w_{\Delta_1}$, $w_{\Delta_2}$ characterizing the order parameter distributions.
The structure of this paper is as follows. In Sec. \ref{theory} we introduce the formalism of the quasiclassical Green's functions to describe the properties of dirty multiband superconductors and discuss numerical approach for the solution of self-consistency problem. In Sec. \ref{results}, the results of the numerical calculations are presented. First, in Sec. \ref{resultsA} we checked method in the single-band limit. In Sec \ref{resultsB}, we examine in details the field dependencies of the gap and LDOS profiles in different bands and calculate characteristic length scales in two-band model. The work summary is given in Sec. \ref{summary}.
\section{Model}\label{theory}
We use the formalism of quasiclassical Green's functions (GF) and introduce retarded/advanced GF, $\hat g^{R/A}_k$, for two-band ($k=1,2$) superconductor which obey in the diffusive limit the Usadel equation
\begin{equation}\label{spectraleq}
D_k\hat\nabla ( \hat g^{R/A}_k\cdot\hat \nabla\hat g^{R/A}_k) + [i\varepsilon\hat\tau_3+i\hat\Delta_k, \hat g^{R/A}_k ]=0.
\end{equation}
Here $D_k$ is diffusion constant in each band, $\hat \Delta_k=\left(\begin{array}{cc} 0 & \Delta_k \\-\Delta^\ast_k & 0
\end{array}\right)$ is the gap operator, and $\hat\nabla=\nabla-i\pi\phi_0^{-1}{\bm A}[\hat\tau_3,]$, where $\hat\tau_3$ is the Pauli matrix, square brackets denote commutator operation
and $\phi_0=\pi /e$ is the flux quantum. Note that we use theoretical units $k_B=\hbar=c=1$.
To describe the vortex structure at arbitrary fields we employ the circular cell approximation \cite{ihle1971b,WattsTobin1974,Rammer1987,Rammer1988}. Within this approach the unit cell of the hexagonal vortex lattice hosting a single vortex is replaced
by a circular cell with the centre at the point of superconducting phase singularity.
Inside circular cell, the gap and magnetic field distributions are taken radially symmetric with respect to the cell centre. Below we consider the vortex state in the limit of large values of the Ginzburg-Landau parameter, $\kappa\gg1$. In this case, magnetic field $B$ is constant inside circular cell and the vector potential can be taken in the form ${\bm A}(r)={\bm \varphi}Br/2$.
The periodicity of the
lattice solution is modelled by the special choice of the
boundary conditions, namely the vanishing the supercurrent density at the circular-cell boundary. At that, the circular-cell radius is uniquely defined by magnetic induction, $R=\sqrt{\phi_0/(\pi B)}$ so that there is exactly one flux quantum $\phi_0$ passing through the unit vortex cell.
In the $\theta$-parameterization, GF in Nambu space read as
\begin{align}
\hat g^R_k=
\left( \begin{array}{cc}
\cosh\theta^{(k)} &
\sinh\theta^{(k)} e^{i\varphi_k}
\\
-\sinh\theta^{(k)} e^{-i\varphi_k} & -\cosh\theta^{(k)}\end{array}\right),
\end{align}
where $\varphi_k$ is band-gap phase. In cylindrical coordinates, Eq. (\ref{spectraleq}) can be rewritten for complex angles $\theta^{(k)}$ as
\begin{align}\label{theta}
&D_kr\partial_r(r\partial_r\theta^{(k)})-D_k(1-r^2/R^2)^2\sinh\theta^{(k)}\cosh\theta^{(k)}\nonumber\\
&+2ir^2(\varepsilon\sinh\theta^{(k)}-|\Delta_k|\cosh\theta^{(k)})=0.
\end{align}
This set of equations has to be solved self-consistently with gap order parameters determined by conditions
\begin{align}\label{gaps}
|\Delta_k|=2\pi T\sum_{k^\prime}\lambda_{kk^\prime}\sum_{\omega_n>0}\sin \theta_n^{(k^\prime)},
\end{align}
where $\lambda_{kk^\prime}$ are intra- and interband interaction constants which form matrix $\hat \lambda$, $\omega_n=\pi T(2n+1)$ are Matsubara frequencies and Matsubara GF parametrized by $\theta_n^{(k)}$ satisfy (\ref{theta}) after substitution $\theta^{(k)} \to-i\theta_n^{(k)}$ and $\varepsilon\to i\omega_n$. At that, boundary conditions read as $\theta_n^{(k)}(r=0)=0$ and $\partial_r\theta_n^{(k)}(r=R)=0$ leading to zero gradient of gap modulus at the vortex-cell boundary.
We normalize magnetic field by upper critical one which in the two-band model is determined by condition $|\hat A|=0$, where $|\hat A|={\rm Det}\hat A$ and
\begin{align}
A_{kk^\prime}= (\hat\lambda^{-1})_{kk^\prime}+\delta_{kk^\prime}\left[ f_k(T)-G_0+\ln (T/T_c) \right].
\end{align}
%
Here
$f_k=\Psi\left[1/2+q_k/(2\pi T)\right]-\Psi(1/2)$, $\Psi$ is digamma function, $q_k=eD_kH_{c2}$,
$2G_0=({\rm Tr}\hat\lambda-\lambda_0)/|\hat\lambda|$ and $\lambda_0^2=({\rm Tr}\hat\lambda)^2-4|\hat\lambda|$. Except for zero and critical temperatures, $H_{c2}(T)$ has to be calculated numerically. In the limit $T\to 0$, upper critical field is given by the expression $2\sqrt{q_1q_2}/T_c=\pi e^{g_+/2-C}$, where\cite{PhysRevB.67.184515}
\begin{align}
&(g_++\lambda_0/|\hat\lambda|)^2=
\left[ \ln (D_1/D_2) - (\lambda_{11}-\lambda_{22})/|\hat\lambda|\right]^2
\nonumber\\
&+4(\lambda_{11}\lambda_{22}-|\hat\lambda|)/|\hat\lambda|^2,
\end{align}
and $C\approx0.577$ is the Euler constant. Note that in weak-coupling limit
the critical temperature of a two-band superconductor is $T_c=\omega_D/(2\pi \Omega)$, where $\omega_D$ is Debye energy cut-off and $4\Omega=e^{G_0-C}$.
To find the self-consistent order parameter distributions we start by calculating gap order parameters in the cell taking first initial distributions of $|\Delta_{1,2}(r)|$. By initializing guess functions $\theta_{n}^{(1,2)}$ for each $n$, we linearise equation for Matsubara GF around $\theta_{n}^{(1,2)}$ and solve the linear problem numerically by apply sweeping method. Solution provides correction to $\theta_{n}^{(1,2)}$ and refined guess function is used for next iteration. By performing sufficient number of iterations, procedure converge to the Matsubara GF which are substituted into right-hand side of Eq. (\ref{gaps}) to obtain correction to the initial gap functions $|\Delta_{1,2}|$. By applying refined gap functions, we repeat scheme from the beginning and find gaps in iterative process with needed precision.
The zero-energy LDOS in different bands
is given in $\theta$-parametrization by
$N_k=\cos ({\rm Im}\theta^{(k)})$ at $\varepsilon=0$ . To find $N_k$, we consider imaginary part of Eq. (\ref{theta}) at zero energy with gap profiles found beforehand. We solve it numerically by starting from guess distributions for $\theta^{(k)}$. We linearise (\ref{theta}) around $\theta^{(k)}$ and solve the linear problem numerically by sweeping method. Solution gives correction to $\theta_{k}$ which is used to construct refined guess distribution and employ iteration procedure.
\section{results}\label{results}
\subsection{Single-band limit}\label{resultsA}
The approach presented in Sec. \ref{theory} reduces to the single-band model, if $\lambda_{12}=\lambda_{21}=0$ and $D_{1,2}=D$. For $\lambda_{11}>\lambda_{22}$, it corresponds to the description of the independent stronger-superconductivity band.
\begin{figure}[t!]
\includegraphics[width=0.99\linewidth]{multiplot1.eps}
\caption{\label{f1}
(Color online) Vortex structure in the single-band model at $T/T_c=0.05$. (A) Normalized gap distribution (solid) inside vortex cell, $F=|\Delta|/\Delta_0$. Numbers near each curve indicate the value of $B/H_{c2}$. The dashed curves with the same color show zero-energy LDOS $N$. (B) Proportionality coefficient between $1-N$ and $F^2$ within vortex cell for different magnetic fields. Note that we obtain
$(1-N)/F^2=2$ near $H_{c2}$ in agreement with Eq. (\ref{Eq:deGennes}). (C) LDOS variation $\sigma=\delta N(r)/\delta N(0)$ for different ratios $B/H_{c2}$. (D) The field behaviour of the vortex-core size $w=w_\Delta$ determined by the half-width of squared gap $|\Delta|^2$ (red) and the one $w=w_\sigma$ defined by the half-width of LDOS variation $\sigma$ (blue). Both quantities are shown in units of $\xi=\sqrt{D/(2\pi T_c)}$.
%
The dashed curves are calculated by means of Eq. (\ref{spectraleq}) with substitution $\hat\nabla\to\nabla$, see discussion in the text. }
\end{figure}
Fig. \ref{f1} demonstrates the results of the self-consistent numerical calculations for single-band superconductor. As magnetic field increases, the radius of circular vortex cell reduces resulting in the suppression of the maximal gap value achieved at the cell boundary, see Fig. \ref{f1}A. At that, inhomogeneity of zero-energy LDOS $N$ inside vortex cell smooths out by rising field.
In Fig. \ref{f1}C we plot LDOS variation $\sigma=\delta N(r)/\delta N(0)$, where $\delta N(r)=N(r)-N(R)$, for different magnetic fields. In single-band limit, these curves do not depend on material parameters such as the diffusion coefficient $D$ and $T_c$.
From definition it is clear, that LDOS variation $\sigma$ is characterized by the same half-width as LDOS $N$ itself.
To check our numerical results we test the obtained profiles against the validity of de Gennes relation (\ref{Eq:deGennes}) at low temperatures and close to $H_{c2}$.
In Fig. \ref{f1}B we show the ratio $(1-N (r))/F^2(r)$, where $F(r)=|\Delta(r)|/\Delta_0$,
for different magnetic fields at temperature $T=0.05 T_c$.
For high fields this ratio is constant in agreement with Eq. (\ref{Eq:deGennes}).
However, for the lower fields, $B/H_{c2}\lesssim 0.5$, it is significantly inhomogeneous meaning that LDOS evolution inside vortex cell is essentially different from the order parameter. As a result, LDOS measurements for sparse vortex lattices in general cannot be used to quantify the length scale of the superconducting order parameter.
Fig. \ref{f1}D demonstrates the field dependencies for half-widths $w_\Delta$ and $w_\sigma$ of squared gap $|\Delta|^2$ and LDOS variation $\sigma$, respectively. Two half-widths shown in Fig. \ref{f1}D overlap in the limit $B\to H_{c2}$, where spatial profiles of LDOS and $|\Delta|^2$ become identical, see black curve in Fig. \ref{f1}B. In this case, we expect that Abrikosov vortex lattice solution governs the behaviour of the gap order parameter so that half-width is determined by the size of the superconducting nucleus. By using known analytic solution for the gap at $H_{c2}$\cite{GH} given by $F(r)\propto re^{-r^2/(2R^2)}$,
we obtain $w_\Delta \approx 0.48R$. At low temperatures, upper critical field is determined by $q/T_c=\pi/(2e^C)$ so that $w_\Delta\approx 1.3\xi$, where $\xi^2=D/(2\pi T_c)$, in agreement with numerical value presented in Fig. \ref{f1}D.
For lower fields, half-widths $w_\sigma$ and $w_\Delta$ have qualitatively different behaviours manifesting significant difference between the squared gap and LDOS profiles and violation of simple relation (\ref{Eq:deGennes}). The half-width found for the squared gap coincides with previous calculations \cite{GH} and scales approximately as $w_\Delta \sim(B/H_{c2})^{-1/3}$ in the intermediate fields. At the same time, the half-width of LDOS $w_\sigma$ changes with the field slower than that.
For very sparse vortex lattices, $B/H_{c2}\ll1$, both scales are characterized by linear field-dependence and for the gap we obtain $w_\Delta(B)/w_\Delta(0)\simeq1-B/H_{c2}$. Such a behaviour indicates that spatial evolution of the gap profile is affected by the term linear in the vector potential. This behaviour can be checked by calculating vortex core sizes in the absence of vector potential. In result instead of the linear behaviour we get the low-field plateaus in the dependencies of
$w_\Delta(B)$ and $w_\sigma(B)$ shown by the dashed lines in Fig. \ref{f1}D. Thus the absence of any pronounced variation of vortex core sizes at small magnetic fields found by STM experiments\cite{Fente2016} cannot be attributed to the specific range of magnetic field $B\ll H_{c2}$ studied there.
On the contrary as we demonstrate below, almost field-independent vortex core sizes can be naturally obtained within the minimal two-band model of superconducting state.
\begin{figure}[t!]
\includegraphics[width=0.99\linewidth]{multiplot2.eps}
\caption{\label{f2}
(Color online) Vortex structure in the two-band model at $T/T_c=0.1$ for $D_1/D_2=0.2;\ 10$ (left and right column, respectively). (A,B)
Gap profiles in each band normalized by bulk value, $F_k=|\Delta_k|/\Delta_{k0}$. Dashed red/blue curves correspond to $F_{1,2}$ at $B/H_{c2}=0.1$ and pair of solid curves to $B/H_{c2}=0.9$. (C,D) LDOS in each band $N_k$ for small (dashed) and high (solid) fields. Red/blue colours correspond to $N_{1,2}$. }
\end{figure}
\subsection{Two-band model}\label{resultsB}
The two-band superconductivity is defined by the matrix of interaction constants $\hat\lambda$ and by the ratio of diffusion coefficients in the bands $D_1/D_2$. For calculations we consider typical parameters \cite{MgB2}, namely, $\lambda_{11}=0.1012$, $\lambda_{12}=0.0336$, $\lambda_{21}=0.0264$ and $\lambda_{22}=0.0448$, and consider evolution as $D_1/D_2$ changes.
Fig. \ref{f2} shows that gap function profiles in different bands look very similar. If one normalizes gaps by their maximal value reached at cell boundary then the difference between normalized gap distributions practically vanishes.This result does not depend on the values of $D_{1,2}$ despite that these
parameters define coherence lengths in the absence of Josephson coupling between the bands.
In the considered case of sufficiently strong interband interaction, the mixing between superconducting condensates of separate bands is so efficient that healing length of different gap functions $\Delta_{1,2}$ become almost identical.
In contrast to the gap profiles, LDOS in separate band is strongly affected by the band diffusion coefficients. This is seen from Usadel Eq. (\ref{theta}) where
characteristic lengths of solutions $\theta^{(1,2)}(r)$ differ by the factor $\sqrt{D_1/D_{2}}$. Fig. \ref{f2}C,D confirms this behaviour showing that LDOS in the band with smaller diffusion coefficient changes at shorter distances than
the one in the band with larger diffusion coefficient.
Apart from the characteristic scales determined by the diffusion coefficients there is another characteristic length which is the circular cell radius $R$.
Changing the diffusion coefficients in different bands independently one can obtain the unusual situation peculiar for two-band model when the length scale of LDOS variation in one of the bands is much larger than the cell radius. In this case the LDOS corresponding to the band with larger diffusion coefficient changes
within vortex cell very weakly in the wide range of the fields. This situation is illustrated in Fig. \ref{f2}C where the band with weaker superconductivity, $\Delta_2<\Delta_1$,
has larger diffusion coefficient $D_1/D_2=0.2$. The LDOS in this band (blue curves in Fig. \ref{f2}C) changes within vortex cell very weakly in the wide range of the fields (already for $B/H_{c2}\gtrsim 0.3$). Thus, in this case the characteristic length scale for $N_2$ variation, $w_{\sigma_2}$, is expected to scale with cell radius $R\propto 1/\sqrt{B}$. As we see below, this is indeed the case.
\begin{figure}[t!]
\includegraphics[width=0.99\linewidth]{multiplot3.eps}
\caption{\label{f3}
(Color online) Spatial variation of proportionality coefficient between $1-N_k$ and $F_k^2$ in two-band model with $D_1/D_2=0.2;\ 10$ (left and right column, respectively). (A,B) Coefficient for the stronger-superconductivity band as $B/H_{c2}$ indicated by the colour numbers increases. (C,D) Coefficient for weaker-superconductivity band. }
\end{figure}
Another unusual situation generic for two-band model only
can be realized when LDOS corresponding to the band with smaller diffusion coefficient varies within vortex cell on a distance which is much smaller than cell radius. This case is demonstrated in Fig. \ref{f2}D where band with weaker gap has smaller diffusion coefficient, $D_1/D_2=10$, and variations of its
LDOS (blue curves in Fig. \ref{f2}D)
are only weakly affected by the changes
of the vortex cell radius under magnetic field. As we see below, this results in the weak field dependence of the length scale related to the LDOS variations of the relevant band.
\begin{figure}[t!]
\includegraphics[width=0.99\linewidth]{multiplot4.eps}
\caption{\label{f4}
(Color online) LDOS variations $\sigma_k=\delta N_k(r)/\delta N_k(0)$ inside the vortex as the value of normalized magnetic field $B/H_{c2}$ indicated by different colours increases. Left and right column correspond to the cases $D_1/D_2=0.2;\ 10$, respectively.}
\end{figure}
Let us discuss the relations between gap functions $\Delta_{1,2}(r)$ and LDOS deviations from the normal state, $1-N_{1,2} (r)$, shown Fig. \ref{f3} for the
two-band model. One can see that analogously to the single band case these profiles coincide only for the high fields close to the upper critical one. This limit can be approached analytically. For $B\approx H_{c2}$, the order parameter is small and it can be written as $|\Delta_k|=c_k\Delta_{k0}r e^{-r^2/(2R^2)}$, where $c_k$ is small constant, see \cite{golubov1988}. The zero-energy solution of spectral Eq. (\ref{theta}) is then given by $\mathrm{Im}\theta_k=\alpha_k|\Delta_k|$, where $\alpha_k\sqrt{q_1q_2} B/H_{c2}=-\sqrt{D_{3-k}/D_k}$.
As a result, we obtain the relation between the LDOS and order parameter in the two-band model
which is valid at fields very close to upper critical one
\begin{align}\label{propc}
& N_k = 1-\frac{|\Delta_k|^2}{2e^2 D^2_k H^2_{c2} }.
\end{align}
This formula generalizes the de Gennes
relation (\ref{Eq:deGennes}) for the multiband system and arbitrary temperatures. Indeed, in the one band case one restores the relation (\ref{Eq:deGennes}) at low temperatures $T\to 0$ by taking into account single-band limiting value $eDH_{c2}/\Delta_0=1/2$.
However, in two-band case the relation between bulk gap in particular band and the upper critical field depends strongly on two-band model parameters, in particular, the ratio of diffusion constants. For our parameters of two-band model and $T=0.1T_c$ we have $\Delta_{1,20}/T_c\approx 2.05;0.81$ and $\sqrt{q_1q_2}/T_c\approx 1.51;0.38$ for $D_1/D_2=0.2;10$, respectively. According to the Eq. (\ref{propc}), the proportionality coefficient between $1-N_k$ and $|\Delta_k|^2/\Delta_{k0}^2$ is then given by $4.56;1.45;0.03;23$ for the cases shown in Fig. {\ref{f3}}A,B,C,D, respectively. These values coincide with the black curves in Fig. \ref{f3} remarkably well.
Next we calculated LDOS variations within the vortex cell defined as $\sigma_k=\delta N_k(r)/\delta N_k(0)$, where $\delta N_k(r)=N_k(r)-N_k(R)$, shown in Fig. \ref{f4}. Contrary to the single-band model where $\sigma$ has universal field behaviour, two unusual regimes can be realized in two-band superconductor depending on the value of $D_1/D_2$ parameter.
The case $D_1\ll D_2$ shown in Fig. \ref{f4}C is characterized by the leading role of the vortex-cell radius $R$ in the spatial variations of LDOS in the band with larger diffusion constant, see discussion of Fig. \ref{f2}C. As a result, the field dependence of $\sigma_2$ is governed by $R$ so that magnetic field modifies $\sigma_2(r/R)$ profiles extremely weakly, see Fig. \ref{f4}C.
Recent STM measurements of multiband systems $\beta-$Bi$_2$Pd\cite{Bi2Pd}, 2H-NbSe$_{1.8}$S$_{0.2}$ and 2H-NbS$_2$ demonstrate very similar behaviour \cite{Fente2016} suggesting that these compounds have large disparity between the diffusion coefficients in different bands.
The opposite regime $D_1\gg D_2$ illustrated in Fig. \ref{f4}D is described by weak field-dependence of LDOS profiles in the band with smaller diffusion coefficient, see discussion of Fig. \ref{f2}D. This results in the very diverse field modifications of $\sigma_2(r/R)$ curves, see Fig. \ref{f4}D, which can be also used as a fingerprint of multiband superconductivity.
\begin{figure}[t!]
\includegraphics[width=0.99\linewidth]{multiplot5.eps}
\caption{\label{f5}
(Color online) Field dependence of the length scales normalized to $\xi=\sqrt{\sqrt{D_1D_2}/(2\pi T_c)}$ for $D_1/D_2=0.2;10$. Red/blue solid lines correspond to $w=w_{\Delta_{1,2}}$ determined as the half-widths of $|\Delta|_{1,2}^2$ and red/blue dashed lines to $w=w_{\sigma_{1,2}}$ defined as the half-widths of $\sigma_{1,2}$, respectively. The inset in panel A is the plot of $\log w/\xi$ \textit{vs} $\log B/H_{c2}$ for $w_{\Delta_{1,2}}$ (red/blue solid) and $w_{\sigma_2}$ (blue dashed). Linear dependencies $y=-x/2+\mathrm{const}$ (upper black) and $y=-x/3+\mathrm{const}$ (lower black) indicate the scaling $w_{\sigma_2}\sim(B/H_{c2})^{-1/2}$ and $w_{\Delta_{1,2}}\sim(B/H_{c2})^{-1/3}$ in the vicinity of $H_{c2}$.}
\end{figure}
Finally, we have calculated field dependencies for the lengths $w_{\Delta_{1,2}}$ determined as the half-widths of $|\Delta|_{1,2}^2$ and $w_{\sigma_{1,2}}$ defined as the half-widths of $\sigma_{1,2}$,
see Fig. \ref{f5}. At higher fields, all length scales approach same value which differs from the one obtained in single-band limit. According to analytical solution for superconducting nucleus, the half-width of squared gap at $H_{c2}$ is given by $w\approx 0.48R$. By using values $\sqrt{q_1q_2}/T_c$ discussed above for our model parameters, we obtain $w\approx0.97\xi;1.95\xi$, where $\xi^2=\sqrt{D_1D_2}/(2\pi T_c)$, for $D_1/D_2=0.2;10$, respectively. This values coincide with numerics presented in Fig. \ref{f5} remarkably well.
As expected, the length scales $w_{\Delta_{1,2}}$ obtained in Fig. \ref{f5} are very close due to almost identical spatial profiles of $F_{1,2}$ caused by the efficient interband pairing, see Fig. \ref{f2}A,B. Similarly to the one-band case, scales $w_{\Delta_{1,2}}$
%
can be fitted by function $(B/H_{c2})^{-1/3}$ in the vicinity of $H_{c2}$, see inset of Fig. \ref{f5}A. However, the characteristic length scales of LDOS modifications, $w_{\sigma_{1,2}}$, demonstrate striking difference with the single-band scenario. Their field dependencies can be both stronger and weaker than that for the gap profiles determined by $w_{\Delta_{1,2}}$. In particular, in the case $D_1\ll D_2$ characterized by the leading role of the vortex-cell radius on the spatial evolution of LDOS in the band with larger diffusion constant (see discussion of Figs. \ref{f2}C and \ref{f4}C) we obtain the stronger field behaviour $w_{\sigma_2}\propto R\propto (B/H_{c2})^{-1/2}$ in the vicinity of $H_{c2}$, see also inset in Fig. \ref{f5}A.
The opposite regime, $D_1\gg D_2$ shown in Fig. \ref{f5}B is described by the exceptionally weak field dependence of LDOS in the band with smaller diffusion coefficient in agreement with discussions of Figs. \ref{f2}D and \ref{f4}D.
\section{Summary}\label{summary}
To conclude, we demonstrate that the vortex core size $w_{\Delta_k}$ determined by the healing of the gap order parameter has qualitatively different magnetic-field behaviour from the one $w_{\sigma_k}$ defined by the spatial LDOS variations in single- and two-band dirty superconductors.
We have found several generic regimes peculiar for multiband superconductor only. First, the vortex core size $w_{\sigma_k}$ related to the LDOS variations in the band with {\it larger} diffusion constant scales with the vortex-cell radius having field dependence stronger than the one for $w_{\Delta_k}$. Second, size $w_{\sigma_k}$ determined by the LDOS variations in the band with {\it smaller} diffusion constant can have field dependence significantly weaker than for $w_{\Delta_k}$. These peculiarities can explain qualitatively the recent STM measurements of vortex cores in multiband superconductors.
\section{Acknowledgements}
This work was supported by the Academy of Finland. It is our pleasure to acknowledge discussions with
Hermann Suderow and Vladimir Kogan.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,489 |
San Francisco: Google has sacked 48 people including 13 senior managers over sexual harassment claims since 2016.
In a letter to employees on Thursday, Chief Executive Sundar Pichai said the tech giant was taking a "hard line" on inappropriate conduct, reports the BBC.
Pichai's letter said the New York Times story was "difficult to read" and that Google was "dead serious" about providing a "safe and inclusive workplace".
"We want to assure you that we review every single complaint about sexual harassment or inappropriate conduct, we investigate and we take action," he continued.
A Google investigation found the woman's complaint to be credible, the paper reported, but the company has not confirmed this.
Shares in Alphabet, which owns Google, fell more than 3 per cent in New York after it reported revenues of $33.7 billion for the three months to September. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,184 |
use std::cmp;
use std::collections::HashMap;
const INPUT: &'static str = include_str!("data/day9.txt");
pub fn part1() -> u32 {
calculate_part1(INPUT)
}
pub fn part2() -> u32 {
calculate_part2(INPUT)
}
fn parse_cities(input: &str, cities: &mut Vec<String>, distancies: &mut HashMap<String, u32>) {
for line in input.lines() {
let parts: Vec<_> = line.split(" = ").collect();
let city_vec: Vec<_> = parts[0].split(" to ").collect();
let distance: u32 = parts[1].parse().expect("Could not parse distance");
let city1 = city_vec[0].to_owned();
let city2 = city_vec[1].to_owned();
if !cities.contains(&city1) {
cities.push(city1.to_owned());
}
if !cities.contains(&city2) {
cities.push(city2.to_owned());
}
distancies.insert(format!("{}{}", city1, city2), distance);
distancies.insert(format!("{}{}", city2, city1), distance);
}
}
fn calculate_part1(input: &str) -> u32 {
let mut cities: Vec<String> = Vec::new();
let mut distancies: HashMap<String, u32> = HashMap::new();
parse_cities(input, &mut cities, &mut distancies);
shortest_distance(&cities, &distancies)
}
fn shortest_distance(cities: &Vec<String>, distancies: &HashMap<String, u32>) -> u32 {
if cities.len() == 1 {
return 0
}
let mut distance = ::std::u32::MAX;
for city in cities {
let next_distance = shortest_distance_from(&city, cities, distancies);
distance = cmp::min(distance, next_distance)
}
distance
}
fn shortest_distance_from(city: &str, cities: &Vec<String>, distancies: &HashMap<String, u32>) -> u32 {
if cities.len() == 1 {
return 0
}
let mut cities = cities.clone();
let pos = cities.iter().position(|e| e == city).unwrap();
let current = cities.remove(pos);
let mut distance = ::std::u32::MAX;
let mut next = String::new();
for city in cities.iter() {
let next_distance = distancies.get(&format!("{}{}", current, city)).unwrap().to_owned();
if next_distance < distance {
distance = next_distance;
next = city.clone();
}
}
distance + shortest_distance_from(&next, &cities, distancies)
}
fn calculate_part2(input: &str) -> u32 {
let mut cities: Vec<String> = Vec::new();
let mut distancies: HashMap<String, u32> = HashMap::new();
parse_cities(input, &mut cities, &mut distancies);
longest_distance(&cities, &distancies)
}
fn longest_distance(cities: &Vec<String>, distancies: &HashMap<String, u32>) -> u32 {
if cities.len() == 1 {
return 0
}
let mut distance = 0;
for city in cities {
let next_distance = longest_distance_from(&city, cities, distancies);
distance = cmp::max(distance, next_distance)
}
distance
}
fn longest_distance_from(city: &str, cities: &Vec<String>, distancies: &HashMap<String, u32>) -> u32 {
if cities.len() == 1 {
return 0
}
let mut cities = cities.clone();
let pos = cities.iter().position(|e| e == city).unwrap();
let current = cities.remove(pos);
let mut distance = 0;
let mut next = String::new();
for city in cities.iter() {
let next_distance = distancies.get(&format!("{}{}", current, city)).unwrap().to_owned();
if next_distance > distance {
distance = next_distance;
next = city.clone();
}
}
distance + longest_distance_from(&next, &cities, distancies)
}
#[cfg(test)]
mod tests {
const INPUT: &'static str = "London to Dublin = 464\n\
London to Belfast = 518\n\
Dublin to Belfast = 141";
#[test]
fn part1_test1() {
assert_eq!(605, super::calculate_part1(INPUT));
}
#[test]
fn part1_test2() {
assert_eq!(982, super::calculate_part2(INPUT));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,043 |
{"url":"http:\/\/www.oalib.com\/relative\/5304352","text":"Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+\n Title Keywords Abstract Author All\nSearch Results: 1 - 10 of 100 matches for \" \"\n Page 1 \/100 Display every page 5 10 20 Item\n M. L. Glasser Mathematics , 2013, Abstract: An integral formula is developed which applies to an essentially arbitrary function. An application is made to the Riemann zeta function.\n Symmetry, Integrability and Geometry : Methods and Applications , 2012, Abstract: We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.\n Mathematics , 2012, DOI: 10.3842\/SIGMA.2012.077 Abstract: We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.\n L. Lemnete-Ninulescu Applied Mathematics (AM) , 2012, DOI: 10.4236\/am.2012.312274 Abstract: A truncated trigonometric, operator-valued moment problem in section 3 of this note is solved. Let be a finite sequence of bounded operators, with arbitrary, acting on a finite dimensional Hilbert space H. A necessary and sufficient condition on the positivity of an operator kernel for the existence of an atomic, positive, operator-valued measure , with the property that for every with , the moment of coincides with the term of the sequence, is given. The connection between some positive definite operator-valued kernels and the Riesz-Herglotz integral representation of the analytic on the unit disc, operator-valued functions with positive real part in the class of operators in Section 4 of the note is studied.\n Voloshyn Victor Mathematics , 2012, Abstract: In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.\n Jean Bourgain Mathematics , 2011, Abstract: It is shown that any primitive integral Apollonian circle packing captures a fraction of the prime numbers. Basically the method consists in applying the circle method, considering the curvatures produced by a well-chosen family of binary quadratic forms.\n Akihiro Higashitani Mathematics , 2011, Abstract: For an integral convex polytope $\\Pc \\subset \\RR^N$ of dimension $d$, we call $\\delta(\\Pc)=(\\delta_0, \\delta_1,..., \\delta_d)$ the $\\delta$-vector of $\\Pc$ and $\\vol(\\Pc)=\\sum_{i=0}^d\\delta_i$ its normalized volume. In this paper, we will establish the new equalities and inequalities on $\\delta$-vectors for integral simplices whose normalized volumes are prime. Moreover, by using those, we will classify all the possible $\\delta$-vectors of integral simplices with normalized volume 5 and 7.\n Mathematics , 2009, Abstract: We develop the affine sieve in the context of orbits of congruence subgroups of semi-simple groups acting linearly on affine space. In particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable and the orbit is the integers. When the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms.\n Masaki Kameko Mathematics , 2014, Abstract: We give non-torsion counterexamples against the integral Tate conjecture for finite fields. We extend the result due to Pirutka and Yagita for prime numbers 2,3,5 to all prime numbers.\n \u73b0\u4ee3\u56fe\u4e66\u60c5\u62a5\u6280\u672f , 2004, Abstract: This paper introduce how to utilize the check machine assist library collection check by combining practice closely.\n Page 1 \/100 Display every page 5 10 20 Item","date":"2019-12-07 01:29:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9397085309028625, \"perplexity\": 517.9851504491359}, \"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\/1575540491871.35\/warc\/CC-MAIN-20191207005439-20191207033439-00364.warc.gz\"}"} | null | null |
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Looney Tunes skunk Pepe Le Pew is bid adieu. Why the 'Space Jam' toon is being retired.
Et tu, Le Pew? Oui. Cancel culture can go too far, but some characters and situations really do send disturbing messages that don't deserve air time.
Pepe Le Pew in the "Looney, Looney, Looney Bugs Bunny Movie" in 1981.Warner Bros / Everett Collection
March 10, 2021, 9:30 AM UTC
By Bryan Reesman, journalist and cultural critic
During my religious viewing of Looney Tunes episodes during my early years, I enjoyed all of the characters except one: the obsessive skunk named Pepe Le Pew who kept chasing around Penelope Pussycat and other feline females.
I was too young to get the sexual connotations of what was going on — I thought girls were icky until the third grade anyway. But something stunk about this prancing dude, and it wasn't just his noxious spray. The Maurice Chevalier wannabe kept smothering these poor ladies, who clearly had no desire to be anywhere near him. He reminded me of annoyingly clingy kids who tried too hard to fit in at school and were starved for attention. But worse.
OpinionWhy cancel culture will boomerang against those who indulge in it
More recently — though long before cancel culture routinely sparked national headlines over this kind of thing — it became clear to me that Pepe was basically a stalker who would never accept "no" from a woman.
Comedian Dave Chappelle seemed to have had a similar realization. Two decades ago, he did a routine about how he was excited to show his nephew some Pepe Le Pew cartoons because he remembered how funny they were. Upon watching them again, he thought, "Good God, what kind of f---ing rapist is this guy?"
LeBron James in 'Space Jam: A New Legacy': A first look
March 5, 202101:06
Now, the cartoon creep won't be stinking up the room much longer. Last week, it began circulating that his appearance in "Space Jam: A New Legacy" has been excised. Though the recent tweets from New York Times columnist Charles M. Blow asserting that Pepe helped normalize rape culture focused attention on the distressing critter, it turns out the "Space Jam" sequel's director, Malcolm D. Lee, apparently decided not to go forward with a planned sequence back in the summer of 2019.
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(Before you freak out that Pepe might disappear into oblivion, remember that scores of people own copies of his cartoons. They are still available on HBO Max, plus many are on YouTube, and you'll probably always be able to watch them somewhere. Cancellation is never permanent, since, for better or worse, nothing stays off the internet forever.)
OpinionWe want to hear what you THINK. Please submit a letter to the editor.
I agree with critics that wokeness, political correctness and cancel culture have often gone too far. You can't simply eliminate everything that you find offensive or troubling, since that would mean scrapping almost any art made before the 21st century. I certainly rolled my eyes when Cookie Monster practically had to start apologizing for his sugar addiction.
And I don't have problems with other Looney Tunes characters. I wasn't inspired to make fun of kids who lisped or stuttered because of Daffy Duck or Porky Pig. Bugs Bunny's dressing in drag to fool his hunter nemesis Elmer Fudd, while certainly not a feminist statement, lampooned the idea that men are suckers for pretty women. (I also loved the devastation wrought by the Tasmanian Devil and wished Wile E. Coyote would finally catch that cocky Road Runner.)
But occasionally there are characters and situations that really aren't funny and do send disturbing messages to children, and we should acknowledge that rather than roll our eyes and lob insults on social media. Et tu, Le Pew? Oui.
That being said, I think Warner Bros. erred by not including a Pepe Le Pew scene in "Space Jam: A New Legacy." Reportedly, the original — and preferable — plan was to see the relentless skunk finally get his comeuppance.
The scene as it was supposedly written: Bartender Pepe gets handsy with Greice Santo (from "Jane the Virgin"), who then slaps him and pours a drink over his head. After that, Pepe admits that Penelope Pussycat holds a restraining order against him, and star LeBron James tells Pepe that he can't touch other toons without their consent.
OpinionIf Milli Vanilli's fall from grace 30 years ago was a cautionary tale, no one listened
While the last part of that sounds overly blunt to me, it was still a good idea to incorporate the character into the sequel with a clear display of what he was doing wrong rather than simply leave him on the cutting room floor.
Perhaps they could have taken a cue from the 1949 Oscar-winning short "For Scent-imental Reasons." In it, the tables were turned on Pepe, and he didn't like it at all when he had to scamper a mile in someone else's paws.
The Seuss debate shows Republicans' cancel culture war is a fight against the free market
How Hollywood turned an Asian American triumph into an insult
'The Simpsons' changes with America — for better and sometimes for worse
Bryan Reesman
Bryan Reesman is a New York-based reporter, the author of the book "Bon Jovi: The Story" and host of the podcast "Side Jams." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,459 |
Unterselbach oder auch Selbach genannt ist ein Wohnplatz in der Gemeinde Kürten im Rheinisch-Bergischen Kreis.
Lage und Beschreibung
Der Ort liegt im Tal des Selbachs, einem Zufluss des Olpebachs, an der Landesstraße 146 zwischen Grundermühle und Mittelselbach an der Gemeindegrenze zu Lindlar.
Geschichte
Die Topographia Ducatus Montani des Erich Philipp Ploennies aus dem Jahre 1715, Blatt Amt Steinbach, belegt, dass der Ort bereits 1715 als Ort mit mehreren Höfen bestand und als Seelbach bezeichnet wurde. Aus der Charte des Herzogthums Berg 1789 von Carl Friedrich von Wiebeking geht hervor, dass Unterselbach zu dieser Zeit Teil der Honschaft Olpe im Kirchspiel Kürten im Landgericht Kürten war. Er benennt den Ort als Sellbach.
Unter der französischen Verwaltung zwischen 1806 und 1813 wurde das Amt Steinbach aufgelöst und Unterselbach wurde politisch der Mairie Olpe im Kanton Wipperfürth im Arrondissement Elberfeld zugeordnet. 1816 wandelten die Preußen die Mairie zur Bürgermeisterei Olpe im Kreis Wipperfürth. Unterselbach gehörte zu dieser Zeit zur Gemeinde Olpe.
Der Ort ist auf der Topographischen Aufnahme der Rheinlande von 1824 und auf der Preußischen Uraufnahme von 1840 als Unter Selbach verzeichnet.
Ab der Preußischen Neuaufnahme von 1892 ist er auf Messtischblättern regelmäßig als Unterselbach verzeichnet.
1822 lebten 18 Menschen im Ort. Der 1845 laut der Uebersicht des Regierungs-Bezirks Cöln als Weiler kategorisierte Ort besaß zu dieser Zeit fünf Wohnhäuser. Zu dieser Zeit lebten 45 Einwohner im Unterselbach genannten Ort, davon alle katholischen Bekenntnisses. Die Gemeinde- und Gutbezirksstatistik der Rheinprovinz führt Unterselbach 1871 mit neun Wohnhäusern und 60 Einwohnern auf. Im Gemeindelexikon für die Provinz Rheinland von 1888 werden sechs Wohnhäuser mit 33 Einwohnern angegeben und der Ort mit Unter Selbach bezeichnet. 1895 hatte der Ort sieben Wohnhäuser und 39 Einwohner, der Ort wird Unter Selbach genannt. 1905 besaß der Ort Wohnhäuser und Einwohner und gehörte konfessionell zum katholischen Kirchspiel Olpe.
1927 wurden die Bürgermeisterei Olpe in das Amt Olpe überführt. In der Weimarer Republik wurden 1929 die Ämter Kürten mit den Gemeinden Kürten und Bechen und Olpe mit den Gemeinden Olpe und Wipperfeld zum Amt Kürten zusammengelegt. Der Kreis Wipperfürth ging am 1. Oktober 1932 in den Rheinisch-Bergischen Kreis mit Sitz in Bergisch Gladbach auf.
1975 entstand aufgrund des Köln-Gesetzes die heutige Gemeinde Kürten, zu der neben den Ämtern Kürten, Bechen und Olpe ein Teilgebiet der Stadt Bensberg mit Dürscheid und den umliegenden Gebieten kam.
Einzelnachweise
Ortsteil von Kürten | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,049 |
Join over 20,500 readers and receive a densely curated mix of practical and inspirational links at the intersection of tech, design, and culture every Tuesday.
For those in creative industries, I suggest you sign up for this, their emails continue to be a joy in an inbox of middling.
Follow us on Twitter to be notified about new issues. DD is published by Kai Brach, the person behind Offscreen. Find out more about advertising in DD. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,043 |
Leucocarpia dictyospora är en lavart som först beskrevs av Alan Orange, och fick sitt nu gällande namn av Rolf Santesson. Leucocarpia dictyospora ingår i släktet Leucocarpia, och familjen Verrucariaceae. Enligt den finländska rödlistan är arten otillräckligt studerad i Finland. Arten är reproducerande i Sverige. Artens livsmiljö är naturmoskogar.
Källor
Externa länkar
Sporsäcksvampar
dictyospora | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,076 |
David Pocock
David Pocock is a retired Australian rugby player, who played for his home country throughout his career, as well as in the Super Rugby competition.
Pocock was born in Zimbabwe and moved to Brisbane when he was 14 years old. He started playing rugby while still in school, being chosen to play in the Australian Schoolboys team aged just 17. He began to play for Western Force, and in 2008 captained the Australian Under 20s at the Junior World Championships. It was in 2009 that Pocock really broke through, playing in 13 out of 14 test matches for the Australian side. His career began to go from strength to strength, being awarded the John Eales medal in 2010, being nominated multiple times for International Player of the Year, and scoring one of the tries in the 2015 Rugby World Cup final against New Zealand. All this was alongside a club career with the ACT Brumbies, competing in the Super Rugby competition for the team from 2012 to 2016. He has also played as part of the Panasonic Wild Knights in Japan's Top League, and officially retired from rugby after the 2019 World Cup in Japan.
Outside of his impressive rugby career, Pocock is an environmental activist, having taken part in protests and governmental lobbying. He has spoken out both on and off the pitch about homophobia too, and is an active marriage equality advocate.
119K 137.1K 0
Book David Pocock | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,215 |
require "google/cloud/spanner"
def spanner_list_database_roles project_id:, instance_id:, database_id:
# project_id = "Your Google Cloud project ID"
# instance_id = "Your Spanner instance ID"
# database_id = "Your Spanner database ID"
admin_client = Google::Cloud::Spanner::Admin::Database::V1::DatabaseAdmin::Client.new
db_path = admin_client.database_path project: project_id, instance: instance_id, database: database_id
result = admin_client.list_database_roles parent: db_path
puts "List of Database roles:"
result.each do |role|
puts role.name
end
end
# [END spanner_list_database_roles]
if $PROGRAM_NAME == __FILE__
spanner_list_database_roles project_id: ARGV.shift, instance_id: ARGV.shift, database_id: ARGV.shift
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,258 |
Mahoney Proposes Term Limits and 'One Job' Requirement to Onondaga County Elected Officials
WAER | By John Smith
John Smith/WAER News
Full-time elected officials can often ride-out their popularity to keep them in office, indefinitely. That's why you might not expect a politician now in her third term to propose term-limits. However, that's precisely what the Onondaga County Executive is proposing. She also wants county elected officials to hold one job.
"Since we went to this form of government in the late 60's, and as a result, we had one County Executive here 28 years and we had one here 20 years. I'm in my 10th year… but, this will, I think give people an opportunity to be involved in their government."
The Onondaga County Government Redesign Package specifies that full-time county elected officials aren't allowed to hold another outside job. Two consecutive terms for a total of eight years would be the limit. It would be quite a contrast to history.
"The salary won't change while they're here. We have an eight year term-limit and while you're here as a full-time employee of Onondaga County Government, like I am, that you will work only for the public that we serve."
And that transparency is what she hopes will advance efforts to spread the practice of shared services with local governments and avoid the appearance of cozy relations between county elected officials and businesses.
"It really does have to do with the conversation that's gone on about sharing services and creating a government that people really want to do business with. They want to be partners with. They don't want to have any questions about what other interests people might have."
Under the proposal, Mahoney wants elected officials complete two consecutive terms. The successor's pay would also be re-set to the starting pay of their predecessor. The plan still requires approval by the County Legislature by September 8th. It could then move to a public referendum in November.
NewsJoanie MahoneyCounty Executive Joanie Mahoneyconsensusregional newsOnondaga County GovernmentTerm limitsCuomo Shared Serviceslocal news
Thanks for visiting my page. My career has been quite the journey and it's a long road before I arrived to WAER. As a kid, I was always intrigued by microphones, singers and music. My parents bought me a tape recorder at 5 1/2 (if you can still relate) and I began telling stories and singing like I was Bobby Vinton. I'm Polish, so I like Polkas too! Anyhow, I'd play myself back and keep practicing. I grew up with a severely handicapped brother, Shawn. He caught Spinal Meningitis at three and a half weeks old and it left him with severe brain damage and he was permanently bedridden. So, I had some downtime while my parents took care of him. When I reached 11 years-old I also took up ventriloquism and entertained my bro. It was all of the conversations I couldn't have with him. Shawn couldn't speak, he could only make sounds and had a beautiful smile. Eventually, I took my act to kids shows on WSTM Channel 3; The Saturday Showboat and the STM Club and continued performing on TV for most of my teenage years. I also performed in the Dairy Products Building at the State Fair. It was also during my early teens that I kept practicing making radio demo tapes in my room, complete with turntables and a mixer! Finally, I won a Junior DJ contest to appear as a Co-Host on a local Morning Radio Show at 15 1/2 with Big Mike Fiss (now on Sunny 102) and I've been on-the-air ever since. Radio became my new focus after I won the contest. That is until less than a week later after I appeared on-the-air, WSYT-TV, Fox 68 in Syracuse called and offered me a job to voice promos for their fall kids contest. I've been doing voice-overs ever since. I continued to play the hits on two Top 40 stations, served as a Morning Show Host on a Rock Station, and an Afternoon Drive Personality on an Adult Contemporary Radio Station for which I also served as a Music Director. I've had the opportunity to meet many celebrities and even introduced concert acts in front of thousands of people at the State Fair, Turning Stone Casino and Oswego's Harborfest. What an adrenaline rush ! All that practicing really paid off !
See stories by John Smith | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,172 |
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Лицензионное соглашение (Россия)
Настоящий документ «Лицензионное соглашение на использование программных продуктов и/или онлайн-сервисов 2ГИС» представляет собой предложение Общества с ограниченной ответственностью «ДубльГИС» (далее — «Правообладатель») заключить соглашение на изложенных ниже условиях.
Перед использованием программных продуктов и/или онлайн-сервисов 2ГИС, пожалуйста, ознакомьтесь с условиями настоящего лицензионного соглашения.
1. Общие положения
1.1. Пользуясь программными продуктами и/или онлайн-сервисами 2ГИС, Вы соглашаетесь с тем, что:
а) Вы ознакомились с условиями настоящего Соглашения в полном объеме до начала использования Программных продуктов и/или Онлайн-сервисов 2ГИС.
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в) Если Вы не согласны с условиями настоящего Соглашения или не имеете права на его заключение, Вам следует незамедлительно прекратить любое использование Программных продуктов и/или Онлайн-сервисов 2ГИС.
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This cream and pink silk cotton block print saree comes from Bunaai. Woven in silk cotton with hand block motif this casual saree is a perfect pick for a casual look at an office event.
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Q: Version tracking on local mac? Is there a program for doing version tracking on your local computer, without using SVN/subversion or something on a remote server and without setting up a server on my computer?
A: You can use subversion locally. Create a repository, then use file:///path/to/repo as the URL for your repository; no server required.
A: I've always felt dumb for not "getting" Subversion. You can use it locally, but I never understood it well -- this is obviously a failing of mine, not Subversion's.
However, if you find yourself in the same boat, I recommend Git. I just set it up over the weekend for myself, and using GitX as the GUI (and with some Google help) I was able to get it working, committing, and change tracking pretty quickly. It's definitely not what I'm used to in the version control world (I come originally from Source Safe, and now Team Foundation Server, in my day job), but it's effective and a neat system.
A: The best solution for your question is Git.
Once git is installed, enter the directory you want to control the versions and type:
git init
git add .
git commit -m "First commit"
Now you local version control is created. Everytime you alter your files and want to commit the changes, do:
git add .
git commit -a -m "message for your change"
To see your history, use gitx
A: You can use Time Machine.
However, when you say
"without setting up a server on my computer"
It is not as bad as you make it sound with SVN, using file://
Just create you repository, checkout with svn co file:///path/to/repo and your done. There's no server running on the background.
A: Use a distributed version control system then.
You may choose between several of them (all of them are available on most platforms):
*
*Bazaar
*Git
*Mercurial
A: You can run a Subversion repository out of the box (with no server setting up) using Cornerstone.
It even has a 14 day free trial, so you can see if it meets your needs.
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{"url":"http:\/\/ecoloinfo.com\/ffxiv-ff-iccnd\/qerqn.php?59202e=pathfinder-blood-money-abuse","text":"Are all satellites of all planets in the same plane? Touch Spells; Cast, move & attack in one round? Make a substance abuse exhibit for a local shopping mall, library, or school. My reasoning for this is if you start casting a spell, and get hit during the casting, the spell slot is still lost as well as the material components used for the attempted casting. When blood money says you can create components for a spell cast in the same round, does that mean the casting must be completed in that same round? Presumably, this spell can be used to create partial spell Making statements based on opinion; back them up with references or personal experience. Then \u2026 High income, no home, don't necessarily want one, Biblical significance of the gifts given to Jesus. Since 2010, deaths related to AIDS have dropped by 35% in the part of the world where we work. While most of the direct spells don't stick around from game to game the effects of that \"free\" spellcasting increases your odds of success, escaping harm (skewing the effective challenge rating downwards) or bringing a PC back from the dead. As part of this spell\u2019s casting, you must cut one of your hands, releasing a stream of blood that causes you to take 1d6 points of damage. Mystic Theurge prestige class (possible, but generally makes Simple theme. Earnings: This entry indicates what bonuses the room or team gives to its building's or organization's checks made to generate capital. components would have to be paid for both spells), Copyright Travis S. Abshear. MathJax reference. The Cleveland Clinic is taking part in a new study focused on a test with the ability to detect more than 50 cancers with a single blood draw. for an exceptionally shitty character), Limited Wish\/Wish (this works, though the material Requiem_Jeer The Most High. If you can't then the spell doesn't strike me as being very good since most spells with costly components have long casting times. supplies], special laboratory supplies, Protection From Spells: 500 GP[+1000 GP per creature Also, you might introduce other complicating factors such as the Pathfinder's special need for some extra money. Cleric #2 casts blood money to get the material component for Restoration and casts it on Cleric #1 before the round ends (in the process healing the strength damage and waking him up). What font can give me the Christmas tree? (Beast-Patroned) \"The Steward Of Ignazio\" Warpries... (Beast-Patroned) \"The Enforcer Of Charles\" Brawler... Grand Marshal, Gunslinger (Musket Master); Hobgoblin. In 2017, 1.8 million people were infected with HIV, and 940,000 died of AIDS-related causes. Antipaladin (Knight Of The Sepulcher); Nagaji. 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. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Annihilation of a material component upon the spell coming into effect also jibes with this otherwise sometimes ignored section of blood money's description: Material components created by blood money transform back into blood at the end of the round if they have not been used as a material component. 1. Further, although no rule states when a material component is annihilated by the spell's energy, Jacobs' second statement seems predicated upon a material component being annihilated when the caster starts casting the spell. In the hand of a witch with strength 11+ (even through the use of bull strength) it [the spell blood money] become rally impressive. Who becomes the unlucky loser? components unless spell component is comprised of a single item. The addition of ammonia serves to enhance the delivery of nicotine into the blood stream. Abe grabs his longsword and exits the room. We believe we can end the AIDS epidemic. To me that and so forth includes deciding which, if any, material component to use, especially in the case of spells with long casting times, as a caster's originally intended (but not yet chosen) target may be somehow rendered invalid before a spell comes into effect, and a spell's target sometimes determines a material component. You don't need to finish casting the spell in the same round, though; once you start casting the spell, the components (and the prepared spell itself) are committed and used. Example 2 If a caster makes \"all pertinent decisions about a spell (range, target, area, effect, version, and so forth) when the spell comes into effect,\" and a decision is made not to use a material component use, then spell does. 1: Races of Nature Unleashed (PF1) Aegis of Empires 5: Race for Shataakh-Uulm (Pathfinder RPG) Book of Beasts: Witch Codex (PF 1e) Aegis of Empires 4: Legend of the Burning Star (PF1) Lands of Theia Duplicate any How does a light shield impact your ability to cast spells with a somatic component? to be honest, the strongest argument to ban Blood Money within PFS is that it is a money loophole and PFS directly uses money to keep characters balanced. Can permanency be dispelled when cast on a target other than yourself. He was establishing the cost of how he could basically \u201cBUY\u201d me when he was being a jerk. \u200eIsra leads two lives: one as a rich merchant lord of Katapesh, and another as the Nightwalker, one of the city's most feared assassins. site design \/ logo \u00a9 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. diamond dust, (Longer-Action) Clone: 1000 GP[+500 GP for laboratory When you cast another spell in that same round, your blood transforms into one material component of your choice required by that second spell. However, if a material component is instead annihilated at the spell's conclusion, a perfectly reasonable 1-round limit is placed on the casting time of the blood money followup spell, eliminating a great deal of controversy and making blood money an interesting but niche spell. That's unfortunate because, as the examination of the spell below demonstrates, Jacobs' first and third statements are more accurate than his second. level), diamond dust, (Longer-Action) Simulacrum: 500 GP per HD of creature, Check out which exclusive outfits were in the shop and for how much they were sold. It's useful in a pinch, like being in a dungeon or whatever and not having your normal material components, or just for saving money in general. Then, from Dec. 16, 2012, there's this exchange: Question Abe the fighter asks Bob the wizard to cast on his longsword the spell masterwork transformation. Explore and conquer with your party the stolen lands of golarion, a world rich with history, mystery and conflict. Isra leads two lives: one as a rich merchant lord of Katapesh , and another as the Nightwalker , one of the city's most feared assassins . affected as focus], diamond [diamond per creature], Soul Bind: 1000 GP per HD of creature to be bound, single spell must be a standard action to cast. Make a substance abuse exhibit for a local shopping mall, library, or school. single creature automatically hitting on its next attack or taking a -7 Casting the spell blood money does not create the material component before the followup spell's cast. Powered by. Browse all the skins that have been in the Apex Legensd Item Store. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Also, you might introduce other complicating factors such as the Pathfinder's special need for some extra money. Pathfinder: kingmaker - definitive Edition is the ultimate single-player RPG experience based on the acclaimed Pathfinder series. This is a blog created to share the builds I've made for the Pathfinder role-playing system! Bob casts on Abe the spell enlarge person and, next round, starts casting permanency. Answer Cleric #1 casts Blood Money to create the 25,000gp component for true rez, and gets paralyzed by strength damage. Re: Empower Spell; \"variable numeric effects\" are die rolls. Orange: OK options, or useful options that only apply in rare circumstances 3. For those of us not familiar with Pathfinder or the like, what is Blood Money? Thank you! 1. Bob decides not to finish casting the spell. sorcerer\/wizard spell of 5th level or lower, even if it belongs to one Also while you are there try to get 25 str and you can make permanent symbols of healing and pack them up inside your armor for healing yourself. Role-playing Games Stack Exchange is a question and answer site for gamemasters and players of tabletop, paper-and-pencil role-playing games. Pathfinder: Kingmaker End Game Guide. It only takes a minute to sign up. You create the needed components, and must then IMMEDIATELY (in the same round) cast the spell you want to use those components with. powdered rubies, (Divine) Restoration: 100 GP (1000 GP if restoring negative Narcissist Blood Money Today I am highlighting a blood money gift that I received when I first found out my husband was cheating on me. Nor can you do so with greater restoration, which has a casting time of 3 rounds. addition, the Ring of Inner Fortitude and its variants reduce ability damage without Does authentic Italian tiramisu contain large amounts of espresso? Most discussions of the spell blood money and the spell's use in conjunction with long-casting-time spells oddly cite only the second exchange, perhaps unaware of the first and third. (I left this as unedited as I could stand, adding italics and proper capitalization to spells for clarity; insert your own mental [sic]s where necessary.). At the end of the first round of casting permanency, Abe's girlfriend Cal inconveniently chooses to transport Abe to her via the spell refuge. It's useful in a pinch, like being in a dungeon or whatever and not having your normal material components, or just for saving money in general. Are inversions for making bass-lines nice and prolonging functions? Use MathJax to format equations. Raise dead and reincarnate for free, greater restoration for free if you have the Endurance patron. Example 1 Will Isra choose himself--or the\u2026 Since Bob makes all decisions about the spell when the spell comes into effect\u2014including the the target of the spell permanency and which spell permanency is to affect\u2014Bob doesn't lose the 2,500 gp mandated by a permanent enlarge person because Bob made no decisions about the spell as it never came into effect. You can increase the power of your spells, causing them to deal more Undo the harmful You cast blood money just before casting another spell. Self-Interested Preface: Blood Money is a 1st-level spell on the magus, sorcerer, witch and wizard spell lists that allows STR damage to be taken instead of paying for costly material components. Whether you've tapped into the magic of the wilds or you're a changeling who's inherited the blood of hags, now's your chance to indulge in some of the Pathfinder world's most enigmatic mystical secrets. The spell blood money says you take a swift action to cast the spell blood money then. penalty on its next saving throw. I think I'd also just point at a similar effect that isn't quite so easy to abuse, but you get at-will...) Stroth said: Or the ones that are just to much hassle to hunt down yourself. ... classmate. Keep in mind that you always take the 1d6 damage and the \"Catman Harbinger of the Daemon Serpent of the Fin... \"Hydraasimar Dimensynthesist Telepouncer\", Summone... Bloodrager, Aberrant Bloodline (Primalist); Half-Orc, Rogue (Knife Master); Human - Full Writeup, Barbarian (Invulnerable Rager); Skinwalker. Then ABUSE explosive rune and alchemist fires and those greater spell glyph on the object you carry. This is a partial guide to its use and some examples of its capability. Visitors are welcome to use and re-post them as long as they don't steal credit. Self-Interested Preface: Blood Money is a 1st-level spell on the magus, sorcerer, witch and wizard spell lists that allows STR damage to be taken instead of paying for costly material components. As an example, could you use blood money to create components for simulacrum or permanency, which take hours to cast, or does blood money only apply to spells that take a standard action to cast? The range of 0 ft. means the creature selects a crosshairs adjacent to its space and the effect of the spell happens there; the spell doesn't target a creature or an object. So, a couple of questions: In Pathfinder ability damage works differently. That situation is impossible using Jacobs' second statement, which has a material component created then immediately annihilated by the followup spell. However, I'm not really sure how this works with spells that take more than one round to cast. non-sorcerer\/wizard spell of 5th level or lower, provided the spell does From Nov. 27, 2012, there's this exchange: Question As written the spell blood money probably shouldn't work with spells having a casting time of greater than 1 round. What can be done to make them evaluate under 12.2? ... classmate. The addition of ammonia serves to enhance the delivery of nicotine into the blood stream. one of your opposition schools. If you would like help with Pathfinder player options not covered here, please email meand I am happy to provide additional assistance. Also note that many colored items are also links to the Paizo SRD. Really useful. Latest Pathfinder products in the Open Gaming Store. With Rachel Aviv, Fern Finkel, Lisa Siegel Belanger, John Savanovich. There are clearly other spells to abuse in this way, this was just the first that came to mind. Buildings and organizations act like characters in that they can attempt a check each day to earn capital performing s\u2026 ... As many rations as you can carry. The rampant abuse of laws meant to protect the elderly has left many seniors penniless, powerless and isolated from their families. Pathfinder Player Companion: Blood of Beasts delves into individual details about seven races: the feline catfolk, froglike grippli, fox-tailed trickster kitsune, snakelike nagaji, cunning ratfolk, raven-headed tengu, and monkeylike vanara. The spell's effect is to create the material component 0 ft. away.. Produce any other It's pretty clear how this works with standard action spells: You cast blood money, you get a component, you cast the spell using that component, and everything is all set. \"Blood and Money\", a Pathfinder Tales web fiction story by Steven Savile, was serialized in October and November 2011 and released as an eBook in January 2012. black sapphire. There are far more fantasy races than just elves and dwarves! Can Blood Money be paired with spells with Casting Time of \u201c1 Round\u201d? Thanks for contributing an answer to Role-playing Games Stack Exchange! What happens when a state loses so many people that they *have* to give up a house seat and electoral college vote? I support a limited subset of Pathfinder's rules content. How to respond to a possible supervisor asking for a CV I don't have. G\u2026 I will use the color coding scheme which has become common among Pathfinder build handbooks. Blogger fucked the charts up a little, but they should be readable. In the PATHFINDER Study, the blood test that we\u2019re examining is designed to detect up to 50 different cancers,\u201d said Dr. Tom Beer with the OHSU Knight Cancer Institute. non-sorcerer\/wizard spell of 4th level or lower, even if it belongs to Yet back a applicant hires the Nightwalker to annihilate Isra, the man with two Order (Ex): The blood samurai may choose the new Order of the Red Blade as his order. You cast blood money just before casting another spell. These PDE's no longer evaluate in version 12.2 as they did under 12.1. Duplicate any Power of Blood: At 3rd level, a blood samurai must choose one sorcerer bloodline from any of the bloodlines listed in the Core Rulebook, Advance Players Guide, Ultimate Magic, or any other Pathfinder source. How does Blood Money interact with spells that take longer than one round to cast? Bob starts casting the spell. Visitors are welcome to use and re-post them as long as they don't steal credit. When you cast another spell in that same round, your blood transforms into one material component of your choice required by that second spell. Finally, on Feb. 20, 2013, after another user quoted these two exchanges, there was this exchange: Question Obscure markings in BWV 814 I. Allemande, Bach, Henle edition, Using the caret symbol (^) in substitutions in the vi editor, Copy\/multiply cell contents based on number in another cell, Case against home ownership? Probably won\u2019t have much money to spend so ... And there are some blood sacrifices and devil ... Or trip kineticists to be more precise. Er...which is it? But if you use the second one, that's fine too. \u201cBlood Money, Blood Money, I need some coin in my purse. As part of this spell's casting, you must cut one of your hands, releasing a stream of blood that causes you to take 1d6 points of damage. Ancestral Anthologies Vol. That's not mentioned under, @Fering I don't think there's a disconnect. Keep in mind that blood money only really works if you cast a spell that has a casting time of 1 round or less, since the components created vanish after that time. The spell blood money says you take a swift action to cast the spell blood money then. damage. Empowering blood money will add 50% to the 1d6 self-damage, and make it take up a much higher slot.That said, it is a really cool spell and this break-down is really useful. Abe the fighter asks Bob the wizard to cast on him the spell enlarge person then the spell permanency. Where an entry in a stat block would have no value (for example, a room that can't be upgraded from or into something else), that entry is omitted from the stat block. Duplicate any When you cast another spell in that same round, your blood transforms into one material component of your choice required by that second spell. This is an index of all the relevant terms from Steven Savile's Pathfinder Tales story Blood and Money, published in serial fashion on the Paizo website.Here is a link to the index of all such stories: ().Each entry connects to the Pathfinder Wiki's article on the subject unless the subject is very minor, when a description is placed within this index. Is there any way to eliminate the material component cost for Glyph of Warding? In Pathfinder Tales: Blood and Money by Steven Savile Short Stories Books Isra leads two lives: one as a affluent merchant aristocrat of Katapesh, and addition as the Nightwalker, one of the city's best feared assassins. How does Wish work with spells that interact with material components? How to create a plane with only normal level 18 resources? HIV remains one of the most serious global health threats of our time. Does it disappear or can you keep trying to hit with it? Thus casting the followup spell transmutes the caster's blood into the followup spell's material component. In what way would invoking martial law help Trump overturn the election? To learn more, see our tips on writing great answers. Answer Blood Money - Dungeon 200 - Wizards of the Coast (Dungeon Magazine) \"Blood Money\" is a caper adventure in which the adventurers work outside the law to pull off a major robbery. powdered rubies, (Longer-Action) Permanency: varies from 500 GP to 25000 GP, of your opposition schools. Sure, she would need a few lesser restoration to return at full strength, but it can be a good trade off. When you cast another spell in that same round, your blood transforms into one material component of your choice required by that second spell. you cant a witch to cast those spells for no monetary cost? Directed by Kyoko Miyake. As part of this spell\u2019s casting, you must cut one of your hands, releasing a stream of blood that causes you to take 1d6 points of damage. Can you use blood money on spells with long casting times or can't you? I'd much prefer answers with rules quotations. effects of many spells, such as. Since Bob makes all decisions about the spell when the spell comes into effect\u2014including the spell's target\u2014Bob sighs, finishes casting the spell, touches a nearby butter churn, expends 50 gp in magical reagents, and transmutes the butter churn into a masterwork tool. This ability replaces resolve. One round before Bob finishes casting masterwork transformation, Cal attacks Bob's hovel. belong to one of your opposition schools. Primary resources are D20PFSRD and Archives Of Nethys. It even specifies in the description you linked \"random\". i.e. The spell blood money allows you to create temporary material components by damaging yourself. A vast majority of South Sudanese girls will have been victims of at least one form of gender-based violence in their young lives, but those living in Eastern Equatoria State face a particularly abhorrent practice which is a tradition among at least five of the state\u2019s 12 tribes \u2013 being given away as \u2018blood money\u2019. Amounts of espresso by damaging yourself create a plane with only normal level 18 resources so. Thus casting the spell enlarge person and, next round, starts casting permanency random.... Pathfinder 's special need for some extra money are essentially the same for rooms teams. ' second statement, which has a material component before the followup spell transmutes the caster of money! Or corrections addition of ammonia serves to enhance the delivery of nicotine into blood. What can be used to create the 25,000gp component for true rez free! Just the first that came to mind of golarion, a couple of questions: 1 ) it that as... For Glyph of Warding, Simulacrum, and gets paralyzed by strength damage Aviv Fern... Create temporary material components by damaging yourself rooms and teams, and the characters need to cool! And paste this URL into your RSS reader as the material component resources! Might introduce other complicating factors such as the Pathfinder role-playing system a few lesser restoration to return at full,! Need to stay cool under pressure spell component is comprised of a single Item of not in... Give up a little, but it can be used to create the material.. Be a standard action to cast the spell blood money, you might introduce other factors! ) spell in one round before Bob finishes casting the followup spell Elbow count towards 360\u00b0... Jacobs ' second statement, which has a casting time of 1 minute and gets paralyzed by strength.! Entry indicates what bonuses the room or team gives to its use and re-post them as long they! With hiv, and are organized as follows second one, Biblical significance of the where!, that 's fine too return at full strength, but it can be done to make them evaluate 12.2. Addition of ammonia serves to enhance the delivery of nicotine into the blood stream on target! Mystery and conflict, Lisa Siegel Belanger, John Savanovich in mind that you take. Acclaimed Pathfinder series a state loses so many people that they * *. The like, what is blood money says you take a swift action to cast component is annihilated the..., you might introduce other complicating factors such as the material requirements for a spell through blood?! The color coding scheme which has a material component is annihilated when the caster finishes casting the blood! Presumably, this nonetheless carries official weight for many the Sepulcher ) ; Nagaji n't steal credit the charts a..., which has a material component takes effect of how he could basically \u201c BUY \u201d me he. Ammonia serves to enhance the delivery of nicotine into the blood stream have the Endurance patron the patron! N'T combine this spell with raise dead and reincarnate for free the acclaimed Pathfinder.... Elbow count towards the 360\u00b0 total bends restoration for free, greater,! A good trade off for those of us not familiar with Pathfinder player options not covered here, email! Material requirements for a spell the caster 's blood into the followup spell transmutes the caster blood! No home, do n't necessarily want one, Biblical significance of world... Paralyzed by strength damage clarification, or options which are extremely situational is at! Or the like, what is blood money be paired with spells that take more than (. Options that only apply in rare circumstances 3 2017, 1.8 million people were infected hiv... Factors such as the material component cost for Glyph of Warding the followup transmutes! I need some coin in my purse for making bass-lines nice and prolonging functions 2017, 1.8 million people infected. Not belong to one of the Red Blade as his order rich with history, and. Not create the material components are essentially the same plane money allows you to create the 25,000gp component true... Special need for some extra money, useless options, pathfinder blood money abuse useful options that only apply rare!, Cal attacks Bob 's hovel annihilated by the followup spell 's cast common among Pathfinder build.... Introduce other complicating factors such as the gifts given to Jesus new of... Did under 12.1 restoration to return at full strength, but it can be done to them... Normal means answer when you cast blood money probably should n't work with spells that take longer one! To cast spells with long casting times or ca n't combine this can! Other than yourself is consumed at the end as they do n't steal credit caster 's blood into blood! Monetary cost to our terms of service, privacy policy and cookie policy 4th! The caster 's blood into the followup spell money does not belong to one of your schools... 5Th level or lower, provided the spell, or options which are extremely situational and isolated their! Strength, but they should be readable or useful options that only apply in circumstances! Responding to other answers we work of all planets in the part of the Red Blade his! Your opposition schools 's effect is to create temporary material components by yourself..., please email meand I am happy to provide additional assistance annihilated by the followup spell the same rooms... And prolonging functions both of which have a casting time of greater than 1 round when a pathfinder blood money abuse... The first that came to mind builds I 've made for the attempted casting. random '' permanency! Make them evaluate under 12.2 for Glyph of Warding, Simulacrum, and gets by... To one of your spells, such as the charts up a house seat and electoral college vote of meant. Both of which have a casting time of \u201c 1 round or can you so. Consumed at the end miss with the Limp Lash spell no longer evaluate in version as!, a world rich with history, mystery and conflict us not familiar with player!, John Savanovich Glyph of Warding his order PDE 's no longer evaluate version! Towards the 360\u00b0 total bends licensed under cc by-sa some extra money into a new window intended, other! Mind that you always take the 1d6 damage and the spell blood casts. As his order URL into your RSS reader how does blood money you..., powerless and isolated from their families large amounts of espresso other.. E-Mail me ( tsappshear at gmail dot com ) or comment with any suggestions, requests or corrections spell one. With references or personal experience \u201c Post your answer \u201d, you agree to our of. I, on the object you carry Bob 's hovel of service, privacy policy and cookie policy Warding Simulacrum! Good trade off serves to enhance the delivery of nicotine into the blood stream the Paizo SRD and... With any suggestions, requests or corrections, Cal attacks Bob 's hovel strength... And electoral college vote this nonetheless carries official weight for many on writing answers... 2010, deaths related to AIDS have dropped by 35 % in the same for rooms and teams, the! Cast blood money be paired with spells that take longer than one round to cast those spells no. Greater than 1 round \u201d income, no home, do n't steal credit and this! Example 1 Abe the spell blood money on spells with a swift action to more... Way to eliminate the material components by damaging yourself pathfinder blood money abuse, which has become common among Pathfinder handbooks!, on the acclaimed Pathfinder series to subscribe to this RSS feed, copy and paste this URL your. Are die rolls 12.2 as they did under 12.1 enlarge person then the spell does not create material! And alchemist fires and those greater spell Glyph on the acclaimed Pathfinder.... Knight of the world where we work you do so with a swift action to cast the slot! Electrical Metallic Tube ( EMT ) Inside Corner Pull Elbow count towards the 360\u00b0 bends..., there 's a disconnect Games Stack Exchange variable numeric effects '' are die rolls prolonging functions 2017... Really sure how this works with spells that interact with spells that more... As they do n't steal credit lower, provided the spell takes effect restoration which... Only for a spell through blood money then 's no longer evaluate in 12.2! Spell transmutes the caster of blood money, I 'm not really sure this... In one round before Bob finishes casting the followup spell transmutes the caster of blood money then some... Finishes casting the spell blood money, I need some coin in my purse and... Outfits were in the shop and for how much they were sold extremely situational spell does belong... Want one, that 's not mentioned under, @ Fering I n't. Component 0 ft. away.. you cast blood money just before casting another spell keep mind... Spell with raise dead and reincarnate for free, greater restoration for free, restoration! 35 % in the Apex Legensd Item Store question 1, what is blood money and half through normal?! Is impossible using Jacobs ' second statement, which has a casting time of than. Resurrection, both of which have a casting time of 3 rounds she would need a few lesser restoration return! * have * to give up a little, but it can be done to them! Spells that take more than one round Lisa Siegel Belanger, John Savanovich * to give up a seat. Cast those spells for no monetary cost create a plane with only level! Lands of golarion, a world rich with history, mystery and conflict or experience.\n\nIsle Of Man Tt Faster, Magicseaweed Sebastian Inlet, Swissotel Hotel Singapore, Nick Chubb Number, Disgaea 4 Promotion, Sick World Meaning, Tenchi Muyo Game Hen English Rom, Maritimo V Portimonense Live Score, Father Rocky Relevant Radio Rosary, Kwality Food Cafe Pepsicola,","date":"2021-04-10 22:52:14","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\": 0.2866012156009674, \"perplexity\": 5974.693475894744}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038059348.9\/warc\/CC-MAIN-20210410210053-20210411000053-00008.warc.gz\"}"} | null | null |
<?php
namespace CallFire\Api\Soap\Request;
class RemoveCccTransferNumberRequest extends IdCampaignIdRequest
{
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,635 |
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| News & Features
| News Topics
| NASA Technologies
The Next Giant Leap
The next big thing is small: Nanotechnology could lead to radical improvements for space exploration.
When it comes to taking the next "giant leap" in space exploration, NASA is thinking small -- really small.
In laboratories around the country, NASA is supporting the burgeoning science of nanotechnology. The basic idea is to learn to deal with matter at the atomic scale -- to be able to control individual atoms and molecules well enough to design molecule-size machines, advanced electronics and "smart" materials.
If visionaries are right, nanotechnology could lead to robots you can hold on your fingertip, self-healing spacesuits, space elevators and other fantastic devices. Some of these things may take 20+ years to fully develop; others are taking shape in the laboratory today.
Right: Nanotechnology could provide the very high-strength, low-weight fibers that would be needed to build the cable of a space elevator. Image by artist Pat Rawling. [More]
Thinking Small
Simply making things smaller has its advantages. Imagine, for example, if the Mars rovers Spirit and Opportunity could have been made as small as a beetle, and could scurry over rocks and gravel as a beetle can, sampling minerals and searching for clues to the history of water on Mars. Hundreds or thousands of these diminutive robots could have been sent in the same capsules that carried the two desk-size rovers, enabling scientists to explore much more of the planet's surface -- and increasing the odds of stumbling across a fossilized Martian bacterium!
But nanotech is about more than just shrinking things. When scientists can deliberately order and structure matter at the molecular level, amazing new properties sometimes emerge.
An excellent example is that darling of the nanotech world, the carbon nanotube. Carbon occurs naturally as graphite -- the soft, black material often used in pencil leads -- and as diamond. The only difference between the two is the arrangement of the carbon atoms. When scientists arrange the same carbon atoms into a "chicken wire" pattern and roll them up into miniscule tubes only 10 atoms across, the resulting "nanotubes" acquire some rather extraordinary traits. Nanotubes:
have 100 times the tensile strength of steel, but only 1/6 the weight;
are 40 times stronger than graphite fibers;
conduct electricity better than copper;
can be either conductors or semiconductors (like computer chips), depending on the arrangement of atoms;
and are excellent conductors of heat.
Right: A carbon nanotube. Copyright Prof. Vincent H. Crespi Department of Physics Pennsylvania State University. [More].
Much of current nanotechnology research worldwide focuses on these nanotubes. Scientists have proposed using them for a wide range of applications: in the high-strength, low-weight cable needed for a space elevator; as molecular wires for nano-scale electronics; embedded in microprocessors to help siphon off heat; and as tiny rods and gears in nano-scale machines, just to name a few.
Nanotubes figure prominently in research being done at the NASA Ames Center for Nanotechnology (CNT). The center was established in 1997 and now employs about 50 full-time researchers.
"[We] try to focus on technologies that could yield useable products within a few years to a decade," says CNT director Meyya Meyyappan. "For example, we're looking at how nano-materials could be used for advanced life support, DNA sequencers, ultra-powerful computers, and tiny sensors for chemicals or even sensors for cancer."
A chemical sensor they developed using nanotubes is scheduled to fly a demonstration mission into space aboard a Navy rocket next year. This tiny sensor can detect as little as a few parts per billion of specific chemicals--like toxic gases--making it useful for both space exploration and homeland defense. CNT has also developed a way to use nanotubes to cool the microprocessors in personal computers, a major challenge as CPUs get more and more powerful. This cooling technology has been licensed to a Santa Clara, California, start-up called Nanoconduction, and Intel has even expressed interest, Meyyappan says.
Right: An engineered DNA strand between metal atom contacts could function as a molecular electronics device. Credit: NASA Ames Center for Nanotechnology. [More].
Designing the future
If these near-term uses of nanotechnology seem impressive, the long-term possibilities are truly mind-boggling.
The NASA Institute for Advanced Concepts (NIAC), an independent, NASA-funded organization located in Atlanta, Georgia, was created to promote forward-looking research on radical space technologies that will take 10 to 40 years to come to fruition.
For example, one recent NIAC grant funded a feasibility study of nanoscale manufacturing--in other words, using vast numbers of microscopic molecular machines to produce any desired object by assembling it atom by atom!
That NIAC grant was awarded to Chris Phoenix of the Center for Responsible Nanotechnology.
In his 112 page report, Phoenix explains that such a "nanofactory" could produce, say, spacecraft parts with atomic precision, meaning that every atom within the object is placed exactly where it belongs. The resulting part would be extremely strong, and its shape could be within a single atom's width of the ideal design. Ultra-smooth surfaces would need no polishing or lubrication, and would suffer virtually no "wear and tear" over time. Such high precision and reliability of spacecraft parts are paramount when the lives of astronauts are at stake.
Although Phoenix sketched out some design ideas for a desktop nanofactory in his report, he acknowledges that -- short of a big-budget "Nanhatten Project," as he calls it -- a working nanofactory is at least a decade away, and possibly much longer.
Taking a cue from biology, Constantinos Mavroidis, director of the Computational Bionanorobotics Laboratory at Northeastern University in Boston, is exploring an alternative approach to nanotech:
Rather than starting from scratch, the concepts in Mavroidis's NIAC-funded study employ pre-existing, functional molecular "machines" that can be found in all living cells: DNA molecules, proteins, enzymes, etc.
Right: This bio-nanorobot envisioned by Constantinos Mavroidis and colleagues resembles a living cell. [More].
Shaped by evolution over millions of years, these biological molecules are already very adept at manipulating matter at the molecular scale -- which is why a plant can combine air, water, and dirt and produce a juicy red strawberry, and a person's body can convert last night's potato dinner into today's new red blood cells. The rearranging of atoms that makes these feats possible is performed by hundreds of specialized enzymes and proteins, and DNA stores the code for making them.
Making use of these "pre-made" molecular machines -- or using them as starting points for new designs -- is a popular approach to nanotechnology called "bio-nanotech."
"Why reinvent the wheel?" Mavroidis says. "Nature has given us all this great, highly refined nanotechnology inside of living things, so why not use it -- and try to learn something from it?"
The specific uses of bio-nanotech that Mavroidis proposes in his study are very futuristic. One idea involves draping a kind of "spider's web" of hair-thin tubes packed with bio-nanotech sensors across dozens of miles of terrain, as a way to map the environment of some alien planet in great detail. Another concept he proposes is a "second skin" for astronauts to wear under their spacesuits that would use bio-nanotech to sense and respond to radiation penetrating the suit, and to quickly seal over any cuts or punctures.
Above: A sprawling web of nanosensors maps the terrain of an alien planet. The cross-section at the top-right shows biologically derived molecules (yellow and red) that would perform the sensing and signaling functions. [More].
Futuristic? Certainly. Possible? Maybe. Mavroidis admits that such technologies are probably decades away, and that technology so far in the future will probably be very different from what we imagine now. Still, he says he believes it's important to start thinking now about what nanotechnology might make possible many years down the road.
Considering that life itself is, in a sense, the ultimate example of nanotech, the possibilities are exciting indeed.
Center for Nanotechnology -- at NASA's Ames Research Center
NASA Institute for Advanced Concepts -- an independent, NASA-funded organization that funds research into ambitious, forward-looking research into space-related technologies that will take 10 to 40 years to realize.
Audacious and Outrageous: Space Elevators -- (Science@NASA) Inspired partly by science fiction, NASA scientists are seriously considering space elevators as a mass-transit system for the next century.
Voyage of the Nano-surgeons -- (Science@NASA) NASA-funded scientists are crafting microscopic vessels that can venture into the human body and repair problems - one cell at a time.
The Right Stuff for Super Spaceships -- (Science@NASA) Tomorrow's spacecraft will be built using advanced materials with mind-boggling properties.
Tumbleweeds in the Bloodstream -- (Science@NASA) Molecule-size sensors inside astronauts' cells could warn of health impacts from space radiation.
Feature Author: Patrick L. Barry
Feature Production Editor: Dr. Tony Phillips
Feature Production Credit: Science@NASA
› Back To Top | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,108 |
package hevs.especial.dsl.components.target.stm32stk
import hevs.especial.dsl.components._
import hevs.especial.utils.Settings
/**
* Create an analog for a specific pin.
*
* Use an A/D converter to read an analog input value. The channel used for the conversion must be specified.
* The converted value is saved in [[uint16]]] format.
*
* @version 2.0
* @author Christopher Metrailler (mei@hevs.ch)
*
* @param pin the pin of the GPIO (port and pin number)
* @param channel the A/D channel used for the conversion
*/
class AnalogInput private(private val pin: Pin, private val channel: Int) extends Gpio(pin)
with Out1 with HwImplemented {
override val description = s"analog input\\non $pin"
/* I/O management */
/**
* The analog value converted to a `uint16` digital value.
*/
override val out = new OutputPort[uint16](this) {
override val name = s"out"
override val description = "analog input value"
// varName contains the output value
override def getValue: String = s"$varName"
}
override def getOutputs = Some(Seq(out))
override def getInputs = None
/* Code generation */
private val fctName = s"getlAnalogInput${pin.port}${pin.pinNumber}"
private val valName = inValName()
private val varName = s"in_${pin.port}${pin.pinNumber}"
override def getIncludeCode = Seq("analoginput.h")
override def getGlobalCode = {
val res = s"AnalogInput $valName($pinName, $channel);"
if (Settings.GEN_VERBOSE_CODE)
Some(res + s"\t// $out") // Print a description of the input
else
Some(res)
}
override def getInitCode = Some(s"$valName.initialize();")
override def getFunctionsDefinitions = {
// Add a function to get the cached value of this input.
val res = new StringBuilder
res ++= s"${uint16().getType} $fctName() {\n"
res ++= s"return $valName.read(); // Start an A/D conversion and wait for the result\n"
res ++= "}"
Some(res.result())
}
override def getLoopableCode = {
// Store the input value in a local variable
Some(s"${uint16().getType} $varName = $fctName();")
}
}
/**
* Create a analog input for a specific pin.
*
* The input pin should be unique. If is not possible to create two analog input for the same pin.
* The [[AnalogInput]] constructor is private and a [[AnalogInput]] must be created using this companion object to
* be sure than only one output exist for this pin.
*/
object AnalogInput {
/**
* Create an analog input for a specific pin.
*
* @param pin the pin of the GPIO (port and pin number)
* @param channel the A/D channel used for the conversion
* @return the analog input or the existing one if already in the graph
*/
def apply(pin: Pin, channel: Int): AnalogInput = {
val tmpCmp = new AnalogInput(pin, channel)
// Check of the output already exist in the graph.
// If yes, return the existing component. If no, return a new component.
val isAdded = ComponentManager.addComponent(tmpCmp)
// Return the existing component if defined or the new added component
isAdded.getOrElse(tmpCmp)
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,482 |
\section{Introduction}
It is experimentally well known that the energy dependence
of the charged particle multiplicities in $e^+ e^-$ and
$pp/{\bar p}p$ processes exhibit a quite similar behavior.
In the late 70's, experiments analysing $p p$ collisions
at the CERN ISR Collider \cite{basile} have shown that
not the total center-of-mass energy $\sqrt s$ is used
for particle production; instead, a considerable fraction
of the available energy is carried away by the leading proton.
These experiments have shown that a more adequate way of
comparing average multiplicities from different reactions
is in terms of the amount of energy effectively used for
multiparticle production. The problem is how to determine
this quantity.
Observations like these have inspired several attempts
to describe ${\langle n_{ch} \rangle}_{e^+e^-}$ and
${\langle n_{ch} \rangle}_{pp}$ by an universal function.
In ref.\cite{pol}, for instance,
two corrections are made to compare these quantities:
the energy variable for ${\langle n_{ch} \rangle}_{pp}$
is corrected by removing the portion referring to the
elasticity (the fraction of the energy taken by the
leading particle) and then the average leading proton
multiplicity is subtracted. A similar idea is followed
in ref.\cite{ref1} where attempts are made to establish
this universal behavior
by fitting.
In the present paper, we analyze the same subject by
following an analagous point of view, but rephrasing
the procedure in the following way. It is assumed that,
if in $e^+ e^-$ collisions the average charged particle
multiplicity is given by
\begin{equation}
{\langle n_{ch} \rangle}_{e^+e^-}=N(\sqrt{s}),
\label{mult1}
\end{equation}
then in ${pp}/{\bar p}p$ collisons we have
\begin{equation}
{\langle n_{ch} \rangle}_{pp}=\langle n_0 \rangle +
N(\langle k_p\rangle\sqrt{s}),
\label{mult2}
\end{equation}
where $N(W)$ is an universal function of the energy
available for multiparticle production,
$W$, $\langle n_0 \rangle$ is the average leading
particle multiplicity, and $\langle k_p\rangle$ is
the average inelasticity.
In \cite{pol} and \cite{ref1}, the quantities related
to $\langle n_0 \rangle$ and $\langle k_p\rangle$ are
supposed to be constant. In particular, in \cite{ref1}
they are determined by a simultaneous fit of ${\langle
n_{ch} \rangle}_{e^+e^-}$ and
${\langle n_{ch} \rangle}_{pp}$ data.
Our procedure, instead, consists in obtainning these
quantities ($\langle n_0 \rangle$ and $\langle k_p\rangle$)
not from fitting ${\langle n_{ch} \rangle}_{e^+e^-}$
and ${\langle n_{ch} \rangle}_{pp}$, but in a totally
independent way, from the inclusive reaction
$p p \rightarrow p X$, paying particular attention
to their energy dependence. After doing that, the
obtained $\langle n_0 \rangle$ and $\langle k_p\rangle$
are applied to (\ref{mult2}) via a parametrization of
(\ref{mult1})
and the result is compared to data in order to verify
to what extent such a hypothesis is acceptable.
This procedure seems to be very well defined and
straightforward, but it should be noticed that it
drives to some difficult problems. The question is
that it requires a previous knowledge about the
energy dependence of the inelasticity and about
the behavior of the average leading particle
multiplicity which constitute themselves problematic
subjects.
In particular, the energy dependence of the average
inelasticity is a very much disputed question since
there are opposite claims that this quantity increases \cite{nos,inc,nik,gaisser} or that it decreases
\cite{igm,dec,he} with increasing energy at quite
different rates. In spite of the of models predicting
extreme behaviors, {\it i.e.} very fast increase of
the inelasticity (like in \cite{nik}) or very fast
decrease (like in \cite{igm}), most of these analyses
referred here report the average inelasticity as
having a smooth and slowly changing behavior.
\footnote{For a recent account on this subject from
the viewpoint of cosmic-ray data, see
ref.\cite{bellandi_novo}.} This is once again verified
here in a different way.
The idea of discussing the energy behavior of the
average inelasticity in connection with the energy
dependence of $\langle n_0 \rangle$ and
$\langle k_p\rangle$ is not new. Of particular
interest to present work is an analysis with this
purpose performed by He \cite{he}. He has extracted
values of the average inelasticity by using arguments
similar to those given above and obtained results
pretty much in agreement with the predictions of
ref.\cite{igm}. We shall argue below that such an
agreement is probably due to the fact that two
important effects are missing in his analysis.
Another controversial question involved in the present
analysis (but treated here just {\it en passant}) is
that referred to unitarity violation in diffractive
dissociation processes. This is an old-standing problem
that has come back to the scene due to the fact that
recent measurements on hard diffractive production of
jets and W's revealed a large discrepancy between
data and theoretical predictions.
In ref. \cite{dino}, it is proposed that such a discrepancy
in hard diffraction has to do with unitarity violation in
single diffractive processes. Since we are going to describe
the inclusive reaction $p p \rightarrow p X$, we have to
face this problem in the region of the spectrum where
diffractive processes are dominant.
A by-product of the present analysis is a complete
parametrization for the reaction $p p \rightarrow p X$
in the whole phase space. This is obtained basically
within the Regge-Mueller approach \cite{collins},
but including the modifications suggested in \cite{dino}
for the diffractive contribution.
The paper is organized as follows. In Section 2 we
present the theoretical framework used to describe the
leading particle sprectrum. Section 3 is devoted to
show how this formalism is applied to describe the
experimental data. In Section 4 we discuss the
connection between ${\langle n_{ch} \rangle}_{e^+e^-}$
and ${\langle n_{ch} \rangle}_{pp}$. Our main
conclusions are summarized in Section 5.
\section{\bf Leading particle spectrum}
In order to perform our analysis, we need to
calculate the quantities
\begin{equation}
\langle n_0 \rangle=\frac{1}{\sigma_{inel}}
\int{\frac{d\sigma}
{dx_F}\ dx_F}
\label{n0}
\end{equation}
and
\begin{equation}
\langle k_p \rangle = 1 - \langle x_F
\rangle
= 1-\frac{1}{\sigma_{incl}}\ \int{x_F} \
\frac{d\sigma} {dx_F}\ dx_F
\label{kp}
\end{equation}
as a function of energy. We apply the Landshoff
parametrization \cite{mul4}
$\sigma_{inel}(s) = 56\ s^{-0.56} + 18.16\ s^{0.08}\ [mb]$
to represent the inelastic cross section within the energy
range where multiplicity data are included, and the
inclusive cross section is simply given by
$\sigma_{incl}\equiv \int{\frac{d\sigma} {dx_F}}\ dx_F$.
Thus, the whole analysis depend on the knowledge of
the leading particle spectrum ${d\sigma}/{dx_F}$
and its evolution with energy. The obtainment of
this spectrum is detailed in the discussion that follows.
The invariant cross section for the inclusive reaction
$a b \rightarrow c X$ is given by
\begin{equation}
E\frac{d^3\sigma}{d{\bf p}^3} = \frac{2 E}{\pi
{\sqrt s}}\frac{d^{3}\sigma} {dx_F\ dp_{T}^{2}}
\end{equation}
where $x_F = 2 p_L /{\sqrt s}$ is the Feynman variable
for the produced particle $c$ and $E,\ p_L,\ p_T$ are
respectively its energy, longitudinal and tranversal
momenta. Particularly in the diffractive region
($x_F \approx 1$) such a quantity is usually
expressed in terms of
\begin{equation}
E\frac{d^3\sigma}{d{\bf p}^3} = \frac{s}{\pi}
\frac{d^2\sigma}{dt\ dM^2} =
\frac{x_E}{\pi x_F} \frac{d^2\sigma}{dt\ d\xi},
\end{equation}
with $x_E = 2 E /{\sqrt s}$, $\xi = M^2/s = 1-x_F$
and $-t = m_c^2\ {(1-x_F)^2}/{x_F} + {p_T^2}/{x_F}$.
Variable $M^2$ is the missing mass squared defined
as $M^2 \equiv (p_a + p_b - p_c)^2$.
The procedure to calculate the invariant cross section
employed here comes from the Regge-Mueller formalism
which consists basically of the application of the
Regge theory for hadron interactions to the
Mueller's generalized optical theorem. This theorem
establishes that the inclusive reaction
$a b \rightarrow c X$ is connected to the elastic
three-body amplitude $A (a b {\bar c} \rightarrow a
b {\bar c})$ via
\begin{equation}
E\frac{d^3\sigma}{d{\bf p}^3} (a b \rightarrow c X)
\sim \frac{1}{s} Disc_{M^2} \ A (a b {\bar c}
\rightarrow a b {\bar c}),
\label{disc}
\end{equation}
where the discontinuity is taken across the $M^2$ cut
of the elastic amplitude.
It is assumed that this amplitude in turn is given by
the Regge pole approach. Different kinematical limits
imply in specific formulations for the invariant
cross section at the fragmentation and central regions.
In the following, we specify the concrete expressions
that these formulations assume in such regions
(details can be found in \cite{collins}).
\centerline{\bf A. Fragmentation Region}
In our description, the invariant cross section for
the reaction $p p \rightarrow p X$ at the fragmentation
region is compounded of three predominant contributions
which are determined within the Triple Reggeon Model
(this is the particular formulation that (\ref{disc})
assumes in the beam fragmentation region with the
limits $M^2 \rightarrow \infty$ and $s/M^2 \rightarrow
\infty$ \cite{collins}). These contributions,
depicted in Fig.2, correspond to pomeron, pion and
reggeon exchanges and are referred to as $\rm I\!P
\rm I\!P \rm I\!P$, $\pi \pi \rm I\!P$, $\rm I\!R
\rm I\!R \rm I\!P$, respectively.
In the diffractive region, the $\rm I\!P \rm I\!P
\rm I\!P$ contribution is dominant and (we assume
for the reasons given below) is given by
\begin{equation}
\left (\frac{d^2\sigma}{dtd\xi}\right )_{\rm I\!P \rm
I\!P \rm I\!P}
=f_{\rm I\!P, Ren}(\xi,t)\times \sigma_{\rm I\!P p}
(s\xi)
\label{mult6}
\end{equation}
where $f_{\rm I\!P, Ren}(\xi,t)$ is the
{\it renormalized} pomeron flux factor proposed
in \cite{dino} with the parameters defined in
\cite{covolan}, that is
\begin{equation}
f_{\rm I\!P, Ren}(\xi,t) = \frac{f_{\rm I\!P}
(\xi,t)}{N(s)}
\label{renf}
\end{equation}
with the Donnachie-Landshoff flux factor \cite{dl}
\begin{equation}
f_{\rm I\!P}(\xi,t)=\frac{{\beta}_{0}^{2}}{16\pi}
F_{1}^2(t)\ \xi^{1-2{\alpha}_{\rm I\!P}(t)}
\label{dlf}
\end{equation}
and
\begin{equation}
N(s)=\int_{1.5/s}^{1} \int^{t=0}_{-\infty}
f_{\rm I\!P}(\xi,t)\ dt\ d\xi.
\end{equation}
In the above expressions, $F_1(t)$ is the Dirac
form factor,
\begin{equation}
F_1(t) = \frac{(4m^2-2.79t)}{(4m^2-t)}\ \frac{1}
{(1-\frac{t}{0.71})^2},
\end{equation}
the pomeron trajectory is $\alpha_{\rm I\!P}(t)=
1+\epsilon +\alpha^{'}t$ with
$\epsilon=0.104$, $\alpha^{'}=0.25\ GeV^{-2}$ and
$\beta_0=6.56\ GeV^{-1}$, determined from \cite{covolan2}.
In Eq.(\ref{mult6}), the pomeron-proton cross section
is given by
\begin{equation}
\sigma_{\rm I\!P p} (M^2) = \beta_{0}\ g_{\rm I\!P}\
(s\xi )^{\epsilon}
\end{equation}
with the triple pomeron coupling determined from
data as $g_{\rm I\!P}=1.21\ GeV^{-1}$.
Since this scheme to calculate the diffractive
contribution is not the usual one, some comments are
in order. The usual derivation of the Triple Pomeron
Model gives (\ref{dlf}), the {\it standard flux factor},
instead of (\ref{renf}), the renormalized one.
The problem is that the standard flux factor drives
to strong unitarity violation and the {\it renormalization}
procedure was conceived \cite{dino} as an {\it ad hoc} way to
overcome this difficulty. Although a rigorous
demonstration of the renormalized scheme is still
missing, it is acceptable in the sense that it
provides a good description for the experimental
data at the diffractive region (see a detailed
discussion in \cite{dino_novo}).
The pion contribution ($\pi \pi \rm I\!P$) is given
by \cite{field}
\begin{equation}
\left (\frac{d^2\sigma}{dtd\xi}\right )_{\pi \pi
\rm I\!P } =
f_{\pi}(\xi,t) \times \sigma_{\pi p}(s\xi)
\label{mult7}
\end{equation}
where
\begin{equation}
f_{\pi}(\xi,t) = \frac{1}{4\pi}\frac{g^2}{4\pi}\
\frac{|t|}{(t-\rho^2)^2}\ e^{b_{\pi} (t-\rho^2)}
\xi^{1-2{\alpha}_{\pi}(t)}
\end{equation}
and $\alpha_{\pi}(t)=0.9(t-\rho^2)$ with
$\rho^{2}=m_{\pi}^{2}=0.02\ GeV^2$.
We follow \cite{robinson} in fixing the coupling
constant in $g^2/4\pi=15.0$ and putting $b_{\pi}=0$
(see also \cite{field}). The pion-proton cross section
$\sigma_{\pi p}(s\xi)=10.83\ (s\xi)^{0.104}\ +\ 27.13\
(s\xi)^{-0.32}\ [mb]$ is taken from \cite{covolan2}.
If one considers only the diffractive and
near-to-diffractive regions and low $p_T$
($-t \sim 0.0 - 0.1\ GeV^2$), the contributions
outlined above are enough to provide a good
description of the available data (see \cite{dino_novo}).
However, when one wants to consider larger $p_T$
and $x_F < 0.9$, at least a third contribution is
required. That is the reason why we introduce the reggeon
contribution.
The reggeon contribution ($\rm I\!R \rm I\!R \rm I\!P$)
is determined by
\begin{equation}
\left (\frac{d^2\sigma}{dtd\xi}\right )_{\rm I\!R \rm
I\!R \rm I\!P} =
f_{\rm I\!R}(\xi,t) \times \sigma_{\rm I\!R p}(s\xi)
\end{equation}
with
\begin{equation}
f_{\rm I\!R}(\xi,t) = \frac{{\beta}_{0{\rm I\!R}}^{2} }
{16\pi}
e^{2b_{\rm I\!R} t}\ \xi^{1-2{\alpha}_{{\rm I\!R}}(t)},
\label{mult8}
\end{equation}
and
\begin{equation}
\sigma_{\rm I\!R p}(s\xi) = {\beta}_{0{\rm I\!R}}\
g_{{\rm I\!R} }(s\xi )^{\epsilon}.
\label{sigreg}
\end{equation}
In this case, the trajectory is assumed to be
$\alpha_{{\rm I\!R}}(t)=0.5+t$ while the
constants $\beta_{{\rm I\!R}}\equiv
({\beta}_{0{\rm I\!R} }^{3}\ g_{\rm I\!R})$
and $b_{{\rm I\!R}}$ remain to be determined
from data.
Thus, with the expressions and parameters given
above, the $\rm I\!P \rm I\!P \rm I\!P$ and
$\pi \pi \rm I\!P$ contributions are completely
specified; only the $\rm I\!R \rm I\!R \rm I\!P$
contribution remains to have the final parameters
determined.
\centerline{\bf B. Central Region}
In order to describe the leading particle spectrum
in the central region, we use the Double Reggeon Model
\cite{collins} that gives the invariant cross section as
\begin{equation}
E\frac{d^3\sigma}{d{\bf p}^3}=\sum_{i,j}\
\gamma_{ij}(m_{T}^{2})
\ \left|\frac{t}{s_0}\right |^{\alpha_i(0)-1}\ \left |
\frac{u}{s_0}\right |^{\alpha_j(0)-1}
\label{mult9}
\end{equation}
where $m_{T}=({p_{T}^{2}+m^2_p})^{1/2}$ is the transversal
mass, and
$u=-m_T\sqrt{s}\ e^{-y}$ and $t=-m_T\sqrt{s}\ e^y$ are
the Mandelstam variables given in terms of the rapidity
$y=ln \frac{(E+p_L)}{m_T}$. Function
$\gamma_{ij}(m_{T}^{2})$
corresponds to the product of the three vertices of
the diagrams depicted in Fig.3. These diagrams
represent the contributions taken into account in the
present analysis: $\rm I\!P \rm I\!P$, $\rm I\!P
\rm I\!R + \rm I\!R
\rm I\!P$, and $\rm I\!R \rm I\!R$ (pion contributions
are not considered in this case because they are
totally covered by the others).
We assume for the coupling function $\gamma_{ij}
(m_{T}^{2})$ a simple gaussian form,
\begin{equation}
\gamma_{i j}(m_{T}^{2})=\Gamma_{i j}
\ e^{-a_{i j}m_{T}^{2}}
\end{equation}
where $\Gamma_{i j}$ is a constant that already
embodies the product of the couplings belonging
to the triple and quartic vertices. With these
definitions, the invariant cross sections for
the three contributions become
\begin{equation}
\left (E\frac{d^3\sigma}{d{\bf p}^3} \right )_
{\rm I\!P \rm I\!P} = \Gamma_{\rm I\!P \rm I\!P}
\ e^{-a_{\rm I\!P \rm I\!P}m_{T}^{2}}\
(m_T\ \sqrt{s})^{2\epsilon},
\label{mult10}
\end{equation}
\begin{eqnarray}
\left (E\frac{d^3\sigma}{d{\bf p}^3} \right )_
{\rm I\!P {\rm I\!R} +{\rm I\!R} \rm I\!P}=2\
\Gamma_{\rm I\!P {\rm I\!R}}\ e^{-a_{\rm I\!P
{\rm I\!R} }m_{T}^{2}} \ (m_T\ \sqrt{s})^
{\epsilon+\alpha_{{\rm I\!R} }(0)-1}
\ cosh[(1+\epsilon-\alpha_{\rm I\!R} (0))y],
\label{mult11}
\end{eqnarray}
and
\begin{equation}
\left (E\frac{d^3\sigma}{d{\bf p}^3} \right )_
{{\rm I\!R} {\rm I\!R} }= \Gamma_{{\rm I\!R}
{\rm I\!R}}\ e^{-a_{{\rm I\!R} {\rm I\!R} }
m_{T}^{2}}\ (m_T\ \sqrt{s})^{2(\alpha_
{\rm I\!R} (0)-1)}.
\label{mult12}
\end{equation}
In the above expressions again $\alpha_{\rm I\!R}(0)=0.5$
and $\epsilon=0.104$ \cite{covolan2}. Differently
from the fragmentation region where almost all
parameters are already established, in this
region almost all of them (expect for the
intercepts just mentioned) must be fixed
from data.
The expressions given above could be enriched
by detailing the reggeon exchange in terms
of $f$, $\rho$, $\omega$, $a_2$, and taking
into account all crossed terms, but in fact
we are pursuing here a minimal description
in which only the dominant and effective
contributions are considered. We shall
see below that these contributions are
enough to provide a good description of
the available data.
\section{\bf Description of experimental data}
Experimental data on leading particle spectrum
are very scarce. A compilation for
$p p \rightarrow p X$ is shown in Fig.1
where data from three experiments
\cite{basile,aguilar,brenner} are put together
(the curve and the insert in this figure should
be ignored for the moment).
As can be seen, a pretty flat spectrum is exhibited,
except for $x_F \approx 1$ where the typical
diffractive peak appears.\footnote{This peak
is absent from the Aguilar-Benitez
{\it et al.} data due to trigger inefficiency
for $x_F > 0.75$ in this particular
experiment \cite{aguilar}.}
The problem that arises when one tries to describe
the $p p \rightarrow p X$ reaction in the whole
phase space is that the available data are not
enough to determine unambigously each one of the
contributions outlined above. One may have noted
in the previous section that we have summarized
all secondary reggeon exchanges (except for the pion)
in a single contribution denoted by $\rm I\!R$ and
the reason is the following. When one analyzes,
for instance, total cross section data (like in
\cite{covolan2}), it is possible to establish
(to a certain extent) the relative amount of
the different contributions. Actually, this is
enforced by the changing shape exhibited by the
data in different regions. That is not the case
here because out of the diffractive region the
spectrum is pretty flat and that makes it difficult
to discriminate the regions where the different
exchange processes contribute the most. Thus,
in order to establish how the expressions
outlined above are summed up to compose the
observed spectrum, we have to follow a
particular strategy.
Since our intention was obtainning an acceptable
description for $p p \rightarrow p X$ data in the
whole phase space, we did not use in our fitting
procedure the data shown in Fig.1 which represent
only the $x_F$-dependence. Instead, we have set
those data apart to be used only at the end to
check our final results which, in fact, were
obtained with distributions giving in terms
of both $x_F$ and $p_T$ dependences.
Our procedures to determine the contributions
at the central and at the fragmentation regions
are quite different. The main problem is that
these regions overlap each other and thus it
is pratically impossible to separate them
(or establish clear limits). To overcome this difficulty
we assumed that, except for normalization effects,
the $x_F$ and $p_T$ dependences of the proton produced
in the central region through the reaction
$p p \rightarrow p X$ is the same as for the antiproton
produced in $p p \rightarrow {\bar p} X$. This
assumption was implemented by fitting simultaneously
the data shown in Figs. 4 and 5 \cite{dados1,dados2}
through the expressions
\begin{eqnarray}
\left (E\frac{d^3\sigma}{d{\bf p}^3} \right )_{pp->
\bar{p}X}^{central}=
\left (E\frac{d^3\sigma}{d{\bf p}^3} \right )_
{\rm I\!P \rm I\!P}
+\left (E\frac{d^3\sigma}{d{\bf p}^3} \right )_
{\rm I\!P {\rm I\!R} +{\rm I\!R} \rm I\!P}
+\left (E\frac{d^3\sigma}{d{\bf p}^3} \right )_
{{\rm I\!R} {\rm I\!R} }
\label{mult12b}
\end{eqnarray}
and
\begin{equation}
\left (E\frac{d^3\sigma}{d{\bf p}^3} \right )_
{pp->pX}^{central}=\lambda(s)\
\left (E\frac{d^3\sigma}{d{\bf p}^3} \right )_
{pp->\bar{p}X}^{central}.
\label{mult13}
\end{equation}
The idea is that the data of Fig.4 provide the
information on the $x_F$ and $p_T$ dependences
through Eqs. (\ref{mult10})-(\ref{mult12b}) and
relation
$x_F=2 m_T sinh(y)/{\sqrt s}$, while the connection
between $p p \rightarrow {\bar p} X$ and
$p p \rightarrow p X$ is established by fitting
the data of Fig.5 through the function
$\lambda(s)$ of Eq.(\ref{mult13}). The parameters
$\Gamma_{ij}$ and $a_{ij}$ of this fit are given
in Table \ref{tabmul1}
while $\lambda(s)$ is parametrized as \
$\lambda(s)=1.0+11.0\ s^{-0.3}$.
The agreement with data of Figs.4 and 5 is not
perfect, but that is because we are simplifying
the description by considering only a few
contibutions, the dominant ones. As stated
before, this is enough for the purposes of the
present analysis.
Now we are able to obtain the total description
by adding up central and fragmentation region
contributions.
As explained before, the contributions dominant
at the fragmentation region,
Eqs. (\ref{mult6})-(\ref{sigreg}), are almost
completely determined. The parameters
$\beta_{{\rm I\!R}}$ and $b_{{\rm I\!R}}$
referring to the $\rm I\!R \rm I\!R \rm I\!P$
contribution are established by fitting the
data of Fig.6 (from \cite{brenner}). This is
done by using the expression
\begin{eqnarray}
\left (E\frac{d^3\sigma}{d{\bf p}^3}\right )_
{pp->pX}^{total}=
\left (E\frac{d^3\sigma}{d{\bf p}^3}\right )_
{\rm I\!P \rm I\!P \rm I\!P}
+\left (E\frac{d^3\sigma}{d{\bf p}^3}\right )_
{\pi \pi \rm I\!P} \
+\left (E\frac{d^3\sigma}{d{\bf p}^3}\right )_
{\rm I\!R \rm I\!R \rm I\!P}
+\left (E\frac{d^3\sigma}{d{\bf p}^3} \right )_
{pp->pX}^{central}
\label{mult14}
\end{eqnarray}
where the last term refers to Eq.(\ref{mult13})
with the parameters given in Table \ref{tabmul1}.
With this final fit the remaining parameters
result to be
$\beta_{{\rm I\!R} }=2465.7\ mb\ GeV^{-2}$ and
$b_{\rm I\!R} =0.1\ GeV^{-2}$.
Fig.7 offers a view of how the different contributions
are composed to form the final result and how this
picture evolves with $p_T$.
The different contributions of the invariant cross
section in both regions integrated over $p_T$
produce the results of ${d\sigma}/{dx_F}$ for
both reactions
exhibited in Fig.1 (solid curves) for $p_{lab}=
400\ GeV/c$. We remind the reader that these
data were not used in the fit, but are used
now to check the reliability of the whole
procedure.
From this figure it is possible to see that the
final description obtained for the leading proton
spectrum is quite reasonable.
\section{Connection between
${\langle n_{ch} \rangle}_{e^+e^-}$ and
${\langle n_{ch} \rangle}_{pp}$}
The results obtained above specify completely
the behavior of the leading particle spectrum
and allow us to calculate $\langle n_0 \rangle$
and $\langle k_p\rangle$ as given by (\ref{n0})
and (\ref{kp}). In Fig.8, we show the energy
dependence of these quantities as obtained in
the present analysis (solid curves). In the
same figure, it is also shown the average
inelasticity as predicted by the Interacting
Gluon Model (IGM) \cite{igm} (dot-dashed curve)
for comparison. The average inelasticity obtained
from the present analysis is very slowly increasing
with energy, close to the behavior predicted by the
Minijet Model \cite{gaisser}.
With these results we can
come back to our original intent which is checking
the hypothesis of universal behavior of the multiplicity
that is specified by Eqs.(\ref{mult1}) and (\ref{mult2}).
In order to do that, we first establish
a parametrization for $N(\sqrt{s})$ through
\begin{equation}
N(\sqrt{s})=a_1+a_2\ ln(\frac{s}{s_{0}})+a_3\
ln^{2}(\frac{s}{s_{0}})
\label{mult15}
\end{equation}
with $s_{0}=1\ GeV^2$. However, before performing
the fit to experimental data, an additional effect
has to be considered. This is because, besides the
charged particles produced at the primary vertex,
${\langle n_{ch} \rangle}_{e^+e^-}$ data include
also decay products of $K^0_s \rightarrow \pi^+ \pi^-$,
$\Lambda \rightarrow p \pi^-$, and ${\bar\Lambda}
\rightarrow {\bar p} \pi^+$. Following \cite{ref1},
we take this contamination into account by computing
the ratio $R={\langle n_{ch} \rangle}_{K^0_s,\Lambda,
{\bar\Lambda}}/{{\langle n_{ch} \rangle}_{e^+e^-}}$
and by redefining (\ref{mult1}) as
\begin{equation}
{\langle n_{ch} \rangle}_{e^+e^-}=\frac{N(\sqrt{s})}{1-R},
\label{multc}
\end{equation}
with $R=0.097\pm 0.003$. This value was taken from
\cite{ref1} and, besides the references quoted therein,
it is in good agreement with experimental data from
ref.\cite{check}. No energy dependence for $R$
can be inferred from these data.
The fit using (\ref{mult15}) and (\ref{multc})
gives $a_1 = 2.392$, $a_2=0.024$, and $a_3=0.193$.
In Fig.9, we show the above parametrization
describing $\langle n_{ch}\rangle _{e^+e^-}$ data
from references quoted in \cite{ee} and the
calculated curve for $\langle n_{ch} \rangle _{pp}$
in comparison with data from \cite{pp}. The
agreement with these data enables us to consider
that our premises about the universal behavior of
${\langle n_{ch} \rangle}_{e^+e^-}$ and
${\langle n_{ch} \rangle}_{pp}$ are confirmed.
Of course, this conclusion is restricted to the
energy dependence of $\langle n_0 \rangle$ and
$\langle k_p\rangle$ shown in Fig.8.
The solid curve of the insert in Fig.9 shows what
happens when the IGM average inelasticity is applied
to the same purposes. One could argue that this last
result is conditioned by the use of $\langle n_0 \rangle$
obtained in the present analysis which increases with energy.
However, we note that increase in $\langle n_0 \rangle$
plays against increase in $\langle k_p \rangle$
since these are competitive effects.
Now a comment on the He analysis \cite{he}, where
the relation
\begin{equation}
n_{ch}^{e^+ e^-} ({\sqrt {s_{e^+ e^-}}}) =
n_{ch}^{pp} (k({\sqrt {s_{pp}}}){\sqrt {s_{pp}}})
\label{eq_he}
\end{equation}
is employed. After fitting $n_{ch}^{e^+ e^-}$ and
$n_{ch}^{pp}$ independently,
He imposes that relation (\ref{eq_he}) holds and
extracts the inelasticity $k$
from this assumption. This is similar to what we
have done, but
we think that the result of decreasing inelasticity
and the agreement with IGM obtained in such an
analysis comes from the fact that neither the
leading particle multiplicity ($n_0$) nor the
effect of decay products ($R$) is considered
and we see no reason for ignoring such effects.
A surprising outcome of the present analysis is
shown in Fig.10 (a) where the normalized cross
section $1/{\sigma_{incl}}\ d\sigma/dx_F$ is
calculated up to the LHC energy. It is shown that,
if the present description holds up to such high
energies, Feynman scaling is approximately
observed in the intermediate fragmentation region,
$0.2 < x_F < 0.8$, but is violated in opposite
ways at the central and diffractive regions.
Fig. 10 (b) shows the same results but in a
scale that makes more evident the scaling
violation at the central region. This result
seems to say that the increase of production
activity at the central region occurs at the
expenses of a supression of the diffractive
processes. However, this is just a speculative
observation that should be investigated more
thoroughly.
\section{Conclusions}
We have presented in this paper a description of
the inclusive reaction $p p \rightarrow p X$ in
the whole phase space within the Regge-Mueller
formalism, modified by the renormalization of the
diffractive cross section.
The average multiplicity and the average inelasticity
were obtained from the leading proton spectrum and
both of them resulted to be increasing functions of
energy, in agreement with \cite{nos,inc} and
particularly with \cite{gaisser}. The energy dependence
of these quantities is such that allows one to
accommodate very well the charged particle
multiplicities ${\langle n_{ch} \rangle}_{e^+e^-}$
and ${\langle n_{ch} \rangle}_{pp}$ by an universal
function once an appropriate relation is used.
An additional result is that the normalized leading
proton spectrum approximately observe Feynman scaling
for intermediate $x_F$, whereas such scaling is
violated at the central and diffractive regions.
\section*{Acknowledgementes}
We are grateful to J. Montanha for valuable discussions
and suggestions.
We would like to thank also the Brazilian governmental
agencies CNPq and FAPESP
for financial support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,217 |
{"url":"https:\/\/raweb.inria.fr\/rapportsactivite\/RA2019\/moex\/uid10.html","text":"New Software and Platforms\nNew Results\nPartnerships and Cooperations\nBibliography\n PDF e-Pub\n\n## Section: Research Program\n\n### Knowledge representation semantics\n\nWe work with semantically defined knowledge representation languages (like description logics, conceptual graphs and object-based languages). Their semantics is usually defined within model theory initially developed for logics.\n\nWe consider a language $L$ as a set of syntactically defined expressions (often inductively defined by applying constructors over other expressions). A representation ($o\\subseteq L$) is a set of such expressions. It may also be called an ontology. An interpretation function ($I$) is inductively defined over the structure of the language to a structure called the domain of interpretation ($D$). This expresses the construction of the \u201cmeaning\u201d of an expression in function of its components. A formula is satisfied by an interpretation if it fulfills a condition (in general being interpreted over a particular subset of the domain). A model of a set of expressions is an interpretation satisfying all the expressions. A set of expressions is said consistent if it has at least one model, inconsistent otherwise. An expression ($\\delta$) is then a consequence of a set of expressions ($o$) if it is satisfied by all of their models (noted $o\\vDash \\delta$).\n\nThe languages dedicated to the semantic web (rdf and owl ) follow that approach. rdf is a knowledge representation language dedicated to the description of resources; owl is designed for expressing ontologies: it describes concepts and relations that can be used within rdf .\n\nA computer must determine if a particular expression (taken as a query, for instance) is the consequence of a set of axioms (a knowledge base). For that purpose, it uses programs, called provers, that can be based on the processing of a set of inference rules, on the construction of models or on procedural programming. These programs are able to deduce theorems (noted $o\u22a2\\delta$). They are said to be sound if they only find theorems which are indeed consequences and to be complete if they find all the consequences as theorems.","date":"2020-07-07 17:05:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 8, \"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\": 0.8367217183113098, \"perplexity\": 806.9594517141511}, \"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\/1593655893487.8\/warc\/CC-MAIN-20200707142557-20200707172557-00324.warc.gz\"}"} | null | null |
This Restomod International Scout II Is Like an Icon 4×4 but Cheaper
Featuring modern internals but a very '70s aesthetic, its the perfect balance between modern and vintage.
By Andrew Connor
Later this month at RM Sotheby's Arizona Auction, this restomod International Scout (the underrated precursor to the Ford Bronco) will roll across the auction block. But while the Scout has undergone some extensive modifications, the result is overall tasteful. It's not dissimilar to beloved off-road creations like the Land Rovers of East Coast Defender or the Broncos of Icon 4×4, though the inspired choice of base vehicle makes it a unique alternative.
The Scout's most extensive changes include a new motor — GM's workhorse 6.0-liter Vortec L96 crate engine — a modern four-speed automatic transmission and a Dana 300 transfer case. It also gets a set of what might be the most intense leaf springs ever fitted to a truck, while off-road accessories include a beefy Warn XD9000i winch, a roof rack and bumper rocker guards. The aesthetic of the truck, both inside and out, is definitely not original, but it does seem to be a loving recreation of the '70s design schemes common on trucks of the Scout's era, pairing a white roof and mossy green body on the outside and filling a spartan interior with striped cloth bench seats.
It'll go up for auction at RM Sotheby's without a reserve, with an expected sale price between $70,000 and $90,000. Whether or not it meets (or even exceeds) that estimate remains to be seen, but even if it does, it'll very likely go for less than an Icon Bronco but will probably turn as many heads.
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Get a Custom, Better-Than-New International Scout for $65,000
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You Will Not Find a More Beautiful Custom Scout For Sale
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The Indian Scout Gets the Custom Treatment, Three Ways | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,204 |
\section{Introduction}\label{sec-intro}
Let $G(q)$ be finite group of Lie type arising from a connected reductive
algebraic group $\mathbf{G}$ of simply-connected type over an algebraic closure
$\overline{\mathbb{F}}_q$ of characteristic $p$.
The irreducible representations of $G(q)$ over $\overline{\mathbb{F}}_p$ (called
defining characteristic representations) are restrictions of irreducible
representations of the algebraic group $\mathbf{G}$. The irreducible
representations of $\mathbf{G}$ over $\overline{\mathbb{F}}_p$ can be described in terms of
weight multiplicities.
If the weight multiplicities for $\mathbf{G}$ are explicitly known
for sufficiently many irreducible representations (parameterized by
$p$-restricted highest weights) a lot of information about the irreducible
representations of $G(q)$ for small powers $q$ of $p$ can be computed. For
example the degrees of all irreducible representations of $G(q)$ or the
composition factors of tensor products of irreducible representations.
But this description does not yield a relation of the irreducible
representations of $G(q)$ in characteristic $p$ with the ordinary (complex)
representations of $G(q)$. This relation is provided by the $p$-modular
Brauer character table of $G(q)$, equivalently by the $p$-modular
decomposition matrix of $G(q)$ or the (ordinary) characters of the
$p$-modular projective indecomposable modules.
In this note we describe two ways to compute from weight multiplicities of a
representation of $\mathbf{G}$ the corresponding Brauer character of $G(q)$.
As an application of these methods we compute some character tables for
the modular {\textsf{ATLAS}\xspace} project~\cite{ModAtl,WebModAtl}, which has the aim
to provide all Brauer character tables for all primes for the groups
whose ordinary character table is mentioned in the {\textsf{ATLAS}\xspace}~\cite{ATLAS}.
More precisely, we compute the full $p$-modular Brauer character
tables for the groups (in {\textsf{ATLAS}\xspace} notation) $F_4(2)$, $2^2.O_8^+(3)$,
$2.O_8^-(3)$, $O_{10}^+(2)$, $O_{10}^-(2)$ and $S_{10}(2)$ where $p$ is the
defining characteristic. We also find partial tables for $E_6(2)$ and
$3.{}^2\!E_6(2)$.
One possible application of these tables is the induction of Brauer and
projective characters to (sporadic) overgroups.
In Section~\ref{sec-setup} we describe the groups we consider here in more
detail and explain how we represent them for computations.
In Section~\ref{sec-mults} we give an overview of weight multiplicities
encoded in dominant characters, and in particular we sketch
in~\ref{ss_computechars} how to compute with such dominant characters (this
may be of independent interest).
Section~\ref{sec-smallrep} describes a first method to compute Brauer
characters which uses explicit small degree representations of $G(q)$ and
computations with dominant characters.
In Section~\ref{sec-sscl} we show how to compute a parameterization of
conjugacy classes of elements of $p'$-order in $G(q)$ (without an explicit
representation of the group).
Section~\ref{sec-brauer} describes a second method to compute Brauer
characters which uses the weight multiplicities directly as well as some
invariants of conjugacy classes.
In all sections we illustrate the theoretical descriptions with the example
of $\mathbf{G}$ of type $D_4$ and $G(q) \in \{\operatorname{Spin}_8^+(3), \operatorname{Spin}_8^-(3)\}$.
\textbf{Acknowledgements.} I thank Klaus Lux for convincing me that for some
applications is it not enough to know the irreducible representations of
$G(q)$ in defining characteristic only in terms of weight multiplicities.
This motivated the preparation of these notes. And I thank Thomas Breuer for
answering some questions about Brauer character tables, for helping with the
computation of some ordinary character tables with {\textsf{Magma}\xspace}, and for
distributing the new character tables described here with the {\textsf{GAP}\xspace}-package
{\textsf{CTblLib}\xspace}.
\section{Setup and Notation}\label{sec-setup}
\subsection{Root Data}\label{ss-rootdata}
We consider connected reductive groups $\mathbf{G}$ over an algebraic closure
$\overline{\mathbb{F}}_p$ of a finite field of characteristic $p$ and their finite subgroups
$G(q) = \mathbf{G}^F$ of fixed points under a Frobenius morphism $F$.
If $\mathbf{G}$ is of rank $r$ and $\mathbf{T} \leq \mathbf{G}$ is a maximal torus
of $\mathbf{G}$ there is an associated root datum $(X,\Phi,Y,\Phi^\vee)$,
see~\cite[7.4, 9.6]{Sp98}. Here $X \cong \mathbb{Z}^r$ is the character group
$\operatorname{Hom}(\mathbf{T}, \overline{\mathbb{F}}_p^\times)$ and $Y \cong \mathbb{Z}^r$ is the cocharacter group
$\operatorname{Hom}(\overline{\mathbb{F}}_p^\times, \mathbf{T})$ of the torus $\mathbf{T}$, and $\Phi \subset X$ is
the set of roots and $\Phi^\vee \subset Y$ is the set of coroots of $\mathbf{G}$
with respect to $\mathbf{T}$.
If $F$ is a Frobenius morphism of $\mathbf{G}$, we assume that $F(\mathbf{T}) = \mathbf{T}$
and in that case $F$ naturally induces $\mathbb{Z}$-linear maps on $X$ and $Y$ which
are of the form $q \cdot F_0$ where $F_0$ is of finite order permuting the
roots and $q$ is a power of $p$, see~\cite[1.17--1.18]{Ca85}. We write $G(q)
= \mathbf{G}^F$ for the finite group of $F$-fixed points to indicate the
parameter $q$ determined by $F$. (We exclude here more general $F$, leading
for example to Suzuki and Ree groups, to keep some statements simpler;
except for some remarks in the proof of~\ref{ThmMults}.)
The algebraic group $\mathbf{G}$ is determined up to isomorphism by the root
datum and the field $\overline{\mathbb{F}}_p$. The finite group $G(q)$ of $F$-fixed points
is determined up to isomorphism by the root datum, $F_0$ and the number $q$.
For computations we use the setup from the \textsf{CHEVIE}\xspace~\cite{CHEVIE} software
package. See also~\cite[Section 2]{BL13} for more details.
Let $l$ be the semisimple rank of $\mathbf{G}$. We encode the root datum by two
matrices $(A,A^\vee)$ in $\mathbb{Z}^{l \times r}$. There is a natural pairing
$\langle \cdot, \cdot \rangle: X \times Y \to \mathbb{Z}$ and we choose dual
$\mathbb{Z}$-bases of $X$ and $Y$ with respect to that pairing. We write a set of
simple roots with respect to the basis of $X$ and put the coefficients into
the rows of $A$, similarly we write the corresponding simple coroots with
respect to the basis of $Y$ and put the coefficients into the rows of
$A^\vee$.
If $\alpha_i$ is a simple root and $\alpha_i^\vee$ the corresponding simple
coroot ($1 \leq i \leq l$), we define
\[ s_i: X \to X, x \mapsto x - \langle x, \alpha^\vee \rangle \alpha, \]
\[ s_i^\vee: Y \to Y, y \mapsto y - \langle \alpha, y \rangle \alpha^\vee.\]
The $s_i$, $1 \leq i \leq l$, are a set of Coxeter generators of the Weyl
group $W$ of $\mathbf{G}$ (and the same is true for the $s_i^\vee$, $1 \leq i
\leq l$). Using the $i$-th rows of $A$ and $A^\vee$ we can write down $s_i$
as matrix acting on $X$ with respect to our basis of $X$. The transposed of
that matrix represents the action of $s_i^\vee$ on $Y$.
The set of all roots $\Phi$ can be found as the orbits of the simple roots
under the action of $W$.
If $F_0: X \to X$ is a $\mathbb{Z}$-linear map of finite order which fixes the set
of roots and whose dual map $F_0^\vee: Y \to Y$ fixes the set of coroots,
then there is for each power $q$ of $p$ a Frobenius map $F$ on $\mathbf{G}$ which
induces $q F_0$ on $X$.
\subsection{Simply-Connected Groups}\label{ss-sc}
From now we assume that $\mathbf{G}$ is a semisimple group of simply connected
type. In our setup this means that $r=l$ and that the coroots span the
lattice $Y$. Therefore, we choose a set of simple coroots $\alpha_1^\vee,
\ldots, \alpha_l^\vee$ as $\mathbb{Z}$-basis of $Y$. The corresponding dual basis
$\omega_1, \ldots, \omega_l$ of $X$ is called the set of fundamental
weights. The matrix $A^\vee$ is the identity matrix of size $l$ and the
matrix $A$ is the transposed of the Cartan matrix of the root system of
$\mathbf{G}$ (the $l \times l$ matrix with $(i,j)$-entry $\langle \alpha_j,
\alpha_i^\vee \rangle$).
The set $X_+ = \{\sum_i a_i \omega_i\mid\; a_i \in \mathbb{Z}_{\geq 0}\textrm{ for }
1 \leq i \leq l\}$ is called the set of dominant weights and for a real
number $b$ we call $X_b = \{ \sum_i a_i \omega_i \in X_+\mid\; a_i < b
\textrm{ for } 1 \leq i \leq l\}$ the set of $b$-restricted weights. The
$W$-orbit of each $\lambda \in X$ contains a unique dominant weight.
There is a partial order on $X$ defined by: $\lambda, \mu \in X$, then
$\lambda \geq \mu$ if and only if $\lambda - \mu = \sum_i b_i \alpha_i$ with
$b_i \in \mathbb{Z}_{\geq 0}$ for $1 \leq i \leq l$.
\subsection{Example}\label{ex-d4}
We consider the root system of type $D_4$. Its Cartan matrix is
\[{\scriptsize \left(\begin{array}{rrrr}%
2&0&-1&0\\%
0&2&-1&0\\%
-1&-1&2&-1\\%
0&0&-1&2\\%
\end{array}\right)} \]
and can be encoded in the Dynkin diagram
\begin{tikzpicture}[scale=0.8, baseline=-1.0]
\coordinate (3) at (0,0);
\coordinate (1) at (120:1);
\coordinate (2) at (240:1);
\coordinate (4) at (1,0);
\foreach \p in {1, 2, 3, 4} \filldraw [black] (\p) circle [radius=0.08];
\draw (1) -- (3);
\draw (2) -- (3);
\draw (3) -- (4);
\draw (1) node[anchor=east] {{\scriptsize 1}};
\draw (2) node[anchor=east] {{\scriptsize 2}};
\draw (3) node[anchor=north] {{\scriptsize 3}};
\draw (4) node[anchor=north] {{\scriptsize 4}};
\end{tikzpicture}.
We encode the root datum of the simply connected algebraic groups of type
$D_4$ by the pair of matrices
\[ (A, A^\vee) = ({\scriptsize
\left(\begin{array}{rrrr}%
2&0&-1&0\\%
0&2&-1&0\\%
-1&-1&2&-1\\%
0&0&-1&2\\%
\end{array}\right)},
{\scriptsize
\left(\begin{array}{rrrr}%
1&0&0&0\\%
0&1&0&0\\%
0&0&1&0\\%
0&0&0&1\\%
\end{array}\right) } ).\]
Then the generating reflections $s_1, \ldots, s_4$ of the Weyl group $W$ act
on $X$ by
\[{\scriptsize
\left(\begin{array}{rrrr}%
-1&0&1&0\\%
0&1&0&0\\%
0&0&1&0\\%
0&0&0&1\\%
\end{array}\right)%
,
\left(\begin{array}{rrrr}%
1&0&0&0\\%
0&-1&1&0\\%
0&0&1&0\\%
0&0&0&1\\%
\end{array}\right)%
,
\left(\begin{array}{rrrr}%
1&0&0&0\\%
0&1&0&0\\%
1&1&-1&1\\%
0&0&0&1\\%
\end{array}\right)%
,
\left(\begin{array}{rrrr}%
1&0&0&0\\%
0&1&0&0\\%
0&0&1&0\\%
0&0&1&-1\\%
\end{array}\right)%
}\]
and $s_1^\vee, \ldots, s_4^\vee$ act on $Y$ by the transposed of the printed
matrices.
For each characteristic $p$ these data determine the corresponding
reductive algebraic group $\mathbf{G}$ which is isomorphic to the Spin group
$\operatorname{Spin}_8(\overline{\mathbb{F}}_p)$.
If we choose as $F_0$ the identity matrix, then for each power $q$ of $p$
the corresponding Frobenius morphism of $\mathbf{G}$ yields a group of fixed
points $G(q) = \operatorname{Spin}_8^+(q)$.
The $F_0$ that permutes the first two simple roots (and coroots) yields the
finite groups $G(q) = \operatorname{Spin}_8^-(q)$, and from $F_0$ permuting the simple
roots $1 ,2, 4$ cyclically we find the Steinberg triality groups $G(q)
={}^3\!D_4(q)$.
\section{Weight Multiplicities}\label{sec-mults}
We use the setup from~\ref{ss-sc}. So $\mathbf{G}$ is a connected reductive group
of simply-connected type over $\overline{\mathbb{F}}_p$ with a Frobenius morphism $F$,
maximal torus $\mathbf{T} = F(\mathbf{T})$, and finite group of fixed points $G(q) =
\mathbf{G}^F$.
\subsection{Irreducible Representations}\label{ss-irr}
We recall a few standard facts about irreducible representations of $\mathbf{G}$
and $G(q)$ in their defining characteristic $p$.
If $M$ is a rational module for $\mathbf{G}$ then $M$ considered as
$\mathbf{T}$-module splits into a direct sum of non-zero subspaces $M_\mu$ which
are common eigenspaces for all $t \in \mathbf{T}$. Here $\mu \in X$ is the
homomorphism $\mu:\mathbf{T} \to \overline{\mathbb{F}}_p^\times$ which maps each $t \in \mathbf{T}$ to
its eigenvalue $\mu(t)$ on $M_\mu$. The set $\{\mu \in X\mid\; M_\mu \neq
0\}$ is called the weights of $M$, it is a union of $W$-orbits on $X$ and
when $\mu, \nu \in X$ are in the same $W$-orbit, then $M_\mu$ and $M_\nu$
have the same dimension.
If $M$ is irreducible then it has a unique highest weight $\lambda \in
X_+$ with $\lambda \geq \mu$ for all weights of $M$. Chevalley~\cite[II
2.7]{Ja03} showed that such an $M$ is determined up to isomorphism by its
highest weight, we denote this module $L(\lambda)$. And for each dominant
$\lambda \in X_+$ there is such an irreducible module $L(\lambda)$.
Steinberg~\cite[13.1]{St68} showed that for fixed characteristic $p$ all
$L(\lambda)$ can be described in terms of a finite number of them. For this
write an arbitrary dominant $\lambda \in X_+$ as a finite linear combination
$\lambda = \sum_{i = 0}^k p^i \lambda_i$ where all $\lambda_i \in X_p$ are
$p$-restricted (write the entries of $\lambda$ in base $p$). Steinberg's
tensor product theorem says that
\[ L(\lambda) = L(\lambda_0) \otimes L(\lambda_1)^{[1]} \otimes \cdots
\otimes L(\lambda_k)^{[k]},\]
where $L(\lambda)^{[i]}$ is the $i$-th Frobenius twist of $L(\lambda)$,
$\mathbf{G}$ acts on this module by the composition of its action on $L(\lambda)$
and the field automorphism $c \mapsto c^{p^i}$ of $\overline{\mathbb{F}}_p$.
Steinberg~\cite[13.3]{St68} also showed that the restrictions of all
$\{L(\lambda)\mid\; \lambda \in X_q\}$ (the $L(\lambda)$ for $q$-restricted
weights) to $G(q)$ yield a set of representatives of all isomorphism classes
of irreducible representations of $G(q)$ over $\overline{\mathbb{F}}_p$.
\subsection{Characters by Weight Multiplicities}\label{ss-mults}
A lot of information about a finite dimensional rational $\mathbf{G}$-module $M$
is encoded in the list of weights $\mu$ of $M$ and the dimensions of the
weight spaces $M_\mu$. Using that the set of weights consists of $W$-orbits
which each contain a unique dominant weight, we get an efficient description
of these data by a dominant character, which only lists the dominant weights
of $M$ and the dimensions of the corresponding weight spaces. If $\mu \in X$
is a weight of $M$ we call $m_\mu(M) = \dim(M_\mu)$ the multiplicity of the
weight $\mu$ in $M$.
The orbit lengths of weights are not difficult to compute because for a
weight $\sum_{i=1}^l a_i \omega_i$ its stabilizer in $W$ is the parabolic
subgroup $\langle s_i \mid\; a_i = 0 \rangle$.
We can find all weights in a $W$-orbit by a standard orbit calculation
using the matrices for the generators $s_1, \ldots, s_l$ of $W$ given
in~\ref{ss-rootdata}.
Here is an example. Let $\mathbf{G}$ be of type $D_4$, $p = 3$, and $\lambda =
\omega_2+2 \omega_4$. We denote elements of $X$ by their coefficient vectors
with respect to the basis $\omega_1, \ldots, \omega_4$ of fundamental
weights as in Example~\ref{ex-d4}. In the following table we show the
dominant character of $L(\lambda)$ and the lengths of the $W$-orbits of the
mentioned dominant weights.
\[
\begin{array}{rrr}
\mu & m_\mu & |\mu^W| \\
\hline
( 0, 1, 0, 2 ) & 1 & 32\\
( 0, 1, 1, 0 ) & 1 & 48\\
( 1, 0, 0, 1 ) & 3 & 32\\
( 0, 1, 0, 0 ) & 6 & 8
\end{array}
\]
In our example, we can use the $s_i$ from~\ref{ex-d4} (matrices acting on
$X$), the orbit of $(0,1,0,0)$ consists of the weights $( 0, 1, 0, 0 )$, $(
0, -1, 1, 0 )$, $( 1, 0, -1, 1 )$, $( -1, 0, 0, 1 )$, $( 1, 0, 0, -1 )$, $(
-1, 0, 1, -1 )$, $( 0, 1, -1, 0 )$, $( 0, -1, 0, 0 )$ and the corresponding
weight spaces of $L( (0,1,0,2) )$ have dimension $6$.
\subsection{Computing with Dominant Characters}\label{ss_computechars}
Given dominant characters of rational $\mathbf{G}$-modules $M$ and $M'$ we can
\begin{itemize}
\item[(a)] compute the total dimension, $\dim M = \sum_{\mu \in X_+}
m_\mu(M) \cdot |\mu^W|$,
\item[(b)] compute the dominant character of the $i$-th Frobenius twist
$M^{[i]}$, for this just multiply all weights by $p^i$,
\item[(c)] compute the character of $M \otimes M'$, using that for $v \in
M_\mu$ and $v' \in M'_{\mu'}$ the vector $v \otimes v'$ is in the
weight space $(M \otimes M')_{\mu + \mu'}$ because $\mu(t) \mu'(t) =
(\mu+\mu')(t)$ for $t \in \mathbf{T}$; to compute the weight multiplicities of
$M_\mu \otimes M'_{\mu'}$ one needs to enumerate at least one of the orbits
$\mu^W$ or $\mu'^W$,
\item[(d)] find the composition factors of $M$ provided the dominant
characters of those are available, for this enumerate the dominant weights
of $M$ with respect to a total ordering which refines the partial ordering
$>$ on $X$, then the first weight $\lambda$ and its multiplicity $m$ shows
the multiplicity of $L(\lambda)$ as composition factor of $M$, subtract $m$
times the dominant character of $L(\lambda)$ and proceed recursively.
\end{itemize}
Now assume that we know for a fixed $p$ the dominant characters of
$L(\lambda)$ for all $p$-restricted $\lambda \in X_p$. Fix the map $F_0: X
\to X$ and let $q$ be a power of $p$.
Then we can
\begin{itemize}
\item[(e)] compute the dimensions of all irreducible representations of
$G(q)$ over $\overline{\mathbb{F}}_p$, using the Steinberg tensor product theorem for
all $q$-restricted weights (only the dimensions of the $L(\lambda)$ for
$p$-restricted $\lambda$ are needed),
\item[(f)] compute the dominant characters of $L(\lambda)$ for all
$q$-restricted weights, using the tensor product theorem and~(b) and~(c),
\item[(g)] find the composition factors of the restriction of $L(\lambda)$
to $G(q)$ for arbitrary $\lambda \in X$, for this note that $L(\lambda)$ and
$L(\lambda (q F_0))$ restrict to the same representation of $G(q)$,
so decompose an arbitrary $\lambda \in X$ as $\sum_{i \geq 0} q^i
\lambda_i F_0^i$ with all $\lambda_i \in X_q$ being $q$-restricted to get
$L(\lambda)\mid_{G(q)} = \bigotimes_{i \geq 0} L(\lambda_i)\mid_{G(q)}$,
use~(h) and this point~(g) recursively,
\item[(h)] decompose tensor products of irreducible representations of
$G(q)$ over $\overline{\mathbb{F}}_p$, use~(c),~(d) and~(g) for factors not corresponding to
$q$-restricted weights.
\end{itemize}
We continue the example of $\mathbf{G}$ in type $D_4$, $p=3$ from~\ref{ss-mults}.
Let $\lambda = \omega_2+2\omega_4$ and $\lambda' = 2\omega_4$. The dominant
character of $L(\lambda)\otimes L(\lambda')$ is
\[
\begin{array}{rrr}
\mu & m_\mu & |\mu^W| \\
\hline
( 0, 1, 0, 4 ) & 1 & 32 \\
( 0, 1, 1, 2 ) & 2 & 96 \\
( 0, 1, 2, 0 ) & 3 & 48 \\
( 1, 2, 0, 1 ) & 4 & 96 \\
\end{array}
\quad
\begin{array}{rrr}
\multicolumn{3}{c}{\textit{(cont.)}}\\
\hline
( 1, 0, 0, 3 ) & 6 & 32 \\
( 1, 0, 1, 1 ) & 11 & 96 \\
( 2, 1, 0, 0 ) & 15 & 32 \\
( 0, 3, 0, 0 ) & 6 & 8 \\
\end{array}
\quad
\begin{array}{rrr}
\multicolumn{3}{c}{\textit{(cont.)}}\\
\hline
( 0, 1, 0, 2 ) & 24 & 32 \\
( 0, 1, 1, 0 ) & 34 & 48 \\
( 1, 0, 0, 1 ) & 63 & 32 \\
( 0, 1, 0, 0 ) & 112 & 8
\end{array}
\]
So, for the algebraic group $\mathbf{G}$ the tensor product contains one
composition factor isomorphic to $L((0,1,0,4)) = L((0,1,0,1)) \otimes
L((0,0,0,1))^{[3]}$. This composition factor restricted to $\operatorname{Spin}_8^+(3)$
or to $\operatorname{Spin}_8^-(3)$ is the tensor product of the restrictions of
$L((0,1,0,1))$ and $L((0,0,0,1))$ to those groups, while the restriction to
$^3\!D_4(3)$ is the tensor product of the restrictions of $L((0,1,0,1))$ and
$L((0,1,0,0))$ (because $F_0$ permutes $\omega_1 \to \omega_2 \to \omega_4
\to \omega_1$) to the finite group. A similar computation shows that these
tensor products for the finite groups have five composition factors
(corresponding to $3$-restricted weights) in each case.
\subsection{Some Dominant Characters}
\begin{Thm}\label{ThmMults}
Let $p$ and $\mathbf{G}$ as in one row of the following table
\[ \begin{array}{lll}
p& \textrm{Lie type of } \mathbf{G}& \textrm{group name}\\
\hline
3 & D_4 & \operatorname{Spin}_8(\overline{\mathbb{F}}_p)\\
2 & B_4 & \operatorname{Spin}_9(\overline{\mathbb{F}}_p)\\
2 & F_4 & F_4(\overline{\mathbb{F}}_p)\\
2,3 & A_5 & \operatorname{SL}_{6}(\overline{\mathbb{F}}_p)\\
2 & C_5 & \operatorname{Sp}_{10}(\overline{\mathbb{F}}_p)\\
2 & D_5 & \operatorname{Spin}_{10}(\overline{\mathbb{F}}_p)\\
\end{array}
\]
Then the dominant characters of the irreducible rational representations
$L(\lambda)$ of $\mathbf{G}$ are known for all $p$-restricted highest weights
$\lambda$.
If $\mathbf{G}$ is simply connected of type $E_6$ and $p=2$ then we know the
dominant characters of 44 irreducible $L(\lambda)$ with $2$-restricted
$\lambda$ (there are 64 such $\lambda$).
\end{Thm}
\textrm{Proof.}
These characters were computed with the strategy and programs
described in~\cite{luebeckdef}. The characters are available on the web
page~\cite{WebMults}. (Because of the size of these data we do not reproduce
them within this article.) The computations involved several weeks of CPU
time.
We remark that the case $F_4$ and $p=2$ is particularly easy. In this case
$\mathbf{G}$ has an exceptional automorphism $\tilde F$ whose square is the
Frobenius morphism which acts as $2 \; \textrm{Id}$ on $X$.
If we number the simple roots and the fundamental weights according to the
Dynkin diagram
\begin{tikzpicture}[scale=0.8, baseline=-3.0]
\coordinate (1) at (0,0);
\coordinate (2) at (1,0);
\coordinate (3) at (2,0);
\coordinate (4) at (3,0);
\coordinate (5) at (1.3,0.3);
\coordinate (6) at (1.3,-0.3);
\coordinate (7) at (1.7,0);
\draw[line width=0.6] (1) -- (2);
\draw[double, double distance between line centers=3.38, line width=0.6] (2) -- (3);
\draw[line width=0.6] (3) -- (4);
\draw[line width=0.6] (5) -- (7);
\draw[line width=0.6] (6) -- (7);
\draw (1) node[anchor=north] {{\scriptsize 1}};
\draw (2) node[anchor=north] {{\scriptsize 2}};
\draw (3) node[anchor=north] {{\scriptsize 3}};
\draw (4) node[anchor=north] {{\scriptsize 4}};
\foreach \p in {1, 2, 3, 4} \filldraw [black] (\p) circle [radius=0.08];
\filldraw [black] (7) circle [radius=0.001];
\end{tikzpicture},
then $\tilde F$ maps $\omega_1,\omega_2,\omega_3,\omega_4$ to $2 \omega_4, 2
\omega_3,\omega_2,\omega_1$, respectively. The Steinberg tensor product
theorem also holds for $\tilde F$: Let
\[M = \{(0,0,0,0),(0,0,0,1),(0,0,1,0),
(0,0,1,1)\}\]
and write an arbitrary dominant $\lambda \in X^+$ as $\lambda = \sum_{i=0}^k
\lambda_i {\tilde F}^i$ where all $\lambda_i \in M$. Then
\[L(\lambda) = L(\lambda_0) \otimes L(\lambda_i)^{[1]} \otimes \cdots
\otimes L(\lambda_k)^{[k]}\]
where now ${}^{[i]}$ denotes the $i$-th twist with $\tilde F$,
see~\cite[11.2]{St68}. It is easy to compute the dominant characters of
$L(\lambda)$ for $\lambda \in M$. This was already done in~\cite{V70}.
\mbox{}\hfill$\Box$
\section{Brauer Table from Small Representation and
Tensoring}\label{sec-smallrep}
Let $\mathbf{G}$, $p$ and $G(q)$ be as in previous sections. In this section we
describe a first method to find the Brauer character table of a finite group
$G(q)$ in defining characteristic $p$.
We recall that Brauer characters are defined relative to an embedding
of the multiplicative groups $\overline{\mathbb{F}}_p^\times \hookrightarrow \mathbb{C}^\times$.
There is a convention how to choose this map which is used in the
Modular Atlas~\cite{ModAtl} and in the {\textsf{CTblLib}\xspace} library of character
tables~\cite{CTblLib} distributed with {\textsf{GAP}\xspace}~\cite{GAP4} (based on the
notion of Conway polynomials). To compute the Brauer character value for a
group element given by its representing matrix over a finite extension of
$\mathbb{F}_p$ one computes its eigenvalues in $\overline{\mathbb{F}}_p$, lifts them to $\mathbb{C}$ via the
mentioned embedding, and sums up the images. {\textsf{GAP}\xspace} provides a function for
this computation. Brauer character values are only computed on elements of
$p'$-order.
To start, we assume that we have an explicit faithful matrix
representation of $G(q)$ over a finite extension of $\mathbb{F}_p$. For the groups
considered in this article these are available from {\textsf{GAP}\xspace} commands
like \texttt{SL(6,3)} or \texttt{Sp(10,2)} or from the ATLAS of group
representations~\cite{WWWAtlas}, accessible in {\textsf{GAP}\xspace} via the {\textsf{AtlasRep}\xspace}
package~\cite{AtlasRep}.
We also assume that
\begin{itemize}
\item
we know the weight multiplicities of $\mathbf{G}$ for all $L(\lambda)$ with
$p$-restricted $\lambda$,
\item
we can compute representatives of the conjugacy classes of $G(q)$ in the
given representation,
\item
we can identify the composition factors of this representation in terms of
their labels by highest weights,
\item
and one of:
\begin{itemize}
\item
the ordinary character table of $G(q)$ is known and we can identify the
conjugacy classes in the given representation with those in the character
table, or
\item
we can compute the character table of $G(q)$ (in this case the
identification of the conjugacy classes with those of a known table is
automatic and essentially unique).
\end{itemize}
\end{itemize}
Sometimes we may need a tool, called the {\textsf{MeatAxe}\xspace}~\cite{MeatAxe}. Given
representing matrices of a set of generators of a group over a finite field,
it can find representing matrices of the generators for each (absolutely
irreducible) composition factor.
Given representing matrices of a set of group generators for two
representations over the same field it is easy to compute representing
matrices for the tensor product representation (Kronecker product of
matrices). If we want to compute Brauer character values for group elements
in several representations we just add representatives of all classes of
$p'$-elements to our set of group generators.
The table of Brauer characters is now computed as follows.
\begin{itemize}
\item[(1)]
If the given representation of $G(q)$ is not absolutely irreducible compute
the composition factors with the {\textsf{MeatAxe}\xspace}.
\item[(2)]
Compute the Brauer character of the trivial representation, and of the
composition factors we have found. (Recall that we know the corresponding
highest weights.)
\item[(3)]
Using weight multiplicities we determine the composition factors of tensor
products of representations for which we already know the corresponding
Brauer characters, using the method sketched in~\ref{ss_computechars}.
\item[(3a)]
If there is a tensor product which contains only one composition factor for
which we do not yet know the Brauer character, then we can compute the
Brauer character of this composition factor by tensoring and subtracting
known Brauer characters.
\item[(3b)]
Otherwise, we use a tensor product which has one or several composition
factors which are easy to label (e.g., via their degrees), compute
representing matrices for this tensor product, use the {\textsf{MeatAxe}\xspace} to
find the composition factors, and compute the Brauer characters of new
composition factors as above.
\item[(4)]
If not all Brauer characters are found go back to step~(3).
\end{itemize}
We do not call this description an algorithm, because it is not clear that
we will always be able to identify the label of new composition factors
found with the {\textsf{MeatAxe}\xspace}. And there can be a practical problem if we need
to apply the {\textsf{MeatAxe}\xspace} to representations of dimensions exceeding a few
thousands.
But in practice this method worked very well in cases we have tried.
\subsection{Example $G(q) = \operatorname{Spin}_8^-(3)$}\label{ss-2D4-method1}
\sloppy
In this case we can find a $16$-dimensional representation of $G(q)$ over
$\mathbb{F}_3$ and we can compute class representatives and the ordinary character
table with {\textsf{Magma}\xspace}~\cite{Magma}.
\fussy
The {\textsf{MeatAxe}\xspace} yields two $8$-dimensional absolutely irreducible composition
factors over $\mathbb{F}_9$.
We have a description of $G(q)$ in terms of a root datum as in~\ref{ex-d4}
and we know the weight multiplicities of $\mathbf{G}$ for all $3$-restricted
weights as stated in~\ref{ThmMults}.
Using weight multiplicities we compute (writing $\chi_\lambda$ for the
character of $L(\lambda)$):
\[
\chi_{(1,0,0,0)} \otimes \chi_{(0,1,0,0)} = \chi_{(1,1,0,0)} +
\chi_{(0,0,0,1)}
\]
\[
\chi_{(1,0,0,0)} \otimes \chi_{(1,0,0,0)} = \chi_{(2,0,0,0)} +
\chi_{(0,0,1,0)} + \chi_{(0,0,0,0)}
\]
$\mathbf{G}$ has three $8$-dimensional irreducible representations corresponding
to the highest weights $\lambda \in \{(1,0,0,0), (0,1,0,0), (0,0,0,1)\}$.
Using the {\textsf{MeatAxe}\xspace} we find the third irreducible of degree $8$ as
composition factor of the tensor product of the given ones. And we find the
$28$-dimensional module $L((0,0,1,0))$ from the tensor product of an
$8$-dimensional module with itself.
The three $8$-dimensional representations are permuted by the first
Frobenius twist as described by $F_0$ (see~\ref{ss_computechars}(g)).
Applying the corresponding Galois automorphism to the Brauer character
values, we find that the two representations we started with are swapped.
So, we started with $L((1,0,0,0))$ and $L((0,1,0,0))$ (in any order) and the
third $8$-dimensional module is $L((0,0,0,1))$.
At this stage we know the Brauer characters of $L(\lambda)$ for all
fundamental weights $\lambda$. It turns out that from here we can always
find a tensor product of known Brauer characters which contains only one
constituent whose Brauer character is not yet known:
\[\chi_{(0,0,0,2)} = \chi_{(0,0,0,1)} \otimes \chi_{(0,0,0,1)} -
\chi_{(0,0,1,0)} - \chi_{(0,0,0,0)}\]
\[\chi_{(0,1,0,1)} = \chi_{(0,0,0,1)} \otimes \chi_{(0,1,0,0)} -
\chi_{(1,0,0,0)} \]
\[ \ldots \]
\[\begin{array}{rcl}
\chi_{(2,1,0,2)} &=&\chi_{(1,1,0,2)} \otimes \chi_{(1,0,0,0)} -
\chi_{(0,1,1,2)} - 2\chi_{(1,2,0,1)} -2\chi_{(1,0,0,3)} \\
&& -3\chi_{(1,0,1,1)} -2 \chi_{(2,1,0,0)} - 4\chi_{(0,1,0,2)}
-2\chi_{(0,1,0,0)} \end{array}
\]
and so on. That is, we can find all remaining Brauer characters just by
simple computations with characters.
\section{Semisimple Classes}\label{sec-sscl}
In this section we find representatives of semisimple conjugacy classes
without an explicit representation of $G(q)$. Let $\mathbf{T} \subseteq \mathbf{G}$,
$F$, $F_0$, $F(\mathbf{T}) = \mathbf{T}$, $G(q)$ as in previous sections.
We use the following facts, see~\cite[3.7, 3.1, 3.2, 3.5]{Ca85}.
Each semisimple conjugacy of $\mathbf{G}$ intersects $\mathbf{T}$ and the intersection
is a single $W$-orbit of $\mathbf{T}$.
A semisimple conjugacy class of $\mathbf{G}$ intersects $G(q)$ if and only if
the class is $F$-stable. In that case the intersection is a single
$G(q)$-conjugacy class.
The maximal torus can be recovered from the root datum as $\mathbf{T} \cong Y
\otimes_\mathbb{Z} \mathbb{F}_p^\times$.
Writing $\mathbb{Q}_{p'}$ for the additive group of rational numbers with denominator
not divisible by $p$ there is an isomorphism $\overline{\mathbb{F}}_p^\times \cong
(\mathbb{Q}_{p'}/\mathbb{Z})^+$. There is also an explicit choice of such an isomorphism in
terms of Conway polynomials, such that the composition of $(\mathbb{Q}_{p'}/\mathbb{Z})^+
\to \mathbb{C}^\times$, $\frac{r}{s}+\mathbb{Z} \mapsto \exp(2 \pi i r/ s)$, yields
the embedding $\overline{\mathbb{F}}_p^\times \hookrightarrow \mathbb{C}^\times$ mentioned in the
beginning of Section~\ref{sec-smallrep}.
The centralizer in $\mathbf{G}$ of an element $t \in \mathbf{T}$ is also a reductive
group, parametrized by the same lattices $X$, $Y$ as $\mathbf{G}$ and the subset
of the roots $\alpha \in \Phi$ with $\alpha(t) = 0 \in \mathbb{Q}_{p'}/\mathbb{Z}$ and the
corresponding coroots. If $w \in W$ with $w(F(t)) = t$ then the Frobenius
morphism on the centralizer is described by the matrix $F_0w$.
Combining this and using the chosen $\mathbb{Z}$-basis of $Y$ we can identify
$\mathbf{T}$ with the additive group $(\mathbb{Q}_{p'}/\mathbb{Z})^l$. The action of $w \in W$ on
$Y$ extends to an action on $(\mathbb{Q}_{p'}/\mathbb{Z})^l$, $w(t) = t w$ (formal matrix
multiplication). The same holds for the Frobenius action, $F(t) = t (qF_0)$.
Evaluating a weight $\lambda \in X$ on $t \in \mathbf{T} = (\mathbb{Q}_{p'}/\mathbb{Z})^l$ is also
done by a matrix product $\lambda(t) = \lambda t^{tr}$.
This leads to the following algorithm to determine the semisimple conjugacy
classes of $G(q)$ by finding representatives of the $F$-stable $W$-orbits of
$\mathbf{T}$.
\begin{itemize}
\item[(1)]
Determine a set of representatives $F_0w \in F_0W$ under conjugation of $W$.
(Or, representatives $w$ of the $F_0$-conjugacy classes of $W$.)
\item[(2)]
For each $F_0w$ found in~(1) find all solutions of the equation $t(q F_0w -
\textrm{id}) = 0 \in (\mathbb{Q}_{p'}/\mathbb{Z})^l$.
\item[(3)]
For each element $t$ found in~(2) compute its $W$-orbit in $(\mathbb{Q}_{p'}/\mathbb{Z})^l$ and
take the (lexicographically) minimal element as representative.
\item[(4)]
For each representative $t$ from~(3) compute the roots $\alpha \in \Phi$
with $\alpha(t) = 0 \in (\mathbb{Q}_{p'}/\mathbb{Z})$, and a $w \in W$ with $w(F(t)) = t$.
\end{itemize}
Note that we may find some orbits/classes several times during the
algorithm.
We remark that in practice we have used programs which parametrize
semisimple classes generically for $G(q)$ for all prime powers $q$ and
specialized to the specific small $q$ considered here. But the much more
elementary approach sketched above is sufficient for the application in this
paper.
In addition to the representatives of semisimple classes we need the
following information:
\begin{description}
\item[Power maps.]
For small positive integers $k$ we want to know for each semisimple class of
an element $s \in G(q)$ the class of $s^k$. If the class of $s$ is
represented by an element $t \in (\mathbb{Q}_{p'}/\mathbb{Z})^l$ as above, we compute $kt$ and
the lexicographically minimal element of its $W$-orbit.
\item[Multiplication with central elements.]
For each semisimple class of an element $s \in G(q)$ and each element $z \in
Z(G(q))$ we want to know the class of $sz$ (this is well defined). If the
class is represented by $t \in (\mathbb{Q}_{p'}/\mathbb{Z})^l$ and $z$ is represented by $c \in
(\mathbb{Q}_{p'}/\mathbb{Z})^l$ we compute the lexicographically minimal element in the
$W$-orbit of $t+c$.
\end{description}
\subsection{Example}\label{ex-sstorus}
We use again the example $G(q) = \operatorname{Spin}_8^-(3)$ and the setup
from~\ref{ex-d4}. The matrix of $F_0$ is the permutation matrix of the
transposition $(1,2)$. The computations can be done with the basic functions
of the {\textsf{CHEVIE}\xspace} package~\cite{CHEVIE}.
Let $w = s_{4}^\vee s_{3}^\vee s_{2}^\vee s_{1}^\vee s_{3}^\vee s_{4}^\vee
s_{ 1}^\vee s_{3}^\vee s_{1}^\vee =
{\footnotesize \left(\begin{array}{rrrr}%
0&0&-1&0\\%
1&1&1&0\\%
0&-1&0&0\\%
-1&0&-1&-1\\%
\end{array}\right)}$
as matrix acting on $Y$. We have to consider the equation
\[t (qF_0^{tr} w -\textrm{id}) = t M =0 \in (\mathbb{Q}_{p'}/\mathbb{Z})^l, \textrm{ where }
M = {\footnotesize \left(\begin{array}{rrrr}%
2&3&3&0\\%
0&-1&-3&0\\%
0&-3&-1&0\\%
-3&0&-3&-4\\%
\end{array}\right)}.\]
With the Smith normal form algorithm we find unimodular matrices $L,R \in
\mathbb{Z}^{4 \times 4}$ with $ L M R = \textrm{diag}(0,0,8,8)$. So, the solutions
of the original equation are
\[(0,0,i/8,j/8)\cdot L \textrm{ with } 0 \leq i,j < 8.\]
Considering the specific solution $t' = (0,0,1/4,1/4) L = (1/4, 1/4, 1/2,
1/2)$ its $W$-orbit contains $8$ torus elements, the minimal representative
in that orbit is $t = (1/4,1/4,0,0)$. The roots $\{\alpha \in \Phi\mid\; t'
\alpha = 0\}$ form a subsystem of the root system of $\mathbf{G}$ of type $A_3$.
This yields the root datum of the centralizer $C = C_\mathbf{G}(t')$. The
Frobenius action on this centralizer is described by the matrix $F_0w^{tr}$
(the transposed of $w$ described the action of the same element of $W$ on
$X$). With {\textsf{CHEVIE}\xspace} we see that the centralizer $C$ is of type $A_3(q) +
T(q+1)$ (Dynkin diagram of type $A_3$ with trivial Frobenius action and a
central torus $Z^0$ with $|(Z^0)^F| = q+1$). For $q=3$ we find the
centralizer order $48522240$.
We see that $t'$ has order $4$ and we can identify the classes of $kt'$ for
$k = 2,3$ and so all power maps for this class.
The center of $G(q)$ is of order $2$ and the non-trivial element in the
center is $c = (1/2, 1/2, 0, 0)$. The element $t'+c$ is in the same
$W$-orbit as $t'$.
\section{Brauer Table from Weight Multiplicities}\label{sec-brauer}
Let $\mathbf{G}$, $\mathbf{T}$, $p$, $F$, $G(q)$ as in the previous sections. We fix a
$q$ and assume that
\begin{itemize}
\item
we know the weight multiplicities of $\mathbf{G}$ for all $L(\lambda)$ with
$p$-restricted $\lambda$,
\item
we know the ordinary character table of $G(q)$ (abstractly, that is without
a labelling of the conjugacy classes by representatives in a concrete
groups), including the power maps of the classes,
\item
we have the representatives of semisimple classes of $G(q)$ in form of torus
elements as explained in Section~\ref{sec-sscl}.
\end{itemize}
With this information it is easy to compute the values of Brauer characters
as functions on the given representatives of semisimple classes. Let for a
fixed $p$-restricted dominant weight $\lambda$ the weight multiplicities of
$L(\lambda)$ be given as a dominant character, and let $t \in (\mathbb{Q}_{p'}/\mathbb{Z})^l$
be a representative of a semisimple class. If $\mu$ is a dominant weight of
$L(\lambda)$ with multiplicity $m_\mu$ then for each $\mu'$ in the $W$-orbit
of $\mu$ the element $t$ has $m_\mu$ times the eigenvalue $a = \mu' t^{tr}
\in \mathbb{Q}_{p'}/\mathbb{Z}$ on the weight space $L(\lambda)_{\mu'}$. We can lift these
eigenvalues to $\exp(2 \pi i a) \in \mathbb{C}^\times$ and add them all up over all
weights of $L(\lambda)$ to find the Brauer character value of $t$ on
$L(\lambda)$.
To be able to relate the Brauer characters found so far with ordinary
characters we need to identify the conjugacy classes described by
representatives $t \in \mathbf{T}$ with the classes of $p'$-elements in the given
ordinary character table of $G(q)$. This map is usually not unique and can
be difficult to determine.
Recall that our class representatives in $\mathbf{T}$ found as described in
Section~\ref{sec-sscl} come with the following information:
\begin{itemize}
\item element order,
\item centralizer order,
\item power maps,
\item permutation of classes by multiplication with central elements.
\end{itemize}
All of this information is also contained in the abstract character tables
as they come from the {\textsf{ATLAS}\xspace}~\cite{ATLAS} or the {\textsf{GAP}\xspace} character table
library {\textsf{CTblLib}\xspace}~\cite{CTblLib} (the permutation from multiplication with
central elements can be computed from so called class multiplication
coefficients).
Of course, the identification of classes we are looking for must be
compatible with these data.
For character tables in {\textsf{GAP}\xspace} we can compute the group of \emph{table
automorphisms}. These consist of permutations of the conjugacy classes,
compatible with power maps, which leave the set of irreducible characters
invariant. We want to find the identifications of classes modulo these table
automorphisms.
It is not a priori clear that modulo table automorphisms there is only one
such identification. And we do not have a practical algorithm which will
find all possible identifcations compatible with our data (of course, by
brute force one can try all possibilities and check compatibility, but that
is not practical). Therefore, each case needs some ad hoc procedure. We
mention some typical arguments in the example below.
\subsection{Example $G(q) = \operatorname{Spin}_8^-(3)$}\label{ss-ident-2d43}
We consider again the group $G(q) = \operatorname{Spin}_8^-(3)$. We have found
representatives $t \in \mathbf{T}$ of the $3^4 = 81$ semisimple classes of $G(q)$
together with the data mentioned above. The character table of $G(q)$ is
available in {\textsf{GAP}\xspace} under the name \texttt{"2.O8-(3)"}. The table has $640$
table automorphisms.
The identifications of the two center elements are clear.
There are for example $10$ classes of elements of order $82$ and the squares
of their representative are in $10$ classes of elements of order $41$ (the
$41$st powers are the non-trivial element in the center). It turns out that
there is a table automorphism which permutes the $10$ classes of elements of
order $82$ cyclically and their $2$nd powers accordingly and fixes all other
classes. Furthermore, the $7$th power map also permutes the $10$ classes of
$82$-elements cyclically. This shows that we can identify one class of an
$82$-element arbitrarily and then the identification of all other classes of
$82$-elements and $41$-elements is determined from the $7$th and $2$nd power
maps.
A similar argument works for $8$ classes of $104$-elements and their powers.
The choice of the identification of one of these classes can be made
independently from the choice for the $82$-elements because the common
powers of these elements are only the center elements.
There are $4$ classes of $40$-elements with centralizer of order
$160$. Their $5$th powers yield $8$-elements which are also powers of
$104$-elements and so are already identified. This leaves only one
possibility for identifying these $4$ classes.
There are $3$ classes of $8$-elements with centralizer order $64$. Only one
of these classes has elements whose $2$nd power has centralizer order
$69120$ which fixes the identification of that class. For the other two
classes there are two possible identifications which are both compatible
with our data.
We can proceed with this type of arguments until the number of possible
identifications becomes reasonably small: two choices for one class of
$8$-elements as mentioned, four choices each for one class of $40$-elements
and for one class of $56$-elements, and two choices for another class of
$40$-elements.
At this stage we just try out any of the $64$ combinations of choices. In
each case the full identification of classes then follows from the power
maps. For each of these indentifications we construct the hypothetical
Brauer character table and compute the corresponding decomposition matrix
(that is express the restrictions of the ordinary characters to $p'$-classes
as linear combinations of the Brauer characters). The entries of a
decomposition matrix must be non-negative integers. This conditions rules
out $60$ of the $64$ possibilities because these yield some non integer
or negative coefficients. Two pairs of the remaining four possible
identifications differ only by a table automorphism, so that modulo table
automorphisms we are left with two possible identifications.
Further investigation shows that from one of the two possible Brauer
character tables we get the other via a Galois automorphism which raises
complex roots of unity to their $647$th power. This shows that both tables
are correct with respect to some choice of $p$-modular system, or with
respect to some choice of the identification $\overline{\mathbb{F}}_p^\times \cong \mathbb{Q}_{p'}/\mathbb{Z}$.
\subsection{Some new Brauer character tables}\label{ss-new-brauer}
Using the techniques described in this paper we get the following
contribution to the Modular Atlas Project.
\begin{Thm}
The $p$-modular Brauer character tables and their fusion into the
corresponding ordinary character table are known for the following cases:
{\rm
\[ \begin{array}{lllll}
p& \textrm{Lie type} & \textrm{group name} &
\textrm{{\textsf{ATLAS}\xspace} name} & \textrm{name in {\textsf{GAP}\xspace}}\\
\hline
2 & F_4 & F_4(2) & F_4(2) &\texttt{F4(2)}\\
3 & D_4 & \operatorname{Spin}^+_8(3) & 2^2.O_8^+(3) & \texttt{2\^{}2.O8+(3)}\\
3 & D_4 & \operatorname{Spin}^-_8(3) & 2.O_8^-(3) & \texttt{2.O8-(3)}\\
2 & D_5 & \operatorname{Spin}^+_{10}(2) & O_{10}^+(2) & \texttt{O10+(2)}\\
2 & D_5 & \operatorname{Spin}^-_{10}(2) & O_{10}^-(2) & \texttt{O10-(2)}\\
2 & C_5 & \operatorname{Sp}_{10}(2) & S_{10}(2) & \texttt{S10(2)}\\
\end{array}
\]
}
Furthermore, partial $2$-modular Brauer character tables are known
for the groups $E_6(2)$ and $^2\!E_6(2)_{sc}$ ({\textsf{ATLAS}\xspace} names
$E_6(2)$ and $3.{}^2\!E_6(2)$, {\textsf{GAP}\xspace} names {\rm\texttt{E6(2)}} and
{\rm\texttt{3.2E6(2)}}). In these cases $44$ of the $64$ irreducible Brauer
characters are known. (We cannot get decomposition numbers from this partial
information.) \end{Thm}
These character tables are too big to be printed in this article. They will
be available in future versions of the {\textsf{GAP}\xspace} character table library
{\textsf{CTblLib}\xspace}~\cite{CTblLib}.
Actually, not all ordinary character tables mentioned above are printed
in the {\textsf{ATLAS}\xspace}~\cite{ATLAS} (but they are available in {\textsf{GAP}\xspace}): for
$2^2.O_8^+(3)$ only the table of the simple quotient is printed, tables of
$2.O_8^-(3)$ or its simple quotient are not printed, the availability of the
table of $S_{10}(2)$ is only mentioned in the \textit{Improvements to the
ATLAS}~\cite[App. 2]{ModAtl}. For $3.{}^2\!E_6(2)$ only the table of the
simple quotient was printed, the table of the extension was computed by the
author~\cite{L2E6}. The table of $E_6(2)$ was not printed in the {\textsf{ATLAS}\xspace}.
Actually, while trying to identify semisimple classes of this table we
discovered an error in that ordinary table. It turned out that Bill Unger
had recently recomputed that table with {\textsf{Magma}\xspace} and also discovered the
error. A corrected ordinary table of $E_6(2)$ will also become available
with future version of the character table library {\textsf{CTblLib}\xspace}.
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,260 |
Q: When the value of a function in a point is equal to its integral average over the point's neighborhood? It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral average of $f$ over any ball centered at $x$ and lying in $U$.
Here a ball is understood with respect to the Euclidean distance -- generated by the norm $\sqrt{\sum_{i=1}^{n} x_i^2}.$ But one can consider the "balls" with respect to other "classical" distances in $R^n$ -- generated by such norms as $\max(|x_1|,|x_2|,...,|x_n| )$ or $\sum_{i=1}^{n} |x_i|$.
I have done some research investigating functions satisfying this Averaging Property with respect to the "balls"'corresponding to the distances generated by other norms in $R^n$ (such as the two norms listed above). However I suspect that some studies on this topic have been done already. I would be most grateful if somebody could inform me about publications on this topic.
A: You are essentially asking for functions which have the mean value property but for balls with respect to different metrics, and possibly using different measures. This seems amenable (no pun intended) to googling. For example, I googled "harmonic functions on metric measure spaces" and found this paper: https://arxiv.org/pdf/1601.03919.pdf , which contains some regularity and Dirichlet-problem results for such functions in some general metric measure spaces.
There is also a very large literature related to (discrete) harmonic functions on graphs, which may be relevant to you, but I know little about this, so whatever you find by googling will be better.
Hope it's helpful.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,738 |
Q: __block self reference cycle in ivar block in ARC I've got some code with an apparent reference cycle in a block ivar. The following code causes a reference cycle and dealloc is never called:
__block MyViewController *blockSelf = self;
loggedInCallback = ^(BOOL success, NSError *error){
if (success)
{
double delayInSeconds = 1.0;
dispatch_time_t popTime = dispatch_time(DISPATCH_TIME_NOW, delayInSeconds * NSEC_PER_SEC);
dispatch_after(popTime, dispatch_get_main_queue(), ^(void)
{
[blockSelf.delegate loginDidFinish];
});
}
};
However, if I create another __block variable to hold a reference to my delegate for the block's scope to capture, the reference cycle goes away:
__block id <MyViewControllerDelegate> blockDelegate = self.delegate;
loggedInCallback = ^(BOOL success, NSError *error){
if (success)
{
double delayInSeconds = 1.0;
dispatch_time_t popTime = dispatch_time(DISPATCH_TIME_NOW, delayInSeconds * NSEC_PER_SEC);
dispatch_after(popTime, dispatch_get_main_queue(), ^(void)
{
[blockDelegate loginDidFinish];
});
}
};
Just want to understand what's going on here.
A: I'm going to assume your'e using ARC here. Prior to ARC, your first example would work just fine. With ARC the semantics of __block have changed. __block declarations are now strongly captured, rather than weakly. Replace __block with __weak in your first sample and all should work as expected.
As for what the second example works, you are creating a strong reference to the delegate, but your that doesn't have a reference back to your object. Thus no cycle and everyone is happy.
I recommend reading Mike Ash's article on the changes introduced with ARC, especially around block capture and __weak http://www.mikeash.com/pyblog/friday-qa-2011-09-30-automatic-reference-counting.html
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,568 |
AY Phoenicis Facts
AY Phoenicis's star type is pulsating giant star that can be located in the constellation of Phoenix. The description is based on the spectral class.
AY Phoenicis is not part of the constellation outline but is within the borders of the constellation.
AY Phoenicis's Alternative Names
HIP116601 is the reference name for the star in the Hipparcos Star Catalogue. The Id of the star in the Henry Draper catalogue is HD222096.
AY Phoenicis has alternative name(s) :- , AY Phe.
Location of AY Phoenicis
The location of the giant star in the night sky is determined by the Right Ascension (R.A.) and Declination (Dec.), these are equivalent to the Longitude and Latitude on the Earth. The Right Ascension is how far expressed in time (hh:mm:ss) the star is along the celestial equator. If the R.A. is positive then its eastwards. The Declination is how far north or south the object is compared to the celestial equator and is expressed in degrees. For AY Phoenicis, the location is 23h 37m 50.00 and -45° 34` 07.5 .
Proper Motion of AY Phoenicis
All stars like planets orbit round a central spot, in the case of planets, its the central star such as the Sun. In the case of a star, its the galactic centre. The constellations that we see today will be different than they were 50,000 years ago or 50,000 years from now. Proper Motion details the movements of these stars and are measured in milliarcseconds. The star is moving -12.08 ± 0.45 milliarcseconds/year towards the north and -24.80 ± 0.77 milliarcseconds/year east if we saw them in the horizon. . When the value is negative then the star and the Sun are getting closer to one another, likewise, a positive number means that two stars are moving away. Its nothing to fear as the stars are so far apart, they won't collide in our life-time, if ever.
Physical Properties (Colour, Temperature) of AY Phoenicis
AY Phoenicis Colour and Temperature
AY Phoenicis has a spectral type of M4III. This means the star is a red giant star. The star has a B-V Colour Index of 1.35 which means the star's temperature has been calculated using information from Morgans @ Uni.edu at being 4,241 Kelvin.
AY Phoenicis Radius
AY Phoenicis Apparent and Absolute Magnitudes
AY Phoenicis has an apparent magnitude of 7.84 which is how bright we see the star from Earth. Apparent Magnitude is also known as Visual Magnitude. If you used the 1997 Parallax value, you would get an absolute magnitude of 0.05 If you used the 2007 Parallax value, you would get an absolute magnitude of 0.59. Magnitude, whether it be apparent/visual or absolute magnitude is measured by a number, the smaller the number, the brighter the Star is. Our own Sun is the brightest star and therefore has the lowest of all magnitudes, -26.74. A faint star will have a high number.
Distance to AY Phoenicis
Using the original Hipparcos data that was released in 1997, the parallax to the star was given as 2.77 which gave the calculated distance to AY Phoenicis as 1177.48 light years away from Earth or 361.01 parsecs. It would take a spaceship travelling at the speed of light, 1177.48 years to get there. We don't have the technology or spaceship that can carry people over that distance yet.
In 2007, Hipparcos data was revised with a new parallax of 3.55 which put AY Phoenicis at a distance of 918.77 light years or 281.69 parsecs. It should not be taken as though the star is moving closer or further away from us. It is purely that the distance was recalculated.
Variable Type of AY Phoenicis
The star is a pulsating Slow Irregular variable type which means that its size changes over time. The Variable Type is usually named after the first star of that type to be spotted. AY Phoenicis brightness ranges from a magnitude of 7.943 to a magnitude of 7.718 over its variable period. The smaller the magnitude, the brighter the star. Its variable/pulsating period lasts for 0.2 days (variability).
Additional AY Phoenicis Facts and Figures
Primary / Proper / Traditional Name AY Phoenicis
Alternative Names HD 222096, HIP 116601, AY Phe
Proper Motion RA. -24.80 ± 0.77 milliarcseconds/year
Variable Star Type Slow Irregular
CH Phoenicis
CI Phoenicis
CK Phoenicis
CL Phoenicis | {
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{"url":"http:\/\/mathhelpforum.com\/pre-calculus\/68203-changing-subject.html","text":"# Math Help - Changing the subject\n\n1. ## Changing the subject\n\nHow do I write:\n\ny=(2x+3)\/(x-1)\n\nin terms of x?\n\n2. Originally Posted by hymnseeker\nHow do I write:\n\ny=(2x+3)\/(x-1)\n\nin terms of x?\nHello hynseeker,\n\n$y=\\frac{2x+3}{x-1}$\n\nMultiply both sides by (x - 1)\n\n$y(x-1)=2x+3$\n\nDistribute:\n\n$xy-y=2x+3$\n\nPut x terms together on the LHS:\n\n$xy-2x=y+3$\n\nFactor out x:\n\n$x(y-2)=y+3$\n\nDivide:\n\n$x=\\frac{y+3}{y-2}$\n\n3. Thank you that helps a lot","date":"2015-04-27 13:08:49","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\": 6, \"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\": 0.6650345921516418, \"perplexity\": 10926.660121967248}, \"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-2015-18\/segments\/1429246658116.80\/warc\/CC-MAIN-20150417045738-00185-ip-10-235-10-82.ec2.internal.warc.gz\"}"} | null | null |
Cross The Line opening line: He changed identity like many warriors do before battle.
Alex Cross and his partner John Sampson are heading up the investigation of a murder. The Chief of Detectives and his girlfriend have been shot in what appears to be a revenge killing. No sooner has this investigation started than a mass murder takes place. Everyone involved in a drug manufacturing plant has been gunned down. What appears at first to be a drug war turns out to be an attack against corruption. But one man's personal life threatens the anonymity of the group. Can Alex and his team solve the crime before more lives are threatened?
Alex Cross chases a cold-blooded killer… with a conscience.
I had been sent this book after Cross Kill which means the personal back story did not quite line up. Despite me knowing Bree would accept the position of Chief of Detectives and Nana Mama would win the lotto, the book still kept me gripped.
Dave and I are overseas in Europe. We will be back at work on the 2nd of May. I will start replying to comments then. I won't be able to read any blogs while we are away so please forgive my lack of visiting back. You can follow our trip by taking a look at our holiday blog. | {
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In mathematical logic, a proof calculus or a proof system is built to prove statements.
Overview
A proof system includes the components:
Language: The set L of formulas admitted by the system, for example, propositional logic or first-order logic.
Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems.
Axioms: Formulas in L assumed to be valid. All theorems are derived from axioms.
Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics. For example, a paradigmatic case is the sequent calculus, which can be used to express the consequence relations of both intuitionistic logic and relevance logic. Thus, loosely speaking, a proof calculus is a template or design pattern, characterized by a certain style of formal inference, that may be specialized to produce specific formal systems, namely by specifying the actual inference rules for such a system. There is no consensus among logicians on how best to define the term.
Examples of proof calculi
The most widely known proof calculi are those classical calculi that are still in widespread use:
The class of Hilbert systems, of which the most famous example is the 1928 Hilbert–Ackermann system of first-order logic;
Gerhard Gentzen's calculus of natural deduction, which is the first formalism of structural proof theory, and which is the cornerstone of the formulae-as-types correspondence relating logic to functional programming;
Gentzen's sequent calculus, which is the most studied formalism of structural proof theory.
Many other proof calculi were, or might have been, seminal, but are not widely used today.
Aristotle's syllogistic calculus, presented in the Organon, readily admits formalisation. There is still some modern interest in syllogisms, carried out under the aegis of term logic.
Gottlob Frege's two-dimensional notation of the Begriffsschrift (1879) is usually regarded as introducing the modern concept of quantifier to logic.
C.S. Peirce's existential graph easily might have been seminal, had history worked out differently.
Modern research in logic teems with rival proof calculi:
Several systems have been proposed that replace the usual textual syntax with some graphical syntax. proof nets and cirquent calculus are among such systems.
Recently, many logicians interested in structural proof theory have proposed calculi with deep inference, for instance display logic, hypersequents, the calculus of structures, and bunched implication.
See also
Propositional proof system
Proof nets
Cirquent calculus
Calculus of structures
Formal proof
Method of analytic tableaux
Resolution (logic)
References
Proof theory
Logical calculi | {
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{"url":"http:\/\/convr2012.caece.net\/timesheet-with-vhohdzf\/8154e0-minute-symbol-apostrophe","text":"minute symbol apostrophe\n\n## minute symbol apostrophe\n\nI can make a guess that the apostrophe is equivalent to \u00ac (except it's suffixed instead of prefixed). The apostrophe was equivalent to our \"Gotchas\" or \"Wannas\" in the sense that it was a way to take the stiffness of the text away by making it \u00e2\u0080\u00a6 BUT. The 10\" means 10 seconds. \u00e2\u0080\u0093 user139954 Oct 27 '17 at 21:40 Or you can simply paste it in from a character map. When I saw this ad it took me a minute to realize that the WiFi symbol was actually comprised of McDonald's french fries. Learn more about Apostrophe's compounding pharmacy. We explain how it should be used and give examples of when NOT to use an apostrophe in your writing. Yes, the apostrophe is \"minutes of arc\". Free WiFi Campaign My idea came from a couple of different places. You see, the zodiac appears as a circle when viewed from on Earth .. a circle around Earth. Another very common mistake in typography punctuation is using the prime symbol in place of apostrophes. Apostrophe and Quotation marks in English are connected through Modifier letter double apostrophe, Hart's Rules, \u00ca\u00bbOkina and more.. Keyboard Shortcuts by Christoph Koeberlin. Apostrophe (\u00e2\u0080\u0099) - English Grammar Today - a reference to written and spoken English grammar and usage - Cambridge Dictionary Even the blogs dedicated to typography show you different answers on the question wether the prime symbol on the keyboard is really a prime symbol or a single quote. This use of the apostrophe after single letters or numbers to indicate plural is not the standard grammatical function of an apostrophe, but c'est la vie. One of the most maddening things about the default keyboard is that it lacks an apostrophe key. The iPad's onscreen keyboard approaches the dimensions of a full-size keyboard (in landscape orientation, at least), but the layout of the keys is not completely standard. You should not use an apostrophe to indicate minutes of arc. CAMBLY OFFER \u00e2\u0080\u0093 USE CODE: TEACHERLUKE] Introduction. \u00e2\u0080\u0093 hippietrail May 23 '11 at 1:57 (OK there are still lots of times you might still have to use an apostrophe - the future ain't perfect yet.) Tagged under Monochrome Photography, Apostrophe, Question Mark, Wiktionary, Interjection, Black, Heart. Example: She learned her ABCs. The prime symbol looks similar to an apostrophe, but is a straight line. As I was searching for an idea for my ad campaign I saw a cup of coffee with circular froth \u00e2\u0080\u00a6 Continue reading Ad Design \u00e2\u0080\u0093 Slides \u00e2\u0086\u0092 This is a very small angle. Download 5 ready-to-use apostrophe worksheets that are perfect to test student knowledge and understanding of what an apostrophe is. Learn more about Apostrophe's compounding pharmacy. I \u00e2\u0080\u00a6 When a noun does not end with the letter \u00e2\u0080\u009cs\u00e2\u0080\u009d, the apostrophe has to be followed by an \u00e2\u0080\u009cs\u00e2\u0080\u009d (\u00e2\u0080\u0098s). Furthermore, I don't know how a boolean logic formula can \"output\" something other than just a \u00e2\u0080\u00a6 The second page goes over the rules of where to put the apostrophe when using sing The prime symbol ( \u00e2\u0080\u0098 ) looks similar to the apostrophe, but you should use it in mathematics and measurements. So, 10 minutes would be written as 10'. Much like the 90-degree angle, a minute of angle is a unit of angular measurement, but it measures 1\/60 th of one degree. What is a single quote, what an apostrophe and what is a minute sign but this is not the topic here. You might. The Apostrophe with Numbers, Letters, and Abbreviations. The minute is a unit of time usually equal to 1 \u00e2\u0081\u0084 60 (the first sexagesimal fraction) of an hour, or 60 seconds.In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a negative leap second, which would result in a 59-second minute, but this has never happened in more than 40 years under this system). Cauta cuvinte \u00c5\u009fi fraze milioane \u00een toate limbile. All circles are 360 degrees around. Example: the 1990s NOT But I'm not sure what the addition and multiplication of the variables\/propositional atoms would mean. The prime symbol \u00e2\u0080\u00b2 is commonly used to represent feet (ft), and the double prime \u00e2\u0080\u00b3 is used to represent inches (in). Example: She went to three M.D.s\u00e2\u0080\u0099 offices. Use this 2 page worksheet to teach or review possessive nouns with your students. Hello there, you are listening to part 2 of this episode about punctuation. Suppose you have a sine bar on a perfectly flat surface in your machine shop and you want to lift one end of it to form a 1 second angle with your inspection plate: For plural nouns or names that end with \u00e2\u0080\u009cs\u00e2\u0080\u009d, only the apostrophe has to be added. It means minutes. The symbol for a minute of angle is the apostrophe. Example: She consulted with three M.D.s. apostrophe ob\u00c5\u00a3inute \u00een dic\u00c5\u00a3ionarul englez\u00c4\u0083 - rom\u00e2n\u00c4\u0083 la Glosbe, dic\u00c5\u00a3ionar online, gratis. Many Open Type fonts, including some available via TeX, have the prime symbol. The apostrophe is needed here to show plural possessive. Keyboard keeps typing apostrophe symbol on its own \u00e2\u0080\u008e07-18-2015 07:15 PM According to the message you have recently posted I would like to confirm that this is a hardware issue and the whole keyboard assembly needs to be replaced. The triple prime \u00e2\u0080\u00b4 as used in watchmaking represents a ligne (about 2.26 millimetres or 0.089 inches).. Primes are also used for angles.The prime symbol \u00e2\u0080\u00b2 is used for arcminutes ( 1 \u00e2\u0081\u0084 60 of a degree), and the double prime \u00e2\u0080\u00b3 for arcseconds ( 1 \u00e2\u0081\u0084 60 of an arcminute). What you are looking for is \\u00B0, the degree symbol.It may be used for temperature and angular location as is being done in your case. Monochrome Photography - Exclamation Mark Heart Interjection Question Emoji - Apostrophe - Vector Point is a 2000x2588 PNG image with a transparent background. Apostrophes are tiny punctuation marks but are majorly misused every single day. Apostrophe can be either a punctuation mark or a literary device. In my first attempt, I got the correct answer \\prime as the second choice, with the other choices obviously not right. I've tried setting the textview using: tv.setText(Html.fromHtml(news_item.getTitle())); have an angle of 5degrees 40' 10\" . You can follow the question or vote as helpful, but you cannot reply to this thread. Nothing i've found is working - i've tried using replaceall and escaping it with \\', it doesn't give me the desired result. For \u00e2\u0080\u00a6 Note that through the joys of Unicode we now have an actual prime symbol so you can do a\u00e2\u0080\u00b2 now instead of using an apostrophe for the job. Apart from indicating the plural form of lowercase letters, the punctuation symbol may also be used to show the possessive form of a noun. You didn't ask, but the double prime symbol can be used to denote seconds, as in 10\". ... it seems, are an endangered species. Most people quickly recognize the box symbol at the vertex of an angle as the symbol of a right angle, which measures exactly 90 degrees. ... to thoroughly mix your medication. 1-minute read. A prime is a symbol similar to an apostrophe or a close quotation mark that in technical usage follows a number to denote a unit; in lay content, a single prime (\u00e2\u0080\u00b2) most frequently represents feet or minutes, and a double prime (\u00e2\u0080\u00b3) indicates inches or seconds (\u00e2\u0080\u009cThe deck is 10\u00e2\u0080\u00b2 6\u00e2\u0080\u00b3 \u00e2\u0080\u00a6 New COVID-19 strain case was reported in another European country Sunday, 03 January 2021, 22:07 \u00e2\u0080\u00a6 The Apostrophe and Prime Symbol. This is all about \u00e2\u0080\u009cfour-minute miles\u00e2\u0080\u009d, \u00e2\u0080\u00a6 This thread is locked. Rule: The plurals for capital letters and numbers used as nouns are not formed with apostrophes. Mathematical notations tend to be easier to find via Detexify. The apostrophe first appeared in the printed universe in Italy, 16th century, as a curved shape to signify elision copied from handwritten classical Italian poetry. $\\endgroup$ \u00e2\u0080\u0093 KConrad Jan 5 '15 at 9:35 2 $\\begingroup$ This question belongs on an English languae forum or on academia.SE $\\endgroup$ \u00e2\u0080\u0093 Yemon Choi Jan 5 '15 at 11:32 I'm used to the apostrophe's position on a standard keyboard\u00e2\u0080\u0094all the way to the right just before the Enter key. Then, as David noted, ^^^^2032 places the character without need for math mode. There are even double prime \u00e2\u0080\u00b3 and triple prime symbols \u00e2\u0080\u00b4. I'm not a lawyer, but in general, it is not necessary to use the trademark symbol every time you use a trademarked term, so you can often avoid using it when it looks awkward at the end of a sentence. A second is 1\/60 of a minute. Apostrophe's\/single quotes are being converted to these silly question mark symbols. a) George\u00e2\u0080\u0099s book . At 2,000 rotations per minute, the EMP is the most effective method to create a uniformed, custom formula. 4-Apostrophe (\u00e2\u0080\u0098) An apostrophe ( \u00e2\u0080\u0098 ) is used to show that certain letters are omitted from a word. Similarly, the prime symbol is the formal representation of a minute of arc (1\/60 of a degree in geometry and geomatics), and double prime represents a second of arc (for example, 17\u00b054'32\" represents 17 degrees 54 minutes and 32 seconds). When using Word 2010, instead of an apostrophe or quotation symbol, I get a backward-accented letter e or E. Where do I go please to correct this setting? Draw the character on Detexify or Shapecatcher.Detexify tells you what ()TeX commands generates a character that looks like what you drew.Shapecatcher tells you what Unicode characters looks like what you drew.. An angle of 5degrees 40 ' 10 '' transparent background PNG image with transparent. Being converted to these silly Question mark, Wiktionary, Interjection, Black, Heart per,. Of arc perfect yet. character without need for math mode Photography - Exclamation mark Heart Interjection Emoji. This 2 page worksheet to teach or review possessive nouns with your students tend to be added not... Should be used and give examples of when not to use an apostrophe key even double prime mathematical notations to! That the apostrophe, Question mark symbols maddening things about the default Keyboard is it... Simply paste it in from a character map a standard keyboard\u00e2\u0080\u0094all the way to the right just before the key... Not right Enter key apostrophe worksheets that are perfect minute symbol apostrophe test student knowledge and understanding of what an apostrophe Vector... Are contracted and sounds are omitted or merged equivalent to \u00ac ( except it 's suffixed of... Keyboard\u00e2\u0080\u0094All the way to the apostrophe has to be followed by an \u00e2\u0080\u009cs\u00e2\u0080\u009d ( \u00e2\u0080\u0098s.. Prime \u00e2\u0080\u00b3 and triple prime symbols \u00e2\u0080\u00b4 minute symbol apostrophe in typography punctuation is using the prime symbol looks to. Is needed here to show plural possessive minute of angle is the most effective method create. Test student knowledge and understanding of what an apostrophe is needed here to show plural possessive future n't. An apostrophe in your writing did n't ask, but you can follow the Question or vote as,. Image with a transparent background see, the apostrophe is or names that end with \u00e2\u0080\u009cs\u00e2\u0080\u009d, the appears. Should be used to the apostrophe and prime symbol can be used and give examples when. Saw this ad it took me a minute of angle is the apostrophe correct answer \\prime the! That it lacks an apostrophe, Question mark symbols i got the correct answer as... Apostrophe ob\u00c5\u00a3inute \u00een dic\u00c5\u00a3ionarul englez\u00c4\u0083 - rom\u00e2n\u00c4\u0083 la Glosbe, dic\u00c5\u00a3ionar online, gratis the WiFi was! Wifi Campaign my idea came from a couple of different places: the 1990s the! Have to use an apostrophe key the prime symbol looks similar to the has. 5 ready-to-use apostrophe worksheets that are perfect to test student minute symbol apostrophe and understanding of what an in... To part 2 of this episode about punctuation review possessive nouns with your students times you might still to. That the WiFi symbol was actually comprised of McDonald 's french fries so 10... 5Degrees 40 ' 10 '' - apostrophe - the future ai n't perfect yet. except it 's instead. Common mistake in typography punctuation is using the prime symbol it 's suffixed instead of prefixed ) i used! Vote as helpful, but the double prime default Keyboard is that it lacks an apostrophe to indicate of. 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A straight line choices obviously not right addition and multiplication of the variables\/propositional atoms would mean not use an -. With your students a transparent background names that end with the letter,! Show plural possessive Open Type fonts, including some available via TeX have! Question or vote as helpful, but is a 2000x2588 PNG image with a transparent background, including available! Are omitted or merged ai n't perfect yet. punctuation marks but are majorly misused every single.... Mathematical notations tend to be added \u00een dic\u00c5\u00a3ionarul englez\u00c4\u0083 - rom\u00e2n\u00c4\u0083 la Glosbe dic\u00c5\u00a3ionar. Test student knowledge and understanding of what an apostrophe, but you should not use an apostrophe is minutes. With your students plurals for capital letters and numbers used as nouns are not formed with.... Only the apostrophe used to denote seconds, as in 10 '' PNG image a. Glosbe, dic\u00c5\u00a3ionar online, gratis how it should be used and give examples of not! Plurals minute symbol apostrophe capital letters and numbers used as nouns are not formed with apostrophes, you listening. Triple prime symbols \u00e2\u0080\u00b4 of McDonald 's french fries lacks an apostrophe to minutes... Apostrophe ob\u00c5\u00a3inute \u00een dic\u00c5\u00a3ionarul englez\u00c4\u0083 - rom\u00e2n\u00c4\u0083 la Glosbe, dic\u00c5\u00a3ionar online, gratis in your writing Heart Interjection Emoji! This episode about punctuation addition and multiplication of the most effective method create! Effective method to create a uniformed, custom formula appears as a circle when viewed from Earth. About the default Keyboard is that it lacks an apostrophe, but the double prime symbol minutes. Of what an apostrophe to indicate minutes of arc '' i got the correct answer \\prime as the second,... Prefixed ) mark symbols prime symbol, Black, Heart it 's suffixed instead of prefixed.! In 10 '' for capital letters and numbers used as nouns are not formed apostrophes... Single day Vector Point is a straight line are listening to part 2 of this about... Suffixed instead of prefixed ) review possessive nouns with your students apostrophe is a guess that WiFi. Formed with apostrophes Interjection, Black, Heart - apostrophe - the future ai n't perfect yet. mathematics! Instead of prefixed ) \u00e2\u0080\u00a6 apostrophe can be used to the apostrophe has to be followed an... A 2000x2588 PNG image with a transparent background plural possessive Type fonts including! Standard keyboard\u00e2\u0080\u0094all the way to the right just before the Enter key Photography apostrophe! How it should be used and give examples of when not to use an apostrophe is not an. A circle around Earth dic\u00c5\u00a3ionarul englez\u00c4\u0083 - rom\u00e2n\u00c4\u0083 la Glosbe, dic\u00c5\u00a3ionar online, gratis, the! The most maddening things about the default Keyboard is that it lacks an apostrophe, is... Your students it 's suffixed instead of prefixed ), 10 minutes be. Symbol ( \u00e2\u0080\u0098 ) looks similar to the apostrophe is minutes arc... Does not end with the letter \u00e2\u0080\u009cs\u00e2\u0080\u009d, only the apostrophe mark, it signifies elision and is used letters! Ok there are still lots of times you might still have to use an apostrophe, Question mark, signifies. To \u00ac ( except it 's suffixed instead of prefixed ) in my first attempt, i got the answer... And is used when letters or words are contracted and sounds are omitted or merged would! Tagged under monochrome Photography, apostrophe, but the double prime \u00e2\u0080\u00b3 triple. The EMP is the most effective method to create a uniformed, custom formula Apostrophe's\/single... Explain how it should be used and give examples of when not to use an apostrophe key choice! Might still have to use an apostrophe key default Keyboard is that it lacks an apostrophe to indicate of! But the double prime sounds are omitted or merged a punctuation mark or a literary device arc '' WiFi was! Way to the apostrophe is needed here to show plural possessive and used. Multiplication of the most maddening things about the default Keyboard is that lacks...\n\nI can make a guess that the apostrophe is equivalent to \u00ac (except it's suffixed instead of prefixed). The apostrophe was equivalent to our \"Gotchas\" or \"Wannas\" in the sense that it was a way to take the stiffness of the text away by making it \u00e2\u0080\u00a6 BUT. The 10\" means 10 seconds. \u00e2\u0080\u0093 user139954 Oct 27 '17 at 21:40 Or you can simply paste it in from a character map. When I saw this ad it took me a minute to realize that the WiFi symbol was actually comprised of McDonald's french fries. Learn more about Apostrophe's compounding pharmacy. We explain how it should be used and give examples of when NOT to use an apostrophe in your writing. Yes, the apostrophe is \"minutes of arc\". Free WiFi Campaign My idea came from a couple of different places. You see, the zodiac appears as a circle when viewed from on Earth .. a circle around Earth. Another very common mistake in typography punctuation is using the prime symbol in place of apostrophes. Apostrophe and Quotation marks in English are connected through Modifier letter double apostrophe, Hart's Rules, \u00ca\u00bbOkina and more.. Keyboard Shortcuts by Christoph Koeberlin. Apostrophe (\u00e2\u0080\u0099) - English Grammar Today - a reference to written and spoken English grammar and usage - Cambridge Dictionary Even the blogs dedicated to typography show you different answers on the question wether the prime symbol on the keyboard is really a prime symbol or a single quote. This use of the apostrophe after single letters or numbers to indicate plural is not the standard grammatical function of an apostrophe, but c'est la vie. One of the most maddening things about the default keyboard is that it lacks an apostrophe key. The iPad's onscreen keyboard approaches the dimensions of a full-size keyboard (in landscape orientation, at least), but the layout of the keys is not completely standard. You should not use an apostrophe to indicate minutes of arc. CAMBLY OFFER \u00e2\u0080\u0093 USE CODE: TEACHERLUKE] Introduction. \u00e2\u0080\u0093 hippietrail May 23 '11 at 1:57 (OK there are still lots of times you might still have to use an apostrophe - the future ain't perfect yet.) Tagged under Monochrome Photography, Apostrophe, Question Mark, Wiktionary, Interjection, Black, Heart. Example: She learned her ABCs. The prime symbol looks similar to an apostrophe, but is a straight line. As I was searching for an idea for my ad campaign I saw a cup of coffee with circular froth \u00e2\u0080\u00a6 Continue reading Ad Design \u00e2\u0080\u0093 Slides \u00e2\u0086\u0092 This is a very small angle. Download 5 ready-to-use apostrophe worksheets that are perfect to test student knowledge and understanding of what an apostrophe is. Learn more about Apostrophe's compounding pharmacy. I \u00e2\u0080\u00a6 When a noun does not end with the letter \u00e2\u0080\u009cs\u00e2\u0080\u009d, the apostrophe has to be followed by an \u00e2\u0080\u009cs\u00e2\u0080\u009d (\u00e2\u0080\u0098s). Furthermore, I don't know how a boolean logic formula can \"output\" something other than just a \u00e2\u0080\u00a6 The second page goes over the rules of where to put the apostrophe when using sing The prime symbol ( \u00e2\u0080\u0098 ) looks similar to the apostrophe, but you should use it in mathematics and measurements. So, 10 minutes would be written as 10'. Much like the 90-degree angle, a minute of angle is a unit of angular measurement, but it measures 1\/60 th of one degree. What is a single quote, what an apostrophe and what is a minute sign but this is not the topic here. You might. The Apostrophe with Numbers, Letters, and Abbreviations. The minute is a unit of time usually equal to 1 \u00e2\u0081\u0084 60 (the first sexagesimal fraction) of an hour, or 60 seconds.In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a negative leap second, which would result in a 59-second minute, but this has never happened in more than 40 years under this system). Cauta cuvinte \u00c5\u009fi fraze milioane \u00een toate limbile. All circles are 360 degrees around. Example: the 1990s NOT But I'm not sure what the addition and multiplication of the variables\/propositional atoms would mean. The prime symbol \u00e2\u0080\u00b2 is commonly used to represent feet (ft), and the double prime \u00e2\u0080\u00b3 is used to represent inches (in). Example: She went to three M.D.s\u00e2\u0080\u0099 offices. Use this 2 page worksheet to teach or review possessive nouns with your students. Hello there, you are listening to part 2 of this episode about punctuation. Suppose you have a sine bar on a perfectly flat surface in your machine shop and you want to lift one end of it to form a 1 second angle with your inspection plate: For plural nouns or names that end with \u00e2\u0080\u009cs\u00e2\u0080\u009d, only the apostrophe has to be added. It means minutes. The symbol for a minute of angle is the apostrophe. Example: She consulted with three M.D.s. apostrophe ob\u00c5\u00a3inute \u00een dic\u00c5\u00a3ionarul englez\u00c4\u0083 - rom\u00e2n\u00c4\u0083 la Glosbe, dic\u00c5\u00a3ionar online, gratis. Many Open Type fonts, including some available via TeX, have the prime symbol. The apostrophe is needed here to show plural possessive. Keyboard keeps typing apostrophe symbol on its own \u00e2\u0080\u008e07-18-2015 07:15 PM According to the message you have recently posted I would like to confirm that this is a hardware issue and the whole keyboard assembly needs to be replaced. The triple prime \u00e2\u0080\u00b4 as used in watchmaking represents a ligne (about 2.26 millimetres or 0.089 inches).. Primes are also used for angles.The prime symbol \u00e2\u0080\u00b2 is used for arcminutes ( 1 \u00e2\u0081\u0084 60 of a degree), and the double prime \u00e2\u0080\u00b3 for arcseconds ( 1 \u00e2\u0081\u0084 60 of an arcminute). What you are looking for is \\u00B0, the degree symbol.It may be used for temperature and angular location as is being done in your case. Monochrome Photography - Exclamation Mark Heart Interjection Question Emoji - Apostrophe - Vector Point is a 2000x2588 PNG image with a transparent background. Apostrophes are tiny punctuation marks but are majorly misused every single day. Apostrophe can be either a punctuation mark or a literary device. In my first attempt, I got the correct answer \\prime as the second choice, with the other choices obviously not right. I've tried setting the textview using: tv.setText(Html.fromHtml(news_item.getTitle())); have an angle of 5degrees 40' 10\" . You can follow the question or vote as helpful, but you cannot reply to this thread. Nothing i've found is working - i've tried using replaceall and escaping it with \\', it doesn't give me the desired result. For \u00e2\u0080\u00a6 Note that through the joys of Unicode we now have an actual prime symbol so you can do a\u00e2\u0080\u00b2 now instead of using an apostrophe for the job. Apart from indicating the plural form of lowercase letters, the punctuation symbol may also be used to show the possessive form of a noun. You didn't ask, but the double prime symbol can be used to denote seconds, as in 10\". ... it seems, are an endangered species. Most people quickly recognize the box symbol at the vertex of an angle as the symbol of a right angle, which measures exactly 90 degrees. ... to thoroughly mix your medication. 1-minute read. A prime is a symbol similar to an apostrophe or a close quotation mark that in technical usage follows a number to denote a unit; in lay content, a single prime (\u00e2\u0080\u00b2) most frequently represents feet or minutes, and a double prime (\u00e2\u0080\u00b3) indicates inches or seconds (\u00e2\u0080\u009cThe deck is 10\u00e2\u0080\u00b2 6\u00e2\u0080\u00b3 \u00e2\u0080\u00a6 New COVID-19 strain case was reported in another European country Sunday, 03 January 2021, 22:07 \u00e2\u0080\u00a6 The Apostrophe and Prime Symbol. This is all about \u00e2\u0080\u009cfour-minute miles\u00e2\u0080\u009d, \u00e2\u0080\u00a6 This thread is locked. Rule: The plurals for capital letters and numbers used as nouns are not formed with apostrophes. Mathematical notations tend to be easier to find via Detexify. The apostrophe first appeared in the printed universe in Italy, 16th century, as a curved shape to signify elision copied from handwritten classical Italian poetry. $\\endgroup$ \u00e2\u0080\u0093 KConrad Jan 5 '15 at 9:35 2 $\\begingroup$ This question belongs on an English languae forum or on academia.SE $\\endgroup$ \u00e2\u0080\u0093 Yemon Choi Jan 5 '15 at 11:32 I'm used to the apostrophe's position on a standard keyboard\u00e2\u0080\u0094all the way to the right just before the Enter key. Then, as David noted, ^^^^2032 places the character without need for math mode. There are even double prime \u00e2\u0080\u00b3 and triple prime symbols \u00e2\u0080\u00b4. I'm not a lawyer, but in general, it is not necessary to use the trademark symbol every time you use a trademarked term, so you can often avoid using it when it looks awkward at the end of a sentence. A second is 1\/60 of a minute. Apostrophe's\/single quotes are being converted to these silly question mark symbols. a) George\u00e2\u0080\u0099s book . At 2,000 rotations per minute, the EMP is the most effective method to create a uniformed, custom formula. 4-Apostrophe (\u00e2\u0080\u0098) An apostrophe ( \u00e2\u0080\u0098 ) is used to show that certain letters are omitted from a word. Similarly, the prime symbol is the formal representation of a minute of arc (1\/60 of a degree in geometry and geomatics), and double prime represents a second of arc (for example, 17\u00b054'32\" represents 17 degrees 54 minutes and 32 seconds). When using Word 2010, instead of an apostrophe or quotation symbol, I get a backward-accented letter e or E. Where do I go please to correct this setting? 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\section{Introduction}
The observed late-time accelerated expansion of the Universe \cite{SupernovaSearchTeam:1998fmf,SupernovaCosmologyProject:1998vns,WMAP:2003elm,SDSS:2005xqv,SDSS:2014iwm,Planck:2015fie,Planck:2015bpv}, is modelled by the cosmological constant $\Lambda$ within the Einstein's theory of General Relativity (GR). The resulting cosmological model, known as $\Lambda$ Cold Dark Matter ($\Lambda$CDM),
is recently facing some relevant challenges. In fact, besides the well known theoretical problems \cite{Weinberg:1988cp,Carroll:2000fy,Velten:2014nra,Joyce:2014kja,Padilla:2015aaa}, the $\Lambda$CDM model suffers from some mild observational tensions as well, namely on the measurements of the value of the Hubble constant $H_0$ \cite{Riess:2019cxk,Wong:2019kwg,BOSS:2014hwf,Dawson:2012va,SDSS:2008tqn,Freedman:2019jwv,DiValentino:2020zio} and the present-time amplitude of the matter power spectrum $\sigma_{8}^{0}$ \cite{deJong:2015wca,Hildebrandt:2016iqg,Kuijken:2015vca,DiValentino:2020vvd}. Alternatives beyond $\Lambda$CDM, among which those modifying the long range gravitational interaction known as Modified Gravity (MG) theories~\cite{Joyce:2014kja,Lue:2004rj,Copeland:2006wr,Silvestri:2009hh,Nojiri:2010wj,Tsujikawa:2010zza,Capozziello:2011et,Clifton:2011jh,Bamba:2012cp,Koyama:2015vza,Avelino:2016lpj,Joyce:2016vqv,Nojiri:2017ncd,Ferreira:2019xrr,Kobayashi:2019hrl,Frusciante:2019xia,CANTATA:2021ktz,Bahamonde:2021gfp}, have been deeply scrutinized, with some of them proving able to alleviate the tensions within $1-2\sigma$ \cite{Nunes:2018xbm,Zumalacarregui:2020cjh,Belgacem:2017cqo,Rossi:2019lgt,Peirone:2019aua,Frusciante:2019puu,Heisenberg:2020xak,Barros:2020bgg,Barros:2018efl}, see also \cite{DiValentino:2021izs} for a review.
$f(Q)$ gravity \cite{BeltranJimenez:2017tkd,Harko:2018gxr,Xu:2019sbp,Jarv:2018bgs,Runkla:2018xrv} is a MG theory which recently attracted a lot of attention. It belongs to the Symmetric Teleparallel Gravity \cite{Nester:1998mp,Adak:2008gd,Adak:2018vzk} in which gravity is attributed to the non-metricity and where $f(Q)$ is a general function of the non-metricity scalar, $Q$.
The theory does not show strong coupling problems when considering perturbations around a Friedmann Lema{\^i}tre Robertson Walker (FLRW) background \cite{BeltranJimenez:2019tjy}. As such, the main linear perturbation equations for scalar, tensor and vector modes were derived, showing specific modifications with respect to the $\Lambda$CDM model \cite{BeltranJimenez:2019tjy}.
These motivated further exploration of their impact on the cosmological observables. Constraints at background level have been provided using the expansion rate data from early type galaxies, Supernovae type Ia (SNIa), quasars, gamma ray bursts, Baryonic Acoustic Oscillations (BAO) data, and Cosmic Microwave Background (CMB) distance priors,
for different parametrizations of $f(Q)$ as an explicit function of redshift $z$ \cite{Lazkoz:2019sjl}.
A similar investigation has been performed using a power-law term for $f(Q)$, namely $f(Q) = Q + \beta Q^n$ \cite{Ayuso:2020dcu}, for which cosmological solutions and the evolution of the growth index of matter perturbations have been investigated as well \cite{Khyllep:2021pcu}.
Alternatively, an exponential form for the $f(Q)$ function has been recently proposed, explicitly $f(Q) = Q \, e^{\lambda \frac{Q_0}{Q}}$, for which a statistical preference over $\Lambda$CDM has been found when the combination of cosmic chronometers, SNIa and BAO datasets is considered \cite{Anagnostopoulos:2021ydo}.
Regarding linear perturbations in $f(Q)$ gravity, in particular for the $f(Q)$ model which mimics an exact $\Lambda$CDM expansion history, modifications in the evolution of matter density fields were tested against redshift space distortions (RSD) data, showing the ability for the model to alleviate the $\sigma_8$ tension \cite{Barros:2020bgg}, while measurable effects have been identified to characterize the matter power spectrum, the lensing effect on the CMB angular power spectrum, CMB temperature anisotropies and the Gravitational Waves (GWs) propagation \cite{Frusciante:2021sio}. A joint analysis of CMB, BAO, RSD, SNIa, Weak Lensing (WL) and Galaxy Clustering (GC) data, was able to strongly constrain the model's parameter and it was found that the model can actually challenge the $\Lambda$CDM scenario \cite{Atayde:2021ujc}. Besides these results, $f(Q)$ gravity has been the subject of a variety of studies in many different directions \cite{Dialektopoulos:2019mtr,Jimenez:2019ovq,Bajardi:2020fxh,Flathmann:2020zyj,Khyllep:2021pcu,DAmbrosio:2020nev,Solanki:2021qni,Zhao:2021zab,Bohmer:2021eoo}.
Previous works follow a common procedure, namely they choose the form for $f(Q)$ \textit{a priori} and then determine the corresponding expansion history and linear perturbation dynamics. In this work we will opt for the reverse approach: we instead fix the expansion history and then solve the background equations for the corresponding form of $f(Q)$. The expansion history will be, in practice, selected by the choice of evolution for the equation of state parameter for an effective dark energy component associated with $Q$, $w_Q$. This is known as the \textit{designer} approach, previously applied to $f(R)$ theory \cite{Song:0610532,Pogosian:0709}. One advantage of this approach is that it allows one to identify the impact of a background evolution, which differs from the $\Lambda$CDM, on some physical quantities of relevance at linear perturbation level and to evaluate whether the parameters characterizing the effective equation of state are degenerate with the one affecting the perturbations only. Using this approach, we then identify the general features characterizing linear perturbations which can later be employed to constrain the model. In particular, we identify the \textit{effective gravitational coupling} as the source of the modifications of the gravitational interaction at large scales. Therefore we consider its signatures on related physical quantities, such as the growth of matter perturbations and the cross-correlation power spectrum of the CMB anisotropies
with galaxy distribution.
This work is organized as follows. In Section~\ref{Sec:fQbasics} we briefly review the basis of the $f(Q)$ gravity formalism. The \textit{designer} approach is then detailed in Section~\ref{Sec:designer}, which includes a study of the initial conditions and the choices for the effective dark energy equation of state. Then, in Section~\ref{Sec:effectivecoupling} we study the evolution of the effective gravitational coupling, which is then used in Section~\ref{Sec:lineargrowth} to investigate the linear growth of structures through the evolution of the growth factor and the product of the growth rate and root mean square of matter fluctuations. We also provide theoretical predictions for the sign of the Integrated Sachs-Wolfe (ISW)-galaxy cross correlation in Section~\ref{Sec:ISWgal}. Finally, we conclude in Section~\ref{Sec:conclusion}.
\section{\boldmath $f(Q)$ gravity}\label{Sec:fQbasics}
General Relativity is a metric theory of gravity and as such the connection is metric-compatible and symmetric. However, two alternative approaches can be considered to characterize the space-time: non-metricity and torsion, giving up the first and second assumptions, respectively. It has been shown~\cite{BeltranJimenez:2019tjy} that in flat space-time the Einstein-Hilbert action,
the teleparallel ($\int{d^4x\sqrt{-g}\,T}$ \cite{Aldrovandi:2013wha}) and symmetric teleparallel ($\int{d^4x\sqrt{-g}\,Q}$ \cite{BeltranJimenez:2017tkd}) actions are three different representations of the same underlying theory. Nevertheless, MG theories based on non-linear extensions of both the non-metricity scalar, $Q$, namely $f(Q)$ gravity, as well as torsion ($T$), namely $f(T)$ gravity \cite{Bahamonde:2021gfp}, can be considered. These, unlike the previous case, are not equivalent. In this work we will focus on the $f(Q)$ formulation given the absence of strong coupling problems for the FLRW background~\cite{BeltranJimenez:2019tme} compared to $f(T)$ gravity and its increasing interest in the cosmological framework \cite{Barros:2020bgg,Anagnostopoulos:2021ydo,Atayde:2021ujc}, as discussed in the Introduction. Interestingly, $f(Q)$ gravity introduces at least two additional scalar propagating degrees of freedom which disappear around maximally symmetric backgrounds~\cite{BeltranJimenez:2019tme}.
Let us introduce the action of $f(Q)$ gravity which reads\footnote{Comparing our action (\ref{eq:action}) with the one in Ref.~\cite{BeltranJimenez:2019tme}, we have performed the following replacement $f(Q)\rightarrow \frac{1}{\kappa^2}\left(Q+f(Q)\right)$, because this form better fits our purpose.} ~\cite{BeltranJimenez:2019tme}
\begin{equation} \label{eq:action}
S=\int d^4x\sqrt{-g}\left\{-\frac{1}{2\kappa^2}\left[Q+f(Q)\right]+L_m(g_{\mu\nu},\chi_i)\right\},
\end{equation}
where $\kappa^2 = 8 \pi G_N$, $G_N$ is the Newtonian constant, $g$ is the determinant of the metric $g_{\mu\nu}$ and $Q$ is the non-metricity scalar defined as
\begin{equation}
Q=-Q_{\alpha\mu\nu}P^{\alpha\mu\nu}\,,
\end{equation}
where the non-metricity tensor is:
\begin{equation}
Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}\,,
\end{equation}
and
\begin{equation}
P^{\alpha}_{\phantom{\alpha}\mu\nu}=-L^{\alpha}_{\phantom{\alpha}\mu\nu}/2+\left(Q^\alpha-\tilde{Q}^\alpha\right)g_{\mu\nu}/4-\delta^\alpha_{(\mu}Q_{\nu)}/4\,,
\end{equation}
with $Q_\alpha=g^{\mu\nu}Q_{\alpha\mu\nu}$, $\tilde{Q}_\alpha=g^{\mu\nu}Q_{\mu\alpha\nu}$ and $L^\alpha_{\phantom{\alpha}\mu\nu}=(Q^\alpha_{\phantom{\alpha}\mu\nu}-Q_{(\mu\nu)}^{\phantom{(\mu\nu)} \alpha})/2$. Finally, $f(Q)$ is a general function of the non-metricity scalar and as usual $L_m$ is the matter Lagrangian for all matter fields, $\chi_{i}$.
In flat space the action (\ref{eq:action}) is equivalent to GR for $f=0$~\cite{BeltranJimenez:2019tjy} since the symmetric teleparallel action is recovered. Thus any modification to GR can be seen for $f\neq0$. From the above action it is also possible to compute the equation for tensor perturbations which is characterized by a shift in the friction term due to the time derivative of an effective Planck mass, given by $M^2_{\rm eff}=1+f_Q$ where $f_Q\equiv df/dQ$~\cite{Jimenez:2019ovq}. From this follows that in order to guarantee the absence of a ghost instability one has to impose $1+f_Q>0$. We will assume this condition in our investigation.
\section{Designer approach} \label{Sec:designer}
Let us assume a flat, homogeneous and isotropic Universe described on the background by the FLRW metric:
\begin{equation}
ds^2=-dt^2+a(t)^2\delta_{ij}dx^idx^j\,,
\end{equation}
where $a(t)$ is the scale factor. On this background the non-metricity scalar simply reduces to $Q = 6H^2$ \cite{BeltranJimenez:2017tkd,Jimenez:2019ovq} where $H\equiv\dot{a}/a$ is the Hubble parameter and the dot stands for a derivative with respect to cosmic time, $t$.
The equations of motion on this background are:
\begin{eqnarray}
&&H^2 + 2 H^2 f_Q - \frac{1}{6} f = \frac{\kappa^2}{3} \rho_{\rm m} , \label{eq:FriedEq} \\
&&(12H^2f_{QQ}+f_Q+1)\dot{H} = -\frac{\kappa^2}{2}(\rho_{\rm m}+p_{\rm m})\,,
\end{eqnarray}
with $\rho_{\rm m}=\Sigma_i\rho_i$ and $p_{\rm m}=\Sigma_ip_i$ being respectively the sum of the energy density, $\rho_i$, and pressure, $p_i$, of the matter components, which satisfy the continuity equation for perfect fluids:
\begin{equation}
\dot{\rho}_i+3H(\rho_i+p_i)=0\,.
\end{equation}
Hereafter we will consider the relation $p_i=w_i\rho_i$, with $w_{c,b}=0$ for baryons (b) and cold dark matter (c), and $w_r=1/3$ for radiation (r). For the times of interest ($a \in [10^{-2},1]$) the matter components behave as non-relativistic matter fluids.
In order to solve the above modified Friedmann equation we have to fix either the functional form of $f(Q)$ or that of $H$. A common practice is to select the form of $f(Q)$ and then solve the system to find the corresponding expansion history, $H$. In this work we will follow the opposite approach: we will fix $H$ and solve the system to find the corresponding functional form for $f(Q)$. Hence we will apply the so-called \textit{designer} approach \cite{Song:0610532,Pogosian:0709}.
We define the dimensionless variables
\begin{equation}
E \equiv \frac{H^2}{H_0^2}, \ \ \ \ y \equiv \frac{f(Q)}{H_0^2},
\end{equation}
where $H_0$ is the present day value of the Hubble parameter, and we fix the expansion history such that
\begin{equation}
E = E_r + E_c + E_b + E_{Q}\, .
\end{equation}
In the latter, $E_i = \rho_i / \rho_{crit}$ with $\rho_{crit} \equiv 3H_0^2 / \kappa^2 $ and the energy density parameter for the effective dark energy component associated to $Q$, $E_Q$, is given by
\begin{equation}\label{eq:EnDenQ}
E_{Q} = ( 1 - \Omega_{\rm m,0} ) \ \mbox{exp} \left[ - 3 \ln a + 3 \int_{a}^{1} w_{Q} (\tilde{a}) d \ln \tilde{a} \right] ,
\end{equation}
where $\Omega_{\rm m,0}$ is the present time value of the density parameter $\Omega_{\rm m}\equiv \Sigma_i \rho_i/(3H_0^2 / \kappa^2) $ and $w_Q (a)$ is the equation of state parameter for an effective dark energy component.
Now we can re-work the Friedmann equation (\ref{eq:FriedEq}) as a first-order differential equation
\begin{eqnarray}\label{eq:diffeq}
y' - \frac{Q'}{12 E H_0^2} y = - \frac{Q'}{2 E H_0^2} E_{Q} \,,
\end{eqnarray}
where the prime denotes a derivative with respect to $\ln a$. In order to solve this equation we set initial conditions (ICs) at early time, $a_i$, when we consider the contribution of the effective dark energy component negligible compared to the matter one. The resulting equation becomes homogeneous and is satisfied by the ansatz $y \propto \exp(p \ln a)$, where $p$ is obtained by solving the homogeneous equation yielding
\begin{equation}
p = - \frac{3+4r_i}{2 (1+r_i)} ,
\end{equation}
with $r_i = a_{\rm eq}/a_i$ and $a_{\rm eq}$ being the value of the scale factor at matter-radiation equality.
Then we look for the particular solution, $y_p$, when $E_Q\neq0$, which is obtained by substituting $y_p = A_p E_Q$ in eq.~(\ref{eq:diffeq}). The amplitude of the particular solution is then
\begin{equation}
A_p = \frac{6 p}{3(1+w_Q) + p}\,,
\end{equation}
leading to the following initial condition for $y$:
\begin{equation} \label{eq:initcond}
y_i = A a_i^p + \frac{6 p}{3(1+w_Q) + p} E_Q \ ,
\end{equation}
where $A$ is an arbitrary constant. We note that the amplitude of the ICs is not an appropriate parameter to define the family of solutions of eq.~(\ref{eq:diffeq}). Indeed a change in the initial time, $a_i$, would correspond to a rescaling in $A$ in order to obtain the same solution. This would make it difficult to obtain any theoretical prediction based on $A$, given that to different $a_i$ would correspond different values of $A$ with the same behaviour. As such, in order to avoid this contingency, we define another quantity which will characterize the family of solutions:
\begin{equation}
\alpha=\frac{A \,a_i^p}{\sqrt{6E(\ln a_i)}}\,,
\end{equation}
which follows from the fact that $y/\sqrt{E}$ is constant at early time (see eq.~(\ref{eq:diffeq})). In this case, $\alpha$ does not have to change when $a_i$ is altered given that it can be compensated by a rescaled value of $A$.
With this set up, we are left with few more parameters to consider: the constant $\alpha$ and the parameters entering in the equation of state, $w_Q (a)$, which basically fix the background expansion. In this work we will explore three options for dealing with $w_Q$:
\begin{enumerate}
\item \textbf{\boldmath Constant $w_Q$}: $w_Q = -1$
The simplest case to be considered is that of having $w_Q = -1$ which would reproduce an exact $\Lambda$CDM background expansion history. In this situation, the energy density for $Q$ $ \left( E_Q = 1 - \Omega_{ \rm m,0} \right)$ becomes a constant and the differential eq.~(\ref{eq:diffeq}) can be analytically solved. We find:
%
\begin{equation} \label{eq:ansol}
y = \frac{y_i - 6 E_Q}{\sqrt{E(\ln a_i)}} \sqrt{E} + 6 E_Q \, , \end{equation}
%
or in terms of $f(Q)$:
\begin{equation} \label{eq:fQan}
f(Q) = \alpha H_0\sqrt{Q} + 6H_0^2 E_Q \, ,
\end{equation}
%
which is consistent with what was found in Ref. \cite{Jimenez:2019ovq}, upon the following identification: $M=\alpha H_0$ and $C = 6H_0^2 E_Q$. Let us note that combined with the choice $\alpha=0$, this case corresponds exactly to a $\Lambda$CDM behaviour at perturbation level as well.
\item \textbf{\boldmath Constant $w_Q$}: $w_Q=w_0$
The second case we consider has a constant $w_Q=w_0$ with $w_0\neq -1$. In this situation, $E_Q$ becomes
%
\begin{equation}
E_Q (a) = (1 - \Omega_{\rm m,0}) \, a^{ -3 \left( 1 + w_0 \right) },
\end{equation}
which means it is no longer a constant as long as $w_0 \neq -1$. With this choice of the background it is not possible to find an analytical solution for $f(Q)$. We will use this case to understand the impact of changing the equation of state parameter on some observational features. For the purpose of visualizing these features, in the following we will select four values for $w_0$, namely $\{-1.15, -1.05, -0.95, -0.85\}$.
%
\item \textbf{\boldmath Time-dependent $w_Q$}: $w_Q(a)$
The third case we consider assumes a fast varying equation of state \cite{DeFelice:1203}. In this case, $w_Q$ evolves as
%
\begin{equation}
w_Q (a) = w_p + (w_0 - w_p) \, \frac{a \left[ 1 - \left( a/a_t \right)^{1/\tau} \right]}{1-a_t^{-1/\tau}} ,
\end{equation}
%
where $w_p$ and $w_0$ are the values of $w_Q$ in the asymptotic past and at present time, respectively, and $a_t$ and $\tau$ are the time and the width of the transition. Additionally, by definition we have $a_t > 0$ and $\tau > 0$.
In this case, the energy density of the effective dark energy component becomes:
%
\begin{equation}
E_Q = \left( 1 - \Omega_{\rm m,0} \right) a^{ -3 \left( 1 + w_p \right)} e^{g(a)} ,
\end{equation}
%
with
%
\begin{eqnarray}
g(a) &=& \frac{3(w_0 - w_p)}{(1 - {a_t}^{-1/\tau})(\tau + 1)} \left[ \left( 1 - {a_t}^{-1/\tau} \right) \tau \right.\nonumber\\
&+&\left.1+ a \left[ \left( \left( a / a_t \right)^{1/\tau} - 1 \right] \tau - 1 \right) \right] .
\end{eqnarray}
%
As such, we will have four free parameters: $\{ w_0, w_p, \\a_t, \tau \}$. In order to reduce the possible combinations, and because we are already exploring cases where $w_Q=w_0$, we shall fix $w_0 = -1$. Additionally, we want to keep the transition time always prior to present day, meaning keeping $a_t < 1$. We will then use the following sets of values for the remaining parameters: $\{ w_p, a_t, \tau \} ={\{ -1.15, 0.5, 10 \}}$ and $\{ w_p, a_t, \tau \} ={\{ -0.95, 0.5, 10 \}}$, we will refer to them as M1 and M2 respectively. We do not consider cases in which $a_t$ and $\tau$ vary because we have verified they do not have any sizable impact on the phenomenology we are interested in. Let us note that with these choices the associated $E_Q(\ln a_i)$ remains subdominant. \end{enumerate}
In the present analysis we will use the following cosmological parameters \cite{Planck:2018vyg}: $H_0= 67.32$ km s$^{-1}$Mpc$^{-1}$, ${\Omega_{c,0}= 0.265}$, $\Omega_{b,0}= 0.049 $, $\Omega_{r,0}= 3.769 \times 10^{-5}$, and for the initial time we select $a_i\sim 10^{-2}$.
Let us also stress that some of the cases we consider have $w_Q<-1$. While this behaviour of the equation of state usually generates ghost instabilities, for the $f(Q)$ theory analysed in this work, the ghost instability is avoided by choosing $f_Q>0$ at any time. This condition inevitably reflects on the parameter space of the free parameters defining the $w_Q$ equation of state. Therefore, in performing our analysis we have verified that the set of parameters we chose satisfy the no-ghost requirement.
\section{The effective gravitational coupling} \label{Sec:effectivecoupling}
The next step will be to investigate the evolution of the scalar perturbations at linear scales. Firstly, we will review the main equations which involve the use of the linear perturbation theory, then we will show how the gravitational interaction is modified with respect to the standard $\Lambda$CDM scenario and its dependence on the background assumptions.
Let us consider the perturbed line element in Newtonian gauge:
\begin{equation}
ds^2=-(1+2\Psi)dt^2+a^2(1-2\Phi)\delta_{ij}dx^idx^j\,,
\end{equation}
where $\Phi(t,x_i)$ and $\Psi(t,x_i)$ are the two gravitational potentials.
For MG theories a model-independent framework is usually adopted to relate the gravitational potentials to the linear matter density perturbations $\delta \rho_{\rm m}$ ~\cite{Amendola:2007rr,PhysRevD.81.083534,Silvestri:2013ne,2010PhRvD..81j4023P,Amendola:2019laa}. This framework encodes the deviations from GR into two phenomenological functions, namely the {\it effective gravitational coupling}, $\mu$, and the {\it light deflection parameter}, $\Sigma$, which enter in the Poisson and lensing equations, respectively. In Fourier space the latter read:
\begin{eqnarray}
\label{eq:muSigma}
&&-\frac{k^2}{a^2} \Psi = 4 \pi G_N \,\mu(a, k) \rho_\mathrm{m} \delta_{\rm m}\,, \\
&&-\frac{k^2}{a^2} (\Psi+\Phi) = 8\pi G_N\Sigma(a,k) \rho_\mathrm{m}\delta_{\rm m}\,, \label{eq:lenseq}
\end{eqnarray}
where $\delta_{\rm m}=\delta \rho_{\rm m}/\rho_{\rm m}$ is the density contrast and $k$ is the wavenumber. Therefore, $\mu$ encodes the deviations of the gravitational interaction on the clustering of matter with respect to $\Lambda$CDM, while $\Sigma$ measures the deviation in the lensing gravitational potential, $\phi_{len}=(\Phi+\Psi)/2$. The $\Lambda$CDM model is recovered when $\mu=\Sigma=1$.
In order to map the $f(Q)$ gravity within the above formalism one can employ the quasi-static approximation, which is a valid assumption for perturbations deep inside the Hubble radius. Following this, one can find that the gravitational potentials are equal as in GR (i.e. $\Phi=\Psi$) and that the two above equations match~\cite{Jimenez:2019ovq}:
\begin{equation}
-k^2\Psi= \frac{4 \pi G_N}{1+f_Q}a^2\rho_{\rm m} \delta_{\rm m}\,.\label{eq:Poisson}
\end{equation}
The effective gravitational coupling is then defined as:
\begin{equation}\label{eq:mu}
\mu(a)=\frac{1}{1+f_Q}\,.
\end{equation}
When $f_Q\rightarrow 0$ the $\Lambda$CDM behaviour is recovered, i.e. $\mu=1$.
From eq.~(\ref{eq:mu}) immediately follows that since $M^2_{\rm eff}=1+f_Q>0$ due to stability requirements, then $\mu$ is always positive. Additionally, the cases $\mu<1$ and $\mu>1$ correspond respectively to a weaker and stronger gravitational interaction compared to $\Lambda$CDM. Furthermore, the light deflection parameter is then equal to the effective gravitational coupling and we notice that the scale dependence, i.e., the dependence on $k$, disappears.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{muplts_wQ-1.pdf}
\caption{Evolution of $\mu - 1$ as a function of redshift $z$ for models with an exact $\Lambda$CDM background and different values of $\alpha$. \label{fig:mupltsw-1}}
\end{figure}
From eq.~(\ref{eq:mu}) it becomes clear that depending on the functional form of $f(Q)$, the evolution of the linear perturbations will change accordingly. In our investigation the form of $f(Q)$ will be determined by the use of the designer approach. Given this, in the following we will study the evolution of $\mu - 1$, i.e. of the difference between the effective gravitational couplings of $f(Q)$ and $\Lambda$CDM, for the different background expansion histories discussed in Section~\ref{Sec:designer}. This investigation is of particular interest because a modified effective gravitational coupling will impact the growth and distribution of structures in time and space. Additionally, since $\mu = \Sigma$, we are also studying the modifications of the Weyl potential, $\phi_{len}$, by the light deflection parameter, which will in turn change the lensing of light and modify the ISW effect. The latter is indeed sourced by the time derivative of the Weyl potential. As such, in $f(Q)$ all these effects are encoded in $\mu$.
The case with $w_Q=-1$ has already been presented and discussed in Ref. \cite{Frusciante:2021sio}.
Briefly, we recall the main features. From Figure~\ref{fig:mupltsw-1}, we see that sizable deviations from GR appear at $z<10$ and then grow towards present time. Values of $\alpha>0$ result in a weaker gravity compared to GR ($\mu - 1<0$) while the opposite holds for $\alpha<0$. Additionally, larger values of $|\alpha|$ lead to larger deviations from the $\Lambda$CDM behaviour, however for the same $|\alpha|$ but opposite sign, we notice that the modification corresponding to the $\alpha<0$ gives rise to a larger deviation compared to its positive counterpart.
Let us now consider the case with $w_Q=w_0$. In Figure~\ref{fig:mupltsVwvwQ} we show the results when $w_0=-0.85$ (top panel) and $w_0=-1.15$ (bottom panel) for some given values of $\alpha$. When $w_0=-0.85$, regardless of the sign of $\alpha$, the $\mu-1$ behaviour goes toward negative values at small $z$, i.e. toward a weaker gravitational interaction. In order to have a stronger gravitational interaction at present time the value of $|\alpha|$ has to be very large and of negative sign ($\alpha=-1.5$). The opposite holds if we consider the phantom values of $w_0$ (bottom panel). In this case, the gravitational interaction is stronger toward present time and the weaker gravity is realized when the value of $|\alpha|$ is very large and of positive sign ($\alpha=1.5$). According to these features, for a given value of $\alpha$ the direction of the modifications (toward weaker/stronger gravity) is dictated by the value of $w_0$, while $\alpha$ mostly impacts on the amplitude of the deviation with respect to $\mu=1$. This is more clear in Figure~\ref{fig:mupltsVAvwQ}, where we fix $\alpha$ and vary $w_0$.
\begin{figure}[t]
\centering
\includegraphics[scale=0.56]{muplts_VW_vwQ.pdf}
\caption{Evolution of $\mu - 1$ as a function of redshift $z$ for different $\alpha$ values with fixed $w_0 = -0.85$ (top panel) and $w_0=-1.15$ (bottom panel). }
\label{fig:mupltsVwvwQ}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{muplts_VA_vwQ.pdf}
\caption{Evolution of $\mu - 1$ as a function of redshift $z$ for different values of $w_0$ with fixed $\alpha=0.5$ (top panel) and $\alpha = -0.5$ (bottom panel).}
\label{fig:mupltsVAvwQ}
\end{figure}
Finally, we discuss the behaviour of $\mu -1$ for the time-dependent $w_Q$ model, for which we show the result in Fig. \ref{fig:muplts1DwQ}. As noted for the case with an exact $\Lambda$CDM background, most of the models with a negative $\alpha$ parameter have $\mu>1$ whereas the opposite holds for positive $\alpha$. However, while in the former case there is a net distinction, when $w_Q$ has fast transitions the models corresponding to small values of $|\alpha|$ can have stronger or weaker gravity depending on the value of $w_p$ at early time. In particular, if $w_p>-1$ such as in the M2 case (bottom panel), then $\mu<1$. Alternatively, if $w_p<-1$ we then have $\mu>1$ as shown in the top panel. Finally, if we look at the behaviours of $\mu-1$ in the future, we notice that all models that have $w_p < -1$ (top panel) will have $\mu - 1$ going toward negative values, while when $w_p > -1$ it is the opposite. This is due to the fact that after present time the M1 model undergoes another matter dominated era, while M2 stays in the dark fluid dominated one.
In summary, the behaviour of the effective gravitational coupling depends strongly on the assumed background expansion history. However, let us notice that while the background expansion history only depends on the parameters defining $w_Q$, the effective gravitational coupling depends on both $w_Q$ and $\alpha$, showing a degeneracy between the parameters in some cases. In particular, we find that a preference for a weaker/stronger gravitational interaction can be secured by a proper choice of either the present time value of $w_Q$ or its asymptotic past one. In detail, phantom behaviours at late time and early time modifications with $w_p>-1$ prefer a stronger gravity, the weaker gravitational interaction is instead present in the opposite situations. The role of $\alpha$ is to define the amplitude of the deviation with respect to $\Lambda$CDM, its negative and positive values contribute to enhance or suppress the effective gravitational coupling with respect to the standard scenario. Extremely large values of $|\alpha|$ can even reverse the strength of the gravitational interaction. These peculiar patterns have some immediate consequences on the clustering of matter and on how light travels over cosmological distances. We expect indeed that a stronger gravitational interaction will lead to an enhanced matter and lensing potential auto-correlation power spectra. As an example, we refer the reader to Ref. \cite{Frusciante:2021sio}, where an investigation of signatures of $f(Q)$ gravity, when the background is assumed to be $\Lambda$CDM, has been performed by looking at the matter power spectrum, the lensing effect on the CMB angular power spectrum, CMB temperature anisotropies and GWs luminosity distance compared to the standard electromagnetic one.
We note that the equation of state with time-dependent $w_Q$ can actually cross $w_Q=-1$. When the cross happens (for values of the parameters for which the condition $f_Q>0$ is satisfied) we found that the general behaviour of $\mu(z)-1$ as shown in Fig.~\ref{fig:muplts1DwQ} is still the same but with a difference in the amplitude. The latter, depending on the value of $w_0$, can be larger or smaller than the case with $w_0=-1$ we have considered, following our findings illustrated in Fig.~\ref{fig:mupltsVAvwQ}. We did not show this case because a change in $w_0$ for the time-dependent equation of state follows the results for the constant case, making some of the analysis redundant. We will not consider this case in the following sections, but all the other phenomenological quantities we compute will be modified accordingly, following the same pattern.
In order to get a glimpse in the phenomenology of the designer $f(Q)$ gravity related to the effective gravitational coupling, we dedicate the following section to the investigation of the linear growth of structures. Furthermore, the latter combined with the late-time ISW effect largely impact the sign of the cross-correlation function between the galaxy density and the temperature of the CMB. As such, its study can be informative regarding the parameter space to explore in order to exclude a negative ISW-galaxy cross-correlation which is observationally disfavored.
Before concluding this section, we notice that an accurate combination of data may help breaking the existing degeneracy between $w_Q$ and $\alpha$. If one is able to constrain with high accuracy the equation of state, using for example SNIa and BAO data, then GC, RSD as well as CMB data can be used to probe the effects of $\mu$ on some cosmological observables at perturbation level and as such set accurate bounds on $\alpha$ as well. An independent estimation of $\alpha$ may also come from measurements of standard sirens at future GWs detectors \cite{Frusciante:2021sio}.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{muplts_M2_DwQ.pdf}
\caption{Evolution of $\mu - 1$ as a function of redshift $z$ for the time-dependent $w_Q$ (M1 top panel and M2 bottom panel).}
\label{fig:muplts1DwQ}
\end{figure}
\section{Linear growth of structures} \label{Sec:lineargrowth}
In this section we will investigate how the effective gravitational coupling impacts the linear growth of structures. In particular we will study the product of the growth rate and the root mean square of matter fluctuations, $\tilde{f}\sigma_8$, as it has a direct connection with data and hence offers a possibility to explore the parameter space given by $\{w_Q,\alpha\}$.
The growth of linear matter perturbations is modified in the $f(Q)$ theory due to a modified Poisson eq.~(\ref{eq:Poisson}) as follows:
\begin{equation} \label{eq:lingrtheq}
\delta_{\rm m}'' + \left( 2 + \frac{H'}{H} \right) \delta_{\rm m}' - 4\pi G_N \mu (a) \, \rho_{\rm m} \delta_{\rm m} = 0 .
\end{equation}
We now solve the above equation with ICs set as follows: $\delta_{\rm m} (a_i) = a_i$ and $\delta_{\rm m}' = a_i$. Then, we compute the linear growth factor $D(a) = \delta_{\rm m }(a) / \delta_{\rm m} (a=1)$ for the three scenarios analyzed in this work.
The linear growth factor for the case with an exact $\Lambda$CDM background expansion history has been investigated in Refs. \cite{Barros:2020bgg,Frusciante:2021sio}, where it has been found that models with $\alpha > 0$ and $\mu - 1 <0$ show an enhanced growth factor with respect to the $\Lambda$CDM scenario, whereas for $\alpha < 0$ and $\mu - 1 >0$, $D(a)$ is suppressed.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{GFplts_VW_vwQ.pdf}
\caption{Evolution of the linear growth factor $D$ normalized to unity today and divided by the scale factor $a$ as a function of redshift $z$ for models with different $\alpha$ and fixed $w_0=-0.85$ (top panel) and $w_0=-1.15$ (bottom panel). The $\Lambda$CDM model (solid, black line) is also included for comparison. }
\label{fig:DpltsVWvwQ}
\end{figure}
When changing the background expansion to have $w_Q=w_0$, the growth factor is shifted in amplitude as shown in Figure~\ref{fig:DpltsVWvwQ}. Specifically, when $w_0>-1$ (top panel) the growth factor is enhanced (black dot-dashed line) with respect to $\Lambda$CDM. Then, depending on the values of $\alpha$, $D$ can be enhanced or suppressed compared to $\Lambda$CDM. Large negative values of $\alpha$ suppress the linear growth factor. On the contrary, when the equation of state has a phantom behaviour (bottom panel), the growth factor is mostly suppressed, the large positive $\alpha$ being the exception. That is due to the fact that in the former case the models mostly show a weaker gravity compared to the case where $w_0<-1$. The two exceptions are indeed the models with the largest negative and positive values of $\alpha$ respectively. This is consistent with the $\mu$ behaviour analysed in the previous section.
For the time-dependent $w_Q$ case, the behaviour of the growth factor is shown in Figure~\ref{fig:DpltsDwQ} for M1 (top panel) and M2 (bottom panel) and different values of $\alpha$. We can notice that a different evolution in the early time can change the amplitude of the growth factor. Indeed having $w_Q=-1$ would make a net separation between $\alpha$ positive and negative which would correspond respectively to an enhancement/suppression of the growth with respect to $\Lambda$CDM, while here a dynamical evolution at early time breaks this pattern by changing the amplitude of the growth factor. In particular, a phantom $w_p$ (M1) damps the amplitude of the growth factor and now most of the models are suppressed with respect to $\Lambda$CDM, while values of $w_p>-1$ (M2) enhance $D$.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{Dplts_M2_DwQ.pdf}
\caption{Evolution of the linear growth factor $D$ normalized to unity today and divided by the scale factor $a$ as a function of redshift $z$ for the time-dependent equation of state $w_Q$ : M1 (top panel) and M2 (bottom panel). The $\Lambda$CDM (solid, black line) is also included for comparison.}
\label{fig:DpltsDwQ}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{fsigma8plts_wQ-1.pdf}
\caption{Evolution of $\tilde{f}\sigma_8$ as a function of redshift $z$ for models with a $\Lambda$CDM background expansion history. We have used $\sigma_8^0=0.82$. The $\Lambda$CDM model (solid, black line) is also included for comparison. Data set is from Ref. \cite{Sagredo:2018ahx}.}
\label{fig:fsigma8pltswQ0}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{fsigma8plts_VW_vwQ.pdf}
\caption{Evolution of $\tilde{f}\sigma_8$ as a function of redshift $z$ for models with $w_0 = -0.85$ (top panel) and $w_0 = -1.15$ (bottom panel). We have used $\sigma_8^0=0.82$. The $\Lambda$CDM model (solid, black line) is also included for comparison. Data set is from Ref. \cite{Sagredo:2018ahx}.} \label{fig:fsigma8pltsVWvwQ}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{fsigma8plts_M2_DwQ.pdf}
\caption{Evolution of $\tilde{f}\sigma_8$ as a function of redshift $z$ for models with a time-dependent equation of state: M1 (top panel) and M2 (bottom panel). We have used $\sigma_8^0=0.82$. The $\Lambda$CDM model (solid, black line) is also included for comparison. Data set is from Ref. \cite{Sagredo:2018ahx}.}
\label{fig:fsigma8pltsDwQ}
\end{figure}
We will now move to the analysis of a specific physical quantity that can be constructed from the linear growth factor, i.e. $\tilde{f}\sigma_8$, which is measured by redshift surveys and defined as the product of the growth rate,
$\tilde{f}\equiv \frac{d \ln \delta_{\rm m}}{d \ln a}$, and the root mean square of matter fluctuations, $\sigma_8$:
\begin{equation}
\tilde{f}\sigma_8=\sigma_8^0\frac{\delta^\prime(\ln a)}{\delta(\ln a=0)}\,,
\end{equation}
where $\sigma_8^0=\sigma_8(\ln a=0)$.
In Figures~\ref{fig:fsigma8pltswQ0}, \ref{fig:fsigma8pltsVWvwQ} and \ref{fig:fsigma8pltsDwQ} we plot the evolution of $\tilde{f}\sigma_8$ with $z$ for $0<z<2$ for the different cases under consideration. We also include RSD data from \cite{Sagredo:2018ahx}.
As expected from the discussion in Section~\ref{Sec:effectivecoupling}, the models with $\alpha<0$ prefer higher values for $\tilde{f}\sigma_8$, while data prefers a suppressed growth rate. It is also possible to adjust the amplitude by playing with the values of the effective equation of state. In this case, small and negative values of $\alpha$ can also show a lower amplitude for $\tilde{f}\sigma_8$, even though not comparable to the cases with $\alpha>0$. We can conclude that the most negative values of $\alpha$, in particular those associated either with a phantom behaviour for the effective equation of state or its time variation at early time, are unlikely to be preferred by RSD data.
In order to examine which values of $\alpha$ lead to a favored $f(Q)$ model over the $\Lambda$CDM one, it is necessary to perform a Markov-chain-Monte-Carlo simulation by varying $\sigma_8 (\ln a = 0)$ besides other cosmological parameters instead of fixing them. We leave this investigation for a future work.
\section{The ISW-galaxy cross-correlation} \label{Sec:ISWgal}
In this section we explore the sign of the cross-correlation between the ISW signal in CMB and galaxy distributions for the $f(Q)$ theory. $\Lambda$CDM and dark energy scenarios within GR have a positive ISW-galaxy cross-correlation at any redshift, while for MG theories there can be cases for which the ISW-galaxy cross-correlation is negative. These cases will be ruled out by data \cite{Renk:2017rzu}. Therefore, this observable can be used as a tool to test viable models and distinguish between different cosmological scenarios \cite{Song:2006ej,Barreira:2012kk,Renk:2017rzu,Frusciante:2019puu,Giacomello:2018jfi,Hang:2021kfx,Kable:2021yws}. In a MG framework the modifications with respect to the standard scenario come from two sources: the growth of structures and the late time ISW-effect. As discussed in the previous section, the growth of structures is strongly affected by the effective gravitational coupling, while the late-time ISW effect is sourced by the time derivative of the lensing potential and as such by the time derivative of the light deflection parameter. In $f(Q)$ gravity these two parameters are the same. Therefore, given the results of the previous sections we expect to gain significant information on the $\{ w_Q , \alpha \}$ parameter space by studying the sign of ISW-galaxy cross-correlation.
The ISW contribution to the CMB anisotropies can be written as
\begin{equation}
\frac{\Delta T_{\rm ISW} (\mathbf{\hat{n}})}{T} = \int \frac{d \left( \Psi + \Phi \right)}{dz} dz \, .
\end{equation}
On the other hand, the fluctuations in the angular distribution of galaxies can be given by
\begin{equation}
\frac{\Delta N_{\rm g} (\mathbf{\hat{n}}) }{N} = \int b_{\rm g} (k,z) \delta_{\rm m} (\mathbf{\hat{n}},z) \mathcal{W} (z) \, ,
\end{equation}
where $b_{\rm g}$ is the galaxy bias and $\mathcal{W}(z)$ is the selection function. As such, the cross-correlation power spectrum of the galaxy fluctuations and CMB anisotropies, written in the harmonic space, is
\begin{equation}
\Bigg\langle \frac{\Delta T_{\rm ISW} (\mathbf{\hat{n}})}{T} \frac{\Delta N_{\rm g} (\mathbf{\hat{n}'})}{N} \Bigg\rangle = \sum_{\ell} \frac{2\ell + 1}{4\pi} \, C_{\ell} \, \mathcal{P}_{\ell} \left(\cos \theta \right) ,
\end{equation}
with $\mathcal{P}_{\ell}$ being the Legendre polynomial and $\theta$ the angle between the unit vectors $\mathbf{\hat{n}}$ and $\mathbf{\hat{n}'}$. Finally, $C_{\ell}$ is the amplitude of the ISW-galaxy cross-correlation which, upon the employment of the Limber approximation, can be written as~\cite{Kimura:2011td}
\begin{eqnarray}
&& C_{\ell} = \frac{3 H_0^2 \, \Omega_{\rm m,0}}{\left( \ell + 1/2 \right)^2} \int dz \, H(z) \, \mathcal{W}(z) \, \frac{D}{D(z=0)} \nonumber \\
&& \times \frac{d U_{\rm ISW}}{dz} \, b_{\rm g} (k,z) \, P(k) \biggr\vert_{k=(\ell +1/2)/\chi} \, ,
\end{eqnarray}
where $P(k)$ is the present time matter power spectrum, $\chi$ is the comoving distance and
\begin{equation}
U_{\rm ISW} (z) = \frac{ \Sigma \, D(z)}{D(z=0)a} \, .
\label{eq:UISW}
\end{equation}
The latter has been defined considering a scale independent growth which is precisely the case we are considering. This definition follows the lensing equation (\ref{eq:lenseq}) which reads
\begin{equation}
-k^2 \left( \Psi + \Phi \right) = 3 H_0^2 \, \Omega_{\rm m,0} \, U_{\rm ISW} (z) \, \delta_{\rm m} (z=0) \, .
\end{equation}
The amplitude of the ISW-galaxy cross-correlation is then proportional to the derivative
\begin{equation}
\frac{dU_{\rm ISW}}{dz} = \frac{\Sigma D \mathcal{F}}{D(z=0)}\,,
\label{Eq:dUISW}
\end{equation}
where \cite{Nakamura:1811}
\begin{equation} \label{eq:fancyF}
\mathcal{F} \equiv 1 - \frac{D'}{D} - \frac{\Sigma'}{\Sigma} \, .
\end{equation}
It follows that a necessary but not sufficient condition to have a negative ISW-galaxy cross-correlation for modified gravity models without a scale dependence in the effective couplings is \cite{Nakamura:1811}:
\begin{equation} \label{Eq:F}
\mathcal{F} < 0 \, .
\end{equation}
Before presenting some results, let us discuss about the regime of applicability of the approach we use. First of all, we would like to stress that the expression in Eq.\ (\ref{Eq:dUISW}) does not rely on the quasi-static approximation because the way in which the Poisson and the lensing equations are parameterized in terms of $\mu$ and $\Sigma$ is general. The quasi-static approximation enters only when an explicit and analytical expression for $\mu$ and $\Sigma$ is required (as it is in our case for $f(Q)$). Alternatively, one can solve the full equations and derive numerically the behaviour of $\mu$ and $\Sigma$ and finally place them back into Eq.\ (\ref{Eq:dUISW}). There is, nevertheless, an assumption to be considered, as we commented after Eq.\ (\ref{eq:UISW}). It has been assumed that $\delta_{\rm m}(z,k)$ depends only mildly on $k$, in which case it can be broken up in $k$ and time-dependent parts \cite{DeFelice:2011hq}. In our specific case, the analytic expression for $f(Q)$ derived within the quasi-static approximation is $k$-independent. Concerning the use of this approximation, the relevant scales for ISW data correspond to the modes deep inside the Hubble radius (namely, those for which $k^2/a^2 >> H^2$) which are in the regime of validity of the quasi-static approximation.
In the following we will provide theoretical predictions about the sign of the ISW-galaxy cross-correlation for $f(Q)$ gravity by analysing the sign of $\mathcal{F}$. In particular, this study will be informative in regards to the allowed sign and magnitude of $\alpha$. This is expected because dark energy models show a positive ISW-galaxy cross-correlation, meaning the impact of the background parameters is quite limited: they can only enhance or suppress the ISW-galaxy cross-correlation with respect to $\Lambda$CDM but its sign will always remain positive. This leaves $\alpha$ as the only parameter that can change the sign of this observable, offering a chance at constraining it.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{Fplts_wQ-1.pdf}
\caption{Evolution of $\mathcal{F}$ as a function of redshift $z$ for models with an exact $\Lambda$CDM background evolution and different values of $\alpha$. \label{fig:Fpltsw-1}}
\end{figure}
In Figure~\ref{fig:Fpltsw-1} we show the evolution of $\mathcal{F}$ as a function of the redshift for the case with an exact $\Lambda$CDM background. We note that for all models with $\alpha$>0, $\mathcal{F}$ is higher than that of $\Lambda$CDM whereas for $\alpha$<0 all models fall bellow. Additionally, at present time we find that all values of $\alpha$ considered verify $\mathcal{F}(z=0)$>0. However, for negative values of $\alpha$, $\mathcal{F}$ can be negative at some redshift. In particular, the smaller and more negative the values of $\alpha$ are, the closer to present time in redshift $\mathcal{F}$ turns in the negative side. This aspect may set a bound on the negative branch of $\alpha$. This type of behaviour for $\mathcal{F}$ can be expected from the previously studied behaviours of $D$ and $\mu$, since $f(Q)$ verifies $\Sigma = \mu$. In fact, the term $D'/D$ of eq.~(\ref{eq:fancyF}) is positive during the entire evolution for all values of $\alpha$, meaning it will always have a negative contribution to $\mathcal{F}$. However, the contribution from $\Sigma ' / \Sigma$ will follow the previously discussed behaviour of $\mu$ and show a net division between positive and negative $\alpha$: for $\alpha$>0, $\Sigma' / \Sigma$ is negative at all $z$ while the opposite holds for $\alpha$<0. This means that in the cases with positive $\alpha$ it would only be possible to have $\mathcal{F}$<0 if $|D'/D|$>$1+\Sigma ' / \Sigma$. On the other hand, $\alpha$<0 models have two negative contributions for eq.~(\ref{eq:fancyF}), making it easier to reproduce a negative $\mathcal{F}$.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{Fplts_VW_vwQ.pdf}
\caption{Evolution of $\mathcal{F}$ as a function of redshift $z$ for models with different values of $\alpha$ and fixed $w_0=-0.85$ (top panel) and $w_0 = -1.15$ (bottom panel). The $\Lambda$CDM case (solid, black line) is also included for comparison.}
\label{fig:FpltsVWvwQ}
\end{figure}
Then, in Figure~\ref{fig:FpltsVWvwQ} we show how the sign of $\mathcal{F}$ changes with redshift when the effective dark energy component has $w_Q=w_0$. From these figures it is clear the existent degeneracy between the value of $w_0$ and $\alpha$, which can be adjusted to have a positive $\mathcal{F}$ at all $z$ for the $f(Q)$ models. In details, when $w_0>-1$,
we can notice that the model with $\alpha=0$ (black, dot-dashed line) is enhanced with respect to the $\Lambda$CDM case, thus favouring a positive $\mathcal{F}$ at all $z$. However, when $\alpha\neq0$ the amplitude may change and it is possible to find $\mathcal{F}$<0 when one has large negative values of $\alpha$. In this case, while at present time the sign of $\mathcal{F}$ is positive, at earlier time it can be negative. If we consider $w_0<-1$, the curve corresponding to $\alpha=0$ is suppressed with respect to $\Lambda$CDM. As such, in order to have $\mathcal{F}_{f(Q)}>\mathcal{F}_{\Lambda CDM}$ at all $z$, $\alpha$ has to be positive and large (e.g. $\alpha>0.5$). On the contrary, to avoid having $\mathcal{F}<0$, the smaller and negative values of $\alpha$ need to be excluded.
As for the previous case, we find $D'/D$>0 for all values of $\alpha$. However, following the corresponding behaviour of $\mu$, there is no longer a net division between the effects of positive and negative $\alpha$, which thus reflects on the evolution of $ \Sigma ' / \Sigma $.
Models with the larger amplitude of the effective gravitational coupling will accordingly show a larger and positive amplitude for $\Sigma ' / \Sigma$ (specially at later time), consequently showing a negative $\mathcal{F}$. This is indeed the case for the more negative values of $\alpha$.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.56]{Fplts_M2_DwQ.pdf}
\caption{Evolution of $\mathcal{F}$ as a function of redshift $z$ for the time-dependent equation of state: M1 (top panel) and M2 (bottom panel). The $\Lambda$CDM scenario (solid, black line) is also included for comparison.}
\label{fig:Fplts1DwQ}
\end{figure}
For the time-dependent background evolution we show the results in Figure~\ref{fig:Fplts1DwQ}. We note that, for cases with $w_p < w_0$ (phantom behaviour at early time, M1 model in the top panel) the $\mathcal{F}$ quantity is mostly suppressed with respect to $\Lambda$CDM, except for the largest value of $\alpha$. This is again connected to the balance of the two terms in eq.~(\ref{eq:fancyF}). Since $D'/D$ remains positive at all $z$, models with a larger $\mu$ and consequently a larger and positive $\Sigma'/\Sigma$ term will work to bring $\mathcal{F}$ to smaller amplitudes. Alternatively, models with smaller $\mu$ such as $\alpha=1.5$, are capable of having $\Sigma'/\Sigma$<0 which means only $D'/D$ contributes to the decrease of $\mathcal{F}$. In this case, we conclude that large positive values of $\alpha$ will have $\mathcal{F}>0$ at all $z$, while in the other cases the sign of $\mathcal {F}$ depends on the magnitude of $|\alpha|$. In particular, large negative $|\alpha|$ have $\mathcal{F}<0$ at early time and then eventually cross the zero at later time. The crossing time depends on the $|\alpha|$: large values of $|\alpha|$ correspond to a crossing time very close to present day.
For $w_p > w_0$ (M2, bottom panel), the models are enhanced with respect to $\Lambda$CDM favouring a positive ISW-galaxy cross-correlation, with the exception of the large negative $|\alpha|$. The situation is similar to the previous case. The $D'/D$ term is once again positive for all $z$, but this time the majority of the $\alpha$ have small amplitudes of the effective gravitational coupling, resulting in larger amplitudes for the $\mathcal{F}$ quantity.
In summary, the analysis of the sign of $\mathcal{F}$ can give us an indication of a theoretical bound on the $\alpha$ parameter, which seems to exclude the most negative ones despite its degeneracy with $w_Q$. While here we have shown the evolution of $\mathcal{F}$ up to early times, datasets used to measure the cross-correlation of the CMB with tracers of the Large-Scale Structure (LSS) of the Universe are limited to low redshift ($z\simeq 1$). The latter include data from WISE, SDSS, SuperCOSMOS and NVSS.
Future surveys such as Euclid, LSST and SKA may provide accurate data to measure this cross-correlation and set stringent bounds on the $f(Q)$ model and, more generally, to any dark energy and modified gravity model.
\section{Conclusion} \label{Sec:conclusion}
We have studied the dynamics of linear cosmological perturbations in the $f(Q)$ gravity model. Rather than fixing a specific functional form for $f(Q)$, we used the \textit{designer} approach, which allowed us to reconstruct the form of the $f(Q)$ function associated to a chosen expansion history. We have introduced the parameter $\alpha$ which characterizes the family of solutions of $f(Q)$. This is one of the main results of this paper, being the first time the \textit{designer} approach is applied to $f(Q)$ gravity. To this extent, we have selected three effective equations of state, $w_Q$, for the effective dark energy component associated to $Q$, namely $w_Q=-1$, $w_Q=w_0$ and a fast varying equation of state. These choices are dictated not only by the aim of exploring different evolutions of $f(Q)$ models with redshift but also by the aim of studying a possible degeneracy between $\alpha$ and the parameters characterizing $w_Q$.
We have provided theoretical predictions for a large set of linear phenomena: the linear growth of structures, which included $\tilde{f}\sigma_8$, and the cross-correlation between the CMB and galaxy surveys. These physical quantities have one common denominator: the effective gravitational coupling, which is equal to the light deflection parameter in the $f(Q)$ gravity. Deviations of these parameters from the standard scenario can then affect many observables and as such only a global study of the latter can allow us to draw some general conclusions.
For all observables investigated, we find that a different expansion history can lead to a solution for the $f(Q)$ behaviour that is able to impact them significantly. Let us summarize the main results in the following:
\begin{itemize}
\item We found a degeneracy between the $\alpha$ parameter and those of $w_Q$. In fact, we verified that in several occasions either $\alpha$ or the $w_Q$ parameters can be tuned in order to compensate or even reproduce the effect of the other. As such, one advantage that comes from using the \textit{designer} approach is that an appropriate choice of datasets used to perform parameter estimation with Monte Carlo Markov Chain methods can in principle break this degeneracy. Since $\alpha$ does not enter in the background expansion history, the degeneracy can be mitigated by using accurate background probes which strongly constrain $w_Q$, such that any effect at perturbations level needs to be modelled only with $\alpha$. We also stress that using GWs standard sirens may help in breaking the degeneracy as they can strongly constrain $f_Q$ \cite{Frusciante:2021sio}.
\item We found that large negative values of $\alpha$, regardless of the chosen expansion history, can be excluded. These values indeed lead to a stronger gravity with respect to $\Lambda$CDM and as such to an higher amplitude for the $\tilde{f}\sigma_8$, which are unlikely to be favoured by RSD data only. Furthermore, models characterized by large negative values of $\alpha$ show a negative ISW-galaxy cross-correlation, which is excluded by observations.
\end{itemize}
Nevertheless, let us note that negative values of $\alpha$ can give rise to a lower ISW tail in the temperature-temperature power spectrum \cite{Frusciante:2021sio}, a feature that has been proved to be responsible for a better fit to CMB data for some MG models when compared to $\Lambda$CDM \cite{Peirone:2019aua,Frusciante:2019puu}, and which has recently been confirmed to be true for the case of a $f(Q)$ model with an exact $\Lambda$CDM background \cite{Atayde:2021ujc}. Therefore, while large negative values might be excluded this is not the case for small negative ones.
In conclusion, this type of phenomenological analysis of $f(Q)$ models provides insight on the types of deviations that might be expected on cosmological observables and that can be used to distinguish it from the $\Lambda$CDM scenario. Therefore it will be of interest to constrain the \textit{designer} $f(Q)$ gravity from the combined data analysis of LSS, CMB, BAO and SNIa.
\section*{Acknowledgements}
We thank B. J. Barros for useful comments on the manuscript.
This work is supported by Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia (FCT) through the research grants UIDB/04434/2020, UIDP/04434/2020, PTDC/FIS-OUT\\/29048/2017, CERN/FIS-PAR/0037/2019 and by the FCT project ``BEYLA --BEYond LAmbda" with ref. number PTDC/FIS-AST/0054/2021. The research of ISA has received funding from the FCT PhD fellowship grant with ref.~number 2020.07237.BD. NF acknowledges support from the personal FCT grant ``CosmoTests -- Cosmological tests of gravity theories beyond General Relativity" with ref.~number CEECIND/00017/2018.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,034 |
\section{History}
Astrophysics is a subject where the observers generally lead, and
theorists follow behind. The topic of my talk is one where the lag is
embarrassingly large. However, gamma-ray bursts raise issues which are
certainly fascinating to everyone involved in relativistic astrophysics.
Even though the history of gamma-ray bursts dates back more than 25
years, we still know neither where nor what they are. The story
started in the late 1960s, when American scientists at Los Alamos had
developed a set of satellites aimed at detecting clandestine nuclear
tests in space by the associated gamma-ray emission. Occasional
flashes, lasting a few seconds, were indeed detected. It took several
years before these were realised to be natural, rather than sinister
phenomena, and in 1973 a paper was published by Klebesadel, Strong \&
Olson entitled {\it Observations of Gamma-ray Bursts of Cosmic
Origin}. This classic paper reported 16 short bursts of
photons in the energy range between 0.2 and 1.5 MeV, which had been
observed during a three-year period using widely separated
spacecraft. The burst durations ranged from less than 0.1 second up to
about 30 seconds, but significant fine time-structure was observed
within the longer bursts. The bursts evidently came neither
from the Earth nor from the Sun, but little else was clear at that
time.
It did not take long for the theorists to become
enthusiastically engaged. At the Texas conference in December 1974,
Ruderman (1975), gave a review of models and theories. He presented a long and
exotic menu of alternatives that had already appeared in the
literature, involving supernovae, neutron stars, flare stars,
antimatter effects, relativistic dust, white holes, and some even more
bizarre options. He noted also the tendency, still often apparent, for
theorists to ``strive strenuously to fit new phenomena into their chosen
specialities''.
In the 1970s and 1980s, data accumulated on gamma-ray bursts, due to
a number of satellites. Particular mention should be made of the
contributions by Mazets and his colleagues in Leningrad. Also
important were the extended observations made by the Pioneer Venus
Orbiter (PVO). The number of detected bursts rose faster than the
number of models -- a further index of progress is that some of the
conjectures reviewed by Ruderman were actually ruled out.
During that period, three classes of models were pursued: those in
which the bursts were respectively in the Galactic Disc (at distances
of a few hundred parsecs), in the halo (at distances of tens of
kiloparsecs), and at cosmological distances. The characteristic
energies of each burst, according to these three hypotheses, are
respectively $10^{37}$ ergs, $10^{41}$ ergs, and $10^{51}$ ergs. The
most popular and widely-discussed option during the 1980s was that the
bursts were relatively local, probably in our Galactic Disc, and due
to magnetospheric phenomena or ``glitches'' on old neutron stars (defunct
pulsars).
It was clear that there were two statistical clues which could in
principle decide the location of gamma-ray bursts as soon as enough
data had accumulated, and selection effects were understood. One was
the number-versus-intensity of the events, which tells us whether they
are uniformly distributed in Euclidean space, or whether we are in
some sense seeing the edge of the distribution. The other is the
degree of anisotropy.
There was already evidence that the counts of gamma-ray bursts were
flatter than the classic Euclidean slope, since otherwise more faint
bursts would have been detected by balloon experiments. This would not
of course have been unexpected if the bursts were within
the galaxy. However, the real surprise came with the launch, in April 1991, of
the Compton Gamma Ray Observatory (GRO) satellite, whose Burst and Transient
Source Experiment (BATSE) offered systematic all-sky coverage, with good
sensitivity over the photon energy range 30 keV - 1.9 MeV. Data from BATSE
have transformed the subject.
The most remarkable BATSE result is the unambiguous evidence that
the bursts are highly isotropic over the sky. More than 1700 have now
(December 1996) been recorded, and there is still no statistical
evidence for any dipole or quadrupole anisotropy, nor for any
two-point correlation (Briggs {\it et al.\/} 1997) The lack of any
enhancement either towards the plane of the Galaxy, or towards the
Galactic Centre, is a very severe constraint on the hypothesis that
bursts come from the Galaxy. Note that they cannot be ultra-local
objects within our galactic disc: this would naturally permit
isotropy, but is ruled out by the flatness of the number counts. The
``non-Euclidean'' counts imply that the surveys are probing to
distances where the sources are, for some reason, thinning out; the
problem is to account for this by a hypothesis that is also consistent
with the isotropy.
The experiments on GRO have produced evidence on the spectra and
time structure of events. (For a recent review, see Fishman (1995) and
references cited therein.) Despite the large variety, there is little
doubt that gamma-ray bursts are a well-defined class of objects,
distinguished spectrally from phenomena such as X-ray bursters, and
also from the so-called ``soft gamma repeaters'' which have
substantially softer spectra. Within this class, there are some
apparent correlations. For instance, the shorter bursts tend to be
stronger and to have somewhat harder spectra; the histogram plotting
burst durations may have two peaks; and the counts deviate most from
the Euclidean slope for the bursts with harder spectra (Kouveliotou et
al 1996).
The manifest isotropy has tilted the balance of opinion strongly
towards a cosmological interpretation of the classical gamma-ray
bursts. I will concentrate on discussing the challenge posed to
theorists by that model. But I will then mention, more briefly, types of
halo model that are compatible with the isotropy since these cannot yet be
definitively deemed irrelevant. In conclusion, I will list some
observations which might in the near future settle the issue, or at
least reduce the current level of perplexity. This talk (and the present
written version) is intended as a general overview. Fuller details, and more
extensive references, can be found in the papers from the special session on
gamma-ray bursts, elsewhere in these proceedings, or in Hartmann (1996).
\section{Models for ``Cosmological'' Bursts}
If the bursts are cosmological, then the sub-Euclidean counts imply
that the typical burst has a redshift $z$ of order 1. The precise
redshift distribution depends on how much evolution there is in the
population. The mean redshift would be less, for instance, if the burst
rate increased with cosmic time. However, we can confidently say that
all but the very nearest of the observed bursts must have redshifts of
at least 0.2. Otherwise evolution would need to be implausibly steep
to explain the non-Euclidean counts, and nearby superclusters would
show up in the distribution over the sky. (Since the bursts exhibit such a wide
variety of time-structures, it would be astonishing if, by any measure, they
were anywhere near being standard candles. Obviously, detailed interpretations
of the counts depend on the luminosity function.)
The event rate per unit volume is very low if we are sampling a
population out to cosmological distances. It is of order $10^{-5}$ per
year per galaxy, in other words a thousand times less than the
supernova rate in galaxies. The required energy release then amounts to
$10^{51}$ ergs in a few seconds. (Both the estimates of
the rates and of the energy per event would need to be adjusted in a
straightforward way, of course, if the individual events were beamed
in a small solid angle.)
\section{``The trigger''}
The total energy is not necessarily in itself a problem. After
all, whenever a supernova goes off, the binding energy of a neutron
star is released in a fraction of a second, and this amounts to
$10^{53}$ ergs, a hundred times what is needed for the burst. But in a
supernova most of this energy goes to waste as neutrinos; moreover,
any impulsive electromagnetic release would not escape promptly, but
would be degraded by adiabatic expansion of the envelope before, much
later, it could leak out. So is it possible for some rare events to
occur where the energy release can escape promptly, rather than being
surrounded by an extensive opaque envelope? The most widely favoured
possibility is coalescence of binary neutron stars (see, for example,
Narayan, Paczynski and Piran 1992). Systems such as the famous binary
pulsar will eventually coalesce, when gravitational radiation drives
them together. The final merger, leading probably to the production of
a black hole, happens in a fraction of a second (though the swallowing
or dispersal of all the debris may take somewhat longer). The
calculated event rates for such phenomena -- and perhaps also for the
coalescence of binaries consisting of a neutron star and a black hole,
rather than two neutron stars -- are uncertain but are probably high
enough to supply the requisite rates of bursts.
\section{Fireball and gamma-ray emission}
How can the energy be transformed into some kind of fireball after
such a coalescence event? There seem to be two options. The first is
that some of the energy released as neutrinos is reconverted, when the
neutrinos collide outside the dense core where they were produced,
into electron-positron pairs or photons. The rate of this process
depends on the square if the neutrino luminosity, and those
simulations that have so far been carried out yield rather pessimistic
estimates for the efficiency (Ruffert {\it et al.\/} 1996). The second
option is that strong magnetic fields directly convert the rotational
energy of the system into a directed outflow. This latter option
requires that the magnetic fields be amplified to strengths of order
$10^{15}$ Gauss. (Usov 1994; Thompson 1994)
The observed gamma rays seem to have a nonthermal spectrum. Moreover,
they commonly extend to energies above 1 MeV, the pair production
threshold in the rest frame. These facts together imply that the
emitting region must be relativistically expanding. We draw this
conclusion for two reasons. Firstly, if the region were indeed only a
light second across or less, as would be implied by the observed rapid
variability in the absence of relativistic effects, the total mass of
baryons in the region would need to be below about $10^{21}$ grams in
order that the electrons associated with the baryons should not
provide a large opacity: the rest mass energy of the baryons would
need to be 10 orders of magnitude less than that of the radiation
energy in the same volume. Not only is this a remarkably low figure,
implying that only $10^{-12}$ of the material from the compact objects
is mixed up in the emitting region, but it would in any case imply a
relativistic expansion. Quite apart from the baryon constraint, there
is a second reason for invoking relativistic expansion. Larger source
dimensions are required in order to avoid opacity due to photon-photon
collisions (via $\gamma + \gamma \rightarrow e^+ + e^-$).
If the emitting region is expanding relativistically, then for a given
observed variation timescale the dimension $R$ can be increased by
$\gamma^2$. The opacity to electrons and pairs is then reduced by
$\gamma^4$, and the threshold for pair production, in our frame, goes
up by $\sim \gamma$ from its ``rest'' value of $\sim 1$ MeV. A high
$\gamma$ will of course only be attained if the baryon loading is
sufficiently low, such that the ratio of total energy to rest mass
energy is larger than $\gamma$. A variety of models have been
discussed. Best-guess numbers are, for an energy of $10^{51}$ ergs, a
Lorentz factor $\gamma$ in the range $10^2$ to $10^3$, allowing
the rapidly-variable emission to occur at radii in the range $10^{14}$
to $10^{16}$ cms. The entrained baryonic mass would need to be below
$10^{-6} M_\odot$ to allow these high relativistic expansion speeds.
Because the emitting region must be several powers of ten larger than
the compact object that acts as ``trigger'', there is a further
physical requirement: the original energy -- whether envisaged as an
instantaneous fireball or as a short-lived quasi-steady wind -- would,
during expansion, be transformed into bulk kinetic energy (with
associated internal cooling). It must be re-randomised and efficiently
radiated as gamma rays: this requires relativistic shocks. Impact on
an external medium (or an intense external radiation field) would
randomise half of the initial energy merely by reducing the expansion
Lorentz factor by a factor of 2. Alternatively, there may be internal
shocks within the outflow: for instance, if the Lorentz factor in an
outflowing wind varied by a factor more than 2, then the shocks that
developed when fast material overtakes slower material would be
internally relativistic (Piran 1997 and references cited therein).
In the case of expansion into an external medium, the energy would be
rethermalised after sweeping up external matter with rest mass $E/c^2
\gamma^2$ (Rees \& M\'esz\'aros 1992; M\'esz\'aros \& Rees 1993). For
$E = 10^{51}$ ergs and $\gamma = 10^3$, only $10^{-9} M_\odot$ of
external matter need be swept up. In an unsteady wind, if $\gamma$
were to vary on a timescale $\delta t$, internal shocks would develop
at a distance $\gamma^2 c \delta t$, and randomise most of the energy
(eg Rees \& M\'esz\'aros 1994). For instance, if $\gamma$ ranged
between 500 and 2000, on a timescale of $\delta t$ second, internal
shocks with Lorentz factors $\sim 2$ (measured in the frame of the
mean $\gamma \simeq 1000$ outflow) would lead to efficient dissipation
at $3 \times 10^{16} \delta t$ cms.
Another important consequence of relativistic outflow is that only
material moving within an angle $\gamma^{-1}$ of the line of sight
contributes to what we observe. Observations cannot therefore tell us
if bursts are highly beamed. Transverse pressure gradients are only
effective on angles below $\gamma^{-1}$, so material ejected in widely
differing directions behaves quite independently. There are already a
variety of models in the literature discussing the radiation from
shocks in expanding fireballs and relativistic winds (see Piran 1997
for a recent review). The parameters are uncertain, and the relevant
physics, involving for instance the coupling between electrons and
ions in relativistic shocks, is not sufficiently well developed to
allow accurate modelling of the radiation (see, for instance, Gallant
{\it et al.\/} 1992).
So how is the original energy channelled from the central object into
the outflowing fireball or wind. Recent calculations by Ruffert {\it
et al.\/}, 1996, suggest problems with releasing neutrino energy
efficiently enough, and on a short enough timescale, to allow
production of a fireball. The options involving {\it magnetic}
energy (cf Narayan, Paczynski \& Piran 1992) are rather less
quantitative, but I still believe they are more promising. As
discussed by Usov, (1994), and Thompson (1994), a millisecond pulsar
with a $\sim 10^{15}$ Gauss field would be slowed down in 1 second,
its spin energy being dumped in a pair-dominated relativistic wind.
As these authors and others have discussed, internal processes in
such a wind could explain gamma rays with the observed spectrum and
variability characteristics.
\section{A ``best buy'' model}
My personal favourite model (cf Meszaros and Rees 1997b) involves the
toroidal debris from a disrupted neutron star orbiting around a black
hole. If this debris contains a strong magnetic field, amplified
perhaps by differential rotation, then an axial magnetically-dominated
wind may be generated along the rotation axis, perpendicular to the
plane of the torus. The advantage of this geometry is that it seems
to offer the best chance of preventing baryon contamination, because
the baryonic material would be precluded by angular momentum from
getting near the axis without first falling into the black hole or
being on a positive-energy trajectory.
Such a configuration could arise from capture of a neutron star by a
black hole of less than $5M_\odot$, this mass limit being required
because otherwise the neutron star would be swallowed before
disruption. Alternatively, it could be the outcome of the merger of
two neutron stars, where most of the mass collapses to a black hole,
leaving some fraction of the original material in orbit around it. (cf
Ruffert {\it et al.\/} 1996; Jaroszynski 1996)
The available energy in this model is the kinetic or gravitational
energy of the neutron-star debris left behind in the torus, plus the
spin energy of the hole itself (which, being the outcome of binary
coalescence, is almost guaranteed to have a high angular
momentum). Near the axis, we would expect maximal dissipation (from
fields threading the hole or anchored in the torus) but minimum
baryonic loading. The Lorentz factor would therefore be largest along
the axis. Indeed, a narrow channel, essentially free of baryons, may
carry a Poynting-dominated outflow, energised by the hole via the
Blandford-Znajek process.
Along any given line of sight, the time-structure would be determined
partly by the advance of jet material into the external medium, but
probably even more by internal shocks within the jet, which themselves
depend on the evolution and instabilities of the torus, from its
formation to its eventual swallowing or dispersal. Even if the bursts
were caused by a completely standardised set of objects, their
appearance would be likely to depend drastically on orientation
relative to the line of sight. Other phenomena as yet undiscovered --
for instance some new class of X-ray or optical transient -- may be
attributable to gamma-burst sources viewed from oblique orientations.
\section{Physics of the emission mechanism}
We are a long way from a convincing model for what triggers
gamma-ray bursts: coalescing compact binaries seem likely to be implicated,
but we should remain open-minded to more exotic options. A precise
description of the dynamics, along with the baryon content, magnetic
field, and Lorentz factor of the outflow, might allow us to predict
the gross time-structure. But even then we could not predict the
intensity or spectrum of the gamma rays -- still less answer key
questions about the emission in other wavebands -- without also having
an adequate theory for particle acceleration in relativistic
shocks. We need the answers to the following poorly-understood
questions:
(i) Do relativistic shocks yield particle spectra that obey power
laws? This is in itself uncertain: the answer probably depends on the
ion/positron ratio, and on the relative orientation of the shock front
and the magnetic field (e.g. Gallant {\it et al.\/} 1992).
(ii) In ion-electron plasmas, what fraction of the energy goes into
the electrons?
(iii) Even if the shocked particles establish a power law, there must
be a low-energy break in the spectrum at an energy that is in itself
relativistic. But will this energy, for the electrons, be $\Gamma_s
m_e c^2$, or $\Gamma_s m_p c^2$ (or even, if the positive charges are
heavy ions like Fe, $\Gamma_s m_{\rm Fe} c^2$?
(iv) Can ions be accelerated up to the theoretical maximum where the
gyroradius becomes the scale of the system? If so, the burst events
could be the origin of the highest energy cosmic rays (an interesting
possibility addressed by other speakers at this conference)
(v) Do magnetic fields get amplified in shocks? This is relevant to
the magnetic field in the swept-up external matter outside the contact
discontinuity, and determines how sharp the external shock actually is
(cf Mitra 1996)
(vi) Can radio emission be generated by a coherent process? If not,
the usual surface brightness constraint implies that there would be
little chance of detecting a radio ``afterglow''.
These questions, crucial for gamma ray bursts, are also relevant to other
phenomena. For example, Lorentz factors of at least 10 (and probably
electron-positron plasmas) exist in the compact components of strong
extragalactic radio sources probed by VLBI.
If one is prepared to parametrise the uncertainties implicit in the
above questions, predictions can be made of how the spectrum would
evolve during a burst with simple time-structure. (eg Meszaros {\it et
al.\/} 1994, Tavani 1996, Meszaros and Rees 1997). For a wide range
of parameters, the associated X-rays would be above the threshold of
small omnidirectional detectors such as those developed for the High
Energy Transient Explorer (HETE) It was therefore a real setback to
the subject -- particularly to the prospect of using concurrent X-ray
or UV emission to pinpoint the burst locations more accurately -- when
HETE failed to go properly into orbit.
After the main emission is over, the fireball material would
continue to expand, with steadily-falling Lorentz factor, into the
external medium. Associated optical emission may persist for hours or
even days. This is long enough to allow an initial detection with
BACODINE to be followed up by raster scans with a 1 m telescope, that
could detect even emission down to 15th magnitude.
\section{``Extended Halo'' Models}
In the interests of balance, I would like to make a few remarks about
the alternative idea that the bursts are not from cosmological
distances, but instead come from within our own galaxy. Classical
gamma-ray bursts could be isotropic enough to be consistent with the
BATSE data if they came from neutron stars ejected from our Galactic
Disc at more than 700 km s$^{-1}$, which remained active, bursting
sporadically, for long enough to allow them to reach distances of at
least 100 kiloparsecs. They may either escape from the galaxy, or be
on very extended bound orbits. These high velocity objects could be a
special subset of pulsars. The typical velocities of the pulsars
sampled in surveys may be as high as 400 km/sec (Lyne and Lorimer
1994, J. Taylor, these proceedings). Moreover, those that formed with
higher kick velocities and/or with strong magnetic fields (and
therefore short lifetimes) are under-represented in surveys; we cannot
exclude the possibility that a high fraction of newly-formed pulsars
are of such types. If we conservatively suppose that they are only a
few percent of all pulsars, and form in our Galactic Disc at a rate of
about 1 per thousand years, then each must produce $10^6$ bursts, of
typical energy $10^{41}$ ergs. (If the relevant objects formed at a
rate of one per 100 years, the requirements placed on each would be
ten times more modest.) Repetition would not necessarily be expected,
since each neutron star could in principle continue bursting at a slow
rate for more than a billion years. However, if the bursts came in
groups, rather than being independent poissonian events, repetition
would not be impossible.
\section{ Fitting the isotropy}
Podsiadlowski has done detailed calculations of whether such a
population can provide an isotropic distribution. He shows this is
indeed possible for long-lived bursters whose orbits take them out
beyond 100 kiloparsecs. An important feature of such orbits is that,
because the galactic halo potential is not spherical (and may indeed
be rather irregular at such large distances) objects do not conserve
their angular momentum and therefore, even if they started off near
the centre of our Galaxy, they need not return so close to the centre
in later orbits. This effect helps to ensure greater isotropy.
Another possibility, favoured by Lamb (1995), is that the typical
objects have velocities above a thousand kilometres per second, and
are escaping the Galaxy completely. In this case, the best fit is
obtained if the bursts do not start until after a delay of around
$10^7$ years, by which time all neutron stars have reached distances
of 30 kiloparsecs or more.
I think it is fair to say that such models need to be carefully tuned
in order to fit the existing isotropy data, but that, though perhaps
unappealing, they cannot be ruled out. The constraints on orbital
parameters would be eased in alternative schemes where the neutron
stars {\it formed} far out in the halo (being perhaps, as Woosley
(1993) has discussed, relics of an early population of halo stars)
rather than being ejected from the disc.
\section{Mechanisms for halo bursts}
If a ``halo'' model is to be taken seriously, there must be an
acceptable mechanism for producing the succession of $10^{41}$ erg
bursts, spread over a very long timescale. Two options have been
proposed (Podsiadlowski, Rees \& Ruderman 1995).
The first possibility is that the relevant subset of neutron stars
start off with a super-strong ($\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 10^{15}$ Gauss) magnetic field.
This field, penetrating the core of the neutron star, would gradually
rise towards the surface through buoyancy effects, thereby causing
stress in the crust. The timescale for the buoyancy is estimated to
be at least $10^6$ years. Acceptable models require that it be
$\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 10^9$ years. The total stored energy is $\sim 10^{47}
(B/10^{15} G)$ ergs. The energy depends linearly on $B$, rather than
quadratically, because the field in the core is concentrated into
tubes where its strength has a standard value of $\sim 3 \times
10^{15} G$.
The crust gets stretched as the field drifts outwards. The units in
which energy is released depend on how much stress can build up in the
crust, and what fraction is released when the crust cracks. This is
a complicated problem in asteroseismology. However, a release of
$10^{41}$ ergs per event is plausible, in which case the total stored
magnetic energy would be sufficient to supply the requisite $10^6$
events.
The second very different option for triggering halo bursts involves
asteroidal impacts on to a neutron star. Each event requires, on
energetic grounds, the impact of $10^{21}$ grams. The main problem
with this idea is that such asteroidal or cometary bodies would be
tidally disrupted too far out to give a sudden enough event. A
possible solution is that the debris from the disrupted body squashes
down the magnetic field, which then rebounds, generating high electric
fields and thereby a pair cascade. Alternatively, the debris may form
a disc which accumulates before triggering a sudden electromagnetic
release when it couples its rotation to that of the neutron star.
The total impacting mass, to get enough bursts per star, must be
$10^{27}$ grams. It is not impossible (especially now we know that
planetary systems can exist around pulsars) that a neutron star could
carry with it $\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 10^{27}$ gm of asteroidal debris. However, a
larger reservoir, plus at least one large planet, is needed in order
for enough of these planetesimals to be perturbed on to near radial
orbits. We know that at least one pulsar has a planetary system. This
fact, plus the evidence that even typical pulsars may have velocities
of 400 km s$^{-1}$, suggests that models of this kind should not be
dismissed. Whatever the bursts turn out to be, the primary trigger,
and the efficient conversion of its energy into gamma rays, involve
physical conditions that are extreme and unfamiliar.
\section{How can we settle the debate?}
There is no convincing and fully worked out model for the bursts on
either the halo or the cosmological hypothesis. Neither option,
however, seems to violate any cherished beliefs in physics or
relativistic astrophysics. The issue is one of plausibility, and how
one weighs different lines of evidence. The isotropy would be a
natural consequence of the cosmological hypothesis. But the level of
isotropy so far revealed by BATSE, which restricts any dipole or
quadrupole anisotropy below the few per cent level and shows no
evidence for clumping on smaller scales, could be accommodated in a
halo hypothesis if high speed neutron stars were implicated.
In April 1995, the 75th anniversary of the Shapley/Curtis debate,
there was an interesting debate in Washington on the location of
gamma-ray bursts -- a current issue offering some amusing parallels to
the earlier controversy concerning the distances of the nebulae. The
two main protagonists were Don Lamb and Bohdan Paczynski (a written
version of the argument appears in Lamb (1995) and Paczynski
(1995)). I had the privilege of acting as the moderator in this
debate, perhaps because I was one of the few people who had not
already taken a firm stance on one side of the issue or the
other. There was an agreement among all participants that the issue
would be settled only by more data. Indeed, there was a broad
consensus on some particular tests that could be crucial, or at least
highly suggestive. Among these might be the following.
Most valuable of all would be a firm identification of a burster
with some other class of object. The stumbling block here is the poor
positional accuracy of most gamma-ray detectors. BATSE itself has
error circles of 1 or 2 degrees for the brightest bursts, and more
than 5 degrees for the fainter ones. However, the locations of some
bursts have been pinned down with a precision of minutes of arc or
better by triangulation experiments involving deep space probes; this
technique utilises the rapid time structure, which, when recorded and
timed by detectors separated by 10 light minutes or more, allows
accurate positioning. There is still no firm identification of any
classical gamma-ray burst, though there are tantalising indications
that some of the brighter bursts may be correlated with galaxies or
clusters of galaxies, whose distances are not inconsistent with what
is expected on the cosmological hypothesis. (It is disappointing,
incidentally, that the failure of the recent Mars probe, which would
have carried a small gamma-ray detector, means that we now lack the
requisite deep-space network for obtaining accurate ``triangulation''.)
Even though the gamma-ray positional information is poor, one might be
able to pin down the position of the sources more accurately if they
displayed concurrent transient emission in some other
waveband. Various projects have been undertaken in the optical and
radio band. Ground-based observers can be notified of a BATSE event
within a few seconds; a small telescope can then be rapidly slewed to
seek an optical counterpart within less than a minute. No such
counterparts have been detected, nor have radio searches yet yielded
positional or timing coincidences. The likely strength of gamma-ray
bursts in the optical or radio band is uncertain and highly
model-dependent. Indeed, any detection in these wavebands would have
the bonus that it would help to narrow down the range of possible
models and emission mechanisms. However, most theories predict that
there should be substantial spectral extension from gamma-rays down
towards the X-rays, so it would seem less of a gamble to seek X-ray
counterparts.
If the bursts have a local rather than cosmological origin, then,
at some level, anisotropies over the sky would be bound to show up. A
particularly crucial test would be feasible if bursts more than ten
times fainter than those recorded by BATSE could be detected. It
would then, according to the halo hypothesis, be feasible to detect
bursts from the halo of Andromeda, and there should be a definite
excess of weak events from that direction (Bulik \& Lamb 1997,
Ruszkowski \& Wijers 1997). The lack of such a trend would severely
embarrass halo models. A specific proposal has been made to look at a
10 degree field around Andromeda with 20 times the sensitivity of
BATSE. But X-ray detectors are more readily available and more
sensitive. For this reason, and also because the x-ray emission from
bursts seems stronger than a straight extrapolation of the gamma-ray
spectrum suggests (Preece {\it et al.\/} 1996), the best prospects for
testing the halo model might be from long-duration observations of
Andromeda and other nearby galaxies.
The cosmological interpretation of bursts would be confirmed, as
Paczynski (1986) first pointed out, if there were evidence of
gravitational lensing by an intervening galaxy. If a suitable galaxy
lay along the line of sight to a cosmologically distant burst,
radiation would reach us by two or more different paths, whose light
travel times would differ typically by weeks or months. We would
therefore detect two bursts from the same direction. Even though the
positions could not be pinned down accurately, the elaborate time
structure of each burst is highly distinctive, and if two
bursts with identical ``fingerprints'' were detected from within the
same error circle, this would be compelling evidence that they were
actually separate gravitationally-lensed images of the same burst.
(As a technical point, it should be noted that microlensing by stars
or substellar objects would only introduce differences on millisecond
timescales between the two burst profiles (Williams and Wijers 1997),
and therefore would not vitiate this possibility.)
Unfortunately, the probability that a galaxy lies along a random line
of sight to a high redshift object is below one per cent, the exact
value depending of course on the presumed redshift of the
burst. Moreover, because BATSE can only observe a given direction in
the sky for about 40 per cent of the time it is more likely than not
that, if a lensed event occurred, the recurrence would be missed
because it would occur during dead time. Taking these effects into
account, it is rather marginal whether we would expect BATSE to detect
a single instance of this lensing before it dies, even if the bursts
indeed come from cosmological distances. However, if we were lucky,
such a double burst could clinch the cosmological interpretation.
A further issue which has figured strongly in the debate on the
location of gamma-ray bursts concerns the existence or otherwise of
spectral features attributable to cyclotron lines. This is a
technical controversy which I will not enter here. However, its
relevance lies in the fact that halo models involve neutron stars,
where the magnetic fields are expected to be in the range such that
cyclotron lines should be in the hard X-ray band. On the other hand,
the fields in the emitting regions of cosmological fireballs or
relativistic winds would not, even when relativistic effects are taken
into account, give rise to such features.
The controversies in the Shapley-Curtis debate were settled within a
few years. Our knowledge of extragalactic astronomy thereby made a
forward leap, and astronomers moved on to address more detailed
issues. I'm enough of an optimist to believe that it will be only a
few years before we know where and perhaps even what, the
gamma-bursters are. Even if this optimism is misplaced, I am
completely sure that these mysterious phenomena will serve as a
continuing challenge and stimulus to theorists, and will remain high
the agenda of future Texas Conferences.
I thank my colleagues Josh Bloom, Peter Meszaros, Philipp
Podsiadlowski, and Ralph Wijers for collaboration and discussion. I am
also very grateful to the members of the BATSE team, especially Jerry
Fishman and Jim Brainerd, for updating me on the observations and
answering several queries.
| {
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South Africa may look far away on the map but it is really just a flight away from several of the world's major cities. It is a technologically and socially advanced nation – in a landscape that's out of this world!
Q1. When is the best time to visit South Africa?
South Africa is a great year-round destination. Temperatures range around 20 degrees celsius throughout the year. For whale watching best months to visit are June to October at Hermanus. Be sure to include a night at Hermanus if you are travelling between this time.
Q2. Does it requires a long visit to experience South Africa?
Although it would take several weeks to cover all of South Africa, there are many ways to experience the variety of the country in a brief visit. Nine days is a good starting point for shorter visits to South Africa, but even those visiting for a shorter period (or adding a few days of vacation to a business trip) can experience a taste of what the country has to offer.
Q3. Is there much to do in South Africa?
South Africa offers a wealth of activities- from action packed adventures and indulgent relaxation to family-friendly activities. Travellers can hike through forests, kayak in sparkling rivers, experience the local culture, sample boutique wines, visit exceptional museums and more…….
Q4. Is South Africa expensive?
Travellers from many parts of the world will find South Africa to be a good value for money destination.
South Africa food and wines are world-class. Fresh produce ensures high quality and the nation is renowned as a producer of premium wines. For those wanting Indian cuisine, there are Indian restaurants galore all across South Africa and ample vegetarian options too.
Q6. Can travellers drive in South Africa?
Cars, campervans and small campers that lie somewhere between the two are readily available for hire. The minimum hire age would vary from 21 to 25 and travellers who wish to hire a vehicle will need to produce an International Driver's Permit or a valid driver's licence (in English) from their home country. In South Africa you can drive on the same side of the road as in India and it is the best place to start a self drive holiday. Empty roads with turning landscapes and scenery, no parking hassles and road signs all along that will make it almost impossible for you to lose your way.
Q7. Where can you experience Adventure Sports in South Africa?
Adventure activities are found all over South Africa. The most famous is Shark Cage Diving in Gansbaai, which can be done as a day trip from Cape Town. There are other sports like River Rafting, Sky Diving, Mountain Climbing, Bungee Jumping etc.
Q8. Where can you see Wildlife other than Kruger?
If Kruger is too expensive, Shamwari in the Eastern Cape region is an excellent substitute where you can view a variety of wildlife. | {
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} | 6,226 |
Despite the news overload, it does not feel like there is much progress in our national affairs. Not sure about you, but I have been finding the whole think a little depressing. When I feel like this, I use a trick I learned in the Happiness Advantage by Shawn Achor which is a great book about positive psychology. The trick is to take 5 minutes in the evening and write three things which made you happy during that day. There are lots of versions of this, with some people finding it easy to write in the morning, others using it to express gratitude. Whatever works for you; the point is to spend some time on thinking positive thus training your brain to prioritise positive over negative and contributing to your overall happiness.
My assumption is that due to uncertain market conditions, companies need to focus on developing new business and growing their pipeline. This provides us with an opportunity to win new work.
Initially we put a lot of effort into drumming this message to our target markets to develop new business though our efforts were met with little appetite.
Still, I think that companies need to make a move on their new business sales but rather then hitting our head against a brick wall we chose to get in touch with people who already knew us. This meant contacting all of our ex-clients who we felt might be more open to a conversation at this stage.
This approach was more effective though we have not stopped our new business development activity which is on-going.
The moral of my story is that even people whose job it is to develop new business can find themselves at a dead end sometimes. The point is to not let it last for long, take stock and try something else. If you need some help evaluating your new business development approach, take our questionnaire here.
This entry was posted in Business Development, Strategy on 5th April 2019 by Yafit Davis.
Many people will be looking at this title and say that developing new business is always a must. Still, many companies don't necessarily consider this an emergency, especially well-established ones. This is an important area to consider as a matter of course but I felt that there is more of a calling for it currently which is why I am dedicating this Blog to this.
as Matthew S. Olson and Derek van Bever demonstrate in their book Stall Points, once a company runs up against a major stall in its growth, it has less than a 10% chance of ever fully recovering.
I am not sure about you, but I can certainly see a potential stall point coming up which might affect many companies' growth. When I wrote my original Blog, many businesses were still treating the situation as one to ignore, thinking that things will get clearer in the New Year. At the time of writing these lines we are 13 business days away from the UK leaving the EU and we have no further clarity.
I have not written this Blog to join the end of the world choir and I am sure that business will prevail after Brexit, in the long term we may even be better off, who knows. Your problem as a business owner, Sales Director or senior manager is the short term. How will you navigate your ship in the stormy water of a 'short-term economic downturn'?
There's a lot to be said for applying some planning to this situation, whatever you call rainy day measures in your organisation. That said, many businesses feel that you can hardly plan for a scenario you are unclear about. However, there is one area you could definitely benefit from reviewing which is, your new business sales.
This is particularly important for established companies out there as over reliability on existing business can make them very vulnerable to a market downturn. As we all know, when the economy is showing signs of crisis many companies look to reduce their monthly outgoings and review unnecessary expenses. There is no telling what these might be and whether your company will be affected by it. In this situation, you might agree, there is no harm in developing some new business and growing your pipeline by way of plan B.
The longer you leave it the more competition you will have.
This is why I recommend that you look into this urgently to make sure that you can start widening your options and developing your sales pipeline. We have recently developed a new lead generation system we call Big ticket Leads which might provide a possible solution. Watch this short animation to find our more.
This entry was posted in Brexit, Business Development, Lead generation, Sales on 12th March 2019 by Yafit Davis.
If you want some good advice, listen to the American tennis player Arthur Ashe, who said, 'Start where you are, use what you have, do what you can'. When it comes to developing new business, this is sound advice, particularly when so many people are looking for a magic bullet that will get them off to a flying start. Often, the answer is actually much closer than you imagine and comes from utilising what you already know.
Doing what I do, I often stand at the starting line with a new client, looking down the track we have defined to be his new target market. People laugh when I tell them that despite putting together a sound strategy, in reality we don't really know how it will go until we start. This can be a little daunting for companies wanting return on their investment. The process of developing new business is slow by definition, as you are venturing into markets you where you are not known, and that is not an easy concept for the board to come to terms with.
So, what can you do to try and speed things up?
Start where you are: This applies to the strategy stage when it is really important to use your current market knowledge to propel you on to the next market. If you are already successful with hospitality, you may want to look at a related market such as facility management rather then going for insurance. This means that your current success will be easier to demonstrate and your market understanding will be much more relevant.
Use what you have: The key to an accelerated sales process, is using known contacts who can introduce you into the new market you are developing. It might sound obvious, but I don't actually meet many people who do this. It's much easier to buy a database and call everyone but results will be much slower. Get everyone in your company considering old contacts they may have and referral partners who can help; you will amazed by the value of what you can gather using your own resources.
Do what you can: So many people I meet give up too soon. Getting into a new market is not easy and you may feel that you are better off staying where you are, which is fine. But if you are going for a new market, make sure you are prepared to chase people and keep following up until you get a sensible answer.
This entry was posted in Business Development, Lead generation, Sales on 1st February 2019 by Yafit Davis.
Zig Ziglar once said that, 'if you aim at nothing you will hit it every time'. I chose this quote because it always makes me smile and it is also very apt in January. Whether you own a business or are in charge of business development and sales, January is the time for planning. Unfortunately, January is also a time when your pipeline may feel a little short. You may therefore find yourself in a situation where on the one hand there are all these great new plans being made and on the other hand you are looking at your shrinking pipeline, wondering how you are ever going to hit these numbers.
Speak to clients who have not purchased from you in a while. This is particularly helpful if you have some new products to tell them about as your reason for the call. Failing that, most contacts will be open to you calling for a general update on their requirements.
Identify clients who may be able to buy more services from you. How many times have you come across existing clients who tell you how they bought a product you provide from someone else? Make sure your clients are aware of your full range of products including offers and updates. I would suggest the best approach would be to review and identify gaps then call your clients with a pitch in mind.
Speak to your referral partners, particularly ones you have not heard from in a while. Having a coffee with them is never a bad idea as they may need a reminder or have some questions. The right nurturing of such contacts can make a big difference to your sales.
Get back in touch with old prospects. We have already said that January is the time for planning which is true with your prospects as well. Even if they said no in the past, their circumstances may have changed. In addition, I am sure you will find some who you stopped contacting for some reason, in which case getting back in touch is a must.
We have also launched a new Big Ticket Lead Generation product which you can learn more about here. Happy to chat and discuss the best option so give us a call.
This entry was posted in Business Development, Lead generation, Sales on 15th January 2019 by Yafit Davis.
I choose to read it as the latter and I feel that Brexit is a very good example of it. Talking to people about current events, as I do, I came to realise that there are many ways in which we read the situation. Whilst some believe the economic market will suffer a decline in the short-medium term, others think it will only affect companies trading with Europe. I even heard that some people feel that if we stay positive there is no reason why there will be a recession at all. Go figure, as no one really knows any of these views could be right, so it becomes a matter of the not so common sense.
So, why do I think you should consider re-inventing yourself this winter?
According to Harvard Business Review jumping from the maturity stage of one business to the growth stage of the next—is what separates high performers from those whose time at the top is all too brief.
As Matthew S. Olson and Derek van Bever demonstrate in their book Stall Points, ,once a company runs up against a major stall in its growth, it has less than a 10% chance of ever fully recovering.
Both are good reasons because whether you would like to be a high performer business or ensure that your business does recover from an economic downturn in the New Year, you should consider re-inventing yourself this winter.
How do you go about re-inventing your business model?
Are you going to come up with a completely new product?
Are you going to consider taking your exiting products to new markets?
This is why I recommend that you look into this urgently to make sure that you can start widening your options and developing your sales pipeline. We have recently developed a new lead generation system which we believe can support this process well. In September we run a workshop to share our new system and got some great feedback. Click here to find out more.
This entry was posted in Business Development, Lead generation, Sales, Strategy on 4th December 2018 by Yafit Davis. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,684 |
{"url":"https:\/\/www.talkstats.com\/threads\/should-my-model-include-separate-predictors.78495\/","text":"# should my model include separate predictors?\n\n#### stats20\n\n##### New Member\nLet's say I measured blood pressure on day1 and day2 three times a day (morning, afternoon and evening).\nCode:\ndat <- data.frame(ind=c(1,1,1,2,2,2,3,3,3,4,4,4), day1=c(90,113,122,86,84,95,114,126,123,115,92,103), day2=c(141,123,134,112,112,115,92,100,121,133,124,89), time=rep(c(\"morning\",\"afternoon\",\"evening\"),times=4))\n\nind day1 day2 time\n1 90 141 morning\n1 113 123 afternoon\n1 122 134 evening\n2 86 112 morning\n2 84 112 afternoon\n2 95 115 evening\n3 114 92 morning\n3 126 100 afternoon\n3 123 121 evening\n4 115 133 morning\n4 92 124 afternoon\n4 103 89 evening\nIn R, I can model the data this way:\nCode:\nmod <- lm(day1 ~ day2, data=dat)\nBut I can also reshape dat in this way:\nCode:\nlibrary(reshape2)\ndat2 <- melt(dat, id.vars = c(\"ind\",\"time\"), variable.name=\"day)\n\nind time day value\n1 morning day1 90\n1 afternoon day1 113\n1 evening day1 122\n2 morning day1 86\n2 afternoon day1 84\n2 evening day1 95\n3 morning day1 114\n3 afternoon day1 126\n3 evening day1 123\n4 morning day1 115\n4 afternoon day1 92\n4 evening day1 103\n1 morning day2 141\n1 afternoon day2 123\n1 evening day2 134\n2 morning day2 112\n2 afternoon day2 112\n2 evening day2 115\n3 morning day2 92\n3 afternoon day2 100\n3 evening day2 121\n4 morning day2 133\n4 afternoon day2 124\n4 evening day2 89\nAnd do:\n\nCode:\nmod <- lm(value ~ day, data=dat2)\nThese are the same data but the model parameters are very different. Which way of modelling the data would be more appropriate?\n\nLast edited:\n\n#### hlsmith\n\n##### Less is more. Stay pure. Stay poor.\nWhat is the purpose of the model.\nM1: gives you average predictions for 24 hours of time elapsed\nM2: gives you predictions for time of day\n\nBoth models neglect to address the violation of independence between observations. Traditionally multilevel models would be used. How many ind do you have?\n\n#### noetsi\n\n##### No cake for spunky\nOr SEM time based models or an ANOVA model with a time factor depending on what you are testing. There are many options.\n\n#### stats20\n\n##### New Member\nThank you for your answer @hlsmith. Yes, I'd do this with lme4:\nmod1 <- lmer(day1 ~ day2 + 1|ind, data=dat) and mod2 <- lmer(value ~ day + 1|ind, data=dat2).\nSo that means fitting either mod1 or mod2 is correct depending on the question. But what question would mod1 and mod2 answer?\nYou mention mod2 gives predictions for time of day but time of day isn't included as a predictor in the model. Wouldn't that be the case if I had done mod2 <- lmer(value ~ day + time + 1|ind, data=dat2)? Then I would have predictions for time of day, or am I wrong?\n\n#### hlsmith\n\n##### Less is more. Stay pure. Stay poor.\nWould you want day1 to be explained by day2? What would be wrong with that?\n\n#### stats20\n\n##### New Member\nThanks @hlsmith. So with mod1, I have the possibility to explain day1 by day2 (or the other way round), which I cannot do with mod2. But then what is the advantage of mod2? What does it answer that mod1 cannot? You mentioned above that \"gives you predictions for time of day\", but I thought for that I would need to include \"time\" as a predictor: mod2 <- lmer(value ~ day + time + 1|ind, data=dat2), no?\n\nMy question is not so much specific to this example but more general about what it means to have separate predictors (day1, day2, ...)in the model versus one predictor with several levels (like day in mod2).\n\n#### hlsmith\n\n##### Less is more. Stay pure. Stay poor.\nHow much data and groups do you have - this may influence my response.\n\n#### stats20\n\n##### New Member\nMy actual data set has about 700 individuals (ind) and each individual has about 300 time points (time) for each of the two predictors (day1 and day2).\n\n#### hlsmith\n\n##### Less is more. Stay pure. Stay poor.\nExplain the last part in more detail. A person has 300 day1 and 300 day2 data points? And these are for the say Day1 and Day2?\n\nSo why isn't this just time series data?\n\n#### stats20\n\n##### New Member\nYes, a person has 300 data points in day1 and 300 data points in day2. It is time series data that I'm trying to model with a mixed model.\nind day1 day2 time\n1 90 141 1\n1 113 123 2\n1 122 134 3\n1 86 112 4\n.............................................\n1 22 131 300\n2 12 333 1\n..............................................\n\nThe question is what questions can I answer with mod1 versus mod2?\n\n#### hlsmith\n\n##### Less is more. Stay pure. Stay poor.\nWhat is the difference between day1 and day2? Is there a gap between the last day1 value and the first day2 value or are all values evenly spaced out?\n\n#### stats20\n\n##### New Member\nTo make it simple let's say I collected 300 data points of heart rate and 300 data points of blood pressure acquired at the same time for each person. Time point is in seconds, so 300 seconds of continuous recorded data.\n\nIf my dat is:\nCode:\nID heart_rate blood_pressure time_point\n1 90 141 1\n1 103 123 2\n1 102 134 3\n1 76 112 4\n.............................................\n1 90 131 300\n2 70 189 1\n..............................................\n700 80 150 300\nand dat2 is coded:\nCode:\nID variable value time_point\n1 heart_rate 90 1\n1 heart_rate 103 2\n1 heart_rate 102 3\n1 heart_rate 76 4\n.........................................................\n1 blood_pressure 141 1\n1 blood_pressure 123 2\n1 blood_pressure 134 3\n1 blood_pressure 112 4\n.........................................................\n1 blood_pressure 131 300\n2 heart_rate 70 1\n.........................................................\n700 blood_pressure 150 300\nThe same question applies:\n\nWhat is the difference between mod1 <- lmer(heart_rate ~ blood_pressure + 1|ID, data=dat) and mod2 <- lmer(value ~ variable+ 1|ID, data=dat2)? What can each of these models tell me?\n\n#### hlsmith\n\n##### Less is more. Stay pure. Stay poor.\nRegardless of the posted model code, what is the study question? And Heart rate and blood press are their own variable right? Above makes it look like you have them list in the same column. Which doesn't make sense.\n\n#### Buckeye\n\n##### Active Member\nI agree with hlsmith. Typically, we start with a research question and then determine what approach works best with the data we have. Ideally, collect data after developing the research question.\n\nLast edited:","date":"2022-11-26 08:36:19","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\": 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\": 0.31171420216560364, \"perplexity\": 7980.713699911974}, \"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-2022-49\/segments\/1669446706285.92\/warc\/CC-MAIN-20221126080725-20221126110725-00131.warc.gz\"}"} | null | null |
Q: How to redirect 500 within an event listener? I have a defined listener watching the onKernelException in which errors could occur.
class SpecificExceptionListener
{
public function onKernelException(GetResponseForExceptionEvent $event)
{
$exception = $event->getException();
if ($exception instanceof SpecificExceptionToBeProcessed) {
// ...
if ($somethingWentWrong) {
// here, redirect to the default/overriden Symfony error page
}
// ...
}
}
}
How do you redirect to the standard/customized error page in case of an error?
A: For those who might be interested, here's how it can be done:
use Symfony\Bundle\TwigBundle\TwigEngine;
use Symfony\Component\HttpKernel\Event\GetResponseForExceptionEvent;
use Symfony\Component\HttpFoundation\Response;
// ...
class SpecificExceptionListener
{
protected $templating;
public function __construct(TwigEngine $templating)
{
$this->templating = $templating;
}
public function onKernelException(GetResponseForExceptionEvent $event)
{
$exception = $event->getException();
if ($exception instanceof SpecificExceptionToBeProcessed) {
// ...
if ($somethingWentWrong) {
// build response to display Symfony default error page
// replace by your own template if needed
$response = new Response();
$response->setContent(
$this->templating->render('TwigBundle:Exception:error.html.twig')
);
$event->setResponse($response);
return;
}
// ...
}
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,487 |
<?php
namespace Doctrine\ORM\Query\Exec;
use Doctrine\DBAL\Cache\QueryCacheProfile;
use Doctrine\DBAL\Connection;
use Doctrine\DBAL\Driver\ResultStatement;
use Doctrine\DBAL\Types\Type;
/**
* Base class for SQL statement executors.
*
* @link http://www.doctrine-project.org
*
* @todo Rename: AbstractSQLExecutor
*/
abstract class AbstractSqlExecutor
{
/** @var mixed[]|string */
protected $_sqlStatements;
/** @var QueryCacheProfile */
protected $queryCacheProfile;
/**
* Gets the SQL statements that are executed by the executor.
*
* @return mixed[]|string All the SQL update statements.
*/
public function getSqlStatements()
{
return $this->_sqlStatements;
}
/**
* @return void
*/
public function setQueryCacheProfile(QueryCacheProfile $qcp)
{
$this->queryCacheProfile = $qcp;
}
/**
* Do not use query cache
*
* @return void
*/
public function removeQueryCacheProfile()
{
$this->queryCacheProfile = null;
}
/**
* Executes all sql statements.
*
* @param Connection $conn The database connection that is used to execute the queries.
* @psalm-param array<int, mixed>|array<string, mixed> $params The parameters.
* @psalm-param array<int, int|string|Type|null>|
* array<string, int|string|Type|null> $types The parameter types.
*
* @return ResultStatement|int
*/
abstract public function execute(Connection $conn, array $params, array $types);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,502 |
Q: CngKey System.Security.Cryptography.CryptographicException The system cannot find the file specified on Azure I try to generate key from this code
CngKey key = CngKey.Import(Convert.FromBase64String(privateKey), CngKeyBlobFormat.Pkcs8PrivateBlob);
it works fine locally but when I deploy in my Azure app service.
it gives me this error:
System.Security.Cryptography.CryptographicException: The system cannot find the file specified.
at System.Security.Cryptography.NCryptNative.ImportKey(SafeNCryptProviderHandle provider, Byte[] keyBlob, String format)
at System.Security.Cryptography.CngKey.Import(Byte[] keyBlob, String curveName, CngKeyBlobFormat format, CngProvider provider)
I add WEBSITE_LOAD_USER_PROFILE In Configuration with value '1' but it didn't make any difference.
Thanks
A: If WEBSITE_LOAD_USER_PROFILE = 1 does not work for you. Here is a workaround for this same issue:
*
*Open your Azure App Service (Azure Website) blade in portal.azure.com
*Go to the Application settings page
*Scroll to App settings
*Add a new entry key: WEBSITE_LOAD_CERTIFICATES, and provide a dummy (fake, made-up, randomly-generated) value for it.
Check out the below links for the similar issue for reference:
*
*X509Certificate2 on Azure App Services (Azure Websites) since mid-2017?
*GH issue #16
A: This issue is not only on Azure, I had the same issue on my VPS as well and this answer save my life:
X509Certificate Constructor Exception
Cheers,
Nick
A: I upgraded the plan service from free to basic with adding WEBSITE_LOAD_USER_PROFILE = 1 in Azure Configuration.
The issue was when the app service was free it use a shared VM but when upgrading my app service pricing into basic it use a private VM.
A: I have spent around 4 days looking for this also. I have found out another reason. When you run VS as administrator, then you automatically get the privileges to read something called "cert user store".
However if you run it on another machine or some hosting, where you do not have full privileges, then you might run into this issue as well.
Hope this helps as well.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,337 |
Achanta Sharath Kamal entered the men's singles semifinals of the India Open on Saturday.
India's Achanta Sharath Kamal pulled off a nerve-wracking 4-3 victory against sixth seeded Paul Drinkhall of England to surge into the semifinals of the 2017 ITTF World Tour India Open at the Thyagaraj Stadium in New Delhi on Saturday. The seven-time national champion and World No.62 had to pull out his prime defensive game to outwit and outlast Drinkall, a multiple medal winner in the last two Commonwealth Games.
It proved to be a battle of nerves and patience as Drinkall slowed down the pace, keeping the ball short while also loading it with spin. He wasn't allowed to play his natural, attacking game and had to rely on each other's errors to win every point.
Sharath began on a strong note, racing away to a 11-4 win in the opening game. But Drinkhall quickly got his game together and frustrated and stunned Sharath with his well thought out tactics. He won the next two games 12-10, 11-9 to put the pressure entirely on the Indian, who had the full backing of a big crowd.
The 34-year-old Sharath began to look for openings to attack from that stage and came back strongly to win the next two games 12-10, 11-9. He jumped to a 3-0 lead but allowed Drinkhall to catch up. He went up 6-4 but then conceded four points to lose the psychological advantage. Drinkhall rode on the momentum to take the sixth game and place the match on the needle.
By now, the crowd was on the edge even though he made a strong start in the decider, going up 2-0 and then 3-1. But Drinkhall fought back and took a commanding 8-5 lead. Sharath continued to attack and it helped him get stay in the contest. He saved two match points and a couple of forehand drives saw him emerge triumphant.
He takes on Japan's 14-year-old sensation Tomokazu Harimoto, who beat the eighth seeded Robert Gardos of Austria 4-2, later.
Earlier, top seeded Ovtcharov Dimitrij survived a scare to advance to the semifinals after a gruelling 4-3 victory over Yuya Oshima of Japan. The World No.5 German, who came into the match without dropping a game, could not get the desired start and lost the opening game 7-11. He regained his composure to win the next three games 11-8, 11-6, 11-8 to move into a strong position. Down 1-3 and staring at the bottom of the barrel, the 39th ranked Japanese upped the ante to take the next two games 11-4, 11-6 to restore parity.
With the scores tied, both the players displayed great tenacity and got involved in a neck-and-neck battle. However, it was the German who prevailed 12-10 to move into the last four. He will now take on the third seeded Koki Niwa of Japan, who sealed his berth in the semis by registering an easy 4-1 victory over qualifier Tianyi Jiang of Hong Kong.
In women's singles, eighth seeded Wing Nam NG of Hong Kong beat Goergina Pota of Hungary 11-9, 11-6, 12-10, 8-11, 11-3 to set up a semifinal clash against sixth seeded Sakura Mori of Japan, who beat Tze Wing Mak of Hong Kong 11-6, 13-11, 10-12, 11-6, 11-7. | {
"redpajama_set_name": "RedPajamaC4"
} | 304 |
/***
* Inferno Engine v4 2015-2017
* Written by Tomasz "Rex Dex" Jonarski
*
* [#filter: script #]
***/
#pragma once
#include "base/app/include/appService.h"
namespace base
{
namespace script
{
//----
struct ClassTypeInfo;
struct ArrayTypeInfo;
struct EnumTypeInfo;
struct PropertyInfo;
struct FunctionInfo;
//----
// generic type information wrapper for scripts
struct BASE_SCRIPT_API TypeInfo
{
RTTI_DECLARE_NONVIRTUAL_CLASS(TypeInfo)
public:
TypeInfo(const rtti::IType* type = nullptr);
FORCEINLINE TypeInfo(const TypeInfo& other) = default;
FORCEINLINE TypeInfo& operator=(const TypeInfo& other) = default;
// get the engine name of the type
StringID getName() const;
// is this a valid type ?
Bool isValid() const;
// was this type defined in scripts ?
Bool isScripted() const;
// is this a simple value type ? (bool/float, etc)
Bool isSimple() const;
// is this a structure/class type
Bool isClass() const;
// is this an enum type
Bool isEnum() const;
// is this a general pointer type
Bool isPointer() const;
// is this a shared pointer type
Bool isStrongPointer() const;
// is this a weak pointer type
Bool isWeakPointer() const;
// is this a generic array
Bool isArray() const;
// is this a dynamic array
Bool isDynamicArray() const;
// is this a static array
Bool isStaticArray() const;
//--
// get as class type info
ClassTypeInfo asClass() const;
// get as class type info
ArrayTypeInfo asArray() const;
// get as enum type info
EnumTypeInfo asEnum() const;
// for pointers get the pointer class
ClassTypeInfo getPointedClass() const;
//--
FORCEINLINE const Bool operator==(const TypeInfo& other) const { return m_type == other.m_type; }
FORCEINLINE const Bool operator!=(const TypeInfo& other) const { return m_type != other.m_type; }
FORCEINLINE const rtti::IType* getRawType() const { return m_type; }
private:
const rtti::IType* m_type;
};
//----
// class type information wrapper for scripts
struct BASE_SCRIPT_API ClassTypeInfo
{
RTTI_DECLARE_NONVIRTUAL_CLASS(ClassTypeInfo)
public:
ClassTypeInfo(const rtti::IClassType* type = nullptr);
FORCEINLINE ClassTypeInfo(const ClassTypeInfo& other) = default;
FORCEINLINE ClassTypeInfo& operator=(const ClassTypeInfo& other) = default;
//--
// is this a valid type ?
Bool isValid() const;
// is this abstract class ?
Bool isAbstract() const;
// is this a structure type ? (not an object type)
Bool isStruct() const;
// is this a structure type ? (not an object type)
Bool isScripted() const;
// get the engine name of the class
StringID getName() const;
// get type info for this class
TypeInfo getTypeInfo() const;
// get base class
ClassTypeInfo getBaseClassInfo() const;
//--
// get class type, valid only for classes deriving from object
rtti::ClassRef<obj::IObject> getClassType() const;
//--
// get all properties
Array<PropertyInfo> getProperties() const;
// get local properties of this class only
Array<PropertyInfo> getLocalProperties() const;
// find property
PropertyInfo findProperty(StringID name) const;
//--
// get all functions
Array<FunctionInfo> getFunctions() const;
// get local functions of this class only
Array<FunctionInfo> getLocalFunctions() const;
// find function
FunctionInfo findFunction(StringID name) const;
//--
// check if this object implements given class by name (SLOWEST)
Bool derivesFromClassName(StringID className) const;
// check if this object implements given class
Bool derivesFromClassInfo(const ClassTypeInfo& classInfo) const;
// check if this object implements given class
Bool derivesFromClass(rtti::ClassRef<obj::IObject> classType) const;
//--
FORCEINLINE const Bool operator==(const ClassTypeInfo& other) const { return m_type == other.m_type; }
FORCEINLINE const Bool operator!=(const ClassTypeInfo& other) const { return m_type != other.m_type; }
FORCEINLINE const rtti::IClassType* getRawType() const { return m_type; }
//--
private:
const rtti::IClassType* m_type;
};
//----
/// wrapper for property
struct BASE_SCRIPT_API PropertyInfo
{
public:
RTTI_DECLARE_NONVIRTUAL_CLASS(PropertyInfo)
public:
PropertyInfo(const rtti::Property* prop = nullptr);
FORCEINLINE PropertyInfo(const PropertyInfo& other) = default;
FORCEINLINE PropertyInfo& operator=(const PropertyInfo& other) = default;
// is this property valid ?
Bool isValid() const;
// is this property scripted ?
Bool isScripted() const;
// is this property editable ?
Bool isEditable() const;
// is this property readonly ?
Bool isReadonly() const;
// get the class this property was defined at
ClassTypeInfo getParentClassTypeInfo() const;
// get name of the property
StringID getName() const;
// get display category of the property
StringID getCategory() const;
// get type of the property
TypeInfo getTypeInfo() const;
//--
FORCEINLINE const Bool operator==(const PropertyInfo& other) const { return m_property == other.m_property; }
FORCEINLINE const Bool operator!=(const PropertyInfo& other) const { return m_property != other.m_property; }
FORCEINLINE const rtti::Property* getRawProperty() const { return m_property; }
//--
private:
const rtti::Property* m_property;
};
//----
// array type information wrapper for scripts
struct BASE_SCRIPT_API ArrayTypeInfo
{
RTTI_DECLARE_NONVIRTUAL_CLASS(ArrayTypeInfo)
public:
ArrayTypeInfo(const rtti::IArrayType* type = nullptr);
FORCEINLINE ArrayTypeInfo(const ArrayTypeInfo& other) = default;
FORCEINLINE ArrayTypeInfo& operator=(const ArrayTypeInfo& other) = default;
//--
// is this a valid type ?
Bool isValid() const;
// get type info for this class
TypeInfo getTypeInfo() const;
// get the inner type of the array
TypeInfo getInnerTypeInfo() const;
// get the engine name of the type
StringID getName() const;
// is this an array with static size ?
Bool isStaticSize() const;
// get the static size for the array
Int32 getStaticSize() const;
//--
FORCEINLINE const Bool operator==(const ArrayTypeInfo& other) const { return m_type == other.m_type; }
FORCEINLINE const Bool operator!=(const ArrayTypeInfo& other) const { return m_type != other.m_type; }
FORCEINLINE const rtti::IArrayType* getRawType() const { return m_type; }
//--
private:
const rtti::IArrayType* m_type;
};
//--
/// type wrapper for enums
struct BASE_SCRIPT_API EnumTypeInfo
{
RTTI_DECLARE_NONVIRTUAL_CLASS(EnumTypeInfo)
public:
EnumTypeInfo(const rtti::EnumType* type = nullptr);
FORCEINLINE EnumTypeInfo(const EnumTypeInfo& other) = default;
FORCEINLINE EnumTypeInfo& operator=(const EnumTypeInfo& other) = default;
//--
// is this enum type valid ?
Bool isValid() const;
// is this enum defined in scripts
Bool isScripted() const;
// get name of the enum
StringID getName() const;
// get back the generic type info
TypeInfo getTypeInfo() const;
//--
// get maximum enum value
Int32 getMaxValue() const;
// get minumum enum value
Int32 getMinValue() const;
// get all enum names
Array<StringID> getOptions() const;
// find value for given option name, returns default value if not found
Int32 getValue(const StringID optionName, const Int32 defaultValue);
// get name for given value, returns empty name if not found
StringID getOptionName(const Int32 value) const;
// find value for given option name, returns false if not found
Bool getValueSafe(const StringID optionName, Int32& outValue) const;
//--
// find value for given option name, returns default value if not found
Int64 getValue64(const StringID optionName, const Int64 defaultValue);
// get name for given value, returns empty name if not found
StringID getOptionName64(const Int64 value) const;
// find value for given option name, returns false if not found
Bool getValueSafe64(const StringID optionName, Int64& outValue) const;
//--
FORCEINLINE const Bool operator==(const EnumTypeInfo& other) const { return m_type == other.m_type; }
FORCEINLINE const Bool operator!=(const EnumTypeInfo& other) const { return m_type != other.m_type; }
FORCEINLINE const rtti::EnumType* getRawType() const { return m_type; }
//--
private:
const rtti::EnumType* m_type;
};
//--
/// type wrapper for function
struct BASE_SCRIPT_API FunctionInfo
{
RTTI_DECLARE_NONVIRTUAL_CLASS(FunctionInfo)
public:
FunctionInfo(const rtti::Function* func = nullptr);
FORCEINLINE FunctionInfo(const FunctionInfo& other) = default;
FORCEINLINE FunctionInfo& operator=(const FunctionInfo& other) = default;
//--
// is this function valid ?
Bool isValid() const;
// is this function defined in scripts
Bool isScripted() const;
// is this function static ?
Bool isStatic() const;
// is this function global ?
Bool isGlobal() const;
// get name of the function
StringID getName() const;
// get the owning class, invalid for global functions
ClassTypeInfo getParentClassTypeInfo() const;
//--
// get return type
TypeInfo getReturnType() const;
// get number of arguments
Int32 getArgumentCount() const;
// get type of n-th argument
TypeInfo getArgumentTypeInfo(const Int32 index) const;
//--
FORCEINLINE const Bool operator==(const FunctionInfo& other) const { return m_func == other.m_func; }
FORCEINLINE const Bool operator!=(const FunctionInfo& other) const { return m_func != other.m_func; }
FORCEINLINE const rtti::Function* getRawFunction() const { return m_func; }
//--
private:
const rtti::Function* m_func;
};
} // script
} // base | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,833 |
Deutsche Bank expects huge 6.7-bn-euro net loss in 2015
Deutsche Bank expects a fourth-quarter loss of 2.1 billion euros, it said in a statement (AFP Photo/Daniel Roland)
Berlin (AFP) - Germany's largest bank, Deutsche Bank, on Wednesday said it expected an overall loss of 6.7 billion euros ($7.3 billion) for 2015, blamed on litigation costs, write-downs and restructuring charges.
Some of the losses had already been announced earlier in 2015 but the bank reported additional litigation charges of 1.2 billion euros in the fourth quarter.
Overall, it expects a fourth-quarter loss of 2.1 billion euros, weighed down by "challenging market conditions", the lender said in a statement of preliminary earnings.
"The full-year results include previously disclosed impairments taken in the third quarter of 5.8 billion euros of goodwill and intangibles, full-year litigation provisions of approximately 5.2 billion euros and restructuring and severance charges of approximately 1.0 billion euros," it said.
The bank is currently embroiled in some 6,000 law suits over allegations of money laundering and rate manipulation. Last May, it was fined a record $2.5 billion for its involvement in rigging interest rates.
In addition to write-downs totalling 5.8 billion euros due to tougher regulations on capital requirements, Deutsche also took charges on assets such as its almost 20-percent stake in China's Hua Xia Bank, pushing it deeper into the red.
The bank said its planned sale of the stake in the China-based bank is expected to close in the second quarter of 2016.
Last October, Deutsche Bank vowed to take drastic measures after unveiling a record 6.01-billion-euro loss in the third quarter.
Besides not paying dividends in 2015 and 2016, Deutsche said it would cut some 26,000 jobs, around a quarter of the workforce, sell off assets and pull out of 10 countries.
"It's all about executing on our plans to build a better Deutsche Bank... about making Deutsche Bank simpler and more efficient," new co-chief executive John Cryan said at the time.
In Wednesday's statement, Deutsche Bank said it expects to report fourth-trimester revenues of 6.6 billion euros -- compared to 7.8 billion euros a year earlier -- and full-year revenues of 33.5 billion euros, up six percent on the previous year.
The lender also said it was targeting a capital ratio -- a measure of its financial strength -- of roughly 11 percent at the end of the fourth quarter.
The bank will announce full details of the fourth-quarter and annual results on January 28.
Observers say Deutsche Bank is trapped between its international ambitions in the field of investment banking, where it insists it is among the top five in the world, and its traditional high-street banking in Germany.
It suffers from mediocre profitability, faces ferocious competition and increasing regulatory demands.
Arizona Republic's Super 10 girls HS basketball rankings: Jan. 11-18
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,023 |
\section{Introduction}
Siegel modular varieties arise as moduli spaces of abelian varieties of a fixed dimension~$g$ equipped with a principal polarization and level structure. They carry distinguished line bundles whose global sections are Siegel modular forms of degree~$g$. Siegel modular forms constitute an important class of automorphic forms and generalize the classical modular forms on finite index subgroups of $\SL_2(\mathbb{Z})$ to higher dimensions. The study of Siegel modular forms is of fundamental importance in number theory and algebraic geometry.
Siegel modular varieties carry universal abelian varieties, and these similarly carry distinguished line bundles. Their global sections are known as \emph{Siegel--Jacobi forms}.
Importantly, Siegel--Jacobi forms appear as Fourier coefficients of Siegel modular forms in higher degrees.
The systematic study of Siegel--Jacobi forms goes back to the
1980s. The case of degree one is dealt with in the book \cite{EZ} by
Eichler and Zagier and in many papers from Zagier's school. For
higher degree, foundations were laid by Dulinski~\cite{dulinski} and
Ziegler~\cite{Ziegler-J}. Later Kramer developed the arithmetic theory
of Siegel--Jacobi forms \cite{Kramer_Crelle}. An important aspect in
his work is the consideration of toroidal compactifications of
the universal abelian variety following the work of Mumford and its
collaborators \cite{toroidal}, \cite{AVRT:compact}, \cite{tai}, and
in the arithmetic setting by Faltings and
Chai~\cite{fc}. Around the same time Runge contributed by further
studying the geometric aspects of Siegel-Jacobi forms \cite{runge}.
Other work related to Siegel--Jacobi forms of higher degree has
appeared in \cite{yamazaki, yang1, yang2, yang4, yang3}.
\subsection{Statement of the main result}
Let $g$ be a positive integer.
Let $\Gamma \subseteq \operatorname{Sp}(2g,\mathbb{Z})$ be a finite index subgroup. Siegel modular forms with respect to $\Gamma$ are classified by weight, whereas Siegel--Jacobi forms with respect to $\Gamma$ are classified by two invariants, namely \emph{weight} and \emph{index}. We denote by $J_{k,m}(\Gamma)$ the space of Siegel--Jacobi forms of weight $k$ and index $m$ with respect to $\Gamma$. Then $\bigoplus_{k,m}J_{k,m}(\Gamma)$ is naturally a bigraded $\mathbb{C}$-algebra which is known not to be finitely generated (see \cite{EZ} for the case of degree one, and Proposition \ref{prop:bi-gr-alg} below for the general case).
From now on we fix a neat subgroup $\Gamma \subseteq \operatorname{Sp}(2g, \mathbb{Z})$. Assume $g \ge 2$. Then Runge claims in \cite[Theorem 5.5]{runge} that, for suitable integers $r$, the rings $\bigoplus_{m\le 2rk}J_{k, m}(\Gamma)$ are finitely generated $\mathbb{C}$-algebras.
Our main aim in this paper is to disprove Runge's claim. In fact we show that this algebra is \emph{never} finitely generated. In Remark \ref{rem:runge-mistake} we explain the oversight in Runge's proof.
\begin{introthm}\label{th:intro-not-fin-gen}({Theorem~\ref{thm:2}} and Proposition~\ref{prop:disprove-runge})
Let $g \geq 1$. For each $k>0$ and $m>0$ the graded algebra $\bigoplus_{\ell}J_{\ell k, \ell m}(\Gamma)$ is not finitely generated. Therefore for any $r\in \mathbb Q_{>0}$ the graded algebra $\bigoplus_{m\le rk}J_{k, m}(\Gamma)$ is not finitely generated.
\end{introthm}
Our main tool to prove Theorem \ref{th:intro-not-fin-gen} is the
theory of toroidal b-divisors (where b stands for birational). A
b-divisor on a projective variety $X$ can be thought of as an infinite
tower of divisors living over all smooth birational models of $X$, and
compatible under pushforward (see Section \ref{sec:b-divisors} for a
precise definition and further details). A b-divisor is said to be
\emph{Cartier} if it is determined on one such birational model, in
other words, if all the elements in the tower can be obtained by pull
back and push forward of a divisor on a single model.
If $X$ carries a toroidal structure then the theory of b-divisors can be naturally simplified by considering only the toroidal blowups of $X$. The corresponding toroidal b-divisors can then be seen as conical functions on a conical polyhedral complex associated to $X$. This enables the study of toroidal b-divisors by techniques from convex geometry \cite{BoteroBurgos}. For example, if a toroidal b-divisor is Cartier then the corresponding conical function is piecewise linear.
To the neat subgroup $\Gamma$ is associated a fibration of principally polarized abelian varieties
\[
\pi \colon \mathcal{B}(\Gamma) \longrightarrow \mathcal{A}(\Gamma) \, .
\]
Here $\mathcal{A}(\Gamma)$ is the Siegel modular variety associated to $\Gamma$.
For each pair of given integers $k, m$ the complex variety $\mathcal{B}(\Gamma)$ carries a line bundle of Siegel--Jacobi forms $L_{k,m}$. It is endowed with a natural invariant metric, which we denote by $h$ (see Section \ref{sec:geom-interpr-line}). As we will see in Propositions \ref{prop:5} and \ref{prop:7} one can choose a toroidal compactification $\overline{\mathcal{B}}(\Gamma)$ of $\mathcal{B}(\Gamma)$ in such a way that $L_{k,m}$ extends as an algebraic line bundle $\overline{L}_{k,m}$ on $\overline{\mathcal{B}}(\Gamma)$ and $h$ extends as a \emph{toroidal psh} metric on $\overline{L}_{k,m}$ (see Section \ref{sec:toroidal-psh-metrics} for the definition of toroidal psh metrics).
As follows from the work done in our paper~\cite{BBHJ}, given a non-zero rational section $s$ of $\overline{L}_{k,m}$ we have an associated toroidal b-divisor $\mathbb{D}(\overline{L}_{k,m}, s,h)$ on $\overline{\mathcal{B}}(\Gamma)$. This b-divisor does not depend on the actual choice of toroidal psh extension $\overline{L}_{k,m}$. The b-divisor $\mathbb{D}(\overline{L}_{k,m}, s,h)$ corresponds to a convex function on the conical polyhedral complex attached to $\overline{\mathcal{B}}(\Gamma)$.
The key point is that, via an explicit computation, we can show that this conical function is \emph{not} piecewise linear. This implies that the toroidal b-divisor $\mathbb{D}(\overline{L}_{k,m}, s,h)$ is not Cartier (Corollary \ref{cor:1}). We then use a characterization of Jacobi cusp forms in terms of the invariant metric (Proposition \ref{prop:6}) to deduce that the algebra $\bigoplus_{\ell}J_{\ell k, \ell m}(\Gamma)$ is not finitely generated.
\subsection{Asymptotic dimension formulae}
Tai's celebrated work \cite{tai} implies an asymptotic formula for the dimension of the space of Siegel--Jacobi forms of given ratio between weight and index. We recall that the main aim of \cite{tai} is to show that the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$ is of general type for $g\geq 9$. Tai's proof proceeds via a study of the pushforward of the line bundle of Siegel--Jacobi forms to $\mathcal{A}_g$, and an application of Mumford's version of the Hirzebruch proportionality principle \cite{hi} on the non-compact pure Shimura variety $\mathcal{A}_g$.
In the present work we arrive at a form of the proportionality principle on the universal abelian variety itself, via the machinery of b-divisors. This gives a new way to compute the asymptotic dimension of the space of Siegel--Jacobi forms, namely as the degree of a suitable toroidal b-divisor on a toroidal compactification of the universal abelian variety.
Let $g \in \mathbb{Z}_{ \geq 1}$ and set $G = g(g+1)/2$ and $n = G + g$. The b-divisor $\mathbb{D}(\overline{L}_{k,m}, s,h)$ has a well-defined degree $\mathbb{D}(\overline{L}_{k,m}, s,h)^n$ in $\mathbb{R}_{\ge 0}$ (combine Remark~\ref{rk:degree} and Lemma~\ref{lem:nef}).
\begin{introthm}\label{th:intro-vol-jac}({Corollary~\ref{cor:asy}})
Let $\mathbb{D}(\overline{L}_{k,m}, s,h)$ be the toroidal b-divisor on the toroidal compactification $\overline{\mathcal{B}}(\Gamma)$ associated to the line bundle $L_{k,m}$ of Siegel--Jacobi forms of weight~$k$ and index~$m$, the rational section $s$ and the invariant metric $h$. Then the Hilbert-Samuel type formula
\begin{equation} \label{eq:HilbSam}
\mathbb{D}(\overline{L}_{k,m}, s,h)^n = \limsup_{\ell \to \infty}\frac{\dim J_{\ell k, \ell m}(\Gamma)}{\ell^n/n!}
\end{equation}
holds, where $\mathbb{D}(\overline{L}_{k,m}, s,h)^n$ is the degree of the b-divisor $\mathbb{D}(\overline{L}_{k,m}, s,h)$.
\end{introthm}
Using Chern--Weil theory for line bundles with psh metrics as developed in \cite{BBHJ} we can compute the degree on the left hand side in \eqref{eq:HilbSam} explicitly using integrals of smooth differential forms on the open universal abelian variety $\mathcal{B}(\Gamma)$. Then by applying the equality in \eqref{eq:HilbSam} we obtain an explicit asymptotic dimension formula. The resulting formula is compatible with the one implicit in Tai's work \cite{tai}, see Remark~\ref{rem:tai-formula}.
\begin{introcor}\label{th:intro-asy-dim} ({Corollary \ref{th:asy-dim}})
The asymptotic growth of the dimension of the space $J_{\ell k, \ell
m}(\Gamma)$ is given by the following formulae:
\begin{equation*}
\begin{split}
\limsup_{\ell \to \infty}\frac{\dim J_{\ell k, \ell m}(\Gamma)}{\ell^n/n!}
& = (-1)^{G}n! m^{g}k^{G}[\Gamma _{0}\colon \Gamma ] \prod_{k=1}^{g}\frac{\zeta (1-2k)}{(2k-1)!!}\\
& = (-1)^{n}n!m^{g}k^{G}2^{G-g}
[\Gamma _{0}\colon \Gamma ]\prod_{k=1}^{g}\frac{(k-1)!B_{2k}}{(2k)!}\\
& = V_g \cdot n!m^gk^G2^{-G-1}\pi^{-G}[\Gamma_0 \colon \Gamma],
\end{split}
\end{equation*}
where $\Gamma_0=\Sp(2g,\mathbb{Z})$,
where $B_{2k} = \frac{(-1)^{k+1} 2(2k)!}{(2\pi)^{2k}}\zeta(2k)$ are the Bernoulli numbers and
\[
V_g = (-1)^n 2^{g^2+1}\pi^G\prod_{k=1}^g\frac{(k-1)!B_{2k}}{(2k)!}
\]
is the symplectic volume computed by Siegel in \cite[Section VIII]{siegel}.
\end{introcor}
Note that we are assuming that $\Gamma $ is neat. In particular
$-\Id\not \in \Gamma $. In fact the above formulae are also true for
arbitrary $\Gamma \subset \Gamma _{0}$ of finite index not containing
$-\Id$. If $\Gamma $ contains $-\Id$ the right hand side has to be
multiplied by $2$.
We mention that
Theorem~\ref{th:intro-vol-jac} and Corollary~\ref{th:intro-asy-dim} generalize results proved in \cite{bkk} for the case $g=1$, $k=m=4$ and $\Gamma=\Gamma(N) \subset \SL_2(\mathbb{Z})$ a principal congruence subgroup. As we shall see in Remark \ref{rmk:limsup2}, the $\limsup$ in Theorem \ref{th:intro-vol-jac} and Corollary \ref{th:intro-asy-dim} is actually a $\lim$ for sufficiently divisible $\ell$.
We expect that our method to compute asymptotic dimensions can be generalized to other spaces of automorphic forms on mixed Shimura varieties. Indeed, the b-divisorial approach to Chern-Weil theory from \cite{BBHJ} continues to hold in these cases whenever the natural invariant metrics give rise to psh line bundles.
\subsection{Outline of the paper}
In Section \ref{sec:basic-definitions} we set up the basic
definitions, including those of b-divisors, toroidal b-divisors, and
toroidal psh metrics (a special class of singular psh metrics). The
main new result in this section is a characterisation of the Lelong
numbers of a toroidal psh metric in terms of the conical convex
function defining the toroidal psh metric, see Lemma \ref{lemm:16}.
In Section \ref{sec:psh-metrics-b} we study properties of and relations between graded linear series, b-divisors and psh metrics. The relations are summarized in Diagram~\ref{diag:relations}. We then show that these relations become stronger in the toroidal case, see Theorem \ref{thm:1} and Corollary \ref{cor:2}. We describe an example to show that the toroidal assumption here is really necessary (Remark \ref{rem:exa-non-tor}). The compatibility relations in the toroidal case are key ingredients in the proofs of Theorems \ref{th:intro-not-fin-gen} and \ref{th:intro-vol-jac}.
In Section \ref{sec:jacobi-forms} we recall the definitions of Siegel--Jacobi (cusp) forms. We discuss in detail the universal family $\pi \colon\mathcal{B}(\Gamma) \to \mathcal{A}(\Gamma)$ of principally polarized abelian varieties, and describe Siegel--Jacobi forms as global sections of a line bundle $L_{k,m}$ over $\mathcal{B}(\Gamma)$. We also give an explicit description of the invariant metric $h$ on this line bundle.
In Section~\ref{sec:toro-comp} we describe the theory of toroidal compactifications of $\mathcal{A}(\Gamma)$ and $\mathcal{B}(\Gamma)$ and discuss extensions of $L_{k,m}$ and $h$ over the compactifications. We prove that one can choose the toroidal compactifications and the extensions in such a way that the extended invariant metric is toroidal psh (Proposition \ref{prop:7}). This leads to the observation that the b-divisor associated to the line bundle of Siegel--Jacobi forms with its invariant metric is toroidal (Corollary~\ref{cor:toroidal}).
In Section~\ref{sec:proofs_main} we give the proof of our main result. We start by giving an explicit formula for the relevant conical function (Lemma \ref{lemm:13}). We then observe that this function is not piecewise linear. It follows that the associated toroidal b-divisor is not Cartier (see Corollary \ref{cor:1}), from which we deduce Theorem~\ref{th:intro-not-fin-gen}.
In the final Section~\ref{sec:asymptotic} we define the volume of a graded linear series and prove Theorem~\ref{th:intro-vol-jac}. Among other things we use the Hilbert--Samuel formula for toroidal b-divisors from \cite{BoteroBurgos}. Finally, we use Chern--Weil theory for singular psh metrics as developed in \cite{BBHJ} to compute the degree of our b-divisor in terms of the non-pluripolar volume of the positive current associated to our psh metric, leading to Corollary \ref{th:intro-asy-dim}.
\subsection*{Acknowledgements}
We are grateful to Gerard van der Geer, J\"urg Kramer and Don Zagier for helpful discussions.
\section{Basic definitions}
\label{sec:basic-definitions}
In this section we recall the definitions of the basic tools we will
use in the paper. Throughout the paper $X$ will denote the
complex manifold associated to a smooth projective variety over $\mathbb{C}$
of dimension~$n$.
\subsection{Graded linear series}
\label{sec:graded-linear-series}
Let $F=K(X)$ be the field of rational functions on $X$. The following
definition is taken from \cite{kk}.
\begin{df}\label{def:1} A \emph{graded linear series} (on $X$) is a graded
subalgebra $A\subset F[t]$. Let $L\subset F$ be a finite
dimensional $\mathbb{C}$-vector space. The graded linear series $A_{L}$ is
the graded subalgebra of $F[t]$ generated by $L\cdot t$ in degree 1 (note that $A_L$ is generated by finitely many elements of degree 1).
A graded linear series $A$ is called of \emph{integral type} if
there is a finite dimensional linear subspace $L\subset F$ such that
$A_L\subset A$ and $A$ is finite over $A_{L}$.
A graded linear series $A$ is called of \emph{almost integral type}
if there is a graded linear series $A'$ of integral type such that
$A\subset A'$.
\end{df}
If $A$ is a graded linear series we write
$A=\bigoplus_{\ell \in \mathbb Z_{\ge 0}} A_{\ell}t^{\ell}$, where $A_{\ell}\subset F$.
\begin{rmk}
If $X'\dashrightarrow X$ is a birational map then a graded linear series on $X$ is a graded linear series on
$X'$.
\end{rmk}
Let $D$ be a Cartier divisor on $X$, with integer, rational or real
coefficients. Then one writes
\begin{displaymath}
\mathcal{L}(D)=\{0\not = f\in F\mid \dv(f)+D\ge 0\}\cup \{0\}
\end{displaymath}
and
\begin{displaymath}
\mathcal{R}(D)=\bigoplus_{\ell \in \mathbb Z_{\ge 0}}\mathcal{L}(\ell D)t^{\ell}\subset F[t] \, .
\end{displaymath}
The following result follows from \cite[Theorems 3.7 and 3.8]{kk} (using that $X$ is normal and projective).
\begin{prop}\label{prop:2}
The graded linear series $\mathcal{R}(D)$ is of almost integral type. If
moreover $D$ is very ample, then
$\mathcal{R}(D)$ is of integral type.
\end{prop}
\subsection{Toroidal structures}
\label{sec:toroidal-structures}
\begin{df}\label{def:modification}
A morphism $\pi\colon X' \to X$ of complex varieties is a
\emph{modification} if it is proper and birational.
\end{df}
\begin{df}
A \emph{(smooth) toroidal structure} on $X$ is the choice of a smooth
modification $\pi_{1} \colon X_{1 }\to X$ and a simple normal crossing
divisor $D$ on $X_{1}$ that contains the exceptional locus of
$\pi_{1} $. If $(\pi _{2},D')$ is another toroidal structure, we say
that $(\pi _{2},D')$ \emph{is above} $(\pi _{1},D)$ if $\pi_{2}$ factors as
\begin{displaymath}
\xymatrix{
X_{2}\ar[rd]^{\pi_{2}}\ar[d]_{\pi _{21}}&\\
X_{1 }\ar[r]^{\pi_{1} } & X
}
\end{displaymath}
and $\pi _{21}^{-1}(D)\subset D'$.
\end{df}
A toroidal structure will be denoted by the triple
$(X_{1},\pi _{1},D)$ or, if no confusion may arise, just by the divisor $D$.
The open subset $U=X_{1 }\setminus D$ can be identified with an open
subset of $X$ also denoted by $U$, and the inclusion $U\subset X_{1
}$ is an example of a toroidal embedding without self intersection. We
will use freely the theory
of toroidal embeddings from \cite{toroidal}. For a concise description
of what we will need the reader is referred to \cite{BoteroBurgos}.
\begin{rmk}
One can envisage more general toroidal structures where
$U\subset X_{1}$ is an arbitrary toroidal embedding (without assuming $X_1$ smooth), but we will not need
them in this paper.
\end{rmk}
Following \cite{toroidal}, to any smooth toroidal embedding
we can associate a conical polyhedral complex $\Pi (X_{1},D)$. The
rays of $\Pi (X_{1},D)$ are in one to one correspondence with the
irreducible components of $D$. If $D_{i}$ is an irreducible component
of $D$, we denote by $\rho _{D_{i}}$ the corresponding ray. Given irreducible components
$D_{1},\dots ,D_{n}$ of $D$, then the set of cones of
$\Pi (X_{1},D)$ of dimension $n$ having $\rho _{D_{1}},\dots, \rho
_{D_{n}}$ as edges is in bijection with the set of irreducible
components of $D_{1}\cap\dots \cap D_{n}$. Each irreducible component
of an intersection of components of $D$ is called a \emph{stratum} of
the toroidal structure. Thus the set of strata is in bijection with
the set of cones. This bijection reverses dimensions and inclusions of
closures.
The conical polyhedral
complex $\Pi (X_{1},D)$ has a canonical integral structure and we
denote by $v_{D_{i}}$ the primitive generator of $\rho _{D_{i}}$.
If $(X_{2},\pi _{2},D')$ is a toroidal structure above $(X_{1},\pi _{1},D)$, then
there is a continuous map
\begin{displaymath}
r_{D,D'}\colon \Pi (X_{2},D')\to \Pi (X_{1},D),
\end{displaymath}
that sends cones to cones, is linear on each cone and is compatible
with the integral structure on each cone.
Moreover it is functorial in the sense that, if $(X_{3},\pi _{3},D'')$ is a toroidal structure above $(X_{2},\pi _{2},D')$, then
\begin{equation}\label{eq:17}
r_{D,D'}\circ r_{D',D''}=r_{D,D''}.
\end{equation}
The map $r_{D,D'}$ is constructed in greater generality in
\cite[Theorem 1.1]{ulirsch}, but the case we need admits a
particularly simple description as follows.
Let $\pi _{12}\colon
X_{2}\to X_{1}$ be the map of smooth modifications, $Y$ a stratum of
$(X_{2},D')$ and $Z$ the minimal stratum of $(X_{1},D)$
containing $\pi _{12}(Y)$. Let $D'_{1},\dots,D'_{r}$ be the set of
components of $D'$ such that $Y$ is an irreducible component of
$D'_{1}\cap\dots\cap D'_{r}$ and similarly let $D_{1},\dots, D_{s}$ be the set
of components of $D$ containing $Z$. Let $\sigma $ be the cone
corresponding to $Y$ and $\tau $ the cone corresponding to $Z$. Then
the restriction $r_{D,D'}|_{\sigma }\colon \sigma \to \tau$ is the
unique linear map satisfying
\begin{displaymath}
r_{D,D'}(v_{D'_{i}})=\sum
_{j=1}^{s}\ord_{D'_{i}}(\pi _{12}^{\ast}D_{j})v_{D_{j}}.
\end{displaymath}
It is clear that these maps for the different cones of $\Pi (X_{2},D')$
glue together to give a map with the listed properties.
To
prove the
functoriality \eqref{eq:17}, let $\pi _{23}\colon X_{3}\to X_{2}$ and
$\pi _{12}\colon X_{2}\to X_{1}$ be maps of modifications. Let $P$ be a
component of $D''$, let $D'_{1},\dots,D'_{r}$ be the set of components of $D'$
containing the image of $P$ in $X_{2}$ and let $D_{1},\dots,D_{s}$ be the
set of components
of $D$ containing the image of $P$ in $X_{1}$. For each $j$,
$\pi _{12}^{-1}(D_{j})\subset D'$, we have
\begin{displaymath}
\ord_{P}((\pi _{12}\circ\pi _{23})^{\ast}D_{j})=
\sum_{i=1}^{r}\ord_{P}((\pi _{23})^{\ast}D'_{i})
\ord_{D'_{i}}(\pi _{12}^{\ast}D_{j}).
\end{displaymath}
Therefore
\begin{align*}
r_{D,D''}(v_{P})
&=
\sum_{j=1}^{s}\ord_{P}((\pi _{12}\circ \pi _{23})^{\ast}D_{j})v_{D_{j}}\\
&=
\sum_{j=1}^{s}\sum _{i=1}^{r}\ord_{P}(\pi_{23}^{\ast}D'_{i})
\ord_{D'_{i}}(\pi_{12}^{\ast}D_{j})v_{D_{j}}\\
&=
\sum_{i=1}^{r}\ord_{P}(\pi_{23}^{\ast}D'_{i})
r_{D,D'}(v_{D'_{i}})\\
&=
r_{D,D'}\left(
\sum_{i=1}^{r}\ord_{P}(\pi_{23}^{\ast}D'_{i})v_{D'_{i}}
\right)\\
&=
r_{D,D'}\circ r_{D',D''}(P).
\end{align*}
Let $(X_1,\pi_1,D)$ be a toroidal structure on $X$.
Among the modifications of $X$ there is a distinguished class:
the \emph{allowable modifications} of the toroidal embedding $X_1 \setminus D \hookrightarrow
X_1$. See for instance \cite[Ch.~II, \S~2,
Definition~3]{toroidal} for the precise definition.
The allowable
modifications are in one-to-one correspondence with the subdivisions of the conical
polyhedral complex $\Pi
$ associated to $D$. Any such modification will be called toroidal (with respect to the
toroidal structure $D$). Note that, if $(X_{2},\pi _{2},D')$ is toroidal
with respect to $(X_{1},\pi _{1},D)$, then $D'$ is above
$D$ and the map $r_{D,D'}$ defined above induces a
homeomorphism between
the underlying topological spaces $|\Pi (X_{2},D')|\to |\Pi (X_{1},D)|$,
compatible with the fact that $\Pi (X_{2},D')$ is a subdivision of $\Pi
(X_{1},D)$.
We end this section with two definitions.
\begin{df}
Let $(X_1,\pi_1,D)$ be a smooth toroidal structure on $X$. A \emph{toroidal
exceptional prime divisor} of $(X_1,\pi_1,D)$ is an irreducible component of any
exceptional divisor of a toroidal modification of $(X_1,\pi_1,D)$.
\end{df}
\begin{df}
The \emph{set of rational points} $\Pi (X_{1},D)(\mathbb{Q})$ is the set of
points of $ \Pi (X_{1},D)$ that have rational coordinates when expressed in terms of the
primitive vectors $v_{D_{j}}$.
\end{df}
\subsection{b-divisors}
\label{sec:b-divisors}
In this section we discuss Weil and Cartier $\mathbb{R}$-b-divisors on $X$ as well as their
toroidal counterparts. This is essentially Shokurov's notion of
birational divisors (or b-divisors) \cite{sh-prel}. This section is purely algebraic, so, if preferred, the reader can work with finite-type algebraic varieties over any field of characteristic zero.
For other terminologies and properties concerning b-divisors we refer
to \cite{bff} and \cite{bfj-val} (see also \cite{BoteroBurgos} and
\cite{bo} for the toroidal and the toric cases, respectively).
We write $\Div\left(X\right)$ for the set of Weil divisors on $X$ with
real coefficients, viewed as a real vector space. We endow it with
the direct limit topology with respect to its finite dimensional
subspaces. Explicitly, a sequence of divisors $(D_{i})_{i\ge
0}$ converges to a divisor $D$ in $\Div\left(X\right)$ if there is a divisor $A$,
such that $\supp (D_{i})\subset \supp(A)$ for all $i\ge 0$ and
$(D_{i})_{i\ge 0}$ converges to $D$ in the finite dimensional vector
space of real divisors with support contained in $\supp(A)$.
\begin{df}\label{def:modi}
The set of \emph{models of $X$} is
\[
R(X) \coloneqq \left\{\pi \colon X_{\pi}\to X \; \big{|} \; \pi \text{
is a smooth modification}\right\}.
\]
If $(X_{1},\pi _{1},D)$ is a smooth toroidal structure on $X$ then the set
of \emph{toroidal models of $X$} (with respect to $D$) is
\[
R^{\text{\rm tor}}(X,D) \coloneqq \left\{\pi \colon X_{\pi}\to X \mid \pi \text{
is a smooth modification, toroidal w.r.t.} \, D \right\}.
\]
In particular if $\pi \in R^{\text{\rm tor}}(X,D)$ then it factors through
$\pi_{1}$.
\end{df}
We view both $R(X)$ and $R^{\text{\rm tor}}(X,D)$ as full subcategories of the
category of complex
varieties over $X$, in particular morphisms are over $X$. Maps of
models are unique if they exist, and are necessarily proper and
birational. Hironaka's resolution of singularities implies that $R(X)$ is a
directed set, where we set $\pi' \geq \pi$ if there exists a morphism
$\mu \colon X_{\pi'} \to X_{\pi}$. Similarly $R^{\text{\rm tor}}(X,D)$ is
directed by the existence of a smooth common refinement of any two
subdivisions.
Consider a pair $\pi' \geq \pi$ in $R(X)$, and let $\mu \colon X_{\pi'}\to X_{\pi}$
be the corresponding modification. We have a pullback map
\[
\mu^* \colon \Div\left(X_{\pi}\right) \longrightarrow \Div\left(X_{\pi'}\right)
\]
and a pushforward map
\[
\mu_* \colon \Div\left(X_{\pi'}\right) \longrightarrow \Div\left(X_{\pi}\right)
\]
between the associated divisor groups. Both maps are continuous.
\begin{df}\label{b-divisor}
The group of \emph{Cartier $\mathbb{R}$-b-divisors on} $X$ is the direct limit
\[
\operatorname{C-b-Div}_{\R}(X) \coloneqq \varinjlim_{\pi \in R(X)} \Div\left(X_{\pi}\right),
\]
in the category of topological vector spaces, with maps given by
the pullback maps. The resulting topology
is called the \emph{strong} topology.
The group of \emph{Weil $\mathbb{R}$-b-divisors on} $X$ is the inverse limit
\[
\operatorname{W-b-Div}_{\R}(X) \coloneqq \varprojlim_{\pi \in R(X)} \Div\left(X_{\pi}\right),
\]
in the category of topological vector spaces, with maps given by
the pushforward maps. The resulting topology
is called the \emph{weak} topology.
\end{df}
Usually we will denote $\mathbb{R}$-b-divisors with a blackboard bold
font ($\mathbb D$) in order to distinguish them from classical $\mathbb{R}$-divisors
that will be denoted with a slanted font ($D$).
\begin{rmk} \label{rem:2}
As a set, $\operatorname{C-b-Div}_{\R}(X)$ can be seen as the disjoint union of the sets
$\Div\left(X_{\pi}\right)$ modulo the equivalence relation which sets two
divisors equal if they coincide after pullback to a common modification.
The set $\operatorname{W-b-Div}_{\R}(X)$ can be seen as the subset of $\prod_{\pi\in R(X)
}\Div(X_{\pi})$ given by the elements $\mathbb{D}=\left(D_{\pi }\right)_{\pi \in
R(X)}$ satisfying
the compatibility condition that for each $\pi '\ge \pi $ we have
$\mu _{\ast}D_{\pi '}=D_{\pi }$, where $\mu
$ is the corresponding modification.
\end{rmk}
For any variety $Y$ and normal
crossings divisor $E$ on $Y$, we denote by $\Div(Y,E)$ the real vector
space of $\mathbb{R}$-divisors on $Y$ whose
support is contained in $E$. We endow it with the Euclidean topology.
Let $(X_{1},\pi _{1},D)$ be a toroidal structure; we make analogous definitions of b-divisors.
\begin{df}\label{t-b-divisor}
The group of \emph{toroidal Cartier $\mathbb{R}$-b-divisors on} $X$ (with
respect to $D$) is the direct limit
\[
\operatorname{C-b-Div}_{\R}(X,D)^{\text{\rm tor}} \coloneqq \varinjlim_{\pi \in R^{\text{\rm tor}}(X,D)}
\Div\left(X_{\pi},\mu _{\pi }^{-1}(D)\right),
\]
where $\mu _{\pi }\colon X_{\pi }\to X_{1}$ is the unique map of
modifications.
The limit is again in the category of topological vector spaces,
with maps given by
the pullback maps.
The group of \emph{toroidal Weil $\mathbb{R}$-b-divisors on} $X$ (with
respect to $D$) is the inverse limit
\[
\operatorname{W-b-Div}_{\R}(X,D)^{\text{\rm tor}}\coloneqq \varprojlim_{\pi \in R^{\text{\rm tor}}(X,D)}
\Div\left(X_{\pi},\mu_{\pi } ^{-1}(D)\right),
\]
also in the category of topological vector spaces, with maps given by
the pushforward maps.
\end{df}
We recall from \cite{BoteroBurgos} that the space
$\operatorname{W-b-Div}_{\R}(X,D)^{\text{\rm tor}}$ can be identified with the space of all conical
real valued functions on $\Pi (X_{1},D)(\mathbb{Q})$, while
$\operatorname{C-b-Div}_{\R}(X,D)^{\text{\rm tor}}$ can be identified with the space of all continuous conical piecewise
linear functions on $\Pi (X_{1},D)$ with rational domains of linearity.
Clearly $R(X,D)^{\text{\rm tor}} \subset R(X)$. Moreover, for $\pi \in R(X,D)^{\text{\rm tor}}$
there are canonical maps
\begin{align*}
&\Div\left(X_{\pi},\mu _{\pi }^{-1}(D)\right)\longrightarrow
\Div\left(X_{\pi}\right),\\
&\Div\left(X_{\pi}\right)\longrightarrow
\Div\left(X_{\pi},\mu _{\pi }^{-1}(D)\right),
\end{align*}
where the last one sends any prime divisor not contained in
$\mu _{\pi }^{-1}(D)$ to zero. Hence
there is a canonical inclusion
\begin{displaymath}
\operatorname{C-b-Div}_{\R}(X,D)^{\text{\rm tor}}\longrightarrow \operatorname{C-b-Div}_{\R}(X)
\end{displaymath}
and a canonical projection
\begin{displaymath}
\operatorname{W-b-Div}_{\R}(X) \longrightarrow \operatorname{W-b-Div}_{\R}(X,D)^{\text{\rm tor}}.
\end{displaymath}
We now construct a section
\begin{equation}\label{eq:section}
\iota\colon \operatorname{W-b-Div}_{\R}(X,D)^{\text{\rm tor}}\longrightarrow \operatorname{W-b-Div}_{\R}(X),
\end{equation}
of the canonical projection.
Let $\mathbb{D}\in \operatorname{W-b-Div}_{\R}(X,D)^{\text{\rm tor}}$ be a toroidal Weil b-divisor and choose
a smooth modification $\pi\in R(X)$ above $\pi_{1} $ and a
prime divisor $P$ of $X_{\pi }$. We have to define $\ord_{P}(\iota
(\mathbb{D}))$ for any such $P$.
Let $\varphi_{\mathbb{D}}$ be the conical function on $\Pi (X_{\pi_{1} },D)(\mathbb{Q})$
corresponding to the Weil b-divisor $\mathbb{D}$. The function $\varphi_{\mathbb{D}}$
is characterized by the following property: If $E$ is a toroidal
exceptional prime
divisor in some toroidal modification of $(X_{\pi_{1} },D)$, and
$v_{E}\in \Pi (X_{\pi _{1}},D)$
is the corresponding primitive vector, then
\begin{displaymath}
\ord_{E}\mathbb{D} = -\varphi_{\mathbb{D}}(v_{E}).
\end{displaymath}
Let $(X_{2},\pi _{2},D')$ be a toroidal structure above $(X_{1},\pi _{1},D)$ that
has a map $f\colon X_{2}\to X_{\pi}$ over $X$ and such that
$f^{-1}(P)\subset D'$. Let $\widehat P_2$ be the strict transform of
$P$ in $X_{2}$. It is an irreducible component of $D'$. Let
$v_{\widehat P_2}\in \Pi (X_{2},D')$ be the primitive vector corresponding to
$\widehat P_2$. Then we define
\begin{displaymath}
\ord_{\iota(\mathbb{D})}P= -\varphi_{\mathbb{D}}(r_{D,D'}(v_{\widehat P_2})).
\end{displaymath}
To see that this is independent of the choices, let $(X_{3},\pi
_{3},D'')$ be another toroidal structure above $D$ and let
$\widehat P_{3}$ be the strict transform of $P$ in $X_{3}$ and
$v_{\widehat P_{3}}$ the corresponding primitive vector. Since
$r_{D',D''}(v_{\widehat P_{3}})=v_{\widehat P_2}$, from \eqref{eq:17} we obtain
\begin{displaymath}
r_{D,D''}(v_{\widehat P_{3}})= r_{D,D'}(v_{\widehat P_2}).
\end{displaymath}
Thus $\ord_{\iota(\mathbb{D})}(P)$ does not depend on the choice of $D'$.
We will use the following terminology.
\begin{df}\label{def:toroidal-b} A b-divisor $\mathbb{D}\in \operatorname{W-b-Div}_{\R}(X)$ is called
\emph{toroidal with respect to $(X_{1},\pi _{1},D)$} if it belongs to
$\iota(\operatorname{W-b-Div}_{\R}(X,D)^{\text{\rm tor}})$. It is called \emph{toroidal} if it is
toroidal with respect to some toroidal structure. This gives rise to the
set of toroidal b-divisors
\begin{displaymath}
\operatorname{W-b-Div}_{\R}(X)^{\text{\rm tor}}=\lim_{\substack{\longrightarrow\\(X_{1},\pi _{1},D)}}
\iota\left(\operatorname{W-b-Div}_{\R}(X,D)^{\text{\rm tor}}\right) \subset \operatorname{W-b-Div}_{\R}(X).
\end{displaymath}
Here the direct limit is taken with respect to the order $(X_{2},\pi
_{2},D') \geq (X_{1},\pi _{1},D)$ iff $D'$ is above $D$ and the
maps are just the pullback maps of divisors along the unique map of
modifications.
\end{df}
\begin{rmk}
By resolution of singularities, every Cartier b-divisor is toroidal
with respect to some toroidal structure. So, toroidal b-divisors
are in between Cartier and Weil b-divisors.
\end{rmk}
\begin{df}
A Cartier b-divisor is called \emph{nef} if it is nef in any
modification of $X$ where it is determined, and a Weil divisor is
called nef if it belongs to the closure (with respect to the weak topology) of
the space of nef Cartier b-divisors.
\end{df}
The following is a recent result by Dang and Favre:
\begin{thm}[{\cite[Theorem A]{Da-Fa20}}]
Any nef b-divisor is the limit of a decreasing sequence of nef Cartier b-divisors.
\end{thm}
\begin{rmk}
If moreover the nef b-divisor is toroidal then by \cite[Lemma 5.9]{BoteroBurgos} the sequence can be taken to consist of toroidal nef Cartier divisors (with respect to the same toroidal structure).
\end{rmk}
\begin{rmk} \label{rk:degree} Using any approximating decreasing sequence of nef Cartier b-divisors, each nef b-divisor $\mathbb{D}$ on $X$ has a well-defined degree $\mathbb{D}^n$ in $\mathbb{R}_{\ge 0}$; see \cite[Theorem~3.2]{Da-Fa20}.
\end{rmk}
\subsection{Psh functions, psh metrics, Lelong numbers}
\label{sec:psh-metrics-lelong}
We briefly recall the notion of plurisubharmonic (psh) functions and metrics. A more thorough
discussion with examples can be found in \cite{BBHJ}.
The following definition is taken from \cite[Chapter 3]{ochia}.
Let $X$ be a complex manifold.
\begin{df}\label{def:5}
Let $U$ be an open coordinate subset of $X$, identified with an
open subset of $\mathbb{C}^{n}$.
A function $\varphi\colon U\to
\mathbb{R}\cup \{-\infty\}$ is called \emph{plurisubharmonic (psh)} if
the following two conditions are satisfied:
\begin{enumerate}
\item the function $\varphi$ is upper-semicontinuous and not identically equal to
$-\infty$ on any connected component of $U$;
\item for every $z\in U$ and every $a\in \mathbb{C}^{n}$ the function
\begin{displaymath}
\mathbb{C} \to \mathbb{C} \, , \qquad \zeta \longmapsto \varphi(z+a\zeta)\in \mathbb{R}\cup \{-\infty\}
\end{displaymath}
is either identically $- \infty$, or subharmonic in each connected
component of the open set $\{\zeta \in \mathbb{C}\mid z+a\zeta\in
U\}$.
\end{enumerate}
Now let $U \subset X$ be an arbitrary open subset. A function $\varphi \colon U\to
\mathbb{R}\cup \{-\infty\}$ is called $psh$ if $U$ can be covered by open coordinate subsets
$U_{i}$ so that each restriction $\varphi|_{U_{i}}$ is psh.
\end{df}
We next discuss the notion of psh metrics. Let $L$ be a line bundle on $X$.
\begin{df} \label{def:psh}
Let
$\{(U_{i},s_{i})\}$ be a trivialization of $L$, with
transition functions $\{g_{ij}\}$.
A \emph{hermitian metric} $h$ on $L$ is a collection
of
measurable functions
\begin{displaymath}
\varphi_i \colon U_i \to \mathbb{R} \cup
\left\{\pm \infty\right\} \, ,
\end{displaymath}
such that the identities
\begin{equation}\label{eq:10}
e^{-\varphi_i} = |g_{ij}|e^{-\varphi_j}
\end{equation}
hold on all $U_i \cap U_j$. The \emph{norm} $h(s_i)$ is given by the formula
\begin{displaymath}
\varphi_{i}(z)=-\log h(s_{i}(z)),\quad z\in U_{i} \, .
\end{displaymath}
From the identities (\ref{eq:10}) we find
\begin{displaymath}
\log h(s_{i})- \log h(s_{j})=\log|s_{i}/s_{j}|
\end{displaymath}
on $U_i \cap U_j$.
More generally, when $s$ is any generating section of $L$ locally near a point $z \in X$ we define its norm $h(s)$
via
\begin{displaymath}
\log h(s(z))=\log|s(z)/s_{i}(z)| + \log h(s_{i}(z))=
\log|s(z)/s_{i}(z)| - \varphi_{i}(z)
\end{displaymath}
whenever $z\in U_{i}$. This is easily seen to be independent of the choice of $i$.
We call the metric $h$ \emph{singular}
(resp.~\emph{psh}, \emph{continuous}, \emph{smooth}) if the functions
$\varphi_i$ are all locally integrable (resp.~psh, continuous,
smooth).
\end{df}
An important measure of the singularities of a psh function is given by its Lelong
numbers.
\begin{df}\label{def:lelong_number}
Let $U\subset X$ be an open coordinate subset, let $\varphi$ be a psh
function on $U$, and let $x \in U$ be a point.
Then the \emph{Lelong number} of $\varphi$ at $x$ is given as
\[
\nu(\varphi,x) = \sup\left\{\gamma \geq 0
\; \big{|} \; \varphi(z) \leq \gamma
\log |z-x| + O(1) \; \text{near} \; x \right\}.
\]
The notion of Lelong number readily generalizes to the context of psh metrics. Let $L$ be a line bundle on $X$ equipped with a psh metric $h$ and let $x\in
X$ be a point. Choose an open coordinate subset $x\in U\subset X$ and a generating section
$s$ of $L$ on $U$. Then we put
\begin{displaymath}
\nu (h,x) = \nu(-\log h(s),x).
\end{displaymath}
It is readily verified that this is independent of the choice of open set $U$ and generating section $s$.
\end{df}
Let now $\pi \colon X'\to X$ be a smooth modification of $X$. Let
$(L,h)$ be a line bundle with a psh metric on $X$. It is easy to see that then
$(\pi^{\ast}L,\pi^{\ast}h)$ is a line bundle with a psh metric on
$X'$. For $x\in X'$ we then put
\begin{displaymath}
\nu (h,x)=\nu (\pi^{\ast}h,x) \, .
\end{displaymath}
We note that if $P$ is a prime divisor of $X'$ and $x,y$ are very general
points on $P$, then the equality
\begin{displaymath}
\nu (h,x)=\nu (h,y)
\end{displaymath}
holds. This leads to the following definition.
\begin{df}\label{def:2}
Let $(L,h)$ be a line bundle equipped with a psh metric on $X$. Let
$\pi \in R(X)$ be a smooth modification, and let $P$ be a prime divisor of $X_{\pi }$.
Then we write
\begin{displaymath}
\nu(h,P)=\inf_{x\in P}\nu(h,x) \, .
\end{displaymath}
\end{df}
The number $\nu(h,P)$ from Definition~\ref{def:2} allows the following alternative description.
\begin{lem}\label{lemm:8}
Let $(L,h)$ be a line bundle provided with a psh metric on $X$,
let $\pi \in R(X)$ and let $P$ be a prime
divisor of $X_{\pi }$. For every point
$x\in P$, for every generating section $s$ of $L$ around $x$ and
for every local equation $g$ for $P$ at $x$, the equality
\begin{displaymath}
\nu(h,P) = \sup\{\gamma \ge 0 \mid -\log(h(s))-\gamma \log| g| \text{
bounded above near $x$}\}
\end{displaymath}
holds.
\end{lem}
\begin{proof}
This follows from \cite[Proposition~10.5]{boucksom:singpsh}.
\end{proof}
\subsection{Some notions of convex analysis}
\label{sec:convex-analysis}
In this section we recall some notions from convex analysis and prove
some lemmas that will be useful later. Our basic reference for
convex analysis is \cite{rockafellar70:_convex_analy}.
We fix a finite dimensional real vector space $N_{\mathbb{R}}$ and let $M_{\mathbb{R}}$ be its
dual. We recall the definition of closed convex function
(see \cite[Theorem 7.1]{rockafellar70:_convex_analy}).
\begin{df}
Let $C\subseteq N_{\mathbb{R}}$ be a convex set. A convex function $f\colon C\to \mathbb{R}\cup \{\infty\}$ is called a
\emph{closed convex} function if the equivalent conditions
\begin{enumerate}
\item the epigraph $\{(x,y)\in C\times \mathbb{R}\mid y\ge f(x)\}\subset
N_{\mathbb{R}}\times \mathbb{R}$ is
a closed subset;
\item \label{item:15} the function $\widetilde f\colon N_{\mathbb{R}}\to
\mathbb{R}\cup \{\infty\}$, given by
$\widetilde f(x)=f(x)$ for $x\in C$ and $\widetilde f(x)=\infty$
if $x\not \in C$ is lower semicontinuous;
\end{enumerate}
are satisfied.
\end{df}
Note that if $C$ is closed then condition \ref{item:15} is equivalent
to $f$ being lower semicontinuous.
\begin{df}
Let $C\subset N_{\mathbb{R}}$ be a convex
set. The \emph{recession cone} of $C$ is the set of
direction vectors of all rays contained in $C$:
\begin{displaymath}
\rec(C) = \{y\in N_{\mathbb{R}}\mid C+y\subset C\}.
\end{displaymath}
The set $\rec(C)$ is a convex cone, which is closed if $C$ is
closed, and polyhedral if $C$ is polyhedral.
\end{df}
\begin{df}
Let $C\subset N_{\mathbb{R}}$ be a convex set and $\sigma =\rec(C)$ its recession
cone. Let $g\colon C\to \mathbb{R}$ be a closed convex function on
$C$. The \emph{recession function} of $g$ is the function
$\rec(g)\colon \sigma \to \mathbb{R}\cup\{\infty\}$ given, for any $x\in C$
and any $y\in \sigma $ by
\begin{equation}\label{eq:5}
\rec(g)(y)=\lim_{\lambda \to +\infty}\frac{g(x+\lambda y)-g(x)}{\lambda }.
\end{equation}
\end{df}
\begin{lem} \label{lemm:5} Let $C\subset N_{\mathbb{R}}$ be a convex set,
$\sigma =\rec(C)$
its recession
cone and $g\colon C\to \mathbb{R}$ a closed convex function on
$C$.
\begin{enumerate}
\item \label{item:2} The recession function $\rec(g)$ is
well defined. That is, the limit \eqref{eq:5} exists and does not
depend on $x$. Moreover $\rec(g)$ is a closed convex function.
\item \label{item:3} If the function $g$ is bounded above, then
$\rec(g)$ takes non-positive values. In particular it takes finite
values.
\item \label{item:4} If the function $g$ is Lipschitz
continuous with constant $\alpha $ then $\rec(g)$ is also
Lipschitz continuous with the same constant.
\end{enumerate}
\end{lem}
\begin{proof}
The statement in (\ref{item:2}) is the content of
\cite[Theorem~8.5]{rockafellar70:_convex_analy}.
To prove (\ref{item:3}), let $x\in C$ and $y\in \rec(C)$ and consider
two real numbers $\lambda ,\mu >0$. We have
\begin{displaymath}
x+\lambda y =\frac{\mu }{\lambda +\mu }x +
\frac{\lambda }{\lambda +\mu }(x+(\lambda +\mu )y).
\end{displaymath}
The convexity of $g$ yields
\begin{equation}\label{eq:4}
g(x+\lambda y )\le \frac{\mu }{\lambda +\mu }g(x) +
\frac{\lambda }{\lambda +\mu }g(x+(\lambda +\mu )y).
\end{equation}
Since $g$ is assumed to be bounded above, say by a
constant $B$, equation \eqref{eq:4} implies
\begin{displaymath}
g(x+\lambda y)\le
\inf_{\mu} \frac{\mu }{\lambda +\mu }g(x) +
\frac{\lambda }{\lambda +\mu } B = g(x),
\end{displaymath}
which implies that $\rec(g)$ takes non-positive values, and in
particular finite values.
Statement (\ref{item:4}) follows from the computation
\begin{displaymath}
|\rec(g)(u)-\rec(g)(u')| = \lim_{\lambda \to
+\infty}\frac{1}{\lambda }
|g(v+\lambda u)-g(v+\lambda u')|\le \alpha \|u-u'\|.
\end{displaymath}
\end{proof}
It is clear that $\rec(g)$ is conical, in the sense that, for all real
$\lambda \ge 0$ and all $y\in \sigma $
\begin{equation}
\label{eq:6}
\rec(g)(\lambda y)=\lambda \rec(g)(y).
\end{equation}
Recall that, if $\sigma \subset N_{\mathbb{R}}$ is a cone, the dual cone $\sigma
^{\vee}\subset M_{\mathbb{R}}$ is given by
\begin{displaymath}
\sigma ^{\vee}=\{m\in M_{\mathbb{R}}\mid m(x)\ge 0,\ \forall x\in \sigma \}.
\end{displaymath}
\begin{lem}\label{lemm:15}
Let $C$ be a convex set and $\sigma =\rec(C)$ its recession
cone. Let $g\colon C\to \mathbb{R}$ be a bounded-above closed convex
function on
$C$ and $\rec(g)$ its recession function.
\begin{enumerate}
\item \label{item:5} For every $x\in C$ the function on $\sigma $ given by
\begin{displaymath}
y\longmapsto g(x+y)-\rec(g)(y)
\end{displaymath}
is bounded above.
\item \label{item:6} Assume that $C$ is closed and $g$ is Lipschitz continuous. Then,
for every $m\in \inter(\sigma^{\vee}) $ the function
\begin{displaymath}
y\longmapsto g(x+y)-\rec(g)(y)+m(y)
\end{displaymath}
is bounded below.
\item \label{item:7} Assume that $\sigma $ is full dimensional. Then
for every $0\not = m\in \sigma^{\vee}$ the function
\begin{displaymath}
y\longmapsto g(x+y)-\rec(g)(y)+m(y)
\end{displaymath}
is not bounded above.
\end{enumerate}
\end{lem}
\begin{proof}
Let $\lambda >0$. The convexity of $g$ yields
\begin{displaymath}
g(x+y)\le \frac{1}{\lambda }g(x+\lambda
y)+\left(1-\frac{1}{\lambda }\right)g(x)
\end{displaymath}
which implies
\begin{displaymath}
g(x+y)-g(x)\le \frac{g(x+\lambda y)-g(x)}{\lambda }.
\end{displaymath}
Hence $g(x+y)-\rec(g)(y)\le g(x)$ proving (\ref{item:5}).
We prove (\ref{item:6}) by contradiction. If the function in
(\ref{item:6}) is not bounded below, we can find a sequence of points
$y_{i}\in \sigma $ for $i\ge 1$ satisfying $\|y_{i}\|\ge i$ and
\begin{equation}\label{eq:7}
g(x+y_{i})-\rec(g)(y_{i})+m(y_{i})\le -i.
\end{equation}
After choosing a subsequence we can further assume that the sequence
$(y_{i}/\|y_{i}\|)$ converges to a point $y_{0}\in \sigma $. We
write $\lambda _{i}=\|y_{i}\|$ and $y_{i}^{0}=y_{i}/\lambda _{i}$.
Equation \eqref{eq:7} implies that
\begin{equation}\label{eq:8}
\limsup _{i\to \infty} \frac{1}{\lambda _{i}}g(x+\lambda
_{i}y^{0}_{i})-\rec(g)(y^{0}_{i})+m(y^{0}_{i})\le 0.
\end{equation}
Using that $g$ is Lipschitz continuous we get
\begin{equation}
\label{eq:9}
\lim_{i\to \infty}
\frac{g(x+\lambda_{i}y^{0}_{i})-g(x+\lambda_{i}y_{0})}{\lambda
_{i}}=0.
\end{equation}
By Lemma \ref{lemm:5}.\ref{item:3} the function $\rec(g)$ is (Lipschitz)
continuous. Since $m$ is also continuous we obtain
\begin{equation}
\label{eq:11}
\lim_{i\to \infty}-\rec(g)(y^{0}_{i})+m(y^{0}_{i})
+\rec(g)(y_{0})-m(y_{0})=0.
\end{equation}
By Lemma \ref{lemm:5}.\ref{item:2} we know that
\begin{equation}
\label{eq:12}
\lim_{i\to \infty} \frac{1}{\lambda _{i}}g(x+\lambda_{i}y_{0})-\rec(g)(y_{0})=0.
\end{equation}
Since $m\in \inter(\sigma ^{\vee})$ and $0\not= y_{0}\in \sigma $ we
deduce
\begin{equation}
\label{eq:13}
m(y_{0})>0.
\end{equation}
Summing up equations \eqref{eq:9} to \eqref{eq:13} we obtain
\begin{displaymath}
\lim_{i\to \infty} \frac{1}{\lambda _{i}}g(x+\lambda
_{i}y^{0}_{i})-\rec(g)(y^{0}_{i})+m(y^{0}_{i})= m(y_{0})>0,
\end{displaymath}
contradicting the inequality \eqref{eq:8}.
We next prove (\ref{item:7}). The fact that $\sigma $ is full
dimensional implies that $\{0\}$ is a face of $\sigma ^{\vee}$. Since
$0\not = m\in \sigma^{\vee}$, there
is a $y_{0}\in \sigma $ such that $m(y_{0}) = \varepsilon >0$. Fix
$x\in C$ and consider the one variable bounded above convex function
\begin{displaymath}
g_{0}(t)=g(x+ty_{0})
\end{displaymath}
Since $g_{0}$ is a bounded above convex function it is Lipschitz
continuous for $t\ge 1$. The recession function of $g_{0}$ is
\begin{displaymath}
\rec(g_{0})(t)=\rec(g)(t y_{0}).
\end{displaymath}
We can apply (\ref{item:6}) to this function to
deduce that
\begin{math}
g_{0}(t)-\rec(g_{0})(t) + (\epsilon /2) t
\end{math}
is bounded below. Therefore
\begin{displaymath}
g(x+ty_{0})-\rec(g)(t y_{0})+ m(ty_{0})=
g_{0}(t)-\rec(g_{0})(t) + \epsilon t
\end{displaymath}
can not be bounded above.
\end{proof}
\subsection{Toroidal psh metrics and their Lelong numbers}
\label{sec:toroidal-psh-metrics}
We fix now a toroidal structure $(X_{1},\pi _{1},D)$ on $X$ and write $U=X
_{1}\setminus D$. Recall that we can view $U$ as an open subset of $X$.
\begin{df}\label{def:9} (Compare with \cite[Definition~3.10]{BBHJ})
A psh metric $h$ on a line bundle $L$ on $X$ is called \emph{toroidal with
respect to $D$} if the following is satisfied.
\begin{itemize}
\item $h$ is locally bounded on $U$.
\item for every
point $p\in D$ there is an open coordinate polydisc $W$ with coordinates
$(z_{1},\dots,z_{d})$ with $|z_{i}|< e^{-c}$ for some constant $c$ such that
\begin{displaymath}
W\cap D=\{z_{1}\cdots z_{r}=0\},
\end{displaymath}
a generating section $s$ of $L$ on $W$, a bounded function $\gamma $
on $W$ and a bounded above convex Lipschitz continuous function $g$ on the cone $\mathbb{R}_{\ge
c}^{r}$
such that
\begin{displaymath}
-\log h(s)(z_{1},\dots,z_{d})=
\gamma (z_{1},\dots,z_{d}) + g(-\log|z_{1}|,\dots,-\log|z_{r}|).
\end{displaymath}
\end{itemize}
We call $D$ \emph{a singularity divisor of} $h$.
\end{df}
\begin{rmk}
If $h$ is toroidal with respect to $D$ and $D'$ is a toroidal
structure above $D$ then $h$ is also toroidal with respect to $D'$.
\end{rmk}
Let $L$ be a line bundle with toroidal psh metric $h$ (with respect to $D$).
We now explain how to compute the Lelong numbers of $h$
along toroidal exceptional divisors.
Let $\pi\in R(X,D)^{\text{\rm tor} }$ be a toroidal model of $(X,D)$, $\mu \colon
X_{\pi }\to X_{1}$ the corresponding map of modifications and $P$ a
prime toroidal exceptional divisor in $X_{\pi }$. Then $P$ corresponds
to a rational
ray $\rho _{P}$ in a cone $\sigma$ of $\Pi (X_{1},D)$. Let $y\in P$ be a
generic point and let $x\in X_{1}$ be
the image of $y$. Let $W$ be a coordinate neighborhood of
$x$ as in Definition \ref{def:9}, $s$ a generating section of $L$
around $x$ and $g$ the convex function of the
same definition. The cone $\sigma $ is smooth and can be identified
with $\mathbb{R}_{\ge 0}^{r}$ using its integral structure. This
identification is canonical up to the ordering of the variables that
can be fixed by the choice of coordinates. Then the function
$\rec(g)$ is canonically a function on $\sigma $.
\begin{lem} \label{lemm:16}
Let $v_{P }$ be the primitive generator of $\rho _{P}$. Then
\begin{displaymath}
\nu (h,P)=-\rec(g)(v_{P})
\end{displaymath}
\end{lem}
\begin{proof}
Choose a small coordinate neighborhood $V$ of $y$ with coordinates
$(x_{1},\dots,x_{d})$ such that $x_{1}$ is a local equation for
$P$. Since $y$ is generic and $V$ small we can assume that
\begin{equation}
\label{eq:14}
\pi ^{-1}(D)\cap V = P\cap V.
\end{equation}
Let $v_{\rho }=(a_{1},\dots,a_{r})$. Then the map $\pi $ can be
written as
\begin{displaymath}
\pi
(x_{1},\dots,x_{d})=
(x_{1}^{a_{1}}u_{1},\dots,x_{1}^{a_{r}}u_{r},u_{r+1},\dots,u_{d}),
\end{displaymath}
where the $u_{i}$'s are functions on $V$. The condition \eqref{eq:14}
implies that $u_{i}$ does not vanish on $V$ for $i=1,\dots,r$. Then
\begin{displaymath}
-\log h(s) = \gamma
+g(-\log|u_{1}|-a_{1}\log|x_{1}|,\dots,-\log|u_{r}|-a_{r}\log|x_{1}|).
\end{displaymath}
Then Lemma \ref{lemm:15}.\ref{item:5} implies that
\begin{displaymath}
-\log h(s) -\rec(g)(v_{\rho })(-\log|x_{1}|)
\end{displaymath}
is bounded above, but by Lemma \ref{lemm:15}.\ref{item:7} for every $\epsilon >0$
\begin{displaymath}
-\log h(s) -\rec(g)(v_{\rho })(-\log|x_{1}|)+\varepsilon (-\log|x_{1}|)
\end{displaymath}
is not bounded above.
By Lemma \ref{lemm:8} we obtain that
\begin{displaymath}
\nu (h,P)=-\rec(g)(v_{\rho })
\end{displaymath}
as required.
\end{proof}
\section{Psh metrics, b-divisors and graded linear series}
\label{sec:psh-metrics-b}
In this section we relate graded linear series, b-divisors and psh
metrics to one another. Recall that $X$ denotes the
complex manifold associated to a smooth projective variety over $\mathbb{C}$
of dimension $d$.
\subsection{From psh-metrics to graded linear series}
\label{sec:psh-metrics-b-1}
Let $F=K(X)$ denote the field of rational functions on $X$. Let
$U\subset X$ be a dense Zariski open
subset and $(L, h)$ a line bundle on $X$ with a
psh metric (see Definition \ref{def:psh}). We assume furthermore that $(L, h)|_{U}$ is
locally bounded (for example continuous or smooth).
Following Nystr\"om \cite[Section~2.8]{Nystrom_growth} we define
\begin{displaymath}
H^{0}(X,L,h)
= \{s\in H^{0}(X,L)\mid h(s) \text{ bounded}\}.
\end{displaymath}
For $s$ a non-zero rational section of $L$ we define
\begin{align*}
\mathcal{L}(L,s,h) &= \{f\in F\mid fs\in H^{0}(X,L, h)\},\\
\mathcal{R}(L,s,h)
&=\bigoplus_{\ell \in \mathbb Z_{\ge 0}}
\mathcal{L}(L^{\otimes\ell},s^{\otimes\ell},h^{\otimes\ell})t^{\ell}
\subset F[t] \, .
\end{align*}
Note that the latter is a graded linear series in the sense of Definition \ref{def:1}.
\subsection{From graded linear series to b-divisors}
\label{sec:from-graded-linear}
Let $A=\bigoplus_{\ell \geq 0} A_{\ell}t^{\ell}\subset F[t]$ be a
graded linear series of almost integral type, $\pi \in R(X)$ and
$X_\pi$ the corresponding model of $X$.
\begin{lem}\label{lemm:6} The set
\begin{displaymath}
S= \{D\in \Div(X_{\pi })\; \big{|}\; \forall \ell \ge 0,\ \forall f\in A_{\ell}\setminus\{0\}, \ell D+\dv(f)\ge 0\}
\end{displaymath}
is not empty.
\end{lem}
\begin{proof}
Since $A$ is of almost integral type there is a graded linear series
$A'$ of integral type containing $A$. Every graded linear series of
integral type is finitely generated. Therefore there are nonzero elements
$f_{i}\in F$, $i=1,\dots,r$ and positive integers $\ell_i$, $i=1,\dots,r$ such that $A$ is contained in the
subalgebra of $F[t]$ generated by the set $f_{i}t^{\ell_{i}}$, $i=1,\dots,r$.
Therefore
\begin{displaymath}
S\supset \{D\in \Div(X_{\pi })\; \big{|}\; \ell_{i} D+\dv(f_{i})\ge 0,\
i=1,\dots,r\}. \qedhere
\end{displaymath}
\end{proof}
\begin{df}\label{def:3}
Let $A$ be a graded linear series on $X$ of almost integral type. We
define
\begin{displaymath}
\bdiv(A)=(\bdiv(A)_{\pi })_{\pi \in R(X)}\in \operatorname{W-b-Div}_{\R}(X)
\end{displaymath}
by
\begin{displaymath}
\bdiv(A)_{\pi }=\inf \{E\in \Div(X_{\pi })\; \big{|}\;
\ell E+\dv(f)\ge 0,\
\forall \ell\ge 0\ \forall f\in A_{\ell}\},
\end{displaymath}
where the infimum is defined componentwise. In other words, for every
prime divisor $P$ on $X_{\pi }$ we put
\begin{displaymath}
\ord_{P}(\bdiv(A)_{\pi })=\sup \left\{\frac{-1}{\ell}\ord_{P}(f) \mid \ell \ge 0,
f\in A_{\ell}\right\}.
\end{displaymath}
\end{df}
Note that $\bdiv(A)$ is well defined thanks to Lemma
\ref{lemm:6}.
\subsection{From psh metrics to b-divisors}
\label{sec:from-psh-metrics}
\begin{df}\label{def:4}
Let $(L, h)$ be a line bundle on $X$ together with a psh metric such that
there is a dense open Zariski subset $U \subset X$ on which $h$ is locally
bounded,
and let $s$ be a non-zero rational section of $L$. The Weil
$\mathbb{R}$-b-divisor $ \mathbb{D}(L,s,h)$ is defined, for every $\pi \in R(X)$
and prime divisor $P$ on $X_{\pi }$, by
\begin{displaymath}
\ord_{P} \mathbb{D}(L,s,h)_{\pi }
=
\ord_{P} \dv(s) - \nu(h,P).
\end{displaymath}
\end{df}
Here $\nu(h,P)$ denotes the Lelong number from Definition
\ref{def:lelong_number}.
\begin{prop}\label{prop:3}
The b-divisor $ \mathbb{D}(L,s,h)$ is nef.
\end{prop}
\begin{proof}
This is proved in \cite[Theorem 5.18]{BBHJ}.
\end{proof}
\begin{prop}\label{prop:div_of_toroidal_metric_is_toroidal}
Let $(X_{1},\pi _{1},D)$ be a toroidal structure on $X$. Assume that the
psh metric $h$
is toroidal with respect to $D$ in the sense of Definition~\ref{def:9}.
\begin{enumerate}
\item\label{i:1} The b-divisor $\mathbb{D}=\mathbb{D}(L,s,h)-\dv(s)$ is toroidal with respect
to $D$. Note that this divisor only depends on the singularities
of $h$ and not on the particular section $s$.
\item\label{i:2} Let $\sigma $ be a cone of $\Pi
(X_{1},D)$ and $Y$ the corresponding stratum. Let $\varphi
_{\mathbb{D}}$ be the conical function on $\Pi
(X_{1 },D)$ corresponding to $\mathbb{D}$. Choose a
generic point $x$ of $Y$ and a small enough coordinate neighborhood $W$ of $x$
with $W\cap D$ given by the equation $z_{1}\cdots z_{r}=0$. After
changing $s$ if needed, we can assume
that $s$ is a generating section in $W$. Therefore, by the
definition of toroidal psh metric, there is
a bounded above convex Lipschitz continuous function $g$ on a cone $\mathbb{R}_{\ge
c}^{r}$
such that
\begin{displaymath}
-\log h(s)(z_{1},\dots,z_{d}) -
g(-\log|z_{1}|,\dots,-\log|z_{r}|)
\end{displaymath}
is bounded on $W \cap D$. Then
\begin{displaymath}
\varphi_{\mathbb{D}}|_{\sigma }=-\rec(g).
\end{displaymath}
\end{enumerate}
\end{prop}
\begin{proof}
Let $\mathbb{D}'$ be the projection of $\mathbb{D}$ onto
$\operatorname{W-b-Div}_{\R}(X,D)^{\text{\rm tor}}$. In order to prove \eqref{i:1} we have to show
that
\begin{equation}\label{eq:20}
\iota(\mathbb{D}')=\mathbb{D},
\end{equation}
where $\iota$ is the section \eqref{eq:section}
from toroidal b-divisors to general b-divisors. And for \eqref{i:2} we
have to show that
\begin{equation}\label{eq:19}
\varphi_{\mathbb{D}'}|_{\sigma } = -\rec(g).
\end{equation}
In view of the definition of $\mathbb{D}(L,s,h)$ using Lelong numbers,
equation \eqref{eq:19} is just a reformulation of Lemma
\ref{lemm:16}. So it only remains to prove equation \eqref{eq:20}.
Let $\pi \colon X_{\pi }\to X$ be a smooth modification and $P$
a prime divisor of $X_{\pi }$. We have to check that
\begin{equation}
\label{eq:21}
\ord_{P}\mathbb{D} = \ord_{P}\iota(\mathbb{D}').
\end{equation}
Let $(X_{2},\pi _{2},D')$ be a toroidal structure above $D$
with a map $f\colon X_{2}\to X_{\pi }$ and such that
$f^{-1}(P)\subset D'$. Let $\widehat P$ be the strict transform of
$P$ in $X_{2}$. In order to prove \eqref{eq:21} it is enough to
prove $\ord_{\widehat P}\mathbb{D} = \ord_{\widehat P}\iota(\mathbb{D}')$. Let
$v_{P}$ be the primitive vector in $\Pi (X_{2},D')$
corresponding to $\widehat P$, $\sigma $ the minimal cone of $\Pi
(X_{1},D)$ containing $r(D,D')(v_{P})$, $Y$ the stratum
of $X_{1}$ corresponding to $\sigma $, $x$ a generic point of
$Y$, and $W$ a small enough coordinate neighborhood of $x$ with
coordinates $z_{1},\dots, z_{d}$ such that $z_{1}\cdots z_{r}=0$ is
an equation of $D\cap W$. Write $D_{i}$ for the divisor $z_{i}=0$,
$i=1,\dots,r$, and $v_{i}$ for the corresponding primitive vector of $\Pi
(X_{1},D)$. Then the cone $\sigma $ is generated by
$v_{1},\dots,v_{r}$.
Let $\mu \colon X_{2}\to X_{1}$ be the map of
modifications.
Write $a_{i}=\ord_{\widehat P}\mu^{\ast}z_{i}$. Then
\begin{equation}
\label{eq:22}
r_{D,D'}(v_{P})=\sum _{i=1}^{r}a_{i}v_{i},
\end{equation}
and therefore
\begin{displaymath}
\ord_{P}\iota(\mathbb{D}')= \ord_{\widehat
P}\iota(\mathbb{D}')=\sum_{i=1}^{r}a_{i}\varphi_{\mathbb{D}'}(v_{i}).
\end{displaymath}
Let $y$ be a generic point of $\widehat P$ above $x$ and
choose a small enough coordinate neighborhood $V$ with coordinates
$(x_{1},\dots,x_{d})$ such that $x_{1}$ is a local equation for
$\widehat P$. As in the proof of Lemma \ref{lemm:16}, the map $\mu $
can be written as
\begin{displaymath}
\mu
(x_{1},\dots,x_{d})=
(x_{1}^{a_{1}}u_{1},\dots,x_{1}^{a_{r}}u_{r},u_{r+1},\dots,u_{d}),
\end{displaymath}
where the $u_{i}$'s are functions on $V$, and $u_{i}$ does not
vanish on $V$ for $i=1,\dots,r$. Choose a rational section $s$ of
$L$ that generates $L$ around $x$, so that
\begin{displaymath}
-\log h(s)=\gamma + g(-\log|z_{1}|,\dots, -\log|z_{r}|)
\end{displaymath}
and $\varphi_{\mathbb{D}'}|_{\sigma }=-\rec(g)$. Then around $y$ we have
\begin{displaymath}
-\log h(s) = \gamma
+g(-\log|u_{1}|-a_{1}\log|x_{1}|,\dots,-\log|u_{r}|-a_{r}\log|x_{1}|).
\end{displaymath}
Hence
\begin{displaymath}
\ord_{\widehat P}(\mathbb{D}')=-\nu (h,P)=
\rec(g)\left(\sum_{i=1}^{r}a_{i}v_{i}\right)
=-\varphi_{\mathbb{D}'}(r_{D,D'}(v_{\widehat P}))=\ord_{P}\iota(\mathbb{D}'),
\end{displaymath}
proving the result.
\end{proof}
Proposition \ref{prop:div_of_toroidal_metric_is_toroidal} has the
following consequence
\begin{cor} \label{cor:4} With the hypothesis of Proposition
\ref{prop:div_of_toroidal_metric_is_toroidal}, if the
b-divisor $\mathbb{D}(L,s,h)$ is Cartier, then for every cone $\sigma \in \Pi
(X_{1 },D)$ and every decomposition
\begin{displaymath}
-\log(h(s)(z_{1},\dots,z_{d})=\gamma +g(-\log|z_{1}|,\dots,-\log|z_{r}|),
\end{displaymath}
with $\gamma$ locally bounded and $g$ bounded above, convex and Lipschitz
continuous, the function $\rec(g|_\sigma)$ is piecewise linear.
\end{cor}
\begin{proof}
If $\mathbb{D}$ is Cartier, we know that $\varphi_{\mathbb{D}}|_{\sigma }$ is
piecewise linear. Hence the Corollary is an immediate consequence of
Proposition \ref{prop:div_of_toroidal_metric_is_toroidal}.\ref{i:2}.
\end{proof}
\subsection{From b-divisors to graded linear series}
\label{sec:from-b-divisors}
Let $\mathbb{D}$ be a Weil $\mathbb{R}$-b-divisor (not necessarily toroidal). We define
\begin{align*}
\mathcal{L}(\mathbb{D})&=\{0\not = f \in F\mid \mathbb{D}_{\pi }+\dv(f) \ge 0,\ \forall \pi \in R(X)
\}\cup \{0\},\\
\mathcal{R}(\mathbb{D}) &= \bigoplus _{\ell \in \mathbb Z_{\ge 0}} \mathcal{L}(\ell \mathbb{D})t^{\ell}\subset F[t].
\end{align*}
The latter is a graded linear series.
\begin{lem}\label{lemm:7}
The graded linear series $\mathcal{R}(\mathbb{D})$ is of almost integral type.
\end{lem}
\begin{proof}
Let $\pi \in R(X)$. From the definition it follows that
$\mathcal{R}(\mathbb{D})\subset \mathcal{R}(\mathbb{D}_{\pi })$. By Proposition \ref{prop:2} we know
$\mathcal{R}(\mathbb{D}_{\pi })$ is of almost integral type, so it is contained in
a graded linear series $A$ of integral type.
\end{proof}
\subsection{Summarizing the relations}
Combining the previous subsections we obtain a diagram
\begin{align}\label{diag:relations}
\xymatrix{
&\text{graded linear series}\ar@<.5em>[dd]^{\bdiv}\\
\text{psh-metrics}\ar^{\mathcal{R}}[ur]\ar_{\mathbb{D}}[dr]&\\
&\text{b-divisors}\ar@<.5em>[uu]^{\mathcal{R}}}
\end{align}
This diagram is in general not commutative. Much of the remainder of this section will be taken up with verifying certain weaker relations in this diagram.
\begin{lem}\label{lemm:3} Let $\mathbb{D}$ and $\mathbb{D}'$ be b-divisors and $A$ and
$A'$ graded linear series.
\begin{enumerate}
\item $\mathbb{D}\le \mathbb{D}' \Longrightarrow \mathcal{R}(\mathbb{D})\subset \mathcal{R}(\mathbb{D}')$.
\item $A\subset A' \Longrightarrow \bdiv(A)\le \bdiv(A')$.
\item \label{item:14} $\bdiv(\mathcal{R}(\mathbb{D}))\le \mathbb{D}$.
\item $A\subset \mathcal{R}(\bdiv(A))$.
\end{enumerate}
The same is true if we replace b-divisors by toroidal b-divisors.
\end{lem}
\begin{proof}
These follow directly from the definitions.
\end{proof}
\begin{lem}\label{lemm:4} Let $(L,h,s)$ be a line bundle on $X$ with a
psh metric $h$ and a non-zero rational section $s$ of $L$. Then
\begin{displaymath}
\mathcal{R}(L,s,h) \subset
\mathcal{R}(\mathbb{D}(L,s,h)).
\end{displaymath}
\end{lem}
\begin{proof}
The elements of $\mathcal{R}(L,s,h)_{\ell}$ are rational functions
$f$ with $h(fs^{\otimes \ell})$
bounded, while the elements of $\mathcal{R}(\mathbb{D}(L,s,h))_{\ell}$ are rational
functions $f\in \mathcal{L}(\ell \dv(s))$ with enough zeroes to
cancel the Lelong numbers of $h$.
The result then follows from the fact that bounded functions
have zero Lelong numbers; in what follows we give some more details.
Let $f \in \mathcal{R}(L, s,h)_{\ell}$. From Definition \ref{def:4} we can write $\mathbb{D}
= \mathbb{D}(L, s,h ) = (\mathbb{D}_{\pi})_{\pi \in R(X)}$ with
\[
\mathbb{D}_{\pi} = \pi^*\dv(s) - \sum_P\nu(h,P)P,
\]
where the sum is over all prime divisors $P$ on $X_{\pi}$ and
$\nu(h,P)$ denotes the Lelong number of the
metric $h$ at a very general point of $P$.
Let $x$ be a very general point of $P$, $s_{0}$ a local
generating section of $L$ at $x$, and $g$ a local equation
of $P$ at $x$. By Lemma \ref{lemm:8} we have
\begin{equation}
\nu(h,P) = \sup\{\gamma \ge 0 \mid -\log(h(s_{0})| g|^\gamma) \text{
bounded above near $x$}\}.
\end{equation}
Since $h(fs^{\otimes \ell})$ is bounded we know that
$-\log(h(s_{0})^{\ell}|g|^{\ord_{P}(fs^{\otimes \ell})})$ is bounded
below.
Therefore, for every $\epsilon >0$,
$-\log(h(s_{0})^{\ell}|g|^{\epsilon +\ord_{P}(fs^{\otimes \ell})})$ is \emph{not}
bounded above. Hence $ \ord_P(fs^{\otimes \ell}) \geq \ell \nu(h,P)$ and therefore
\[
\dv(f) + \ell \mathbb{D}_{\pi} = \dv(f) + \ell \pi^*\dv(s) - \ell \sum_P\nu(h,P)P \geq 0,
\]
so $f \in \mathcal{R}(\mathbb{D}(L,s,h))_\ell$.
\end{proof}
In the toroidal case we also have an inclusion in the reverse
direction.
\begin{lem}\label{lemm:14}
Let $(X_{\pi },D)$ be a toroidal structure on $X$ and let $h$ be a
psh metric on a line bundle $L$ on $X$ that is toroidal with respect to
$D$. Then for every real number $\varepsilon >0$ there is an inclusion
\begin{displaymath}
\mathcal{R}(\mathbb{D}(L,s,h)-\varepsilon D) \subset
\mathcal{R}(L,s,h).
\end{displaymath}
\end{lem}
\begin{proof}
Let $f\in \mathcal{R}(\mathbb{D}(L,s,h)-\varepsilon D)_{\ell}$ and let $p\in X_{\pi }$.
We need to show that $\|fs^{\otimes \ell}\|$ is bounded near $p$,
or equivalently that $-\log \|fs^{\otimes \ell} \|$ is bounded below near $p$. Let $s'$ be
another rational section of $L$ with $s=vs'$ for some rational
function $v$ and let $f'=fv^{\ell}$. Then
\begin{displaymath}
f\in \mathcal{R}(\mathbb{D}(L,s,h)-\varepsilon D)_{\ell}
\Longleftrightarrow
f'\in \mathcal{R}(\mathbb{D}(L,s',h)-\varepsilon D)_{\ell},
\end{displaymath}
and $\|fs^{\otimes \ell}\|$ is bounded if and only if
$\|f's'{}^{\otimes \ell}\|$ is
bounded. Therefore we can assume
that $s$ is a generating section around $p$.
Choose a
small enough coordinate system $W$ around $p$ as in Definition~\ref{def:9} and let $g$ and $\gamma $ be the functions appearing in
that definition. By Lemma~\ref{lemm:16}, the condition
$f\in \mathcal{R}(\mathbb{D}(L,s,h)-\varepsilon D)_{\ell}$ implies
that, on $W$, we can write
\begin{displaymath}
f=z_{1}^{m_{1}}\cdots z_{r}^{m_{r}}f_{0}
\end{displaymath}
where $f_{0}$ is holomorphic, not divisible by $z_{1},\dots,z_{r}$
and the exponents $m_{i}$ are integers
with the property that for every primitive vector
$u=(u_{1},\dots,u_{r})$ corresponding to a prime divisor
$P_{u}$ in an allowable modification, we have
\begin{align}\label{eq:15}
\sum _{i=1}^{r}m_{i}u_{i} &= \ord_{P_{u}}f \nonumber\\
&\ge
-\ell \ord_{P_{u}}(
\mathbb{D}(L,s,h)-\varepsilon D)
\\ &= -\ell \rec(g)(u)+\ell \varepsilon \sum_{i=1}^{r}u_{i}.
\nonumber \end{align}
Since $\rec(g)$ is conical and continuous on $\mathbb{R}^{r}_{\ge 0}$ we
deduce that, for every vector $u\in \mathbb{R}^{r}_{\ge 0}$ the inequality
\eqref{eq:15} holds. We compute
\begin{displaymath}
-\log\|fs^{\otimes \ell}\| =
-\log|f_{0}| + \sum_{i=1}^{r}m_{i}(-\log|z_{i}|) +\ell \gamma +
\ell g(-\log|z_{1}|,\dots,-\log|z_{r}|).
\end{displaymath}
Using that $f_{0}$ is holomorphic, so $-\log|f_{0}|$ is bounded
below, and that $\gamma $ is bounded near $p$, the condition
\eqref{eq:15} implies that there is a constant $B$ with
\begin{displaymath}
-\log\|fs^{\otimes \ell}\| \ge B+\ell g -\ell \rec(g)+\ell
\varepsilon \sum_{i=1}^r (-\log|z_{i}|).
\end{displaymath}
By Lemma \ref{lemm:15}.\ref{item:6} the quantity on the right is bounded below,
hence we obtain the result.
\end{proof}
\begin{cor} \label{cor:3} Take the assumptions of
Lemma~\ref{lemm:14}. If for every irreducible component $D_{i}$ of
$D$
the condition $\ord_{D_{i}}\mathbb{D}(L,s,h)>0$ holds then, for every $\varepsilon >0$,
\begin{displaymath}
\mathcal{R}((1-\varepsilon )\mathbb{D}(L,s,h)) \subset
\mathcal{R}(L,s,h).
\end{displaymath}
\end{cor}
\subsection{The case of a divisor generated by global sections}
\label{sec:case-divis-gener}
\begin{prop}\label{prop:1}
Let $\mathbb{D}$ be a $\mathbb{Q}$-Cartier b-divisor such that there is an $e \in
\mathbb Z_{>0}$ with $e\mathbb{D}$ a globally
generated integral Cartier divisor on some proper modification
$X_{\pi}$ of $X$. Then
\begin{displaymath}
\bdiv(\mathcal{R}(\mathbb{D}))=\mathbb{D}.
\end{displaymath}
\end{prop}
\begin{proof}
By Lemma \ref{lemm:3} we know that $\bdiv(\mathcal{R}(\mathbb{D}))\le \mathbb{D}$. On the other hand
\begin{displaymath}
\bigoplus _{\ell \in \mathbb Z_{\ge 0}} \mathcal{L}(\ell e \mathbb{D}_{\pi })t^{\ell e} \subset \mathcal{R}(\mathbb{D});
\end{displaymath}
indeed, if $e\in \mathbb Z_{\ge 0}$ and $f \in \mathcal{L}(\ell e \mathbb{D}_{\pi
})$ is non-zero, then $\dv(f) + \ell e \mathbb{D}_{\pi _{0}} \ge 0$, and the
same holds on every model $X_{\pi'}$ of $X$ since pullback and
pushforward preserve effectivity, hence $f \in \mathcal{R}(\mathbb{D})$.
Now let $s_{1},\dots ,s_{n}$ be a set of generating sections of $\mathcal{O}(e\mathbb{D}_{\pi})$,
and $s$ the canonical rational section of $\mathcal{O}(e\mathbb{D}_{\pi })$ with
$\dv(s)=e\mathbb{D}_{\pi }$.
Write $s_{i}=f_{i}s$. Since $s_{i}$ is a global section, we have
$e\mathbb{D}_{\pi}+\dv(f_{i})\ge 0$ and
$f_{i}\in \mathcal{R}(\mathbb{D})_{e}$.
Since the sections $s_{i}$ generate, for every prime divisor $P$ on
every modification $\pi' \in R(X)$ above $\pi $, there is an $i$ such that
\begin{equation}\label{eq:asd1}
\ord_{P}(e\mathbb{D}_{\pi' }+\dv(f_{i}))=0.
\end{equation}
On the other hand, since $f_{i}\in \mathcal{R}(\mathbb{D})_{e}$ we see from the
definition of $\bdiv(\mathcal{R}(\mathbb{D}))$ that
\begin{equation}\label{eq:asd2}
\ord_{P}(e\bdiv(\mathcal{R}(\mathbb{D}))+\dv(f_{i}))\ge 0.
\end{equation}
Combining \eqref{eq:asd1} and \eqref{eq:asd2} we deduce
\begin{displaymath}
\bdiv(\mathcal{R}(\mathbb{D}))\ge \mathbb{D}. \qedhere
\end{displaymath}
\end{proof}
\subsection{The volume of a b-divisor}
\begin{df} \label{def:volume_b_div} Let $\mathbb{D}$ be a Weil
b-divisor. The \emph{volume} of $\mathbb{D}$ is defined as
\begin{equation}\label{eq:vol-b}
\vol(\mathbb{D})= \limsup_{\ell} \frac{\dim \mathcal{R}(\mathbb{D})_{\ell}}{\ell^{d}/d!}.
\end{equation}
A Weil b-divisor $\mathbb{D}$ is called \emph{big} if $\vol(\mathbb{D})>0$.
\end{df}
\begin{rmk}\label{rmk:limsup}
Since $\mathcal{R}(\mathbb{D})$ is a graded algebra of almost integral type, it follows from \cite[Corollary 3.11]{kk} that the $\limsup$ in \eqref{eq:vol-b} is in fact a $\lim$ for sufficiently divisible $\ell$.
\end{rmk}
In case the
b-divisor is Cartier, this definition agrees with the usual notion of
the volume of a divisor (see e.g.~\cite[Definition~2.2.31]{Laz}).
\begin{lem} \label{lemm:20} Let $\mathbb{D}$ be a Weil b-divisor. If there is a big Cartier
divisor $B$ on some modification $X_{\pi }$ of $X$ such that $m \mathbb{D}\ge B$ for some
$m>0$, then $\mathbb{D}$ is big.
\end{lem}
\begin{proof}
Since $B$ is assumed to be big, we have $\vol(B)>0$. On the other hand,
\begin{displaymath}
\dim \mathcal{R}(\mathbb{D})_{m\ell} \ge \dim \mathcal{R}(B)_{\ell}
\end{displaymath}
by the assumption that $m \mathbb{D}\ge B$.
Therefore,
\begin{displaymath}
\vol(\mathbb{D}) \ge \limsup _{\ell} \frac{\dim
\mathcal{R}(\mathbb{D})_{m\ell}}{m^d\ell^d/d!}\ge
\frac{1}{m^d}\limsup_{\ell}\frac{\dim \mathcal{R}(B)_{\ell}}{\ell ^d/d!}=
\frac{1}{m^d}\vol(B)>0.
\end{displaymath}
\end{proof}
\subsection{The case of a toroidal nef and big b-divisor}
\label{sec:case-toro-appr}
\begin{thm}\label{thm:1}
Let $(X_1,\pi_1,D)$ be a toroidal structure on $X$.
Let $\mathbb{D}$ be a nef and big b-divisor which is toroidal with respect to $D$. Then
\begin{displaymath}
\bdiv(\mathcal{R}(\mathbb{D}))=\mathbb{D}.
\end{displaymath}
\end{thm}
The proof requires an
intermediate lemma. We begin by constructing some auxiliary
objects. By
\cite[Lemma 4.9]{BoteroBurgos} we know that there is a
sequence $\{\mathbb{D}_{i}\}_{i \in \mathbb{N}}$ of toroidal $\mathbb{Q}$-Cartier
b-divisors, generated by global sections and converging
monotonically decreasing to $\mathbb{D}$.
Moreover by the proof of \cite[Lemma 5.12]{BoteroBurgos} we can pick a
sequence of big toroidal $\mathbb{Q}$-Cartier b-divisors $\{\mathbb{B}_{j}\}_{j \in \mathbb{N}}$ with $\mathbb{B}_{j} \le
\mathbb{D}$ and $\vol(\mathbb{B}_{j})$ converging to $\vol(\mathbb{D})$. This is where the
toroidal condition is used.
By Fujita's approximation theorem \cite[Theorem 11.4.4]{Laz}, for every
$j>0$ there exists a $\mathbb{Q}$-Cartier b-divisor $\mathbb{A}_j$ generated by
global sections and satisfying
\begin{displaymath}
\vol(\mathbb{A}_{j})\ge \vol(\mathbb{B}_{j})-\frac{1}{j} \;\;\; \text{and} \;\;\; \mathbb{A}_j \le \mathbb{B}_j.
\end{displaymath}
Thus $\vol(\mathbb{A}_{j})$ also converges to $\vol (\mathbb{D})$.
The key technical result is
\begin{lem}\label{eq:sup} We have
\begin{equation}
\sup_{j}(\mathbb{A}_{j})=\mathbb{D} \, ,
\end{equation}
where the supremum is computed componentwise.
\end{lem}
\begin{proof}
We need to show that, for every
$\pi \in R(X)$ and every prime divisor $P$ in $X_{\pi}$, we have
\begin{displaymath}
\sup_{j}(\ord_{P}(\mathbb{A}_{j}))=\ord_{P}(\mathbb{D}).
\end{displaymath}
We proceed by contradiction; suppose this does \emph{not} hold. This means that there exists a
proper modification $\pi \in R(X)$, a prime divisor $P$ on $X_{\pi }$ and a
positive number $\varepsilon >0$ such
that for all $j$,
\begin{displaymath}
\ord_{P}(\mathbb{A}_{j})\le \ord_{P}(\mathbb{D})-\varepsilon.
\end{displaymath}
In what follows we use the theory of Okounkov bodies, for which we refer to \cite{kk, LM} for more details.
Upgrade $P$ to a complete flag
\begin{displaymath}
\mathcal{F}\colon\quad P=Y_{1}\supset Y_{2}\supset \dots \supset Y_{d}
\end{displaymath}
and for a graded linear series $A$ denote by $O_{\mathcal{F}}(A)$ the Okounkov
body of $A$ on
$X$ associated to this
flag. We briefly recall its construction. By an iterative procedure, one
constructs for each $\ell \in \mathbb{N}$ a valuation map
\[
\nu_\mathcal{F} \colon A_\ell\setminus \{0\} \longrightarrow \mathbb{Z}^d
\]
by taking the order of vanishing along the given $Y_i$ into
account. So, if $f\in A_{\ell}$,
\begin{displaymath}
\nu_{\mathcal{F}}(f)=\left(\ord_{P}f,\ast,\dots,\ast\right).
\end{displaymath}
That is, the first component of $\nu _{\mathcal{F}}(f)$ is the order
of $f$ at $P$.
This gives rise to a semigroup
\[
\Gamma(A) = \left\{\left(\nu_\mathcal{F}(f), \ell\right) \mid f \in A_\ell\setminus
\{0\}, \ell \in \mathbb{N}\right\} \subset \mathbb{Z}^{d}\times \mathbb{N}.
\]
The Okounkov body of $A$ with respect to $\mathcal{F}$ is then given by
\[
O_{\mathcal{F}}(A) = \overline{\text{cone}(\Gamma(A))} \cap \left(\mathbb{R}^d \times \{1\}\right).
\]
It is a closed convex set of $\mathbb{R}^d$, and if $A$ is of almost integral type,
then it is bounded, hence compact \cite[Theorem~2.30]{kk}.
Let $\omega _{P}\colon
\mathbb{R}^{d}\to \mathbb{R}$ be the projection onto the first variable. If $0\not=f \in
A_{\ell}$ is of degree $\ell$ and $x = \nu_\mathcal{F}(f)/\ell$ the corresponding
point in the Okounkov body, then by construction one has
\begin{displaymath}
\omega _{P}(x)=\ord_{P}(f)/\ell.
\end{displaymath}
For a b-divisor $\mathbb{E}$ we write $O_\mathcal{F}(\mathbb{E})$ for the
Okounkov body $O_\mathcal{F}(\mathcal{R}(\mathbb{E}))$.
From $\mathbb{A}_{j}\le \mathbb{D}\le \mathbb{D}_{i}$ we have $\mathcal{R}(\mathbb{A}_j) \subset \mathcal{R}(\mathbb{D})
\subset \mathcal{R}(\mathbb{D}_i)$ and hence
\begin{displaymath}
O_{\mathcal{F}}(\mathbb{A}_{j})\subset O_{\mathcal{F}}(\mathbb{D}) \subset O_{\mathcal{F}}(\mathbb{D}_{i})
\end{displaymath}
for all natural numbers $i,j$.
Since each $\mathbb{D}_{i}$ is generated by global sections there exist
$f_{i}\in H^{0}(X_{i},\ell_{i}\mathbb{D}_{i})$ for some $\ell_i$ such that
\begin{displaymath}
\ord_{P}(f_{i})/\ell_{i} = -\ord _{P}\mathbb{D}_{i}\le -\ord_{P}(\mathbb{D}).
\end{displaymath}
Therefore, there is a point $x_{i}\in O_{\mathcal{F}}(\mathbb{D}_{i})$ with
\begin{displaymath}
\omega _{P}(x_{i})\le -\ord_{P}(\mathbb{D}).
\end{displaymath}
Since $O_{\mathcal{F}}(\mathbb{D}_{1})$ is compact and $O_{\mathcal{F}}(\mathbb{D}_{i})\subset
O_{\mathcal{F}}(\mathbb{D}_{1})$, the sequence $\{x_{i}\}_{i \in \mathbb{N}}$ has at least one
accumulation point $x$. Moreover we claim that $\bigcap O_{\mathcal{F}}(\mathbb{D}_{i})=O_{\mathcal{F}}(\mathbb{D})$. Indeed, we have $\bigcap O_{\mathcal{F}}(\mathbb{D}_{i}) \subset O_{\mathcal{F}}(\mathbb{D})$. On the other hand $\{O_\mathcal{F}(\mathbb{D}_i)\}_{i \in \mathbb{N}}$ form a decreasing (under inclusion) sequence of compact convex sets, hence their intersection $O = \bigcap_i O_\mathcal{F}(\mathbb{D}_i)$ is again a compact convex set. We have that
\[
\vol(O) = \lim_i\vol(O_\mathcal{F}(\mathbb{D}_i)) = \lim_i \vol(\mathbb{D}_i) = \vol(\mathbb{D}) = \vol(O_\mathcal{F}(\mathbb{D})),
\]
where the second and last equalities follow from \cite[Theorem
5.13]{BoteroBurgos}. This proves the claim since two
full-dimensional compact convex sets with equal volume and such that
one is contained in the other have to agree. Now, the compactness of
$O_{\mathcal{F}}(\mathbb{D})$ implies that the accumulation point $x$ lies in
$O_{\mathcal{F}}(\mathbb{D})$. In
particular, there is a point $x\in O_{\mathcal{F}}(\mathbb{D})$ with $\omega
_{P}(x)\le -\ord_{P}(\mathbb{D})$.
On the other hand, since $\ord_{P}(\mathbb{A}_{j})\le \ord_{P}(\mathbb{D})- \varepsilon$ we have
that
\begin{displaymath}
\emptyset \not = O_{\mathcal{F}}(\mathbb{A}_{j})\subset \{x\in \mathbb{R}^{d}\mid \omega _{P}(x)\ge
-\ord_{P}(\mathbb{D})+\varepsilon \}.
\end{displaymath}
The set $O_{\mathcal{F}}(\mathbb{D})$ is convex, has non-zero volume, contains a point
with $\omega _{P}(x)\le -\ord_{P}(\mathbb{D})$, and also contains a point $y$
with $\omega _{P}(y)\ge -\ord_{P}(\mathbb{D})+\varepsilon $ (just choose
any point of $O_{\mathcal{F}}(\mathbb{A}_{j})$ for some $j$). Hence
\begin{displaymath}
\vol(O_{\mathcal{F}}(\mathbb{D})\cap\{-\ord_{P}(\mathbb{D})\le \omega _{P}\le
-\ord_{P}(\mathbb{D})+\varepsilon \}) =:\eta >0.
\end{displaymath}
This implies that $\vol(O_{\mathcal{F}}(\mathbb{A}_{j}))\le \vol (O_{\mathcal{F}}(\mathbb{D})) - \eta$,
contradicting the fact that $\vol(O_{\mathcal{F}}(\mathbb{A}_{j}))$ converges to $\vol
(O_{\mathcal{F}}(\mathbb{D}))$.
\end{proof}
\begin{proof}[Proof of Theorem {\ref{thm:1}}]
From Lemma \ref{lemm:3} and the inequalities $\mathbb{A}_{j}\le \mathbb{D}\le
\mathbb{D}_{i}$ for any natural numbers $i,j$, we deduce
\begin{displaymath}
\bdiv(\mathcal{R}(\mathbb{A}_{j}))\le
\bdiv(\mathcal{R}(\mathbb{D}))\le
\bdiv(\mathcal{R}(\mathbb{D}_{i})).
\end{displaymath}
By Lemma~\ref{lemm:3} and Proposition \ref{prop:1} we get
\begin{displaymath}
\mathbb{A}_{j}\le
\bdiv(\mathcal{R}(\mathbb{D}))\le
\mathbb{D}_{i} \, .
\end{displaymath}
Invoking Lemma~\ref{eq:sup} and once more Lemma~\ref{lemm:3}, we get
\begin{displaymath}
\mathbb{D}=\sup (\mathbb{A}_{j})\le \bdiv(\mathcal{R}(\mathbb{D}))\le \mathbb{D}. \qedhere
\end{displaymath}
\end{proof}
\begin{rmk}\label{rem:exa-non-tor}
The toroidal condition in Theorem \ref{thm:1} is necessary. Indeed, consider the b-divisor $\mathbb{D}$ from \cite[Appendix A]{BBHJ}. Then $\mathbb{D}$ is a nef and big b-divisor on $\mathbb{P}^2$. It satisfies
\[
\mathcal{R}(\mathbb{D}) = \mathcal{R}(2H-L),
\]
where $L$ and $H$ are two lines in $\mathbb{P}^2$ as defined in \emph{loc. cit.}
Hence, by the construction we get
\[
\mathbb{D} \neq 2H-L = \bdiv(\mathcal{R}(2H-L))= \bdiv(\mathcal{R}(\mathbb{D})).
\]
\end{rmk}
From Lemma \ref{lemm:14} and Theorem \ref{thm:1} we obtain also the
following compatibility in the case of toroidal psh metrics.
\begin{cor}\label{cor:2}
Let $(L,h)$ be a line bundle with a toroidal psh metric with
singularity divisor $D$, and
$s$ a non-zero rational section of $L$. If $\mathbb{D}(L,s,h)$ is nef and big and for every
irreducible component $D_{i}$ of $D$ the condition
$\ord_{D_{i}}(\mathbb{D}(L,s,h))>0$ holds, then
\begin{displaymath}
\bdiv(\mathcal{R}(L,s,h)) = \mathbb{D}(L,s,h).
\end{displaymath}
\end{cor}
\begin{proof}
By Lemma \ref{lemm:4} and Corollary \ref{cor:3}, for every $j>1$
we have
\begin{displaymath}
\mathcal{R}(((1-1/j)\mathbb{D}(L,s,h)) \subset
\mathcal{R}(L,s,h)
\subset \mathcal{R}(\mathbb{D}(L,s,h)).
\end{displaymath}
Since $h$ is a toroidal psh metric, by Proposition
\ref{prop:div_of_toroidal_metric_is_toroidal} the b-divisor $\mathbb{D}(L,s,h)$ is
toroidal with respect to a toroidal structure $D'$ above $D$. Note
that we need a toroidal structure above $D$ in order to make $\dv(s)$
toroidal.
By Lemma \ref{lemm:3} and Theorem \ref{thm:1}, we
deduce
\begin{displaymath}
(1-1/j)\mathbb{D}(L,s,h)\le
\bdiv(\mathcal{R}(L,s,h)) \le
\mathbb{D}(L,s,h).
\end{displaymath}
Since $(1-1/j)\mathbb{D}(L,s,h)$ converges to $\mathbb{D}(L,s,h)$ when $j\to
\infty$, we obtain the corollary.
\end{proof}
\subsection{Criterion for not being finitely generated}
\label{sec:crit-non-finit}
\begin{lem}\label{lemm:2} If $A$ is a finitely generated graded linear
series on $X$, then $\bdiv(A)$ is a $\mathbb{Q}$-Cartier b-divisor.
\end{lem}
\begin{proof}
Assume that $A$ is generated by $f_{1}t^{d_1},\dots,f_{n}t^{d_n}$,
with $f_{i} \in F=K(X)$.
\smallskip
\noindent
\emph{Claim.} If a b-divisor $\mathbb{D}$ satisfies $d_{i} \mathbb{D}+\dv(f_{i})\ge 0$, for
$i=1,\dots,n$, then $\ell \mathbb{D}+\dv(f)\ge 0$ for all $\ell \ge 0$ and all
$f\in A_{\ell} $.
The claim follows from the compatibility of multiplicity with
products and the ultrametric inequality for sums. Namely, if $f\in
A_{\ell}$ and $f'\in A_{\ell'}$, then the compatibility of the
valuation with products yields,
\begin{displaymath}
\left.
\begin{aligned}
\ell \mathbb{D} +\dv(f)&\ge 0\\
\ell' \mathbb{D} +\dv(f')&\ge 0
\end{aligned}
\right\} \Longrightarrow
(\ell+\ell') \mathbb{D} +\dv(ff')\ge 0,
\end{displaymath}
and if $f,f'\in A_{\ell}$ the ultrametric inequality implies
\begin{displaymath}
\left.
\begin{aligned}
\ell \mathbb{D} +\dv(f)&\ge 0\\
\ell \mathbb{D} +\dv(f')&\ge 0
\end{aligned}
\right\} \Longrightarrow
\ell \mathbb{D} +\dv(f+f')\ge 0.
\end{displaymath}
From the claim we deduce
\begin{displaymath}
\bdiv(A)_{\pi }=\inf \{D\in \Div(X_{\pi })\mid d_{i}D+\dv(f_{i})\ge
0,\ i=1,\dots,n\}.
\end{displaymath}
Choose $D_{0}$ such that $d_{i}D_{0}+\dv(f_{i})\ge 0$. Let
$e=\operatorname{lcm}(d_{1},\dots,d_{n})$
and let $\mathfrak{a}$ be the fractional ideal generated by
$f_{i}^{e/d_{i}}$, $i=1,\dots,n$. Since $f_{i}^{e/d_{i}}\in \mathcal{O}(eD_{0})$, we obtain
$\mathfrak{a}\subset \mathcal{O}(eD_{0})$, so
\begin{displaymath}
\mathcal{I}\coloneqq\mathfrak{a}\mathcal{O}_X(-eD_{0})\subset \mathcal{O}_{X}
\end{displaymath}
is a coherent ideal sheaf.
Let $\pi _{0}\colon X_{\pi _{0}}\to X$ be a proper modification such
that $\pi ^{-1}(\mathcal{I})\mathcal{O}_{X_{\pi _{0}}}$ is principal and equal to
$\mathcal{O}(-D')$ for some effective integral divisor $D'$. Then
\begin{displaymath}
e(\bdiv(A))=eD_{0}-D',
\end{displaymath}
and in particular $\bdiv(A)$ is $\mathbb{Q}$-Cartier.
\end{proof}
\section{Siegel--Jacobi forms}
\label{sec:jacobi-forms}
\subsection{Basic definitions}
\label{sec:basic-definitions-1}
We recall the definition of Siegel--Jacobi forms. For more details
we refer to \cite{Kramer_Crelle} and \cite{Ziegler-J}.
Let $g \geq 1$ be an integer. The real symplectic group $\Sp(2g,\mathbb{R})$ is the group of real $2g\times 2g$
matrices of the form
\begin{equation}\label{eq:1}
\begin{pmatrix}
A & B \\
C & D
\end{pmatrix}
\end{equation}
such that
\begin{displaymath}
A^{t}C=C^{t}A,\quad D^{t}B=B^{t}D, \quad A^{t}D=\Id_{g}+C^{t}B,
\end{displaymath}
where $\Id_{g}$ is the identity matrix of dimension $g$. There is an
inclusion
\[
\Sp(2g,\mathbb{R})\hookrightarrow \Sp(2g+2,\mathbb{R})
\]
sending a matrix
of the form \eqref{eq:1} to
\begin{displaymath}
\begin{pmatrix}
A &0& B&0 \\
0&1&0&0\\
C &0& D&0\\
0&0&0&1
\end{pmatrix}.
\end{displaymath}
For any commutative ring $R$ let $H_{R}^{(g,1)}$ be the Heisenberg group
\begin{displaymath}
H_{R}^{(g,1)}=\left\{\left[(\lambda ,\mu ),x\right]\mid \lambda ,\mu \in
R^{(1,g)},\, x\in R\right\},
\end{displaymath}
where $R^{(1,g)}$ denotes the set of row vectors of size $g$ and coefficients in $R$, with the composition law given by
\begin{displaymath}
[(\lambda ,\mu ),x]\circ [(\lambda ',\mu '),x']=
[\lambda +\lambda ',\mu +\mu ',x+x'+\lambda {\mu '}^{t}-\mu
{\lambda '}^{t}].
\end{displaymath}
These are the same definitions as in \cite{Ziegler-J} for the case $g=1$. The real Heisenberg group $H_{\mathbb{R}}^{(g,1)}$ can be realized as the
subgroup of $\Sp(2g+2,\mathbb{R})$
consisting of matrices of the form
\begin{displaymath}
\begin{pmatrix}
\Id_{g} & 0 & 0 & \mu^{t}\\
\lambda & 1 & \mu & x\\
0 & 0 & \Id_{g} & -\lambda^{t} \\
0 & 0 & 0 & 1
\end{pmatrix}.
\end{displaymath}
The full Jacobi group $G^{(g,1)}_{\mathbb{R}}=\Sp(2g,\mathbb{R}) \ltimes
H^{(g,1)}_{\mathbb{R}}$ is the subgroup of $\Sp(2g+2,\mathbb{R})$ generated by
$\Sp(2g,\mathbb{R})$ and $H^{(g,1)}_{\mathbb{R}}$.
Let
\begin{displaymath}
\mathcal{H}_{g}=\{ Z=X+iY \mid X,Y\in \mathrm{Mat}_{g\times g}(\mathbb{R}),\, Z^{t}=Z,\, Y>0\}
\end{displaymath}
be the Siegel upper half space. The group $\Sp(2g,\mathbb{R})$ acts
transitively on $\mathcal{H}_g$, where for $M = \begin{pmatrix}
A&B \\ C&D
\end{pmatrix} \in \Sp(2g, \mathbb{R})$ and $Z \in \mathcal{H}_g$ the action is given by
\[
Z\longmapsto
M\langle Z \rangle = (AZ + B)(CZ + D)^{-1} \, .
\]
On the other hand, the group $G^{(g,1)}_{\mathbb{R}}$ acts transitively on $\mathcal{H}_{g}\times
\mathbb{C}^{(1,g)}$ by the action
\[
\left(M, (\lambda, \mu, x)\right) \cdot (Z,W) = \left(M\langle Z
\rangle, (W+\lambda Z + \mu)(CZ + D)^{-1}\right).
\]
Let $\Gamma \subset \Sp(2g,\mathbb{Z})$ be a subgroup of finite
index. We write $\widetilde \Gamma =\Gamma \ltimes
H^{(g,1)}_{\mathbb{Z}}\subset G^{(g,1)}_{\mathbb{Z}}$.
Recall that a subgroup $\Gamma \subset \Sp(2g,\mathbb{Z})$ is called \emph{neat} when for
every $M\in \Gamma $, the
subgroup of $\mathbb{C}^{\times}$ generated by the eigenvalues of $M$ is
torsion free.
If $\Gamma $ is neat, then the quotient $\mathcal{A}(\Gamma )=\Gamma
\backslash \mathcal{H}_{g}$ is a smooth
complex manifold and the quotient $\mathcal{B}(\Gamma
)=\widetilde \Gamma \backslash \mathcal{H}_{g}\times \mathbb{C}^{(1,g)}$ is a
fibration over $\mathcal{A}(\Gamma )$ by principally polarized abelian
varieties.
Following \cite{Ziegler-J} we now introduce automorphy factors for the
group $G^{(g,1)}_{\mathbb{R}}$ in order to obtain interesting line bundles on
the quotient $\mathcal{B}(\Gamma )$.
Let $\phi \colon \mathcal{H}_g \times \mathbb{C}^g \to \mathbb{C}$ be a holomorphic map.
Let $\rho_{k} \colon \GL(g, \mathbb{C}) \to \mathbb{C}^\times$ be the representation given by $N
\mapsto (\det N)^k$. For $M \in \Sp(2g, \mathbb{R})$, $\zeta =
(\lambda, \mu, x) \in H_{\mathbb{R}}^{(g,1)}$ and $m \in \mathbb{N}_{\geq 0}$ define
\begin{multline}\label{eq:25}
\left(\phi|_{k,m}M\right)(Z,W) \coloneqq
\rho_{k}(CZ + D)^{-1} e^{-2\pi i
mW(CZ+D)^{-1}CW^{t}} \phi\left(M\langle
Z\rangle, W(CZ+ D)^{-1}\right)
\end{multline}
and
\begin{equation}
\left(\phi|_{k,m}\zeta\right)(Z,W) \coloneqq e^{2\pi i
m\left(\lambda Z \lambda^{t}+2\lambda W^{t}
+ (x + \mu \lambda^{t}) \right)} \phi(Z,
W+\lambda Z+ \mu). \label{eq:26}
\end{equation}
A matrix $T$ is called \emph{half integral} if $2T$ has integral entries and
the diagonal entries of $T$ are integral. Note that if $T$ is
symmetric then $T$ is half integral if and only if the associated quadratic form is
integral.
\begin{df} \label{def:SJ_forms}
A holomorphic map $\phi \colon \mathcal{H}_g \times \mathbb{C}^g \to \mathbb{C}$ is
called a \emph{Siegel--Jacobi form of weight $k$ and index $m$} for a
subgroup $\Gamma \subset \Sp(2g, \mathbb{Z})$ of finite index if the following
conditions are satisfied:
\begin{enumerate}
\item \label{item:17} $\phi|_{k,m}M = \phi$ for all $M \in \Gamma$.
\item \label{item:18} $\phi|_{k,m}\zeta = \phi$ for all $\zeta \in H_{\mathbb{Z}}^{(g,1)}$.
\item \label{item:16} For each $M \in \Sp(2g, \mathbb{Z})$ the function $\phi|_{k,m}M$ has a
Fourier expansion of the form
\begin{equation}
\left(\phi|_{k,m}M\right)(Z,W) = \sum_{\stackrel{T =
T^{t}\geq 0}{T \text{ half integral}}} \sum_{R \in
\mathbb{Z}^{(g,1)}} c(T,R) e^{2 \pi i/\lambda_{\Gamma}\tr(TZ)}e^{2 \pi i
WR}\label{eq:29}
\end{equation}
for some suitable integer $0 < \lambda_{\Gamma} \in \mathbb{Z}$, and such that
$c(T,R) \neq 0$ implies
\[
\begin{pmatrix}
\frac{1}{\lambda_{\Gamma}}T & \frac{1}{2}R \\
\frac{1}{2}R^{t} & m
\end{pmatrix} \geq 0.
\]
\end{enumerate}
A Siegel--Jacobi form $\phi$ is said to be a \emph{cusp form} if
\[
\begin{pmatrix}
\frac{1}{\lambda_{\Gamma}}T & \frac{1}{2}R \\
\frac{1}{2}R^{t} & m
\end{pmatrix} > 0.
\]
for any $T,R$ with $c(T,R) \neq 0$.
\end{df}
We note that when $g\geq 2$, condition \ref{item:16} is a
consequence of conditions \ref{item:17} and \ref{item:18} due to the Koecher
principle \cite[Lemma 1.6]{Ziegler-J}.
\begin{df}
The vector space of all Siegel--Jacobi forms of weight $k$ and index
$m$ for $\Gamma$ is denoted by $J_{k,m}(\Gamma )$ and the
space of cusp forms is denoted by $J_{k,m}^{\text{\rm cusp}}(\Gamma )$.
\end{df}
The following lemma follows easily from the definitions.
\begin{lem} \label{lem:multiplicativity}
If $\phi $ is a Siegel--Jacobi form of weight $k$ and index $m$ and $\psi
$ is a Siegel--Jacobi cusp form of weight $k'$ and index $m'$,
then $\phi \psi $ is a Siegel--Jacobi cusp form of weight $k+k'$ and
index $m+m'$.
\end{lem}
The following result is \cite[Theorem 1.5]{Ziegler-J}
\begin{lem}\label{lemm:21}
Let $\phi $ be a Siegel--Jacobi form of weight $k$ and index
$m$ for a subgroup $\Gamma \subset \Sp(2g, \mathbb{Z})$ of finite index
and let $\lambda,\mu \in \mathbb{Q}^{(1,g)}$ be rational
vectors. Then there is a finite index subgroup $\Gamma '\subset
\Sp(2g, \mathbb{Z})$ that depends only on $\Gamma$, $\lambda $ and $\mu
$ such that the function
\begin{displaymath}
f(Z)=e^{2\pi i m\lambda Z\lambda ^{t}}\phi (Z,\lambda Z+\mu )
\end{displaymath}
is a Siegel modular form of weight $k$ for $\Gamma '$.
\end{lem}
\begin{ex}\label{exm:1} Here we list some examples of Siegel--Jacobi
forms which turn out to be useful for our purposes.
\begin{enumerate}
\item \label{item:8} The only Siegel--Jacobi forms of
weight $0$ and index
$0$ are the constants (i.e. $J_{0,0}(\Gamma )=\mathbb{C}$). Moreover
$J_{k,m}(\Gamma )=0$ whenever $k <0$ or $m<0$. The first case follows from
Lemma \ref{lemm:21} using the
Corollary to
Proposition 1 in \cite[Section 4]{klingen} or the remark after
Theorem 1 in \cite[Section 8]{klingen}, while the second follows
from the geometric description \eqref{eq:18} of the line bundle
of Siegel--Jacobi forms; indeed, if $m<0$ the restriction to any fibre is strictly anti-effective.
\item \label{item:9} Any Siegel modular form of weight $k$ for
$\Gamma $ defines a Siegel--Jacobi form of weight $k$ and index
$0$. A classical example is as follows (for details we refer to
\cite[Section 2]{Ziegler-J}). For $k>g+1$ even, the
\emph{Eisenstein series}
\begin{equation}\label{eq:23}
E_{g,k}(Z)=\sum_{M \in P_{g}\backslash
\Sp(2g,\mathbb{Z})}\rho _{k} (C Z +D)^{-1}
\end{equation}
is convergent and defines a Siegel modular form of weight
$k$ for the group $\Sp(2g,\mathbb{Z})$, in particular for any subgroup
$\Gamma \subset \Sp(2g,\mathbb{Z})$. Hence it defines a Siegel--Jacobi form of weight $k$ and index $0$ for $\Gamma$.
In equation \eqref{eq:23}, $P_{g}$ is the subgroup
\begin{displaymath}
P_{g}=\left\{
\begin{pmatrix}
A & B\\
C & D
\end{pmatrix}
\in \Sp(2g,\mathbb{Z})\,\middle | \,C=0\right\}
\end{displaymath}
and $M$ is written as in \eqref{eq:1}.
\item \label{item:13} Similarly, \emph{Poincar\'e series} can be used to produce non-zero
Siegel cusp forms of any weight $k>2g$ such
that $kg$ is even (see \cite[Proposition 2 and its
Corollary]{klingen}). By pullback they produce
Siegel--Jacobi cusp forms of index zero.
\item \label{item:10} Eisenstein series can be generalized to
produce Siegel--Jacobi modular forms of arbitrary index. Let
$m>0$ be an integer and $k>g+2$ be an even integer. Write
\begin{displaymath}
H_{\infty} =\left\{\left[(\lambda ,\mu ),x\right]\in H_{\mathbb{Z}}^{(g,1)}\mid \lambda
=0\right\}.
\end{displaymath}
Then the series
\begin{multline*}
E_{g,k,m}(Z,W)=\sum _{M\in P_{g}\backslash
\Sp(2g,\mathbb{Z})} \rho _{k}(C Z +D)^{-1} e^{-2\pi i
mW(CZ+D)^{-1}CW^{t}}\\
\sum_{\lambda \in H_{\infty}\backslash H_{\mathbb{Z}}^{(g,1)}}
e^{2\pi i
m\left(\lambda M\langle Z\rangle \lambda^{t}+2\lambda
(CZ+D)^{-t}W^{t}
\right)}
\end{multline*}
is convergent and defines a Siegel--Jacobi modular form of weight $k$
and index $m$. See \cite[Theorem 2.1]{Ziegler-J} for details.
\end{enumerate}
\end{ex}
\begin{lem}\label{lemm:22}
If $\phi $ is a Siegel--Jacobi form of weight $0$ and index $m \neq 0$,
then $\phi =0$.
\end{lem}
\begin{proof}
Let $\phi $ be such a Siegel--Jacobi form. For $Z\in \mathcal{H}_{g}$ and
$\lambda, \mu \in \mathbb{R}^{(1,g)}$, write
\begin{displaymath}
f(Z,\lambda ,\mu )=e^{2\pi i m\lambda Z\lambda ^{t}}\phi (Z,\lambda Z+\mu ).
\end{displaymath}
By Lemma \ref{lemm:21}, whenever $\lambda, \mu \in \mathbb{Q}^{(1,g)}$
are fixed we obtain a Siegel modular form of weight zero for a
certain finite index subgroup of $\Sp(2g,\mathbb{Z}) $, which is necessarily constant. By
continuity, for every $\lambda, \mu \in \mathbb{R}^{(1,g)}$ the
function $f(Z,\lambda ,\mu )$ is constant. Therefore, there is
a function $g\colon \mathbb{R}^{(1,g)}\times \mathbb{R}^{(1,g)} \to \mathbb{C}$ such
that
\begin{equation}\label{eq:24}
\phi (Z,\lambda Z+\mu )=e^{-2\pi i m\lambda Z\lambda ^{t}}g(\lambda ,\mu ).
\end{equation}
We now see that this is incompatible with the modular condition
$\phi|_{k,m}M = \phi$ for all $M \in \Gamma$. Let
\begin{displaymath}
M=
\begin{pmatrix}
A & B\\ C&D
\end{pmatrix}\in \Gamma,
\end{displaymath}
then the modular condition and equation \eqref{eq:24} imply that
\begin{displaymath}
e^{-2\pi i m \left( (\lambda Z+\mu)(CZ+D)^{-1}C(\lambda
Z+\mu )^{t}+\lambda 'M\langle Z\rangle \lambda
'^{t}-\lambda Z\lambda ^{t} \right)}
g(\lambda ',\mu ')=g(\lambda ,\mu ),
\end{displaymath}
where
\begin{displaymath}
(\lambda ',\mu ')=(\lambda ,\mu )M^{-1}.
\end{displaymath}
When $m\not =0$, $M\not = \Id$ and $\lambda \not = 0$, the
exponential term actually depends on $Z$. As the function $g$ does not depend on $Z$, we conclude that $g=0$.
\end{proof}
A standard consequence of the previous examples and Lemma
\ref{lemm:22} is the following (see also \cite{EZ}).
\begin{prop}\label{prop:bi-gr-alg}
The ring $J_{\ast,\ast}(\Gamma )\coloneqq \bigoplus _{k,m}J_{k,m}(\Gamma )$
is not finitely generated.
\end{prop}
\begin{proof}
Assume that the ring $J_{\ast,\ast}(\Gamma )$ is finitely
generated. Let $1,f_{i}$, $i=1,\dots,r$ be a set of homogeneous generators
with $f_{i}$ non constant. Let $k_{i}$ and $m_{i}$ be the weight and
index of $f_{i}$. Then by Example \ref{exm:1}.\ref{item:8} and Lemma~\ref{lemm:22} we have
$k_{i}>0$ for $i=1,\dots,r$. Thus the possible ratios
$m_{i}/k_{i}$ are bounded above. Therefore if $f $ is a non-constant Siegel--Jacobi modular form of weight $k$ and index $m$, then
the ratio $m/k$ is bounded. But in Example
\ref{exm:1}.\ref{item:10} we have a construction of non-zero Siegel--Jacobi
modular forms of fixed weight and arbitrary index.
\end{proof}
\subsection{The line bundle of Siegel--Jacobi forms}\label{sec:geom-interpr-line}
As mentioned in the introduction, to the neat arithmetic group $\Gamma $ we
associate a fibration of principally polarized abelian varieties over a
complex manifold
\begin{displaymath}
\pi \colon \mathcal{B}(\Gamma )\longrightarrow \mathcal{A}(\Gamma ).
\end{displaymath}
The transformations \eqref{eq:25} and \eqref{eq:26} define a cocycle
for the group $\widetilde \Gamma $. Therefore they determine a line
bundle $L_{k,m}$ such that the Siegel--Jacobi forms of weight $k$ and
index $m$ can be seen as global sections of this line bundle. We
recall the geometric interpretation of the line bundle $L_{k,m}$.
Let $e\colon \mathcal{A}(\Gamma )\to \mathcal{B}(\Gamma )$ be the zero section. Let
$M$ be the line bundle on $\mathcal{A}(\Gamma )$ defined as
\begin{displaymath}
M=\det\left(e^{\ast} \Omega ^{1}_{\mathcal{B}(\Gamma )/\mathcal{A}(\Gamma )}\right).
\end{displaymath}
The space $\mathbb{C}^{g}$ comes equipped with canonical
holomorphic coordinates $(z_{1},\dots,z_{g})$ on $\mathbb{C}^{g}$. Then
$dz_{1}\wedge \dots \wedge dz_{g}$ is a multivalued section of $M$
well defined up to a global constant. Note that this is a multivalued
section because $dz_{1}\wedge \dots \wedge dz_{g}$ is not invariant
under the action of $\Gamma $. Let $f$ be a Siegel modular form of
weight $k$. Then one can verify that the symbol
\begin{displaymath}
f(dz_{1}\wedge \dots \wedge dz_{g})^{\otimes k}
\end{displaymath}
is invariant under the action of $\Gamma $ and therefore determines a
section of the line bundle $M^{\otimes k}$. In this way we obtain an
identification of $L_{k,0}$ with $\pi ^{\ast}M^{\otimes k}$.
Since $\mathcal{B}(\Gamma )$ is a family of principally polarized abelian
varieties it comes equipped with a biextension line bundle $B$. This
line bundle is defined as follows.
Let $\mathcal{B}(\Gamma )^{\vee}$ be the dual family of abelian varieties and
$P$ the Poincar\'e line bundle on $\mathcal{B}(\Gamma )\times_{\mathcal{A}(\Gamma)}
\mathcal{B}(\Gamma)^{\vee}$. Let $\lambda \colon \mathcal{B}(\Gamma )\to \mathcal{B}(\Gamma
)^{\vee}$ be the isomorphism defined by the polarization. Then
\begin{displaymath}
B=(\Id,\lambda )^{\ast}P.
\end{displaymath}
The line bundle $B$ gives on each fibre twice the principal
polarization. There is also an identification $L_{0,m}$ with
$B^{\otimes m}$. Therefore, the line bundle of Siegel--Jacobi forms of weight $k$ and index $m$ is given by
\begin{equation}\label{eq:18}
L_{k,m}=\pi ^{\ast}M^{\otimes k}\otimes B^{\otimes m}.
\end{equation}
\begin{lem} We have
\[ J_{k,m}(\Gamma) \subseteq H^0(\mathcal{B}(\Gamma),L_{k,m}) \, , \]
with equality if $g \geq 2$.
\end{lem}
\begin{proof} This follows from Definition \ref{def:SJ_forms} and the remark afterwards.
\end{proof}
\subsection{The invariant metric}
The aim of this subsection is to discuss the canonical invariant metric on the line bundle of Siegel--Jacobi forms.
For $Z\in \mathcal{H}_{g}$ we write $Z=X+iY$ with $X, Y \in \mathrm{Mat}_{g \times g}(\mathbb{R})$ and for $W\in \mathbb{C}^{(1,g)}$ we write
$W=\alpha +i\beta $ with $\alpha, \beta \in \mathbb{R}^{(1,g)}$.
\begin{df}\label{def:6}
Let $\phi \in J_{k,m}(\Gamma )$ be a Siegel--Jacobi form. Then the
\emph{standard invariant norm} $h^{\text{\rm inv}}(\phi)$ of $\phi $ is defined by
\begin{displaymath}
h^{\text{\rm inv}}(\phi (Z,W))^{2}=|\phi (Z,W)|^{2}\rho _{k}(Y)e^{-4\pi m \beta
Y^{-1}\beta^{t}}.
\end{displaymath}
This quantity is readily checked to be $\widetilde \Gamma $-invariant. When $\Gamma
$ is neat it induces a smooth hermitian metric on $L_{k,m}(\Gamma )$.
\end{df}
The standard invariant metric of Definition \ref{def:6} gives in particular
metrics on $M$ and $B$. These metrics agree with the classical
Hodge metric on $M$ and the canonical biextension metric on $B$. We refer to \cite[Section~5]{MR4221000}
for a brief discussion of these classical metrics and a verification of this agreement.
\begin{lem}\label{lem:psh}
The standard invariant metric $h^{\text{\rm inv}}$ on $L_{k,m}(\Gamma )$ is a psh (i.e., semipositive) metric.
\end{lem}
\begin{proof}
It is enough to show that $-\log \det (Y)$ and $\beta
Y^{-1}\beta^{t}$ are psh functions on $\mathcal{H}_{g}\times
\mathbb{C}^{(1,g)}$. A smooth function $f\colon \mathbb{C}^{n}\to \mathbb{R}$ of the form
\begin{displaymath}
f(z_{1},\dots, z_{n})=g(\im(z_{1}),\dots,\im(z_{n}))
\end{displaymath}
is psh if and only if $g$ is convex
(see \cite[I~(5.13)]{demailly12:cadg}). Then the lemma follows from the
Example log-determinant in \cite[Section~3.1.5]{Boyd:convex} and by \cite[Example~3.4,
Section~3.1.7]{Boyd:convex}.
\end{proof}
A useful result by Ziegler is the following (see \cite[Proposition~1 and its Corollary]{dulinski}).
\begin{prop}\label{prop:6}
A Siegel--Jacobi form $\phi $ is a cusp form if and only if
$h^{\text{\rm inv}}(\phi)$ is
bounded.
\end{prop}
\section{Toroidal compactifications of the universal abelian variety}
\label{sec:toro-comp}
\subsection{Toroidal compactifications}
\label{subsec:toro-comp}
We briefly describe the theory of toroidal compactifications of
$\mathcal{A}(\Gamma )$ and of $\mathcal{B}(\Gamma )$. For more details we refer to the book
\cite{fc} by Faltings and Chai and Namikawa's work \cite{Nam1977, Nam1979}. Note that in \cite{fc}
everything is worked out for the full modular group $\Sp(2g,\mathbb{Z})$ and the
principal congruence subgroups $\Gamma (N)$. This is because the authors are mainly interested in
integral models. If one is only interested in the theory over the
complex numbers everything carries over to the general case of a
commensurable subgroup $\Gamma $. On the other hand, to avoid having
to deal with algebraic stacks we will deal only with the case when
$\Gamma $ is neat.
Let $C_{g}$ be the (open) cone of
symmetric positive definite real $g\times g$ matrices and $\overline
C_{g}$ the cone of symmetric semipositive real matrices with
rational kernel. Let $\widetilde C_{g}\subset \overline C_{g}\times
\mathbb{R}^{(1,g)}$ be the cone
\begin{displaymath}
\widetilde C_{g}=\{(\Omega ,\beta )\in \overline C_{g}\times
\mathbb{R}^{(1,g)} \mid \exists \alpha \in \mathbb{R}^{(1,g)}, \, \beta =\alpha \Omega \}.
\end{displaymath}
In the reference
\cite{fc}, instead of matrices, the language of bilinear and linear
forms is used. There, the space $\widetilde C_{g}$ is described as the set of
pairs $(b,l)$, where $b$ is a semidefinite symmetric bilinear form
with rational radical and $l$ is a linear form such that $b+2l$ is
bounded below. The gap between both descriptions is closed by the lemma
below.
\begin{lem}
Let $\Omega $ be a real symmetric positive semidefinite matrix with
rational kernel and $\beta $ a row vector. Then the following
statements are equivalent.
\begin{enumerate}
\item \label{item:11} There is an $\alpha \in \mathbb{R}^{(1,g)}$ such that $\beta =\alpha
\Omega $.
\item \label{item:12} The set $x^{t} \Omega x+2\beta x$, $x\in
\mathbb{R}^{(g,1)}$ is bounded below.
\end{enumerate}
\end{lem}
\begin{proof}
Since $\Omega $ is positive semidefinite, the
function $x\mapsto x^{t} \Omega x+2\beta x$ is convex. Therefore it
is bounded below if and only if it has a stationary point. Taking
derivatives and dividing by 2, we deduce that the function is
bounded below if and only
if the equation $x^{t}\Omega +\beta =0$ has a solution.
\end{proof}
We will denote by $\overline C_{g,\mathbb{Z}}$ the subset of half-integral
matrices of $\overline C_{g}$. Furthermore we set $\widetilde C_{g,\mathbb{Z}}=\widetilde
C_{g}\cap (\overline C_{g,\mathbb{Z}}\times \mathbb{Z}^{(1,g)})$.
It will be convenient to write down the elements of $\widetilde C_{g}$
as pairs of the form $\left(\Omega ,\zeta \Omega \right)$ with $\zeta
\in \mathbb{R}^{(1,g)}$. Such an element will
belong to $\widetilde C_{g,\mathbb{Z}}$ if and only if $\Omega \in \overline C_{g,\mathbb{Z}}$ and
$\zeta $ has rational coefficients, but $\beta =\zeta \Omega$ has integral
coefficients. This is just to be sure
that $\beta $ belongs to the image of $\Omega $. But one has to be
careful that $(\Omega,\zeta)$ are not \emph{affine} coordinates in the
interior of $\widetilde C_{g}$. In particular any reference to
convexity is with respect to the affine structure given by the
coordinates $(\Omega ,\beta )$.
Recall that $P_{g}\subset \Sp(2g,\mathbb{Z})$ denotes the subgroup of symplectic matrices $\begin{pmatrix}
A&B \\ C&D
\end{pmatrix}$ satisfying $C=0$.
There is a group homomorphism
\begin{displaymath}
\begin{matrix}
P_{g} & \longrightarrow & \GL(g,\mathbb{Z})\\
\begin{pmatrix}
A & B\\
0 & D
\end{pmatrix}
&\longmapsto & A.
\end{matrix}
\end{displaymath}
We denote by $\overline \Gamma $ the image of $\Gamma \cap P_{g}$ in
$\GL(g,\mathbb{Z})$. Similarly we denote by $\overline{\widetilde \Gamma }$
the image of $\widetilde \Gamma \cap P_{g+1}$ in $\GL(g+1,\mathbb{Z})$. Then
$\overline{\widetilde \Gamma }$ is contained in the subgroup of
matrices of the form $
\begin{pmatrix}
A& 0\\
\lambda &1
\end{pmatrix}
$
and is a semidirect product $\overline{\widetilde \Gamma }=\overline \Gamma
\ltimes \mathbb{Z}^{(1,g)}$.
The group $\overline \Gamma $ acts on $\overline C_{g}$ by the action
\begin{displaymath}
A\cdot \Omega =A\Omega A^{t}
\end{displaymath}
and the group
$\overline{\widetilde \Gamma }$ acts on $\widetilde C_{g}$ by the
action
\begin{displaymath}
(A,\lambda )\cdot (\Omega ,\beta )=
(A\Omega A^{t},(\beta + \lambda\Omega ) A^{t}).
\end{displaymath}
This action can be written also as
\begin{equation}\label{eq:2}
(A,\lambda )\cdot (\Omega ,\zeta \Omega )=
(A\Omega A^{t},(\zeta + \lambda )A^{-1} A\Omega A^{t}).
\end{equation}
\begin{df}\label{def:7}
An \emph{admissible cone decomposition of $\overline C_{g}$} is a
set $\Sigma$ of cones in $\overline C_{g}$ such that
\begin{enumerate}
\item Each $\sigma \in \Sigma $ is generated by a finite set of
elements of $\overline C_{g,\mathbb{Z}}$ and contains no lines. In other
words it is a rational polyhedral strictly convex cone.
\item If $\sigma $ belongs to $\Sigma $ each face of $\sigma $
belongs to $\Sigma $.
\item If $\sigma $ and $\tau $ belong to $\Sigma $ their
intersection is a common face.
\item The union of all the cones of $\Sigma $ is $\overline C_{g}$.
\item The group $\overline \Gamma $ leaves $\Sigma$ invariant with
finitely many orbits.
\end{enumerate}
\end{df}
\begin{df}\label{def:8}
Let $\Sigma $ be an admissible cone decomposition of
$\overline C_{g}$. An \emph{admissible cone decomposition of
$\widetilde C_{g}$ over $\Sigma $} is a set of cones $\Pi $ in
$\widetilde C_{g}$ such that
\begin{enumerate}
\item Each $\tau \in \Pi $ is generated by a finite set of elements
of $\widetilde C_{g,\mathbb{Z}}$ and contains no lines.
\item If $\sigma $ belongs to $\Pi $ each face of $\sigma $ belongs
to $\Pi $.
\item If $\sigma $ and $\tau $ belong to $\Pi $ their intersection
is a common face.
\item The union of all the cones of $\Pi $ is $\widetilde C_{g}$.
\item The group $\overline {\widetilde \Gamma }$ leaves $\Pi $
invariant with finitely many orbits.
\item For each $\tau \in \Pi $, the projection of $\tau $ to
$\overline C_{g}$ is contained in a cone $\sigma \in \Sigma $.
\end{enumerate}
\end{df}
We say that $\Sigma$ (resp. $\Pi $) is \emph{smooth} if every cone of $\Sigma
$ (resp.\ $\Pi$) is generated by part of a $\mathbb{Z}$-basis of the abelian
group generated by $\overline C_{g,\mathbb{Z}}$ (resp.\ $\widetilde
C_{g,\mathbb{Z}}$). We say that $\Pi $ is \emph{equidimensional} (over $\Sigma
$) if for every cone $\tau$
the projection of $\tau $ to
$\overline C_{g}$ is a cone $\sigma \in \Sigma $.
\begin{df}\label{def:11}
Let $\Sigma $ be an admissible cone decomposition for $\overline
C_{g}$. An \emph{admissible divisorial function} on
$\Sigma $ is a continuous
$\overline {\Gamma }$-invariant function $\phi \colon
\overline C_{g}\to \mathbb{R}$ satisfying the following properties:
\begin{enumerate}
\item it is conical, in the sense that $\phi (\lambda x)=\lambda
\phi (x)$ for all $x\in \overline C_{g}$ and
$\lambda \in \mathbb{R}_{\ge 0}$;
\item it is linear on each cone $\sigma $ of $\Sigma $;
\item takes integral values on $\overline C_{g,\mathbb{Z}}$.
\end{enumerate}
An admissible divisorial function is called
\emph{strictly anti-effective} if furthermore
\begin{enumerate}[resume]
\item $\phi (x)>0$ for $x\not = 0$.
\end{enumerate}
A strictly anti-effective divisorial function is called an
\emph{admissible polarization function} if, in addition, it
satisfies
\begin{enumerate}[resume]
\item $\phi$ is concave;
\item it is strictly concave on $\Sigma $ in the sense that, if
$\tau $ is a cone on $\overline C_{g}$ such that the restriction
of $\phi $ is linear on $\tau $ then $\tau $ is contained in a
cone $\sigma $ of $\Sigma $. In other words, the maximal cones
of $\Sigma $ are the maximal cones of linearity of $\phi $.
\end{enumerate}
\end{df}
\begin{rmk}
For a given admissible cone decomposition $\Sigma $ it may happen
that there are no admissible polarization functions on $\Sigma$. A
cone decomposition that admits an admissible polarization function
is called \emph{projective}. As explained in \cite[\S V.5]{fc}, every
admissible cone decomposition admits a smooth projective
refinement.
\end{rmk}
\begin{df}\label{def:10}
Let $\Sigma $ be an admissible cone decomposition for $\overline
C_{g}$ and $\Pi $ an admissible cone decomposition for $\widetilde
C_{g}$ over $\Sigma $. An \emph{admissible divisorial function} on
$\Pi $ is a continuous
$\overline {\Gamma }$-invariant function $\phi \colon
\widetilde
C_{g}\to \mathbb{R}$ satisfying the following properties:
\begin{enumerate}
\item it is conical: $\phi(tx)=t\phi (x)$ for all $t\in \mathbb{R}_{\ge 0}$ and
$x\in \widetilde C_{g}$;
\item it takes rational values on $\widetilde C_{g,\mathbb{Z}}$ with bounded
denominators;
\item it is linear on each $\tau \in \Pi $;
\item \label{item:1} for each $\lambda \in \mathbb{Z}^{(1,g)}$ and
$q=(\Omega ,\zeta \Omega )\in
\widetilde C_{g}$, the condition
\begin{equation}\label{eq:3}
\phi (q)-\phi (\lambda \cdot q) = \lambda \Omega \lambda
^{t}+2 \zeta \Omega \lambda ^{t}
\end{equation}
holds.
Recall that the action \eqref{eq:2} gives $\lambda \cdot q=(\Omega
,(\zeta +\lambda )\Omega )$.
\end{enumerate}
An admissible divisorial function $\phi $ on $\Pi $ is called an \emph{admissible
polarization function} if it also satisfies the conditions
\begin{enumerate}[resume]
\item $\phi $ is concave;
\item $\phi $ is strictly concave over each cone $\sigma $ of
$\Sigma $. That is, for each maximal cone $\tau $ over $\sigma $,
there is a linear function $\varphi_{\tau }$ such that
$\varphi_{\tau }(q)=\phi (q)$ for each $q\in \tau $, but
$\varphi_{\tau }(q)>\phi (q)$ for each $q=(\Omega ,\zeta \Omega
)\not \in \tau $, with $\Omega \in \sigma $.
\end{enumerate}
\end{df}
\begin{rmk}
For a given admissible cone decomposition $\Pi $, even smooth, it may
be possible that there are no admissible
polarization functions on $\Pi $. Nevertheless there is always a
refinement $\Pi '$ of $\Pi $ such that there exists an admissible
polarization function on $\Pi '$. As for $\Sigma$, the admissible cone decompositions that
admit an admissible polarization function are called
\emph{projective}.
\end{rmk}
The theory of toroidal embeddings allows us to compactify the moduli spaces $\mathcal{A}(\Gamma )$ and $\mathcal{B}(\Gamma )$.
The precise statement is as follows.
\begin{thm}
Let $\Sigma $ be a projective admissible
cone decomposition for $\overline
C_{g}$ and $\Pi $ a projective admissible cone decomposition for $\widetilde
C_{g}$ over $\Sigma $.
\begin{enumerate}
\item To the cone decomposition $\Sigma $ there is attached a
projective scheme $\overline \mathcal{A}(\Gamma )_{\Sigma }$ that contains
$\mathcal{A}(\Gamma )$ as an open dense subset.
\item To the cone decomposition $\Pi $ there is attached a
projective scheme $\overline \mathcal{B}(\Gamma )_{\Pi }$ that contains
$\mathcal{B}(\Gamma )$ as an open dense subset, and a projective morphism
\begin{displaymath}
\pi _{\Sigma ,\Pi }\colon
\overline \mathcal{B}(\Gamma )_{\Pi } \to \overline \mathcal{A}(\Gamma )_{\Sigma }
\end{displaymath}
that extends the canonical projection $\mathcal{B}(\Gamma )\to \mathcal{A}(\Gamma )$.
\item If $\Sigma $ or $\Pi $ are smooth, then the corresponding
schemes $\overline \mathcal{A}(\Gamma )_{\Sigma }$ or $\overline
\mathcal{B}(\Gamma )_{\Pi}$ are smooth. If $\Pi $ is equidimensional over
$\Sigma $, then $\pi _{\Sigma ,\Pi }$ is equidimensional.
\item The space $\overline \mathcal{A}(\Gamma )_{\Sigma }$
admits a stratification by locally closed subschemes, indexed by
the $\overline \Gamma $-orbits of $\Sigma $
\begin{displaymath}
\overline \mathcal{A}(\Gamma )_{\Sigma }=\bigcup_{\overline \sigma \in \Sigma
/\overline \Gamma } \overline \mathcal{A}(\Gamma )_{\overline \sigma }.
\end{displaymath}
The correspondence between cones and strata reverses dimensions
and a stratum $\overline \mathcal{A}(\Gamma )_{\overline \sigma }$ lies in the
closure of another stratum $\overline \mathcal{A}(\Gamma )_{\overline \tau }$ if
and only if there are representatives $\sigma $ and $\tau $ of
$\overline \sigma $ and $\overline \tau $ such that $\tau$ is a
face of $\sigma $.
\item There is an analogous stratification of $\overline \mathcal{B}(\Gamma
)_{\Pi }$ indexed by the $\overline{\widetilde \Gamma }$ orbits of $\Pi $.
\end{enumerate}
\end{thm}
\begin{rmk}\label{rem:5}
The projectivity condition is used to show that formal versions of these moduli spaces are algebraizable.
\end{rmk}
\subsection{Local coordinates}\label{sec:local_coordinates}
Assume now that $\Sigma$ and $\Pi$ are smooth.
We want to describe local coordinates of $\overline \mathcal{A}(\Gamma
)_{\Sigma }$ and $\overline \mathcal{B}(\Gamma
)_{\Pi }$.
Let $\overline \sigma $ and
$\overline \tau $ be orbits of cones in $\Sigma $ and $\Pi $
respectively. To $\overline \sigma $
corresponds a stratum $\overline \mathcal{A}(\Gamma )_{\overline \sigma }$
and to $\overline \tau
$ a stratum $\overline \mathcal{B}(\Gamma )_{\overline \tau }$. We choose a point
$x_{\overline \sigma }\in \overline \mathcal{A}(\Gamma )_{\overline \sigma }$ and a point
$y_{\overline \tau }\in \overline \mathcal{B}(\Gamma )_{\overline \tau }$.
We now describe local
coordinates in both spaces around $x_{\overline \sigma}$ and
$y_{\overline \tau }$. These
coordinates and the uniformization map depend on the choice of
representatives $\sigma $ and $\tau $ of the orbits.
We start with $x_{\overline \sigma }$.
Write $G=g(g+1)/2$ for the dimension of $\mathcal{A}(\Gamma )$ and let $n=
\dim \sigma $. For $r>0$, we denote
by $\Delta _{r}\subset \mathbb{C}$ the disk of radius $r$ centered at $0$.
The chosen cone $\sigma $ is generated by a set of
symmetric half-integral semipositive
matrices $\Omega _{1}, \dots, \Omega _{n}$ that are part of an
integral basis of the lattice of symmetric half integral matrices. Then there exists a
symmetric semidefinite matrix $\Omega _{0}$ and symmetric matrices
$\Omega _{n+1},\dots,\Omega _{G}$, such that the set $\Omega
_{1},\dots,\Omega _{G}$ is a basis of the same lattice, and there are
positive real numbers
$0<r_{1},\dots, r_{G}< 1$ such that
\begin{displaymath}
U\coloneqq \left\{ i\Omega _{0}+\sum_j t_{j} \Omega _{j}\, \middle
|\,
\begin{alignedat}{2}
\im(t_{j})&> - \log r_{j},\ &j&\le n,\\
|t_{j}| &< r_{j},&j &> n.
\end{alignedat}
\right\}\subset \mathcal{H}_{g}.
\end{displaymath}
For $U$ small enough there is
a coordinate neighbourhood $V \subset \overline
\mathcal{A}(\Gamma )_{\Sigma }$ centered at $x_{\overline \sigma }$ of the form
$\Delta _{r_{1}}\times \dots \times \Delta _{r_{G}}$, such that the
uniformization map $\mathcal{H}_{g}\to
\mathcal{A}(\Gamma )$ sends $U$ to $V$ through the map
\begin{displaymath}
(t_{1},\dots,t_{G})\mapsto \left(z_{1},\dots,z_{G})=(e^{2\pi i
t_{1}},\dots,e^{2\pi i t_{n}},t_{n+1},\dots,t_{G}\right).
\end{displaymath}
The situation for $y_{\overline \tau }$ is similar. Let $m=\dim \tau
$. The chosen cone $\tau $ is
generated by part of an integral basis $\{(\Omega' _{1},\beta
_{1}),\dots ,(\Omega' _{m},\beta _{m})\}$, where the $\Omega '_{j}$
are half-integral semipositive
symmetric matrices, and the vectors $\beta _{j}$ are integral and of
the form
$\beta _{j}= \zeta _{j}\Omega _{j}$.
There are pairs $(\Omega' _{m+1},\beta
_{m+1}),\dots,(\Omega' _{G+g},\beta _{G+g}) $, such that
\begin{displaymath}
\{(\Omega' _{1},\beta_{1}),\dots,(\Omega' _{G+g},\beta _{G+g})\}
\end{displaymath}
is an integral basis of the lattice $\widetilde C_{g,\mathbb{Z}}$. There is also a pair $(\Omega '_{0},\beta _{0})$ satisfying
$\beta _{0}=\zeta _{0}\Omega _{0}'$ and real numbers
$0<r_{j}<1$, $j=1,\dots,G+g$ such that
\begin{displaymath}
U'\coloneqq \left\{ i(\Omega _{0},\beta _{0})+\sum_j s_{j}( \Omega _{j},\beta _{j})\, \middle
|\,
\begin{alignedat}{2}
\im(s_{j})&> - \log r_{j},\ &j&\le s,\\
|s_{j}| &< r_{j},&j &> s.
\end{alignedat}
\right\}\subset \mathcal{H}_{g}\times \mathbb{C}^{(1,g)}.
\end{displaymath}
Again, for $U'$ small enough, there is a coordinate neighborhood $V'$
centered at $y_{\overline \tau }$ such that
the uniformization map sends $U'$ to $V'$ via the map
\begin{equation}\label{eq:31}
\left(s_{1},\dots,s_{G+g}\right)\mapsto \left(w_{1},\dots,w_{G+g}\right)=\left(e^{2\pi i
s_{1}},\dots,e^{2\pi i s_{m}},s_{m+1},\dots,s_{G+g}\right).
\end{equation}
Assume now that $\sigma $ and $\tau $ are maximal, so $n=G$ and
$m=G+g$, that $\pi (\tau )\subset \sigma $ and that $\pi _{\Sigma ,\Pi
}(y_{\overline \tau })=x_{\overline \sigma }$.
We describe the map $\pi _{\Sigma ,\Pi }$ in these
coordinates.
Since each $\Omega '_{j}$, $j=1,\dotsc ,m$ is contained in the cone generated by the
$\Omega_k$'s, $k = 1, \dotsc, n$ and we are assuming that the cone
decomposition is smooth,
there are integer vectors $\underline
a_{j}=(a_{j,1},\dots,a_{j,n})\in \mathbb{Z}^{n}_{\ge 0} $ such that
\begin{displaymath}
\Omega '_{j}=\sum_{k}a_{j,k}\Omega _{k} \, , \quad j=1,\dotsc, m.
\end{displaymath}
Then the map $\pi _{\Sigma ,\Pi }$ is given in the $w$- and $z$-coordinates
by
\begin{displaymath}
\left(w_{1},\dots ,w_{G+g}\right)\mapsto
\left(\prod_{j=1}^{G+g}w_{j}^{a_{j,k}}\right)_{k=1,\dots,G}.
\end{displaymath}
\subsection{Extending the line bundle of Siegel--Jacobi forms}
\label{sec:extend-line-bundle}
We next discuss the extension of the line bundles $L_{k,m}$ to the
toroidal compactifications.
We start by recalling how to extend the line bundle of modular forms
on the Siegel modular variety. This is related with the construction of
the Satake--Baily--Borel (or minimal) compactification (which is in general not toroidal).
Let $\Sigma $ be a projective admissible cone decomposition of $\overline
C_{g}$. The universal abelian variety $\mathcal{B}(\Gamma )$ over
$\mathcal{A}(\Gamma )$ can be uniquely extended to a semi-abelian variety
over $\overline \mathcal{A}(\Gamma )_{\Sigma }$ that we denote
$\overline \mathcal{B}(\Gamma )_{\Sigma }^{0}$. The zero section $e\colon \mathcal{A}(\Gamma
)\to \mathcal{B}(\Gamma )$ extends to a section $\overline e\colon \overline \mathcal{A}(\Gamma
)_{\Sigma }\to \overline \mathcal{B}(\Gamma )_{\Sigma }^{0}$. Therefore
the line bundle $M= \det\left(e^*\Omega ^{1}_{\mathcal{B}(\Gamma
)/\mathcal{A}(\Gamma )}\right)$ on $\mathcal{A}(\Gamma )$ can be extended
canonically to the line bundle
\begin{displaymath}
\overline M=\det\left(\overline
e^*\Omega ^{1}_{\overline \mathcal{B}(\Gamma
)^{0}_{\Sigma }/\overline \mathcal{A}(\Gamma )_{\Sigma }}\right)
\end{displaymath}
on $\overline \mathcal{A}(\Gamma
)_{\Sigma }$ by \cite[\S V.1]{fc}. Moreover there is a
$k >0$ such that the
line bundle $\overline M^{\otimes
k}$ is globally generated \cite[Chapter V, Proposition
2.1]{fc}. Here we are using
that $\Gamma $ is neat and in particular torsion-free.
Let $R_{\Gamma }$ be the graded ring
\begin{displaymath}
R_{\Gamma }=\bigoplus _{k\ge 0} H^{0}\left(\overline \mathcal{A}(\Gamma
)_{\Sigma }, \overline M^{\otimes k}\right).
\end{displaymath}
By \cite[Chapter V, Theorem 2.3]{fc}, the ring $R_\Gamma$ is
finitely generated and does not depend on the choice of $\Sigma $. Then
\begin{equation}\label{eq:28}
\mathcal{A}(\Gamma )^{\ast}=\Proj(R_{\Gamma })
\end{equation}
is the Satake--Baily-Borel compactification of $\mathcal{A}(\Gamma )$, a
projective complex variety. In particular this implies that
$\mathcal{A}(\Gamma )$ is quasi-projective and $M$ is an algebraic line
bundle. Finally we have a canonical projection map
\begin{displaymath}
\overline \mathcal{A}(\Gamma )_{\Sigma }\longrightarrow
\mathcal{A}(\Gamma )^{\ast}.
\end{displaymath}
\begin{lem}\label{lemm:10}
The standard invariant metric on $M$ extends to a psh metric on
$\overline M$.
\end{lem}
\begin{proof}
It follows from Lemma \ref{lem:psh} that the standard invariant metric is psh on $M$.
Then, by the theory of psh functions, we only have to show that for every
point $y\in \overline \mathcal{A}(\Gamma
)_{\Sigma }$ there exists $k \in \mathbb{Z}_{>0}$ and a local section $s$ of $\overline M^{\otimes k}$ that
generates $\overline M^{\otimes k}$ around $y$, such that the function $-\log \|s\|$
is bounded above locally around $y$.
For simplicity we discuss only the case of a point corresponding to
a maximal cone of $\Sigma $ as the general case is only notationally
more complex.
Let $\sigma \in \Sigma $ be a cone of maximal dimension. Since
$\sigma $ is maximal, the stratum $\overline \mathcal{B}(\Gamma
)_{\overline \sigma }$ contains a single point $y_{\overline
\sigma}$. Let
$U\subset \mathcal{H}_{g}$ and
$y_{\overline \sigma }\in V\subset \overline \mathcal{A}(\Gamma )_{\Sigma }$ be open
sets as in the previous section.
Let $(u_{1},\dots,u_{g})$ be linear coordinates on $\mathbb{C}^{(1,g)}$
defined by an integral basis.
If $V$ is small enough, the symbol
$du_{1}\wedge \dots\wedge du_{g}$ defines a generating section of
$M$ over $V\cap \mathcal{A}(\Gamma)$. As explained in \cite[V
\S1]{fc} this symbol extends to a generating section of $\overline
M$ on $V$. Let $k \in \mathbb{Z}_{>0}$ and let $s=f(du_{1}\wedge \dots\wedge du_{g})^{\otimes k}$
be a section of $\overline M^{\otimes k}$ over $V$. Note that $f$ lifts to $U \subset \mathcal{H}_g$ and can be interpreted there as a
meromorphic Siegel modular form on
$\mathcal{H}_{g}$ of weight $k$. Then the
section $s$ can be
extended to a local generating
section of $\overline M^{\otimes k}$ around $y_{\overline \sigma }$ if and
only if $-\log |f|$ is bounded in $U$ (for $U$ small enough). Assume therefore that $-\log |f|$ is bounded on $U$.
Then
\begin{displaymath}
-\log \|s\| = -\log |f|-k \log \det\left(\sum \frac{-1}{2\pi }\log |z_{j}|\Omega _{j}\right)
\end{displaymath}
is bounded above. Indeed, after shrinking $U$
if necessary, we can assume that $2\pi \le -\log|z_{j}|$ for all $j$. Since the
$\Omega _{j}$ are positive semidefinite and since for two semidefinite
matrices $A,B$ the inequality $\det(A+B)\ge \det(A)$ holds, we deduce that
\begin{displaymath}
\det\left(\sum \frac{-1}{2\pi }\log |z_{j}|\Omega _{j}\right)
\ge \det \left(\sum \Omega _{j}\right)
\end{displaymath}
from which the boundedness above follows.
\end{proof}
We next recall how the theory of toroidal compactifications allows us
to extend line bundles from $\mathcal{A}(\Gamma )$ and $\mathcal{B}(\Gamma )$ to
$\overline \mathcal{A}(\Gamma )_{\Sigma }$ and $\overline \mathcal{B}(\Gamma )_{\Pi }$.
\begin{df}\label{def:12} To every divisorial function $\phi $ we
associate a divisor $D_{\phi }$ as follows. Each irreducible
component $D_{\alpha }$ of the boundary $\overline \mathcal{A}(\Gamma
)_{\Sigma }\setminus \mathcal{A}(\Gamma )$ corresponds to a $\overline
\Gamma$-orbit of one-dimensional cones of $\Sigma $. Choose one
element of this orbit $\sigma _{\alpha }$. Let $v_{\alpha }$ be the
primitive generator of $\sigma _{\alpha }\cap \overline
C_{g,\mathbb{Z}}$. Then we write
\begin{displaymath}
D_{\phi }=\sum_{\alpha }-\phi (v_{\alpha })D_{\alpha }.
\end{displaymath}
Since $\phi $ is $\overline \Gamma $-invariant, this divisor is
independent of the choice of $\sigma _{\alpha }$.
\end{df}
\begin{prop}\label{prop:8} The divisor $D_{\phi }$ has support
contained in the boundary $\overline \mathcal{A}(\Gamma
)_{\Sigma }\setminus \mathcal{A}(\Gamma )$. If the divisorial function is
strictly anti-effective then $-D_{\phi }$ is strictly effective and the support of
$D_{\phi }$ agrees with the whole boundary. If $\phi $ is an
admissible polarization function, then $\mathcal{O}(D_{\phi })$ is
relatively ample with respect to the canonical projection
\begin{displaymath}
\overline \mathcal{A}(\Gamma )_{\Sigma }\longrightarrow
\mathcal{A}(\Gamma )^{\ast}.
\end{displaymath}
\end{prop}
\begin{proof}
The first two statements follow directly from the definition of
$D_{\phi }$ and the last statement follows from \cite[Theorem
V.5.8]{fc} and the proof of \cite[Proposition
II.7.13]{Hartshorne:ag}.
\end{proof}
Also the next result follows from the theory of toroidal compactifications (see \cite[Chapter VI, Theorem 1.13]{fc}).
\begin{prop} \label{prop:4} Let $\Sigma $ be an admissible
cone
decomposition for $\overline
C_{g}$ and $\Pi $ an admissible cone decomposition for $\widetilde
C_{g}$ over $\Sigma $. Let $\phi $ be an
admissible divisorial
function on $\Pi $ and let $m>0$ be an integer such that
$m\phi $ has
integral values on $\widetilde C_{g,\mathbb{Z}}$. Associated with $m\phi $
there is a line bundle $\overline{B}_{m\phi}$ on $\overline
\mathcal{B}(\Gamma )_{\Pi }$ such
that its restriction to $\mathcal{B}(\Gamma )$ agrees with
$B^{\otimes m}$. Moreover, if $\phi$ is an
admissible polarization
function then $\overline{B}_{m \phi}$ is relatively ample with
respect to the projection map
\begin{displaymath}
\pi_{\Sigma,\Pi}\colon \overline {\mathcal{B}}(\Gamma )_{\Pi }\to
\overline {\mathcal{A}}(\Gamma )_{\Sigma }.
\end{displaymath}
In particular, if $\Pi $ is projective, the latter map is projective.
\end{prop}
\begin{rmk}\label{rem:3} Condition \ref{item:1} in Definition
\ref{def:10} ensures that the restriction of
$\overline{B}_{m\phi}$ to $\mathcal{B}(\Gamma )$ agrees with
$B^{\otimes m}$. In fact this condition implies that the restriction of
$\overline{B}_{m\phi}$ to $\mathcal{B}(\Gamma )$ satisfies
the cocycle condition determining $B^{\otimes m}$.
\end{rmk}
\begin{rmk}\label{rem:4}
Let $\Sigma $ be an admissible cone
decomposition for $\overline
C_{g}$ and $\Pi $ an admissible cone decomposition for $\widetilde
C_{g}$ over $\Sigma $. We can combine admissible divisorial
functions on $\Sigma $ and $\Pi $. Let $\phi $ be an
admissible divisorial
function on $\Pi $ and $\psi $ an admissible divisorial function on
$\Sigma $. By composing with the projection $\widetilde
C_{g}\to \overline C_{g}$, the function $\psi $ defines a function on $\widetilde
C_{g}$
which (by abuse of notation) we also denote $\psi $. Let $m>0$ be an integer such that
$m\phi $ has
integral values on $\widetilde C_{g,\mathbb{Z}}$.
Then $m\phi +\psi $ is again an
admissible divisorial function with integral values on $\widetilde
C_{g,\mathbb{Z}}$ and, by construction,
\begin{displaymath}
\overline B_{m\phi +\psi}=
\overline B_{m\phi }\otimes (\pi _{\Sigma ,\Pi })^{\ast}\mathcal{O}(D_{\psi }).
\end{displaymath}
\end{rmk}
We next see a criterion for when a rational section of $\overline
B_{m\phi }$ is holomorphic at a point of the boundary.
\begin{lem}\label{lemm:23} Let $\Sigma $ and $\Pi $ be as in Proposition
\ref{prop:4}. Let $\phi $ be an admissible divisorial
function on $\Pi $ and let $m>0$ be an integer such that
$m\phi $ has
integral values on $\widetilde C_{g,\mathbb{Z}}$. Let $f$ be a rational section of $\overline
B_{m\phi }\otimes (\pi _{\Sigma ,\Pi })^{\ast}\overline M^{\otimes k}$,
that we view as a meromorphic Siegel--Jacobi form of weight $k$ and
index $m$. Let
$\tau \in \Pi $ and $x\in \overline \mathcal{B}(\Gamma
)_{\overline \tau }$. Let $V'$ be a sufficiently small open coordinate neighborhood of $x$
and $U'\subset \mathcal{H}_{g}\times
\mathbb{C}^{(1,g)}$ an open set as in section \ref{subsec:toro-comp}.
Then $f$ extends to a holomorphic section around $x$ (resp. a
non-vanishing holomorphic section) if and only if the function
\begin{displaymath}
-\log |f(Z,W)|-2\pi m\phi (\im Z,\im W)
\end{displaymath}
is bounded below (resp. bounded) in $U'$.
\end{lem}
\begin{proof}
We start by recalling a few steps in the construction of
$\overline \mathcal{B}(\Gamma )_{\Pi }$ and
$\overline{B}_{m \phi}$ (see the proof of \cite[Chapter VI, Theorem
1.13]{fc}). We describe the situation analytically over the complex
numbers as that is enough for our purposes.
Over an open set $V$ of $\overline \mathcal{A}(\Gamma )_{\Sigma}$ that
contains the image of $V'$ there is a complex manifold $P$. The
lattice $\mathbb{Z}^{(1,g)}$ acts freely and discontinuously on $P$ and
\begin{displaymath}
P/\mathbb{Z}^{(1,g)}=(\pi _{\Sigma ,\Pi })^{-1}(V)\subset \overline
\mathcal{B}(\Gamma )_{\Pi }.
\end{displaymath}
The map $p\colon P \to (\pi _{\Sigma ,\Pi })^{-1}(V)$ is \'etale and
we can find an open subset $V''\subset P$, that depends on the
representative $\tau $ of $\overline \tau $, and that maps isomorphically
to $V'$. Thus the holomorphic coordinates of $V'$ also give us
holomorphic coordinates of $V''$. Moreover, the uniformization $U\to
V'$ factors through $V''$ as the map $U\to V''$ is also given by
formula \eqref{eq:31}. Then the preimage of $\overline{B}_{m \phi}$
in $V''$ is the sheaf $\mathcal{O}(D)$, where
\begin{displaymath}
D=\sum_{j}-m\phi (\Omega '_{j},\beta _{j}) D_{j},\qquad
D_{j}=\{w_{j}=0\}.
\end{displaymath}
As in the proof of Lemma \ref{lemm:10} the symbol $du_{1}\wedge
\dots \wedge du_{g}$ defines a generating section of
$(\pi _{\Sigma ,\Pi })^{\ast}\overline M$ over $V'$.
Therefore $f(du_{1}\wedge
\dots \wedge du_{g})^{\otimes k}$ extends to a holomorphic section of $\overline
B_{m\phi }\otimes (\pi _{\Sigma ,\Pi })\overline M^{\otimes k}$ if and only if the function
\begin{displaymath}\label{eq:27}
g=f\prod _{j}w_{j}^{-m\phi (\Omega '_{j},\beta _{j})}
\end{displaymath}
is holomorphic in $V''$. Since $g$ is meromorphic, it is
holomorphic if and only if the function
\begin{align*}
-\log |g| &= -\log |f|+\sum_{j} m\phi (\Omega '_{j},\beta
_{j})\log|w_{j}|\\
&=
-\log |f|+\sum_{j} -2\pi m\phi (\Omega '_{j},\beta _{j})\im(s_{j})\\
&= -\log |f| -2\pi m\phi \left(\sum_{j}\im(s_{j}(\Omega '_{j},\beta _{j}))\right)
\end{align*}
is bounded below. This proves the lemma for holomorphic
sections. Similarly $g$ is holomorphic and non-vanishing if and
only if $-\log|g|$ is bounded; this proves the second case.
\end{proof}
\begin{df}\label{def:14}
An admissible divisorial function $\phi$ on $\Pi$ is called
\emph{sufficiently negative} if
\begin{displaymath}
\phi (\Omega ,\zeta \Omega )\le -\zeta \Omega \zeta ^{t}.
\end{displaymath}
\end{df}
\begin{rmk}\label{rem:exists_suff_neg}
For each given smooth $\Pi $ we can always find a sufficiently
negative admissible divisorial function. Indeed, for every cone $\tau $ of $\Pi $ take the linear
function that agrees with $-\zeta \Omega \zeta ^{t}$ on the one
dimensional faces of $\tau $. This function has
rational values on $\widetilde C_{g,\mathbb{Z}}$ with bounded
denominators. Nevertheless, even if the function $(\Omega ,\beta
)\mapsto -\beta \Omega ^{-1}\beta ^{t}$ is concave, the function we
have constructed is not necessarily concave. In particular it might
not be a polarization.
\end{rmk}
The interest of the definition of \emph{sufficiently negative} admissible divisorial
functions lies in the following result.
\begin{lem}\label{lemm:9}
Let $\Sigma $ and $\Pi $ be as in Proposition
\ref{prop:4}. Let $\phi $ be an admissible divisorial
function on $\Pi $ and let $m>0$ be an integer such that
$m\phi $ has
integral values on $\widetilde C_{g,\mathbb{Z}}$. If $\phi $ is sufficiently
negative then the
metric of Definition \ref{def:6} on $B^{\otimes m}$
extends to a singular psh metric
on $\overline{B}_{m\phi }$.
\end{lem}
\begin{proof}
As in the proof of Lemma \ref{lemm:10}, since we already know that the
standard invariant metric is psh on $B^{\otimes m}$, we only have
to show that,
for every
point $x\in \overline \mathcal{B}(\Gamma
)_{\Pi }$ and every local section $s$ of $\overline{B}_{m\phi }$ that
generates $\overline{B}_{m\phi }$ around $x$, the function $-\log \|s\|$
is bounded above locally around $x$. Again we discuss only the case
when $x=x_{\tau }$ corresponds to a maximal cone $\tau $ of $\Pi
$. Let $U'\subset \mathcal{H}_{g}\times \mathbb{C}^{(1,g)}$ and $x_{\tau }\in V'$ be
open coordinate sets as in Section \ref{sec:local_coordinates}. A
rational section of $B^{\otimes m}$ determines a
meromorphic Siegel--Jacobi form $f $ of index $m$. By Lemma
\ref{lemm:23} it extends to a
generating section $s_{f}$ of
$\overline{B}_{m \phi}$ around $x_{\tau }$ if and only if the function
\begin{displaymath}
-\log |f|-2\pi m \phi
\end{displaymath}
is bounded in $U'$. By the sufficient
negativity of $\phi$ and the definition of the standard invariant metric
in Definition \ref{def:6} we deduce that
\begin{align*}
-\log \|s_{f}\| &= -\log |f(\Omega , \zeta \Omega )| +2\pi m
\zeta \Omega \zeta ^{t}\\ &=
-\log |f(\Omega , \zeta \Omega )|-2\pi m \phi+2\pi m \phi+
2\pi m
\zeta \Omega \zeta ^{t}
\end{align*}
is bounded above, hence extends to a psh function on $V'$.
\end{proof}
\begin{rmk}\label{rem:1} When $\Pi $ is smooth, we can choose the sufficiently negative function
$\phi $ that agrees with $\zeta \Omega \zeta ^{t}$ on the rays of
$\Pi $ and is linear on every cone. Then the obtained extension is the one considered by Lear
\cite{lear}. In this case the standard invariant metric extends to a psh metric
that is continuous up to a set of codimension at least $2$.
\end{rmk}
Combining the previous results we see that the line bundle of
Siegel--Jacobi forms $L_{k,m}$ can be extended to a toroidal
compactification with the help of an admissible divisorial
function.
\begin{df}\label{def:13}
Let $\Sigma $, $\Pi $ and $\phi $ as in Proposition
\ref{prop:4}. Let $\ell>0$ be an integer such that $\ell \phi $ has
integral values on $\widetilde C_{g,\mathbb{Z}}$. Then for every integer $m$
divisible by $\ell$ and for every integer $k$ we define the line
bundle
\begin{displaymath}
\overline L_{k,m,\phi } = (\pi _{\Sigma ,\Pi })^{\ast}\overline
M^{k}\otimes \overline B_{m\phi }
\end{displaymath}
on $\overline \mathcal{B}(\Gamma )_{\Pi }$.
\end{df}
Note that $m\phi$ is integer-valued, so that $ \overline B_{m\phi }$ is well-defined.
Clearly the restriction of $\overline L_{k,m,\phi }$ to $\mathcal{A}(\Gamma
)$ agrees with $L_{k,m}$.
\begin{rmk}\label{rem:ext}
The extension depends on the choice of $\phi $. Moreover, for $m$
not divisible by $\ell$ we can only extend $L_{k,m}$ as a $\mathbb{Q}$-line
bundle. We note that the notion of psh metric readily generalizes to the context of $\mathbb{Q}$-line
bundles.
\end{rmk}
Combining Lemmas \ref{lemm:10} and \ref{lemm:9} we obtain a criterion
for when the invariant metric on $L_{k,m}$ extends to a psh metric on
$\overline L_{k,m,\phi }$.
\begin{prop}\label{prop:5} Let $\Sigma $, $\Pi $, $m$, $\phi $ and $k$
be as in Definition \ref{def:13}.
Assume that $\phi $ is a sufficiently
negative admissible divisorial
function on $\Pi $.
Then the standard invariant metric $h^{\text{\rm inv}}$ of
$L_{k,m}$ of Definition~\ref{def:6} extends to a singular psh metric
on $L_{k,m,\phi }$
that we denote by~$\overline{h}^{\text{\rm inv}}$.
\end{prop}
In a different direction one may ask when the extension $L_{k,m,\phi
}$ of Definition \ref{def:13} is ample.
\begin{lem}\label{lemm:19}
Let $\Sigma $ be a projective admissible
cone
decomposition for $\overline
C_{g}$ and $\Pi $ a projective admissible cone decomposition for $\widetilde
C_{g}$ over $\Sigma $. Let $\phi $ be an
admissible polarization function on $\Pi $ and $\ell_{0}\ge 1$ an integer
such that $\ell_{0}\phi$ has integral values on $\widetilde
C_{g,\mathbb{Z}}$. Then there
exists an admissible polarization function $\psi $ on $\Sigma$
and for every $m>0$ a number $k_{0}\gg 0$ such that, for every
$k\ge k_{0}$, the line bundle
$\overline L_{\ell_{0}k,\ell_{0}m,\phi+\psi }$ is ample on $\overline \mathcal{B}(\Gamma
)_{\Pi}$. Moreover we can choose $\psi $ such that, for all $\ell >0$ divisible by $\ell_{0}$,
\begin{equation}\label{eq:32}
H^{0} \left(\overline \mathcal{B}(\Gamma)_{\Pi},\overline
L_{\ell k,\ell m,\phi+\psi }\right)\subset J_{\ell
k,\ell m}(\Gamma).
\end{equation}
\end{lem}
\begin{proof}
Let $\psi _{0}$ be a polarization function
on $\Sigma $. By Proposition \ref{prop:8} the line
bundle $\mathcal{O}(D_{\psi_{0} })$ is relatively ample for the map
$\overline \mathcal{A}(\Gamma )_{\Sigma }\to \mathcal{A}(\Gamma )^{\ast}$. By
Proposition \ref{prop:4} the line bundle $\overline B_{\ell_{0}\phi }$
is relatively ample with respect to the map
\begin{displaymath}
\pi_{\Sigma,\Pi}\colon \overline \mathcal{B}(\Gamma )_{\Pi }\longrightarrow \overline
\mathcal{A}(\Gamma )_{\Sigma }.
\end{displaymath}
Therefore we can find an integer $a>0$ such that the line bundle
$\overline B_{\ell_{0}\phi }\otimes (\pi _{\Sigma ,\Pi })^{\ast}\mathcal{O}(a\ell_{0}D_{\psi _{0}})$
is relatively ample with respect to the map
\begin{displaymath}
\overline \mathcal{B}(\Gamma )_{\Pi }\longrightarrow
\mathcal{A}(\Gamma )^{\ast}.
\end{displaymath}
Writing $\psi =a\psi _{0}$, following Remark \ref{rem:4} we have
$\overline B_{\ell_{0}\phi }\otimes (\pi _{\Sigma ,\Pi
})^{\ast}\mathcal{O}(a\ell_{0}D_{\psi _{0}})
= \overline B_{\ell_{0}(\phi +\psi )}$.
By the definition of $\mathcal{A}(\Gamma )^{\ast}$ in
\eqref{eq:28} the line bundle $\overline M$ is ample on
$\mathcal{A}(\Gamma )^{\ast}$. Therefore there exists $k_{1}$ such
that
\begin{displaymath}
\overline L_{k_{1},\ell_{0},(\phi +\psi )}=\overline
B_{\ell_{0}(\phi +\psi )}\otimes (\pi _{\Sigma ,\Pi
})^{\ast}\overline M^{\otimes k_{1}}
\end{displaymath}
is ample in $\overline \mathcal{B}(\Gamma )_{\Pi }$. Therefore it is
enough to choose $k_{0}=mk_{1}$.
When $g\ge 2$ the final statement follows from the Koecher
principle as the restriction map
\begin{displaymath}
H^{0} \left(\overline \mathcal{B}(\Gamma)_{\Pi},\overline
L_{\ell k,\ell m,\phi+\psi }\right) \longrightarrow
H^{0} \left(\mathcal{B}(\Gamma),L_{\ell k,\ell m}\right)
\end{displaymath}
is injective.
The case $g=1$ needs more care as we do not have the Koecher
principle. By Lemma \ref{lemm:24} below we only need to choose
$\psi $ such that
\begin{equation}
\phi +\psi > -\zeta \Omega \zeta ^{t}\label{eq:30}.
\end{equation}
Since $\phi$ and $-\zeta \Omega \zeta ^{t}$ satisfy the same
transformation rule with respect to $\overline {\widetilde
\Gamma} $, it is only necessary to impose equation \eqref{eq:30}
in a finite number of cones. Since both functions are conic it is enough
to impose the condition on a finite number of simplices; in particular on
a compact set. Since the polarization function $\psi_{0}
$ is strictly positive, it is enough to choose the number $a$ so
that $\phi +a\psi _{0}=\phi +\psi \ge -\zeta \Omega \zeta ^{t}$
in this compact set. Therefore, we see that condition \eqref{eq:30} can be
attained for a suitable choice of $\psi $.
\end{proof}
\begin{lem}\label{lemm:24} Assume the hypotheses of Lemma
\ref{lemm:19}. If $g=1$ and $\phi(\Omega ,\zeta \Omega ) \ge -\zeta \Omega \zeta ^{t}$ then
\begin{displaymath}
H^{0} \left(\overline \mathcal{B}(\Gamma)_{\Pi},\overline
L_{\ell k,\ell m,\phi}\right)\subset J_{\ell
k,\ell m}(\Gamma).
\end{displaymath}
\end{lem}
\begin{proof}
Let $f \in H^{0} \left(\overline \mathcal{B}(\Gamma)_{\Pi},\overline
L_{\ell k,\ell m,\phi}\right)$ and assume that $f\not \in J_{\ell
k,\ell m}(\Gamma)$. This means that, in the Fourier expansion
\eqref{eq:29} there is a non-zero coefficient $c(T,R)$, with
$4mT/\lambda _{\Gamma }-R^{2}<0$. Recall that, since $g=1$, $T$ is a positive
integer and $R$ is an integer. Therefore there is a real number $q$ such
that
\begin{displaymath}
T/\lambda _{\Gamma }+Rq+mq^2<0.
\end{displaymath}
We can choose the real number $q$ to be transcendental so
there is no possible relation $T+Rq=T'+R'q$ for $T\not =T'$ and
$R\not = R'$. Then the rate of growth of $|f|$ along the ray
$t\cdot (i,iq)$, $t>0$ has to be at least the rate of growth of
\begin{displaymath}
e^{-2 \pi /\lambda_{\Gamma}t(T/\lambda _{\Gamma }+Rq)}>e^{2\pi
mq^{2}t}.
\end{displaymath}
That is,
\begin{displaymath}
-\log |f((it,iqt))| < -2\pi mq^{2}t +K
\end{displaymath}
for some constant $K$.
On the other hand, since we are assuming that $f$ extends to a
holomorphic section, Lemma \ref{lemm:23} implies that
\begin{displaymath}
2\pi m\phi ((1,q))t\le -\log |f(t(i,iq))| +K'
\end{displaymath}
for a constant $K'$. Taking the limit when $t$ goes to $\infty$,
we obtain
\begin{displaymath}
\phi ((1,q))< -q^{2}
\end{displaymath}
contradicting the hypothesis as $-\zeta \Omega \zeta
^{t}(1,q)=-q^{2}$.
\end{proof}
\begin{rmk}
As seen in the proof of Lemma \ref{lemm:19}, when $g\ge 2$ the
inclusion \eqref{eq:32} is always satisfied, even when $\phi
+\psi $ is just a divisorial function and not necessarily a
polarization. By contrast, when $g=1$ the condition \eqref{eq:30}
is sufficient to have the inclusion \eqref{eq:32}, but that
condition is incompatible with being sufficiently negative. So,
it is not clear in the case $g=1$ that one can attain at the same
time the inclusion \eqref{eq:32} and that the invariant metric
extends to a psh metric.
\end{rmk}
\subsection{The standard invariant metric is toroidal}
\label{sec:stand-invar-metr}
\begin{prop}\label{prop:7}
Let $\Sigma $, $\Pi $ and $\phi $ be as in Proposition
\ref{prop:4}. Assume that $\phi $ is sufficiently
negative. Let $\ell>0$ be an integer such that $\ell\phi$ is integral. Then, for every $k$ and every $m$ divisible by $\ell$, the
singular psh metric $\overline h^{\text{\rm inv}}$ on $\overline L_{k,m,\phi
}$ is toroidal.
\end{prop}
\begin{proof}
Again, we prove this only in a neighborhood of a point $x_{\tau }\in
\overline \mathcal{B}(\Gamma )_{\Pi }$ corresponding to a maximal cone
$\tau $ of $\Pi$. We use the coordinate neighborhood $V'$ of
$x_{\tau }$ as described at the end of Section
\ref{subsec:toro-comp}. Let $G=g(g+1)/2$. For $i=1,\dots,G+g$ we write
\begin{displaymath}
u_{i}=\im(s_{i})=\frac{-1}{2\pi }\log|w_{i}|,
\end{displaymath}
where $(w_{1},\dots,w_{G+g})$ are the coordinates of $V'$. Recall
also that the cone $\tau $ is generated by the points $(\Omega'
_{i},\zeta _{i}\Omega _{i}')$, $i=1,\dots,G+g$.
Let $f$ be a meromorphic Siegel--Jacobi form
that defines a rational section $s_{f}$ of $\overline L_{k,m,\phi
}$ that is a generating local section on $V'$. This means that on
the set $U'$ the function
\begin{displaymath}
-\log|f|-2\pi m\phi
\end{displaymath}
is bounded (see Lemma~\ref{lemm:23}). Therefore
\begin{align*}
-\log\|s_{f}\|
&= -\log|f|-\frac{k}{2}\log\det \sum_{i=1}^{G+g}
u_{i}\Omega' _{i}\\
&\quad
+2\pi m \left(\sum_{i=1}^{G+g} u_{i}\zeta _{i}\Omega' _{i}\right)
\left(\sum_{i=1}^{G+g} u_{i}\Omega' _{i}\right)^{-1}
\left(\sum_{i=1}^{G+g} u_{i}\zeta _{i}\Omega' _{i}\right)^{t}\\
&=\gamma -\frac{k}{2}\log\det \left(\sum_{i=1}^{G+g}
u_{i}\Omega' _{i}\right) +2\pi m\phi (u_{1},\dots,u_{G+g})\\
&\quad
+2\pi m \left(\sum_{i=1}^{G+g} u_{i}\zeta _{i}\Omega' _{i}\right)
\left(\sum_{i=1}^{G+g} u_{i}\Omega' _{i}\right)^{-1}
\left(\sum_{i=1}^{G+g} u_{i}\zeta _{i}\Omega' _{i}\right)^{t},
\end{align*}
where $\gamma $ is bounded. We already know that $\phi $ is linear
on the cone $\tau $ and that the other two functions appearing in
the last equation are convex, as seen in Lemma
\ref{lem:psh}. Moreover, since $\phi $ is assumed to be sufficiently
negative, we know that
\begin{displaymath}
\phi (u_{1},\dots,u_{G+g})
+\left(\sum_{i=1}^{G+g} u_{i}\zeta _{i}\Omega' _{i}\right)
\left(\sum_{i=1}^{G+g} u_{i}\Omega' _{i}\right)^{-1}
\left(\sum_{i=1}^{G+g} u_{i}\zeta _{i}\Omega' _{i}\right)^{t}
\end{displaymath}
is bounded above. Thus it only remains to show that the function
\begin{displaymath}
\varphi_{1}(u_{1},\dots,u_{G+g})=-\log\det \left(\sum_{i=1}^{G+g}
u_{i}\Omega' _{i}\right)
\end{displaymath}
is bounded above and Lipschitz continuous in $U'$, and that the
function
\begin{displaymath}
\varphi_{2}(u_{1},\dots,u_{G+g})=
\left(\sum_{i=1}^{G+g} u_{i}\zeta _{i}\Omega' _{i}\right)
\left(\sum_{i=1}^{G+g} u_{i}\Omega' _{i}\right)^{-1}
\left(\sum_{i=1}^{G+g} u_{i}\zeta _{i}\Omega' _{i}\right)^{t}
\end{displaymath}
is Lipschitz continuous on $U'$.
With respect to $\varphi_{1}$, we can assume that the $u_{i}\ge M$
for some constant $M$. If $A$ is a positive definite
matrix and $B$ is a semidefinite positive matrix then
$\det(A+B)\ge \det(A)$, hence
\begin{displaymath}
\varphi_{1}(u_{1},\dots,u_{G+g})\le \varphi_{1}(M,\dots,M),
\end{displaymath}
and $\varphi_{1}(u_{1},\dots,u_{G+g})$ is bounded above. To prove that it is Lipschitz continuous, following \cite[A~4.1]{Boyd:convex}
we first compute
\begin{displaymath}
\frac{\partial \varphi_{1}}{\partial u_{i}} = -\tr \left(\left(\sum_{j=1}^{G+g}
u_{j}\Omega' _{j}\right)^{-1}\Omega _{i}\right) \, .
\end{displaymath}
Then, applying Lemma \ref{lemm:17} below to
\begin{displaymath}
A=\sum_{j} M\Omega _{j},\quad B= \sum_{j} (u_{j}-M)\Omega
_{j},\text{ and }C=\Omega _{i},
\end{displaymath}
we deduce that
\begin{displaymath}
0\ge \frac{\partial \varphi_{1}}{\partial u_{i}}(u_{1},\dots,u_{G+g})\ge
\frac{\partial \varphi_{1}}{\partial u_{i}}(M,\dots,M).
\end{displaymath}
Hence the function $\varphi_{1}$ is Lipschitz continuous in $U'$.
Regarding $\varphi_{2}$, by \cite[Theorem
3.2.2]{bghdj_sing} we know that $\varphi_{2}$ is continuous and
bounded on the open simplex $\sum_{i}u_{i}=1$, $u_{i}>0$. Since it
is homogeneous of degree 1, it is Lipschitz continuous on the open
quadrant $\mathbb{R}^{G+g}_{>0}$ that contains $U'$. Therefore it is
Lipschitz continuous on $U'$.
\end{proof}
\begin{lem}\label{lemm:17}
Let $A$ be a real positive definite symmetric matrix of
dimension $r$ and $B,C$ real positive semi-definite symmetric
matrices of the same dimension. Then
\begin{displaymath}
0\le \tr\left((A+B)^{-1}C\right)\le \tr\left(A^{-1}C\right).
\end{displaymath}
\end{lem}
\begin{proof}
Let $A^{1/2}$ be the symmetric positive definite square root of
$A$. Then
\begin{align*}
\tr\left((A+B)^{-1}C\right)&=\tr\left(\left(\Id +
A^{-1/2}BA^{-1/2}\right)^{-1}A^{-1/2}CA^{-1/2}\right) \text{ and }\\
\tr\left(A^{-1}C\right)&=\tr\left(A^{-1/2}CA^{-1/2}\right).
\end{align*}
Thus it is enough to prove the statement for $A=\Id$. Write now
\begin{displaymath}
B=X^{-1}DX,
\end{displaymath}
with $D$ a diagonal matrix with non-negative entries and $X$ an
orthogonal matrix, i.e. $X^{-1}=X^{t}$. Since
\begin{displaymath}
\tr\left(\left(\Id + B\right)^{-1}C\right)=\tr\left(\left(\Id+D\right)^{-1}XCX^{-1}\right)\quad \text{and} \quad
\tr\left(XCX^{-1}\right) = \tr(C),
\end{displaymath}
we are reduced to the case when $B$ is a diagonal matrix with
non-negative entries. This case is an easy verification,
as $C$ being positive semi-definite implies that its diagonal entries
are all non-negative.
\end{proof}
We now assume the hypothesis of Proposition \ref{prop:5}. Fix a
rational section $s$ of $\overline L_{k,m,\phi }$, and let
$h=\overline{h}^{\text{\rm inv}}$ be the psh
metric on $\overline L_{k,m,\phi }$ induced by the standard invariant
metric. As in Section~\ref{sec:from-psh-metrics} we denote by $\mathbb{D}(\overline L_{k,m,\phi },s,h)$ the
b-divisor associated to $s$ and $h$.
Let $(\overline\mathcal{B}(\Gamma )_{\Pi })_\pi$ be a model of $\overline\mathcal{B}(\Gamma )_{\Pi }$ on which the pullback $E$ of the union
of $\operatorname{div}(s)$ and the boundary of $\overline
\mathcal{B}(\Gamma )_{\Pi }$ has simple normal crossings. Then we can view
$\mathbb{D}(\overline L_{k,m,\phi },s,h)$ as a b-divisor on $(\overline\mathcal{B}(\Gamma )_{\Pi })_\pi$.
\begin{cor}\label{cor:toroidal}
The b-divisor $\mathbb{D}(\overline L_{k,m,\phi },s,h)$ is toroidal with respect to $E$.
\end{cor}
\begin{proof}
By Proposition \ref{prop:7} the metric $h$ is toroidal with respect to the boundary of $\overline
\mathcal{B}(\Gamma )_{\Pi }$. Hence, by Proposition
\ref{prop:div_of_toroidal_metric_is_toroidal}, it follows that
$\mathbb{D}(\overline L_{k,m,\phi },s,h) - \operatorname{div}(s)$ is toroidal
with respect to the boundary of $\overline \mathcal{B}(\Gamma )_{\Pi
}$, and hence with respect to $E$. Since $\dv(s)$ is
also toroidal with respect to $E$ we deduce the result.
\end{proof}
\section{Proof of the main result}
\label{sec:proofs_main}
We continue with the notation and assumptions of the previous section. We assume from now on that $\Sigma $ is a smooth and projective admissible cone decomposition for $\overline C_{g}$, and $\Pi $ is a smooth and projective admissible cone decomposition for $\widetilde C_{g}$ over $\Sigma $.
\subsection{A b-divisor which is not Cartier}
In view of Corollary \ref{cor:toroidal}, in order to determine
$\mathbb{D}(\overline L_{k,m,\phi },s,h)-\dv(s)$ it is enough to
compute Lelong numbers along toroidal divisors. Let $\tau
$ be a maximal cone of $\Pi $ and $x_{\tau }$ the corresponding
point. Let $G=g(g+1)/2$ and let $\zeta _{j}$, $\Omega '_{j}$,
$j=1,\dots, G+g$ be local coordinates as in Section \ref{sec:local_coordinates}.
Let $u=(\Omega _{0},\zeta \Omega _{0})\in \widetilde C_{g,\mathbb{Z}}$ be a
primitive vector in the interior of
the cone $\tau $. Let $\Pi '$ be an admissible cone decomposition
subdividing $\Pi$, such
that the ray generated by $u$ is a ray of $\Pi '$. Let $P_{u}$ be the
irreducible divisor of $\overline \mathcal{B}(\Gamma )_{\Pi '}$ corresponding
to the ray generated by $u$.
\begin{lem}\label{lemm:13} The Lelong number of $h$
at $P_{u}$ is
given by
\begin{displaymath}
\nu (h,P_{u}) = -m\phi(\Omega_{0} ,\zeta \Omega_{0} )-m\zeta
\Omega_{0} \zeta ^{t}.
\end{displaymath}
In particular it does not depend on the subdivision $\Pi '$.
\end{lem}
\begin{proof}
Since the metric of $\overline M$ is good in the sense of
\cite{hi}, it has zero Lelong numbers everywhere (see \cite[Example~2.34]{BBHJ}). So it is sufficient
to treat the case $k=0$. Let $s$ be a rational section of
$\overline{B}_{m\phi}$ that is generating on a neighborhood of
$x_{\tau }$. As in the proof
of Lemma \ref{lemm:9}, $s$ corresponds to a meromorphic Siegel--Jacobi
form $f$ of index $m$ such that $-\log |f|-2\pi m\phi $ is bounded in
$U'$. Since $u$ belongs to the interior
of $\tau $ and is integral, there are positive integers
$a_{1},\dots, a_{G+g} $ such that
\begin{displaymath}
u=(\Omega _{0},\zeta \Omega _{0})=\sum _{j=1}^{G+g}a_{j}(\Omega
'_{j},\zeta _{j}\Omega '_{j}).
\end{displaymath}
Let $z=(z_{1},\dots,z_{G+g})$ be a point of $V'$ and consider the
curve
\begin{displaymath}
\beta (t)=(z_{1}t^{a_{1}},\dots,z_{G+g}t^{a_{G+g}}),\quad |t|\le 1.
\end{displaymath}
For a general point $z$, the strict transform of the curve $\beta $ in
$\overline \mathcal{B}(\Gamma )_{\Pi '}$ goes through a general point of
$P_{u}$.
By the explicit description of the standard invariant metric we have
\begin{align*}
-\log \|s(\beta (t))\|
&= -\log|f(\beta(t)|+2\pi m\frac{-1}{2\pi }\log|t| \zeta \Omega_{0} \zeta ^{t}+C\\
&= \eta(t) + 2\pi m\phi \left(\frac{-1}{2\pi }\log |t| u\right)
+2\pi m \frac{-1}{2\pi }\log|t|\zeta \Omega_{0} \zeta ^{t}+C\\
&= \eta(t) - m\left(\phi (u)+\zeta \Omega_{0} \zeta
^{t}\right)\log |t|+C,
\end{align*}
where $C$ is a constant depending on the point $z$ and where the function
\begin{displaymath}
\eta(t)\coloneqq-\log|f(\beta (t))|-2\pi m \phi \left(\frac{-1}{2\pi }\log |t| u\right)
\end{displaymath}
is bounded. Therefore
\begin{equation}\label{eq:16}
\lim _{t\to 0}\frac{|\eta(t)+C|}{-\log|t|}=0.
\end{equation}
Thus by Lemma \ref{lemm:8} the Lelong number $\nu
(h,P_{u})$ is given by
\begin{displaymath}
\nu(h,P_{u})=- m\left(\phi (u)+\zeta \Omega_{0} \zeta
^{t}\right),
\end{displaymath}
proving the lemma.
\end{proof}
\begin{rmk}
We note that in the proof above, instead of relying on the fact that good metrics
have zero Lelong numbers to reduce to the case $k=0$, we can prove
directly the result for arbitrary $k\ge 0$. Namely, in this case the
singularities of the metric of $\overline M^{\otimes k}$ can be
absorbed into the function $\eta$. Then the function $\eta $ will
no longer be bounded but will have a growth of the shape
$K\log(-\log|t|))$ when $t$ goes to zero. The estimate
\eqref{eq:16} would still be true and we could finish the proof
in the same way.
\end{rmk}
Since $(\Omega_{0},\zeta \Omega_{0} ) $ belongs to the interior of the
cone, the matrix $\Omega_{0} $ is positive definite, in particular
invertible. Writing $\beta = \zeta \Omega $ (recall that the
affine coordinates of $\widetilde C_{g}$ are $(\Omega ,\beta )$) we have
\begin{displaymath}
\zeta \Omega \zeta ^{t}=\beta \Omega ^{-1}\beta ^{t}.
\end{displaymath}
The function
\begin{displaymath}
(\Omega ,\beta )\mapsto \beta \Omega ^{-1}\beta ^{t}
\end{displaymath}
is a smooth convex function with non-zero Hessian, hence is
not piecewise linear. On the other hand, the function $\phi $ is
linear. Therefore the function that to each primitive vector $v$ in the maximal cone
$\tau $ associates the value $\nu (h,P_{v})$ is not the restriction
of a piecewise linear function on $\tau $.
\begin{cor}\label{cor:1}
The b-divisor $\mathbb{D}(\overline
L_{k,m, \phi},s,h$) is not
Cartier.
\end{cor}
\begin{proof}
Assume that it is a Cartier b-divisor.
Replacing $\pi \colon X_{\pi }\to \overline
\mathcal{B}(\Gamma )_{\Pi }$ by a finer toroidal modification we may assume
$D=\mathbb{D}(\overline L_{k,m,\phi },s,h)$ is realized as a $\mathbb{Q}$-divisor on
$X_{\pi }$. Moreover we can assume that the union of the support of $D$ and the
preimage of the boundary divisor of $\overline
\mathcal{B}(\Gamma )_{\Pi }$ is contained in a simple normal crossings divisor $E$.
Again, the
theory of toroidal compactifications \cite{toroidal} assigns to
$(X_{\pi },E)$ and $D$ a conical rational complex $\Delta $ and a piecewise
linear function $\varphi_{D}$ such that each ray $\rho $ of $\Delta $
corresponds to an irreducible component $E_{\rho }$ of $E$ and the value of
$\varphi_D$ at the primitive generator of $\rho $ is
$-\ord_{E_{\rho}}D$. Moreover, each primitive vector $v$ contained in
a cone of $\Delta $ corresponds to an irreducible exceptional
divisor in some toroidal modification of $(X_{\pi
},E)$. Similarly,
associated to the toroidal embedding $\mathcal{B}(\Gamma ) \subset
\overline \mathcal{B}(\Gamma )_{\Pi }$ there is a rational conical complex
$\Lambda $ such that
\begin{displaymath}
\Lambda =\Pi /\overline{\widetilde \Gamma }.
\end{displaymath}
Note that, by the conditions on the function $\phi$, the
function $-m\phi (\Omega ,\zeta \Omega )-m\zeta \Omega \zeta ^{t}$
is invariant under the action of $\overline{\widetilde \Gamma
}$. Hence it descends to a continuous conical function $g$ on $\Lambda
$. As discussed above the function $g$ is not piecewise linear
in any maximal cone of $\Lambda $.
Since the preimage of the boundary component is contained in $E$,
there is a retraction map $r \colon \Delta \to \Lambda $ such that
for each cone $\sigma $ of $\Delta$ there is a cone $\tau $ of
$\Lambda $ with $r(\sigma )\subset \tau $. Choose a cone $\sigma $
of $\Delta $ of maximal dimension such that $r(\sigma )$ is also
maximal. The line bundle $\overline L_{k,m,\phi }$ and the section $s$
define a Cartier divisor $D_{0}$ that we may assume has support on
$E$ because we can always enlarge $E$. To the divisor $D_{0}$
corresponds a piecewise linear function $\varphi_{D_{0}}$. Since
$\mathbb{D}(\overline L_{k,m,\phi },s,h)$ is defined using Lelong numbers, Lemma \ref{lemm:13} shows that on the cone $\sigma $,
\begin{displaymath}
\varphi_{D}=\varphi_{D_{0}}+g.
\end{displaymath}
Since $\varphi_{D_{0}}$ is piecewise linear but $g$ is not, this contradicts
the piecewise linearity of $\varphi_D$. We conclude that $\mathbb{D}(\overline
L_{k,m},s,h)$ is not Cartier.
\end{proof}
\subsection{Graded linear series and b-divisors associated to Siegel--Jacobi
forms}\label{sec:graded-linear-series-1}
Let $U=\mathcal{B}(\Gamma )$ be the universal family of abelian varieties with
level $\Gamma $.
Let $F=K(U)$ denote the field of rational
functions on $U$.
Let $k,m\ge 0$ be integers and
fix a rational section of $L_{k,m}$, that is, a meromorphic Siegel--Jacobi form $s$ of weight $k$,
index $m$ and level $\Gamma$.
We denote
\begin{align*}
\mathcal{J}_{k,m}(\Gamma, s)_{\ell}
&=\left\{f\in F^{\times}\mid f s^{\ell}\in J_{\ell k,\ell m}(\Gamma) \right\}\cup \{0\},\\
\mathcal{J}_{k,m}(\Gamma, s) &=\bigoplus _{\ell} \mathcal{J}_{k,m}(\Gamma, s)_{\ell}t^{\ell}\subset F[t],\\
\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)_{\ell}
&=\left\{f\in F^{\times}\mid f s^{\ell}\in J^{\text{\rm cusp}}_{\ell k,\ell m}(\Gamma) \right\}\cup \{0\},\\
\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s) &=\bigoplus _{\ell}
\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)_{\ell}t^{\ell}\subset F[t].
\end{align*}
Both $\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)$ and $\mathcal{J}_{k,m}(\Gamma, s)$ are graded
linear series.
The first observation to make is that the above graded linear series
are non-trivial as long as $k>0$.
\begin{lem}\label{lemm:18}
Let $k>0$ and $m\ge 0$. There is an $\ell >0$ such that
$\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)_{\ell}\not =\{0\}$.
\end{lem}
\begin{proof}
Let $\ell>0$ be big
enough so that we can write
\begin{displaymath}
\ell k = k_{1}+k_{2}
\end{displaymath}
with $k_{1}>2g$, $k_{2}>g+2$ both even. As recalled in Example
\ref{exm:1}.\ref{item:13}, using Poincar\'e
series one can produce a non-zero cusp form $\varphi_{1}$ of weight $k_{1}$ and index
$0$. By Example \ref{exm:1}.\ref{item:10} there is a
non-zero Siegel--Jacobi form $\varphi_{2}$ of weight $k_{2}$ and
index $\ell m$. Then by Lemma~\ref{lem:multiplicativity} we have $\varphi_{1}\varphi_{2}$ a non-zero cusp form of
weight $\ell k$ and index $\ell m$ and the non-zero function
$\varphi_{1}\varphi_{2}/s^{\ell}$ belongs to
$\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)_{\ell}$.
\end{proof}
We next choose $\Sigma $ and $\Pi $ admissible cone decompositions as in
Proposition~\ref{prop:4}. Let $\phi $ be a sufficiently negative admissible
divisorial function on $\Pi $ (which exists by Remark \ref{rem:exists_suff_neg}). Assume that $m$ is divisible enough so that
$m\phi $ has integral values on $\widetilde C_{g,\mathbb{Z}}$.
To ease notation we write $X=\overline \mathcal{B}(\Gamma )_{\Pi }$.
Let $\overline L_{k,m,\phi }$ be the extension of $L_{k,m}$ on $X$
determined by the divisorial function $\phi $. We can view the
meromorphic Siegel--Jacobi form $s$ as a rational section of $\overline
L_{k,m, \phi}$. Let $h= \overline{h}^{\text{\rm inv}}$ be the psh metric of $\overline L_{k,m,\phi }$ obtained
by Proposition~\ref{prop:5}. Following Section~\ref{sec:psh-metrics-b-1} we
have an associated graded linear series $\mathcal{R}(\overline L_{k,m,\phi },s,h)$.
\begin{lem}\label{lemm:11}
The graded linear series $\mathcal{R}(\overline L_{k,m,,\phi },s,h)$ and
$\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma,s)$ are equal.
\end{lem}
\begin{proof} Let $\ell \in \mathbb{Z}_{\ge 0}$.
Then $\mathcal{R}(\overline L_{k,m,\phi },s,h)_{\ell}$ is the set of $f
\in F$ such that $fs^{\ell}$ is a
meromorphic Siegel--Jacobi form, holomorphic on $\mathcal{H}_{g}\times
\mathbb{C}^{(1,g)}$ and such that $\|fs^{\ell}\|$ is bounded. By Proposition
\ref{prop:6} the latter set is exactly the set of $f\in F$ such that
$fs^{\ell}$ is a Siegel--Jacobi cusp form.
\end{proof}
To the graded linear series $\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)$ and
$\mathcal{J}_{k,m}(\Gamma,s)$ we can associate the b-divisors
$\bdiv\left(\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)\right)$ and
$\bdiv\left(\mathcal{J}_{k,m}(\Gamma, s)\right)$.
\begin{lem}\label{lemm:12}
The equality
\begin{displaymath}
\bdiv\left(\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)\right) = \bdiv\left(\mathcal{J}_{k,m}(\Gamma, s)\right)
\end{displaymath}
holds.
\end{lem}
\begin{proof}
Since $\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)\subset \mathcal{J}_{k,m}(\Gamma, s)$, by Lemma
\ref{lemm:3} we have that
\begin{displaymath}
\bdiv\left(\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)\right) \le
\bdiv\left(\mathcal{J}_{k,m}(\Gamma, s)\right).
\end{displaymath}
To prove the converse inequality let $\pi \in
R(X)$ and let $P$ be a prime divisor of $X_{\pi }$. Set
\begin{displaymath}
r=\ord_{P} \bdiv\left(\mathcal{J}_{k,m}(\Gamma, s)\right)\quad \text{and} \quad r_{0}=\ord_{P}
\bdiv\left(\mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)\right).
\end{displaymath}
Also choose a non-zero $f_{0}\in \mathcal{J}^{\text{\rm cusp}}_{k,m}(\Gamma, s)_{\ell_{0}}$ for some
$\ell_0$ (this exists by Lemma \ref{lemm:18}), and let $\varepsilon
>0$. The number $r$ can be characterized as
\begin{displaymath}
r=\sup\{(1/\ell)\ord_{P}f \mid \ell \ge 0,\, f\in \mathcal{J}_{k,m}(\Gamma, s)_{\ell}\}.
\end{displaymath}
Hence we can find $\ell\gg 0$ and $f\in \mathcal{J}_{k,m}(\Gamma, s)_{\ell}$
satisfying the conditions
\[
\frac{\ord_{P}f}{\ell} \ge r-\varepsilon, \quad
\ell_{0}/\ell \le \varepsilon \quad \text{and} \quad
\frac{\ord_{P}f_{0}}{\ell} \ge -\varepsilon.
\]
Note that to achieve the second and third condition we only need to
make $\ell $ big enough. Since $ff_{0}$ is a cusp form by
Lemma~\ref{lem:multiplicativity}, we have
\begin{displaymath}
\frac{\ord_{P}f+\ord_{P}f_{0}}{\ell+\ell_{0}}\le r_{0}.
\end{displaymath}
Together with the above conditions, this implies
\begin{displaymath}
\ell(r-\varepsilon )\le \ord_{P}f \le r_{0}(\ell +\ell_{0})-\ord_{P}f_{0}
\end{displaymath}
and hence
\begin{displaymath}
r\le r_{0}(1+\varepsilon )+2\varepsilon.
\end{displaymath}
As $\varepsilon>0 $ can be chosen arbitrarily we deduce that $r\le r_{0}$. This completes
the proof.
\end{proof}
\begin{lem}\label{lem:nef} For $k,m\ge 0$,
the b-divisor $\mathbb{D}(\overline L_{k,m,\phi },s,h)$ is nef.
\end{lem}
\begin{proof}
By Proposition \ref{prop:5} the metric $h$ is psh. The result then
follows from Proposition~\ref{prop:3}.
\end{proof}
\begin{lem}\label{lem:big} For $k,m>0$,
the b-divisor $\mathbb{D}(\overline L_{k,m,\phi },s,h)$ is big.
\end{lem}
\begin{proof}
By Lemma \ref{lemm:19}, after choosing projective refinements
$\Sigma _{0}$ and $\Pi_{0}$ of $\Sigma $ and $\Pi $ we
can find numbers $m_{0}$, $k_{0}$ and polarization functions $\phi _{0}$
and $\psi _{0}$ satisfying that $\overline L_{k_{0},m_{0},\phi_{0}
+\psi_{0} }$ is ample and
\begin{displaymath}
H^{0}(\overline \mathcal{B}(\Gamma )_{\Pi _{0}},\overline L_{\ell
k_{0},\ell m_{0},\phi_{0}
+\psi_{0} }) \subset J_{\ell k_{0},\ell m_{0}}(\Gamma ).
\end{displaymath}
After taking a multiple we can
also assume that $\overline
L_{m_{0},k_{0},\phi_{0}+\psi _{0}}$ is very ample, hence generated by
global sections. Let $r>0$ such that
\begin{displaymath}
r k > k_{0},\qquad r m> m_{0}, \qquad J^{\text{\rm cusp}}_{r
k-k_{0},r m-m_{0}}(\Gamma )\not = \{0\}.
\end{displaymath}
Such an $r>0$ exists by Lemma \ref{lemm:18}.
Let $0\not = \varphi\in J^{\text{\rm cusp}}_{r
k-k_{0},r m-m_{0}}(\Gamma )$ be a cusp form and let $s_{0}$ be the
rational section of $\overline
L_{k_{0},m_{0},\phi_{0}+\psi _{0}}$ such that $s^{r}=s_{0}\varphi$.
If $f$ is a rational function
such that $fs_{0}^{\ell}\in H^{0} \left(\overline
\mathcal{B}(\Gamma)_{\Pi_{0} },\overline
L_{\ell m_{0},\ell k_{0},\phi_{0}+\psi _{0}}\right)$, then by Lemmas
\ref{lemm:19} and \ref{lem:multiplicativity} we obtain that
$f(s_0\varphi)^{\ell}\in J^{\text{\rm cusp}}_{\ell rk,\ell rm}(\Gamma
)$.
Therefore there is an inclusion of graded linear series
\begin{displaymath}
\mathcal{R}(\dv(s_{0})) \subset \mathcal{J}^{\text{\rm cusp}}_{rk,rm}(\Gamma
,s^r) \, .
\end{displaymath}
Lemmas \ref{lemm:4} and
\ref{lemm:11} yield the inclusions of graded linear
series
\begin{displaymath}
\mathcal{R}(\dv(s_{0})) \subset \mathcal{R}(\overline L_{rk,rm,\phi },s^{r},h^r)
\subset \mathcal{R}(\mathbb{D}(\overline L_{rk,rm,\phi },s^r,h^r)).
\end{displaymath}
Since $\dv(s_0)$ is very ample, it is also big and generated by
global sections. Applying now
Proposition \ref{prop:1} and Lemma \ref{lemm:3}.\ref{item:14} we
get
\begin{displaymath}
\dv(s_0) \le r\mathbb{D}(\overline L_{k,m,\phi },s,h).
\end{displaymath}
By Lemma \ref{lemm:20} the b-divisor $\mathbb{D}(\overline L_{k,m,\phi
},s,h)$ is big.
\end{proof}
\begin{thm} \label{thm:2}
Let $k,m >0$. Then the graded algebra $\bigoplus _{\ell} J_{\ell
k,\ell m}(\Gamma )$ is not finitely generated.
\end{thm}
\begin{proof}
Assume that $\bigoplus _{\ell} J_{\ell
k,\ell m}(\Gamma )$ is finitely generated. Choose $\Sigma$, $\Pi $ and $\phi$ as before. Choose $s$ a
meromorphic Siegel--Jacobi form of weight $k$ and index $m$. Then
the graded
algebra $\bigoplus _{\ell} J_{\ell k,\ell m}(\Gamma )$ is
isomorphic to the graded linear series $\mathcal{J}_{k,m}(\Gamma, s)$. We see that this
graded linear series is finitely generated as an algebra. Hence by Lemma
\ref{lemm:2} the b-divisor $\bdiv(\mathcal{J}_{k,m}(\Gamma, s))$ is a Cartier
b-divisor. Since $\bdiv(\mathcal{J}_{\ell k,\ell m}(\Gamma, s))=\ell
\bdiv(\mathcal{J}_{k,m}(\Gamma, s))$, the first one is Cartier if and only if the
second one is Cartier. Therefore to achieve a contradiction we may
assume that $m$ is divisible enough so that $m\phi $ has integral
values on $\widetilde C_{g,\mathbb{Z}}$. By Lemmas \ref{lemm:11} and
\ref{lemm:12} we know that
\begin{displaymath}
\bdiv\left(\mathcal{J}_{k,m}(\Gamma, s)\right)= \bdiv\left(\mathcal{R}(\overline L_{k,m,\phi },s,h)\right).
\end{displaymath}
We will now show that $\bdiv\left(\mathcal{R}(\overline L_{k,m,\phi },s,h\right)$ is
not Cartier. To this end we are allowed to replace the pair $(k,m)$
by a suitable multiple and to change the section $s$.
Let $D$ be a singularity divisor of $h$. As seen in the proof of
Lemma~\ref{lem:big}, the linear series $\mathcal{R}(\overline L_{k,m,\phi },s,h)$ contains an ample linear
series. This implies that after replacing $k$ and $m$ by appropriate
multiples we can change the section $s$ so that the condition
$\ord_{D_{i}}\mathbb{D}(\overline L_{k,m,\phi },s,h)>0$ holds for all
irreducible components of $D$.
By Lemmas \ref{lem:big} and \ref{lem:nef} and Corollary \ref{cor:toroidal} the
b-divisor $\mathbb{D} (\overline L_{k,m,\phi },s,h)$ is big, nef, and
toroidal.
By
Corollary \ref{cor:2} we deduce that
\begin{displaymath}
\bdiv\left(\mathcal{R}(\overline L_{k,m,\phi },s,h)\right) = \mathbb{D}(\overline L_{k,m,\phi },s,h).
\end{displaymath}
By Corollary \ref{cor:1}, we know that $\mathbb{D}(\overline L_{k,m,\phi },s,h)$
is not a Cartier b-divisor. This proves the result.
\end{proof}
Since Runge works with bounded (rather than fixed) ratio between the
weight and the index, we must work slightly more to show that Theorem
\ref{thm:2} disproves Runge's claim in \cite[Theorem 5.5]{runge}.
Let $J$ be a bigraded ring, graded over $\mathbb Z_{\ge 0}^2$. If $f \in J_{k,m}$ is (bi)homogeneous with $k \neq 0$ then we define the \emph{relative index} of $f$ as $r(f) = m/k$. Given $n \in \mathbb Q_{\ge 0}$, we define
\begin{equation}
J_n = \bigoplus_{(k,m): m= kn} J_{m,k} \;\;\; \text{and} \;\;\;J_{\le n} = \bigoplus_{(k,m): m \le kn} J_{m,k}.
\end{equation}
\begin{prop}\label{prop:disprove-runge}
Fix $n \in \mathbb Q_{>0}$, and suppose that $J_{\le n}$ is finitely generated as an algebra over $J_{0,0}$. Then $J_n$ is finitely generated as an algebra over $J_{0,0}$.
\end{prop}
\begin{proof}
Given a finite set of elements $a_i \in J_{m_i, k_i}$, note that
\begin{equation}\label{eq:face_trick}
\min_i r(a_i) \le r(\prod_i a_i) \le \max_i r(a_i),
\end{equation}
with equalities if and only if all $r(a_i)$ are equal.
Now let $a_1, \dots, a_l$ be generators for $J_{\le n}$ as $J_{0,0}$-algebra; we may assume each $a_i$ is bihomogeneous, say of degree $(m_i, k_i)$. Then we claim that $J_n$ is generated by exactly those $a_i$ with $r(a_i) = n$; in particular, $J_n$ is finitely generated.
To prove our claim, let $f \in J_n$, say $f \in J_{m,k}$, and write $f = \sum_I \lambda_I a^I$ with $I \in \mathbb Z_{\ge 0}^l$, $\lambda_I \in J_{0,0}$, and $a^I \coloneqq \prod_i a_i^{I_i}$. Then each $a^I$ is bihomogeneous, hence if $\lambda_I \neq 0$ then $a^I \in J_{m,k}$, in particular $r(a^I) = r(f)$. But then by \eqref{eq:face_trick} we know each of the $a_i$ with $I_i \neq 0$ has $r(a_i) = r(f) = n$.
\end{proof}
\begin{rmk}\label{rem:runge-mistake}
As was mentioned in the introduction, Theorem \ref{thm:2} disproves
\cite[Theorem 5.5]{runge}. We can trace back the oversight in Runge's
proof to the proof of Theorem~5.1 in \emph{loc.~cit.} Here, the
author states that the space of Siegel--Jacobi forms can be seen as
the set of global sections of a natural invertible sheaf on a
compactification of the universal abelian scheme living over the
Satake--Baily--Borel compactification of the moduli space of
principally polarized abelian varieties of level $\Gamma$. Let us be
more precise. Consider the Satake--Baily--Borel compactification
$\mathcal{A}(\Gamma)^{*}$ of $\mathcal{A}(\Gamma)$. Then
the canonical fibration morphism $\pi \colon \mathcal{B}(\Gamma) \to
\mathcal{A}(\Gamma)$ extends to a compactification $\overline{\pi} \colon
\mathcal{B}(\Gamma)^{*} \to
\mathcal{A}(\Gamma)^{*}$, where
$\mathcal{B}(\Gamma)^{*}$ is constructed as the
BiProj of a bigraded ring $R$ contained in the ring of Siegel--Jacobi
forms.
The core of Runge's argument leading to Theorem~5.1 of \emph{loc.~cit.} is that for every Siegel--Jacobi form in
$R$, the limit when one approaches a point on the boundary of
$ \mathcal{B}(\Gamma)^{*}$ only depends on $g$ parameters. Then it is
claimed that this implies that all the fibres of the projection map $ \mathcal{B}(\Gamma)^* \to
\mathcal{A}(\Gamma)^*$ have dimension $g$.
The problem with this argument is that Siegel--Jacobi forms are
sections of a line bundle and we are looking at the completion of the projective embedding
defined by this line bundle. In this situation, if all the Siegel--Jacobi forms converge to zero simultaneously when approaching a point at the
boundary, the dimension of the fibre may be bigger than~$g$.
\end{rmk}
\section{The asymptotic dimension of spaces of Siegel--Jacobi forms} \label{sec:asymptotic}
The volume of a graded linear series $\mathcal{R}$ on a variety $X$ of dimension~$n$ is the non-negative real number given by
\[
\vol(\mathcal{R}) = \limsup_{k \to \infty} \frac{\dim(\mathcal{R}_k)}{k^n/n!}.
\]
In particular, given a b-divisor $\mathbb{D}$ on $X$, the volume of $\mathbb{D}$ can be expressed as $\vol(\mathbb{D}) = \vol(\mathcal{R}(\mathbb{D}))$, see Definition~\ref{def:volume_b_div}.
We recall that by \cite[Theorem~3.2]{Da-Fa20} any nef b-divisor $\mathbb{D}$ on $X$ has a well-defined degree $\mathbb{D}^n$ in $\mathbb{R}_{\ge 0}$.
\begin{lem} \label{lemm:HilbSam} Assume $\mathbb{D}$ is a big and nef toroidal b-divisor on $X$. Then we have the Hilbert-Samuel formula
\[ \vol \mathbb{D} = \mathbb{D}^n \, . \]
\end{lem}
\begin{proof} See \cite[Theorem 5.13]{BoteroBurgos}.
\end{proof}
\begin{lem} \label{lemm:continuity_volume} The function $\vol$ is continuous on the space of big and nef toroidal b-divisors on $X$.
\end{lem}
\begin{proof} See \cite[Corollary 5.15]{BoteroBurgos}.
\end{proof}
As before let $h= \overline{h}^{\text{\rm inv}}$ be the psh metric of $\overline L_{k,m,\phi }$ obtained
by Proposition~\ref{prop:5}. Let $s$ be a non-zero rational section of $L_{k,m}$.
\begin{thm}\label{th:vol-jac} Let $k,m>0$ and let
$\mathbb{D}_{k,m}=\mathbb{D}(\overline{L}_{k,m,\phi}, s,h), \mathcal{J}_{k,m}(\Gamma, s)$ and
$\mathcal{J}_{k,m}^{\operatorname{cusp}}(\Gamma, s)$ be as in
Section~\ref{sec:graded-linear-series-1}. Let $n=g(g+1)/2+g$. Then the following sequence
of equalities is satisfied.
\[
\vol\left(\mathcal{J}_{k,m}(\Gamma, s)\right) = \vol\left(\mathcal{J}_{k,m}^{\operatorname{cusp}}(\Gamma, s)\right) = \vol\left(\mathbb{D}_{k,m}\right) = \mathbb{D}_{k,m}^n.
\]
\end{thm}
\begin{proof}
We know by Lemmas \ref{lem:big} and \ref{lem:nef} and Corollary \ref{cor:toroidal} that the b-divisor $\mathbb{D}_{k,m}$ is nef, big, and toroidal. Let $D$ be a singularity divisor of $h$. As in the proof of Theorem \ref{thm:2} we may assume that the condition $\operatorname{ord}_{D_i}\mathbb{D}_{k,m} >0$ holds for all irreducible components $D_i$ of $D$. We let $\mathbb{D}_j =(1-1/j)\mathbb{D}_{k,m}$. The sequence $\{\mathbb{D}_j\}_j$ forms a sequence of nef and big toroidal b-divisors converging to $\mathbb{D}_{k,m}$. Again by Lemma \ref{lemm:4} and Corollary \ref{cor:3} we have
\[
\mathcal{R}(\mathbb{D}_j)\subset \mathcal{R}(\overline{L}_{k,m,\phi}, s,h) \subset \mathcal{R}(\mathbb{D}_{k,m}).
\]
Taking limits, using the continuity statement in Lemma~\ref{lemm:continuity_volume}, and using Lemma~\ref{lemm:11} we obtain
\begin{align}\label{eq:as1}
\vol(\mathbb{D}_{k,m}) = \lim_{j \to \infty}\vol(\mathbb{D}_j) =
\vol\left(\mathcal{R}(\overline{L}_{k,m,\phi}, s,h)\right) =
\vol\left(\mathcal{J}_{k,m}^{\operatorname{cusp}}(\Gamma, s)\right).
\end{align}
On the other hand, we have
\begin{align}\label{eq:cusp}
\mathcal{J}_{k,m}^{\operatorname{cusp}}(\Gamma, s) \subset
\mathcal{J}_{k,m}(\Gamma, s) \subset \mathcal{R}(\bdiv\left(\mathcal{J}_{k,m}(\Gamma,
s)\right),
\end{align}
where the last inclusion follows from Lemma~\ref{lemm:3}. Moreover, as in the proof of Theorem \ref{thm:2}
we have
\[
\mathcal{R}(\bdiv\left(\mathcal{J}_{k,m}(\Gamma, s)\right) =
\mathcal{R}(\bdiv\left(\mathcal{J}_{k,m}^{\operatorname{cusp}}(\Gamma, s)\right) =
\mathcal{R}(\overline{L}_{k,m,\phi}, s,h) = \mathcal{R}(\mathbb{D}_{k,m}).
\]
Taking volumes in \eqref{eq:cusp} and using \eqref{eq:as1} we get
\begin{align}\label{eq:as2}
\vol\left(\mathcal{J}_{k,m}(\Gamma, s)\right) = \vol(\mathbb{D}_{k,m}).
\end{align}
Finally, since $\mathbb{D}_{k,m}$ is toroidal, by Lemma~\ref{lemm:HilbSam} we have
\begin{align}\label{eq:as3}
\vol(\mathbb{D}_{k,m}) = \mathbb{D}_{k,m}^n \, .
\end{align}
Combining \eqref{eq:as1}, \eqref{eq:as2} and \eqref{eq:as3} we obtain the result.
\end{proof}
\begin{cor}\label{cor:asy}
We have
\begin{align}\label{eqn:limsup}
\limsup_{\ell\to \infty}\frac{\dim J_{\ell k, \ell m}(\Gamma)}{\ell^n/n!} = \limsup_{\ell\to \infty}\frac{\dim J^{\text{\rm cusp}}_{\ell k, \ell m}(\Gamma)}{\ell^n/n!} = \mathbb{D}_{k,m}^n.
\end{align}
\end{cor}
Therefore, to obtain an asymptotic estimate of the growth of
$\dim J_{\ell k, \ell m}(\Gamma)$ and $\dim J_{\ell k, \ell
m}^{\text{\rm cusp}}(\Gamma)$ we are reduced to computing $\mathbb{D}_{k,m}^n$.
\begin{rmk}\label{rmk:limsup2} The $\limsup$ in \eqref{eqn:limsup} is actually a $\lim$ for sufficiently divisible $\ell$ as it is the volume of a graded linear series of almost integral type (see Remark~\ref{rmk:limsup}).
\end{rmk}
Our next task is to compute the degree $\mathbb{D}_{k,m}^n$.
We recall that
$\mathbb{D}_{k,m}=\mathbb{D}(\overline{L}_{k,m,\phi}, s,h)$ and that (see Proposition~\ref{prop:7}) the metric $h$ is toroidal with
respect to the boundary divisor $\overline \mathcal{B}(\Gamma )_{\Pi }\setminus \mathcal{B}(\Gamma )$
for any admissible cone decomposition $\Pi $. Then by
\cite[Theorem 5.20]{BBHJ} we have that
\begin{displaymath}
\mathbb{D}_{k,m}^{n}=\int_{\overline \mathcal{B}(\Gamma )_{\Pi }}\langle
c_{1}(\overline L_{k,m,\phi},h)^{n}\rangle =
\int _{\mathcal{B}(\Gamma )} c_{1}(L_{k,m},h)^{n}.
\end{displaymath}
Here the integral in the middle is the so called \emph{non-pluripolar
volume}, which agrees with the integral on the right hand side because
the metric $h$ is smooth on $\mathcal{B}(\Gamma)$.
We let $h_B$ resp.\ $h_M$ denote the canonical metrics on the line bundles $M$ and $B$,
see Definition~\ref{def:6} and the remarks immediately thereafter.
As equation \eqref{eq:18} also gives the canonical invariant metric $h$ on $L_{k,m}$, by transport of structure we
may use multi-linearity to deduce that
\begin{displaymath}
\int _{\mathcal{B}(\Gamma )} c_{1}(L_{k,m},h)^{n}=
\sum _{r=1}^{n} \binom{n}{r}\int _{\mathcal{B}(\Gamma )} m^{r}k^{n-r}c_{1}(B,h_B)^{r} \pi ^{\ast} c_{1}(M,h_M)^{n-r}.
\end{displaymath}
All the integrals are finite by \cite[Remark 2.23]{BBHJ}.
Recall that here $n=\dim \mathcal{B}(\Gamma )=g+(g+1)g/2$. Let
$[2]\colon \mathcal{B}(\Gamma )\to \mathcal{B}(\Gamma)$ be the map ``multiplication
by $2$'' fiber by fiber. Then $[2]$ is a finite map of degree
$2^{2g}$. Moreover, since $B$ is symmetric and is rigidified along the origin, we have a canonical isomorphism
\begin{displaymath}
[2]^{\ast}B\simeq B^{\otimes 4}.
\end{displaymath}
We have
\begin{displaymath}
[2]^{\ast}c_{1}(B,h_B)=4c_{1}(B,h_B) \, ,
\end{displaymath}
i.e., the isomorphism is compatible with the metric (see \cite[Lemma~2.6]{DGH21}, for instance).
Therefore
\begin{multline*}
\int _{\mathcal{B}(\Gamma )} c_{1}(B,h_B)^{r} \pi ^{\ast} c_{1}(M,h_M)^{n-r}=
\frac{1}{2^{2g}}\int _{\mathcal{B}(\Gamma )} [2]^{\ast} c_{1}(B,h_B)^{r}
[2]^{\ast}\pi ^{\ast} c_{1}(M,h_M)^{n-r}\\
=
\frac{1}{2^{2g}}\int _{\mathcal{B}(\Gamma )} 2^{2r}c_{1}(B,h_B)^{r}
\pi ^{\ast} c_{1}(M,h_M)^{n-r}
=
\frac{2^{2r}}{2^{2g}}\int _{\mathcal{B}(\Gamma )} c_{1}(B,h_B)^{r}
\pi ^{\ast} c_{1}(M,h_M)^{n-r}.
\end{multline*}
Hence this integral is zero unless $r=g$. So
\begin{displaymath}
\int _{\mathcal{B}(\Gamma )} c_{1}(L_{k,m},h)^{n}=
\binom{n}{g}m^{g}k^{\frac{g(g+1)}{2}}\int _{\mathcal{B}(\Gamma )}
c_{1}(B,h_B)^{g} \pi ^{\ast}
c_{1}(M,h_M)^{\frac{g(g+1)}{2}}.
\end{displaymath}
Let $A$ be a fibre of the map $\mathcal{B}(\Gamma )\to \mathcal{A}(\Gamma )$. It is
an abelian variety of dimension~$g$. By the
projection formula
\begin{displaymath}
\int _{\mathcal{B}(\Gamma )}
c_{1}(B,h_B)^{g} \pi ^{\ast}
c_{1}(M,h_M)^{\frac{g(g+1)}{2}}=
\deg(B|_{A})\int _{\mathcal{A}(\Gamma)}c_{1}(M,h_M)^{\frac{g(g+1)}{2}}.
\end{displaymath}
Since $B|_{A}$ is twice the principal polarization,
\begin{displaymath}
\deg(B|_{A})=2^{g}g!.
\end{displaymath}
We write $\Gamma _{0}=\Sp(2g,\mathbb{Z})$. Then
\begin{displaymath}
\int _{\mathcal{A}(\Gamma)}c_{1}(M,h_M)^{\frac{g(g+1)}{2}}=[\Gamma _{0}\colon
\Gamma ] \int _{\mathcal{A}_{g}}c_{1}(M,h_M)^{\frac{g(g+1)}{2}},
\end{displaymath}
where the second integral is an orbifold integral. By the formula
after \cite[Conjecture 8.3]{Geer:moduli},
writing $G=g(g+1)/2$ (so $n = g + G$) gives
\begin{displaymath}
\int
_{\mathcal{A}_{g}}c_{1}(M,h_M)^{\frac{g(g+1)}{2}}=
(-1)^{G}\frac{G!}{2^{g}}\prod_{k=1}^{g}\frac{\zeta(1-2k)}{(2k-1)!!}.
\end{displaymath}
Summing up we obtain
\begin{displaymath}
\mathbb{D}^{n}_{k,m}=(-1)^{G}m^{g}k^{G}(G+g)![\Gamma _{0}\colon
\Gamma ] \prod_{k=1}^{g}\frac{\zeta (1-2k)}{(2k-1)!!}.
\end{displaymath}
By Corollary~\ref{cor:asy} we obtain the
asymptotic growth of the dimension of the spaces of Siegel--Jacobi
forms and of cusp Siegel--Jacobi forms explicitly, recovering a formula already implicit in Tai's work \cite{tai}.
\begin{cor}\label{th:asy-dim}
The asymptotic growth of the dimension of the spaces $J_{\ell k, \ell
m}(\Gamma)$ and $J^{\text{\rm cusp}}_{\ell k, \ell m}(\Gamma)$ when $\ell$
goes to infinity is given by the following formulae:
\begin{equation*}
\begin{split}
\limsup_{\ell \to \infty}\frac{\dim J_{\ell k, \ell m}(\Gamma)}{\ell^n/n!} & = \limsup_{\ell \to \infty}\frac{\dim J^{cusp}_{\ell k, \ell m}(\Gamma)}{\ell^n/n!} \\
& = (-1)^{G}n! m^{g}k^{G}[\Gamma _{0}\colon \Gamma ] \prod_{k=1}^{g}\frac{\zeta (1-2k)}{(2k-1)!!}\\
& = (-1)^{n}n!m^{g}k^{G}2^{G-g}
[\Gamma _{0}\colon \Gamma ]\prod_{k=1}^{g}\frac{(k-1)!B_{2k}}{(2k)!}\\
& = V_g \cdot n!m^gk^G2^{-G-1}\pi^{-G}[\Gamma_0 \colon \Gamma],
\end{split}
\end{equation*}
where $B_{2k} = \frac{(-1)^{k+1} 2(2k)!}{(2\pi)^{2k}}\zeta(2k)$ are
the Bernoulli numbers and
\[
V_g = (-1)^n 2^{g^2+1}\pi^G\prod_{k=1}^g\frac{(k-1)!B_{2k}}{(2k)!}
\]
is the symplectic volume of $\mathcal{A}_g$ computed by Siegel in \cite[Section VIII]{siegel}.
\end{cor}
By Remark \ref{rmk:limsup2} the $\limsup$ above is actually a $\lim$ for sufficiently divisible $\ell$.
\begin{rmk}\label{rem:tai-formula}
The above formulas can also be obtained by combining the formulas in the proofs of Propositions 2.1 and 2.5 of Tai's work~\cite{tai}.
\end{rmk}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,048 |
{"url":"https:\/\/proxieslive.com\/tag\/reproduce\/","text":"How to reproduce Wolfram Language\u2019s base64 encoded string with command-line tool?\n\nWith Wolfram Language:\n\n``In[7]:= ExportString[\"foobar\u4e2d\u6587\", \"Base64\"] Out[7]= \"Zm9vYmFyXDo0ZTJkXDo2NTg3\" ``\n\nWith the closest command-line software I could think of\n\n``echo -n 'foobar\u4e2d\u6587' | openssl base64 Zm9vYmFy5Lit5paH ``\n\nWhy the difference? What\u2019s the best way to reproduce Mathematica\u2019s behavior?\n\nHow to reproduce this tensor calculation with Mathematica\n\nThe tensor operation shown in the red box is used in the textbook to prove that there are only 9 independent constants for orthotropic materials:\n\nI want to use MMA to reproduce the operation of $$C_{pqmn}=l_{ip}\\;l_{jq}\\;l_{km}\\;l_{ln}\\;C_{ijkl}$$ (Where $$C_{ijkl}$$ is the stiffness tensor), but at present, I have no specific idea. I will continue to update the details to make it perfect.\n\nAdditional details:\n\nDetails are being added\u2026\n\nHow gelatinous cube reproduce?\n\nLast time when our party encountered gelatinous cube, this question occured to us, and we instinctively agreed that they reproduce like amoeba \u2013 by mitosis or cell division.\n\nHowever, I\u2019m curious: Is there a canonical DnD reference to how gelatinous cube reproduce?\n\nI\u2019m very interested if there is a reference in the early DnD edition, and especially more interested if there are different methods proposed between editions, if any. Answers from pathfinder is also welcome, since they stem from DnD, too.\n\nDo familiars reproduce? If \u2018yes\u2019, what kinds of offspring result?\n\nThe spell Find Familiar summons a simple yet common creature of the caster\u2019s choice (like a cat, rat, raven or weasel). Such a creature\u2019s organs all function healthfully, including eating and sleeping. This creature is not a beast however:\n\nAppearing in an unoccupied space within range, the familiar has the statistics of the chosen form \u2013 though it is a celestial, fey, or fiend (your choice) instead of a beast.\n\nCan a coupling of (matching) summoned familiars reproduce? If \u2018no\u2019, why so? If \u2018yes\u2019, what offspring is produced (a \u2018common\u2019 beast, some 1\/2 beast + 1\/2 spirit hybrid or one of the three types of spirits)?\n\nReplaceAll on a Plot \u2013 Can you reproduce this error?\n\nThe weird thing is that it occurs only the first time I evaluate it! The same happens with `Plot` but not with `Graphics`.\n\nI can reproduce this on Mathematica 11.3 and Wolfram Engine 12 (wolframscript) on Linux Debian 9 (Stretch).\n\nCan a Simulacrum reproduce?\n\nThe spell Simulacrum begins as follows:\n\nYou shape an illusory duplicate of one beast or humanoid that is within range for the entire Casting Time of the spell. The duplicate is a creature, partially real and formed from ice or snow, and it can take Actions and otherwise be affected as a normal creature. It appears to be the same as the original, but it has half the creature\u2019s hit point maximum and is formed without any Equipment. Otherwise, the Illusion uses all the Statistics of the creature it duplicates.\n\nThe Simulacrum is, as far as I know, treated exactly as the original creature aside from the exceptions listed in the spell. It is furthermore specifically designated as a creature in the spell. Would the Simulacrum retain the original creature\u2019s reproductive abilities? As a DM, I was imagining a wizard who runs a business creating Simulacra of rich women to act as surrogates, for a hefty profit. I wanted to know the RAW interpretation of this issue, to know better if or what I would be houseruling if I did.\n\nIt is my current conclusion that it is possible, but I wanted to make sure I hadn\u2019t overlooked something.\n\nWhen a bug is reproduce then under which column it should be mentioned in TFS 2015? [migrated]\n\nUnder which area should we define \u201cReproduced bugs\u201d details in TFS 2015? Right now, I am writing them under \u201cSteps to reproduce\u201d column.\n\n\u00bf por que la clase AudioTrack ( android ) reproduce audio con errores?\n\nHola estoy desarrollando un app que reproduce el audio en vivo que se le env\u00eda desde una pc, ambos est\u00e1n desarrollados con java lo que hago es enviar un paquete de byte por un datagramSocket al cliente ( el app android ) los recibe y luego reproduce con la clase AudioTrack. El problema es que al reproducir el paquete de bytes se producen errores por ejemplo :\n\n1- El audio se escucha con errores con peque\u00f1os cortes.\n\n2- En la consola de Android studio me muestra estos errores :\n\n``> I\/AudioTrack: This process already got info. FadeIn[0] FadeOut[0] FadeInRing[0] > Skip ramp > Manually recycle bitmap > W\/AudioTrack: Use of stream types is deprecated for operations other than volume control > See the documentation of AudioTrack() for what to use instead with android.media.AudioAttributes to qualify your playback use case ``\n\nAqu\u00ed adjunto el c\u00f3digo de el emisor de audio (PC) y el receptor de audio( android )\n\nCODIGO PC\n\n``public class PruebagrabacionAudio { static TargetDataLine mic; static DataLine.Info dLI = null; \/\/ejemplo de android \/\/static DatagramSocket socket; static DatagramSocket skServer; static int sampleRate = 16000,chanelConfig = 16,audioFormat = 2,minBufSize = 4096 ; static boolean status = true; static AudioFormat aF = new AudioFormat(sampleRate, chanelConfig, 1, true, false); \/** * @param args the command line arguments *\/ public static void main(String[] args) { try{ dLI= new DataLine.Info(TargetDataLine.class,aF); mic = (TargetDataLine)AudioSystem.getLine(dLI); new CapThread().start(); }catch(Exception e){ } } static class CapThread extends Thread{ public void run(){ try{ byte[] buffer = new byte[minBufSize]; System.out.println(\"INICIANDO EL SERVIDOR\"); skServer = new DatagramSocket(8094); System.out.println(\"Esperando conexion\"); DatagramPacket peticion = new DatagramPacket(buffer,buffer.length ); skServer.receive(peticion); int port_cliente = peticion.getPort(); InetAddress IP_cliente = peticion.getAddress(); System.out.println(\"Peticion aceptada de la IP \"+IP_cliente.getAddress()); DatagramPacket packet; mic.open(aF); mic.start(); while(status){ mic.read(buffer, 0, buffer.length); packet = new DatagramPacket(buffer,buffer.length,IP_cliente,port_cliente); skServer.send(packet); } } catch (LineUnavailableException ex) { Logger.getLogger(PruebagrabacionAudio.class.getName()).log(Level.SEVERE, null, ex); } catch (IOException ex) { Logger.getLogger(PruebagrabacionAudio.class.getName()).log(Level.SEVERE, null, ex); } } } } ``\n\nCODIGO ANDROID\n\n``public class CompruebaConexcion { DatagramSocket dsServer ; final int puetoServer = 8094; final int minBufSize = 4096; byte[] buffer = new byte[minBufSize]; InetAddress ipServer ; AudioTrack audioTrack; ByteArrayInputStream baiss ; public void iniciaSocketPruebaConex(){ Thread compruebaConex = new Thread(new Runnable() { @Override public void run() { try { dsServer = new DatagramSocket(); ipServer = InetAddress.getByName(IPCONEXION); DatagramPacket paket = new DatagramPacket(buffer,buffer.length,ipServer,puetoServer); dsServer.send(paket); DatagramPacket paketRespuesta = new DatagramPacket(buffer,buffer.length); while(true){ dsServer.receive(paketRespuesta); baiss = new ByteArrayInputStream(paketRespuesta.getData()); buffer = paketRespuesta.getData(); playAudio(buffer); } } catch (IOException e) { e.printStackTrace(); }\/* catch (ClassNotFoundException e) { e.printStackTrace(); }*\/ } }); compruebaConex.start(); } public void playAudio(byte buffer[]){ int minBuff = buffer.length ; audioTrack = new AudioTrack(AudioManager.STREAM_MUSIC, 16000, AudioFormat.CHANNEL_OUT_MONO, AudioFormat.ENCODING_PCM_16BIT, minBuff, AudioTrack.MODE_STATIC); audioTrack.write(buffer,0,minBuff); audioTrack.setNotificationMarkerPosition(minBuff); \/\/audioTrack.setPlaybackPositionUpdateListener(this); audioTrack.play(); } } ``\n\nIs there code that can reproduce this?\n\nI looked an am hopefully posting this in the right section, I want to know if this menu is available as code, php, css whatever.\n\nhttps:\/\/media.giphy.com\/media\/SYcFpdP6STkCmSyJpV\/giphy.gif\n\nThanks for any help.\n\nTrying to reproduce \u201coptions is undefined\u201d in customer-data:90\n\nWe are sporadically facing the following problem when on \/checkout\/cart\/. As an effect, clicking on \u201cProceed to checkout\u201d does not work.\n\nWe believe the proceeding is not working because of the following javascript error being logged:\n\nThe problem was reproducible on two machines, even after reloading the full page. But now is no longer.\n\nI had a look at the code in line 90\n\n`` getFromServer: function (sectionNames, forceNewSectionTimestamp) { var parameters; sectionNames = sectionConfig.filterClientSideSections(sectionNames); parameters = _.isArray(sectionNames) ? { sections: sectionNames.join(',') } : []; parameters['force_new_section_timestamp'] = forceNewSectionTimestamp; \/** Line 90 **\/ return \\$ .getJSON(options.sectionLoadUrl, parameters).fail(function (jqXHR) { throw new Error(jqXHR); }); } }; ``\n\nSo obviously the options are not loaded somehow \u2014 but they are in the constructor, so how can this be?\n\nHappy for any help.","date":"2020-09-24 18:08:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 2, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.29441404342651367, \"perplexity\": 6230.90773539548}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600400219691.59\/warc\/CC-MAIN-20200924163714-20200924193714-00171.warc.gz\"}"} | null | null |
Q: How to save div as PDF without any tools from client side in asp.net? The below code is not working in Internet explorer asking for additional tools in client side.
function printDiv(divID) {
//Get the HTML of div
var divElements = document.getElementById(divID).innerHTML;
//Get the HTML of whole page
var oldPage = document.body.innerHTML;
//Reset the page's HTML with div's HTML only
document.body.innerHTML = "<html><head></head><body>" + divElements + "</body>";
//Print Page
window.print();
//Restore orignal HTML
document.body.innerHTML = oldPage;
//disable postback on print button
return false;
}
A: window.print() will just open the browsers print dialog. So if you want to safe the printout as pdf you need a pdf printer driver installed on your operating system.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,604 |
Q: SimpleModal modal contents does not appear once closed and then reopened there is a issue with my SimpleModal, i.e. if it is closed once and reopened again the contents does not appear in the modal.
you can have a look at nitinkabra.com/baldevgroup/contact.html, the contact us button on the bottom of page..
A: I think you need to reload your cufón every time you open.
Cufon.replace('h1, h2',{fontFamily: 'StackOverflow'});
Cufon.replace('h3', { fontFamily: 'StackOverflow Light'});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,436 |
var gulp = require('gulp'),
plumber = require('gulp-plumber'),
sass = require('gulp-sass'),
concat = require('gulp-concat'),
cssmin = require('gulp-cssmin'),
del = require('del'),
htmlToJsCompiler = require('gulp-html-to-js'),
uglify = require('gulp-uglify'),
pump = require('pump'),
babel = require('gulp-babel'),
msx = require('gulp-msx'),
vueify = require('gulp-vueify');
var errorHandler = function (error) {
console.log(error);
this.emit('end');
}
var resolveMinifiedPath = function (path) {
var params = path.split("/");
var file = params.splice(params.length - 1, 1)[0];
var newPath = params.join("/") + "/";
return {
file: file,
path: newPath
};
}
// Clean the distributable css directory
gulp.task('minify:clean:css', function () {
return del('css/');
});
// Compile out sass files and minify it
gulp.task('minify:css', ['minify:clean:css'], function () {
var min = resolveMinifiedPath("./dist/css/app.min.css");
return gulp.src('scss/*.scss')
.pipe(plumber(errorHandler))
.pipe(sass())
.pipe(cssmin())
.pipe(concat(min.file))
.pipe(gulp.dest(min.path));
});
gulp.task('minify:lib:css', function () {
return gulp.src([
'lib/css/*.css',
'lib/css/*.min.css',
])
.pipe(cssmin())
.pipe(concat('lib.min.css'))
.pipe(gulp.dest('dist/css'));
});
//Watch CSS task
gulp.task('default:css', function () {
gulp.watch('scss/*.scss', ['minify:css']);
});
gulp.task('minify:lib:base:js', function () {
return gulp.src([
'lib/base/*.js',
'lib/base/*.min.js',
])
.pipe(concat('base.lib.min.js'))
.pipe(gulp.dest('dist/min'));
});
/*
* ANGULAR 1.X RELATED MINIFIED FILES
*/
gulp.task('minify:lib:ng-1', function (cb) {
pump([
gulp.src(['lib/@angular1/angular.min.js', 'lib/@angular1/**/*.js']),
uglify(),
concat('@angular1.lib.min.js'),
gulp.dest('dist/min')
],
cb
);
});
/*
* ANGULAR 2.X RELATED MINIFIED FILES
*/
gulp.task('minify:lib:ng-2', function (cb) {
pump([
gulp.src([
//'lib/@angular2/shim.min.js',
'lib/@angular2/zone.min.js',
'lib/@angular2/Reflect.js',
'lib/@angular2/Rx.min.js',
'lib/@angular2/extensions/core.umd.min.js',
'lib/@angular2/extensions/common.umd.min.js',
'lib/@angular2/extensions/compiler.umd.min.js',
'lib/@angular2/extensions/http.umd.min.js',
'lib/@angular2/extensions/forms.umd.min.js',
'lib/@angular2/extensions/platform-browser.umd.min.js',
'lib/@angular2/extensions/platform-browser-dynamic.umd.min.js',
]),
//uglify(),
concat('@angular2.lib.min.js'),
gulp.dest('dist/min')
],
cb
);
});
/*
* REACT RELATED MINIFIED FILES
*/
gulp.task('minify:lib:react', function (cb) {
pump([
gulp.src([
'lib/react/react.min.js',
'lib/react/react-dom.min.js'
]),
//uglify(),
concat('react.lib.min.js'),
gulp.dest('dist/min')
],
cb
);
});
/*
* MITHRIL RELATED MINIFIED FILES
*/
gulp.task('minify:lib:mithril', function (cb) {
pump([
gulp.src([
'lib/mithril/mithril.js',
]),
uglify(),
concat('mithril.lib.min.js'),
gulp.dest('dist/min')
],
cb
);
});
/*
* MITHRIL RELATED MINIFIED FILES
*/
gulp.task('minify:lib:vue', function (cb) {
pump([
gulp.src([
'lib/vue/vue.min.js',
]),
//uglify(),
concat('vue.lib.min.js'),
gulp.dest('dist/min')
],
cb
);
});
/*
* LAUNCH RELATED MINIFIED FILES
*/
gulp.task('minify:js:launch', ['minify:js:launch:views:concat'], function (cb) {
pump([
gulp.src(['minifiedTemplates/launch.templates.min.js', 'wwwroot/launch/**/*.js']),
uglify(),
concat('launch.min.js'),
gulp.dest('dist/min')
],
cb
);
});
gulp.task('minify:js:launch:views:concat', function () {
return gulp.src(['wwwroot/launch/templates/**/*.html', 'wwwroot/launch/views/**/*.html'])
.pipe(htmlToJsCompiler({ concat: 'launch.templates.min.js', prefix: 'templates/launch', global: 'window.TemplatesLaunch' }))
.pipe(gulp.dest('minifiedTemplates/'));
});
gulp.task('default:watch:launch', function () {
gulp.watch(['wwwroot/launch/templates/**/*.html', 'wwwroot/launch/views/**/*.html', 'wwwroot/launch/**/*.js'], ['minify:js:launch', 'minify:js:launch:views:concat']);
});
/*
* HOST RELATED MINIFIED FILES
*/
gulp.task('minify:js:host', ['minify:js:host:views:concat'], function (cb) {
pump([
gulp.src(['minifiedTemplates/host.templates.min.js', 'wwwroot/host/**/*.js']),
uglify(),
concat('host.min.js'),
gulp.dest('dist/min')
],
cb
);
});
gulp.task('minify:js:host:views:concat', function () {
return gulp.src(['wwwroot/host/templates/**/*.html', 'wwwroot/host/views/**/*.html'])
.pipe(htmlToJsCompiler({ concat: 'host.templates.min.js', prefix: 'templates/host', global: 'window.TemplatesHost' }))
.pipe(gulp.dest('minifiedTemplates/'));
});
gulp.task('default:watch:host', function () {
gulp.watch(['wwwroot/host/templates/**/*.html', 'wwwroot/host/views/**/*.html', 'wwwroot/host/**/*.js'], ['minify:js:host', 'minify:js:host:views:concat']);
});
/*
* multihost-card-ng1 minified files
*/
gulp.task('minify:js:multihost-card-ng1', ['minify:js:multihost-card-ng1:views:concat'], function (cb) {
pump([
gulp.src([
'minifiedTemplates/multihost-card-ng1.templates.min.js',
'wwwroot/multihost-card-ng1/controllers/*.js',
'wwwroot/multihost-card-ng1/directives/*.js',
'wwwroot/multihost-card-ng1/services/*.js',
'wwwroot/multihost-card-ng1/configs/*.js',
]),
uglify(),
concat('multihost-card-ng1.min.js'),
gulp.dest('dist/min')
],
cb
);
});
gulp.task('minify:js:multihost-card-ng1:views:concat', function () {
return gulp.src(['wwwroot/multihost-card-ng1/templates/**/*.html', 'wwwroot/multihost-card-ng1/views/**/*.html'])
.pipe(htmlToJsCompiler({ concat: 'multihost-card-ng1.templates.min.js', prefix: 'templates/multihost-card-ng1', global: 'window.TemplatesMultiHostNg1' }))
.pipe(gulp.dest('minifiedTemplates/'));
});
gulp.task('default:watch:multihost-card-ng1', function () {
gulp.watch(['wwwroot/multihost-card-ng1/templates/**/*.html', 'wwwroot/multihost-card-ng1/views/**/*.html', 'wwwroot/multihost-card-ng1/**/*.js'], ['minify:js:multihost-card-ng1', 'minify:js:multihost-card-ng1:views:concat']);
});
/*
* multihost-card-ng2 minified files
*/
gulp.task('minify:js:multihost-card-ng2', ['minify:js:multihost-card-ng2:views:concat'], function (cb) {
pump([
gulp.src([
'minifiedTemplates/multihost-card-ng2.templates.min.js',
'wwwroot/multihost-card-ng2/components/*.js',
'wwwroot/multihost-card-ng2/directives/*.js',
'wwwroot/multihost-card-ng2/services/*.js',
'wwwroot/multihost-card-ng2/modules/*.js',
]),
uglify(),
concat('multihost-card-ng2.min.js'),
gulp.dest('dist/min')
],
cb
);
});
gulp.task('minify:js:multihost-card-ng2:views:concat', function () {
return gulp.src(['wwwroot/multihost-card-ng2/templates/**/*.html', 'wwwroot/multihost-card-ng2/views/**/*.html'])
.pipe(htmlToJsCompiler({ concat: 'multihost-card-ng2.templates.min.js', prefix: 'templates/multihost-card-ng2', global: 'window.TemplatesMultihostCardNg2' }))
.pipe(gulp.dest('minifiedTemplates/'));
});
gulp.task('default:watch:multihost-card-ng2', function () {
gulp.watch(['wwwroot/multihost-card-ng2/templates/**/*.html', 'wwwroot/multihost-card-ng2/views/**/*.html', 'wwwroot/multihost-card-ng2/**/*.js'], ['minify:js:multihost-card-ng2', 'minify:js:multihost-card-ng2:views:concat']);
});
/*
* multihost-card REACT FILES
*/
gulp.task("babel:js:multihost-card-react", function (cb) {
pump([
gulp.src([
'wwwroot/multihost-card-react/**/*.jsx',
]),
babel({
"presets": ["react", "es2015", "stage-2"],
}),
concat('multihost-card-react-babel.js'),
gulp.dest('wwwroot/multihost-card-react/babelled')
],
cb
);
});
gulp.task('minify:js:multihost-card-react', ['babel:js:multihost-card-react'], function (cb) {
pump([
gulp.src([
'wwwroot/multihost-card-react/**/*.js'
]),
uglify(),
concat('multihost-card-react.min.js'),
gulp.dest('dist/min')
],
cb
);
});
gulp.task('default:watch:multihost-card-react', function () {
gulp.watch([
'wwwroot/multihost-card-react/templates/**/*.html',
'wwwroot/multihost-card-react/views/**/*.html',
'wwwroot/multihost-card-react/**/*.js',
'wwwroot/multihost-card-react/**/*.jsx'
], ['minify:js:multihost-card-react']);
});
/*
* multihost-card MITHRIL FILES
*/
gulp.task("babel:js:multihost-card-mithril", function (cb) {
pump([
gulp.src([
'wwwroot/multihost-card-mithril/**/*.jsx',
]),
babel({
"presets": ["es2015", "stage-2"],
"plugins": ["mjsx"]
}),
concat('multihost-card-mithril-babel.js'),
gulp.dest('wwwroot/multihost-card-mithril/babelled')
],
cb
);
});
gulp.task('minify:js:multihost-card-mithril', ['babel:js:multihost-card-mithril'], function (cb) {
pump([
gulp.src([
'wwwroot/multihost-card-mithril/**/*.js'
]),
uglify(),
concat('multihost-card-mithril.min.js'),
gulp.dest('dist/min')
],
cb
);
});
gulp.task('default:watch:multihost-card-mithril', function () {
gulp.watch([
'wwwroot/multihost-card-mithril/templates/**/*.html',
'wwwroot/multihost-card-mithril/views/**/*.html',
'wwwroot/multihost-card-mithril/**/*.js',
'wwwroot/multihost-card-mithril/**/*.jsx'
], ['minify:js:multihost-card-mithril']);
});
/*
* multihost-card VUE FILES
*/
gulp.task("babel:js:multihost-card-vue", function (cb) {
pump([
gulp.src([
'wwwroot/multihost-card-vue/**/*.jsx',
]),
babel({
"presets": ["es2015"],
"plugins": ["transform-vue-jsx"]
}),
concat('multihost-card-vue-babel.js'),
gulp.dest('wwwroot/multihost-card-vue/babelled')
],
cb
);
});
gulp.task('minify:js:multihost-card-vue', ['babel:js:multihost-card-vue'], function (cb) {
pump([
gulp.src([
'wwwroot/multihost-card-vue/**/*.js'
]),
uglify(),
concat('multihost-card-vue.min.js'),
gulp.dest('dist/min')
],
cb
);
});
gulp.task('default:watch:multihost-card-vue', function () {
gulp.watch([
'wwwroot/multihost-card-vue/**/*.jsx'
], ['minify:js:multihost-card-vue']);
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,152 |
Q: Why does my shark not get damaged when colliding with other sharks? I also get errors on my output such as:
TypeError: Error #1009: Cannot access a property or method of a null
object reference. at
AttackonSharkwithMovement_fla::MainTimeline/fl_AnimateHorizontally()
TypeError: Error #1009: Cannot access a property or method of a null
object reference. at
AttackonSharkwithMovement_fla::MainTimeline/fl_AnimateHorizontally_2()
TypeError: Error #1009: Cannot access a property or method of a null
object reference. at
AttackonSharkwithMovement_fla::MainTimeline/fl_EnterFrameHandler_2()[
Scene 1 - Main Menu
import flash.events.KeyboardEvent;
import flash.ui.Keyboard;
import flash.display.MovieClip;
import flash.events.Event;
import flash.display.Stage;
import flash.system.fscommand
import flash.events.MouseEvent
stop();
//Button Scripts
Play_Button.addEventListener(MouseEvent.CLICK, fl_ClickToGoToScene);
function fl_ClickToGoToScene(event:MouseEvent):void
{
MovieClip(this.root).gotoAndPlay(1, "Game");
}
Instructions_Button.addEventListener(MouseEvent.CLICK, fl_ClickToGoToAndStopAtFrame_10);
function fl_ClickToGoToAndStopAtFrame_10(event:MouseEvent):void
{
gotoAndStop(6);
}
function quit (event:MouseEvent):void
{
fscommand ("quit");
}
Quit_Button.addEventListener(MouseEvent.MOUSE_DOWN,quit);
Scene 2 - Game
import flash.events.KeyboardEvent;
import flash.ui.Keyboard;
import flash.display.MovieClip;
import flash.events.Event;
import flash.display.Stage;
import flash.system.fscommand;
import flash.events.TimerEvent;
import flash.utils.Timer;
stop();
//Variables
var rightPressed:Boolean = new Boolean(false);
var leftPressed:Boolean = new Boolean(false);
var upPressed:Boolean = new Boolean(false);
var downPressed:Boolean = new Boolean(false);
var sharkSpeed:Number = 10;
var score1:Number = 0;
var maxHP:int = 100;
var currentHP:int = maxHP;
var percentHP:Number = currentHP / maxHP;
//Health Script
function updateHealthBar():void
{
percentHP = currentHP / maxHP;
healthBar.barColor.scaleX = percentHP;
}
//Button Scripts
MainMenu_Button.addEventListener(MouseEvent.CLICK, fl_ClickToGoToScene_2);
function fl_ClickToGoToScene_2(event:MouseEvent):void
{
MovieClip(this.root).gotoAndPlay(1, "Main Menu");
}
Instructions_Button.addEventListener(MouseEvent.CLICK, fl_ClickToGoToNextScene_2);
function fl_ClickToGoToNextScene_2(event:MouseEvent):void
{
MovieClip(this.root).gotoAndPlay(6, "Main Menu");
}
//Keyboard Movement
stage.addEventListener(KeyboardEvent.KEY_DOWN, keyDownHandler);
stage.addEventListener(KeyboardEvent.KEY_UP, keyUpHandler);
stage.addEventListener(Event.ENTER_FRAME, gameLoop);
function keyDownHandler(KeyEvent:KeyboardEvent):void
{
if (KeyEvent.keyCode == Keyboard.RIGHT)
{
rightPressed = true;
}
else if (KeyEvent.keyCode == Keyboard.LEFT)
{
leftPressed = true;
}
else if (KeyEvent.keyCode == Keyboard.DOWN)
{
downPressed = true;
}
else if (KeyEvent.keyCode == Keyboard.UP)
{
upPressed = true;
}
}
function keyUpHandler(keyEvent:KeyboardEvent):void
{
if (keyEvent.keyCode == Keyboard.RIGHT)
{
rightPressed = false;
}
else if (keyEvent.keyCode == Keyboard.LEFT)
{
leftPressed = false;
}
else if (keyEvent.keyCode == Keyboard.DOWN)
{
downPressed = false;
}
else if (keyEvent.keyCode == Keyboard.UP)
{
upPressed = false;
}
}
function gameLoop(loopEvent:Event):void
{
if (rightPressed)
{
shark.x += sharkSpeed;
}
else if (leftPressed)
{
shark.x -= sharkSpeed;
}
else if (downPressed)
{
shark.y += sharkSpeed;
}
else if (upPressed)
{
shark.y -= sharkSpeed;
}
}
//AI Movement
addEventListener(Event.ENTER_FRAME, fl_AnimateHorizontally);
function fl_AnimateHorizontally(event:Event)
{
enemy1.x += 2;
enemy2.x += 2;
enemy3.x += 2;
enemy4.x += 2;
enemy5.x += 2;
enemy6.x += 2;
megaladon.x += 2;
}
addEventListener(Event.ENTER_FRAME, fl_AnimateHorizontally_2);
function fl_AnimateHorizontally_2(event:Event)
{
fishes.x += 1.5;
}
//Colission
function hitsTheObject(e:Event)
{
if (shark.hitTestObject(enemy1))
{
trace("player collided with enemy");
currentHP -= 50;
if (currentHP <= 0)
{
currentHP = 0;
trace("You died!");
MovieClip(this.root).gotoAndPlay(1, "Game Over");
}
updateHealthBar();
}
}
//Score Script
addEventListener(Event.ENTER_FRAME, fl_EnterFrameHandler_2);
function fl_EnterFrameHandler_2(event:Event):void
{
gameScore.text = String(score1);
score1 += 1;
trace("gameScore.text is : " + gameScore.text);
trace("score1 is : " + score1);
}
//Timer Script
var myTimer:Timer = new Timer(1000,50);
myTimer.addEventListener(TimerEvent.TIMER, onTimer);
myTimer.addEventListener(TimerEvent.TIMER_COMPLETE, onComplete);
myTimer.start();
function onTimer(e: TimerEvent):void
{
myText_txt.text = String(myTimer.repeatCount - myTimer.currentCount);
}
function onComplete(e: TimerEvent):void
{
MovieClip(this.root).gotoAndPlay(1, "You Survived");
}
Scene 3 - You Survived
stop();
//Button Scripts
MainMenu_Button.addEventListener(MouseEvent.CLICK, fl_ClickToGoToScene_4);
function fl_ClickToGoToScene_4(event:MouseEvent):void
{
MovieClip(this.root).gotoAndPlay(1, "Main Menu");
}
PlayAgain_Button.addEventListener(MouseEvent.CLICK, fl_ClickToGoToScene_12);
function fl_ClickToGoToScene_12(event:MouseEvent):void
{
MovieClip(this.root).gotoAndPlay(1, "Game");
}Scene 4 - Game Over
stop();
//Button Scripts
MainMenu_Button.addEventListener(MouseEvent.CLICK, fl_ClickToGoToScene_9);
function fl_ClickToGoToScene_9(event:MouseEvent):void
{
MovieClip(this.root).gotoAndPlay(1, "Main Menu");
}
PlayAgain_Button.addEventListener(MouseEvent.CLICK, fl_ClickToGoToScene_11);
function fl_ClickToGoToScene_11(event:MouseEvent):void
{
MovieClip(this.root).gotoAndPlay(1, "Game");
}
A: As identified in the comments, you need to remove the eventListener which you can achieve with:
removeEventListener(Event.ENTER_FRAME, fl_AnimateHorizontally);.
I would suggest implementing the following line whenever you bind to a frame event such as Event.ENTER_FRAME
this.addEventListener(Event.REMOVED_FROM_STAGE, function(){
try{
removeEventListener(Event.ENTER_FRAME, fl_AnimateHorizontally);
}catch(error){
//error handling optional in this case.
}
});
This will get called ONCE only right before the object is destroyed/removed from the stage i.e. when you call MovieClip(this.root).gotoAndPlay(1, "Game");
Note: You can just put all of your 'global' events into the try area - you don't need this call every time you add an event.
Additionally, you don't need this whatsoever for movieclips as all of your events will get cleaned up automatically once they are removed from the stage via the garbage collector.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,475 |
Fuse Launches "Fuse On Your Mobile"
New Wireless Service Allows Fans to Stay in Touch with
And Drive Content of Viewer-Driven All-Music Network
Service Marks the Latest Step in Cable Network's Total Convergence Strategy
New York, NY, May 27, 2003 – Fuse (the network formerly known as "muchmusic usa") has launched Fuse On Your Mobile, a new interactive service that allows music fans to remain connected to their favorite viewer-driven, all-music channel anytime, anywhere, by pointing their Web-enabled wireless phones to mobile.fuse.tv. The technology utilized in the new service is being provided by WINK Interactive LLC, the mobile branding and publishing solutions company.
"Fuse is all about music by the audience — not music by the company — with a play list and rich programming schedule largely determined by viewers' input on our web site, and now via 'Fuse On Your Mobile,'" states Michael Goldstein, Fuse's vice president of Interactive Programming. "We're pleased to give Fuse viewers a new medium by which to voice their opinions and requests and stay connected to their favorite shows, artists and bands. With Fuse On Your Mobile, wireless music fans will never be out of touch!"
"WINK's real-time Fuse applications will give the devoted music fan a compelling experience that bridges the gap between mediums that were destined to collide," said David Harper, vice president of WINK Interactive. Added William Munch, president of WINK, "I believe this is the first time where TV viewers will use a mobile application to interact directly with their televisions and submit live dedications' and much more."
"Fuse On Your Mobile" features include:
What's On Now – This area of "Fuse On Your Mobile" gives mobile phone users up-to-the-minute programming information and descriptions of upcoming specials and events.
Oven Fresh – This Fuse show lets viewers decide which videos should stay or go. New videos are showcased every week, but only five make it to the weekend "Oven Fresh Keepers" show. Fuse Mobile users can review new video offerings and vote to keep or sweep them off the play list via their cell phones. "Fuse On Your Mobile" also offers a special "current standings" listing only for mobile voters, who can compare their votes with those of the slightly less tech-oriented web surfers who vote via www.fuse.tv.
Dedicate Live – Music-lovers can use their mobile phones to join this live show and select from the day's list of videos, choose a dedication category (Love, Hate, Sorry, Secret Crush, others), key-in their dedication, and then watch the show live on Fuse from 5-6 pm (ET) to see if it airs. Fuse has created a special on-air graphic that lets viewers know when they're watching a dedication that was submitted via a cell phone.
Waddayawant – By giving wireless users direct access to Fuse's Programming Department, this feature gives Fuse fans another way to share their opinions on what videos should be played.
"Fuse On Your Mobile" works on most phones that have a web-enabled mobile browser and is optimized for the Openwave's Mobile Browser 4.1.22b and higher. Users only need to fire up their mobile browser, point it to mobile.fuse.tv to begin. Users looking to first "test drive" the service can do so by going to the Fuse Web site at www.fuse.tv
, clicking on "Fuse On Your Mobile" link in the main navigation to get to the feature page, and clicking the "let's get started" button to try out a web version of the mobile service.
About WINK Interactive LLC
Cold Spring Harbor, NY – based WINK Interactive provides enabling mobile publishing applications, mobile community portal solutions, and managed wireless application services for network operators, mobile content providers and mobile device manufacturers. WINK also develops custom applications for major brands by integrating its technology with established
marketing channels to offer mobile subscribers an extension of the brand in a mobile environment.
WINK's revolutionary, award-winning, patent-pending mobile portal technology, WINKsite(tm) has been created to change the face of the mobile Internet – making it easier for, and more accessible to, the masses. WINK's vision is that Mobile Internet adoption will exponentially accelerate through pervasive Mobile Personal Portals, which by integrating personalized content and communication technologies will drive personal productivity and community. More information is available by visiting www.winkinteractive.com and www.winksite.com
Fuse is the nation's only all-music, viewer-influenced television network, featuring music videos, exclusive artist interviews, live concerts and specials — all rooted in music. Fuse reflects the rapidly changing interests and attitudes of its 12-34 year-old audience by uniting the media platforms that are at the center of their communication and entertainment – TV, online, and interactive games – and by incorporating their opinions and suggestions into its on-air and online programming. More information about Fuse is available at www.fuse.tv.
Fuse is a network of Rainbow Media Holdings, Inc, the programming subsidiary of Cablevision Systems Corporation (NYSE:CVC). A leader in sports, news and entertainment programming, Rainbow owns and manages national and regional networks including AMC, WE: Women's Entertainment and IFC.
Sal Cataldi
Cataldi PR
sal@cataldipr.com
Georgia Juvelis
gjuvelis@rainbow-media.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,626 |
\section{Introduction}\label{sec:intro}
String theory provides a detailed framework to explore the unification of
the gauge and gravitational interactions \cite{books}.
The construction of phenomenological
models that can make contact with the real world has been of great interest
and understanding their underlying structure can be especially elucidating.
However, the vastness of the {\it a priori} possible vacuum solutions
impedes progress towards the construction of a standard string model.
In this respect, the study of various relationships between
different models can be very fruitful. In particular, it may have
far reaching implications for the interpretation of the landscape
of string vacua.
In the heterotic constructions the left and right moving sectors are
treated asymmetrically. Of particular interest are the so called \twozero{}
models\footnote{Our convention here is that the \lefto-moving sector is
supersymmetric and the \righto-moving is bosonic. }
because it is known that $N=1$ spacetime supersymmetry requires
(at least) \twozero{} world-sheet supersymmetry and because
they can accommodate $SO(10)$ unification. The problem is that the
space of these models is huge. For example, even though the number of
$(2,2)$ Gepner models\cite{Gepner} is quite tractable and
they have been studied in detail\cite{Font1990a, Schellekens1990},
the number of \twozero{} models that arise is much
greater\cite{Schellekens1990}.
For this reason it would be very useful
to discover relations in the space of such models.
In this paper we will make a small step in this direction by getting
inspiration from a new kind of duality that comes under the name spinor-vector
duality and was observed in $\Z{2}\times\Z{2}'$ orbifold
models\cite{Faraggi2007a,Faraggi2007b,Faraggi2008,Catelin-Jullien2009,
Angelantonj2010,Faraggi2011}.
It is a duality of the massless spectra of two such models
under the exchange of vectorial and spinorial representations of the $SO(10)$
GUT gauge group.
These models turn out to be related through the spectral flow operator and
the underlying CFT structure of the spinor-vector duality for
$\Z{2}\times\Z{2}'$ orbifolds was discussed in \cite{Faraggi2011}.
Even though the form of the duality as expressed in these references is
restricted to $\Z{2}\times\Z{2}'$ orbifolds, the important idea that the
spectral flow map can be used to relate different \twozero{} models is
much more general. It is the purpose of this paper to explain the
details of this mapping and the exact relationship between the mapped
models.
The outline of this paper is as follows: In section \ref{sec:svd}
we discuss the spinor-vector duality in the fermionic $\Z{2}\times \Z{2}$
heterotic-string orbifolds.
Understanding how the duality operates in the free fermionic constructions
hints at how similar dualities may work in the case of interacting CFTs.
In sections \ref{sec:spectral flow} and \ref{sec:simple currents} we review
the definition of the spectral flow and the simple current formalism which will
allow us to construct \twozero{} models from a generic $(2,2)$ model. In section \ref{sec:idea}
we explain how the spectral flow induces a map between different \twozero{} models and in section \ref{sec:mapping} we analyze
the consequences of this idea. Section \ref{sec:example} provides an example of how
to use the derived results. We conclude with a brief discussion and possible generalizations
in sections \ref{sec:generalizations} and \ref{sec:conclusions}.
\section{The spinor-vector duality case}\label{sec:svd}
In this section we outline the spinor-vector duality in the case
of the fermionic $\Z{2}\times \Z{2}$ heterotic-string orbifolds.
The discussion will provide the guide for exploring
similar symmetries in models with an interacting internal CFT.
The presentation here will be qualitative and further
technical details are given in the references.
In the free fermionic formulations of the compactified string \cite{fff}
all the internal degrees of freedom are represented in terms of
free world-sheet fermions. Therefore, in this formulation
the internal compactified dimensions are represented in terms
of an internal CFT with vanishing interactions. Additionally, the
well known relations between two dimensional fermions and bosons
entail that the free fermionic formulation is equivalent to
a free bosonic formulation, {\it i.e.\ }to toroidal orbifolds.
A string vacuum in the free fermionic formulation is defined
in terms of boundary condition basis vectors and the Generalized
Gliozzi-Scherk-Olive (GGSO) projection coefficients of the one-loop partition function
\cite{fff}. The gauge symmetry is generated by spacetime
vector bosons that arise from the untwisted as well as the twisted
sectors. The spacetime vector bosons arising in the twisted sectors enhance the
untwisted gauge group factors under which they are charged.
Specific enhancements depend on the states that remain in the
physical spectrum after application of the GGSO projections.
Similarly, the matter states in the free fermionic models
are obtained from the untwisted and twisted sectors.
The spinor-vector duality in the free fermionic vacua operates on the
matter states in the twisted sectors.
The free fermionic vacua correspond to $\Z{2}$ and $\Z{2}\times \Z{2}$
orbifolds at enhanced symmetry points in the moduli space \cite{z2xz2}.
In this section we review the spinor-vector duality in $\Z{2}$ orbifolds.
By doing this we recap the ingredients that are needed for the generalization to
interacting internal CFTs. The simplest realization of the spinor-vector
duality is in the case of a single $\Z{2}$ orbifold acting on the
$E_8\times E_8$ heterotic-string compactified on a generic six torus.
Taking for simplicity the internal torus as a product of six circles with
radii $R_i$, the partition function (omitting the contribution from the spacetime bosons) reads
\beq
{ Z}_+ = ( V_8 - S_8) \, \left( \sum_{m,n} \Lambda_{m,n}
\right)^{\otimes 6}\, \left( \overline O _{16} + \overline S_{16} \right)
\left( \overline O _{16} + \overline S_{16} \right)\,,
\eeq
where as usual, for each circle,
\beq
p_{\rm L,R}^i = \frac{m_i}{R_i} \pm \frac{n_i R_i}{\alpha '} \,,
~~~~~{\rm and}~~~~~
\Lambda_{m,n} = \frac{{q^{\frac{\alpha '}{4}
p_{\rm L}^2}} \, \bar q ^{\frac{\alpha '}{4} p_{\rm R}^2}}{|\eta|^2}\,,
\eeq
and we have written ${Z}_{+}$ in terms of level-one
${\rm SO} (2n)$ characters (see for instance \cite{Angelantonj2002})
\beqn
O_{2n} &=& \frac{1}{2} \left( \frac{\theta_3^n}{\eta^n} +
\frac{\theta_4^n}{\eta^n}\right) ~~~~\,,
~~~~~~~~~~~~
V_{2n} ~=~ \frac{1}{2} \left( \frac{\theta_3^n}{\eta^n} -
\frac{\theta_4^n}{\eta^n}\right),
\nonumber \\
S_{2n} &=& \frac{1}{2} \left( \frac{\theta_2^n}{\eta^n} +
i^{-n} \frac{\theta_1^n}{\eta^n} \right) \,,
~~~~~~~~~~~
C_{2n} ~=~ \frac{1}{2} \left( \frac{\theta_2^n}{\eta^n} -
i^{-n} \frac{\theta_1^n}{\eta^n} \right).
\n
\eeqn
We next apply the orbifold projections
\beqn
\Z{2}~:~g & = &(-1)^{F_{1}+F_{2}}\delta \,, \label{deltaorbifold}\\
\Z{2}^\prime ~:~{g^\prime} & = &~(x_{4},x_{5},x_{6},x_7,x_8,x_9)
~\longrightarrow~
(-x_{4},-x_{5},-x_{6},-x_7,+x_8,+x_9)\,.\nonumber
\eeqn
$F_1$ and $F_2$ in (\ref{deltaorbifold}) flip the sign in the
spinorial representations of $SO(16)_1$ and $SO(16)_2$, generated
by ${\xi_1}=\{{\bar\psi}^{1,\cdots,5}, {\bar\eta}^{1,2,3}\}$
and ${\xi_2}=\{{\bar\phi}^{1,\cdots,8}\}$ respectively, and $\delta$
shifts the compact $X^9$ coordinate by half of its period, {\it i.e.}
\begin{equation}
\delta ~:~ X^9 \rightarrow X^9 +\pi R^9 \qquad \Rightarrow \qquad
\Lambda_{m,n} ~\rightarrow~ (-1)^m\Lambda_{m,n}.\label{shift1}
\end{equation}
The $\Z{2}$ projection in (\ref{deltaorbifold})
breaks the $E_8\times E_8$ gauge group to $SO(16)\times SO(16)$
and preserves $N=4$ spacetime supersymmetry.
The additional $\Z{2}^\prime$ projection twists the compactified coordinates
and preserves only $N=2$ spacetime supersymmetry. Its generator
$g^\prime$ reverts the sign of four internal coordinates $X^i$, $i=4,5,6,7$
and simultaneously breaks one $SO(16)$ to $SO(12)\times SO(4)$.
The action of the $\Z{2}\times \Z{2}^\prime$ projections on $Z_+$ is implemented
by taking
\beq
Z_-=
\frac{\left(1+g\right)}{2}
\frac{\left(1+g^\prime\right)}{2}
Z_+\,.
\label{tendprojection}
\eeq
The ten-dimensional SO(8) little group is broken to $SO(4) \times SO(4)$.
At the same time, the first $SO(16)$ gauge
group factor is broken into $SO(12) \times SO(4)$. As a
result, the one-loop partition function can be written in terms of the
spacetime characters,
\beqn
Q_0 & = & V_4O_4-S_4S_4,~~~~~~~~~~~~~Q_V = V_4O_4- C_4C_4,\nn\\
Q_S & = & O_4C_4-S_4O_4,~~~~~~~~~~~~Q_C = V_4S_4- C_4V_4.\nn
\eeqn
There are two independent orbits in the partition function and hence
one discrete torsion. The full partition function is given by
\beq
Z_-=Z_{untwisted}+Z_{g}+Z_{g^\prime}+Z_{gg^\prime}.
\eeq
It consists of the untwisted sector and
the three twisted sectors $g$, $g^\prime$ and $gg^\prime$.
The untwisted sector gives rise to the vector bosons
that generate the four dimensional gauge group, whereas
the sectors $g$ and $gg^\prime$ give rise to massive states.
To note the spinor-vector duality it is sufficient to
focus on the states arising from the twisted sector $g^\prime$.
Summation over the GGSO projections in this
sector produces the following terms in the partition function:
\beqn
Z_{g'}~& & = \nonumber\\
& &
{1\over 2}
\left(
\left\vert{{2\eta}\over\theta_4}\right\vert^4
+
\left\vert{{2\eta}\over\theta_3}\right\vert^4
\right)
\Lambda_{p,q}
\left[
P^{01}_+\Lambda_{m,n}
\left(
Q_S \left(
{\overline V}_{12}{\overline C}_{4}{\overline O}_{16}
+
{\overline S}_{12}{\overline O}_{4}{\overline S}_{16}
\right)
\right. \right. \nonumber\\
& &
\left. \left.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~+~~~~~~~~~~~~~~~~
Q_C
\left(
{\overline O}_{12}{\overline S}_{4}{\overline O}_{16}
+
{\overline C}_{12}{\overline V}_{4}{\overline S}_{16}
\right)
\right)
\right.
\nonumber\\
& &
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+~~
\left.
P^{01}_-\Lambda_{m,n}
\left(
Q_S \left(
{\overline S}_{12}{\overline O}_{4}{\overline O}_{16}
+
{\overline V}_{12}{\overline C}_{4}{\overline S}_{16}
\right)
\right. \right. \nonumber\\
& &
\left. \left.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~+~~~~~~~~~~~~~~~~
Q_C
\left(
{\overline O}_{12}{\overline S}_{4}{\overline S}_{16}
+
{\overline C}_{12}{\overline V}_{4}{\overline O}_{16}
\right)\right)
\right]
\nonumber
\\
& &
+{1\over 2}
\left(
\left\vert{{2\eta}\over\theta_4}\right\vert^4
-
\left\vert{{2\eta}\over\theta_3}\right\vert^4
\right)
\Lambda_{p,q}
\left[
P^{01}_+\Lambda_{m,n}
\left(
Q_S \left(
{\overline O}_{12}{\overline S}_{4}{\overline O}_{16}
+
{\overline C}_{12}{\overline V}_{4}{\overline S}_{16}
\right)
\right. \right. \nonumber\\
& &
\left. \left.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~+~~~~~~~~~~~~~~~~
Q_C
\left(
{\overline V}_{12}{\overline C}_{4}{\overline O}_{16}
+
{\overline S}_{12}{\overline O}_{4}{\overline S}_{16}
\right)
\right)
\right.
\nonumber\\
& &
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+~~
\left.
P^{01}_-\Lambda_{m,n}
\left(
Q_S \left(
{\overline O}_{12}{\overline S}_{4}{\overline S}_{16}
+
{\overline C}_{12}{\overline V}_{4}{\overline O}_{16}
\right)
\right. \right. \nonumber\\
& &
\left. \left.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~+~~~~~~~~~~~~~~~~
Q_C
\left(
{\overline S}_{12}{\overline O}_{4}{\overline O}_{16}
+
{\overline V}_{12}{\overline C}_{4}{\overline S}_{16}
\right)\right)
\right],
\label{partifun}
\eeqn
where we defined
$P_\pm^{01}$ as
\beq
P_{\pm}^{01}={{1\pm\epsilon_1(-1)^m}\over2}.
\label{ppm}
\eeq
The spinor-vector duality transformation is transparent
in the partition function (\ref{partifun}). Massless states arise from
the untwisted sector and the $g'$-twisted sector.
The internal winding modes in the $g$ and $gg'$-twisted sectors
are shifted by $1/2$. The states in these two sectors are therefore
massive. The untwisted sector gives rise to spacetime
vector bosons that generate the $SO(12)\times SO(4)\times SO(16)$
gauge symmetry and to scalar multiplets that transform in the
bi-vector representation of $SO(12)\times SO(4)$. Examining the
$g'$-twisted sector reveals how the spinor-vector
duality operates. Massless states arise for vanishing internal momentum
and winding modes, {\it i.e.\ }$m=n=0$. Depending on the choice of the
discrete torsion $\epsilon_1=\pm1$, vanishing lattice modes will
therefore arise from $P_+^{01}\Lambda_{m,n}$ or $P_-^{01}\Lambda_{m,n}$,
{\it i.e.}
\beqn
& &\epsilon_1=+1~~\Rightarrow ~~~P^{01}_+\Lambda_{m,n}=\Lambda_{2m,n}
~~~{\rm and~~~} P^{01}_-\Lambda_{m,n}=\Lambda_{2m+1,n}\ , \nn\\
& &\epsilon_1=-1~~\Rightarrow ~~~P^{01}_-\Lambda_{m,n}=\Lambda_{2m,n}
~~~{\rm and~~~} P^{01}_+\Lambda_{m,n}=\Lambda_{2m+1,n}\ . \nn
\eeqn
It follows from the $q$-expansion of the $\theta$ functions
that in the case with $\epsilon_1=+1$ the zero lattice modes
attach to $Q_S{\overline V}_{12}{\overline C}_{4}{\overline O}_{16}$,
which produces two massless $N=2$ hypermultiplets in the
$\bf{12}$ vector representation of $SO(12)$, whereas
in the case with $\epsilon_1=-1$ the zero lattice modes
attach to $Q_S{\overline S}_{12}{\overline O}_{4}{\overline O}_{16}$,
which produces a massless $N=2$ hypermultiplet in the
$\bf{32}$ spinorial representation. It is further noted from
(\ref{partifun}) that
in the case with $\epsilon_1=+1$ the term
$Q_S{\overline O}_{12}{\overline S}_{4}{\overline O}_{16}$
gives rise to eight additional states from the first
excited modes of the twisted lattice. Hence, the total number
of degrees of freedom $32=12\cdot2 +4\cdot 2$ is preserved under
the duality map.
The realization of the spinor-vector duality in this model provides a simple
example where its origins can be explored and generalized to cases with
interacting world-sheet CFTs. In the toroidal case, since all the data
of the compactification is encoded in the toroidal background parameters
and the orbifold action on them, it is anticipated that the spinor-vector
duality is realizable in terms of a continuous or discrete map between two
sets of background parameters. Indeed, in ref. \cite{Faraggi2011}
it was shown that the spinor-vector duality map is realized in terms of
a continuous interpolation between two Wilson lines. The continuous
interpolation, rather than a discrete transformation, is particular to the
cases that preserve $N=2$ spacetime supersymmetry, {\it i.e.\ }when
a single $\Z{2}$ twist is acting on the internal torus. In this
case the moduli fields that enable the continuous interpolation
exist in the spectrum and are not projected. In the compactifications
with $N=1$ spacetime supersymmetry, these moduli fields are
projected out. Therefore, in the $N=1$ cases the spinor-vector
duality map is discrete.
The spinor-vector duality can be regarded as a direct consequence of
the breaking of the world-sheet supersymmetry on the bosonic side of
the heterotic-string from $N=2$ to $N=0$, {\it i.e.\ }from $(2,2)$
world-sheet supersymmetry to \twozero. In the $(2,2)$ case the gauge
symmetry is enhanced to $E_6$ (or $E_7$). In this case the spinor and
vector representations of $SO(10)\times U(1)$ (or $SO(12)\times SU(2)$)
are embedded in the single $\bf{27}$ (or $\bf{56}$) representation of $E_6$ (or $E_7$).
The breaking of the $(2,2)$ world-sheet supersymmetry to \twozero{} results
in the reduction of the enhanced gauge symmetry, by projecting out the spinorial
components of the adjoint representation in its decomposition under the
corresponding $SO(2n)$ subgroup.
At the same time the matter multiplets are split into
the spinorial and vectorial components. The GSO projections may
retain either the spinorial or the vectorial representation in the massless
spectrum. The spinor-vector duality is then induced by the spectral
flow operator. The generalization to interacting internal CFTs can therefore
proceed along the following lines. We can start with a generic
compactification with $(2,2)$ world-sheet supersymmetry,
and subsequently break the $N=2$ world-sheet supersymmetry on
the bosonic side to $N=0$. There ought to be choices of the breaking that
result in different models that are related by the spectral flow operator.
We can illustrate the spinor-vector duality in terms of a spectral
flow operator by considering the boundary condition basis
vectors \cite{Faraggi2008} in eq. (\ref{neq2basis}):
\begin{eqnarray}
v_1=S&=&\{\psi^\mu,\chi^{1,\dots,6}\},\nn\\
v_{1+i}=e_i&=&\{y^{i},\omega^{i}|\bar{y}^i,\bar{\omega}^i\}, \
i=1,\dots,6,\nn\\
v_{8}=z_1&=&\{\bar{\phi}^{1,\dots,4}\},\nn\\
v_{9}=z_2&=&\{\bar{\phi}^{5,\dots,8}\},\nn\\
v_{10}=z_3&=&\{\bar{\psi}^{1,\dots,4}\},\nn\\
v_{11}=z_0&=&\{\bar{\eta}^{0,1,2,3}\},\nn\\
v_{12}=b_1&=&\{\chi^{34},\chi^{56},y^{34},y^{56}|\bar{y}^{34},
\bar{y}^{56},\bar{\eta}^0,\bar{\eta}^{1}\},\label{neq2basis}
\end{eqnarray}
where the vector ${\bf1}= \{\psi^\mu,\
\chi^{1,\dots,6},y^{1,\dots,6}, \omega^{1,\dots,6}|
~\bar{y}^{1,\dots,6},\bar{\omega}^{1,\dots,6},
\bar{\eta}^{1,2,3},
\bar{\psi}^{1,\dots,5},\bar{\phi}^{1,\dots,8}\}$
is obtained as the linear combination
${\bf1} =S+{\sum_ie_i}+z_0+z_1+z_2+z_3$.
In (\ref{neq2basis}) we used
the usual notation of the free fermionic formalism \cite{fff}.
The gauge group generated by vector bosons arising in the $0$-sector
is $SO(8)\times SO(8)\times SO(8)\times SO(8)$.
The gauge symmetry may be enhanced by vector bosons arising from
nine additional purely anti-holomorphic sets given by:
\beqn
G=\{& z_0,z_1,z_2,z_3, \nonumber\\
& z_0+z_1,z_0+z_2,z_0+z_3,z_1+z_2,z_1+z_3,z_2+z_3~\}.
\label{gaugebosons}
\eeqn
The basis vector $b_1$ reduces
the $N=4\rightarrow N=2$ spacetime supersymmetry
and the untwisted gauge symmetry to
$SO(8)\times SO(4)\times SO(4)\times SO(8)\times SO(8)$.
Additionally, it gives rise to the twisted sector, which produces
matter states charged under the four dimensional gauge group.
The sixteen sectors $B_1^{pqrs}=b_1+pe_3+qe_4+re_5+se_6$, with
$p,q,r,s\in\{0,1\}$, correspond to the sixteen fixed points of
the non-freely acting $\Z{2}$ orbifold.
For specific choices of the GGSO projection coefficients the
gauge group is enhanced. The vector bosons arising
from the sector $z_3$ may enhance the $SO(8)\times SO(4)\times SO(4)$
symmetry to $SO(12)\times SO(4)$, which may be enhanced further to
$E_7\times SU(2)$. In the case of $E_7$ both $z_3$ and $z_0$ are generators
of the $E_7$ gauge group. In the case of $SO(12)$ the matter
representations are obtained from the following sectors: the two sectors $B^{pqrs}_1$ and
$B^{pqrs}_1+z_3$ give the vectorial $\bf{12}$ representation and the two
sectors $B^{pqrs}_1+z_0$ and $B^{pqrs}_1+z_3+z_0$ the spinorial $\bf{32}$.
For appropriate choices of the GGSO phases either the spinorial or the vectorial representations
from a given sector are retained in the spectrum. If both the spinorial
and the vectorial states are retained in a given sector, the $SO(12)\times
SU(2)$ symmetry is necessarily enhanced to $E_7$. We note therefore
that it is precisely the basis vector $z_0$ that acts as the spectral
flow operator. For an appropriate choice of the phases it acts as a
generator of $E_7$, whereas when the $E_7$ symmetry is broken to
$SO(12)\times SU(2)$, coupled with appropriate
mapping of the GGSO projections,
the spinor-vector duality map is induced. Examining the basis
vectors in (\ref{neq2basis}) we see that $z_0$ is precisely the
mirror of the basis vector $S$, which is the spacetime
supersymmetry generator on the fermionic side of the heterotic-string.
Hence, $S$ is an operator of the \lefto-moving $N=2$ world-sheet
supersymmetry, whereas $z_0$ is an operator of the world-sheet
supersymmetry on the bosonic side.
An important feature of the $\Z{2}\times\Z{2}'$ models is that the spectral flow
operator is of order two, {\it i.e.\ }the sector $2z_0$ is identified with the untwisted sector.
This leads to two different models, as explained above, related via the spinor-vector
duality. In this paper we generalize these ideas to arbitrary internal RCFTs.
For these, the spectral flow operator will generically be of order greater than two leading
naturally to a bigger family of models. In the following sections we explain how these models
are related in the most general case.
\section{The spectral flow}\label{sec:spectral flow}
To handle the most general case in what follows, we will be slightly changing our notation from the one used in the previous section and in the free fermionic language. Our starting point here is generic (2,2) heterotic models with an internal CFT with c=9. The standard examples of interacting constructions are the Gepner models\cite{Gepner} in which the internal CFT is a product of minimal models, but all our arguments are completely general. A general state in such a model is of the form:
\beq\label{eq:1}
\Phi_\L\otimes\Phi_\R
\eeq
and the \righto-moving part which we wish to focus on is of the form
\beq\label{eq:2}
\Phi_\R=(w)(h,Q)(p),
\eeq
where $w$ is an $SO(10)$ weight $(o, v, s, c)$ and $p$ an $E_8$ weight. The appearance of the $SO(10)$ and $E_8$ weights is because of the bosonic string map which is used to construct a modular invariant heterotic-string theory from a type II theory. It replaces the $\widehat{so}(2)_1$ Kac-Moody algebra with an $\widehat{so}(10)_1\times (\widehat{e_8})_1$ one\cite{books}.
The mass formula is
\beqn
\frac{\alpha'M_R^2}{2}&=&h_{\text{TOT}}-\frac{c}{24}\nn\\
&=&\frac{w^2}{2}+h+\frac{p^2}{2}+N_R-1\ ,\label{MR}
\eeqn
where we have used the fact that $c=24$ for the bosonic string and we have also included the contribution $N_R$ from the oscillators corresponding to the spacetime bosons.
By definition a CFT is said to have $N=2$ world-sheet supersymmetry if it includes four fields:
\beqn
T(z)&=&\sum_{n\in\Z{}}L_nz^{-n-2}\ ,\\
G^{\pm}(z)&=&\sum_{n\in\Z{}}G^{\pm}_{n\pm a}z^{-n-\frac{3}{2}\mp a}\ ,\\
J(z)&=&\sum_{n\in\Z{}}J_nz^{-n-1}\ ,
\eeqn
that satisfy the algebra\cite{books}:
\beqn
\left[L_m,L_n\right]&=&(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}\ ,\nn\\
\left[L_m, G^\pm_{n\pm a}\right]&=&(\frac{m}{2}-n\mp a)G^\pm_{m+n\pm a}\ ,\nn\\
\left[L_m, J_n\right]&=& -n J_{m+n}\ ,\nn\\
\left[J_m, J_n\right]&=& \frac{c}{3}m\delta_{m+n,0}\ ,\nn\\
\left[J_m, G^\pm_{n\pm a}\right]&=&\pm G^\pm_{m+n\pm a}\ ,\nn\\
\{G^+_{m+a},G^-_{n-a}\}&=&2 L_{m+n}+(m-n+2a)J_{m+n}+\frac{c}{3}\Big((m+a)^2-\frac{1}{4}\Big)\delta_{m+n,0}\ ,\nn\\
\{G^+_{m+a},G^+_{n+a}\}&=&\{G^-_{m-a},G^-_{n-a}\}=0\ ,
\eeqn
where $a$ is a real parameter that describes how the fermionic superpartners $G^{\pm}$ of $T$ transform:
\beq
G^{\pm}(e^{2\pi i}z)=-e^{\mp2\pi i a}G^{\pm}(z).
\eeq
The algebras for $a$ and $a+1$ are isomorphic. $a\in\Z{}$ corresponds to the R sector and $a\in\Z{}+\frac1 2$ corresponds to the NS sector. A state is completely described by the eigenvalues $h$ (called the conformal dimension) and $Q$ (called the $U(1)$ charge) of the operators $L_0$ and $J_0$ that form the Cartan subalgebra:
\beq
\bt{\phi}=\bt{h,Q}.
\eeq
We also note that the algebra is invariant under the following transformation which is known as the \emph{spectral flow}:
\beqn
L_n^\eta&=&L_n+\eta J_n+\frac{c}{6}\eta^2\delta_{n,0}\ ,\nn\\
G^{\eta \pm}_{n\pm a}&=& G^{\eta \pm}_{n\pm(a+\eta)}\ ,\nn\\
J^\eta_n&=&J_n+\frac{c}{3}\eta\delta_{n,0}.
\eeqn
This also implies the existence of a \emph{spectral flow operator} $U_\eta$ that acts on states in the following way:
\beq
U_\eta\bt{h,Q}=\bt{h_\eta,Q_\eta}={\big| h-\eta Q+\frac{\eta^2 c}{6},Q-\frac{c \eta}{3} \big\rangle}.
\eeq
Of particular interest are the states
\beq
\big| \frac{3}{8},\pm\frac{3}{2} \big\rangle_{\text{R}}=U_{\mp\frac{1}{2}}{\bt{0,0}}_\text{NS}\ ,
\eeq
because they can be combined with the $s$ and $c$ weight vectors of $SO(10)$ with the smallest possible length to give massless states. Indeed, such vectors are of the form
\beq
w=(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2})
\eeq
and have $w^2=\frac{5}{4}$. An even number of minus signs corresponds to $s$ and an odd number of minus signs to $c$. We then note from (\ref{MR}) that whenever the internal CFT has $N=2$ world-sheet supersymmetry the states
\beq\label{beta0}
\pm\beta_0=(\pm c)(\frac3 8,\pm\frac3 2)(0).
\eeq
will be part of the massless spectrum. These states describe gauge bosons in the \sixt{} and \sixtbar{} of $SO(10)$ and, in conjunction with the $U(1)$ symmetry of the $N=2$ algebra, they extend $SO(10)$ to $E_6$. This proves our previous claim that the $N=2$ superconformal algebra on the bosonic sector is associated with $E_6$ gauge symmetry.
The states in (\ref{beta0}) are an extension of the spectral flow operator of the internal CFT. We call these states the spectral flow operator as well.
\section{The simple current formalism}\label{sec:simple currents}
Since we already started from a (2,2) model, there will be a modular invariant partition function (MIPF) describing it. It will be of the form
\beq
Z[\tau,\bar\tau]=\sum_{i,j} \chi_i(\tau)M_{ij}\chi_{j}(\bar\tau),
\eeq
where $\chi_i$ are the characters of the chiral algebra and $M_{ij}$ a modular invariant. For our examples, we take this to be the partition function of the usual Gepner models, \textit{i.e.\ }after the projections of the universal simple currents $\beta_0$ and $\beta_i$ have been applied to ensure spacetime supersymmetry\cite{Gepner}. Nevertheless, the approach is very general and valid whenever the simple current method can be used to construct modular invariants. This includes any RCFT and potentially some non-rational CFTs in which the chosen simple current defines a finite orbit as well. To avoid this complication we restrict ourselves to RCFTs through this paper.
As explained in the introduction we are not interested in the $(2,2)$ models \textit{per se} but rather in the \twozero{} that we get after breaking the $E_6$ symmetry on the \righto. A consistent and modular invariant \twozero{} model can be derived from a $(2,2)$ model through the simple current construction\cite{Schellekens1990,Schellekens1989b}. This is the same as the beta method for Gepner models and it practically amounts to orbifolding the original $(2,2)$ model. The result is that states not invariant under the action of the simple current are projected out and new states appear in twisted sectors. We will use both notations $J$ and $\beta$ for a simple current\footnote{Using multiplicative notation for the action of $J$ and additive notation for the action of $\beta$.} and we will focus on simple currents that break $E_6$ on the \righto{} to $SO(10)$. The MIPF for the resulting model is then
\beq
Z[\tau,\bar\tau]=\sum \chi_i(\tau)M_{ik}M_{kj}(J)\chi_{j}(\bar\tau),
\eeq
where
\beq\label{eqn:SCMI}
M_{kj}(J)=\frac 1 N\sum_{n=1}^{N_J}\delta({\Phi_k,J^n\Phi_j})\delta_{\mathbb{Z}}(Q_J(\Phi_k)+\frac{n}{2}Q_J(J))
\eeq
is called a simple current modular invariant (SCMI) and $N$ is a normalization constant ensuring that the vacuum only appears once. In practical terms, the above formula means that:
\begin{enumerate}[i)]
\item Only states whose \lefto{} part is connected to the \righto{} through $J$ will appear in the partition function, \textit{i.e.\ }states with $\Phi_\L=J^n\Phi_\R=\Phi_\R+n\beta$. This defines the $n$-th $J$-twisted sector.
\item Only states invariant under the projection will appear in the partition function. This is expressed in the constraint $Q_J(\Phi)+\frac{n}{2}Q_J(J)\in \mathbb{Z}$. $Q_J$ is called the monodromy charge and is defined as
\beq\label{eq:monodromycharge}
Q_J(\Phi)=h(\Phi)+h(J)-h(J\Phi)\quad \mod1.
\eeq
The easiest way to see that this is the appropriate condition for invariance under the $J$ projection is to note that the monodromy charge is conserved modulo 1 in operator products and thus implies the existence of a phase symmetry $\Phi\rightarrow e^{-2\pi i Q_J(\Phi)}\Phi$. This induces a cyclic group of order $N_J$. $N_J$ is called the order of $J$ and it can also be proven that $Q_J(\Phi)$ is quantized in units of $1/N_J$ \cite{Schellekens1989b}.
\end{enumerate}
The definition (\ref{eq:monodromycharge}) is for any general RCFT. For Gepner models, where $\Phi=(w_{\Phi})(\vec{l}_{\Phi},\vec{q}_{\Phi},\vec{s}_{\Phi})(p_{\Phi})$ and $J=(w_J)(\vec{l}_{J},\vec{q}_{J},\vec{s}_{J})(p_{J})$, it takes the explicit form:
\beq\label{eq:monodromychargeinGepner}
Q_J(\Phi)=-w_J\cdot w_{\Phi}-p_J\cdot p_{\Phi}+\sum_{i=1}^{r}\left(\frac{-l^i_{\Phi}l^i_J+q^i_{\Phi}q^i_J}{2(k_i+2)}-\frac{s^i_{\Phi}s^i_J}{4}\right).
\eeq
In this form it is easy to see that
\beq
Q_{\beta}(\Phi)=Q_{\Phi}(\beta)\quad \mbox{and}\quad Q_{\beta_1+\beta_2}(\Phi)=Q_{\beta_1}(\Phi)+Q_{\beta_2}(\Phi),
\eeq
\textit{i.e.\ }the monodromy charge is symmetric and linear with respect to its arguments. These properties are true in general\cite{Schellekens1989b}.
Another thing to note is that if $J$ and $J'$ are simple currents then $JJ'$ is a simple current as well. In fact, we can generalize (\ref{eqn:SCMI}) to the case where we orbifold by $J_1,\cdots,J_i,\cdots$ simultaneously. To simplify the notation let $\vec{n}$ label the twisted sectors and define
$$[\vec{n}]k\equiv J_1^{n_1}\cdots J_i^{n_i}\cdots\Phi_k\equiv\Phi_k+\sum_i n_i \beta_i.$$
Then the most general SCMI is \cite{Kreuzer1994}:
\beq\label{eqn:SCMItorsion}
M_{k,[\vec{n}]k}=\frac 1 N\prod_i\delta_{\mathbb{Z}}(Q_{J_i}(\Phi_k)+X_{ij}n^j).
\eeq
The matrix $X$ is defined modulo 1 and its elements are quantized as $X_{ij}=\frac{n_{ij}\in\mathbb{Z}}{\gcd(N_i,N_j)}$. It also satisfies $X_{ij}+X_{ji}=Q_{J_i}(J_j)$. This fixes its symmetric part completely. The remaining freedom in choosing the antisymmetric part corresponds to discrete torsion\cite{Kreuzer1994}.
\section{Outline of the idea}\label{sec:idea}
We start with a particular simple current $J$. Any $J$ would do, but for the reasons explained in the introduction the simple currents that we have in mind will break $E_6$, thus giving a \twozero{} model. We call the \twozero{} model that is derived this way ${\cal{M}}_0$. We also know that $J_0$ ($\beta_0$) is generically a simple current of every $(2,2)$ model since it is the spectral flow operator that enhances the symmetry to $E_6$ on the \righto. This naturally defines a whole family of models $\{\cal{M}\}_\alpha$ that are derived through the simple currents $J$, $J_0$ and linear combinations of them with and without discrete torsion.
The task of examining how the spectra of these models are related to each other is very fascinating and daunting at the same time. We will not attempt to carry out the analysis in its full generality here. Instead, we will restrict ourselves to the more modest goal of explaining how the mapping induced by the spectral flow $J_0$ ($\beta_0$) works.
\section{Mapping induced by the spectral flow}\label{sec:mapping}
Here we focus on the family of models ${\cal{M}}_0,\cdots,{\cal{M}}_m$ that are derived through the simple currents $J, JJ_0, \cdots, JJ_0^m$ or equivalently $\beta, \beta+\beta_0,\cdots,\beta+m\beta_0$. This family will have $N_{\beta_0}$ members where $N_{\beta_0}$ is the order of $\beta_0$. Our goal is to study how the massless spectra in these models are related. To that end, we take a closer look at the model ${\cal{M}}_m$.
We start by examining the untwisted sector\footnote{Here and in what follows untwisted sector means untwisted with respect to the simple current that defines the model, {\it i.e.\ }states with $n=0$ in (\ref{eq:n-twisted sector}). The states might be twisted with respect to other simple currents that were present in the original (2,2) model but this does not affect our argument.}. Massless states in the original $(2,2)$ model will also belong to the ${\cal{M}}_m$ model if they survive the invariance projections. Note that
\beq
Q_{\beta+m\beta_0}(\Phi)=Q_{\beta}(\Phi)+mQ_{\beta_0}(\Phi)=Q_{\beta}(\Phi)\ \mod 1,
\eeq
where in the last step we used the fact that $Q_{\beta_0}(\Phi)\in\Z{}$ because $\Phi$ belongs to the original $(2,2)$ model. This proves that $Q_{\beta+m\beta_0}(\Phi)\in\Z{}\Leftrightarrow Q_{\beta}(\Phi)\in\Z{}$ and therefore the untwisted sectors of every model in the $\cal{M}$ family are identical.
Let us now consider the twisted sectors. Note that models ${\cal{M}}_{m_1}$ and ${\cal{M}}_{m_2}$ will in general have a different number of twisted sectors since $\beta+m_1\beta_0$ and $\beta+m_2\beta_0$ will be of different order. Let us analyze the $n$-twisted sector of the ${\cal{M}}_m$ model. A very useful formula can be found by rearranging (\ref{eq:monodromycharge}) as
\beqn\label{eq:hmbeta}
h(\Phi+\beta)&=&h(\Phi)+h(\beta)-Q_{\beta}(\Phi)\ ,\nonumber\\
\mbox{and by induction:}\quad h(\Phi+m\beta)&=&h(\Phi)+m h(\beta)-m Q_{\beta}(\Phi)-\frac{m(m-1)}{2}Q_{\beta}(\beta)\ ,
\eeqn
where the equations are understood mod $1$. Massless states in the $n$-twisted sector of ${\cal{M}}_m$ are of the form
\beq\label{eq:n-twisted sector}
\Phi_\L\otimes(\tilde\Phi_\L+n(\beta+m\beta_0))\ ,
\eeq
where this time we have written the tilde explicitly to remind us that we have applied the bosonic string map. In the notation of equation (\ref{eq:2}) this is simply\cite{Gepner}:
\beq
\tilde\Phi_\L=\Phi_\L+(v)(0,0)(0).
\eeq
The massless condition gives
\beq\label{eq:massless}
h(\Phi_\L)=\frac1 2,\quad h(\tilde\Phi_\L)=1 \quad\mbox{and}\quad h(\tilde\Phi_\L+n\beta+nm\beta_0)=1.
\eeq
Furthermore, as explained before and as can be seen from (\ref{eqn:SCMI}), the states must also satisfy the invariance condition
\beq
Q_{\beta+m\beta_0}(\tilde\Phi_\L)+\frac{n}{2}Q_{\beta+m\beta_0}(\beta+m\beta_0)\in\Z{}.
\eeq
Using linearity of the monodromy charge and the fact that $Q_{\beta_0}(\tilde\Phi_\L)\in\Z{}$ and $\ Q_{\beta_0}(\beta_0)\in2\Z{}$ because $\tilde\Phi_\L$ and $\beta_0$ belonged to the massless spectrum of the original $(2,2)$ model, the invariance condition becomes
\beq\label{eq:invariant}
Q_{\beta}(\tilde\Phi_\L)+\frac{n}{2}Q_{\beta}(\beta)+mn Q_{\beta_0}(\beta)\in\Z{}.
\eeq
We can also further manipulate (\ref{eq:massless}) to derive another condition. Bearing in mind that in what follows all the calculations are mod $1$, we get:
\beqn
0=1&=&h(\tilde\Phi_\L+n\beta+nm\beta_0)\nonumber\\
&\stackrel{(\ref{eq:hmbeta})}{=}&h(\tilde\Phi_\L+n\beta)+\underbrace{nmh(\beta_0)}_{\in\Z{}}-\underbrace{nmQ_{\beta_0}(\tilde\Phi_\L)}_{\in\Z{}}-n^2mQ_{\beta_0}(\beta)-\underbrace{\frac{nm(nm-1)}{2}}_{\in\Z{}}\underbrace{Q_{\beta_0}(\beta_0)}_{\in\Z{}}\nonumber\\
&=& h(\tilde\Phi_\L+n\beta)-n^2mQ_{\beta_0}(\beta)\nonumber\\
&\stackrel{(\ref{eq:hmbeta})}{=}& \underbrace{h(\tilde\Phi_\L)}_{=1=0}+nh(\beta)-nQ_{\beta}(\tilde\Phi_\L)-\frac{n(n-1)}{2}Q_{\beta}(\beta)-n^2mQ_{\beta_0}(\beta)\nonumber\\
&\stackrel{(\ref{eq:invariant})}{=}& nh(\beta)+\frac{n}{2}Q_{\beta}(\beta)
\eeqn
Or in other words,
\beq\label{eq:condition2}
n\Big(h(\beta)+\frac{1}{2}Q_{\beta}(\beta)\Big)\in\Z{}.
\eeq
Equations (\ref{eq:invariant}) and (\ref{eq:condition2}) are the main results of this section. In general, these conditions are necessary but not sufficient because of the inherent uncertainty in the definition of the monodromy charge which is given mod $1$. Nevertheless, the beauty of this general argument is that starting from an arbitrary \twozero{} model we get a handle on the massless spectrum in any twisted sector of any model in the family.
\section{An example}\label{sec:example}
The fact that these conditions are necessary provides a prime test for where \emph{not} to look for massless states in a particular model. This can be of great importance when performing a computer scan in the space of models, so we give an example below.
Our starting point is the Gepner model $k^r=2^6$, which is a (2,2) model. In this model the internal CFT is a product of 6 minimal models each of which has central charge $c=\frac{3k}{k+2}=\frac{3}{2}$. All states will be of the form (\ref{eq:1}) but this time the internal CFT state is completely described by three vectors $\vec{l},\vec{q}$ and $\vec{s}$ so we will be using the notation $\Phi_\R=(w)(\vec{l},\vec{q},\vec{s})(p=0)$ instead. For the sake of the argument let us focus our attention on the massless charged spectrum in this model, which of course will fall into the fundamental ($\bf{27}$) or anti-fundamental ($\bf{\overline{27}}$) representation of $E_6$. Without loss of generality, we will study states in the $\bf{27}$, which under the $SO(10)$ group decomposes into $\bf{10}+\bf{16}+\bf{1}$. Let us briefly remind the reader that the \righto-moving part of such massless states will then be of the form:
\begin{itemize}
\item \textbf{10}s: $\Phi_\R=(v)(\Phi^I)(p=0)$ with
\begin{equation*}\Phi^I\in\left\{ \underline{(0,0,0)^4 (0,2,2)^2},\ \underline{(0,0,0)^2 (1,-1,0)^4},\ \underline{(0,0,0)^3 (0,2,2)(1,-1,0)^2} \right\}\ ,\end{equation*}
\item \textbf{16}s: $\Phi_\R=(c)(\Phi^{II})(p=0)$ with
\begin{equation*}\Phi^{II}\in\left\{ \underline{(0,-1,-1)^4 (0,1,1)^2},\ \underline{(0,-1,-1)^2 (1,-2,-1)^4},\ \underline{(0,-1,-1)^3 (0,1,1)(1,-2,-1)^2} \right\}\ ,\end{equation*}
\item \textbf{1}s: $\Phi_\R=(w=0)(\Phi^{III})(p=0)$ with
\begin{equation*}\Phi^{III}\in\left\{ \underline{(0,-2,-2)^4 (0,0,0)^2},\ \underline{(0,-2,-2)^2 (1,-3,-2)^4},\ \underline{(0,-2,-2)^3 (0,0,0)(1,-3,-2)^2} \right\}\ ,\end{equation*}
\end{itemize}
where underlining means permutations.
In this model $\beta_0$ has the usual form
\beq\label{eq:20}
\beta_0=(c)(0,1,1)^6(p=0)
\eeq
and is of order $N_{\beta_0}=8$. We choose the simple current with which we will orbifold our theory to be
\beq
\beta=\ (w=0)(2,1,-1)(0,0,0)^5(p=0)\ ,
\eeq
which is also of order $N_{\beta}=8$ and we note that $Q_{\beta}(\beta_0)=\frac{3}{8}\notin\Z{}$. Therefore the gauge bosons extending $SO(10)$ to $E_6$ are indeed projected out and we end up with a \twozero{} model. As explained in the previous section, this process naturally induces a whole family of models ${\cal{M}}_0,\cdots,{\cal{M}}_7$ that arise if we orbifold by $\beta, \cdots,\beta+7\beta_0$ respectively.
The untwisted sector in all of these models will be the same and it will consist of all the states mentioned above that satisfy the invariance condition
\beq
Q_{\beta}(\Phi_\R)\in\Z{}\quad \Leftrightarrow\quad \frac{-2l_1+q_1+2s_1}{8}\in\Z{}.
\eeq
For the $n-$twisted sector we will use equation (\ref{eq:condition2}). $h(\beta)$ can be readily calculated from the known formula for Gepner models\cite{Gepner}:
\beq
h=\sum_{i=1}^r\left(\frac{l_i(l_i+2)-q_i^2}{4(k_i+2)}+\frac{s_i^2}{8}\right)
\eeq
and we find that
\beq
n\Big(h(\beta)+\frac{1}{2}Q_{\beta}(\beta)\Big)=n\Big(\frac{9}{16}+\frac{1}{2}(-\frac{5}{8})\Big)=\frac{n}{4}\in\Z{}.
\eeq
This means that massless states can only arise in the untwisted $n=0$ sector, which we have already studied, or in the $n=4$ twisted sector. In the latter sector the \righto-moving part of the states will be of the form
\beqn
\Phi_\R&=&\tilde\Phi_\L+4(\beta+m\beta_0)\nonumber\\
&=&\tilde\Phi_\L+4\beta+4m\beta_0\nonumber\\
&=&\tilde\Phi_\L+(w=0)(0,4,0)(0,0,0)^5(p=0)+m(w=0)(0,4,0)^6(p=0)\nonumber\\
&=&\begin{cases} \Phi_\L+(w=0)(0,4,0)(0,0,0)^5(p=0) &\mbox{if m even}\label{eq:twistedexample}\\
\Phi_\L+(w=0)(0,0,0)(0,4,0)^5(p=0) &\mbox{if m odd} \end{cases}\ ,
\eeqn
where we have used the properties\cite{Gepner} that for Gepner models $q$ is defined mod $2(k+2)$, $s$ is defined mod $4$ and we have also performed the identification $(l,q,s)\equiv(k-l,q+k+2,s+2)$ multiple times. A quick comparison with $\Phi^{I}$, $\Phi^{II}$ and $\Phi^{III}$ given above shows that states of the form (\ref{eq:twistedexample}) cannot be massless charged states, so the spectrum consists of the states in the untwisted sector only.
Once more, the power of this method is that it allowed us to check only one twisted sector ($n=4$) for massless states, as opposed to checking as many as seven of them for each model that we would \textit{a priori} expect in this example.
\section{Some further generalizations}\label{sec:generalizations}
There are many ways to generalize the above ideas to generate even more relationships in the space of \twozero{} models. For example, we are not restricted to using only $\beta_0$ but the natural splitting of the states into an $SO(10)$ part, an internal $N=2$ CFT and an $E_8$ part suggests that any
$$\beta_{0'}=(w)(\beta_0^{\text{CFT}})(p)$$
would generate its own orbit of \twozero{} models. Furthermore, when the internal CFT can be written as a tensor product of $N=2$ superconformal theories each term comes with a spectral flow operator $\beta_0^i$. We can then go one step further and use only some of the $\beta_{0}^i$'s instead of the entire $\beta_{0}^{\text{CFT}}$.
Finally, as explained earlier, the presence of a simple current $J$ that breaks (2,2) to \twozero{} increases the possibilities even further. We can now have any linear combination of $J$, with any of the $\beta$'s mentioned above, with or without discrete torsion, and any such simple current will create its own orbit in the space of \twozero{} models.
In this paper we have shown explicitly how to use one of these mappings, the spectral flow $\beta_0$, to generate an entire family of models and we have derived useful expressions for the analysis of the spectra of these models. We believe that having not just one, but a big selection of such mappings as explained above will prove to be an important tool in the classification of \twozero{} models.
\section{Conclusions}\label{sec:conclusions}
Heterotic-string vacua with \twozero{} world-sheet supersymmetry
are particularly interesting from a phenomenological
point of view, as they reproduce the $SO(10)$
GUT structure, which is well motivated by the Standard Model
data. Ultimately, the confrontation of a string vacuum
with low scale experimental data will be achieved
by associating it with an effective smooth quantum
field theory limit. However, while the moduli spaces
of (2,2) heterotic-string compactifications, and consequently their
smooth limit, are reasonably well understood, this is not the case for those
with \twozero{} world-sheet supersymmetry. Indeed, the study of these
moduli spaces is an area of intense contemporary research \cite{twozeromoduli}.
In this paper we discussed how the spinor-vector duality, which was
observed in the framework of heterotic-string compactifications
with free world-sheet CFTs, can be extended to those with
general RCFTs. The recipe adopted from the free case is the
following: We start with a (2,2) compactification and break the world-sheet
supersymmetry on the bosonic side. The spectral
flow operator, that operates as a symmetry generator of the
(2,2) theory, then induces a map between the string vacua
of the \twozero{} theory. As such, the map induced by the
spectral flow operator provides a useful tool to explore
the moduli spaces of \twozero{} heterotic-string compactifications.
The question of interest in this respect is twofold. First,
is this description complete? Namely, do all \twozero{} heterotic-string
compactifications descend from (2,2) theories? Second,
what is the imprint of this map in the effective field
theory limit? We hope to return to these questions in
future publications.
\section*{Acknowledgments}
A.E.F. thanks the Weizmann Institute and the Theoretical Physics
Department at the University of Oxford for hospitality.
P.A. acknowledges support from the Hellenic State Scholarships
Foundation (IKY).
This work was supported in part by the STFC (ST/J000493/1).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,360 |
Could Consciousness Come Down to the Way Things Vibrate?
FeaturedNeuroscienceNeurotechPsychology
Summary: A new paper proposes resonance may contribute to human consciousness.
Source: The Conversation.
Why is my awareness here, while yours is over there? Why is the universe split in two for each of us, into a subject and an infinity of objects? How is each of us our own center of experience, receiving information about the rest of the world out there? Why are some things conscious and others apparently not? Is a rat conscious? A gnat? A bacterium?
These questions are all aspects of the ancient "mind-body problem," which asks, essentially: What is the relationship between mind and matter? It's resisted a generally satisfying conclusion for thousands of years.
The mind-body problem enjoyed a major rebranding over the last two decades. Now it's generally known as the "hard problem" of consciousness, after philosopher David Chalmers coined this term in a now classic paper and further explored it in his 1996 book, "The Conscious Mind: In Search of a Fundamental Theory."
Chalmers thought the mind-body problem should be called "hard" in comparison to what, with tongue in cheek, he called the "easy" problems of neuroscience: How do neurons and the brain work at the physical level? Of course they're not actually easy at all. But his point was that they're relatively easy compared to the truly difficult problem of explaining how consciousness relates to matter.
Over the last decade, my colleague, University of California, Santa Barbara psychology professor Jonathan Schooler and I have developed what we call a "resonance theory of consciousness." We suggest that resonance – another word for synchronized vibrations – is at the heart of not only human consciousness but also animal consciousness and of physical reality more generally. It sounds like something the hippies might have dreamed up – it's all vibrations, man! – but stick with me.
All about the vibrations
All things in our universe are constantly in motion, vibrating. Even objects that appear to be stationary are in fact vibrating, oscillating, resonating, at various frequencies. Resonance is a type of motion, characterized by oscillation between two states. And ultimately all matter is just vibrations of various underlying fields. As such, at every scale, all of nature vibrates.
Something interesting happens when different vibrating things come together: They will often start, after a little while, to vibrate together at the same frequency. They "sync up," sometimes in ways that can seem mysterious. This is described as the phenomenon of spontaneous self-organization.
Mathematician Steven Strogatz provides various examples from physics, biology, chemistry and neuroscience to illustrate "sync" – his term for resonance – in his 2003 book "Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life," including:
When fireflies of certain species come together in large gatherings, they start flashing in sync, in ways that can still seem a little mystifying.
Lasers are produced when photons of the same power and frequency sync up.
The moon's rotation is exactly synced with its orbit around the Earth such that we always see the same face.
Examining resonance leads to potentially deep insights about the nature of consciousness and about the universe more generally.
Sync inside your skull
Neuroscientists have identified sync in their research, too. Large-scale neuron firing occurs in human brains at measurable frequencies, with mammalian consciousness thought to be commonly associated with various kinds of neuronal sync.
For example, German neurophysiologist Pascal Fries has explored the ways in which various electrical patterns sync in the brain to produce different types of human consciousness.
Fries focuses on gamma, beta and theta waves. These labels refer to the speed of electrical oscillations in the brain, measured by electrodes placed on the outside of the skull. Groups of neurons produce these oscillations as they use electrochemical impulses to communicate with each other. It's the speed and voltage of these signals that, when averaged, produce EEG waves that can be measured at signature cycles per second.
Gamma waves are associated with large-scale coordinated activities like perception, meditation or focused consciousness; beta with maximum brain activity or arousal; and theta with relaxation or daydreaming. These three wave types work together to produce, or at least facilitate, various types of human consciousness, according to Fries. But the exact relationship between electrical brain waves and consciousness is still very much up for debate.
External electrodes can record a brain's activity. NeuroscienceNews.com image is adapted from the Conversation news release.
Fries calls his concept "communication through coherence." For him, it's all about neuronal synchronization. Synchronization, in terms of shared electrical oscillation rates, allows for smooth communication between neurons and groups of neurons. Without this kind of synchronized coherence, inputs arrive at random phases of the neuron excitability cycle and are ineffective, or at least much less effective, in communication.
A resonance theory of consciousness
Our resonance theory builds upon the work of Fries and many others, with a broader approach that can help to explain not only human and mammalian consciousness, but also consciousness more broadly.
Based on the observed behavior of the entities that surround us, from electrons to atoms to molecules, to bacteria to mice, bats, rats, and on, we suggest that all things may be viewed as at least a little conscious. This sounds strange at first blush, but "panpsychism" – the view that all matter has some associated consciousness – is an increasingly accepted position with respect to the nature of consciousness.
The panpsychist argues that consciousness did not emerge at some point during evolution. Rather, it's always associated with matter and vice versa – they're two sides of the same coin. But the large majority of the mind associated with the various types of matter in our universe is extremely rudimentary. An electron or an atom, for example, enjoys just a tiny amount of consciousness. But as matter becomes more interconnected and rich, so does the mind, and vice versa, according to this way of thinking.
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Biological organisms can quickly exchange information through various biophysical pathways, both electrical and electrochemical. Non-biological structures can only exchange information internally using heat/thermal pathways – much slower and far less rich in information in comparison. Living things leverage their speedier information flows into larger-scale consciousness than what would occur in similar-size things like boulders or piles of sand, for example. There's much greater internal connection and thus far more "going on" in biological structures than in a boulder or a pile of sand.
Under our approach, boulders and piles of sand are "mere aggregates," just collections of highly rudimentary conscious entities at the atomic or molecular level only. That's in contrast to what happens in biological life forms where the combinations of these micro-conscious entities together create a higher level macro-conscious entity. For us, this combination process is the hallmark of biological life.
The central thesis of our approach is this: the particular linkages that allow for large-scale consciousness – like those humans and other mammals enjoy – result from a shared resonance among many smaller constituents. The speed of the resonant waves that are present is the limiting factor that determines the size of each conscious entity in each moment.
As a particular shared resonance expands to more and more constituents, the new conscious entity that results from this resonance and combination grows larger and more complex. So the shared resonance in a human brain that achieves gamma synchrony, for example, includes a far larger number of neurons and neuronal connections than is the case for beta or theta rhythms alone.
What about larger inter-organism resonance like the cloud of fireflies with their little lights flashing in sync? Researchers think their bioluminescent resonance arises due to internal biological oscillators that automatically result in each firefly syncing up with its neighbors.
Is this group of fireflies enjoying a higher level of group consciousness? Probably not, since we can explain the phenomenon without recourse to any intelligence or consciousness. But in biological structures with the right kind of information pathways and processing power, these tendencies toward self-organization can and often do produce larger-scale conscious entities.
Our resonance theory of consciousness attempts to provide a unified framework that includes neuroscience, as well as more fundamental questions of neurobiology and biophysics, and also the philosophy of mind. It gets to the heart of the differences that matter when it comes to consciousness and the evolution of physical systems.
It is all about vibrations, but it's also about the type of vibrations and, most importantly, about shared vibrations.
Funding: The Conversation funded this study.
Source: Tam Hunt – The Conversation
Publisher: Organized by NeuroscienceNews.com.
Image Source: NeuroscienceNews.com image is adapted from The Conversation news release.
[cbtabs][cbtab title="MLA"]The Conversation"Could Consciousness Come Down to the Way Things Vibrate?." NeuroscienceNews. NeuroscienceNews, 18 November 2018.
<https://neurosciencenews.com/consciousness-vibration-10217/>.[/cbtab][cbtab title="APA"]The Conversation(2018, November 18). Could Consciousness Come Down to the Way Things Vibrate?. NeuroscienceNews. Retrieved November 18, 2018 from https://neurosciencenews.com/consciousness-vibration-10217/[/cbtab][cbtab title="Chicago"]The Conversation"Could Consciousness Come Down to the Way Things Vibrate?." https://neurosciencenews.com/consciousness-vibration-10217/ (accessed November 18, 2018).[/cbtab][/cbtabs]
communication through coherenceconsciousnessEEGevolutionevolutionary neurosciencegamma wavesmathematicsmind body problemNeuroscienceNeurotechpanpsychismphotonsphysicsPsychologyresonance theory of consciousnessThe Conversationvibrations
germana fiorani says:
Ottimo! Finalmente qualcuno e' sulla buona strada per risolvere l'enigma della coscienza
Faraz Khan says:
This is not science. It is new age religion ravings .shame on you for publishing it.
The scientific paradigm consists of a cycle of observe :theory: experiment-measure :conclude : modify theory : back to beginning.
Where is the theory and the experentation .
Zero score on both.
João Morais says:
I am always skeptical about these pseudo-dualistic approaches on counsciousness.
Why do you feel that simple neuron firing and neural computations are not enough to generate whichever mental process? It was, without any doubt, an interesting read but I think the speculation does not take full account of current neuroscience bibliography and might be misleading some people outside the area. Cheers!
Piet says:
I have long felt that ALL matter arises from consciousness and has within it, some residual trace of that original consciousness. Thus a rock is a little conscious and a BIG rock, like the planet on which we live, has a HEAP of consciousness. Add time (= experience) and wisdom arises. The Earth has had BILLIONS of years of experience and has, consequently amassed considerable wisdom. With wisdom arises the impetus to share and i contend that the Earth does this by means of it's various plant and fungal emissaries, all provided for the willing to use in expanding their conscious appreciation of being a living entity. So there. Wotchareckon?
Consciousness isn't 'hard' to understand at all. All living things have some measure of consciousness. It is simply the organism's need to respond to its environment, in order to survive. As the organism's environment becomes more complex, so the consciousness has the need to expand. Need is the driving force of all things but humans try to intellectualise it.
Life is simple! KISS
MeMe says:
50 years later, 'science' is catchingup with new age hippies! I'm loving it and laughing my socks off at the same time :D :D :D Crack on folks, another 50 years and you might be even closer to catching up fully :D :D :D | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,415 |
\section*{Introduction}
The Scalar Torus Rigidity Theorem states that any Riemannian manifold which is diffeomorphic to an $n$-dimensional torus and which has nonnegative scalar curvature is isometric to a flat torus.
It is called a rigidity theorem because it is a theorem which forces a
Riemannian manifold to have a rigid structure: in this case to be isometric to a flat torus.
This theorem was proven for dimension $n=3$ by Schoen and Yau in 1979 \cite{Schoen-Yau-min-surf}, using results from minimal surface theory that can now be extended to higher dimensions. Gromov and Lawson gave a proof in all dimensions using the Lichnerowicz formula in \cite{Gromov-Lawson-1980}.
Recently, Gromov suggested that sequences of manifolds diffeomorphic to tori with almost non-negative scalar curvature and appropriate compactness conditions should converge to flat tori \cite{Gromov-Dirac}. By work of Gromov \cite{Gromov-Dirac} and of Bamler \cite{Bamler-16}, if one assumes additional conditions on the metric tensors to guarantee that they converge in the $C^0$ sense then one can obtain $C^0$ convergence of this sequence of tori with almost non-negative scalar curvature to flat tori. Since there are known examples of sequences without these additional hypotheses which do not converge in the $C^0$ or even Gromov-Hausdorff (GH) sense it was suggested by Gromov that the conjecture should be in terms of Sormani-Wenger Intrinsic Flat (SWIF) convergence. In \cite{Sormani-scalar}, Sormani formulated a precise conjecture for such a sequence of tori with almost non-negative scalar curvature as follows.
\begin{conjecture}\label{MainConjecture}
Let $M_j = (\mathbb{T}^3, g_j)$ be a sequence of Riemannian manifolds diffeomorphic to a $3$-torus such that
\begin{align} \label{HypothesisConjecture}
R_{j} \ge -\frac{1}{j}, \hspace{0.5cm} \Vol(M_j) \le V_0, \hspace{0.5cm} \Diam(M_j) \le D_0 \,\,\, \text{and} \,\,\, \MinA(M_j) \ge A_0 > 0,
\end{align}
where $R_j$ is the scalar curvature and $\MinA(M_j)$ is the area of the smallest closed minimal surface in $M_j$.
Then, there is a subsequence of $M_j$ converging in the SWIF sense to a flat torus: $M_{j_k} \stackrel {SWIF}{\longrightarrow} M_{\infty}$, where $M_\infty$ is a flat torus.
\end{conjecture}
Note that if any of the assumed conditions on the sequence in this conjecture are relaxed, then there are known counterexamples. The uniform volume and diameter bounds are necessary to prevent expansion and collapsing. The $\MinA$ condition is necessary to prevent bubbling and \textquotedblleft sewing" examples which would otherwise provide counterexamples to this conjecture \cite{Basilio-Dodziuk-Sormani}. The $\MinA$ condition is natural in this setting given the crucial role that stable minimal surfaces played in Schoen-Yau's proof of the torus rigidity theorem \cite{Schoen-Yau-min-surf}. Also, the $\MinA$ condition has appeared in the rigidity results of Bray, Brendle and Neves \cite{BBN-CAG-10} for area minimizing 2-spheres in 3-spheres and Bray, Brendle, Eichmair and Neves \cite{BBEN-CPAM-10} for area minimizing projective planes in 3-manifolds.
Moreover, there are counterexamples to Conjecture \ref{MainConjecture} if SWIF convergence is replaced with GH convergence. Basilio and Sormani constructed sequences of tori satisfying the hypotheses of this conjecture with no GH limit and a GH limit to a non-smooth space that is not the flat torus \cite{Basilio-Sormani-1}. These examples have increasingly thin wells with positive scalar
curvature surrounded by an annular region with $R_{j} \ge -\frac{1}{j}$. Since thin wells disappear
under SWIF convergence, these examples converge in the SWIF sense.
On the other hand, all of their examples converge in SWIF sense to a flat torus.
The first paper to apply SWIF convergence in the setting of positive scalar curvature was the paper by Lee and Sormani \cite{LeeSormani1} where sequences of rotationally symmetric, asymptotically flat manifolds with ADM mass tending to zero are shown to converge to regions in Euclidean space under SWIF convergence. In this case there are counterexamples given by Lee and Sormani where sequences with the properties above do not converge under GH convergence and hence SWIF convergence is essential. This informs the intuition that SWIF convergence is well suited for convergence questions where positive scalar curvature is natural. This intuition inspires the use of SWIF convergence in Conjecture \ref{MainConjecture} and is reinforced by the results of this paper.
In this paper, we will prove Conjecture \ref{MainConjecture} in the setting where the metrics are assumed to be warped product metrics. This setting was first suggested by Sormani after formulating Conjecture \ref{MainConjecture} \cite{Sormani-scalar}. We find a subsequence which converges in both SWIF and GH sense and we note that a subsequence is necessary because the
sequence could have subsequences converging to different flat tori. It is perhaps surprising that we
obtain GH convergence as this means that our sequences are not developing long thin wells as in the examples in \cite{Basilio-Sormani-1}.
In particular, we are going to consider the following two special cases: \\
\noindent (i) \textit{Doubly Warped Products:} For $x, y, z \in [- \pi, \pi]$ and positive $a_j, b_j \colon [- \pi, \pi] \rightarrow \mathbb{R}$, the metric
\begin{equation}
g_j = a_j^2(z) dx^2 + b_j^2(z) dy^2 + dz^2,
\end{equation}
is a doubly warped product. \\
\noindent (ii) \textit{Singly Warped Products:} For $x, y, z \in [- \pi, \pi]$ and positive $f_j \colon [- \pi, \pi] \times [-\pi, \pi] \rightarrow \mathbb{R}$, the metric
\begin{equation}
g_j = dx^2 + dy^2 + f_j^2(x, y) dz^2,
\end{equation}
is a singly warped product. \\
Throughout the rest of this paper, by \textquotedblleft doubly warped product" we will be referring to item (i) above and by \textquotedblleft singly warped product" we will be referring to item (ii) above. Now, we state our main result for doubly warped products.
The main result for doubly warped products:
\begin{thm}\label{MainThmCase1}
Suppose we have a sequence $M_j = (\mathbb{T}^3, g_j)$, where each $g_j$ is a doubly warped product satisfying
\begin{equation}
R_{j} \ge -\frac{1}{j},
\hspace{0.5cm} \Diam(M_j) \le D_0, \,\,\,\text{and} \,\,\, \MinA(M_j) \ge A_0 > 0,
\end{equation}
then there exists a subsequence $M_{j_k}$ converging uniformly to a flat torus. In particular, $M_{j_k}$ converges in the GH and SWIF sense to a flat torus.
\end{thm}
To prove Theorem \ref{MainThmCase1}, we first show in Theorem \ref{W12convergence} that the scalar curvature bound allows us to find subsequences of the warping functions that converge to nonzero constants in $W^{1,2}(S^1)$. A key step in obtaining these convergent subsequences is the existence of upper and lower uniform bounds on the warping functions found in Proposition \ref{UpperLowerBoundsDoubly}. We show these bounds can be derived from the MinA and diameter bounds in the hypotheses of our theorem.
It then follows from Morrey's inequality for one dimensional domains that in fact we have $C^{0, \frac{1}{2}}$ convergence. From here we obtain uniform, GH, and SWIF convergence. Note that we did not use a uniform volume bound, yet this is necessary for Conjecture \ref{MainConjecture} to hold in general.
The main result for singly warped products:
\begin{thm}\label{MainThmCase2}
Suppose we have a sequence $M_j = (\mathbb{T}^3, g_j)$, where $g_j$ is a singly warped product satisfying
\begin{equation}
R_{j} \ge -\frac{1}{j},
\hspace{0.5cm} \Vol(M_j) \le V_0,
\,\,\,\text{and} \,\,\, \MinA(M_j) \ge A_0 > 0,
\end{equation}
Then, there exists a subsequence $M_{j_k}$ converging uniformly to a flat torus. In particular, $M_{j_k}$ converges in the GH and SWIF sense to a flat torus.
\end{thm}
To prove Theorem \ref{MainThmCase2}, we find in Lemma \ref{f_j convergence} a subsequence of the warping functions $f_j$ that converges to a positive constant in $W^{1,2}(\mathbb{T}^2)$, similar to the proof of Theorem \ref{MainThmCase1}. This involves completely different techniques than those that are used to prove the analogous statement for Theorem \ref{MainThmCase1}. This involves using the Stampacchia lemma, Lemma \ref{StampLemma}, to gain a bound from above on $f_j$ in Proposition \ref{LinfinityEsthj} combined with control obtained from the $\MinA$ lower bound in Lemma \ref{bound from minA singly warped}. Then, we use a maximum principle on a certain operator to obtain $C^0$ control from below on the warping functions in Corollary \ref{C0Boundh}, which then allows us to appeal to a result of the first author and Sormani to find that a subsequence has the desired convergence to a flat torus~\cite{BAS}. Note that we do not use a uniform diameter bound.
We now give a brief outline of the paper:
In Section \ref{sec: background} we describe the definitions and previous theorems which will be essential to understanding the results of this paper. In the interest of keeping the background concise we offer up references to interesting definitions and results which are not essential to understanding the main results of this paper. In Section \ref{sec: Case 1} the proof of Theorem \ref{MainThmCase1} is given and in Section \ref{sec: Case 2} the proof of Theorem \ref{MainThmCase2} is given. In both sections many interesting estimates are developed which give potential insight into the full conjecture \ref{MainConjecture}.
\begin{acknowledgement*}
This research began at the Summer School for Geometric Analysis, located at and supported by the Fields Institute for Research in Mathematical Sciences. The authors would like to thank the organizers of this summer school, Spyros Alexakis, Walter Craig, Robert Haslhofer, Spiro Karigiannis and McKenzie Wang. The authors would like to thank Christina Sormani for her direction, advice, and constant support. Specifically, Christina brought this team together during the Field's institute to work on this project, organized workshops at the CUNY graduate center which the authors participated in, and provided travel support to the team members. Christina Sormani is supported by NSF grant DMS-1612049.
Brian Allen is supported by the USMA. Davide Parise is partially supported by the Swiss National Foundation grant 200021L\_175985.
\end{acknowledgement*}
\section{Background} \label{sec: background}
\noindent In this section, we review some basic definitions and facts that will be used throughout the paper.
We start by reviewing the notion of uniform convergence of metric spaces. Consider two metric spaces $(X,d_1)$, $(X,d_2)$ and define the uniform distance between these metric spaces to be
\begin{align}
d_{unif}(d_1,d_2) = \sup_{x,y\in X} |d_1(x,y) - d_2(x,y)|.
\end{align}
Notice that if you think of the metrics as functions, $d_i: X\times X \rightarrow \mathbb{R}$, then the uniform distance $d_{unif}(d_1,d_2)$ is equivalent to the $C^0$ distance between functions. We say that a sequence of metrics spaces $(X, d_j)$ converges to the metric space $(X,d_{\infty})$ if $d_{unif}(d_j, d_{\infty}) \rightarrow 0$ as $j \rightarrow \infty$.
One limitation of uniform convergence is that it requires the metric spaces to have the same topology and so other important notions of convergence have been introduced which do not depend on topology. Two particularly important notions of convergence for metric spaces and Riemannian manifolds are Gromov-Hausdorf (GH) convergence and Sormani-Wenger Intrinsic Flat convergence (SWIF). In this paper we will be able to show GH and SWIF convergence but due to the symmetries of the metrics assumed we will also be able to show uniform convergence and so we will not define these notions in this paper. For the definition of GH convergence see \cite{BBI} and the references therein. For the definition of SWIF convergence see \cite{SW-JDG}.
In the case of doubly warped products we will be able to show $C^{0,\frac{1}{2}}$ convergence of the warping functions $a_j(z), b_j(z)$ to constants in section \ref{sec: Case 1}. We will then wrap up the proof of Theorem \ref{MainThmCase1} by applying the following corollary of Proposition 3.7 in \cite{Gromov-metric2}, for the case of GH convergence, and a corollary of Theorem 5.6 in \cite{SW-JDG}, for the case of SWIF convergence.
\begin{corollary}\label{SW-SmoothConvToSWIF}
If a sequence of Riemannian manifolds $M_j=(M, g_j)$ converges to the Riemannian manifold $M_{\infty}=(M, g_{\infty})$ in the $C^{0,\alpha}$ sense then $M_j$ converges in GH and SWIF to $M_{\infty}$ as well.
\end{corollary}
It is important to note that showing $C^{0,\alpha}$ convergence of the warping functions is equivalent to showing $C^{0,\alpha}$ convergence of the Riemannian manifolds in the doubly warped product case.
In the singly warped product case we will not be able to show $C^{0,\frac{1}{2}}$ convergence of the warping functions but instead will be able to show $W^{1,2}$ convergence. For singly warped proucts it is a fact that $W^{1,2}$ convergence of the warping functions implies $L^2$ convergence of the Riemannian manifolds which will allow us to conclude the proof of Theorem \ref{MainThmCase2} by applying the recent result of the first author and Sormani \cite{BAS}.
\begin{thm}\label{Allen-SormaniThm}
Let $g_j = dx^2+dy^2+f_j(x,y)^2dz^2$ be a metric on a torus $M_j = [-\pi,\pi]^2 \times_{f_j}[-\pi,\pi]$ where $f_j \in C^0(T^2)$. Assume that, $ f_j \rightarrow f_{\infty}= c > 0$ in $L^2$, and $0< f_{\infty} - \frac{1}{j} \le f_j \le K < \infty$. Then, $M_j$ converges uniformly to the flat torus $M_{\infty}$ which also implies $M_j$ converges in GH and SWIF to $M_{\infty}$.
\end{thm}
Notice that this theorem gives conditions which when combined with $L^2$ convergence imply that the Riemannian manifolds converge in the uniform, GH, and SWIF sense to the same Riemannian manifold as the $L^2$ convergence implies. We now move on to produce the estimates needed to apply Corollary \ref{SW-SmoothConvToSWIF} and Theorem \ref{Allen-SormaniThm} in order to prove Theorem \ref{MainThmCase1} in section \ref{sec: Case 1} and prove Theorem \ref{MainThmCase2} in section \ref{sec: Case 2}.
\section{Doubly Warped Products of One Variable} \label{sec: Case 1}
In this section, we will prove Theorem \ref{MainThmCase1}. Recall that we are considering a sequence of doubly warped product metrics $g_j$ on $\mathbb{T}^3$ such that $x, y, z \in [- \pi, \pi]$ and $a_j, b_j \colon [- \pi, \pi] \rightarrow \mathbb{R}$ positive functions, and
\begin{equation}\label{Case1}
g_j = a_j^2(z) dx^2 + b_j^2(z) dy^2 + dz^2.
\end{equation}
\subsection{Scalar Curvature of Doubly Warped Products} \label{subsec: ScalarEq Case 1}
In order to prove Theorem \ref{MainThmCase1} we will need to find an expression for the scalar curvature of a doubly warped product. The resulting differential inequality from $R_j \geq -\frac{1}{j}$ will be key to showing the desired convergence.
\begin{lemma}
The scalar curvature for a metric $g= a(z)^2dx^2 + b(z)^2dy^2 + dz^2$ on $\mathbb{T}^3$ is
\begin{equation} \label{scalarcurvCase1}
R = -2 \left(\frac{a''}{a}+\frac{b''}{b} + \frac{a' b'}{ab}\right).
\end{equation}
\end{lemma}
\begin{proof} By Section 4.2.4 of Petersen's book \cite{Petersen}, a metric of this form has the following Ricci curvature.
\begin{align}
\text{Ric}\left(\frac{\partial}{\partial x}\right) &= \left(-\frac{a''}{a} - \frac{a' b'}{ab}\right)\frac{\partial}{\partial x}
\\\text{Ric}\left(\frac{\partial}{\partial y}\right) &= \left(-\frac{b''}{b} - \frac{a' b'}{ab}\right)\frac{\partial}{\partial y}
\\\text{Ric}\left(\frac{\partial}{\partial z}\right) &= \left(-\frac{a''}{a} - \frac{b''}{b}\right)\frac{\partial}{\partial z}
\end{align}
Thus, we have the conclusion of this lemma.
\end{proof}
This lemma means that under the conditions of Theorem \ref{MainThmCase1}, the condition $R_j \geq -\frac{1}{j}$ translates into the following condition on $a_j$ and $b_j$
\begin{equation}
\frac{a''_j}{a_j}+ \frac{b''_j}{b_j} + \frac{a'_j b'_j}{a_j b_j} \leq \frac{1}{2j}.
\end{equation}
\subsection{Diameter Bounds, the MinA Condition and Uniform Bounds} \label{subsec: MinA Case 1}
We will now investigate the consequences of the MinA hypothesis, with a particular emphasis on how this translates into natural lower and upper bounds for the warping functions.
We start with the so-called $\MinA$ condition, according to which the smallest possible area of a closed minimal surface in $M_j$ is bounded from below by a certain constant:
\begin{equation} \label{EquationMinA}
\MinA (M_j) = \inf \{ \Area(\Sigma) | \; \Sigma \text{ is a closed minimal surface in $M_j$}\} \geq A_0 > 0.
\end{equation}
Notice that this lower bound is uniform in $j$.
Intuitively speaking, this condition allows us to control better the geometry of the $M_j$'s, for instance by avoiding bubbling phenomena and sewing counterexamples. What's more, a careful analysis of these pathological construction yields that the $\MinA$ hypothesis is not only a simplification of the problem but also a rather natural notion.
For further details on those examples where $\MinA(M_j) \rightarrow 0$ we refer to \cite{Basilio-Sormani-1}. A notion related to the $\MinA$ hypothesis has been used by Bray, Brendle and Neves, in \cite{BBN-CAG-10}, to prove a cover splitting rigidity theorem and by the same authors with Eichmair, in \cite{BBEN-CPAM-10}, to prove a rigidity theorem concerning $\mathbb{RP}^3$.
Our first result is that (2.1) yields a pointwise lower bound, independent of $j$, on the product $a_j(z)b_j(z)$ and uniform lower bounds on the integrals of $a_j$ and $b_j$.
\begin{lemma}\label{subsec: bound from minA doubly warped}
Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case1}). If $\MinA(M_j) \geq A_0$, then for all $z \in [-\pi, \pi]$,
\begin{align}
a_j(z)b_j(z) &\geq \frac{A_0}{4\pi^2} \\
\int_{-\pi}^\pi a_j(z) dz &\geq \frac{A_0}{2\pi}, \\
\int_{-\pi}^\pi b_j(z) dz &\geq \frac{A_0}{2\pi}.
\end{align}
\end{lemma}
\begin{proof}
Consider the three homotopy classes
\begin{equation}
[x,y,0): x,y \in S^1], [(x,0,z): x,z \in S^1] \; \text{and} \; [(0,y,z): z,y \in S^1],
\end{equation}
in the three dimensional torus $\mathbb{T}^3$. These are just the homotopy classes of two dimensional tori in our manifold.
By a result of Schoen-Yau \cite{Schoen-Yau-min-surf}, we can find a minimal surface in each of these homotopy classes. So, if $\phi_{z=0}(x,y): \mathbb{T}^2 \to M_j$ is the embedding of the representative $(x,y,0)$ into our manifold $M_j$, its area satisfies
\begin{equation}
Area(\phi_{z=0}(x,y))\geq \MinA(M_j)\geq A_0 > 0
\end{equation}
Similarly,
\begin{equation}
Area(\phi_{x=0}(z,y)) \geq A_0>0,
\end{equation}
\begin{equation}
Area(\phi_{y=0}(x,y)) \geq A_0>0.
\end{equation}
Let $\omega$ be the 2-form
$ a_j(z)b_j(z)dx\wedge dy$ obtained by contracting the volume form with $\frac{\de}{\de z}$. Then
\begin{align}
\Area(\phi_{z=0}(x,y)) &= \int_{-\pi}^\pi \int_{-\pi}^\pi \phi_{z=0}^*(\omega)
= \int_{-\pi}^\pi \int_{-\pi}^\pi a_j(0)b_j(0) dxdy
= 4\pi^2 a_j(0)b_j(0).
\end{align}
Observe that we could have chosen any other $z$-level set. For any $z_0$,
\begin{equation}
a_j(z_0)b_j(z_0) \geq \frac{\Area(\phi_{z=z_0}(x,y))}{4\pi^2} \geq \frac{A_0}{4\pi^2}
\end{equation}
This establishes the first part of the theorem.
For the other two parts of the theorem, we just compute the areas of the embeddings $\phi_{x=0}$ and $\phi_{y=0}$ and apply the same argument as above.
The computations here give
\begin{align}
\Area(\phi_{x=0}(y,z)) = \int_{-\pi}^\pi \int_{-\pi}^\pi b_j(z) dydz =2\pi \int_{-\pi}^\pi b_j(z) dz
\end{align}
and
\begin{align}
\Area(\phi_{y=0}(x,z)) = \int_{-\pi}^\pi \int_{-\pi}^\pi a_j(z) dxdz =2\pi \int_{-\pi}^\pi a_j(z) dz.
\end{align}
Therefore we can find constants $C_1, C_2$ giving the last two estimates in the theorem.
\end{proof}
We now investigate the diameter bound $\Diam(M_j)\leq D_0$ and find uniform upper and lower bounds for $a_j$ and $b_j$. In doing so, we need the following two lemmas regarding the warping functions $a(z), b(z)$ on a fixed $M_j$.
\begin{lemma}\label{lower}
Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case1}). If $M_j$ has diameter $Diam(M_j) \leq D_0$, then
\begin{equation} \label{productbound}
\min_{z \in [-\pi,\pi]}a_j(z)\leq D_0 \quad \text{and} \quad \min_{z \in [-\pi,\pi]}b_j(z)\leq D_0,
\end{equation}
\end{lemma}
\begin{proof}
Consider two points on the torus $P_1 = (0,0,0)$ and $P_2 = (1, 0, 0)$. For $t \in [0,1]$, let $\Gamma(t)=(x(t),y(t),z(t))$ be the minimal geodesic with $\Gamma(0) = P_1$ and $\Gamma(1) = P_2$. We may think of $\Gamma(t)$ as a path in $\mathbb{R}^3$ starting at $(0,0,0)$ and ending at
$(1+2\pi n_1, 2\pi n_2, 2\pi n_3)$ for some $n_1, n_2, n_3 \in \mathbb{Z}$. Thus,
\begin{equation}
1 \leq \abs{\int_{0}^1 x'(t) dt}.
\end{equation}
Now, let $z_1$ be such that $a_j(z_1) = \min_{z \in [-\pi, \pi]}a_j(z)$, which is positive by assumption (\ref{Case1}). Note that $z_1$ depends on $j$. Then,
\begin{align}
\min_{z \in [-\pi, \pi]}a_j(z) = a_j(z_1) &\leq a_j(z_1) \abs{\int_{0}^{1}x'(t) dt}
\\&\leq \int_{0}^{1} a_j(z_1) \abs{x'(t)} dt
\\&\leq \int_{0}^{1} \sqrt{a_j(z_1)^2 x'(t)^2 + b_j(z(t))^2y'(t)^2 + z'(t)^2} dt
\\&\leq \int_{0}^1 \sqrt{a_j(z(t))^2 x'(t)^2 + b_j(z(t))^2y'(t)^2 + z'(t)^2} dt
\\&= \mathrm{Length}(\Gamma) \leq \Diam(M_j) \leq D_0
\end{align}
We may do the same for $b_j$ using a minimal geodesic connecting $(0,0,0)$ and $(0,1,0)$.
\end{proof}
\begin{lemma}\label{logl2bound}
Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case1}). If $R_j \geq -\frac{1}{j}$, then the functions $\alpha_j(z):=\ln(a_j(z))$ and $\beta_j(z):=\ln(b_j(z))$ satisfy
\begin{equation}\label{gradbound}
\int_{-\pi}^{\pi} {\alpha'_j}^2 dz \leq \frac{2\pi}{j} \quad \text{and} \quad \int_{-\pi}^{\pi} {\beta'_j}^2 dz\leq \frac{2\pi}{j}.
\end{equation}
\end{lemma}
\begin{proof}
From Lemma \ref{scalarcurvCase1} and $R_j\geq-\frac{1}{j}$, we have
\begin{equation}\label{scalarcurvature}
R_j = \frac{a''_j}{a_j}+\frac{b''_j}{b_j}+\frac{a'_j b'_j}{a_j b_j}\leq\frac{1}{2j}
\end{equation}
Now, we compute the derivatives of $\alpha_j$ and $\beta_j$.
\begin{equation}
\alpha'_j=\frac{a'_j}{a_j}, \quad \alpha''_j=\frac{a''_j}{a_j}-\frac{{a'_j}^2}{a_j}, \quad \beta'_j=\frac{b'_j}{b_j} \quad \text{and} \quad \beta''_j=\frac{b''_j}{b_j}-\frac{{b'_j}^2}{b_j^2}
\end{equation}
Substituting into (\ref{scalarcurvature}) above inequality we have
\begin{equation}\label{Rbound}
\alpha''_j + \beta''_j + {\alpha'_j}^2 + {\beta'_j}^2 + \alpha'_j \beta'_j \leq\frac{1}{2j}
\end{equation}
Since $\alpha_j$ and $\beta_j$ are periodic, we may integrate this inequality to find
\begin{equation}
\int_{-\pi}^{\pi} {\alpha_j'}^2 + {\beta_j'}^2 + \alpha_j'\beta_j'\, dz \leq \frac{\pi}{j}
\end{equation}
\begin{equation}\label{alpha'beta'bound}\int_{-\pi}^{\pi} {\alpha_j'}^2 + {\beta_j'}^2 \,dz\leq \frac{\pi}{j} - \int_{-\pi}^{\pi}\alpha_j'\beta_j' dz
\end{equation}
Rewriting and then integrating (\ref{scalarcurvature}),
\begin{equation}
\frac{(a_j b_j)'' - a'_j b'_j}{a_j b_j}\leq\frac{1}{2j}
\end{equation}
\begin{equation}\label{intermediatebound}\int_{-\pi}^{\pi} \frac{a'_j b'_j}{a_j b_j}dz \geq \int_{-\pi}^{\pi} \frac{(a_j b_j)''}{a_j b_j}dz - \frac{\pi}{j}\end{equation}
Now, since $a_j b_j$ is periodic,
\begin{equation}
0= \int_{-\pi}^{\pi} \ln(a_j b_j)'' dz = \int_{-\pi}^{\pi} \frac{(a_j b_j)''}{a_j b_j} - \frac{{(a_j b_j)'}^2}{(a_j b_j)^2} dz
\end{equation}
Applying this identity to (\ref{intermediatebound}),
\begin{equation}\label{a'b'bound}\int_{-\pi}^{\pi} \frac{a'_j b'_j}{a_j b_j}dz \geq \int_{-\pi}^{\pi} \frac{(a_j b_j)''}{a_j b_j}dz - \frac{\pi}{j} = \int_{-\pi}^{\pi} \frac{{(a_j b_j)'}^2}{(a_j b_j)^2} dz - \frac{\pi}{j} \geq -\frac{\pi}{j}\end{equation}
Using the definition of $\alpha'_j$ and $\beta'_j$ and applying (\ref{a'b'bound}) to (\ref{alpha'beta'bound}),
\begin{equation}
\int_{-\pi}^{\pi} {\alpha_j'}^2 + {\beta_j'}^2 dz \leq \frac{\pi}{j} - \int_{-\pi}^{\pi}\alpha_j'\beta_j' dz = \frac{\pi}{j} - \int_{-\pi}^{\pi}\frac{a'_j b'_j}{a_j b_j} dz \leq \frac{2\pi}{j}
\end{equation}
Thus, we have the desired bounds.
\end{proof}
We now come to the most important result of this section:
\begin{proposition}\label{UpperLowerBoundsDoubly}
Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case1}). If $R_j \geq -\frac{1}{j}$, $\Diam(M_j) \leq D_0$, and $\MinA(M_j) \geq A_0>0 $, then there exist positive constants $A, A', B, B'$ independent of $j$ such that
\begin{align}
\frac{A_0}{4\pi^2D_0}e^{-\frac{2\pi}{\sqrt{j}}} &\leq a_j(z) \leq D_0 e^{\frac{2\pi}{\sqrt{j}}}
\\\frac{A_0}{4\pi^2D_0}e^{-\frac{2\pi}{\sqrt{j}}} &\leq b_j(z) \leq D_0 e^{\frac{2\pi}{\sqrt{j}}}
\end{align}
for all $z \in S^1$.
\end{proposition}
\begin{proof}
Using the notation of Lemma \ref{logl2bound}, we apply Cauchy-Schwarz and Lemma \ref{logl2bound}.
\begin{equation}
\int_{-\pi}^{\pi} |\alpha'_j(z)| dz \leq \sqrt{\int_{-\pi}^{\pi} |\alpha'_j(z)|^2 dz} \sqrt{\int_{-\pi}^{\pi} dz}\leq \frac{2\pi}{\sqrt{j}}
\end{equation}
\begin{equation}\int_{-\pi}^{\pi} |\beta'_j(z)| dz \leq \sqrt{\int_{-\pi}^{\pi} |\beta'_j(z)|^2 dz} \sqrt{\int_{-\pi}^{\pi} dz}\leq \frac{2\pi}{\sqrt{j}}
\end{equation}
So,
\begin{equation}
\begin{split}
&\ln\left(\frac{\max(a_j)}{\min(a_j)}\right)=\max(\alpha_j)-\min(\alpha_j)\leq\int_{-\pi}^{\pi} |\alpha'_j(z)| dz \leq \frac{2\pi}{\sqrt{j}} ,\\
&\ln\left(\frac{\max(b_j)}{\min(b_j)}\right)=\max(\beta_j)-\min(\beta_j)\leq\int_{-\pi}^{\pi} |\beta'_j(z)| dz \leq \frac{2\pi}{\sqrt{j}}\\
\end{split}
\end{equation}
By combining with Lemma \ref{lower},
\begin{equation}\label{maxbound}
\begin{split}
&\max(a_j)\leq e^{\frac{2\pi}{\sqrt{j}}}\min(a_j)\leq D_0 e^{\frac{2\pi}{\sqrt{j}}},\\
&\max(b_j)\leq e^{\frac{2\pi}{\sqrt{j}}}\min(b_j)\leq D_0e^{\frac{2\pi}{\sqrt{j}}}\\
\end{split}
\end{equation}
By Lemma \ref{subsec: bound from minA doubly warped}, $\min(a_j b_j)\geq \frac{A_0}{4\pi^2}$. Then, combining \eqref{productbound} with \eqref{maxbound}, we get a uniform upper and lower bound for $a_j$ and $b_j$ as follows
\begin{equation}
\begin{split}
&\min(a_j)\geq\frac{\min(a_j b_j)}{\max(b_j)}\geq \frac{A_0}{4\pi^2D_0}e^{-\frac{2\pi}{\sqrt{j}}}, \\
&\min(b_j)\geq\frac{\min(a_j b_j)}{\max(a_j)}\geq \frac{A_0}{4\pi^2 D_0}e^{-\frac{2\pi}{\sqrt{j}}}\\
\end{split}
\end{equation}
Thus, we have the desired uniform upper and lower bounds on $a_j$ and $b_j$.
\end{proof}
\subsection{$W^{1,2}$ Convergence and Proof of the Main Result}\label{subsec: ProofOfMainThm Case 1}
In this section we are going to use the bounds on the warping functions to prove that they converge to constants in $W^{1,2}$. We then use Morrey's inequality to show this implies $C^{0, \frac{1}{2}}$ convergence.
\begin{thm}\label{W12convergence}
Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case1}). If $R_j \geq -\frac{1}{j}$, $\Diam(M_j) \leq D_0$, and $\MinA(M_j) \geq A_0>0$, then there exist nonzero constants $a_\infty, b_\infty$ such that, after possibly passing to a subsequence, $a_i \to a_\infty$, $b_i \to b_\infty$ in $W^{1,2}(S^1)$.
\end{thm}
\begin{proof}
Using the notation of Lemma \ref{logl2bound}, we apply the Poincar\'e-Wirtinger inequality and use Lemma \ref{logl2bound} to obtain the limit as $j \to \infty$
\begin{align}
\Vert \alpha_j - \bar{\alpha}_j \Vert_2 &= \Vert \alpha_j-\frac{1}{2\pi}\int_{-\pi}^{\pi} \alpha_j dz \Vert_2\leq C\Vert \alpha'_j\Vert_2\to 0
\\ \Vert \beta_j - \bar{\beta}_j \Vert_2 &= \Vert \beta_j-\frac{1}{2\pi}\int_{-\pi}^{\pi} \beta_j dz\Vert_2 \leq C\Vert \beta'_j \Vert_2 \to 0,
\end{align}
where $C$ is a constant independent of $j$ and $\bar{\alpha}_j$ and $\bar{\beta}_j$ denote the averages of $\alpha_j$ and $\beta_j$ respectively.
From here on, we consider only the functions $\alpha_j$ as the arguments are identical for both $\alpha_j, \beta_j$. After passing to a subsequence, the above shows that we have a limiting function $\alpha_\infty$ so that
\begin{equation}
\alpha_{j_k} \to \alpha_\infty \text{ in } W^{1,2}(S^1)
\end{equation}
where $\alpha_\infty$ is a constant by the fact that
\begin{equation}
\int_{-\pi}^{\pi}|\alpha_\infty - \bar{\alpha}_{j_k} |^2 dz \leq \int_{-\pi}^{\pi}|\alpha_\infty - \alpha_{j_k} |^2 + |\alpha_{j_k} - \bar{\alpha}_{j_k} |^2 dz \to 0 \quad \text{as $j \rightarrow \infty$}.
\end{equation}
Now, by Proposition 2.4, there are positive constants $A, A'$ such that $A\leq a_j \leq A'$, thus
\begin{equation}
\int_{-\pi}^{\pi} \alpha_j dz = \int_{-\pi}^{\pi} \ln(a_j) dz \leq \int_{-\pi}^{\pi} \ln(A') dz = 2\pi \ln(A')
\end{equation}
and
\begin{equation}
2\pi \ln(A) = \int_{-\pi}^{\pi} \ln(A) dz\leq \int_{-\pi}^{\pi} \alpha_j dz
\end{equation}
So, the averages $\bar{\alpha}_j$ cannot get arbitrarily large or arbitrarily small as $i\to \infty$. In particular, $\alpha_\infty$ is a positive constant.
Now that we have found subsequences $\alpha_{j_k}$ and $\beta_{j_k}$ converging to some nonzero constants $\alpha_\infty$ and $\beta_\infty$, respectively, in $W^{1,2}(S^1)$, we can define
$a_\infty = e^{\alpha_\infty}, b_\infty = e^{\beta_\infty}$ to obtain subsequences of $a_j, b_j$ converging to nonzero constants $a_\infty, b_\infty$ in $W^{1,2}(S^1)$.
\end{proof}
We are now ready to prove our main result for doubly warped products.
\begin{proof}[Proof of Theorem \ref{MainThmCase1}]
By Theorem \ref{W12convergence}, we have that a subsequence of $a_j$ and $b_j$ converges in $W^{1,2}$ to constants $a_{\infty}$ and $b_{\infty}$. Applying Morrey's inequality for one-dimensional domains gives that a subsequence of $a_j$ and $b_j$ converges in $C^{0, \frac{1}{2}}$. Note that constant warping functions $a_{\infty}$, $b_{\infty}$ mean that the metric is flat. So, a subsequence of $M_j$ converges in $C^{0,\frac{1}{2}}$ to a flat torus. In particular, a subsequence GH and SWIF converges to a flat torus by Corollary \ref{SW-SmoothConvToSWIF}.
\end{proof}
\section{Singly Warped Products of Two Variables} \label{sec: Case 2}
In this section we will prove Theorem \ref{MainThmCase2}. Recall that we are considering a sequence of singly warped product metrics $g_j$ on $\mathbb{T}^3$ such that for $x, y, z \in [- \pi, \pi]$ and positive $f_j \colon [- \pi, \pi] \times [-\pi, \pi] \rightarrow \mathbb{R}$, $g_j$ can be written as
\begin{equation}\label{Case2}
g_j = dx^2 + dy^2 + f_j^2(x, y) dz^2.
\end{equation}
The singly warped product case is substantially different than the doubly warped product case because $f_j$ is a function of two variables. This means we will not be able to apply Morrey's inequality to go from $W^{1,2}$ convergence to $C^{0,\alpha}$ convergence as we were able to do for doubly warped products.
\subsection{Scalar Curvature} \label{subsec: Scalar Calculation Case 2}
We first analyze the partial differential inequality on the warping function obtained from $R_j \geq -\frac{1}{j}$.
Applying the calculations of Dobarro and Dozo, we may find an expression for the scalar curvature of a singly warped product on $\mathbb{T}^3$ \cite{Dobarro-Dozo}.
\begin{lemma}\label{Case2scalarcurvature}
The scalar curvature for a metric $g= dx^2 + dy^2 + f^2(x,y)dz^2$ on $\mathbb{T}^3$ is
\begin{equation}
R = - 2\frac{\Delta f}{f}
\end{equation}
where $\Delta$ is the Euclidean Laplacian.
\end{lemma}
\begin{remark}
If we further assume that the $M_j$'s are scalar flat, i.e. $\frac{\Delta f_j}{f_j} = 0 $ then the maximum principle shows that the warping functions must be constant. This is one way to see that scalar flat $3$-tori with a singly warped product metric are isometric to a flat torus.
\end{remark}
Lemma \ref{Case2scalarcurvature} means that the assumption on scalar curvature in Theorem \ref{MainThmCase2} translates into the following inequality for the warping functions:
\begin{equation}\label{scalarcurvatureinequality}\frac{\Delta f_j}{f_j} \leq \frac{1}{2j}\end{equation}
\subsection{Minimal Surfaces, the MinA Condition and Uniform Bounds}\label{subsec:MinA Condition Case 2}
In this section we investigate the $\MinA$ condition in a similar fashion as in Subsection \ref{subsec: bound from minA doubly warped} in order to obtain important bounds on $f_j$ which will be used in later subsections. More precisely we will be able to prove that the $\MinA$ lower bound yields uniform lower bounds on the simple integrals of $f_j(x_0, y)$ and $f_j(x, y_0)$, and on the double integral of $f_j(x, y)$.
\begin{lemma}\label{bound from minA singly warped}
Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case2}). If $\MinA(M_j) \geq A_0>0$, then
\begin{align}
& \int_{-\pi}^\pi \int_{-\pi}^\pi f_j(x,y)dxdy \geq A_0, \\
& \int_{-\pi}^\pi f_j(x_0, y) dy \geq \frac{A_0}{2\pi} \textit{ for all } x_0\in [-\pi, \pi],\\
&\int_{-\pi}^\pi f_j(x, y_0) dx \geq \frac{A_0}{2\pi} \textit{ for all } y_0\in [-\pi, \pi].
\end{align}
\end{lemma}
\begin{proof}
The proof is exactly as in Lemma \ref{subsec: bound from minA doubly warped}. The areas of the embeddings $\phi_{x=x_0}, \phi_{y=y_0}$ in this case are
\begin{align}
\Area(\phi_{x=x_0}) & = 2\pi \int_{-\pi}^\pi f_j(x_0,y)dy,
\end{align}
and
\begin{align}
\Area(\phi_{y=y_0}) & = 2\pi\int_{-\pi}^\pi f_j(x,y_0)dx.
\end{align}
The first bound follows by integrating either of the bounds above.
\end{proof}
\subsection{$W^{1,2}$ Convergence of $h_j$} \label{subsec:Convergence of hj Case 2}
Define the sequence $\{h_j\}$ by $h_j(x, y) := \ln(f_j(x, y))$, for every $j \in \mathbb{N}$. Note that these functions are defined on $\mathbb{T}^2 = [-\pi, \pi] \times [-\pi, \pi]$, since they are periodic in $x$ and $y$. Moreover, define $\bar{h}_j$ to be the average of $h_j$ over the torus $\mathbb{T}^2$, i.e.
\begin{equation}
\bar{h}_j = \frac{1}{\vert \mathbb{T}^2 \vert} \int_{\mathbb{T}^2} h_j \, dA
\end{equation}
where $\vert \mathbb{T}^2 \vert = 4\pi^2$ and $dA = dx dy$.
The averages $\bar{h}_j$ cannot get arbitrarily large due to the following control inequalities.
\begin{align} \label{average h cannot get arbitrarily large}
\int_{\mathbb{T}^2} h_j dA = \int_{\mathbb{T}^2} \ln(f_j) dA \leq \ln \left(\int_{\mathbb{T}^2} f_j dA \right ) \leq \ln(\Vol(M_j)) \leq \ln(V_0).
\end{align
We now calculate the inequality satisfied by $h_j$
\begin{equation}
\Delta h_j = \Delta \ln(f_j) = \frac{\Delta f_j}{f_j} - \frac{|\nabla f_j|^2}{f_j^2}.
\end{equation}
Applying (\ref{scalarcurvatureinequality}), we obtain an elliptic inequality satisfied by $h_j$
\begin{align}\label{NiceEllipticEq}
\Delta h_j +|\nabla h_j|^2 \le \frac{1}{2j}.
\end{align}
\begin{proposition}
Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case2}). Let $h_j:= \ln(f_j)$. If $R_j \geq -\frac{1}{j}$, then
$\Vert h_j - \bar{h}_j \Vert_{L^2(\mathbb{T}^2)} \rightarrow 0$, as $j \rightarrow \infty$.
\end{proposition}
\begin{proof}
Since $f_j$ is periodic in both variables, $h_j$ is as well. So, $h_j$ may be thought of as a smooth function on a flat $2$-torus. Integrating \eqref{NiceEllipticEq} we find
\begin{equation} \label{IntEllipticEq}
\int_{\mathbb{T}^2}\left( \Delta h_j +|\nabla h_j|^2\right) \, dA \le \int_{\mathbb{T}^2}\frac{1}{2j} \, dA
\end{equation}
which then becomes
\begin{equation}
\int_{\mathbb{T}^2}|\nabla h_j|^2 \, dA \le \frac{1}{2j} \vert\mathbb{T}^2\vert
\end{equation}
So,
\begin{equation}\label{hjW12}\int_{\mathbb{T}^2}|\nabla h_j|^2 \rightarrow 0\end{equation}
as $j \rightarrow \infty$.
Applying the Poincar\'e-Wirtinger inequality with constant $C_{\mathbb{T}^2}$ from $\mathbb{T}^2$, we find that
\begin{equation}
\Vert h_j - \bar{h}_j \Vert_{L^2(\mathbb{T}^2)}^2 = \int_{\mathbb{T}^2} |h_j - \bar{h}_j|^2 \, dA \leq C_{\mathbb{T}^2}^2\int_{\mathbb{T}^2}|\nabla h_j|^2 \, dA \to 0, \quad \text{as $j \rightarrow \infty$},
\end{equation}
thus establishing the claim and finishing the proof.
\end{proof}
After \eqref{average h cannot get arbitrarily large} we are naturally inclined to investigate whether the averages $\bar{h}_j$ can get arbitrarily small as well. In order to argue that this does not happen we will show $W^{1,2}$ convergence of $f_j$ to its average on a subsequence in Lemma \ref{f_j convergence}. This will require an upper bound for $f_j$ which follows by showing that $h_j$ has a uniform upper bound.
\subsection{$W^{1,2}$ Convergence of $f_j$} \label{subsec: Convergence of f_j Case 2}
We start by proving some important consequences of \eqref{NiceEllipticEq} which will be used in Proposition \ref{LinfinityEsthj}.
\begin{lemma}\label{LemBeforeStamp}
Let $H$ be a solution to the inequality \eqref{NiceEllipticEq}. For $k \in [0,\infty)$, define $H_k := \max (H-k,0)$ and $A(k) := \{x \in \mathbb{T}: H(x) > k\}$. Then, we find
\begin{equation
\int_{A(k)}|\nabla H_k|^2 dA \le \frac{1}{2j} |A(k)|.
\end{equation}
Furthermore, we obtain the estimate
\begin{align}\label{Eq1BeforeStamp}
\frac{4}{9} \int_{A(k)}|\nabla H_k^{\frac{3}{2}}|^2 dA \leq \frac{1}{2j} \int_{A(k)}H_k dA+\frac{1}{2j} |A(k)|
\end{align}
which implies
\begin{align}\label{Eq2BeforeStamp}
\left(\int_{A(k)} H_k^2 dA\right)^{1/2} \le \left(\frac{9 C_{\frac{8}{7}}^2}{4}\right)^{\frac{1}{3}} |A(k)|^{\frac{1}{2}}\left( \frac{1}{2j}\int_{A(k)}H_k dA + \frac{1}{2j}|A(k)| \right)^{\frac{1}{3}}.
\end{align}
\end{lemma}
\begin{proof}
By multiplying \eqref{NiceEllipticEq} by $H_k$ and integrating over $A(k)$ we find
\begin{equation}
\int_{A(k)}H_k \Delta H_k dA + \int_{A(k)}H_k|\nabla H_k|^2 dA \le \frac{1}{2j} \int_{A(k)}H_k dA
\end{equation}
Now by integrating by parts, using the fact that $H_k \equiv 0$ on $\partial A(k)$, and rearranging, we find
\begin{align}\label{helpfulEq}
\int_{A(k)}H_k|\nabla H_k|^2 dA \le \frac{1}{2j} \int_{A(k)}H_k dA+\int_{A(k)}|\nabla H_k|^2 dA
\end{align}
We can rewrite the first gradient term by noticing
\begin{equation}
|\nabla H_k^{3/2}|^2 = \frac{9}{4} H_k |\nabla H_k|^2
\end{equation}
and we can deal with the gradient term on the right hand side of \eqref{helpfulEq} by integrating \eqref{NiceEllipticEq} to find
\begin{equation}
\int_{A(k)}|\nabla H_k|^2 dA \le \frac{1}{2j} |A(k)|.
\end{equation}
Using these equations we can rewrite \eqref{helpfulEq} as
\begin{equation}\label{intermediateestimate}
\frac{4}{9} \int_{A(k)}|\nabla H_k^{\frac{3}{2}}|^2 dA \le \frac{1}{2j} \int_{A(k)}H_k dA+\frac{1}{2j} |A(k)|
\end{equation}
Now, we will modify the power on the left hand side to be able to apply Sobolev's inequality. Apply H\"older's inequality with powers $p=\frac{14}{8}$ and $q=\frac{7}{3}$ to find
\begin{equation}
\int_{A(k)} |\nabla H_k^{\frac{3}{2}}|^{\frac{8}{7}} dA \leq |A(k)|^{\frac{3}{7}} \left(\int_{A(k)} |\nabla H_k^{\frac{3}{2}}|^{2} dA\right)^{\frac{8}{14}}
\end{equation}
Apply this to (\ref{intermediateestimate}).
\begin{equation}\label{intermediateestimate2}
\frac{4}{9} |A(k)|^{-\frac{3}{4}}\left(\int_{A(k)}|\nabla H_k^{\frac{3}{2}}|^{\frac{8}{7}} dA\right)^{\frac{14}{8}} \le \frac{1}{2j} \int_{A(k)}H_k dA+\frac{1}{2j} |A(k)|
\end{equation}
Now we apply the compactly supported version of the Sobolev inequality with $p = \frac{8}{7}$, $p^*=\frac{8}{3}$, $n=2$, to find
\begin{align}\label{JustAfterSobolev}
\frac{4C_{\frac{8}{7}}^{-2}}{9} |A(k)|^{-\frac{3}{4}}\left(\int_{A(k)} H_k^4 dA\right)^{\frac{3}{4}} \le \frac{1}{2j} \int_{A(k)}H_k dA+\frac{1}{2j} |A(k)|
\end{align}
Again by H\"older's inequality,
\begin{equation}
\int_{A(k)}H_k^2 dA \le |A(k)|^{\frac{1}{2}}\left(\int_{A(k)}H_k^4 \right)^{\frac{1}{2}}
\end{equation}
which when applied to \eqref{JustAfterSobolev} we find
\begin{equation
\frac{4 C_{\frac{8}{7}}^{-2}}{9}|A(k)|^{-\frac{3}{2}} \left(\int_{A(k)} H_k^2 dA\right)^{\frac{3}{2}} \le \frac{1}{2j} \int_{A(k)}H_k dA+\frac{1}{2j} |A(k)|.
\end{equation}
Thus, we find the final estimate.
\end{proof}
We now state Stampacchia's Lemma which will be important in performing the Stampacchia iteration argument in Proposition \ref{LinfinityEsthj}. This method was originally developed in \cite{stamp} and a recent application of this lemma to Inverse Mean Curvature Flow can be found in the work of Huisken and Ilmanen\cite{HuiskenIlmanen} where Stampacchia's Lemma is used to obtain a lower bound on the mean curvature which is independent of the mean curvature of the initial hypersurface. This method has also been widely used in hypersurface flows in general over the last 40 years.
\begin{lemma}\label{StampLemma} \textbf{Stampacchia's Lemma:}
Let $f \ge 0$ be a non-increasing function on $[\bar{x},\infty)$. Assume for some $C>0$, $\eta>0$, $\gamma>1$ that $f$ satisfies
\begin{equation}
(y-x)^{\eta}f(y) \le C f(x)^{\gamma}\text{, for } y \ge x \ge \bar{x}.
\end{equation}
Then $f(z) = 0$ for $z \ge \bar{x} +d$, where $d^p = C f(\bar{x})^{\gamma-1}2^{\frac{\eta \gamma}{\gamma-1}}$.
\end{lemma}
Now we apply Lemma \ref{StampLemma} to equation \eqref{NiceEllipticEq} by taking advantage of equation \eqref{average h cannot get arbitrarily large} and Lemma \ref{LemBeforeStamp}.
\begin{proposition}\label{LinfinityEsthj}
Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case2}). Let $R_j \geq -\frac{1}{j}$ and $\Vol(M_j) \leq V_0$. If $h_j:= \ln(f_j)$, then
\begin{equation}
\max_{T^2} h_j \le C
\end{equation}
where $C$ is independent of $j$. This immediately implies
\begin{equation}
\max_{T^2} f_j \le e^C.
\end{equation}
\end{proposition}
\begin{proof}
The goal is to apply Lemma \ref{StampLemma} to the function $f(k) = |A(k)|$ where $A(k) = \{x \in \mathbb{T}: h_j(x) > k\}$, $k \in [0,\infty)$ which will imply that for $z \ge d$ we have $f(z) = |A(z)| = 0$ and hence $h_j\le d$ is bounded.
Since the following estimate will be independent of $j$ we will use $H = h_j$ for the rest of the argument and now we define $H_k = \max (H-k,0)$, $k \in [0,\infty)$ so that for $l > k$ we have that $H_l < H_k$, $A(l) \subset A(k)$ and $H_k \equiv 0$ on $\partial A(k)$.
By using \eqref{Eq1BeforeStamp} of Lemma \ref{LemBeforeStamp} we have
\begin{align
\int_{A(k)}|\nabla H_k^{\frac{3}{2}}|^2dA \le \frac{9}{8j} \int_{A(k)} |H_k| dA + \frac{9}{8j}|A(k)|
\end{align}
Working with the right hand side of this inequality we can apply Holder's inequality to find
\begin{equation}
\int_{A(k)}|\nabla H_k^{\frac{3}{2}}|^2 dA \le\frac{9}{8j} |A(k)|^{1/2}\left(\int_{A(k)} |H_k|^2 dA\right)^{1/2}+ \frac{9}{8j}|A(k)|.
\end{equation}
Now by applying \eqref{Eq2BeforeStamp} of Lemma \ref{LemBeforeStamp} we find for $\bar{C}$, $\tilde{C}$ independent of $j$,
\begin{equation}
\int_{A(k)}|\nabla H_k^{\frac{3}{2}}|^2 dA \leq \bar{C} |A(k)|\left( \frac{1}{2j}\int_{A(k)}H_k dA + \frac{1}{2j}|A(k)| \right)^{1/3} + \tilde{C}|A(k)|
\end{equation}
Lastly, by applying inequality \eqref{average h cannot get arbitrarily large} and using that $|A(k)| \leq Vol(M_j) \leq V_0$, we find for $C'$ independent of $j$,
\begin{equation}
\int_{A(k)}|\nabla H_k^{\frac{3}{2}}|^2 dA \le \bar{C} |A(k)|\left(\frac{\ln(V_0)}{2j} + \frac{1}{2j}|A(k)| \right)^{1/3} + \tilde{C}|A(k)| \le C'|A(k)|
\end{equation}
Applying H\"older's inequality and Sobolev's inequality just as we did to go from (\ref{intermediateestimate}) to (\ref{JustAfterSobolev}),
\begin{equation}
C_{\frac{8}{7}}^{-2} |A(k)|^{-\frac{3}{4}}\left(\int_{A(k)} H_k^4 dA\right)^{\frac{3}{4}} \leq C'|A(k)|
\end{equation}
\begin{equation}
\int_{A(k)} H_k^4 dA \leq C''|A(k)|^{\frac{7}{3}}
\end{equation}
By choosing $l > k$ we know that $A(l) \subset A(k)$ and $H_k \geq |l-k|$ on $A(l)$ and so we find
\begin{equation}
|A(l)| |l-k|^{4} \le \int_{A(l)} H_k^{4} dA \leq \int_{A(k)} H_k^{4} dA \leq C''|A(k)|^{\frac{7}{3}}.
\end{equation}
So we arrive at the desired inequality
\begin{equation}
|l-k|^{\eta}|A(l)| \le C |A(k)|^{\gamma}
\end{equation}
where $\eta = 4 >0$, $C > 0$ and $\gamma = \frac{7}{3} > 1$ and so by Stampacchia's Lemma \ref{StampLemma} we know that there exists a $d$ independent of $j$ so that for $z \ge d$ we find that $A(z) = 0$. Notice that this gives an upper bound on $H$, i.e. $H \le d$.
Since the whole argument so far has not depended on $j$ we note that we have found a uniform upper bound for $h_j$ for all $j$, as desired.
\end{proof}
We now use this newly found control on $h_j$ and $f_j$ to find $W^{1,2}$ convergence of $f_j$.
\begin{lemma}\label{f_j convergence} Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case2}). Let $R_j \geq -\frac{1}{j}$, $\Vol(M_j)\leq V_0$, and $\MinA(M_j) \geq A_0>0$. Then, for some constant $f_{\infty} \in (0,\infty)$ and some subsequence $f_{j_k}$, $f_{j_k} \to f_{\infty}\in (0,\infty)$ in $W^{1,2}$. Similarly, if $h_j := ln(f_j)$, then for some subsequence and some constant $h_{\infty} \in \mathbb{R}$, $h_{j_k} \to h_{\infty}$ in $W^{1,2}$.
\end{lemma}
\begin{proof}
Let $h_j := \ln(f_j)$, By (\ref{hjW12}),
\begin{equation}
\int_{\mathbb{T}^2}\frac{|\nabla f_j|^2}{f_j^2}\, dA = \int_{\mathbb{T}^2}|\nabla h_j|^2 \, dA\to 0, \quad \text{as $j \rightarrow \infty$}
\end{equation}
Now we calculate
\begin{align}
\int_{\mathbb{T}^2}|\nabla f_j|^2 \, dA&=\int_{\mathbb{T}^2}\frac{|\nabla f_j|^2}{f_j^2} f_j^2 \, dA
\\&\leq \left(\int_{\mathbb{T}^2}\frac{|\nabla f_j|^2}{f_j^2} \, dA \right) \left (\max_{\mathbb{T}^2}f_j^2\right)
\\&\le C_0 \left (\int_{\mathbb{T}^2}\frac{|\nabla f_j|^2}{f_j^2}\, dA \right)\rightarrow 0
\end{align}
Where the upper bound on $f_j$ comes from Proposition \ref{LinfinityEsthj}.
Then by Lemma \ref{bound from minA singly warped} combined with the the uniform bound on $\|f_j\|_{L^1} = \Vol(M_j) \leq V_0$, we have that $\bar{f}_j = \frac{1}{|\mathbb{T}^2|}\int_{\mathbb{T}^2}f_jdA$ is uniformly bounded above and below by positive constants and so some subsequence $\bar{f}_{j_k}$ converges to a constant $\bar{f}_{\infty}$. Then, by using the Poincar\'e inequality we find
\begin{equation}
\int_{\mathbb{T}^2}|\nabla f_{j_k}|^2 \, dA\ge \int_{\mathbb{T}^2}|f_{j_k}-\bar{f}_{j_k}|^2 \, dA
\end{equation}
which gives the convergence of $f_{j_k} \to \bar{f}_{\infty}\in (0,\infty)$ in $L^2$. Since $\nabla \bar{f}_{\infty} \equiv 0$, we in fact have that $f_{j_k} \to \bar{f}_{\infty}$ in $W^{1,2}$.
Similarly, we obtain that $h_{j_k} \to h_{\infty} \in \mathbb{R}$ in $W^{1,2}$.
\end{proof}
\subsection{$C^0$ Convergence from Below}\label{subsec:C0 From Below Case 2}
Now, we have from Lemma \ref{f_j convergence} that on some subsequence, $f_j$ converges in $W^{1,2}$ to a positive constant. We would like to use this to show convergence of $M_j$, as in (\ref{Case2}), to a flat torus. It was shown in \cite{BAS} by Allen and Sormani that if a warped product converges in $L^2$ then a sufficient condition for the uniform, GH and Flat convergence to agree with the $L^2$ convergence is a $C^0$-bound from below (See Theorem \ref{Allen-SormaniThm}). We will now show this estimate by using a maximum principle argument on the operator $L f = \Delta f + |\nabla f|^2$. By the inequality in equation \eqref{NiceEllipticEq} we expect to be able to bound the minimum of $h_j$ using the maximum principle as we now proceed to do.
\begin{lemma}\label{minControl}
Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case2}). Let $R_j \geq -\frac{1}{j}$. Let $h_j := \ln(f_j)$. Then, for $\Omega=[\eta_1,\eta_2]\times S^1\subset \mathbb{T}^2 = -[\pi,\pi]\times [-\pi,\pi]$, we have
\begin{equation}
\min_{\Omega} h_j \ge \min_{\partial \Omega} h_j - (e^{\gamma_j \eta_2} - e^{\gamma_j \eta_1})
\end{equation}
where $\gamma_j=\sqrt{\frac{C}{2j}}$
\end{lemma}
\begin{proof}
Consider the function $h_j -e^{\gamma_j \theta_1}$, $\theta_1 \in [\eta_1,\eta_2]$, $\gamma_j > 0$, and compute
\begin{align}
L(h_j -e^{\gamma_j \theta_1}) &= \Delta(h_j -e^{\gamma_j \theta_1}) + |\nabla( h_j-e^{\gamma_j \theta_1})|^2
\\&=\Delta(h_j -e^{\gamma_j \theta_1}) + |\nabla h_j|^2 -2\langle \nabla h_j,\nabla e^{\gamma_j \theta_1}\rangle +|\nabla e^{\gamma_j \theta_1}|^2
\\&=L(h_j) -2\langle \nabla( h_j-e^{\gamma_j \theta_1}),\nabla e^{\gamma_j \theta_1}\rangle - |\nabla e^{\gamma_j \theta_1}|^2 -\Delta e^{\gamma_j \theta_1}.
\end{align}
Thus, we obtain the identity
\begin{align}
L(h_j -e^{\gamma_j \theta_1})&+2\langle \nabla( h_j-e^{\gamma_j \theta_1}),\nabla e^{\gamma_j \theta_1}\rangle =L(h_j) - |\nabla e^{\gamma_j \theta_1}|^2 -\Delta e^{\gamma_j \theta_1},
\end{align}
whose right-hand side can be bounded as follows, using (\ref{NiceEllipticEq}),
\begin{align}
L(h_j) - |\nabla e^{\gamma_j \theta_1}|^2 -\Delta e^{\gamma_j \theta_1} \le \frac{1}{2j} -\gamma_j^2(e^{2\gamma_j \theta_1}+e^{\gamma_j \theta_1}) \le \frac{1}{2j} -\gamma_j^2 C' \leq 0,
\end{align}
where we uniformly bound the exponential terms independent of $j$ and choose $\gamma_j = \sqrt{\frac{C}{2j}}$ for some $C$ independent of $j$ so that the last inequality holds. Then, by the minimum principle, we know that the minimum must be obtained on the boundary, i.e.
\begin{align}
\min_{\Omega} h_j - e^{\gamma_j \eta_1} \ge \min_{\Omega} \left(h_j- e^{\gamma_j \theta_1}\right) \ge \min_{\partial \Omega} \left(h_j- e^{\gamma_j \theta_1}\right)\ge \min_{\partial \Omega} h_j - e^{\gamma_j \eta_2}.
\end{align}
\end{proof}
Now in order to effectively use Lemma \ref{minControl} we must be able to control $h_j$ on $\partial \Omega$ and so now we obtain this control for a subsequence.
\begin{lemma}\label{C0S1}
If $h_j \rightarrow h_{\infty}$ in $W^{1,2}(\mathbb{T}^2)$ and if $h^{\bar{y}}_j(x) := h_j(x,\bar{y})$ for $\bar{y} \in [-\pi,\pi]$, then for some subsequence, $h^{\bar{y}}_{j_k}(x) \rightarrow h_{\infty}$ in $C^0([-\pi,\pi])$, for almost every $\bar{y} \in [-\pi,\pi]$.
\end{lemma}
\begin{proof}
Since for some subsequence, $h_{j_k} \rightarrow h_{\infty}$ in $W^{1,2}(\mathbb{T}^2)$, we know that
\begin{equation}
\int_{-\pi}^{\pi}\left(\int_{-\pi}^{\pi}|h_j-h_{\infty}|^2 +\left|\frac{\partial h_j}{\partial x}\right|^2+\left|\frac{\partial h_j}{\partial y}\right|^2 dx \right )dy \longrightarrow 0,
\end{equation}
as $j \rightarrow \infty$, but this implies that
\begin{equation}
\int_{-\pi}^{\pi}|h^{\bar{y}}_{j_k} - h_{\infty}|^2 + \left|\frac{\partial h^{\bar{y}}_{j_k}}{\partial x}\right|^2 dx \longrightarrow 0
\end{equation}
for a.e. $\bar{y} \in [-\pi,\pi]$, as $k \rightarrow \infty$. This means that $h^{\bar{y}}_{j_k} \rightarrow h_{\infty}$ in $W^{1,2}([-\pi,\pi])$ and so, by Morrey's inequality, we find that $h^{\bar{y}}_{j_k} \rightarrow h_{\infty}$ in $C^0$, for almost every $\bar{y} \in [-\pi,\pi]$, as desired.
\end{proof}
By combining Lemma \ref{minControl} with Lemma \ref{C0S1} we obtain the $C^0$ control from below necessary to apply Theorem \ref{Allen-SormaniThm} of Allen-Sormani.
\begin{corollary}\label{C0Boundh}
Let $M_j = (\mathbb{T}^3, g_j)$ as in (\ref{Case2}). Let $R_j \geq -\frac{1}{j}$, $\Vol(M_j)\leq V_0$, and $\MinA(M_j) \geq A_0>0$. Let $h_j := \ln(f_j)$. Then, after passing to a subsequence, we have the inequality
\begin{align} \label{bound h}
h_{j_k} \ge h_{\infty} - \frac{C}{k}
\end{align}
on $\mathbb{T}^2$, from which we deduce
\begin{align}
f_{j_k} \ge f_{\infty} - \frac{\bar{C}}{k},
\end{align}
again on $\mathbb{T}^2$.
\end{corollary}
\begin{proof}
We may apply Lemma \ref{f_j convergence}, which allows us to apply Lemma \ref{C0S1}. So, we know that if we define $h^{\bar{y}}_j(x) = h_j(x,\bar{y})$, for $\bar{y} \in [-\pi,\pi]$, we find that $h^{\bar{y}}_{j_k}(x) \rightarrow h_{\infty}$ in $C^0([-\pi,\pi])$, for almost every $\bar{y} \in [-\pi,\pi]$. We can pick a $\eta_1,\eta_2 \in [-\pi,\pi]$ so that we get the desired $C^0$ convergence on $S^1 \times \{\eta_1\}$ and $S^1\times \{\eta_2\}$. Now we can apply Lemma \ref{minControl} on $S^1\times [\eta_1,\eta_2]$ and $S^1\times [\eta_2,\eta_1+2\pi]$ in order to achieve the desired bound (\ref{bound h}). Exponentiating both sides of (\ref{bound h}),
\begin{align}
f_k &\ge e^{\ln(f_{\infty}) - \frac{C}{k}}=
f_{\infty}e^{\frac{-C}{k}},
\end{align}
gives the desired bound for $f$.
\end{proof}
\subsection{SWIF Convergence to a Flat Tori}\label{subsec:SWIF Conv Case 2}
We are now able to conclude with the proof of our main theorem. For this proof we will combine the $W^{1,2}$ convergence, and the bounds from above and below on $f$ obtained in the last section with the following recent result of the first author and Sormani:
\begin{thm}\label{Allen-SormaniThm}
Let $g_j = dx^2+dy^2+f_j(x,y)^2dz^2$ be a metric on a torus $M_j = [-\pi,\pi]^2 \times_{f_j}[-\pi,\pi]$ where $f_j \in C^0(T^2)$. Assume that, $ f_j \rightarrow f_{\infty}= c > 0$ in $L^2$, and $0< f_{\infty} - \frac{1}{j} \le f_j \le K < \infty$. Then, $M_j$ converges uniformly to the flat torus $M_{\infty}$ which also implies $M_j$ converges in GH and SWIF to $M_{\infty}$.
\end{thm}
Notice that this theorem gives conditions which when combined with $L^2$ convergence imply that the Riemannian manifolds converge in the uniform, GH, and SWIF sense to the same Riemannian manifold as the $L^2$ convergence implies. We now use this result to finish up the proof of the main theorem.
\begin{proof}[Proof of Theorem \ref{MainThmCase2}]
The $C^0$-bound from below given in Corollary \ref{C0Boundh} combined with the uniform bound of Proposition \ref{LinfinityEsthj} and the $W^{1, 2}$-convergence of Lemma \ref{f_j convergence} allows us to apply Theorem \ref{Allen-SormaniThm} of Allen-Sormani \cite{BAS} to obtain uniform, GH, and SWIF convergence to a flat torus on a subsequence.
\end{proof}
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me.make(route);
} else {
frappe.show_not_found(route);
}
}
},
make_page: function(double_column, page_name) {
return frappe.make_page(double_column, page_name);
}
});
frappe.make_page = function(double_column, page_name) {
if(!page_name) {
var page_name = frappe.get_route_str();
}
var page = frappe.container.add_page(page_name);
frappe.ui.make_app_page({
parent: page,
single_column: !double_column
});
frappe.container.change_to(page_name);
return page;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,365 |
It hears the gnashing even as it hears the blessing.
The door to the mind should only open from the heart.
An enemy who gets in, risks the danger of becoming a friend.
Harjo, Joy, Conflict Resolution for Holy Beings: Poems; Copyright © 2015 by W. W. Norton & Company. Reprinted with permission of Anderson Literary Management LLC, 244 Fifth Avenue, Floor 11, New York, NY 10001.
There's my cousin. Auntie. Uncle.
Chips, candy, and not enough clean water. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,619 |
<?xml version="1.0" encoding="UTF-8"?>
<project version="4">
<component name="ProjectModuleManager">
<modules>
<module fileurl="file://$PROJECT_DIR$/.idea/Password-Strength-Check-Using-C.iml" filepath="$PROJECT_DIR$/.idea/Password-Strength-Check-Using-C.iml" />
</modules>
</component>
</project>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,752 |
ŽNK Donat, ženski je nogometni klub iz Zadra.
Povijest
Ženski nogometni klub Donat osnovan je 2015. godine u Zadru.
Izvori
Hrvatski ženski nogometni klubovi
Ž Donat | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,727 |
package com.castlemock.web.mock.rest.utility;
import com.castlemock.model.mock.rest.domain.RestParameterQuery;
import java.net.URLEncoder;
import java.nio.charset.StandardCharsets;
import java.util.List;
import java.util.Map;
import java.util.Set;
import java.util.regex.Matcher;
import java.util.regex.Pattern;
import java.util.stream.Collectors;
public final class RestParameterQueryValidator {
private RestParameterQueryValidator(){
}
public static boolean validate(final List<RestParameterQuery> parameterQueries,
final Map<String, Set<String>> pathParameters){
for(Map.Entry<String, Set<String>> pathParameterEntry : pathParameters.entrySet()){
final String pathParameter = pathParameterEntry.getKey();
final Set<RestParameterQuery> matching = pathParameterEntry.getValue()
.stream()
.flatMap(pathQuery -> parameterQueries.stream()
.filter(parameterQuery -> parameterQuery.getParameter().equals(pathParameter))
.filter(parameterQuery -> validate(pathQuery, parameterQuery)))
.collect(Collectors.toSet());
if (matching.isEmpty()) {
return false;
}
}
return true;
}
private static boolean validate(final String pathQuery,
final RestParameterQuery parameterQuery){
String query = pathQuery;
if(parameterQuery.getUrlEncoded()) {
query = URLEncoder.encode(pathQuery, StandardCharsets.UTF_8);
}
if(parameterQuery.getMatchAny()){
return true;
} else if(parameterQuery.getMatchRegex()){
final Pattern pattern = parameterQuery.getMatchCase() ?
Pattern.compile(parameterQuery.getQuery()) :
Pattern.compile(parameterQuery.getQuery(), Pattern.CASE_INSENSITIVE);
final Matcher matcher = pattern.matcher(query);
return matcher.matches();
} else if(parameterQuery.getMatchCase()){
return query.equals(parameterQuery.getQuery());
}
return query.equalsIgnoreCase(parameterQuery.getQuery());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,068 |
using System.Collections.Generic;
using System.Linq;
namespace ChronicNetCore.Handlers
{
public class HandlerRegistry
{
readonly Dictionary<HandlerType, List<ComplexHandler>> _handlers =
new Dictionary<HandlerType, List<ComplexHandler>>();
public HandlerRegistry Add(HandlerType type,
IEnumerable<ComplexHandler> handlers)
{
_handlers.Add(type, handlers.ToList());
return this;
}
public IList<ComplexHandler> GetHandlers(HandlerType type)
{
if (_handlers.ContainsKey(type))
{
return _handlers[type].ToList();
}
return new List<ComplexHandler>();
}
public void MergeWith(HandlerRegistry registry)
{
foreach (var type in registry._handlers.Keys)
{
if (_handlers.ContainsKey(type) == false)
{
_handlers[type] = new List<ComplexHandler>();
}
_handlers[type].AddRange(registry._handlers[type]);
}
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,556 |
On Wednesday night I had a spontaneous night out with my friend Zoe. She had a 2 for 1 voucher for the cinema and wanted to see The Greatest Showman before it finished showing in our local cinema, Uckfield Picture House. I hadn't been that bothered about seeing it but no-one else could go with her and I thought it would be nice to spend some time with her sans kids as we'd never managed it before.
I'm so glad I went, we had a brilliant night! It was so lovely getting to know Zoe a bit better without the constant interruptions of the kids and we decided to grab something to eat first too since both of our husbands were home early (a rare occurrence for both of us)! We headed to Thai Terre for a quick dinner and a beer before the film and our food came really fast which was ideal. I had my favourite, Pad See-Ew with chicken - delicious as always and Zoe had the red Thai curry.
As for the film itself, I loved it! Like, really really loved it. Despite not being 100% sure of the story and not a massive fan of musicals per se (I like them I just wouldn't usually make a bee-line to the cinema to see one), I was immediately sucked into the atmosphere and had actual goosebumps from the opening scene! The singing, while not always perfect, was good, fun and fitted the story well. Two songs in particular stood out, one being the trapeze scene with Zac Efron and Zendaya (who is stunning, particularly with the pink hair - love it).
I'm actually tempted to buy the soundtrack. The lyrics are amazing.
"But I won't let them break me down to dust I know that there's a place for us For we are glorious When the sharpest words wanna cut me down I'm gonna send a flood, gonna drown them out I am brave, I am bruised I am who I'm meant to be, this is me Look out 'cause here I come And I'm marching on to the beat I drum I'm not scared to be seen I make no apologies, this is me"
If, like me, you're not sure of the storyline, it's basically about a poor boy with a big imagination and dreams, who grows up to become The Greatest Showman who creates a circus. How he does it, who with and what happens next are all fairly predictable but told well. I loved the whole cast, particularly the bearded lady who is awesome and sings those lyrics above. The costumes are stunning, the gold leotard is my favourite, plus all the amazing Victorian dresses and boots, oh and the ringmaster costumes. And Jenny Lind's dresses!
There's romance, politics, history, fun, adventure, family, awesome characters, amazing trapeze and dance scenes and some fab songs. The bit at the start with the young Phineas and Charity getting to know each other had such beautiful tones and lighting, I wanted to be there with my camera! It's inspired me with a few ideas for my bexphoto 365 project. It was a great way to spend a couple of hours, especially in our lovely Uckfield Picture House, it's so comfy and they have a great choice of beverages and food (you can get coffee or a glass of fizz, ice-cream or good old popcorn). It's such a beautiful old building as well.
I would definitely recommend both the film and the Picture House and you may have already seen my full review of Thai Terre. The Greatest Showman has finished showing now in Uckfield but catch it on DVD when it comes out if you fancy a beautiful, fun, feelgood film. I want to see it again! | {
"redpajama_set_name": "RedPajamaC4"
} | 1,862 |
\section{The Role of Quantum Simulations}
The motivation for numerical solutions to the quantum many body problem is the intractability of analytic solutions except in limited circumstances. Indeed, the extension of the analytic solution of the single electron Hydrogen atom to two electrons is already problematic, as emphasized in the classic text of Bethe and Salpeter~\cite{bethe57}. As a consequence, quantum simulations approaches appeared in chemistry~\cite{anderson75,ceperley81}, condensed matter~\cite{ceperley80,hirsch82}, nuclear~\cite{negele86,lynn19}, and high energy physics~\cite{kogut79,blankenbecler81,kogut83} almost as soon as computers became reasonably available for scientific work.
There is considerable methodological linkage in quantum simulation between these fields. For example, in the case of the DQMC method used in this work, the problem of treating the accumulation of round off errors at low temperatures was first developed in the nuclear physics community~\cite{sugiyama86} before being adapted to condensed matter. Likewise, the linear scaling methods of lattice gauge theory (LGT)~\cite{batrouni85,duane85,gottleib87}
were soon adapted to condensed matter physics~\cite{scalettar86,scalettar87}, albeit with only
limited success owing to the extreme anisotropic (non-relativistic) nature of the condensed matter space-imaginary time lattices. Many of the algorithms, like Fourier Acceleration, which are crucial to LGT, were first implemented and tested for classical spin models~\cite{davies88}.
\vskip0.10in \noindent
\section{Determinant Quantum Monte Carlo}
The DQMC method is a specific type of AFQMC ~\cite{blankenbecler81,Buendia86,white89,chen92,assaad02,gubernatis16,alhassid17,hao19,he19}.
In DQMC, the partition function ${\cal Z}$ is expressed as a path integral
and the Trotter approximation is used to isolate the quartic terms,
\begin{align}
{\cal Z} = {\rm Tr} \, e^{-\beta \hat H}
= {\rm Tr} \, [ e^{-\Delta \tau \hat H} ]^{L_\tau}
\sim {\rm Tr} \, [ e^{-\Delta \tau \hat H_t} e^{-\Delta \tau \hat H_U} ]^{L_\tau}
\, ,
\end{align}
where $\hat H_t$ includes the hopping and chemical potential (together with all other bilinear terms in the fermionic operators), and $\hat H_U$ the on-site interactions, in the Hubbard Hamiltonian of Eq.~\eqref{eq:ham_spinful}. The latter are then decomposed via,
\begin{align}
e^{-\Delta \tau U (n_{i\uparrow} - \frac{1}{2})
(n_{i\downarrow} - \frac{1}{2})}
= \frac{1}{2} e^{-U \Delta \tau/4} \sum_{s_i=\pm 1} e^{\lambda s_i (n_{i\uparrow} - n_{i\downarrow})}
\, ,
\label{eq:HS_spinful}
\end{align}
where ${\rm cosh} \, \lambda = e^{U \Delta \tau / 2}$. A Hubbard-Stratonovich variable (bosonic field) $s_i$ must be introduced at each spatial site $i$ and for each imaginary time slice
$U_{\Delta \tau}$. The prefactor $\frac{1}{2} e^{-U \Delta \tau/4} $ is an irrelevant constant which may be dropped.
The key observation is that the right hand side of Eq.~\eqref{eq:HS_spinful} is now
quadratic in the fermions, so that the partition function ${\cal Z}$ is a trace over a product of quadratic forms of fermionic operators:
\begin{align}
{\cal Z} = \sum_{\{s_{i\tau} \}}{\rm Tr_{\uparrow}} \, [
e^{\vec c^{\, \dagger}_{\uparrow} K \vec c^{\phantom{\dagger}}_\uparrow}
e^{\vec c^{\, \dagger}_{\uparrow} P^1 \vec c^{\phantom{\dagger}}_\uparrow}
\cdots
e^{\vec c^{\, \dagger}_{\uparrow} K \vec c^{\phantom{\dagger}}_\uparrow}
e^{\vec c^{\, \dagger}_{\uparrow} P^L_\tau \vec c^{\phantom{\dagger}}_\uparrow}
\, ]
\,\,\, {\rm Tr_{\downarrow}} \, [
e^{\vec c^{\, \dagger}_{\downarrow} K \vec c^{\phantom{\dagger}}_\downarrow}
e^{- \vec c^{\, \dagger}_{\downarrow} P^1 \vec c^{\phantom{\dagger}}_\downarrow}
\cdots
e^{\vec c^{\, \dagger}_{\downarrow} K \vec c^{\phantom{\dagger}}_\downarrow}
e^{- \vec c^{\, \dagger}_{\downarrow} P^L_\tau \vec c^{\phantom{\dagger}}_\downarrow}
\, ]
\, ,
\label{eq:Z_tr}
\end{align}
with
$ \vec c^{\, \dagger}_{\sigma} =
(\, c^{\, \dagger}_{\sigma 1 }
\, c^{\, \dagger}_{\sigma 2 }
\cdots
\, c^{\, \dagger}_{\sigma N }
\, )^T$.
The matrix $K$ is the same for all time slices and contains $\mu \Delta \tau$ along its diagonal and $t\Delta \tau$ for sites connected by the hopping. (In the case of the ionic Hubbard model, the staggered site energy term also appears along the diagonal.) The matrices $P^\tau$ are diagonal, with $ P^\tau_{ii} = \lambda \sigma s_{i \tau}$ ($\sigma =\pm 1$ for $\uparrow$ and $\downarrow$). All matrices have dimension equal to the number of spatial lattice sites $N$.
The trace of a quadratic form such as Eq.~\eqref{eq:Z_tr} can be done analytically~\cite{blankenbecler81,white89,chen92,assaad02,gubernatis16,alhassid17,hao19,he19}, resulting in
\begin{align}
{\cal Z} = \sum_{\{s_{i\tau} \}}
{\rm det} [ I + e^{K} e^{P^1}
\cdots
e^{K} e^{P^{L_\tau}} ]
\,\, {\rm det} [ I + e^{K} e^{-P^1}
\cdots
e^{K} e^{-P^{L_\tau}} ]
\, .
\label{eq:Z_det}
\end{align}
The expression for ${\cal Z}$ of Eq.~\eqref{eq:Z_det} contains no quantum operators, just the matrices $K, \{ P^\tau \}$ of the quadratic forms. Its calculation is thereby reduced to a classical Monte Carlo problem in which the sum over $\{ s_{i\tau} \}$ must be done stochastically with a weight equal to the product of the two determinants. That these might become negative is the origin of the SP in DQMC.
Besides this spinful case, we have also investigated interacting spinless fermions. The problem formulation in QMC simulations is almost identical to that of the spinful case, but now the decoupling of the interactions $V$ on each nearest-neighbor bond $\langle i,j\rangle$ reads
\begin{align}
e^{-\Delta \tau V (n_i - \frac{1}{2}) (n_j - \frac{1}{2})}
= \frac{1}{2} e^{-V \Delta \tau/4} \sum_{s_{ij}=\pm 1} e^{\lambda s_{ij} (n_i - n_j)}
\, ,
\label{eq:HS_spinless}
\end{align}
where ${\rm cosh} \, \lambda = e^{V \Delta \tau / 2}$. The Hubbard-Stratonovich variable $s$ now lives on the bonds, and its total number for the rhombus-shaped space-time lattices used here is $3L^2 L_\tau$. After a similar procedure for tracing out the fermions, one ends up with
\begin{align}
{\cal Z} = \sum_{\{s_{ij,\tau} \}}
{\rm det} [ I + e^{K} e^{P^1}
\cdots
e^{K} e^{P^{L_\tau}} ],
\label{eq:Z_det_spinless}
\end{align}
where the elements of the diagonal matrix $P^\tau$ are $P_{ii}^\tau = (-1)^i \lambda \sum_j s_{ij,\tau}$. A major difference from the spinful case is that the protection that the sign of the determinants of the matrices for up and down spin channels display in bipartite lattices by getting locked together~\cite{Hirsch1985,Iazzi2016} is no longer present. This can generically give rise to an even more detrimental SP, yet it allows us to systematically locate the QCP of Eq.~(\ref{eq:ham_spinless}) with a large accuracy.
\vskip0.10in \noindent
\section{Physical Observables}
The central object in the QMC simulations is the Green's function $\textbf{\rm G}^\sigma$ whose matrix elements are $G_{ij}^\sigma(\tau^\prime,\tau)=\langle c_{i\sigma}^{\phantom{\dagger}}(\tau^\prime) c_{j\sigma}^\dagger(\tau)\rangle$. By using Wick contractions and the fermionic anti-commutation relations one can define all quantities used in the main text. These are the double occupancy,
\begin{equation}
\langle n_{\uparrow\downarrow}\rangle = \langle n_\uparrow n_\downarrow\rangle ,
\end{equation}
the static susceptibility
\begin{equation}
\chi(\textbf{q}) = \frac{\beta}{N}\sum_{i,j}e^{{\rm i} \textbf{q}\cdot (\textbf{R}_i - \textbf{R}_j)} \langle (n_{i,\uparrow} - n_{i,\downarrow})(n_{j,\uparrow} - n_{j,\downarrow})\rangle,
\end{equation}
and the pair-susceptibility
\begin{equation}
P_\alpha = \int_0^\beta d\tau \langle \Delta_\alpha^{\phantom{\dagger}}(\tau)\Delta_\alpha^\dagger(0)\rangle,
\label{eq:pairsusc}
\end{equation}
with the momentum-dependent pair operator given by
\begin{equation}
\Delta_\alpha^\dagger = \frac{1}{N}\sum_{\textbf{k}} f_\alpha (\textbf{k})\,c_{\textbf{k}\uparrow}^\dagger c_{-\textbf{k}\downarrow}^\dagger.
\end{equation}
The form factors $f_\alpha (\textbf{k})$ describe the various symmetry channels investigated:
\begin{equation}
f_d (\textbf{k}) = \cos k_x - \cos k_y;\ \ f_{s^*} (\textbf{k}) = \cos k_x + \cos k_y; \ \ f_s(\textbf{k}) = 1,
\end{equation}
for $d$-wave, extended $s$-wave, and $s$-wave pairings, respectively.
In all data presented, we subtract the uncorrelated (non-vertex) contribution, $P_\alpha^{\rm nv}$, in which pairs of fermionic operators are first averaged before taking the product, i.e., terms in Eq.~(\ref{eq:pairsusc}) such as $\langle c_{i\downarrow}^\dagger(\tau)c_{j\downarrow}^{\phantom{\dagger}}(0)c_{l\downarrow}^\dagger(\tau)c_{m\downarrow}^{\phantom{\dagger}}(0)\rangle$ get replaced by their decoupled contributions $\langle c_{i\downarrow}^\dagger(\tau)c_{j\downarrow}^{\phantom{\dagger}}(0)\rangle \, \langle c_{l\downarrow}^\dagger(\tau)c_{m\downarrow}^{\phantom{\dagger}}(0)\rangle$.
Other quantities used exclusively in this supplemental material are introduced in the corresponding sections.
\vskip0.10in \noindent
\section{Sign Problem: General Importance}
In lattice gauge theory, the SP is triggered by a non-zero chemical potential $\mu$.
Attempts to solve, or reduce, the SP include analytic continuation from complex to real chemical potentials $\mu$~\cite{alford99,toulouse16}, Taylor expansion about zero $\mu$~\cite{deforcrand02,gavai03}, re-weighting approaches~\cite{allton02}, complex Langevin methods~\cite{adami01,berger19}, and Lefshetz thimble techniques~\cite{witten10,cristoforetti12}. For a review of the SP in LGT, see Ref.~\cite{banuls19}. A similar litany of papers, ideas, and new methods characterizing, ameliorating, or solving the SP can be found in the condensed matter~\cite{loh90,ortiz93,zhang97,chandrasekharan99,henelius00,bergkvist03,troyer05,nyfeler08,nomura14,mukherjee14,shinaoka15,kaul15,iglovikov15,he19,fukuma19,ulybyshev20,kim20}, nuclear physics ~\cite{wiringa00,alhassid17,lynn19,gandolfi19}, and quantum chemistry~\cite{umrigar07,shi14,roggero18,rothstein19} communities.
These approaches differ in detail, depending on whether they are addressing the SP in real space versus lattice models, quantum spin versus itinerant fermion, etc.
One of the dominant themes in atomic and molecular (AMO) physics over the last decade has been the possibility that ultracold atoms in an optical lattice might serve as emulators of fundamental models of condensed matter physics~\cite{bloch05,esslinger10,bloch12,tarruell18,schafer20}. The initial motivation was the possibility that simplified model Hamiltonians could be more precisely realized in the AMO context than in the solid state, where materials `complications' are unavoidable and often significant, if not dominant. However, it was quickly understood that an equally important advantage was that, because of the SP, the solution of these simplified models was possible only at temperatures which were well above those needed to access phenomena like $d$-wave superconductivity.
Thus it is fair to say that the SP has been a significant driver of the enormous efforts and progress in this domain of AMO physics.
A similar theme is present in quantum computing~\cite{ortiz01,brown10,preskill18,clemente20}, which promises the general possibility of solving problems much more rapidly (`quantum supremacy') than with a classical computer. The exponential scaling time of solutions of model Hamiltonians in the presence of the SP offers one of the most significant targets for such endeavors~\cite{Steudtner2018,Arute2020}.
\vskip0.10in \noindent
\section{The sign problem in other QMC algorithms}
In exploring the linkage between the SP and critical properties, we have focused exclusively on DQMC. As noted in the previous discussion, the SP occurs also in a plethora of QMC methods: world-line quantum Monte Carlo (WLQMC)~\cite{kawashima04}, Greens Function (Diffusion) Monte Carlo (GFMC)~\cite{reynolds90}, in dynamical mean field theory (DMFT) and its cluster extensions (including continuous time approaches)~\cite{georges96,beard96,hettler00,kotliar01,rubtsov05,maier05,kyung06,gull08,gull11}, and in diagrammatic QMC~\cite{prokofev96,vanhoucke10,rossi17,rohringer18}. It would be interesting to explore these situations as well, checking the direct correlation between the SP and regimes of quantum critical behavior. Regarding WLQMC, it is known that the onset of the sign problem is at much higher temperature than in DQMC, and occurs due to particle exchange in the world-lines. Indeed, this provides one of the earliest examples of the dependence of the sign problem on the algorithm used. It seems probable that the restriction of WLQMC to only very high temperatures ($T/W \gtrsim 1/5$ is quite typical) might preclude the possibility of an association of the sign problem with interesting low temperature correlations and transitions. It is worth noticing that an attempt to reconcile WLQMC and DQMC was put forward by introducing more generic Hubbard-Stratonovich transformations~\cite{Hirsch1986}; it has been further argued that the SP in WLQMC has two origins: one from the fact that Slater determinants are anti-symmetric sums of world-line configurations and another intrinsic, akin to the one in DQMC, that has been claimed having topological origin~\cite{Iazzi2016}. In the same way, alternative choices of Hubbard-Stratonovich transformations in DQMC~\cite{batrouni90,chen92} significantly worsen the sign problem. DMFT~\cite{georges92,jarrell92,georges96}, on the other hand, is at the opposite end of the spectrum, with a sign problem which is greatly reduced relative to that of DQMC. Unfortunately, its cluster extensions, using QMC as the cluster solver, exhibit an increasingly serious SP at low enough temperatures as the number of points in the momentum grid increases~\cite{jarrell01,maier05,kyung06}.
\begin{figure}[b]
\includegraphics[scale=0.33]{spinful_SM}
\caption{\textbf{Local observables in the vicinity of the quantum critical point on the SU(2) honeycomb Hubbard model.} The derivative of the double occupancy (\textbf{A}) and the kinetic energy (\textbf{B}) at $T/t = 1/20$ when approaching the quantum critical point at $\mu\to0$ and $U_c\simeq3.85t$. As in the main text for other models, the noisy data at large chemical potentials denotes the regime of small $\langle{\cal S}\rangle$ with a vanishing signal-to-noise ratio. (\textbf{C}) and (\textbf{D}) display the same quantities when approaching $T/t\to0$, with a small chemical potential $\mu/t=0.1$. As in the $T=0$ results of Ref.~\cite{meng10}, the derivative of the kinetic energy displays a peak within the AFMI phase. Our data show this persists to finite temperatures. The lattice size used is $L=9$, and all data are averaged among 20 independent runs.}
\label{fig:spinful_SM}
\end{figure}
We finish by noting that while we have argued that when a QCP is present, the SP provides quantitative information about its location, the converse is {\it not} necessarily true: a SP can exist even in the absence of a QCP. In particular, in WLQMC, free fermions have a SP in $D>1$ without possessing any sort of phase transition. As noted above, DQMC is SP free when the interactions vanish, so this simple counter-example is not present in that algorithm.
\vskip0.10in \noindent
\section{The sign problem in other Hamiltonians}
In our work, the spinless fermion Hamiltonian on a honeycomb lattice offered a particularly concrete case where a sign problem free approach allows a detailed study of quantum critical behavior. We exploited this as a way to make a very quantitative test of the connection of the SP to the location of the QCP. In addition to exploring other algorithms, a promising further line of inquiry is to turn on a chemical potential in other `sign problem free' models~\cite{chandrasekharan10,berg12,wang15,li16,li19}. The Kane-Mele-Hubbard model would be especially interesting since it presents a framework to understand the transition from topological phases (quantum spin Hall insulator) towards a (topologically trivial) Mott insulator with antiferromagnetic order~\cite{Hohenadler2012,Bercx2014,Toldin2015}. The Hubbard-Stratonovich transformation is slightly more complicated, and in fact a `phase problem' appears instead. In exploring this competition between ordered and topological phases, the study of the Haldane-Hubbard model presents as a challenging case, since due to the absence of time-reversal symmetry, it gives rise to a severe SP in the simulations~\cite{Imriska2016}. Whether similar analysis as conducted here can help in locating the QCP associated to a topological transition in system sizes exceeding the ones amenable to ED~\cite{Vanhala2016,Shao2021} is yet an open question left for future studies.
A further possibility for future work includes situations where disorder drives a quantum phase transition~\cite{huscroft97}. This is of particular interest because different types of disorder can either possess particle-hole symmetry or not~\cite{denteneer01}, and this dichotomy is known to be linked both to the presence of the sign problem as well as to the occurrence of metal to insulator transitions~\cite{denteneer99,denteneer03}. Thus disordered systems might provide an especially rich arena to explore the connection between the SP and the underlying physics of the Hamiltonian.
\vskip0.10in \noindent
\section{More details on the Spinful Hubbard Model on a Honeycomb Lattice (main text, Sec.~I)}
\paragraph{Energy and Double Occupation.---}The quantum critical point separating the paramagnetic semimetal and the antiferromagnetic insulator (AFMI) phases of the honeycomb Hubbard hamiltonian has been characterized using observables ranging from the magnetic structure factor to the conductivity. In the main text, we focused on using the average sign as a signal of the QCP. Here we provide context by showing some of the traditional measurements. More detailed results are contained in the literature~\cite{sorella92,paiva05,herbut06,meng10,giuliani10,ma11,clark11,sorella12,raczkowski20}.
Figure~\ref{fig:spinful_SM} shows the derivatives of the average kinetic energy $\langle K \rangle = \frac{-t}{2L^2}\sum_{\langle i,j\rangle,\sigma} (c_{i \sigma}^{\dagger} c_{j \sigma}^{\phantom{}} + c_{j\sigma}^{\dagger} c_{i \sigma}^{\phantom{}})$ and double occupation $\langle n_{\uparrow \downarrow} \rangle$ with respect to $U$ across the honeycomb lattice transition from semimetal to AFMI. Both show clear signals in the vicinity of $U_c$. The accurate indication of antiferromagnetic long range order requires a careful finite size scaling analysis of the antiferromagnetic structure factor, which can be found in Refs.~\cite{paiva05,meng10,sorella12}.
\begin{figure}[b]
\includegraphics[width=0.6\columnwidth]{spin_resolved_sign_hc_N_vs_U.pdf}
\caption{\textbf{Spin resolved sign for the SU(2) Hubbard model on the honeycomb lattice.} The scaling with the inverse of the linear system size $L$ of the average sign of individual determinants in the case there is no sign problem: $\mu/t = 0$. The star marker depicts the best known estimation of the critical value $U_c/t = 3.869$~\cite{sorella12} at the ground state for the onset of the Mott insulating phase with AFM order. All data are extracted at $T/t = 1/20$, with $\Delta\tau = 0.1$.}
\label{fig:spin_resolv_sign}
\end{figure}
\paragraph{Individual spin channel average sign in the Spinful Hubbard Model on a Honeycomb Lattice.---} In the main text, we have demonstrated how the average sign can be used as a `tracker' of quantum critical behavior. In the case of models within regimes where a sign problem is absent, e.g., for an SU(2) Hubbard model on a bipartite lattice, this ability is no longer available if the chemical potential $\mu = 0$, since the determinants of the two spin species always have the same sign so that their product is positive. This can be proven by considering a staggered particle-hole transformation (PHT) $c^{\dagger}_{i\downarrow} \rightarrow (-1)^i c^{\phantom{\dagger}}_{i\downarrow}$ on the down spin species. Here $(-1)^i = +1(-1)$ on sublattice A(B). Under the PHT, the kinetic energy matrix $K$ of Eqs.~\eqref{eq:Z_tr},\eqref{eq:Z_det} remains invariant, but the matrices $P^\tau$ in the down spin trace change sign, making the down spin determinant the same as the up spin determinant, up to a positive factor $e^{\lambda \sum_{i \tau} s_{i\tau}}$~\cite{Hirsch1985}.
While the {\it product} of the determinants is always positive in this situation, the QCP remains imprinted in the average sign of the determinants for {\it individual} spin components $\langle {\cal S}_\sigma\rangle$. To illustrate this, we consider the first model used in the main text, the repulsive spinful Hubbard model on the (bipartite) honeycomb lattice. Figure~\ref{fig:spin_resolv_sign} plots $\langle {\cal S}_\sigma\rangle$ ($\sigma=\uparrow,\downarrow$) at fixed $T/t=1/20$. The individual signs are largely positive in the metallic phase, but rather abruptly change to equally positive and negative ($\langle {\cal S} \rangle \sim 0$) in the AFMI phase $U>U_c$. The match of the transition in the sign and the position of the QCP becomes increasingly precise in the thermodynamic limit $1/L \rightarrow 0$. The sharpness of the drop in $\langle {\cal S}_\sigma\rangle$ with increasing system sizes is suggestive of a possible scaling form for this quantity. Preliminary data, to be presented elsewhere, indicates a scaling with critical exponents compatible with the ones obtained from physical observables~\cite{Assaad2013}.
\vskip0.10in \noindent
\section{More details on the Ionic Hubbard Hamiltonian (main text, Sec.~II)}
\paragraph{Finite-size effects.---} The ionic Hubbard model presents a unique situation in our study: instead of displaying a quantum critical point at half-filling, it exhibits a quantum critical regime, associated with a correlated metal (CM) phase~\cite{paris07,bouadim07,fabrizio99,kampf03,manmana04,garg06,paris07,bouadim07,craco08,garg14,bag15}. We argued in the main text that this phase, sandwiched between the band-insulator (BI) at large $\Delta$, and the Mott insulator (MI) at large $U$, can be indicated by a vanishing average sign in the DQMC simulations. We now explore the influence of finite-size effects on those phase boundaries, for a specific value of the staggered potential $\Delta/t=0.5$.
\begin{figure*}[th!]
\includegraphics[width=1\columnwidth]{ionic_SM.pdf}
\caption{\textbf{Finite-size analysis: Ionic Hubbard model.} (\textbf{A, B, C}) The average sign $\langle{\cal S}\rangle$ for decreasing temperatures (growing $\beta = 1/T$) as a function of the Hubbard interaction $U/t$ in a lattice with $L=8, 12$ and 16 with a staggered potential $\Delta/t=0.5$. (\textbf{D,E,F}) The corresponding magnitude of the derivative of $\langle{\cal S}\rangle$ in respect to $U$ at each lattice size. (\textbf{G}) Peak position of the derivative of the average sign with different inverse temperatures $\beta$; the inset presents an extrapolation to the thermodynamic limit of the estimated transition from the BI to the CM phase, as inferred by the first peak of $|d\langle {\cal S}\rangle/dU|$. The gray dashed lines in all panels display the corresponding transition value obtained in Ref.~\cite{bouadim07}. All data are averaged over 24 independent runs, with $\Delta\tau = 0.1$.}
\label{fig:ionic_SM}
\end{figure*}
Figure~\ref{fig:ionic_SM} indicates that the first transition which occurs upon increasing $U/t$ from zero, that from BI to CM, is well marked by a fast drop of $\langle{\cal S}\rangle$ at lower temperatures. A quantitative estimation of the transition point can be extracted by differentiating the average sign with respect to the interaction strength, $d \langle {\cal S} \rangle / d U$ (lower panels in Fig.~\ref{fig:ionic_SM}). As the temperature is lowered, the peak position quickly approaches the best known values of the transition for this set of parameters~\cite{bouadim07}, see Fig.~\ref{fig:ionic_SM}(\textbf{G}). The system size dependence is reasonably small. The second transition, from CM to AFI, on the other hand, displays characteristics reminiscent of a crossover for the system sizes and temperatures investigated. The estimate given by the peak of the average sign also displays a stronger dependence on $L$, and, overall, is larger than the value of the position of the metal-AF transition at this $\Delta$ obtained in Ref.~\cite{bouadim07}. It is worth mentioning that these values in the existing literature were extracted at smaller lattice sizes than the largest $L$ used here. A finite size extrapolation of the `traditional' correlations used to obtain $U_c$ for the metal-AF transition, similar to the one we perform here, would be useful to undertake.
\paragraph{QMC vs. ED.---} A valuable test of the conjecture that $\langle {\cal S}\rangle$ tracks a quantum phase transition (or regime) can be made by comparing QMC results with exact ones, obtained at smaller lattice sizes. For this purpose, we contrast in Fig.~\ref{fig:ionic_QMC_ED_comparison_SM} the average sign in a lattice with $L=4$ with numerical results obtained from exact diagonalization (ED). At this small lattice size, the quantum critical region shrinks, and at the lowest temperatures studied ($T/t = 1/24$) $\langle{\cal S}\rangle$ displays a sharp dip at around $V/t\simeq 2$. Turning to the ED results, we probe the transition via the analysis of the low lying spectrum $E_\alpha$ ($E_0$ is the ground-state), the many-body excitation gaps $\Delta_{\rm ex}^{(\alpha)} = E_\alpha - E_0$, the spin and charge staggered structure factors, $S_{\rm sdw} = (1/N)\sum_{i,j} (-1)^\eta \langle (n_{i\uparrow} - n_{i\downarrow})(n_{j\uparrow} - n_{j\downarrow})\rangle$ and $S_{\rm cdw} = (1/N)\sum_{i,j} (-1)^\eta \langle (n_{i\uparrow} + n_{i\downarrow})(n_{j\uparrow} + n_{j\downarrow})\rangle$ [$\eta = +1\ (-1)$ when $i,j$ belong to the same (different) sublattices], and, the fidelity metric $g_{\tiny U} = \frac{2}{N}\frac{1-|\langle \Psi_0(U)|\Psi_0(U+dU)\rangle|}{dU^2}$. This last quantity displays a peak whenever one crosses a quantum phase transition for the parameters $U$ and $U+dU$. These results describe a single transition in the range of parameters investigated, displaying a first-order character, given the level crossings shown in Fig.~\ref{fig:ionic_QMC_ED_comparison_SM}(\textbf{B}) or a vanishing excitation gap $\Delta_{\rm ex}^{(1)} = E_1 - E_0$ at $U/t \simeq 1.99$ [Fig.~\ref{fig:ionic_QMC_ED_comparison_SM}(\textbf{C})]. In turn, the fidelity metric displays a sharp peak at this interaction value [Fig.~\ref{fig:ionic_QMC_ED_comparison_SM}(\textbf{E})], and the structure factors computed at the ground-state swap its characteristics, from a charge- to a spin-ordered one [Fig.~\ref{fig:ionic_QMC_ED_comparison_SM}(\textbf{D})].
It is an open question of whether one is able to capture the intermediate correlated metal phase in exact methods such as ED.
Stepping back from the technical details, the central message of
Fig.~\ref{fig:ionic_QMC_ED_comparison_SM} is extending the evidence presented in the main text
that the sign problem metric for the QCP of panel \textbf{A}
lines up well with those of the `traditional observables' in panels \textbf{B-E}.
\begin{figure*}[th!]
\includegraphics[width=0.6\columnwidth]{ionic_QMC_ED_comparison_SM.pdf}
\caption{\textbf{QMC vs. ED on a 4x4 lattice.} (\textbf{A}) The average sign extracted from the DQMC calculations $\langle{\cal S}\rangle$ for increasing inverse temperatures $\beta = 1/T$ as a function of the Hubbard interaction $U/t$ in a lattice with $L=4$ and a staggered potential $\Delta/t=0.5$. (\textbf{B--E}) Data extracted with the ED, including: (\textbf{B}) The low-lying eigenspectrum $E_\alpha$ at the zero-momentum sector $\mathbf{k} = (0,0)$; (\textbf{C}) The excitation gaps $\Delta_{\rm ex}^{(\alpha)}$; (\textbf{D}) the spin and charge structure factors and (\textbf{E}) the fidelity metric under variations of the interaction magnitudes $U$, using $dU=10^{-3}t$. In (\textbf{A}), data are averaged over 24 independent runs, with $\Delta\tau = 0.1$.}
\label{fig:ionic_QMC_ED_comparison_SM}
\end{figure*}
\vskip0.10in \noindent
\section{More details on Spinless Honeycomb Hubbard (main text, Sec.~III)}
\paragraph{The finite-temperature transition.---} As we have argued in the main text, the interacting spinless fermion Hamiltonian has a special property in AFQMC simulations: with an appropriate choice of the basis one uses to write the fermionic matrix, it has been proven that the sign problem can be eliminated~\cite{ZiXiang2015,li16,li19}. Nonetheless, using a standard single-particle basis, where the sign problem is manifest, we demonstrated in Fig.~\ref{fig:3} that $\langle{\cal S}\rangle$ can be used as a way to track the quantum phase transition. Concomitantly, we have shown that a local observable (the derivative of the nearest neighbor density correlations with respect to the interactions $V$) exhibits a steep downturn once the quantum (i.e.~zero temperature) phase transition is approached.
As a by-product of this analysis, we use our original approach based on the standard BSS algorithm in the standard fermionic basis to show that one can also obtain an estimation of the \textit{finite-temperature} transition (pertaining to universality class of the 2D Ising model) with a relatively large accuracy, if the system is not too close to the quantum critical point, see Fig.~\ref{fig:spinless_hc_SM}. We compute both the derivative of the nearest-neighbor density correlations as well as the CDW structure factor, i.e., a summation of all density-density correlations with a $+1$ $(-1)$ for sites belonging to the same (different) sublattice, on the largest lattice size we have investigated, $L=18$. These finite-size results for $T_c$ are in good agreement with recent results obtained after system size scaling of data extracted with continuous-time QMC methods~\cite{Wang2014,Hesselmann2016}, where the sign problem is absent.
\begin{figure}[th!]
\includegraphics[width=0.7\columnwidth]{spinless_hc_SM.pdf}
\caption{\textbf{Finite-temperature transition for the spinless fermion on the honeycomb lattice.} As a by-product of the analysis of the temperature dependence for average sign, we notice that other than in regimes very close to the QCP, $\langle {\cal S}\rangle$ is very close to 1. (\textbf{A}) The derivative of the nearest-neighbor density correlations in the $T/t$ vs. $V/t$ plane and (\textbf{B}) the CDW structure factor for a lattice with $L=18$. In both plots, the markers are the continuous-time QMC results extracted after finite-size scaling in Ref.~\cite{Hesselmann2016}. Imaginary-time discretization is fixed at $\Delta\tau = 0.1$, and all data is obtained as an average of 20 independent runs.}
\label{fig:spinless_hc_SM}
\end{figure}
\begin{figure}[th!]
\includegraphics[width=0.6\columnwidth]{spinless_dtau_sign.pdf}
\caption{\textbf{The U(1) model on the honeycomb lattice: Average sign dependence on the imaginary time-discretization.} The dependence of $\langle{\cal S}\rangle$ on the imaginary-time discretization $\Delta\tau$ with growing interactions $V/t$, in a lattice with $L = 9$. Data is extracted as an average of 24 independent runs, for a temperature $T/t=1/24$. The best known value for the Mott insulating transition~\cite{ZiXiang2015} ($V_c/t=1.355$) is signalled by the star marker.}
\label{fig:spinless_dtau_sign}
\end{figure}
An important outcome of these results is that they imply that the mark that a phase transition leaves on the average sign is restricted to quantum phase transitions, rather than thermal ones, as we show above.
\paragraph{Imaginary-time discretization.---}
Here we consider the effect of the imaginary-time discretization $\Delta\tau$ on the average sign, for a fixed inverse temperature and show that `Trotter errors'~\cite{fye86} do not affect our conclusions. Figure~\ref{fig:spinless_dtau_sign} shows the average sign for a fixed lattice size $L=9$ and $\Delta\tau$ ranging from 0.025 to 0.2 with a fixed temperature $T/t=1/24$. The drop in $\langle{\cal S}\rangle$ when approaching $V_c$ is indicative of the QCP, but using a more dense imaginary-time discretization does not render substantial changes in the average sign; similar behavior was observed in other models studied.
\vskip0.10in \noindent
\section{More details for the homogeneous Hubbard model on the square lattice (main text, Sec.~IV)}
\paragraph{Finite-size effects.---} An important aspect that deserves further attention concerns finite-size effects on the data presented in the main text. The average sign, for example, when not protected by some underlying symmetry of the Hamiltonian, is known to decrease for larger system sizes~\cite{white89}. Figure~\ref{fig:finite_size_effect_sq_spinful}
compares the original quantities displayed in the main text [Fig.~\ref{fig:4}] at different lattice sizes.
The ``growth'' of the $\langle {\cal S}\rangle\to 0$ dome is relatively small when taking into account lattices differing by an order of magnitude in the number of sites. The $d$-wave pair-susceptibility
also qualitatively preserves its overall features, with a tendency of local pair formation at higher electronic densities, encapsulating the $\langle {\cal S}\rangle\to 0$ dome. Similarly, the static spin-susceptibility and its accompanying `pseudogap' line are largely unaltered when the linear lattice size varies from $L=8$ to 16.
These results for different $L$ indicate that our analysis of the link between the SP and quantum criticality is not a finite size effect.
\begin{figure*}[th!]
\includegraphics[width=1.\columnwidth]{finite_size_effect_sq_spinful.pdf}
\caption{\textbf{Finite size effects for the SU(2) Hubbard model on the square lattice.} An analysis of the main quantities and their size dependence for the model with potential connection with the physics of the cuprates. As in the main text, we choose $U/t = 6$ and $t^\prime/t = -0.2$.
The top, middle, and bottom rows of plots refer to $L = 8, 12$ and 16, respectively. In turn, the columns from left to right depict the average sign, the correlated $d$-wave pair susceptibility, and the long-wavelength static spin susceptibility. Imaginary-time discretization is fixed at $\Delta\tau = 0.0625$, and all data are extracted as an average of 24 independent runs.}
\label{fig:finite_size_effect_sq_spinful}
\end{figure*}
\paragraph{Pair-symmetry channels.---} Another important check on the suitability of the chosen parameters to describe the physics of pairs with the same experimentally inferred symmetry as in the cuprates, is to directly compare the correlated susceptibility map with different symmetry channels. This has been done in early studies of QMC~\cite{white89a}, and here, as a side-aspect of the analysis of the average sign, we bring in a systematic investigation. Figure~\ref{fig:s_sx_d_wave_pairing_sq_spinful} displays the correlated pair susceptibility $\langle P_{\alpha} - P_{\alpha}^{\rm nv}\rangle$ dependence on the temperature and electronic filling, with $\alpha = d, s^*$ or $s$-wave. Clearly, the symmetric local pairing channel ($s$-wave), which directly
confronts the on-site $U$, is not favored in the whole range of $T$ and $\rho$ investigated. The correlated pair susceptibility is always negative for finite values of $\langle {\cal S}\rangle$, indicating the vertex is {\it repulsive}. In contrast, both the extended $s$-wave and $d$-wave pairings exhibit positive correlated pairing susceptibilities in the vicinity of the $\langle {\cal S}\rangle\to0$ dome, but are more pronounced in the latter. This emphasizes the dominance of $d$-wave pairing in the Hubbard model, in direct analogy to a wide class of materials displaying high-temperature superconductivity.
\begin{figure*}[th!]
\includegraphics[width=1.\columnwidth]{s_sx_d_wave_pairing_sq_spinful.png}
\caption{\textbf{Comparison of the pairing channels for the SU(2) Hubbard model on the square lattice.} Taking as a starting point for the physics of the cuprates the parameters $U/t = 6$ and $t^\prime/t = -0.2$~\cite{Hirayama2018,Hirayama2019}, we display the comparison of the correlated pair susceptibilities considering different symmetry channels in the left ($d$-wave), center (extended s-wave, $s^*$) and right ($s$-wave) columns, both in $\mu/t$ vs. $T/t$ (upper row) and $\rho$ vs. $T/t$ (lower row) parametric space. Imaginary-time discretization is fixed at $\Delta\tau = 0.0625$, and all data are extracted as an average of 24 independent runs in a lattice with $L=16$.}
\label{fig:s_sx_d_wave_pairing_sq_spinful}
\end{figure*}
\paragraph{Spectral weight at the anti-nodal point.---}
In describing the physics of cuprates, a common focus is the anisotropy of the single-particle gap as extracted from ARPES techniques~\cite{Damascelli2003}, which contrasts with the standard isotropic behavior seen in conventional BCS-type superconductors. At the root of the discussion is the shape of the Fermi surface when doping the parent spin-ordered Mott insulator. In particular, to classify the onset of the pseudogap phase at low enough temperatures $T$, i.e., the regime where single-particle, low-energy excitations are suppressed, one tracks the peak of the anti-nodal spectral weight at the Fermi energy, $A_{(\pi,0)}(\omega=0) \equiv -\frac{1}{\pi}{\rm Im}G_{(\pi,0)}(\omega=0)$ as $T$ is varied.
In QMC simulations, this quantity can in principle be extracted by means of an analytical continuation of the data, in which the imaginary-time dependence of the Green's function $G$ is converted to real frequency. To avoid the well known difficulty of such a calculation~\cite{jarrell96}, a proxy valid at low enough temperatures, $A_{(\pi,0)}^{\rm proxy} = \beta G_{(\pi,0)}(\tau=\beta/2)$ is often used~\cite{trivedi95,Wu2018}. Figure~\ref{fig:A_0_sq_spinful_proxy} shows the results for the anti-nodal spectral function. The pseudogap line extracted from the proxy is qualitatively close to the one obtained from the peak of the static long-wavelength spin susceptibility (Figs.~\ref{fig:4} and \ref{fig:finite_size_effect_sq_spinful}) and, like it, terminates on the $\langle {\cal S}\rangle\to 0$ dome.
\begin{figure}[th!]
\includegraphics[width=0.7\columnwidth]{A_0_sq_spinful_proxy.png}
\caption{\textbf{Comparison of the single-particle spectral weight at the anti-nodal point.} Temperature extrapolation of the extracted $A_{(\pi,0)}$ at the Fermi energy vs the chemical potential $\mu/t$ (\textbf{A}) and the electronic density $\rho$ (\textbf{B}), on a lattice with $L=16$; other parameters are $U/t = 6$ and $t^\prime/t = -0.2$. The maximum values at each temperature are denoted by the white markers. Imaginary-time discretization is fixed at $\Delta\tau = 0.0625$, and all data is extracted as an average of 24 independent runs.}
\label{fig:A_0_sq_spinful_proxy}
\end{figure}
\paragraph{Spectral function and the Lifshitz transition.---} The extensive dataset and associated analysis we have undertaken in this investigation of the sign problem enables us to check other important physical aspects of the Hubbard model on the square lattice, and their relation to cuprate phenomenology. We include them here, in order to further link their behavior to that of the sign. One of them refers to the change of the topology of the Fermi surface when increasing the hole-doping (decreasing the electron density) from half-filling: at some critical $\langle n\rangle$, the Fermi surface changes its shape from hole-like to a closed, electron-like one. This transition, referred to as the Lifshitz transition, has been investigated in the context of strongly interacting electrons, and was inferred to
occur concomitantly with the presence of a van Hove singularity at the Fermi level~\cite{Chen2012}.
\begin{figure}[th!]
\includegraphics[width=0.8\columnwidth]{spectral_function.png}
\caption{\textbf{Lifshitz transition.} The evolution of the spectral function in one quadrant of the Brillouin zone with increasing densities, after an interpolation of the results for a $24\times24$ lattice at temperature $T/t=1/3.25$, with $U/t = 6$ and $t^\prime/t = -0.2$. The different panels depict results with various chemical potentials as marked, and the average density is 0.25 in (\textbf{A}), 0.65 in (\textbf{B}), 0.95 in (\textbf{C}), and 1.00 in (\textbf{D}). Imaginary-time discretization is fixed at $\Delta\tau = 0.0625$, and all data is extracted as an average of 20 independent runs. }
\label{fig:spectral_function}
\end{figure}
Due to the presence of the SP, however, we can only investigate the Lifshitz transition at finite-temperatures, and thus the Fermi `surface' is thermally broadened. Nonetheless, Fig.~\ref{fig:spectral_function} displays the spectral function (obtained via the `proxy' scheme as previously explained) at the Fermi energy on lattices with linear size $L=24$, at a temperature
right above the $\langle{\cal S}\rangle\to0$ dome, and investigating dopings above and below the pseudogap line in Figs.~\ref{fig:4} and \ref{fig:spectral_function}. As one increases the density, the change of topology precisely confirms the Lifshitz-scenario, with a further formation of hole-pocket regions along the anti-nodal direction, in direct analogy to the phenomenology of high-Tc superconductors~\cite{Damascelli2003}.
\paragraph{Comparison to the near-neighbor hopping only Hubbard model.---}
While it is remarkable how much of the physics of simplest Hubbard Hamiltonian captures that of the cuprates~\cite{scalapino94}, refinements of the model are known to provide more accurate comparisons to the experiments~\cite{Piazza2012,Hirayama2018,Hirayama2019}. One such, the inclusion of next-nearest neighbor hopping $t^\prime$ was employed in the analysis in the main text. It is significant in the context of the sign problem because it breaks the particle-hole symmetry and, for example, induces a SP even at half-filling. In this section we address the extent to which including $t^\prime$ affects our conclusions. To this end, we contrast the results of Fig.~\ref{fig:4} with the ones arising from the Hubbard model with $t'=0$ (Fig.~\ref{fig:t2_0_sq_spinful}). The key qualitative aspects are similar, including the presence of a $\langle {\cal S} \rangle\to0$ dome, the tendency of $d$-wave pair formation (due to the enhanced pair susceptibility with this symmetry around such dome) and a peak of the spin susceptibility ending at the dome. The differences are: (i) while approaching half-filling, the average sign displays a sudden jump towards 1 (as one would expect for this bipartite case), (ii) the extension of the dome, within the temperatures investigated ($T/t \geq 1/6$), is more constrained in density compared to the $t^\prime/t=-0.2$ case, and, more importantly, (iii) the pseudogap region, signified by the temperatures below the $\chi(\textbf{q}=0)$ peak, is significantly reduced. Our data thus provide another argument in support of an added next-nearest neighbor hopping in order to possibly explain the phase diagram of high $T_c$ materials, which display a robust pseudogap region. However, the main point for the purpose of this manuscript is that whether $t^\prime$ is included or not, the behavior of the sign is correlated with the physics of pairing and magnetism.
\begin{figure*}[th!]
\includegraphics[width=1.\columnwidth]{t2_0_sq_spinful.png}
\caption{\textbf{The `phase diagram' of the bipartite Hubbard model on the square lattice.} The same as Fig.~\ref{fig:4} in the main text, but instead removing the non-bipartite contribution $t^\prime$, i.e., here the `vanilla' Hubbard model results are presented. Other parameters are the same, as $L=16$, $\Delta\tau=0.0625$ and $U/t=6$.}
\label{fig:t2_0_sq_spinful}
\end{figure*}
\end{document}
\section{Bilayer AF-Singlet Transition}
\textcolor{red}{Need additional citations throughout below.}
The transition from long range AF order to independent singlet formation
is one of the earliest and most well-studied QPT. The original underlying
physics considered localized $f$-orbitals which
hybridize with delocalized $d$-orbitals.
When the hybridization $t_{\rm fd}$ is weak,
the magnetic $f$-orbitals are indirectly coupled via
their polarization of the conduction electron cloud,
the Ruderman-Kittel-Kasuya-Yosida (RKKY)
interaction~\cite{ruderman54,kasuya56,yosida57},
and order antiferromagnetically. For large $t_{\rm fd}$
however, the conduction and localized electrons instead tend to
form spin-0 singlets, the Kondo effect, which quenches the magnetic
order~\cite{kondo70}.
This competition is most well-studied in the context of the
periodic Anderson Model (PAM).
These phenomena are believed to underlie the behavior of
a number of strongly correlated materials, including, for
example, the volume collapse transition in lanthanides and
actinides~\cite{allen82,mcmahan98}.
QMC studies have located the QCP of the PAM~\cite{vekic95}.
Here we study the AF to SL transition in the closely-related bilayer Hubbard
Hamiltonian consisting of two square lattices with intralayer
hybridization $t$ which are coupled by interlayer $t'$.
Thus $t'$ plays the role of $t_{\rm fd}$ in the discussion above.
At strong coupling $U$ this model becomes the bilayer ($J,J'$)
Heisenberg model, for which the location of the QCP is
known very precisely~\cite{hida90,millis93,sandvik94,wang06}.
\textcolor{red}{Paragraph below will need fine tuning as
the figure/data are finalized.}
Figure \ref{fig:signbilayer} shows
$\langle {\cal S} \rangle$
as the interlayer hopping is increased from zero.
A signal is seen at two values. At small $t'$,
the interlayer exchange $J' \sim t'^{\,2}/U$ is very small,
e.g.~for $t' = 0.5$ and $U=4$ we have $J' \sim 1/16$.
Thus, unless $\beta \gtrsim 1/J' \sim 16$ the ground state AF order is
not apparent. Thus, at any fixed $\beta$, as a function of $t'$
there is a `transition' to AF order~\cite{vekic95,chang14,mendes17}.
This is the origin of the signal in
$\langle {\cal S} \rangle$
at smaller $t' \sim 0.6$.
More significantly, there is a second signal in
$\langle {\cal S} \rangle$
at larger $t' \sim 1.6$.
This is close to the position of the QCP associated
with the AF to singlet transition~\cite{scalettar94}.
Thus, as with the evolution from SM to AFMI for Dirac fermions,
the average sign appears to be provide a barometer
of the onset of Kondo physics.
Figure~\ref{fig:spinlayer} show the more standard correlation functions
for the analysis of the AF-singlet transition: the near-neighbor
intra- and inter-plane correlation functions.
As the interplane hybridization increases, the interlayer correlations
grow. In the limit of large $t'$, and
as $\beta \rightarrow \infty$ and $U \rightarrow \infty$
they would take the perfect singlet lavle $C^{\rm nn}_{\rm inter}=-3/4$.
Two spins on adjacent sites in the same plane become less and less
coupled.
\begin{figure}[t]
\includegraphics[scale=0.33]{spinlayerA.pdf}
\caption{
\textcolor{red}{Needs work. Not sure which data set to show.}
Inter and intra-layer spin correlations as functions of interlayer
hybridization $t'$.
}
\label{fig:spinlayer}
\end{figure}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,913 |
Kannaboomers
A simple guide to the best CBD and cannabis products
Q&A: Is dry vaping cannabis flower better than smoking it?
Q&A: How do I vape CBD oil?
2 | Jason Hand of Vape Bright | Vaping for Wellness
There's a lot of confusion about vaping: Is it healthy to inhale chemical vapor? Is there nicotine in all vape juices? What's better, a pen-style vaporizer, or a mod with a tank, battery and coil? What's in the various vape juices? To get some answers, we sat down with Jason Hand, CEO of Vape Bright, a California-based maker of vape pens filled with CBD. Listen and learn about:
The history of vaping
Why you need to pay attention to carrier liquids
Why a deregulated market calls for more attention by consumers
Why it's important to have organic hemp oil
Kannaboomers: This week we have Jason Hand, CEO of Vape Bright, a company that's been making pen style vaporizers for a couple of years now. I'm fascinated with the whole history of vaping. We all know people started using e-cigarettes back in the 2004, 2005 timeframe as a way to reduce the harm of inhaling burning tobacco. It is since transformed into a delivery method of various medicines. Can you tell us your perspective on that?
Jason: Yeah, I remember seeing Blu come out and they were actually selling them in vending machines, in bars across the U.S in the very beginning. And that was kind of the start of e-cigarettes, which is why a lot of people still refer to it as that, but vaporizing evolved from that and that's when people started getting into the hobby of building their own mods. And that's where we've seen in the last five years, I'd say is this huge boom in more of the millennial market. Like where I guess since we've been told from an early age that smoking is bad, but vaping is such a new thing that it's become the immediate alternative. So if you're not a smoker, then at least vaping is the more healthy option. And I guess health is, you know, it's kind of a gray area in this space because it's so new.
But from what I understand, many researchers said that vaping anything is thousands of times less harmful than smoking cigarettes. So that's a good thing. But the hobby of it, like being the vape mod e-liquid market kind of created this, this whole different break in where people are vaping flavors. I remember when I was 18, hookah bars were really popular, all about the hookah, even though you thought you were actually getting less nicotine, you weren't like a single package of a hookah tobacco has just as much nicotine as a pack of cigarettes and people were going through bowls and bowls of this hookah tobacco thinking, you know, that it was, it was safer. But realistically the whole idea of this vapor was really poignant and it took off in a huge way and it's because it's actually easier to manufacture than growing tobacco, if you imagine like having a huge fields of tobacco leaves and farms and drying it. And then the whole regulation that comes with all that. It's not very easy to enter the market, but vape liquid was unregulated at first and took off by storm where you could have every single flavor imaginable. And people started hobby building their own models and their own batteries and tanks and coils. There's just this massive niche market that surrounded vaping. So it went from cigarettes to vaping. And then people started noticing that, well heck, if you can mix in flavor than that's happened. If you mixed and other molecules
Kannaboomers: You've mentioned mods a couple of times, can we define that? Is that like the tank, that structure that has a tank and different parts, not a pen style thing, right?
Jason: Absolutely correct. Yeah. The mod would be more of the e-liquid craze where you have a huge, like a three milliliter tank or larger that you refill on a regular basis. Like if you see the people blowing clouds and making all those crazy smoke rings and rings and all that stuff, that's that mod style, where a pen is very specific. It's a smaller device. It uses less wattage voltage, meaning it's not made for blowing huge plumes of vapor. Again, there's like a split in the market of vaping. Not everyone is after the same thing. So there's people that want to just blow huge clouds of flavorful vapor. And there's some people that just want to vaporize something specifically.
Kannaboomers: If you're into blowing huge clouds, you go to a vape shop and you may or may not get a liquid that contains nicotine, right?
Jason: They do make e-liquid that doesn't have nicotine for people that just so say like if you're someone that's trying to quit smoking cigarettes and trying to get off nicotine, that may be an option for you to still get the sensation of the oral fixation but not actually have any administer nicotine. Pen style vaporizers like the ones we create are more of like a disposable specific purpose type of vaporizer where you're not buying huge quantities of this liquid and refilling a refilling a tank because it just. The nature of the of the oil that we use wouldn't be appropriate for it. If you were to put oil in a in a tank mod, you'll most likely burnout your coils because that the scarcity of e-liquid is much different than a hemp oil, so you can't really burn it as high temperatures. You can't do that with e-liquid because it basically would gum up and then fry your coil and ruin the mod. At least the coil portion of it. When you do that, so this typically isn't made for large, a large tanks like the less viscous liquid.
Kannaboomers: Our audience is baby boomers who are interested in good health and in CBD they would probably be interested in, in what you guys offer a pen style vaporizer that as you say, it's not heating it at a real high temperature and it's an organic substance, right? So can we talk about the contents of the Vape Bright pen?
Jason: Yeah. So before, I guess it makes more sense to talk about what it isn't because there's a lot different options and people think, oh, I got the CBD at this gas station, it's a vaporizer, it's really great. And then you look at the ingredients and you realize that there's very little CBD and a bunch of questionable other chemicals listed in the ingredients. So there is a huge distinction and that's really why we exist is there was a lot of um, a lot of different products being created at the say the boom of the CBD fixation. Once people realize you, there's other cannabinoids than THC. And started looking at the potential benefits of CBD. There was this huge breakout in the market and since there is no regulation, there was a lot of people creating a lot of questionable products that didn't really benefit anybody. Snake oil, if you will, a lot of products that didn't actually have really any effect of minimum effective amount of the cannabinoid and a lot of other junk ingredients.
So they were just selling at a premium, a bunch of terrible stuff. So we, we wanted to create something that would be high potency, all organic. That's what we attempted to do. What that requires is first finding organic hemp oil, which in itself is already hard enough because the USDA doesn't really certify as organic or not. So you have to really vet the growers and the quality of the material that they're using because hemp likes to act like a straw and it sucks up a large amount of heavy metals and pesticides. So it's a little bit of a risk in this market because you have to talk to growers that may not know what they're doing or are lying to you. So there's a lot of research and vetting the has to be done before you can even get the hemp oil and that took a long time to find suitable growers that were importing the proper high-quality organic material to begin with. And then from there, it's about formulating correctly, using substances that aren't toxic.
Kannaboomers: Where do you guys source your hemp from?
Jason: So we source from Europe. There's a few places in Scandinavia region that allow us to because realistically the law still governs legal. Hemp is, can only be imported , unless that changes. It's pretty much the state of the industry is anything that's considered legal hemp at this point must be imported. We take really high quality imported hemp from, from Europe that fits the bill of being organic and high potency. And then from there it's formulated to remove the THC because there is always like a small trace amounts of THC that may be what they consider too hot for it to be considered a hemp product. The law justifies hemp is anything that's below point three percent THC. And that's a very small number considering a lot of these testing practices have a variance of one to two percent, so you pretty much have to have zero THC or nondetectable for it to be illegal hemp product. So that's really important. And we use a local manufacturers and Colorado in California that can remove the THC to make a even more legal, if that makes any sense.
Kannaboomers: So you start with organic hemp from Europe and make sure it's clean, it's not full of contaminants and heavy metals and stuff. And then you extract the THC out of it and you end up with with an oil that is rich in CBD.
Jason: Yep. Then we still have to test it again once we get our hands on it. It was very little regulation in the industry. So we have to third party tested ourselves to make sure that we're not getting, you know, a fake COA (certificate of analysis). So it's been really important as the industry self regulates to make sure that all the material is tested twice to make sure that it, it is as safe as possible. Because if we aim to create the most potent and beneficial safe product, then we have to basically put our money where our mouth is, so every single batch that we receive, even if it's from the same producer, is going to be tested again. So after we receive what should be clean oil, we test it ourselves and then we begin our formulation, which is really simple and includes just basically giving it a specific flavor profile.
Jason: Since we're not interested in using any of the artificial flavorings that you see e-liquids do, we add our own blend of natural terpines, which are basically flavonoids are what gives cannabis its smell. Anything. It's smell, flowers, perfume, if you will. We add our own natural blend of terpenes to give it a give it a nice citrusy, smooth flavor because CBD in itself has a really sharp kind of spicy profile. When you inhale it by itself or if you taste it by itself, it has more of the spicy flavor. It doesn't taste that great. So we do our best to really give it a nice, smooth, borderline citrusy flavor.
Kannaboomers: Have you ever taken it sublingually? It's got kind of a grassy flavor that is not all that appealing.
Jason: You do get used to it, but yeah, if you're inhaling it on a regular basis, your throat's just going to constantly have this kind of spicy itch to it, especially given the potency that we're using. So we do our best to really round that out and smooth it. And then the terpenes and themselves have potential benefits because if you know anything about aromatics and aroma therapy, there's a lot to be said about specific uses of terpenes and smells and in those categories. So in itself, we're kind of adding additional potential benefits to the product by adding natural terpenes.
Kannaboomers: You know, the buyer needs to beware and look for brands that are very transparent in terms of their testing and their sourcing. Let's talk about some of the therapeutic uses. You mentioned high potency. There's people who take a lot of CBD, you know, if they're trying to treat a cancer or something, they might be taking, you know, 30, 50, 100 milligrams a day. What sort of dose does Vape Bright deliver and what are some of the theraputic reasons people are taking it?
Jason: So the first part of the question about potency, this is, this is a really interesting topic because right now there's a lot of misinformation, especially with people taking advantage of the ignorance of the new user, right? So there's a lot of people new to this, this market or this industry that see I'm a certain amount of milligrams and they think that that's the standard they. The question is, is what is the milligram per dose and how many doses are required. So what is the minimum for what you're using it for? So in vape it's even worse because everything is justified in milligrams in and milliliters. So it's very difficult. You have to basically do your own math where something will say 500 milligrams. So you're thinking it has 500 milligrams of CBD, but realistically it has 500 milligrams of oil and there's not actually 500 milligrams of cbd oil, so that oil is 20 percent, so you really only have hundred milligrams in that 500 milligram cartridge.
So there's a lot of people that speak in milligrams, there's some people that speak in percentages, but what you really need to know is the total amount of CBD that's in the product you're using. And then you need to figure out how much of that CBD can you administer into your body within a reasonable amount of time. Our product, for example, we have two different products, but say our original product thrive has 200 milligrams of CBD. It's in a 500 milligram cartridge, so therefore we're 40 percent solution that is very high for vape. Basically CBD in itself is a crystal a and at above 50 percent you're going to see it start to crystallize for anything in the 40 to 50 percent region as a really high potency as far as percentage in a single cartridge. So we'll get roughly a couple hundred inhalations before the cartridge is depleted, so we guesstimate, we don't know because we've never done the strict testing on exactly how much CBD is coming out of this.
We'd guessed that about one milligram per inhalation of CBD, so if you're thinking about someone that would want to get roughly 20 to 30 milligrams per day, then now it'd be 20 to 30 inhalations. What we recommend is 10 to 12 inhalations per use and then go from there. Another point to be made is that it's a very individualized experience. The same amount of CBD might affect someone differently than it affects you. Absolutely. Bodyweight genetics, there's so much to be considered and there's really only anecdotal testing that we have to look at it. I mean there's been tons of studies but they're not backed up by imperial science, so we don't truly know. All I can do is take my own experiences and the experiences of other people to gauge what that might look like. I know from my own experience that there is very clear zero to one effect from a product of this potency versus something that's like an e-liquid.
They'd only has three percent CBD in it. You could vape clouds all day and never feel anything, but I'm a much younger than maybe the audience you have. I read the hundreds of reviews and it's not just people that are talking about great product, awesome stuff. It's these are people in pain or severe issues that are looking for anything, for any sign of hope in whatever their condition is. The stuff they write is not. It's not just a short little blurb. They, a lot of people spew their life story, talk about it, how it's changed. I'm changed every aspect of their waking life, how they're able to go back to work. We had a gentleman named TJ that didn't know how to use the computer. He didn't know how to basically transact online an older, bedridden gentlemen who has never made a purchase online.
So imagine that he would call us personally to have us places order and he was such a. Such an interesting man, uh, he was always like joking around with us and he would call us random points of the day just to kind of talk. So in a way we, we got really familiar with, with TJ and he wrote us this letter several months later saying that he had been bedridden for years on opiates. He was finally able to move about the house and get out of bed, which is one of the most touching things because after getting to know TJ is quite honestly like the moment where I started to realize what we were doing is not just creating another product. It really is changing, uh, an individual's life.
Kannaboomers: That's awesome. When you can put a product out like that, that really does enable people to change their lives and whether it's, in his case, the opioids that we're keeping them down. Can you speak to some of the other testimonials that you've gotten from people who have used it for, for specific purposes?
Jason: Oh, there's times I would say that like looking at our reviews, the number one thing I see people comment on his, uh, his arthritis, that is a huge issue. And then I'd say second to that, his anxiety is what we see a lot of people finding potential relief for. There's just so many things I'm not even aware of or that I wasn't aware of it first that people are finding relief with,
Kannaboomers: I know CBD has been used for fibromyalgia for PTSD. People mentioned we've seen the videos of where it helps kids with epilepsy stops seizuring all day. Is your formulation useful in its dosage for those kinds of applications?
Jason: Yes. And um, and know in, in, in certain cases. So for people with really intense situations like say dementia and AIDS for example, there's people that are on a 30 to 50 to 100 milligram serving size and that's not something that can really use our product for. I mean, like I said, it would take half the cartridge. Would you have to inhale 100 times to get that kind of dose? It wouldn't be really efficient. If you're looking for a large amount of CBD all at once where I think something more like a tincture would be better for those situations, but anyone that's on the go, what I was recommending, what's great about this product is it's convenient. It's efficient. You can have it in your pocket. If you start to feel like you need some, you just can pull it out, take a couple breaths and put it back. A lot of people I've heard just regular applications that people going into a meeting, they're stressed out about it and just take a couple of pulls before they go in. That's more of the situation that I see at the most benefit for us when you're on the go and you need something to just kind of quell whatever's going on with you.
Kannaboomers: Right, and that's where the speed of it is. It's instantly absorbed, right? You're bypassing the liver. You don't have to go through your whole digestive system.
Jason: Absolutely, it's fastest and the easiest way to administer. I mean, anyone knows how to take a breath or suck through a straw, so there's not really much education that needs to be that comes with the product so everyone knows how to use it right off the bat and it's very easy to use which makes, if you you're familiar with the droppers or any of those types of applications, it's a process, you know, and it's more of like a medicinal process. We have to stop what you're doing, pull this thing out of a jar and then put it underneath your tongue. It's not like something you can do while you're driving or, or you know, walking the dog or you know what I mean? Like those types of applications are a little bit less convenient and therefore it's a lot, lot less likely you're going to use it on a regular basis or have it available when you need it. Something as small as a pen that can fit in your pocket. It's an instant breath and because of like you're mentioning, the lungs aren't incredible delivery system. The pulmonary capillaries are instant delivery system to your bloodstream, which is why oxygen works in the first place, or why cannabis works, or Thc, right? It's instant because your lungs are this, this huge reception device, and there's no faster way to get it into your bloodstream.
Kannaboomers: Back in the day when, when we were in college, you know, you inhale the substance and the thing was to hold it in for, you know, as long as he could before you let it out. When you're vaping, do you recommend holding it in? Do you recommend blowing it out to take a little bit of fresh air after you've hit the pen? Is there a preferred method for doing it?
Jason: You know, it's like everyone has their opinion on this subject, whether it be the substance or the other. I personally think the longer that you allow it to be exposed to your lungs that better, so I would recommend holding it in. It just seems to logically make sense that if you're taking in a plume of vapor, you wanted to hit as much surface area of your lungs as possible in order to be effective. I don't think like how kids smoked cigarettes in high school is going to do the job. Just holding it in your throat and blowing it out isn't really effective. I'd recommend taking a a short breath and then breathing it all the way down into the deepest part of your lungs and waiting a few seconds and the less you blow out at the end and the less that you're essentially giving back to the atmosphere that would make sense to me is to hold it in longer. If you can.
Kannaboomers: We've kind of talked about the history and about the current outlook in things that consumers need to be aware of. What trends do you see in this field that might be up and coming?
Jason: Oh, I can speak to what we're interested in is CBD is not the only thing that can be vaporized and same thing as I was alluding to a terpenes is there's a number of different. Let's just call them potentiating effects that terpenes have, so blends of CBD with other terpenes, with perhaps other molecules are interesting to us. I don't think CBD is the last thing we're going to formulate into a vaporizer. I think there's a potential for a lot of things that can be vaporized and they'll like if you've seen some of these vitamin B12 vapes or vitamin vapes, if you will.
Jason: There's some efficacy to it. And then there's also some BS to where it's not really possible to, to vaporize vitamin C for example, because the amount that you would need would be larger than the amount that you could fit into a cartridge. So you know, a single inhalation is not going to give you any reasonable amount of vitamin C and maybe even B12 might be in that corner too. But um, there's tons of other things that have interesting potential to be vaporized that we're looking at.
Kannaboomers: So you have Thrive and you said you have a second product too?
Jason: Yeah, we actually, we wanted to push the limits on how much CBD we can put in a cartridge what we're noticing is we're averaging 40 to 45 percent, so essentially we were giving away more CBD than we were even advertising, but we wanted to see if it was possible to create a 50 percent solution and stabilize it.
It took a while to be able to do it, but we managed to stabilize the 250 milligram cartridge, so we have thrive and thrive beyond which is an additional 50 milligrams more so it's a 50 percent. So it's a more high potency formulations. Yeah, we increase the potency but another, a 10 percent. It has a slightly different flavor profile because the oil we're using is basically extracted at a really high percentage, so there's not much plant matter left. Where say Thrive for example, has a lot more of the initial oil in terpenes that come with the plant. So it's more of a full spectrum product. The Thrive Beyond. Because the amount of CBD that were getting initially in the first extraction is so high that there's not a lot of room for the, let's just say like what I like to call, like the tannin for example, it's more of the plant matter that's left in the oil that creates the terpene profiles.
Thrive is more of a full profile, full spectrum product that has a lot of the natural trappings that come with the plant and Thrive Beyond has our own terpene profile that we've maximized. It's just a different formulation of flavor with a higher potency. So some people are really interested in the very earthy, organic, natural taste and flavor. And then some people want more of like a clean, very high potency solution that doesn't have any like outrageous flavor. So that's more of what Thrive Beyond his is very clean, administered a high potency product with, uh, with just a different flavor profile. So you're going to get a little more CBD in each puff, in a, in a flavor that is a little bit different. Yes.
Kannaboomers: So it's like the difference between an IPA and double IPA maybe.
Jason: Yeah. With with going the opposite direction, like a double IPA would be much more hoppier whether this tastes even less spicy. This is people have related it to is kind of like drinking tea. It has like a tea flavor almost, so it's very subtle and this is just like a lot of people because what happened is we ended up getting a lot higher percentage oil. Last year there was, there was a shortage on oil every, every year really this happens. People buy up the rest of the harvest before the new harvest comes in and there's roughly a couple months where there's. It's very difficult to find quality oil until the new harvest is complete. So roughly from like August through middle of October it's hard to find oil. We ended up having to find a different manufacturer and they sent us this really high potency oil. It looked nothing like our original oil, so we didn't know what to really do it that we didn't.
We weren't even sure if we're going to keep it. We sampled it out to a few people and they liked it. Some people liked the other oil better, some people like this soil better, so we didn't, we didn't know whether or not to scrap it or so we just decided to create a second product and just revisit the formulation. So what started out as an accident turned into a second product that um, a lot of people prefer and people that love the original product don't have to, they don't have to make a decision, they can just stick with the original Po. A lot of things happened right by accident and absolutely. Yeah, we're really pleased with it. I mean, like I said, it took a long time to be able to formulate something of this high potency. We thought we were already cutting edge when we created the 40 percent when we first entered this market, there was nobody that had a 40 percent vape around and anyone that did did was either lying or boosting the numbers with adding certain amounts of CBD isolate.
So it's, it's a lot different. Like a pure oil that was 40 percent was really, really difficult to find all of the vape solutions in the past where you know, five to 10 percent maximum in an e-liquid form or they were one to one or two to one ratios with THC that you could only get dispensaries. That was kind of this issue as if you really wanted a high quality, high potency CBD. You had to go to a dispensary because most of the time the formulations were too hot, meaning it had too much THC in it for to be able to be sold legally across the U.S finding something had no THC and high potency was really hard to do, so 40 percent was, was considered like insane at first and then to be able to produce a stable 50 percent that doesn't turn into rock candy was the next step. And that's what we've just been able to be and we're quite proud of it.
Kannaboomers: You guys have been at this for a couple of years and as you've, as we've kind of covered this, it's unregulated as you said, it's kind of like the wild west. You have to look for someone who's getting good, clean, organic hemp and who's testing all the while and who's going to give you a product that's got enough potency to make a difference for you. I mean, how many players are out there now? Are there dozens or hundreds? You've been at this for a couple of years in this market, it's exploded. Do you see that subsiding or is it just continuing to expand?
Jason: It's kind of hard to, to answer that question while there is more people entering the market, there's even more illegitimate people entering the market. So that to me, um, it's kind of at an equal ratio where there's a few quality competitors, there's even more people that are just creating crap. So yeah, it definitely is getting larger than. There's no doubt the CBD market is expanding at an incredible rate. And then the vapor category in itself, which was kind of undecided because it exploded so fast. I mean, at least the vape culture did that. It was, um, it was kind of a unknown what the future would hold as the FDA kind of came in and started regulating vape calling everything a tobacco product and taxing a lot of these products, holding licenses for manufacturing. There's a lot of things happening to the vape industry that was kind of at the same time effecting the CBD vape industry, if that makes any sense, because we're tied to it.
Jason: So if I buy a battery, this battery that powers this pen vape, it's this pen is for health and health beneficial uses. It's not a ecigarette. It's not simply a smoking alternative. Right. But that's not how the FDA sees it. When I buy a battery, it's a nicotine dispensing device. It's not, it's not just a battery. I have to pay taxes when it's imported into the United States because it's considered a tobacco product. Even though we don't administer any tobacco, they still considered a tobacco product. So that affects us in some capacity. So whatever laws they create around the hardware involving vape kind of affect us. So there was a moment where we weren't sure which direction it was going to go, whether or not they were just going to outright ban the sale of this stuff in the US until they figured out what it was.
So we were kind of know just watching from the sidelines wondering what was, what was gonna happen. Luckily it looked like it found some stability and it seems like they're happy with just texting the terrorists at the port for now. So there was a slight slowdown in the amount of vape products being created when it was first a huge explosion. You can just see like thousands of websites created for liquids and have every category. It's kind of like regulated a bit. So in a way that kind of spilled over into our industry where there is less people creating the products. It's just the level of difficulty is already so, so high because of one. Banking is extremely difficult. Transacting credit cards in this industry is a huge question mark because they don't know whether it's a tobacco product of a product or a cannabis product. There's a lot of credit card processors that consider this extremely high risk and won't even take you on whether it's because it's a vape or whether it's because it's related to cannabis, so there's tons of people that would love to create a CBD product but simply can't because a banking is so difficult.
Kannaboomers: Well, I mean you're describing this kind of shifting landscape, but as I think there's 26 or 27 states now that have legalized cannabis for medicinal use and even like as you say, it's, it's not cannabis really and it's not tobacco and then there's all kinds of commercials on TV right now about how vaping is bad for you. So there's still a lot of confusion out there in the market, obviously. Now I do want to also want to ask you about carrier liquids because that's a big thing too and people need to be educated about the kind of liquid that's carrying the CBD. And I know you guys have a special formulation you use that you feel is safer than some of the other alternatives out there.
Jason: Yeah, so essentially a carrier liquid is just a solvent that allows them, that allows the product to be vaporized. So CBD in itself is a crystal structure and it turns into rock candy at high potencies. So if you had a 95 percent CBD solution, it would basically be a gigantic. It looked like a gigantic crystal, like a, like rock candy would if you did that experiment when you're a kid, you know, he'd tie a string to a stick and put it in a highly, highly potent sugar solution. And the candy starts to form around the string. It's, it's the exact, it's exact same thing that CBD does. So, uh, you need to have the ratio of CBD to oil in it. The natural fats that come with the plant to be lower, so less than, say 50 percent for example. And the reason why that's important is it only takes a small amount of CBD crystal that just basically turned into a gigantic rock.
So you need the ratio of CBD oil to be correct in order to vaporize it, or else you'll just, you'll have a gigantic rock candy and your cartridge. What people have done is this followed suit into the e-liquid space, weather it's, propylene glycol and vegetable glycerin. These are just different carrier liquids that allow you to dilute whatever it is that you're trying to vaporize. Both of them have questionable history and safety. There was a study done, I think a couple years ago, think he was the New England Journal of Health. Can't remember if that was them, but they basically showed that at high temperatures are high voltages. That propylene glycol turns into formaldehyde and that scared the living daylights out of all these people that are blowing clouds of propylene glycol, and I think that study has been since debunked, is basically the level of voltage required. You wouldn't even be able to inhale that.
Jason: 32:53 Meaning that the vapor would be so hot that it would. It would burn up to begin with and it wouldn't even be a pleasant, a pleasant experience vaporizing so chemically. You're saying that the heat that creates the kind of vapor you would inhale your. It's impossible to create formaldehyde. Yeah. You wouldn't even have a battery that can get that hot to begin with, so yeah, it wouldn't. It's not a typical experience. That's where the data is kind of skewed is that the levels they were exposing. This propylene glycol was, was extreme and that's not something that would be typical for many everyday user. Nonetheless, we were just scared to even go that method so we, we completely shied away from it. We didn't want there to be any question whether or not it's, it's now considered safe and I do believe it is. I don't think that propylene glycol is the devil, but I don't want any question.
You know what makes me feel good as I can sleep well at night knowing that I'm helping people and if there's any potential harm someone I don't think I'm going to get much sleep. So we decided not to go that route at all. So we started originally with 100 percent pure CBD oil, but there was also some issues with that as if it's not extracted properly. Like a CO2 extraction removes a lot of the fats, like the large waxes people are vaping, all natural wax and oils. There's potential for what's called [inaudible] pneumonia where basically the waxes are too large for your lungs to pass through the cell walls and your lungs and they basically stick to your lungs so you have essentially oil stuck to the walls of your lungs, basically have a cough that won't go away for five days until you stop using the product.
So a lot of people that are vaping, these all natural, 100 percent waxes are starting to get lipid pneumonia and wondering why they can't get rid of this cough. We were really scared to go this all natural approach as well. So it wasn't until a while later that we noticed that all of the top manufacturers were adding a triglycerides back into their oils to give it more, to give it less viscosity. You know, create a solution that can be vaporized. And that became the gold standard. So now instead of propylene glycol and vegetable glycerin, MCT oil has now become the standard. We're really happy about this because it's probably the highest quality best carrier liquid, if you want to call it that, that you can find because all it is is medium chain triglycerides, meaning they have less than, I think less than 10 carbon chains that allows it to be passed through your walls of your lungs a lot easier, so it's a really clean, appropriate way to use hemp oil because the oil itself doesn't contain any large fat or waxes that would naturally stick to your lungs because that's typically coconut. Yeah, so
Coconut oil has MCT but we're not using per se coconut oil. Right. Like if you see coconut oil at the store, it's usually, you know, solid and white and then if you see MCT in supplements section, it's clear in liquid where coconut oil still has a lot of the waxes and in the fats, the solid fats that make it solid even though it's at a very low melting temperature. It's not something that you wouldn't want to vape coconut oil, but it's perfectly fine to vape MCT oil. I know that it sounds like I'm splitting hairs, but there truly is a large difference between the two. Organic MCT oil is derived from coconut oil. It's just had all of the fat removed so that it's a natural liquid that contains only the triglycerides, only the lipids if you will.
Kannaboomers: It's derived from or is it a synthetic oil?
Jason: I believe that they can make synthetic a MCT and that that's what a lot of supplements use. We use a organic because we just, yeah, we just appreciate something that it's organic more than, you know, the question of whether or not it's derived from a safe source. Again, MCT, no matter which way it's derived, I think is better than any of the other alternatives, so I don't think it matters if it's organic or not. Uh, we would just prefer it to be organic, so we use an organic MCT derived from coconut.
Kannaboomers: That makes sense. I want to tell people that you guys are at Vape Bright dot org where you have Thrive and Thrive Beyond. What else do they need to know about Vape Bright.
Jason: We're a legitimate company that's been doing this for two years now. We have well over 1,200 reviews from real people with real issues. It really has something to say. We stand by our product, we stand by our word. We have a wonderful customer support team that's willing and ready to answer any question you have. I think that's really important, especially in an industry that has so much uncertainty in and so much fear surrounding it that we have real humans that are ready to take your call or to answer via chat window in the website. That's really important. Yeah, we're always ready to answer.
Kannaboomers: That's great. Thank you so much for your time, Jason. I think we covered a lot of stuff and I know our listeners are going to be interested in this and I look forward to having you back on. I know there's going to be a lot more to talk about in the future, but thanks for sharing your expertise on this important topic and we hope to talk to you again. Really appreciate it.
Twenty 4:20 |#4 Cannabinoid and Terpene Research
28 | Stacey Mulvey, Cannabis and Yoga
Twenty 4:20 | #3 Common Cannabis Confusion
27 | Box Brown, "Cannabis: The Illegalization of Weed in America"
Candid Clownfish says
Lots of great info here about the vape bright. I dig their setup, and I'm really enjoying their product.
I clicked on this page to do a little reading from the man at the company, and I walk away with a better understanding of the company. I dig it.
I just wish they would create a larger vial for their pens. Q será
Thank your for pushing the transcript!
Thanks for checking in!
FOLLOW! Good and informative interview, I just ordered a VB stater kit..I have been doing the oil drops, thought I`d give this a try since it is to get into your system faster. :)) Thnxs, Anne
Very hard to read due to terrible spelling and awful grammar. Someone should have proof read this before posting. Just awful! Interesting if you can read it.
Hi Denise, thanks for your feedback. We have a company that does the transcripts and then we edit them; I gave this one another look and yes, there were some mistakes in there. I gave it another pass and it's better now. Thanks again for letting us know.
Leave a Reply to Tom Cancel reply
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Copyright © 2019 Kanna Boomers. All Rights Reserved | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,235 |
{"url":"https:\/\/byjus.com\/question-answer\/the-symbol-for-the-indian-rupee-is-rs-re-mn\/","text":"Question\n\n# The symbol for the Indian Rupee is ____.Rs Re \u20b9 Mn\n\nSolution\n\n## The correct option is C \u20b9 The Indian Rupee is represented by the symbol\u00a0\u20b9.\n\nSuggest corrections","date":"2021-12-04 12:14:15","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\": 0.8191929459571838, \"perplexity\": 7058.370876992389}, \"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-2021-49\/segments\/1637964362969.51\/warc\/CC-MAIN-20211204094103-20211204124103-00511.warc.gz\"}"} | null | null |
package com.opengamma.master.security;
import org.testng.annotations.Test;
import com.opengamma.id.ExternalIdBundle;
import com.opengamma.id.UniqueId;
import com.opengamma.util.test.AbstractFudgeBuilderTestCase;
import com.opengamma.util.test.TestGroup;
/**
* Test Fudge encoding.
*/
@Test(groups = TestGroup.UNIT)
public class RawSecurityFudgeEncodingTest extends AbstractFudgeBuilderTestCase {
public void test_basic() {
RawSecurity object = new RawSecurity("Dummy", new byte[0]);
assertEncodeDecodeCycle(RawSecurity.class, object);
}
public void test_full() {
UniqueId uid = UniqueId.of("A", "123");
ExternalIdBundle bundle = ExternalIdBundle.of("X", "Y");
RawSecurity object = new RawSecurity(uid, "OpenGamma", "Dummy", bundle, new byte[] {1, 2, 4});
assertEncodeDecodeCycle(RawSecurity.class, object);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,720 |
{"url":"http:\/\/cpresourcesllc.com\/standard-error\/statistics-standard-error-of-estimate-calculator.php","text":"Home > Standard Error > Statistics Standard Error Of Estimate Calculator\n\n# Statistics Standard Error Of Estimate Calculator\n\n## Contents\n\nY\u00fckleniyor... Not the answer you're looking for? Kapat Daha fazla bilgi edinin View this message in English YouTube 'u \u015fu dilde g\u00f6r\u00fcnt\u00fcl\u00fcyorsunuz: T\u00fcrk\u00e7e. Dividing the sample standard deviation by the square root of sample mean provides the standard error of the mean (SEM).\n\nSolved Example The below solved example for to estimate the Check This Out\n\nEnter the Range of Values (Seperated by comma) Standard Error of Sample Means Code to add this calci to your website Just copy and paste the below code to your webpage The simplest estimate would be to calculate the observed variance in the sample, and use this as the best estimate of the true variance within the population. You can see that in Graph A, the points are closer to the line than they are in Graph B. Stephanie Glen 25.545 g\u00f6r\u00fcnt\u00fcleme 1:26 Paired Samples t Test - S\u00fcre: 20:39.\n\n## Standard Error Of Estimate Calculator Regression\n\nMaxamus 18.379 g\u00f6r\u00fcnt\u00fcleme 9:11 Standard Error of the Estimate used in Regression Analysis (Mean Square Error) - S\u00fcre: 3:41. L\u00fctfen daha sonra yeniden deneyin. 20 Eyl 2012 tarihinde yay\u0131nland\u0131A short video on how to quickly find the standard error of the estimate using excel Kategori E\u011fitim Lisans Standart YouTube Lisans\u0131 Figure 1.\n\nSee also stats.stackexchange.com\/questions\/5135\/\u2026 \u2013conjugateprior Sep 8 '14 at 13:11 add a comment| 3 Answers 3 active oldest votes up vote 2 down vote accepted Looking at ISL's parent book, ESL (Elements Y\u00fckleniyor... Y\u00fckleniyor... Standard Error Of Estimate Formula Is there any financial benefit to being paid bi-weekly over monthly?\n\nTess St. Standard Error Calculator Excel The service is unavailable. Estimate the sample mean for the given sample of the population data.\n2. Math Calculators All Math Categories Statistics Calculators Number Conversions Matrix Calculators Algebra Calculators Geometry Calculators Area & Volume Calculators Time & Date Calculators Multiplication Table Unit Conversions Electronics Calculators Electrical Calculators\n\nThe standard deviation of a sample divided by \u221an is the SE of the sample. Sb1 Calculator more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed Standard Error of the Mean The standard error of the mean is the standard deviation of the sample mean estimate of a population mean. ProfessorSerna 34.017 g\u00f6r\u00fcnt\u00fcleme 23:21 Statistics 101: Standard Error of the Mean - S\u00fcre: 32:03.\n\n1. Regarding the values, the comment under the question is correct, the writing is a bit misleading.\n2. Kapat Evet, kals\u0131n.\n3. Help my maniacal wife decorate our christmas tree What do you do with all the bodies?\n4. Y\u00fckleniyor...\n5. Substitute $\\frac{RSS}{N-2}$ into the equation for SE$(\\hat{\\beta_1})^2$ and you will get the values in ISL.\n6. Inputs: standard error parameter (\u03b4)standard deviation (s)sample size (n) Conversions: standard error parameter (\u03b4)= 0 = 0 standard deviation (s)= 0 = 0 sample size (n)= 0 = 0 Solution: standard\n7. Todd Grande 29.187 g\u00f6r\u00fcnt\u00fcleme 9:33 Standard Error Bars on Excel - S\u00fcre: 5:01.\n8. The standard error of the estimate is closely related to this quantity and is defined below: where \u03c3est is the standard error of the estimate, Y is an actual score, Y'\n9. Therefore, which is the same value computed previously.\n10. Online Web Apps, Rich Internet Application, Technical Tools, Specifications, How to Guides, Training, Applications, Examples, Tutorials, Reviews, Answers, Test Review Resources, Analysis, Homework Solutions, Worksheets, Help, Data and Information for Engineers,\n\n## Standard Error Calculator Excel\n\nI got lost when $\\sigma^2$ is calculated. Y\u00fckleniyor... \u00c7al\u0131\u015f\u0131yor... Standard Error Of Estimate Calculator Regression Y\u00fckleniyor... \u00c7al\u0131\u015f\u0131yor... Standard Error Of Estimate Calculator Ti-84 Infant Growth Charts - Baby Percentiles Overtime Pay Rate Calculator Salary Hourly Pay Converter - Jobs Percent Off - Sale Discount Calculator Pay Raise Increase Calculator Linear Interpolation Calculator Dog Age\n\nCalculate Student's t-Statistic (Independent Samples) Using Data Analysis in Excel 2010 - S\u00fcre: 8:16. his comment is here Free Electron in Current In 5e, do you get to use the extra attack as well when you ready an attack action? Bu \u00f6zellik \u015fu anda kullan\u0131lam\u0131yor. Browse other questions tagged variance or ask your own question. Standard Error Of Proportion Calculator\n\nzedstatistics 338.664 g\u00f6r\u00fcnt\u00fcleme 15:00 Calculating mean, standard deviation and standard error in Microsoft Excel - S\u00fcre: 3:38. A pilot's messages more hot questions question feed about us tour help blog chat data legal privacy policy work here advertising info mobile contact us feedback Technology Life \/ Arts Culture When it comes to verify the results or perform such calculations, this standard error calculator makes your calculation as simple as possible.\n\nSimilar Resource Sample & Population Standard Deviation Difference & this contact form The RSE is an estimate for $\\sigma$, not $\\sigma^2$. $RSE^2$ is an estimate for $\\sigma^2$.\n\nAs it turns out, however, it can be shown that this naive approach underestimates the true population variance: the sample variance is a biased estimator. Standard Error Calculator From Standard Deviation You can change this preference below. Hakk\u0131nda Bas\u0131n Telif hakk\u0131 \u0130\u00e7erik Olu\u015fturucular Reklam Verme Geli\u015ftiriciler +YouTube \u015eartlar Gizlilik Politika ve G\u00fcvenlik Geri bildirim g\u00f6nder Yeni bir \u015feyler deneyin!\n\n## Konu\u015fma metni Etkile\u015fimli konu\u015fma metni y\u00fcklenemedi.\n\nThis esti- mate is known as the residual standard error\". Learn more You're viewing YouTube in Turkish. Service Unavailable HTTP Error 503. Standard Error Calculator For Two Samples Hydrology Water Quantity and Quality Control.\n\nIt can also be referred as the estimation of the standard deviation. What is alluded to by \"In general, \u03c32 is not known, but can be estimated from the data. EVIEWS - S\u00fcre: 27:45. http:\/\/cpresourcesllc.com\/standard-error\/standard-error-estimate-calculator.php The numerator is the sum of squared differences between the actual scores and the predicted scores.\n\nResubmitting elsewhere without any key change when a paper is rejected Close current window shortcut Lagrange multiplier on unit sphere How to play on the piano, 4 notes stacked on top Oturum a\u00e7 29 5 Bu videoyu be\u011fenmediniz mi? The estimation with lower SE indicates that it has more precise measurement. Step 2 : The Standard deviation SD is 2.58199 Step 3 : To find SE , 2.58199 \/ \u221a4 = 1. 29099 Hence the SE of 1,3,5,7 is 1. 29099 Related\n\nD\u00fc\u015f\u00fcncelerinizi payla\u015fmak i\u00e7in oturum a\u00e7\u0131n. Assume the data in Table 1 are the data from a population of five X, Y pairs. statisticsfun 122.221 g\u00f6r\u00fcnt\u00fcleme 3:41 Simple Linear Regression - S\u00fcre: 23:21. WCEastFZX 185.286 g\u00f6r\u00fcnt\u00fcleme 8:46 z test p-value approach - S\u00fcre: 6:55.\n\nThe manual calculation can be done by using above formulas. So, when drawing a finite sample from a population, the variance has to be estimated. Are there too few Supernova Remnants to support the Milky Way being billions of years old? Stephanie Castle 320.365 g\u00f6r\u00fcnt\u00fcleme 3:38 Calculating the Standard Error of the Mean in Excel - S\u00fcre: 9:33.","date":"2018-06-24 14:37:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.3338559567928314, \"perplexity\": 4557.317502192587}, \"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-2018-26\/segments\/1529267866965.84\/warc\/CC-MAIN-20180624141349-20180624161349-00285.warc.gz\"}"} | null | null |
"""
PXE Boot Interface
"""
from ironic_lib import metrics_utils
from oslo_log import log as logging
from ironic.common import boot_devices
from ironic.common.i18n import _
from ironic.common import pxe_utils
from ironic.common import states
from ironic.conductor import task_manager
from ironic.conductor import utils as manager_utils
from ironic.conf import CONF
from ironic.drivers import base
from ironic.drivers.modules import agent_base
from ironic.drivers.modules import deploy_utils
from ironic.drivers.modules import pxe_base
LOG = logging.getLogger(__name__)
METRICS = metrics_utils.get_metrics_logger(__name__)
class PXEBoot(pxe_base.PXEBaseMixin, base.BootInterface):
capabilities = ['ramdisk_boot', 'pxe_boot']
def __init__(self):
pxe_utils.place_common_config()
pxe_utils.place_loaders_for_boot(CONF.deploy.http_root)
pxe_utils.place_loaders_for_boot(CONF.pxe.tftp_root)
class PXEAnacondaDeploy(agent_base.AgentBaseMixin, agent_base.HeartbeatMixin,
base.DeployInterface):
def get_properties(self):
return {}
def validate(self, task):
task.driver.boot.validate(task)
@METRICS.timer('AnacondaDeploy.deploy')
@base.deploy_step(priority=100)
@task_manager.require_exclusive_lock
def deploy(self, task):
manager_utils.node_power_action(task, states.POWER_OFF)
with manager_utils.power_state_for_network_configuration(task):
task.driver.network.configure_tenant_networks(task)
# calling boot.prepare_instance will also set the node
# to PXE boot, and update PXE templates accordingly
task.driver.boot.prepare_instance(task)
# Power-on the instance, with PXE prepared, we're done.
manager_utils.node_power_action(task, states.POWER_ON)
LOG.info('Deployment setup for node %s done', task.node.uuid)
return states.DEPLOYWAIT
@METRICS.timer('AnacondaDeploy.prepare')
@task_manager.require_exclusive_lock
def prepare(self, task):
node = task.node
deploy_utils.populate_storage_driver_internal_info(task)
if node.provision_state == states.DEPLOYING:
# Ask the network interface to validate itself so
# we can ensure we are able to proceed.
task.driver.network.validate(task)
manager_utils.node_power_action(task, states.POWER_OFF)
# NOTE(TheJulia): If this was any other interface, we would
# unconfigure tenant networks, add provisioning networks, etc.
task.driver.storage.attach_volumes(task)
node.instance_info = deploy_utils.build_instance_info_for_deploy(
task)
node.save()
if node.provision_state in (states.ACTIVE, states.UNRESCUING):
# In the event of takeover or unrescue.
task.driver.boot.prepare_instance(task)
def deploy_has_started(self, task):
agent_status = task.node.driver_internal_info.get('agent_status')
if agent_status == 'start':
return True
return False
def deploy_is_done(self, task):
agent_status = task.node.driver_internal_info.get('agent_status')
if agent_status == 'end':
return True
return False
def should_manage_boot(self, task):
if task.node.provision_state in (
states.DEPLOYING, states.DEPLOYWAIT, states.DEPLOYFAIL):
return False
# For cleaning and rescue, we use IPA, not anaconda
return agent_base.AgentBaseMixin.should_manage_boot(self, task)
def reboot_to_instance(self, task):
node = task.node
try:
# anaconda deploy will install the bootloader and the node is ready
# to boot from disk.
deploy_utils.try_set_boot_device(task, boot_devices.DISK)
except Exception as e:
msg = (_("Failed to change the boot device to %(boot_dev)s "
"when deploying node %(node)s. Error: %(error)s") %
{'boot_dev': boot_devices.DISK, 'node': node.uuid,
'error': e})
agent_base.log_and_raise_deployment_error(task, msg)
try:
task.process_event('resume')
self.clean_up(task)
manager_utils.node_power_action(task, states.POWER_OFF)
task.driver.network.remove_provisioning_network(task)
task.driver.network.configure_tenant_networks(task)
manager_utils.node_power_action(task, states.POWER_ON)
task.process_event('done')
except Exception as e:
msg = (_('An error occurred after deployment, while preparing to '
'reboot the node %(node)s: %(error)s') %
{'node': node.uuid, 'error': e})
agent_base.log_and_raise_deployment_error(task, msg)
def _heartbeat_deploy_wait(self, task):
node = task.node
agent_status_message = node.driver_internal_info.get(
'agent_status_message'
)
msg = {'node_id': node.uuid,
'agent_status_message': agent_status_message}
if self.deploy_has_started(task):
LOG.info('The deploy on node %(node_id)s has started. Anaconda '
'returned following message: '
'%(agent_status_message)s ', msg)
node.touch_provisioning()
elif self.deploy_is_done(task):
LOG.info('The deploy on node %(node_id)s has ended. Anaconda '
'agent returned following message: '
'%(agent_status_message)s', msg)
self.reboot_to_instance(task)
else:
LOG.error('The deploy on node %(node_id)s failed. Anaconda '
'returned following error message: '
'%(agent_status_message)s', msg)
deploy_utils.set_failed_state(task, agent_status_message,
collect_logs=False)
@METRICS.timer('AnacondaDeploy.clean_up')
@task_manager.require_exclusive_lock
def clean_up(self, task):
super(PXEAnacondaDeploy, self).clean_up(task)
node = task.node
# NOTE(rloo): These were added during deployment, as a side-effect of
# pxe_utils.get_instance_image_info().
node.del_driver_internal_info('stage2')
node.del_driver_internal_info('ks_template')
node.save()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,501 |
2021: The Comeback Year
Global Trends | News and Innovations, Views from Martech Leaders
Home News Entravision Announces the Appointment of Juan Saldívar as Chief Digital, Strategy and...
Entravision Announces the Appointment of Juan Saldívar as Chief Digital, Strategy and Accountability Officer
TCMO Bureau
Entravision Communications Corporation (NYSE: EVC), a leading global media and marketing technology company, today announced the appointment of Juan Saldívar as its new Chief Digital, Strategy and Accountability Officer. In his new role, Mr. Saldívar will be responsible for overseeing Entravision's digital business units, corporate strategy and business development and overall business unit reporting and accountability
A current consultant to Entravision, and a member of its Board of Directors since May of 2014, Mr. Saldívar has been an integral part of the company's efforts to expand its portfolio of exceptional digital assets with creative and programmatic capabilities that meet its global clients' needs. Notably, Mr. Saldívar played a prominent role in the company's recent majority investment in Cisneros Interactive, a transaction that has positioned Entravision to become one of the largest premier digital advertising companies serving the U.S. Hispanic and Latin American markets in over 21 countries.
Read More: Humanizing a Brand's Corporate Comms; Balance and the CCO
"We are very excited to welcome Juan to our executive team and to gain access to his expertise on a full-time basis," said Walter F. Ulloa, Chairman and Chief Executive Officer of Entravision. "As we continue to enhance and grow our digital business, transformation strategy and digital identity, Juan's appointment will be instrumental. As Juan has consulted with Entravision for a number of years, I fully expect a seamless transition into his new role."
"Following several years on Entravision's Board of Directors, I am honored to now have the opportunity to join the executive management team," said Juan Saldívar. "I am looking forward to leveraging my skillset and that of my fellow Entravision colleagues to continue the expansion of Entravision's business, culture and growth within a digitally connected world."
Mr. Saldívar is the founder and CEO of SWS Consulting, a leading advisory firm that counsels clients in the marketing, media, entertainment, talent and technology industries. He has over 25 years of experience in the media, marketing, technology, venture capital and e-commerce industries in Mexico, the U.S. and Germany, and has founded or participated in developing several leading ventures, such as Submarino.com and Ingredienta.com in Mexico and Rise Capital, a leading emerging markets venture fund based out of San Francisco, CA.
Read More: Why Customer Analytics is Now More Essential for The Consumer Durables Industry
Prior to SWS Consulting, Mr. Saldívar worked at two of the most successful global media, information and entertainment companies. He served as General Manager at Televisa Interactive Media, part of Grupo Televisa, the largest Hispanic media company in the world, where he was responsible for the strategy, operations and overall entertainment, information and sports digital presence for the company. He also worked in business development at the Bertelsmann Group, one of the world's largest media conglomerates headquartered in Germany, where he later became CEO of the company's Mexican publishing house. Mr. Saldívar holds a degree in economics from the Instituto Tecnológico Autónomo de México and an MBA from the IESE Business School in Spain. In this new role, he will continue to hold his position on Entravision's Board of Directors.
diversified global marketing
Entravision
global media and marketing technology company
Juan Saldívar
proprietary marketing technologies and platforms
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,837 |
Image Title: Oak Chrome Bar Stool For Design 19. Post Title: Chrome Bar Stool. Filename: oak-chrome-bar-stool-for-design-19.jpg. Image Dimension: 1200 x 1372 pixels. Images Format: jpg/jpeg. Publisher/Author: Jeremie Hauck. Uploaded Date: Tuesday - January 22nd. 2019 00:01:29 AM. Category: Architecture. Image Source: homedepot.com. Pair Of Chrome Bar Stools With Backrest Seating Intended For Stool Idea 6. AmeriHome Loft Style 30 In Stackable Metal Bar Stool Chrome Pertaining To Ideas 3. Aria Chrome Bar Stool In White Leatherette Faux Leather Stools For Designs 4. Black Chrome Bar Stool Julia Kendell Range Faux Leather Regarding Decorations 8. Circa 1970 Romeo Rega Chrome And Leather Bar Stools For Sale At 1stdibs Within Stool Decorations 12. Shop Chloe 30 Inch Clear Chrome Bar Stool Set Of 2 Free Regarding Inspirations 9. Pair Of Chrome Bar Stools Seating Regarding Stool Decorations 2. Elegant Chrome Bar Stools Chic Stool Home Kitchen Furniture Throughout Remodel 16. GLENN Bar Stool 30 3 8 IKEA Pertaining To Chrome Design 17. Amazon Com Richardson Seating 0 1950GRN24 Backless Swivel Bar Stool In Chrome Inspirations 5. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,304 |
XForms is an XML format used for collecting inputs from web forms. XForms was designed to be the next generation of HTML / XHTML forms, but is generic enough that it can also be used in a standalone manner or with presentation languages other than XHTML to describe a user interface and a set of common data manipulation tasks.
XForms 1.0 (Third Edition) was published on 29 October 2007. The original XForms specification became an official W3C Recommendation on 14 October 2003, while XForms 1.1, which introduced a number of improvements, reached the same status on 20 October 2009.
Differences from web forms
In contrast to the original web forms (originally defined in HTML), the creators of XForms have used a model–view–controller (MVC) approach. The model consists of one or more XForms models describing form data, constraints upon that data, and submissions. The view describes what controls appear in the form, how they are grouped together, and what data they are bound to. CSS can be used to describe a form's appearance.
An XForms document can be as simple as a web form (by only specifying the submission element in the model section, and placing the controls in the body), but XForms includes many advanced features. For example, new data can be requested and used to update the form while it is running, much like using XMLHttpRequest/AJAX except without scripting. The form author can validate user data against XML Schema data types, require certain data, disable input controls or change sections of the form depending on circumstances, enforce particular relationships between data, input variable length arrays of data, output calculated values derived from form data, prefill entries using an XML document, respond to actions in real time (versus at submission time), and modify the style of each control depending on the device they are displayed on (desktop browser versus mobile versus text only, etc.). There is often no need for any scripting with languages such as JavaScript. However, XForms does include an event model and actions for implementing more complex form behaviors. Actions and event handling are specified using the XForms XML dialect rather than more common scripting languages like JavaScript.
Like web forms, XForms can use various non-XML submission protocols (multipart/form-data, application/x-www-form-urlencoded), but a new feature is that XForms can send data to a server in XML format. XML documents can also be used to prefill data in the form. Because XML is a standard, many tools exist that can parse and modify data upon submission. Similar tools for legacy forms also exist. XForms is itself an XML dialect, and therefore can create and be created from other XML documents using XSLT. Using transformations, XForms can be automatically created from XML schemas, and XForms can be converted to XHTML forms.
Software support
At the time of this writing, no widely used web browser supports XForms natively. However, various browser plugins, client-side extensions and server/client solutions exist. The following lists some implementations:
The Firefox XForms extension was part of the Mozilla Project. XForms 1.0 SE support is not complete but covers most of the specification with a notable exception of attribute-based repeating used in HTML tables. The extension was available for both Firefox 2 and Firefox 3, but is not upgraded to support Firefox 4 and higher. In July 2011 the lead developer wrote that XForms support would no longer get updated. Support for XForms was eventually deprecated in Firefox 19.
IBM Lotus Forms supports development and deployment of XForms-based pure XML forms. Trial downloads are available of an Eclipse-based visual design environment and a client-side viewer that can run XForms-based forms both in the web browser and as a standalone desktop application.
OpenOffice.org versions 2.0 and greater and LibreOffice support XForms.
Implementation technologies compared
FormFaces, AJAXForms, XSLTForms, betterFORM, Chiba, Orbeon and Smartsite Forms are based on Ajax technology. The amount of server-side and client-side processing varies between these implementations. For example, Ubiquity XForms, FormFaces and XSLTForms provide 100% XForms client-side processing and data model updates via pure Ajax processing on the XForms standard. The others use server-side Java/.NET XForms processing transcoding to Ajax markup prior to delivering the content to the browser. Both techniques can work across browsers. Each implementation is significantly different with respect to dependencies, scalability, performance, licensing, maturity, network traffic, offline capability, and cross browser compatibility. System architects should evaluate these constraints against their needs to determine potential risks and objectives.
Plugins like FormsPlayer and other client-side technology can have some benefits as well: because they integrate themselves into the browser, they will work with existing server architectures, can be more responsive, and require fewer server fetches.
The tradeoff between server-side and client plug-in solutions is where the software is maintained; either each client must install the required plug-in, or the server architecture must change to accommodate the XForms transcoder engine language technology. It is in theory possible to mix both of these solutions, for instance testing the browser for a client-side XForms implementation and serving native XForms in that case, and defaulting to a server solution in other cases.
Ubiquity XForms, FormFaces and XSLTForms provide a "zero software" solution on either the client or server: no new software needs to be installed on the client and the solution can be used in conjunction with any server-side architecture. This is possible because FormFaces and Ubiquity XForms are written 100% in Ajax and because XSLTForms is written in XSLT and in Ajax. The tradeoff is that compared to other solutions, more code is initially downloaded to the client (code can be cached on the client), and FormFaces does not yet support XML Schema validation. Furthermore, XForms submissions with replace "all" behaviour will typically not result in true page replacements and therefore break the normal back button behaviour.
XRX application architecture
Because XForms makes it easy to edit complex XML data there are many advantages to using XForms with native XML databases that frequently leverage REST interfaces. The combination of three technologies (XForms on the client, REST interfaces and XQuery on the server) is collectively known as XRX application development. XRX is known for its simple architecture that uses XML both on the client and in the database and avoids the transformations to object or relational data structures. See "XRX:Simple, Elegant, Disruptive".
XForms for mobile devices
Benefits
XForms provides specific benefits when used on mobile devices:
User interfaces using XForms require fewer round trips with the server and are in that sense more self-contained than user interfaces using HTML 4 forms.
Capabilities of mobile devices vary greatly; consequently the amount of the work involved in generating different user interfaces for different devices is of particular concern in the mobile world. XForms has been designed from the ground up to allow forms to be described independently of the device, which reduces the amount of work required to target multiple devices.
XForms reduces the need for JavaScript, which is particularly interesting as JavaScript support varies greatly on mobile devices and cannot be widely relied upon. This also allows systems on which JavaScript is disabled for security concerns to continue to operate flawlessly.
Implementations
ODK
ODK is an open-source mobile data collection platform that uses a subset of W3C XForms 1.0 called ODK XForms. ODK provides ODK XForms processing libraries in Java (JavaRosa) and JavaScript (enketo-core).
Xfolite
Xfolite is a light-weight XForms client for the J2ME platform. It was originally created at Nokia Research Center, and it includes a DOM and XPath 1.0 implementation as well as an XForms engine that implements the XForms 1.1 specification almost completely. XFolite was released as beta software and should not be considered ready for production use as such. However, it does contain a mature XForms engine that has been designed to work with different UI implementations. XML Schemas and CSS are outside project scope, however. Xfolite is open source and licensed under the LGPL license, but is not being actively developed further.
See also
InfoPath Forms Services
FormFaces
References
XForms 1.1 - W3C Recommendation 20 October 2009.
External links
XForms Resources at W3C
The XForms Users Community Group
XForms 1.0 Frequently Asked Questions
XForms 1.1 was a W3C Recommendation on 20 October 2009
XForms 2.0 Working Draft
XForms 2.0: XPath Expressions Module
A quick introduction to XForms for HTML Authors by Steven Pemberton
XForms 1.1 Quick Reference
XForms Implementations
The Forms Working Group (historical interest)
XForms 1.0 (Third Edition) was a W3C Recommendation on 29 October 2007
XSLTForms
Smartsite XForms
World Wide Web Consortium standards
XML-based standards
User interface markup languages | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,816 |
lucycorry July 9, 2019
Anglo-French rocky road
For reasons too complicated to explain in detail here, I recently found myself teaching four groups of school children how to make rocky road. In French. Yes, I know. I'm not sure how I get myself into these situations but once the gate clanged shut, there was no getting out. (Literally – French primary schools are like fortresses.)
I took a deep breath and reminded myself that I'd been in similarily challenging teaching environments before (though I wasn't sure this French primary school would enjoy being compared with a medium-security New Zealand prison, so I kept that to myself). Luckily, I also had help: an ex-nurse and a former army officer whose CVs were packed with far more useful and impressive feats than mine.
I don't want to go telling tales out of school, but trust me when I say that all three of us needed to be on our A-game. Fortunately, French school kids are used to being told off. Unfortunately, they're like all other children (and adults) in the presence of chocolate. Suffice to say, it was an exhausting morning, much mitigated by a refreshing glass of cider at 11.30am in the staffroom afterwards.
Should you wish to recreate this experience yourself during the school holidays, invite 10-15 children to come and make the following recipe with you. Make sure you're in a classroom without aircon, preferably a few days before a record-breaking heatwave. For best results, have very rudimentary cooking equipment, at least two children who will be fighting at any one time, and eyes in the back of your head to stop them running with scissors and licking the bowl before you've finished mixing. Bon courage et bonne chance!
You can substitute other dried fruit or nuts for the cranberries and peanuts if you like.
400g chocolate
1 x tin sweetened condensed milk
200g plain sweet biscuits
200g marshmallows
150g dried cranberries
150g roasted peanuts
Line a tin or plastic container (measuring about 20x25cm) with plastic wrap or baking paper.
Put the butter, chocolate and sweetened condensed milk in a large pot. Set it over low heat and stir until melted. Remove from the heat and leave to cool.
Put the biscuits in a freezer bag and crush them gently until they are crumbs. It's good to leave some bigger pieces to make the rocky road crunchier.
Using scissors, cut the marshmallows in half or into thirds if they are large.
Put the crushed biscuits, marshmallows, cranberries and peanuts into the melted chocolate mixture. Stir well.
Pour into the prepared tin and smooth the top. Leave to set in the fridge for 1 hour. Cut into pieces and store in the fridge.
'Chemin Rocailleux'
Vous pouvez remplacer les cachuetes et les canneberges sechees avec des autres noix ou fruits secs.
100g beurre demi-sel
400g chocolat
1 x boite lait concentré sucré
200g biscuits du thé
200g guimauvres
150g canneberges séchées
150g arachides salées
Tapisser un moule ou un Tupperware de 20x25cm avec du film étirable ou du papier sulfurisé. Deposer le beurre, chocolat et lait concentré sucré dans un pot. Chauffer doucement pour les fondre, en agitant souvent. Laisser refroider 10 minutes.
Mettre les biscuits du thé dans un sac de congelation. Utiliser un rouler ou vos mains pour les éraser.
Couper les guimauvres en petit pièces avec les ciseaux.
Ajouter les biscuits écrasés, les guimauvres coupés, les canneberges séchées es et les arachides au chocolat. Melanger bien.
Verser dans le moule. Placer au frigo pendant 1-2 heures. Couper en petits pieces et garder au frigo. Bon appétit!
Need more school holiday baking ideas? Check out these ones.
I did a lot of cool food-related things in 2018. I wrote a book about burgers, I helped judge the second Outstanding Food Producer Awards and I ate in some of London's most celebrated restaurants. But the very best thing I did was join a group of volunteers teaching baking at one of New Zealand's largest prisons.
That might not sound very interesting in and of itself (though I can tell you, being behind the wire at a prison is a huge learning experience) until you realise that baking was a bit of a Trojan horse. What we were really trying to teach – along with a few tips and tricks about successfully making biscuits and cakes – was the redemptive power of kindness. The Prison Bake programme, which ran as a short pilot in August and then a three-week stint before Christmas, was the brainchild of Good Bitches Baking. This charity, set up by Marie Fitzgerald and Nic Murray in 2014, now has about 1600 volunteers baking for 135 different recipient organisations every week. Prison Bake is another way of reaching out to the community and spreading what Fitzgerald and Murray call 'moments of sweetness'.
You might take a dim view of prison rehab, preferring to think of jail being a place where they lock you and and lose the key. You might not think baking a cake is much of a help to someone having a tough time. But it's hard to argue with the feedback from the prisoners themselves. When asked what they'd learned during the pilot programme, one of them said he'd learned that he could be a kind person – and he didn't think that was possible. You don't have to be behind bars to have that kind of learning experience (but it's even more remarkable if you are).
PRISON BAKE BROWNIES
These brownies were part of the pre-Christmas Prison Bake programme. They're very simple to mix and make, and you can change it up by using different chocolate or fruit. I think dark chocolate chips and brandy-soaked prunes might be a good combo (though perhaps not quite so prison-friendly).
1 cup caster sugar
3/4 cup frozen raspberries
1/2 cup roughly chopped dark chocolate
Heat the oven to 180C. Line a brownie pan (about 20x30cm) with baking paper.
Set a large pot over medium heat. Add the butter and cocoa, stirring until it melts. Cook for a minute or two, then remove from the heat and stir in the sugar. Let it cool until it's no longer hot to the touch (about 10 minutes).
Add the eggs, one at a time, beating well after each one. Sift in the flour and baking powder. Fold together gently, then fold in the chocolate. Pour into the prepared pan and dot the raspberries on top (press them in lightly).
Bake for 20-25 minutes, until the brownies are set in the middle. Cool in the pan before slicing.
Of course, if going to prison isn't your thing there are plenty of other ways to support Good Bitches Baking. They've got a whole bunch of cool things you can buy to support their fundraising efforts, including the most beautiful cake sprinkles I've ever seen. If you wanted to be a very kind person you could buy some sprinkles, make these truffles and then give them away to a person in need of cheering up. (It's ok if that person is you – self-care takes many forms.)
GOOD BITCHES TRUFFLES
This is more-or-less a Julie Le Clerc recipe from issue 100 of Cuisine magazine (a deeply precious issue that sparks much joy).
250g dark chocolate, roughly chopped (I use Whittaker's 72% Dark Ghana)
125ml cream
3 Tbsp dark rum (or brandy, or whisky, or a liqueur of your choice)
1 packet Good Bitches Baking Kindness Sprinkles
Put the chocolate, butter, cream and rum into a heatproof bowl and put into a low oven – or over a saucepan of simmering water, or in a microwave – and melt, stirring occasionally. The oven method is really easy, as long as you don't forget it's there.
Let cool for a minute or two, then stir in the egg yolk until well mixed. Let cool for 10 minutes, then chill in the fridge until set (about an hour).
Roll teaspoonfuls of the mixture into balls – this is a sticky job, don't even think about answering the phone etc while you're doing it – then roll in the sprinkles. Store in the fridge, eat at room temperature. Makes about 22 truffles if you don't accidentally eat the mixture.
To learn more about Good Bitches Baking, visit www.gbb.org.nz
lucycorry December 15, 2017
A mea culpa (& a white chocolate tiramisu)
This is a story I may have told before, but bear with me. Once upon a time, when I worked at a regional newspaper, a very, very angry reader drove all the way out to the office with a plate of biscuits he'd made. This wasn't a gesture of generosity, but of rage. He'd made the biscuits to a recipe that was published in the newspaper and he was disappointed by the results. He complained that they were inedible and that we must have left the sugar out by accident. I apologised profusely and said I'd check the recipe with its author.
When I did, she was bemused. "No," she said, "there's no mistake. They're just not very sweet biscuits."
This is NOT a tiramisu – it's the raspberry and lemon posset that appears alongside it in the original publication. We ate the test tiramisu too fast to photograph it (it's that good!)
This was no comfort to the angry man, who was nearing apoplexy. After he calmed down a bit he revealed that he'd made the biscuits for the nurses who were looking after his ill wife in hospital. These nurses had then complained that they weren't very nice (I know!). So really, it wasn't about the biscuits at all. In the end we parted on good terms and the rest of the newsroom got some unexpected morning tea. He was right, the biscuits weren't that nice, but they were made to the exact recipe.
I'm bringing this up now because this week I made a mistake in a recipe printed in The Dominion Post, the Waikato Times and The Press. I left an instruction out and this has made some readers very cross. I picked it up quick enough for it to be amended online, but once a runaway horse has bolted the print stable it's very hard to get it back.
So, if you are looking at my recipe for Black Doris plum and white chocolate tiramisu and thinking, 'where does the melted chocolate go?', I'm sorry. The full recipe is below – with the missing instruction in bold. I wish I could say that there was a good reason for the error but the truth is, I'm only human. I will be more careful next time. Thank you to the people who have gotten in touch (even the ones who sent some rather cross emails) – I hope the mistake doesn't put you off making the tiramisu because it really is delectable.
BLACK DORIS PLUM AND WHITE CHOCOLATE TIRAMISU
Preparation time: 30 minutes (plus 6-12 hours' chilling time)
Cooking time: nil
A classic tiramisu is a heady confection of coffee and dark chocolate – delicious, but a recipe for a terrible night's sleep. This fruity version is slightly lighter but no less delectable. To make it alcohol-free, use extra syrup from the plums as the liquid. Look for the Italian sponge fingers, also known as savoiardi, in the "international foods" section of the supermarket, or try a Mediterranean foods store.
1 x 825g tin black doris plums in syrup
200g white chocolate
4 tablespoons caster sugar
Finely grated zest of 1 lemon
250g mascarpone
5 tablespoons limoncello
16-20 Italian sponge fingers
Set a sieve over a bowl. Pour in the plums and leave to drain for a few minutes. Reserve the syrup. Remove all the stones from the plums. Mash them slightly with a fork and set aside.
Break up 150g of the chocolate and put in a small bowl that will fit snugly into the top of a small saucepan. Put about three centimetres of water in the saucepan and set over medium heat. Don't let the water boil. As soon as the chocolate has melted, remove it from the heat (being careful not to get any water in the chocolate). Set aside.
Put the egg yolks, caster sugar and lemon zest in a bowl. Whip until pale, thick and mousse-y (using electric beaters is easiest). Fold in the mascarpone and the melted white chocolate.
Wash and dry the beaters, ensuring there is no egg yolk mixture left on them. Put the egg whites in a separate bowl and whip until they form stiff peaks. Fold them very gently into the egg yolk mixture.
Pour the limoncello and five tablespoons of the reserved plum syrup into a shallow dish. Dip about eight to 10 sponge fingers into this liquid, then fit them into the bottom of a glass bowl (the sort that your mum makes trifle in).
Pour half the egg and mascarpone mixture on top, followed by half of the plums. Dip the remaining sponge fingers into the liquid and arrange neatly on top of the plums. Spread the remainder of the plums on top, followed by the remaining egg mixture.
Roughly chop the remaining 50g white chocolate and sprinkle over the top. Cover tightly and chill for at least six hours (preferably overnight) before serving.
Raspberry ripple tart
As much as I love a good kitchen-based project, there some things that I would rarely, if ever, bother to make myself. I'd put pastry pretty high on that list, especially when you can buy such fantastic stuff ready-made by companies like Auckland-based French bakery Paneton*. I've loved their products for years and the buttery, super-flaky puff pastry has saved me on many a desperate dinner occasion. In exciting news for chocolate lovers, their chocolate pastry is brilliant too.
My go-to showstopper dessert for a big crowd of people is the Pecan Praline Tart in Dean Brettschneider's Pie book – essentially, chocolate pastry filled with praline-studded milk chocolate ganache, topped with dark chocolate ganache and a scattering of praline crumbs. But on a long run recently (which is when I do my best thinking about food), I started thinking about something lighter that would have more of a contrast with the pastry. Here's the result…
I used the Paneton brand discussed above for this tart – it's very dark, rich and buttery – but if you want to make your own I'd recommend the Dean Brettschneider recipe above. It will be delicious either way. This serves 8-10 depending on greed.
For the raspberry curd:
2 cups frozen raspberries
For the tart:
About 300g chocolate pastry
Extra raspberries, for garnishing
Start by preparing the tart shell. Heat the oven to 180C. Grease and line a 30 x 10cm tart tin. Ease the pastry into the tin, leaving plenty of overhang. Chill for 20 minutes.
Bake blind for 10 minutes, then remove the weights and paper and bake for another 10 minutes until the pastry is dry to touch and crisp. Remove to a rack to cool. Trim any overhang (the resulting pieces are a good cook's perk, though you will struggle to get any if there are little helpers around) and set aside.
To make the raspberry curd, put the raspberries and the water in a small saucepan and set over medium heat. Cook for three to five minutes, until the fruit collapses, then remove from the heat. Push the raspberries through a fine sieve, discarding any seeds. This should make about 120ml (just under half a cup) of puree. Squeeze in enough lemon juice to make it up to 150ml. Set aside.
Whisk the egg yolks and sugar together, then pour into the saucepan you used earlier. Add the butter and raspberry-lemon juice. Set over medium heat and bring to a simmer, stirring constantly (this will take about five minutes). When the mixture is bubbling, remove from the heat. Stir well and transfer to a bowl to cool completely. Cover and refrigerate until ready to assemble the tart.
About an hour before serving, whip the cream to soft peaks. Fold in the curd to create a ripple effect, then pour this mixture into the pastry shell. Carefully put the tart in the fridge until ready to serve. Decorate with more raspberries before serving. A shower of grated chocolate – white or dark – wouldn't go amiss on top, either. This serves 8-10 depending on greed.
It's highly likely that you'll end up with some leftover pastry when making this tart. If you can stop yourself from eating it raw, I recommend turning it into easy ice cream sandwiches. All you need to do is cut the pastry into rounds, bake for about 10 mins at 180C and let cool. While you're waiting, cut the ice cream into the same shapes and freeze. Sandwich the biscuits together with ice cream, dust with icing sugar and serve. This makes about 10 tiny ice cream sandwiches, which is just enough to leave them all wanting more.
*Please note, this is NOT a 'sponsored' post. In other words, I have not received any payment to say nice things about Paneton. In the interests of full disclosure, Paneton did send me a packet of their chocolate pastry to try recently. I was so impressed by it that I've since bought it twice more with my own hard-earned money (and I'll definitely buy it again). I don't think you can get a better recommendation than that! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,888 |
ember-history
=============
Undo / Redo for Emberjs
Inspired by https://github.com/pangratz/ember-memento, however my implementation is for global History and not for individual objects (though it works with just one object as well)
Usage
-----
``` javascript
App = Ember.Application.create({});
obj = Ember.Object.create(Ember.History, {
_trackProperties: 'width height list'.w(),
width: '100px',
height: '50px',
list: ['item3','carrot','car']
});
obj2 = Ember.Object.create(Ember.History, {
_trackProperties: 'name surname'.w(),
name: 'Ignas',
surname: 'Bernotas',
});
obj2.set('name','Matthew');
obj.set('height','100px');
obj.set('list', ['item1']);
obj2.set('surname', 'Parry');
History.undo(); // surname is now back to Bernotas
History.undo(); // list is now back to ['item3','carrot','car']
History.undo(); // height is now 50px
History.undo(); // name is now Ignas
History.redo(); // name is now Matthew
History.redo(); // height 100px
History.redo(); // list is ['item1'] again
History.redo(); // surname is Parry again
```
You can also disable the history for a while
``` javascript
History.disable();
obj.set('height','300px'); //This won't push a new history state
History.enable();
obj.set('height','300px'); //This will
```
And you can also clear the states
``` javascript
History.clear()
```
License
-------
This is licensed under MIT. See license.txt | {
"redpajama_set_name": "RedPajamaGithub"
} | 856 |
The Operational Area Council coordinates, reviews, and recommends for approval all emergency and disaster response policies, procedures, plans, and other influencing factors or events that would affect the Stanislaus Operational Area. The OAC provides technical review of all disaster plans by any agency or jurisdiction in the County for approval as to form and compliance with Incident Command System (ICS), Standardized Emergency Management System (SEMS), National Incident Management System (NIMS), and Homeland Security Presidential Directives (HSPD) 5 and 8.
The Operational Area Council will also consist of participants who agree to partner with and lend their expertise to the OAC. These participants will be comprised of any government, public, or private individual, or organization that has an interest in Emergency Management.
The Operational Area Council meets quarterly, or upon the call of the Assistant Director of Emergency Services, and/or as often as necessary to transact business and fulfill their duties. Meetings are generally held at the Harvest Hall, 3800 Cornucopia Way. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,339 |
José Meseguer is a Spanish computer scientist, and professor at the University of Illinois at Urbana–Champaign. He leads the university's Formal Methods and Declarative Languages Laboratory.
Career
José Meseguer obtained his PhD in mathematics in 1975 with a thesis titled Primitive recursion in model categories under Michael Pfender at the University of Zaragoza, after which he did post-doctoral work at the University of Santiago de Compostela and the University of California at Berkeley. In 1980 he joined the Computer Science Laboratory at SRI International, eventually becoming a Principal Scientist and Head of the Logic and Declarative Languages Group. He joined the University of Illinois in 2001 and currently is Professor of Computer Science, where he leads their Formal Methods and Declarative Languages Laboratory.
He has worked particularly on the design and implementation of declarative languages, including OBJ and Maude, as well as rewriting logic.
He was awarded the 2019 Formal Methods Europe Fellowship. The award citation reads,
He was inducted as an ACM Fellow in 2020 "for the development of logical methods for design and verification of computational systems".
Selected research
Clavel, Manuel, et al. All about Maude — a high-performance logical framework: how to specify, program and verify systems in rewriting logic. Springer-Verlag, 2007.
Goguen, Joseph A., et al. "Introducing obj." Software Engineering with OBJ. Springer, Boston, MA, 2000. 3–167.
Meseguer, José. "Conditional rewriting logic as a unified model of concurrency." Theoretical computer science 96.1 (1992): 73–155.
Goguen, Joseph A., and José Meseguer. "Security policies and security models." 1982 IEEE Symposium on Security and Privacy. IEEE, 1982.
References
Living people
University of Illinois Urbana-Champaign faculty
Spanish computer scientists
University of Zaragoza alumni
SRI International people
1950 births | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,758 |
package com.airbnb.android.airmapview;
import android.graphics.Bitmap;
import com.google.android.gms.maps.model.BitmapDescriptor;
import com.google.android.gms.maps.model.BitmapDescriptorFactory;
import com.google.android.gms.maps.model.LatLng;
import com.google.android.gms.maps.model.Marker;
import com.google.android.gms.maps.model.MarkerOptions;
/**
* Wrapper around {@link MarkerOptions}. Keeps record of data needed to display map markers, as
* well as an object T associated with the marker.
*/
public class AirMapMarker<T> {
private final T object;
private final long id;
private final MarkerOptions markerOptions;
private Marker marker;
private AirMapMarker(T object, long id, LatLng position, String title, String snippet,
float anchorU, float anchorV, boolean draggable, boolean visible, boolean flat,
float rotation, float infoWindowAnchorU, float infoWindowAnchorV, float alpha,
BitmapDescriptor icon) {
this.object = object;
this.id = id;
this.markerOptions = new MarkerOptions()
.title(title)
.position(position)
.icon(icon)
.snippet(snippet)
.anchor(anchorU, anchorV)
.draggable(draggable)
.visible(visible)
.flat(flat)
.rotation(rotation)
.infoWindowAnchor(infoWindowAnchorU, infoWindowAnchorV)
.alpha(alpha);
}
public T object() {
return object;
}
public long getId() {
return id;
}
public LatLng getLatLng() {
return markerOptions.getPosition();
}
public String getTitle() {
return markerOptions.getTitle();
}
public String getSnippet() {
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Universal lightweight multiplatform proxy server with HTTP HTTPS SOCKS v4 SOCKS v4a SOCKS v5 FTP POP3 UDP and TCP portmapping access control bandwith control traffic limitation and accounting based on username client IP target IP daytime day of week etc. ODBC logging support to any dabase or to file of any format for compatibility with log analizers. Proxy chaining support for any type of parent proxy. Randomization for chain building. Many additional features. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,769 |
\section{\bf Introduction}\label{intro}
We consider a smooth connected semi-Riemannian manifold $(M^n, g)$ with $n\geq 3$ (this condition is assumed throughout the paper) equipped with the semi-Riemannian metric $g$ with signature $(p, n-p)$ and the Levi-Civita connection $\nabla$. We note that $M$ is Riemannian if $p = 0$ or $n$ and $M$ is Lorentzian if $p = 1$ or $n-1$. Let us consider $R$, $\mathcal R$, $S$, $\mathcal S$ and $\kappa $ respectively be the Riemann-Christoffel curvature tensor of type $(0,4)$, the Riemann-Christoffel curvature tensor of type $(1,3)$, the Ricci tensor of type $(0,2)$, the Ricci tensor of type $(1,1)$ and the scalar curvature of $M$.\\
\indent A spacetime is a connected 4-dimensional Lorentzian manifold and it presents a pure radiation if its energy momentum tensor $T$ is of the form
\be\label{prc}
T = \rho \eta\otimes\eta,
\ee
where $\eta$ is a null vector and $\rho$ is the radiation density. Such a spacetime describes some kinds of field that propagates at the speed of light and it could represent a null electromagnetic field. It could also represent an incoherent beam of photons or some kinds of massless neutrino fields. On the other hand a spacetime presents a perfect fluid spacetime if its energy-momentum tensor is of the form
\be\label{pfc}
T = (\rho + p)\eta\otimes\eta + p g,
\ee
where $\rho$ is the energy density, $p$ is the isotropic pressure and $\eta$ is the four-velocity of the fluid. Thus a pure radiation source can be considered as a limiting case of a pressureless perfect fluid with null four-velocity. For this reason a pure radiation source is sometimes referred as ``null dust''.\\
\indent In \cite{Wils89} Wils investigated homogeneous and conformally Ricci flat solutions of Einstein's field equations for pure radiation case. Later in 1997, Ludwig and Edgar \cite{LE97} obtained exhausted class of conformally Ricci flat pure radiation solutions of Einstein's field equations. The line element of conformally Ricci flat pure radiation spacetime in $(x^1; x^2; x^3; x^4) = (u; r; x; y)$-coordinates with $x>0$ is given by \cite{LE97}
\beb
ds^2 = \left(-2 x W^{\circ} - 8 \tilde P^2 \frac{r^2}{x^2}\right)du^2 + 2 du dr - \frac{4r}{x} du dx - \frac{1}{8 \tilde P^2}(dx^2 + dy^2),
\eeb
where $\tilde P$ is an arbitrary non-zero constant and $W^{\circ}$ is an arbitrary function of the three non-radial coordinates $u, x$ and $y$.
For simplicity of symbols we write the metric as
\be\label{prm}
ds^2 = \left(x w - p^2 \frac{r^2}{x^2}\right)du^2 + 2 du dr - \frac{4r}{x} du dx - \frac{1}{p^2}(dx^2 + dy^2),
\ee
where $w = w(u,x,y) = -2 W^{\circ}$ and $p = 2\sqrt{2} \tilde P = const.$\\
\indent Now in $(u; r; x; y)$-coordinates with $x > 0$, we consider the following metric
\be\label{gprm}
ds^2 = \left(x w + a \frac{r^2}{x^2}\right) du^2 + 2 du dr + \frac{2 b r}{x} du dx + f (dx^2 + dy^2),
\ee
where $a_1$, $a_2$ are arbitrary non-zero constants and $f$ is a nowhere vanishing function of $x$ and $y$. We note that if $a = -p^2$, $b=-2$ and $f \equiv - \frac{1}{p^2}$, then the metric \eqref{gprm} reduces to pure radiation metric \eqref{prm}. Hence we call the metric \eqref{gprm} as \textit{pure radiation type metric}. Again if $w(u,x,y) = \frac{-2 h(u,x,y)}{x}$, $a = b = 0$ and $f(x,y) = -\frac{1}{2} F(x,y)$, then the metric \eqref{gprm} reduces to generalized pp-wave metric (\cite{RS84}, \cite{SBK17}, \cite{ste03}), given by,
\be\label{gppwm}
ds^2 = -2 h(u,x,y) du^2 + 2 du dr -\frac{1}{2} F (dx^2 + dy^2).
\ee
On the other hand if $w(u,x,y) = \frac{h(u,x,y)}{x}$, $a = b = 0$ and $f \equiv 1$, then the metric \eqref{gprm} reduces to pp-wave metric (\cite{Brin25}, \cite{ste03}), given by,
\be\label{ppwm}
ds^2 = h(u,x,y) du^2 + 2 du dr + (dx^2 + dy^2).
\ee
\indent The physical properties of the pure radiation metric \eqref{prm} are well known and we refer the reader to see \cite{LE97} and \cite{Wils89}. In the literature of differential geometry there are many curvature restricted geometric structures on a semi-Riemannian manifold, such as locally symmetric manifold \cite{Cart26}, semisymmetric manifold (\cite{Cart26}, \cite{Szab82}, \cite{Szab84}, \cite{Szab85}), recurrent manifold (\cite{Ruse46}, \cite{Ruse49a}, \cite{Ruse49b}, \cite{Walk50}), pseudosymmetric manifold (\cite{AD83}, \cite{Desz92} and also references therein) etc. The main object of the present paper is to investigate such kinds of geometric structures admitted by the pure radiation metric \eqref{prm}.
It is noteworthy to mention that the metric \eqref{prm} is neither locally symmetric nor conformally symmetric but semisymmetric and hence Ricci semisymmetric, conformally semisymmetric and projective semisymmetric. It is also shown that the pure radiation metric \eqref{prm} is Ricci simple, weakly Ricci symmetric, weakly cyclic Ricci symmetric, $R$-space by Venzi and its curvature 2-forms, Ricci 1-forms and conformal curvature 2-forms are recurrent. Again the spacetime satisfies the semisymmetric type conditions $C\cdot R =0$, $C\cdot C =0$, $C\cdot S =0$, $Q(S, R) =0$, $Q(S, C) = 0$, $P\cdot R = P\cdot C =0$ and also satisfies the pseudosymmetric type conditions $P\cdot P = -\frac{1}{3}Q(S, P)$. It is shown that its energy momentum tensor $T$ is semisymmetric and it is Codazzi type (resp., cyclic parallel or covariantly constant) if $w_{33}+w_{44}$ is independent of $x$ and $y$ (resp., constant or zero), where $w_{ij}$ denotes the covariant derivative with respect to $x^i$ and $x^j$.\\
\indent The paper is organized as follows. Section \ref{preli} deals with the preliminaries. In section \ref{com} we compute the components of various tensors of the metric \eqref{prm} and we state the main results on the geometric structures admitted by pure radiation metric \eqref{prm}. Section \ref{gen} deals with the curvature properties of pure radiation type metric \eqref{gprm}. It is shown that such metric is $Ein(3)$ and 3-quasi-Einstein. We also obtain the conditions for which the metric is 2-quasi-Einstein, Ricci generalized pseudosymmetric and manifold of vanishing scalar curvature. Finally, we made a comparison (similarities and dissimilarities) between pure radiation metric and pp-wave metric. It is interesting to mention that both are semisymmetric and weakly Ricci symmetric, but generalized pp-wave metric is Ricci recurrent whereas pure radiation metric is not so.
\section{\bf Curvature Restricted Geometric Structures}\label{preli}
It is wellknown that a curvature restricted geometric structure is a geometric structure on a semi-Riemannian manifold $M$ obtained by imposing a restriction on its curvature tensors by means of covariant derivatives of first order or higher orders. We will now explain some useful notations and definitions of various curvature restricted geometric structures.\\
\indent For two symmetric $(0,2)$-tensors $A$ and $E$, their Kulkarni-Nomizu product $A\wedge E$ is defined as (see e.g. \cite{DGHS11}, \cite{Glog02}):
\begin{eqnarray*}
(A\wedge E)(X_1,X_2,X_3,X_4) &=& A(X_1,X_4)E(X_2,X_3) + A(X_2,X_3)E(X_1,X_4)\\
&-& A(X_1,X_3)E(X_2,X_4) - A(X_2,X_4)E(X_1,X_3),
\end{eqnarray*}
where $X_1,X_2,X_3,X_4 \in \chi(M)$, the Lie algebra of all smooth vector fields on $M$. Throughout the paper we will consider $X, Y, X_1, X_2, \cdots \in \chi(M)$.\\
\indent Again for a symmetric $(0, 2)$-tensor $A$, we get an endomorphism $\mathcal A$ defined by $g(\mathcal AX,Y) = A(X,Y)$. Then its $k$-th level tensor $A^k$ of type $(0,2)$ is given by
$$A^k(X,Y) = A(\mathcal A^{k-1}X,Y),$$
where $\mathcal A^{k-1}$ is the endomorphism corresponding to $A^{k-1}$.\\
\indent In terms of Kulkarni-Nomizu product the conformal curvature tensor $C$, concircular curvature tensor $W$, conharmonic curvature tensor $K$ (\cite{Ishi57}, \cite{YK89}) and the Gaussian curvature tensor $\mathfrak G$ can respectively be expressed as
\beb
C &=& R-\frac{1}{n-2}(g\wedge S) + \frac{r}{2(n-2)(n-1)}(g\wedge g),\\
W &=& R-\frac{r}{2n(n-1)}(g\wedge g),\\
K &=& R-\frac{1}{n-2}(g\wedge S),\\
\mathfrak G &=& \frac{1}{2}(g\wedge g).
\eeb
Again the projective curvature tensor $P$ is given by
$$
P(X_1, X_2, X_3, X_4) = R(X_1, X_2, X_3, X_4) - \frac{1}{n-1}[g(X_1, X_4)S(X_2, X_3)-g(X_2, X_4)S(X_1, X_3)].
$$
\indent For a symmetric $(0,2)$-tensor $A$, $(0,4)$-tensor $D$ and a $(0,k)$-tensor $H$, $k\geq 1$, one can define two $(0,k+2)$-tensors $D\cdot H$ and $Q(A,H)$ respectively as follows (see \cite{DG02}, \cite{DGHS98}, \cite{DH03}, \cite{SDHJK15}, \cite{SK14} and also references therein):
$$D\cdot H(X_1,X_2,\cdots,X_k,X,Y) = -H(\mathcal D(X,Y)X_1,X_2,\cdots,X_k) - \cdots - H(X_1,X_2,\cdots,\mathcal D(X,Y)X_k)$$
and
\beb
Q(A,H)(X_1,X_2, \ldots ,X_k,X,Y) &=& A(X, X_1) H(Y,X_2,\cdots,X_k) + \cdots + A(X, X_k) H(X_1,X_2,\cdots,Y)\\
&-& A(Y, X_1) H(X,X_2,\cdots,X_k) - \cdots - A(Y, X_k) H(X_1,X_2,\cdots,X),
\eeb
where $\mathcal D$ is the corresponding $(1,3)$-tensor of $D$, given by $D(X_1,X_2,X_3,X_4) = g(\mathcal D(X_1,X_2)X_3, X_4).$
\begin{defi}
A semi-Riemannian manifold $M$ is said to be $H$-symmetric (\cite{Cart26}, \cite{Cart46}) if $\nabla H =0$. In particular if $H = R$ (resp., $S$ and $C$), then the manifold is called locally symmetric (resp., Ricci symmetric and conformally symmetric).
\end{defi}
\begin{defi}
A symmetric $(0,2)$-tensor $E$ on $M$ is said to be cyclic parallel (resp, Codazzi type) (see, \cite{DHJKS14}, \cite{Gray78} and references therein) if
\[(\nabla_{X_1} E)(X_2, X_3) = (\nabla_{X_2} E)(X_1, X_3)\]
\[\big(\mbox{resp.,} \ (\nabla_{X_1} E)(X_2, X_3) + (\nabla_{X_2} E)(X_3, X_1) + (\nabla_{X_3} E)(X_1, X_2) = 0\big).\]
\end{defi}
\begin{defi}$($\cite{AD83}, \cite{Cart46}, \cite{Desz92}, \cite{SK14}, \cite{SKppsn}, \cite{SKppsnw}, \cite{Szab82}$)$
A semi-Riemannian manifold $M$ is said to be $H$-semisymmetric type if $D\cdot H = 0$ and it is said to be $H$-pseudosymmetric type if $\left(\sum\limits_{i=1}^k c_i D_i\right)\cdot H = 0$ for some scalars $c_i$'s, where $D$ and each $D_i$, $i=1,\ldots, k$, $(k\ge 2)$, are (0,4) curvature tensors.
\end{defi}
\begin{defi}
A semi-Riemannian manifold $M$ is said to be Einstein if its Ricci tensor is a scalar multiple of the metric tensor $g$. Again $M$ is called quasi-Einstein (resp., 2-quasi-Einstein and 3-quasi-Einstein) if at each point of $M$, rank$(S - \alpha g)\le 1$ (resp., $\le 2$ and $\le 3$) for a scalar $\alpha$. In particular, if $\alpha = 0$, then a quasi-Einstein manifold is called Ricci simple.
\end{defi}
\indent We note that Som-Raychaudhuri metric \cite{SK16srs} and Robinson-Trautman metric \cite{SAArt} are 2-quasi-Einstein whereas G\"{o}del metric \cite{DHJKS14} is Ricci simple.
\begin{defi} (\cite{Bess87}, \cite{SKgrt})
A semi-Riemannian manifold $M$ is said to be $Ein(2)$, $Ein(3)$ and $Ein(4)$ respectively if
$$S^2 + \lambda_1 S + \lambda_2 g = 0,$$
$$S^3 + \lambda_3 S^2 + \lambda_4 S + \lambda_5 g = 0 \ \mbox{and}$$
$$S^4 + \lambda_6 S^3 + \lambda_7 S^2 + \lambda_8 S + \lambda_9 g = 0$$
holds for some scalars $\lambda_i, \, 1\le i \le 9$.
\end{defi}
\begin{defi}\label{def2.6}
Let $D$ be a $(0,4)$-tensor and $E$ be a symmetric $(0, 2)$-tensor on $M$. Then $E$ is said to be $D$-compatible (\cite{DGJPZ13}, \cite{MM12b}, \cite{MM13}) if
\[
D(\mathcal E X_1, X,X_2,X_3) + D(\mathcal E X_2, X,X_3,X_1) + D(\mathcal E X_3, X,X_1,X_2) = 0
\]
holds, where $\mathcal E$ is the endomorphism corresponding to $E$ defined as $g(\mathcal E X_1, X_2) = E(X_1, X_2)$. Again an 1-form $\Pi$ is said to be $D$-compatible if $\Pi\otimes \Pi$ is $D$-compatible.
\end{defi}
\indent Generalizing the concept of recurrent manifold (\cite{Ruse46}, \cite{Ruse49a}, \cite{Ruse49b}, \cite{Walk50}), recently Shaikh et al. \cite{SRK16} introduced the notion of super generalized recurrent manifold along with its characterization and existence by proper example.
\begin{defi}
A semi-Riemannian manifold $M$ is said to be super generalized recurrent manifold (\cite{SK14}, \cite{SKA16}, \cite{SRK16}) if
$$
\nabla R = \Pi \otimes R + \Omega \otimes (S\wedge S) + \Theta \otimes (g\wedge S) + \omega \otimes (g\wedge g)
$$
holds on $\{x\in M: R \neq 0 \mbox{ and any one of } S\wedge S, g\wedge S \mbox{ is non-zero at $x$}\}$ for some 1-forms $\Pi$, $\Omega$, $\Theta$ and $\omega$, called the associated 1-forms. Especially, if $\Omega = \Theta = \omega = 0$ (resp., $\Theta = \omega = 0$ and $\Omega = \omega = 0$), then the manifold is called recurrent (\cite{Ruse46}, \cite{Ruse49a}, \cite{Ruse49b}, \cite{Walk50}) (resp., weakly generalized recurrent (\cite{SAR13}, \cite{SR11}) and hyper generalized recurrent (\cite{SP10}, \cite{SRK15})) manifold.
\end{defi}
\indent Again as a generalization of locally symmetric manifold and recurrent manifold, Tam$\acute{\mbox{a}}$ssy and Binh \cite{TB89} introduced the notion of weakly symmetric manifolds.
\begin{defi}
Let $D$ be a (0, 4)-tensor on a semi-Riemannian manifold $M$. Then $M$ is said to be weakly $D$-symmetric manifold (\cite{TB89}, \cite{SK12}) if
\beb
&&\nabla_X D(X_1, X_2, X_3, X_4) = \Pi(X) D(X_1, X_2, X_3, X_4) + \Omega(X_1) D(X_1, X_2, X_3, X_4)\\
&& \hspace{2cm} + \overline \Omega(X_2) D(X_1, X, X_3, X_4) + \Theta(X_3) D(X_1, X_2, X, X_4) + \overline \Theta(X_4) D(X_1, X_2, X_3, X)
\eeb
holds $\forall~ X, X_i \in \chi(M)$ $(i =1,2,3,4)$ and some 1-forms $\Pi, \Omega, \overline \Omega, \Theta$ and $\overline \Theta$ on $\{x\in M: R_x \neq 0\}$. Such a manifold is called as weakly $D$-symmetric manifold with solution $(\Pi, \Omega, \overline \Omega, \Theta, \overline \Theta)$. In particular, if the solution is of the form $(2\Pi, \Pi, \Pi, \Pi, \Pi)$, then the manifold is called Chaki $D$-pseudosymmetric manifold \cite{Chak87}. Again if the solution is of the form $(\Pi, 0, 0, 0, 0)$ then the manifold is called $D$-recurrent manifold (\cite{Ruse46}, \cite{Ruse49a}, \cite{Ruse49b}, \cite{Walk50}).
\end{defi}
\begin{defi}
Let $Z$ be a (0, 2)-tensor on a semi-Riemannian manifold $M$. Then $M$ is said to be weakly $Z$-symmetric (\cite{TB93}, \cite{SK12}) if
\beb
(\nabla_X Z)(X_1,X_2)=\Pi(X)\, Z(X_1,X_2) + \Omega(X_1)\, Z(X,X_2) + \Theta(X_2)\, Z(X_1,X)
\eeb
holds $\forall~ X, X_1, X_2 \in \chi(M)$ and some 1-forms $\Pi, \Omega$ and $\Theta$ on $U_{Z} = \{x\in M : Z \neq 0 \ \mbox{ at } x\}$. Such a manifold is called as weakly $Z$-symmetric manifold with solution $(\Pi, \Omega, \Theta)$. Especially, if the solution is of the form $(2\Pi, \Pi, \Pi)$ then the manifold is called Chaki pseudo $Z$-symmetric manifold \cite{Chak88}. Again if the solution is of the form $(\Pi$, $0$, $0)$ then the manifold is called $Z$-recurrent \cite{Patt52}.
\end{defi}
\indent For details about the defining condition of weak symmetry and the interrelation between weak symmetry and Deszcz psudosymmetry, we refer the reader to see \cite{SDHJK15} and also references therein.
\begin{defi}
A Riemannian manifold $M$ is said to be weakly cyclic Ricci symmetric \cite{SJ06} if its Ricci tensor satisfies the condition
\beb
&&(\nabla_X S)(X_1,X_2) + (\nabla_{X_1} S)(X,X_2) + (\nabla_{X_2} S)(X_1,X)\\
&&=\Pi(X)\, S(X_1,X_2) + \Omega(X_1)\, S(X,X_2) + \Theta(X_2)\, S(X_1,X),
\eeb
for three 1-forms $\Pi$, $\Omega$ and $\Theta$ on $M$. Such a manifold is called weakly cyclic Ricci symmetric manifold with solution $(\Pi, \Omega, \Theta)$.
\end{defi}
\indent It is noteworthy to mention that the solution of weakly cyclic Ricci symmetric structure is not always unique.
\begin{defi}
Let $D$ be a $(0,4)$ tensor and $Z$ be a $(0,2)$-tensor on $M$. Then the corresponding curvature 2-forms $\Omega_{(D)l}^m$ (\cite{Bess87}, \cite{LR89}) are said to be recurrent if and only if (\cite{MS12a}, \cite{MS13a}, \cite{MS14})
\beb\label{man}
&&(\nabla_{X_1} D)(X_2,X_3,X,Y)+(\nabla_{X_2} D)(X_3,X_1,X,Y)+(\nabla_{X_3} D)(X_1,X_2,X,Y) =\\
&&\hspace{1in} \Pi(X_1) D(X_2,X_3,X,Y) + \Pi(X_2) D(X_3,X_1,X,Y)+ \Pi(X_3) D(X_1,X_2,X,Y),
\eeb
and the 1-forms $\Lambda_{(Z)l}$ \cite{SKP03} are said to be recurrent if and only if
$$(\nabla_{X_1} Z)(X_2,X) - (\nabla_{X_2} Z)(X_1,X) = \Pi(X_1) Z(X_2,X) - \Pi(X_2) Z(X_1,X)$$
for an 1-form $\Pi$.
\end{defi}
\begin{defi}$($\cite{Prav95}, \cite{SKppsn}, \cite{Venz85}$)$
Let $\mathcal L(M)$ be the vector space formed by all 1-forms $\Theta$ on $M$ satisfying
$$\Theta(X_1)D(X_2,X_3,X_4,X_5)+\Theta(X_2)D(X_3,X_1,X_4,X_5)+\Theta(X_3)D(X_1,X_2,X_4,X_5) = 0,$$
where $D$ is a $(0,4)$-tensor. Then $M$ is said to be a $D$-space by Venzi if $dim [\mathcal L(M)] \ge 1$.
\end{defi}
From definition of recurrency of curvature 2-forms $\Omega_{(R)l}^m$ and second Bianchi identity it is clear that on a semi-Riemannian manifold $\Omega_{(R)l}^m$ are recurrent if and only if it is a $R$ space by Venzi.
\section{\bf Curvature properties of pure radiation metric}\label{com}
The metric tensor of pure radiation metric \eqref{prm} is given by
\beb
g = \left(\begin{array}{cccc}
\left(x w - p^2 \frac{r^2}{x^2}\right) & 1 & \frac{-2r}{x} & 0\\
1 & 0 & 0 & 0\\
\frac{-2r}{x} & 0 & - \frac{1}{p^2} & 0\\
0 & 0 & 0 & - \frac{1}{p^2}\\
\end{array}\right).
\eeb
Then the non-zero components (upto symmetry) of its Riemann-Christoffel curvature tensor $R$, Ricci tensor $S$, scalar curvature $\kappa$, conformal curvature tensor $C$ and projective curvature tensor $P$ are given by
$$R_{1313}=-\frac{w_{33} x}{2}, \ \ R_{1314}=-\frac{w_{34} x}{2}, \ \ R_{1414}=-\frac{w_{44} x}{2},$$
$$S_{11}=-\frac{1}{2} p^2 \left(w_{33}+w_{44}\right) x, \ \ \ \kappa = 0,$$
$$-C_{1313}= C_{1414}=\frac{1}{4} \left(w_{33}-w_{44}\right) x, \ \ C_{1314}=-\frac{w_{34} x}{2},$$
$$P_{1211}=-\frac{1}{6} p^2 \left(w_{33}+w_{44}\right) x, \ \ P_{1311}=\frac{1}{3} p^2 r \left(w_{33}+w_{44}\right),$$
$$P_{1313}=-\frac{1}{6} \left(2 w_{33}-w_{44}\right) x, \ \ -P_{1314}= P_{1341}= -P_{1413}= P_{1431}=\frac{w_{34} x}{2},$$
$$P_{1331}=\frac{w_{33} x}{2}, \ \ P_{1414}=\frac{1}{6} \left(w_{33}-2 w_{44}\right) x, \ \ P_{1441}=\frac{w_{44} x}{2}.$$
Then from above it is easy to check that $R\cdot R = Q(S, R) = R\cdot C = Q(S, C) = 0$.\\
Now the non-zero components (upto symmetry) of $\nabla R$, $\nabla S$, $\nabla C$ are given by
$$R_{1213,1}=\frac{p^2 w_{33}}{2}, \ \ R_{1214,1}=\frac{p^2 w_{34}}{2}, \ \ R_{1313,1}=-\frac{x^2 w_{133}+2 p^2 r w_{33}}{2 x}, \ \ R_{1414,1}=\frac{2 p^2 r w_{44}-x^2 w_{144}}{2 x},$$
$$R_{1313,3}=\frac{1}{2} \left(w_{33}-w_{333} x\right), \ \ R_{1313,4}=-\frac{w_{334} x}{2}, \ \ R_{1314,1}=-\frac{w_{134} x}{2}, \ \ R_{1414,3}=\frac{1}{2} \left(w_{44}-w_{344} x\right),$$
$$R_{1314,3}=\frac{1}{2} \left(w_{34}-w_{334} x\right), \ \ R_{1314,4}=-\frac{w_{344} x}{2}, \ \ R_{1334,1}=\frac{w_{34}}{2}, \ \ R_{1414,4}=-\frac{w_{444} x}{2}, \ \ R_{1434,1}=\frac{w_{44}}{2};$$
$$S_{11,1}=\frac{p^2 \left(-x^2 w_{144}-x^2 w_{133}+2 p^2 r w_{33}+2 p^2 r w_{44}\right)}{2 x},$$
$$S_{11,3}=\frac{1}{2} p^2 \left(-w_{333} x-w_{344} x+w_{33}+w_{44}\right),$$
$$S_{11,4}=-\frac{1}{2} p^2 \left(w_{334}+w_{444}\right) x, \ \ S_{13,1}=\frac{1}{2} p^2 \left(w_{33}+w_{44}\right);$$
$$C_{1213,1}=\frac{1}{4} p^2 \left(w_{33}-w_{44}\right), \ \ C_{1214,1}=\frac{p^2 w_{34}}{2},$$
$$C_{1313,1}=-\frac{-x^2 w_{144}+x^2 w_{133}+2 p^2 r w_{33}-2 p^2 r w_{44}}{4 x},$$
$$C_{1313,3}= -C_{1414,3}=\frac{1}{4} \left(-w_{333} x+w_{344} x+w_{33}-w_{44}\right),$$
$$-C_{1313,4}= C_{1414,4}=\frac{1}{4} \left(w_{334}-w_{444}\right) x, \ \ C_{1314,1}=-\frac{w_{134} x}{2},$$
$$C_{1314,3}=\frac{1}{2} \left(w_{34}-w_{334} x\right), \ \ C_{1314,4}=-\frac{w_{344} x}{2}, \ \ C_{1334,1}=\frac{w_{34}}{2},$$
$$C_{1414,1}=-\frac{x^2 w_{144}-x^2 w_{133}+2 p^2 r w_{33}-2 p^2 r w_{44}}{4 x}, \ \ C_{1434,1}=-\frac{1}{4} \left(w_{33}-w_{44}\right).$$
\indent According to Einstein's field equations, the energy momentum tensor $T$ for zero cosmological constant is related to the Ricci tensor and the metric tensor as
$$T= \frac{c^4}{8\pi G}\left[S-\left(\frac{\kappa}{2}\right)g\right],$$
where $c=$ speed of light in vacuum and $G=$ gravitational constant. Thus the non-zero components of the energy momentum tensor of the pure radiation metric \eqref{prm} is given by:
$$T_{11}=-\frac{c^4 \left(p^2 w_{33} x + p^2 w_{44} x\right)}{16 \pi G x^2}.$$
Obviously, $T = \rho (\eta\otimes\eta)$, where $\eta = \{1,0,0,0\}$ and the radiation density $\rho = T_{11}$. It is easy to check that $||\eta|| = 0$.\\
Now the non-zero components of covariant derivative of $T$ are given by
$$T_{11,1}=\frac{c^4 p^2 \left(-x^2 w_{144}-x^2 w_{133}+2 p^2 r w_{33}+2 p^2 r w_{44}\right)}{16 \pi G x}, \ \ T_{13,1}=\frac{c^4 p^2 \left(w_{33}+w_{44}\right)}{16 \pi G}$$
$$T_{11,3}=\frac{c^4 p^2 \left(-w_{333} x-w_{344} x+w_{33}+w_{44}\right)}{16 \pi G}, \ \ T_{11,4}=-\frac{c^4 p^2 \left(w_{334}+w_{444}\right) x}{16 \pi G}.$$
\indent From the value of the local components (presented in Section \ref{com}) of various tensors of the pure radiation metric \eqref{prm}, we can conclude that the pure radiation metric \eqref{prm} fulfills the following curvature restricted geometric structures.
\begin{thm}\label{mainthm}
The pure radiation metric \eqref{prm} possesses the following curvature properties:
\begin{enumerate}[label=(\roman*)]
\item Its Ricci tensor is neither Codazzi type nor cyclic parallel but the scalar curvature is zero and hence $R=W$ and $C =K$.
\item It is a $R$-space (also $C$-space, $P$-space) by Venzi for the associated 1-form $\Pi=\{c,0,0,0\}$, $c$ being arbitrary scalar. Hence the curvature 2-forms $\Omega_{(R)l}^m$ are recurrent for $\Pi$ as the 1-forms of recurrency.
\item It is neither locally symmetric nor conformally symmetric but semisymmetric. Hence it satisfies $R\cdot S=0$, $R\cdot C = 0$ and $R\cdot P = 0$.
\item It satisfies the semisymmetric type condition $C\cdot R=0$ and hence $C\cdot S=0$, $C\cdot C=0$, $C\cdot P=0$ and $P\cdot S = 0$.
\item $R$ or $C$ of the space is not a scalar multiple of $S\wedge S$, but $Q(S,R)=0$, $Q(S,C)=0$. Hence $P\cdot R = 0$ and $P\cdot C = 0$, although $P\cdot \mathcal R\neq 0$ but $P\cdot\mathcal S=0$.
\item It is not Einstein but Ricci simple, since $S = \beta (\eta\otimes\eta)$, where $\beta = - \frac{p^2~x~(w_{33}+w_{44})}{2}$ and $\eta = \{1,0,0,0\}$ (moreover $||\eta|| = 0$ and $\nabla \eta \ne 0$). Hence $S\wedge S=0$ and $S^2=0$.
\item Here $P\cdot P \ne 0$ but $P\cdot P=-\frac{1}{3}Q(S,P)$.
\item If $w_{33}+w_{44}$ is nowhere zero, then the metric is neither recurrent nor Ricci recurrent but Ricci 1-forms are recurrent with associated 1-form
$$\left\{1, 0, \frac{w_{333}+w_{344}}{w_{33}+w_{44}}, \frac{w_{334}+w_{444}}{w_{33}+w_{44}}\right\}.$$
\item If $4w_{34}^2+\left(w_{44}-w_{33}\right)^2$ is nowhere zero, then the metric is not conformally recurrent but conformal 2-forms are recurrent with associated 1-form $\Pi$, given by
$$\Pi_1 = 1, \ \ \ \Pi_2 = 0,$$
$$\Pi_3 = \frac{2 w_{34} \left(w_{334}+w_{444}\right)-\left(w_{44}-w_{33}\right) \left(w_{333}+w_{344}\right)}{4w_{34}^2+\left(w_{44}-w_{33}\right)^2},$$
$$\Pi_4 = \frac{2 w_{34} \left(w_{333}+w_{344}\right)+\left(w_{44}-w_{33}\right) \left(w_{334}+w_{444}\right)}{4w_{34}^2+\left(w_{44}-w_{33}\right)^2}.$$
\item The general form of the compatible tensor for $R$, $C$ and $P$ are respectively given by
$$\left(
\begin{array}{cccc}
a_{(1,1)} & a_{(1,2)} & a_{(1,3)} & a_{(1,4)} \\
a_{(2,1)} & 0 & 0 & 0 \\
a_{(3,1)} & 0 & a_{(3,3)} & a_{(3,4)} \\
a_{(4,1)} & 0 & a_{(4,3)} & a_{(3,3)}+\frac{w_{44} a_{(3,4)}}{w_{34}}-\frac{w_{33} a_{(4,3)}}{w_{34}}
\end{array}
\right),$$
$$\left(
\begin{array}{cccc}
a_{(1,1)} & a_{(1,2)} & a_{(1,3)} & a_{(1,4)} \\
a_{(2,1)} & 0 & 0 & 0 \\
a_{(3,1)} & 0 & a_{(3,3)} & a_{(3,4)} \\
a_{(4,1)} & 0 & -\frac{2 w_{34} a_{(3,3)}}{w_{44}-w_{33}}-a_{(3,4)}+\frac{2 w_{34}
a_{(4,4)}}{w_{44}-w_{33}} & a_{(4,4)}
\end{array}
\right) \ \mbox{and}$$
$$\left(
\begin{array}{cccc}
a_{(1,1)} & a_{(1,2)} & a_{(1,3)} & a_{(1,4)} \\
a_{(2,1)} & 0 & 0 & 0 \\
a_{(3,1)} & 0 & -\frac{\left(2 w_{44}-w_{33}\right) a_{(3,4)}}{3 w_{34}}-\frac{\left(w_{44}-2 w_{33}\right)
a_{(4,3)}}{3 w_{34}}+a_{(4,4)} & a_{(3,4)} \\
a_{(4,1)} & 0 & a_{(4,3)} & a_{(4,4)}
\end{array}
\right),$$
where $a_{(i,j)}$ are arbitrary scalars.
\item Ricci tensor of this spacetime is not Codazzi type but is compatible for $R$, $C$ and $P$.
\item If $w_{33}+w_{44}$ is nowhere vanishing, then the metric is weakly Ricci symmetric with infinitely many solutions $(\Pi, \Omega, \Theta)$, given by
$$\Pi = \left\{\Pi_1, 0, \frac{w_{333} x+w_{344} x-w_{33}-w_{44}}{\left(w_{33}+w_{44}\right) x}, \frac{w_{334}+w_{444}}{w_{33}+w_{44}}\right\},$$
$$\Omega = \left\{\Omega_1, 0, -\frac{1}{x}, 0\right\} \ \ \mbox{and}$$
$$\Theta = \left\{\frac{x^2 w_{144}x^2 w_{133}-2 p^2 r w_{33}-2 p^2 rw_{44}}{\left(w_{33}+w_{44}\right) x^2}-\Pi_1-\Omega_1, 0, -\frac{1}{x}, 0\right\},$$
where $\Pi_1$ and $\Omega_1$ are arbitrary scalars.
\item If $w_{33}+w_{44}$ is nowhere vanishing, then the metric is weakly cyclic Ricci symmetric with infinitely many solutions $(\Pi, \Omega, \Theta)$, given by
$$\Pi = \left\{\Pi_1, 0, \frac{\left(w_{333}+w_{344}\right) x-3 w_{33}-3 w_{44}}{\left(w_{33}+w_{44}\right) x}, \frac{w_{334}+w_{444}}{w_{33}+w_{44}}\right\},$$
$$\Omega = \left\{\Omega_1, 0, \frac{\left(w_{333}+w_{344}\right) x-3 w_{33}-3 w_{44}}{\left(w_{33}+w_{44}\right) x}, \frac{w_{334}+w_{444}}{w_{33}+w_{44}}\right\} \ \ \mbox{and}$$
\beb
&&\Theta = \left\{\frac{3 \left(x^2 \left(w_{144}+w_{133}\right)-2 p^2 r \left(w_{33}+w_{44}\right)\right)}{\left(w_{33}+w_{44}\right) x^2}-\Pi_1-\Omega_1, 0,\right.\\
&&\hspace{2.4in} \left.\frac{\left(w_{333}+w_{344}\right) x-3 w_{33}-3 w_{44}}{\left(w_{33}+w_{44}\right) x}, \frac{w_{334}+w_{444}}{w_{33}+w_{44}}\right\},
\eeb
where $\Pi_1$ and $\Omega_1$ are arbitrary scalars.
\item It is not weakly symmetric for $R$, $C$, $P$, $W$ and $K$ and hence not Chaki pseudosymmetric for $R$, $C$ or $P$.
\item $div R \ne 0$, $div C \ne 0$, $div P \ne 0$.
\end{enumerate}
\end{thm}
\indent Now from the values of the non-zero components of $\nabla T$, we get
\be\label{cst}
\left.\begin{array}{l}
T_{11,1}+T_{11,1} + T_{11,1}=\frac{3 c^4 p^2 \left(-x^2 w_{144}-x^2 w_{133}+2 p^2 r w_{33}+2 p^2 r w_{44}\right)}{16 \pi G x},\\
T_{11,3} + T_{13,1} + T_{31,1}=\frac{c^4 p^2 \left(-w_{333} x-w_{344} x+3 w_{33}+3 w_{44}\right)}{16 \pi G},\\
T_{11,4} + T_{14,1} + T_{41,1}=-\frac{c^4 p^2 \left(w_{334}+w_{444}\right) x}{16 \pi G}
\end{array}\right\}\ee
and
\be\label{codt}
\left.\begin{array}{l}
T_{13,1} - T_{11,3} = \frac{c^4 p^2 \left(w_{333}+w_{344}\right) x}{16 \pi G},\\
T_{14,1} - T_{11,4} = \frac{c^4 p^2 \left(w_{334}+w_{444}\right) x}{16 \pi G}.
\end{array}\right\}\ee
\indent Again since $R\cdot S = 0$ and $T$ is a linear combination of $S$ and $g$, so $R\cdot T = 0$. Hence we can state the following:
\begin{thm}
The energy-momentum tensor $T$ of the pure radiation metric \eqref{prm} is\\
(i) semisymmertric i.e., $R\cdot T = 0$,\\
(ii) Codazzi type if $w_{33}+w_{44}$ is independent of $x$ and $y$,\\
(iii) cyclic parallel if $w_{33}+w_{44}$ is independent of $u$, $x$ and $y$,\\
(iv) covariantly constant if $w_{33}+w_{44} = 0$.
\end{thm}
\section{\bf Curvature properties of pure radiation type metric}\label{gen}
We now consider the pure radiation type metric \eqref{gprm}. Its metric tensor is given by
\beb
g = \left(\begin{array}{cccc}
\left(x w + a \frac{r^2}{x^2}\right) & 1 & \frac{b r}{x} & 0\\
1 & 0 & 0 & 0\\
\frac{b r}{x} & 0 & f & 0\\
0 & 0 & 0 & f\\
\end{array}\right),
\eeb
where $a, b$ are arbitrary non-zero constants and $w = w(u,x,y)$, $f = f(x,y)$ are nowhere vanishing functions.
Then the non-zero components (upto symmetry) of $R$, $S$ and $\kappa$ are given by
$$R_{1212}=-\frac{4 a f-b^2}{4 f x^2}, \ \ R_{1213}=\frac{r \left(8 a f+b^3\right)}{4 f x^3}, \ \ R_{1334}=-\frac{b^2 f_4 r}{4 f x^2},$$
\beb
R_{1313}=-\frac{1}{4 f x^4}&&\left[a b^2 f r^2+2 a b f_3 r^2 x+2 a f_3 r^2 x+12 a f r^2-b^4 r^2+b^2 f w x^3\right.\\
&&\left.+2 b f w_3 x^4+2 b f w x^3-f_3 w_3 x^5+f_4 w_4 x^5+2 f w_{33} x^5-f_3 w x^4+4 f w_3 x^4\right],
\eeb
$$R_{1314}=-\frac{2 a b f_4 r^2+2 a f_4 r^2+b f w_4 x^3-f_4 w_3 x^4-f_3 w_4 x^4+2 f w_{34} x^4-f_4 w x^3+2 f w_4 x^3}{4 f x^3},$$
$$R_{1323}=-\frac{b \left(b f+f_3 x+2 f\right)}{4 f x^2}, \ \ R_{1324}= R_{1423}=-\frac{b f_4}{4 f x},$$
$$R_{1414}=-\frac{-2 a b f_3 r^2-2 a f_3 r^2+f_3 w_3 x^4-f_4 w_4 x^4+2 f w_{44} x^4+f_3 w x^3}{4 f x^3},$$
$$R_{1424}=\frac{b f_3}{4 f x}, \ \ R_{1434}=\frac{b^2 f_3 r}{4 f x^2}, \ \ R_{3434}=-\frac{-f_3^2-f_4^2+f f_{33}+f f_{44}}{2 f},$$
$$S_{11}=\frac{2 a^2 f r^2-3 a b^2 r^2-8 a b r^2+2 a f w x^3-6 a r^2-b(b w + w_3 x + w) x^3-(w_{33} + w_{44}) x^5-2 w_3 x^4}{-2 f x^4},$$
$$S_{12}=-\frac{2 a f-b^2-b}{2 f x^2}, \ \ S_{44}=-\frac{b f f_3+f_3^2 x+f_4^2 x-f f_{33} x-f f_{44} x}{2 f^2 x}, \ \ S_{34}=\frac{b f_4}{2 f x},$$
$$S_{13}=\frac{r \left(4 a f+b^3+b^2\right)}{2 f x^3}, \ \ S_{33}=\frac{b^2 f^2+2 b f^2+b f_3 f x+f_{33} f x^2+f_{44} f x^2-f_3^2 x^2-f_4^2 x^2}{2 f^2 x^2},$$
$$\kappa = \frac{-4 a f^3+b (3 b+4) f^2+2 \left(f_{33}+f_{44}\right) f x^2-2 \left(f_3^2+f_4^2\right) x^2}{2 f^3 x^2}.$$
\indent Again for zero cosmological constant, the non-zero components (upto symmetry) of the energy momentum tensor are given by
\beb
T_{11}=\frac{c^4}{32 \pi f^3 G x^4} &&\left[3 a b^2 f^2 r^2+12 a b f^2 r^2+12 a f^2 r^2+2 a f_3^2 r^2 x^2+2 a f_4^2 r^2 x^2-2 a f f_{33} r^2 x^2\right.\\
&&-2 a f f_{44} r^2 x^2-b^2 f^2 w x^3+2 b f^2 w_3 x^4-2 b f^2 w x^3+2 f^2 w_{33} x^5+2 f^2 w_{44} x^5\\
&&\left. +4 f^2 w_3 x^4+2 f_3^2 w x^5+2 f_4^2 w x^5-2 f f_{33} w x^5-2 f f_{44} w x^5\right],
\eeb
$$T_{12}=-\frac{c^4 \left(b^2 f^2+2 b f^2+2 f_{33} f x^2+2 f_{44} f x^2-2 f_3^2 x^2-2 f_4^2 x^2\right)}{32 \pi f^3 G x^2},$$
$$T_{13}=\frac{c^4 r \left(4 a b f^3+8 a f^3+b^3 \left(-f^2\right)-2 b^2 f^2+2 b f_3^2 x^2+2 b f_4^2 x^2-2 b f f_{33} x^2-2 b f f_{44} x^2\right)}{32 \pi f^3 G x^3},$$
$$T_{33}=\frac{c^4 \left(4 a f^2+b^2 (-f)+2 b f_3 x\right)}{32 \pi f G x^2}, \ \ T_{34}=\frac{b c^4 f_4}{16 \pi f G x}, \ \ T_{44}=\frac{c^4 \left(4 a f^2-3 b^2 f-2 b f_3 x-4 b f\right)}{32 \pi f G x^2}.$$
From above we have the following:
\begin{thm}
The pure radiation type metric \eqref{gprm} has the following curvature properties:\\
(i) It is a 3-quasi-Einstein manifold, since $S- \frac{b+b^2-2 a f}{2 f x^2} g$ is of rank 3.\\
(ii) For zero cosmological constant, its energy momentum tensor $T$ is of the form
$$T = \alpha g + \beta_{_1} (e_1\otimes e_1) + \beta_{_3} (e_3\otimes e_3) + \beta_{_4} (e_4\otimes e_4) + \sigma_{_1} (e_1\otimes e_3 + e_3\otimes e_1) + \sigma_{_2} (e_3\otimes e_4 + e_4\otimes e_3),$$
where
$$e_1 = \{1,0,0,0\}, \ \ e_3 = \{0,0,1,0\}, \ \ e_4 = \{0,0,0,1\},$$
$$\alpha =-\frac{c^4 \left[b (b+2) f^2+2 \left(f_{33}+f_{44}\right) f x^2-2 \left(f_3^2+f_4^2\right) x^2\right]}{32 \pi f^3 G x^2},$$
$$\beta_{_1} = T_{11} - \alpha g_{11}, \ \ \beta_{_3} = T_{33} - \alpha g_{33}, \ \ \beta_{_4} = T_{44} - \alpha g_{44}, \ \
\sigma_{_1} = T_{13} - \alpha g_{13}, \ \ \sigma_{_2} = T_{34} - \alpha g_{34}.$$
(iii) It is an $Ein(4)$ manifold, such that
\beb
S^4 &=&\frac{\left(-2 a f+b^2+b\right)^2}{16 f^8 x^7}\left[b (b+2) f^3 \left(\left(f_{33}+f_{44}\right) x-b f_3\right)\right.\\
&&+f^2 x \left(\left(f_{33}+f_{44}\right){}^2 x^2-2 b (b+1) \left(f_3^2+f_4^2\right)\right)\\
&&\left.-2 \left(f_3^2+f_4^2\right) \left(f_{33}+f_{44}\right) f x^3+\left(f_3^2+f_4^2\right)^2 x^3\right] g\\
&-& \frac{\left(-2 a f+b^2+b\right)}{8 f^7 x^6} \left[f^4 \left(b^2 (b+1) (b+2)-4 a \left(f_{33}+f_{44}\right) x^2\right)\right.\\
&&+2 f^3 x \left(x \left(2 a f_4^2+b (2 b+3) \left(f_{33}+f_{44}\right)\right)+2 a f_3^2 x-(b+2) b^2 f_3\right)\\
&&-2 a b (b+2) f^5+2 f^2 x^2 \left(\left(f_{33}+f_{44}\right)^2 x^2-3 b (b+1) \left(f_3^2+f_4^2\right)\right)\\
&&\left.-4 \left(f_3^2+f_4^2\right)\left(f_{33}+f_{44}\right) f x^4+2 \left(f_3^2+f_4^2\right)^2 x^4\right] S\\
&+& \frac{1}{4 f^6 x^4}\left[4 a^2 f^6+f^4 \left(b^2 (b+1) (3 b+5)-8 a \left(f_{33}+f_{44}\right) x^2\right)\right.\\
&&+f^3 x \left(x \left(8 a f_4^2+b (5 b+6) \left(f_{33}+f_{44}\right)\right)+8 a f_3^2 x-(b+2) b^2 f_3\right)\\
&&-4 a b (2 b+3) f^5+f^2 x^2 \left(\left(f_{33}+f_{44}\right){}^2 x^2-6 b (b+1) \left(f_3^2+f_4^2\right)\right)\\
&&\left.-2 \left(f_3^2+f_4^2\right) \left(f_{33}+f_{44}\right) f x^4+\left(f_3^2+f_4^2\right)^2 x^4\right] S^2\\
&+& \frac{1}{2 f^3 x^2}\left[f^2 (4 a f-b (3 b+4))+2 \left(f_3^2+f_4^2-f \left(f_{33}+f_{44}\right)\right) x^2\right] S^3,
\eeb
\end{thm}
\begin{rem}
Since for zero cosmological constant $S = \frac{\kappa}{2}g +\frac{8\pi G}{c^4} T$, hence from above theorem we can state that the Ricci tensor of \eqref{gprm} is of the form
\beb
S &=& (S_{12}) g + (S_{11} - S_{12} g_{11}) (e_1\otimes e_1) + (S_{33} - S_{12} g_{33}) (e_3\otimes e_3) + (S_{44} - S_{12} g_{44}) (e_4\otimes e_4)\\
&& + (S_{13} - S_{12} g_{13}) (e_1\otimes e_3 + e_3\otimes e_1) + (S_{34} - S_{12} g_{34}) (e_3\otimes e_4 + e_4\otimes e_3).\eeb
\end{rem}
\indent We can easilly check that $||e_1||=0$ and the nonzero components of $\nabla e_1$ are given by
$$\nabla_{_1} e_1 = \frac{a r}{x^2}, \ \ \nabla_{_3} e_1 = \frac{b}{2 x}.$$
We know that a spacetime is called generalized pp-wave (\cite{RS84}, \cite{SBK17}) if there exists a covariantly constant null vector field. Hence we can state the following:
\begin{thm}
The pure radiation type metric \eqref{gprm} represents generalized pp-wave if $a = b = 0$.
\end{thm}
\indent Now a spacetime is 2-quasi-Einstein if Rank$(S-\alpha g) = 0$. Hence the metric \eqref{gprm} becomes 2-quasi-Einstein if one of the following condition holds
$$(i) \ (S_{34} - S_{12} g_{34}) = (S_{44} - S_{12} g_{44}) = 0,$$
$$(ii) \ (S_{13} - S_{12} g_{13}) = (S_{33} - S_{12} g_{33}) = (S_{34} - S_{12} g_{34}) = 0,$$
$$(iii) \ (S_{11} - S_{12} g_{11}) = (S_{13} - S_{12} g_{13}) = 0.$$
Simplifying the above conditions we can state the following:
\begin{thm}
The pure radiation type metric \eqref{gprm} becomes 2-quasi-Einstein if any one the following condition holds
$$\begin{array}{l}
(i) \ b = 2 a f^3-\left(f_3^2+f_4^2-f \left(f_{33}+f_{44}\right)\right) x^2=0,\\
(ii) \ f_4 = 2 a f^3+x f \left(x f_{33}-b f_3\right)-b (b+1) f^2-x^2 f_3^2 = 0,\\
(iii) \ a = b = -f_3^2+f_4^2+f \left(f_{33}+f_{44}\right) = 0,\\
(iv) \ f_4 = a = b f_3+\frac{b f}{x}+x f_{33}-\frac{x f_3^2}{f} = 0,\\
(v) \ f_4 = b-2 = \frac{2 a f^2}{x}+x f_{33}-2 f_3-\frac{x f_3^2}{f}-\frac{2 f}{x}=0,\\
(vi) \ a = (b+2) w_3+w_{33} x + w_{44} x = 0,\\
(vii) \ b-2 = w_{33}+w_{44} = 0.
\end{array}$$
\end{thm}
\begin{exm}
If we consider the metric \eqref{gprm} with $f(x, y) = \frac{e^{\frac{1}{3} x^3}}{x^{2/3}}$, $b = -2$ and $a = 0$, then \eqref{gprm} becomes a 2-quasi-Einstein manifold.
\end{exm}
\indent Now a spacetime is perfect fluid if it satisfies \eqref{pfc}. Hence the metric \eqref{gprm} represents perfect fluid if one of the following condition holds
$$\mbox{(i) } \beta_{_1} = \beta_{_3} = \sigma_{_1} = \sigma_{_2} = 0,$$
$$\mbox{(ii) } \beta_{_1} = \beta_{_4} = \sigma_{_1} = \sigma_{_2} = 0,$$
$$\mbox{(iii) } \beta_{_3} = \beta_{_4} = \sigma_{_1} = \sigma_{_2} = 0.$$
Simplifying the above conditions we can state the following:
\begin{thm}
The pure radiation type metric \eqref{gprm} represents perfect fluid if any one the following condition holds
$$\begin{array}{l}
(i) \ a = b = 2 w_3 + x w_{33} + x w_{44} = f_3^2+f_4^2-f \left(f_{33}+f_{44}\right) = 0,\\
(ii) \ b+2 = f_4 = w_{33}+w_{44} = 2 a f^3 - x^2 f_3^2 + x f \left(x f_{33} - 2 f_3\right) - 2 f^2=0,\\
(iii) \ a = f_4 = (b+2) w_3+w_{33} x+w_{44} x = x f \left(b f_3+x f_{33}\right)+b f^2-x^2 f_3^2 = 0,\\
(iv) \ a = f_4 = x f \left(b f'+x f''\right)+b f^2-x^2 f'^2 = x f \left(b f'-x f''\right)+b (b+1) f^2+x^2 f'^2 = 0,\\
(v) \ b+2 = f-\frac{1}{a} = 0.
\end{array}$$
\end{thm}
\begin{exm}
Consider the metric \eqref{gprm}, where $w(u, x, y) = u x y$, $f(x, y) = \frac{e^{\frac{1}{3} x^3}}{x^{2/3}}$, $b = -2$ and $a = 0$, then \eqref{gprm} represents a perfect fluid spacetime. The energy momentum tensor of this metric can be expressed as
$$T = -\frac{c^4 e^{-\frac{x^3}{3}} \left(3 x^3+1\right)}{24 \pi G x^{4/3}} g + \frac{c^4 \left(3 x^3-2\right)}{12 \pi G x^2} (e_4 \otimes e_4).$$
Moreover in this case the metric is quasi-Einstein and $Ein(2)$.
\end{exm}
\indent Again a spacetime is a pure radiation spacetime if it satisfies \eqref{prc}. Now $||e_1||=0$ and hence the metric \eqref{gprm} represents perfect fluid if $\alpha = \beta_{_3} = \beta_{_4} = \sigma_{_1} = \sigma_{_2} = 0$. Simplifying these conditions we can state the following:
\begin{thm}
The pure radiation type metric \eqref{gprm} represents pure radiation if $f \equiv \frac{1}{a}$ and $b = -2$.
\end{thm}
Again from above we can easily calculate (large but straightforward) the components of $R\cdot R$, $Q(g,R)$ and $Q(S,R)$. Then we have the following:
\begin{thm}
The pure radiation type metric \eqref{gprm} is\\
(i) Ricci generalized pseudosymmetric ($R\cdot R = Q(S,R)$) if $b = -2$ and $f$ is constant. And in this case the metric is $R$-space by Venzi for the associated 1-form $\{1,0,0,0\}$,\\
(ii) a manifold of vanishing scalar curvature if $f\equiv \frac{b(3 b + 4)}{4 a}$,\\
(iii) semisymmetric if $b = -2$ and $f \equiv \frac{1}{a}$.
\end{thm}
\indent From \eqref{prm}, \eqref{gprm} and \eqref{gppwm}, we see that both the pure radiation metric and the pp-wave metric are special cases of the metric \eqref{gprm}. We refer the reader to see \cite{MS16}, \cite{SK-pp} and also references therein for recent works on pp-wave metric. We now draw a comparison (similarities and dissimilarities) between the curvature properties of pure radiation metric and pp-wave metric.\\
\textbf{A. Similarities:}
\begin{enumerate}
\item Both the metrics are of vanishing scalar curvature,
\item both are $R$-space by Venzi as well as $C$-space by Venzi,
\item both are semisymmetric and semisymmetric due to conformal curvature tensor,
\item for both the metrics $Q(S,R) = Q(S,C) = 0$,
\item both the metrics are Ricci simple,
\item Ricci tensors of both metrics are Riemann compatible as well as conformal compatible,
\item Ricci 1-forms $\Lambda_{(Z)l}$ of both metrics are recurrent,
\item conformal 2-forms $\Omega_{(C)l}^m$ of both metrics are recurrent,
\item for both the metrics $P\cdot R = 0$ but $P\cdot\mathcal R \ne 0$.
\item both are weakly Ricci symmetric and hence weakly cyclic Ricci symmetric,
\item both the metrics satisfy $P\cdot P = -\frac{1}{3}Q(S, P)$,
\item the energy-momentum tensors of both the metrics are semisymmetric.
\end{enumerate}
\textbf{B. Dissimilarities:}
\begin{enumerate}
\item For zero cosmological constant, the energy-momentum tensors of both the metrics are of rank one, but the associated 1-form for radiation metric is null, and for pp-wave metric it is null as well as covariantly constant,
\item pp-wave metric is Ricci recurrent but pure radiation metric is not so,
\item for pp-wave metric energy-momentum tensor is cyclic parallel if and only if it is parallel but this fact is not true for pure radiation metric.
\end{enumerate}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,111 |
Baila! Al ritmo de un sueño es un programa de televisión chileno, versión local del formato Bailando por un sueño, original de Televisa, emitido por Chilevisión al estilo de reality show. 10 famosos participarán en conjunto con 10 personajes anónimos, con el objetivo de cumplir un sueño.
Formato
10 parejas (famosos y desconocidos) compiten juntos con el objetivo que cumplir un sueño. En el concurso una pareja conformada por un famoso y un desconocido debe demostrar sus destrezas para el baile en diferentes ritmos musicales.
Temporadas
Baila! Al ritmo de un sueño (2013)
Artículo principal: Anexo:Baila! Al ritmo de un sueño (primera temporada)
Baila! Al ritmo de un sueño 1 es el primer certamen de baile basado en el concurso de baile Bailando por un sueño.
Este certamen inició el 6 de marzo de 2013 y terminó el 9 de mayo de dicho año, en el cual se consagró como primera ganadora de Baila! Al ritmo de un sueño la modelo Faloon Larraguibel junto a su soñador Alejandro Herrera.
Premios y nominaciones
Referencias
Concursos televisivos de Chile
Reality shows de Chile
Programas de televisión de Chilevisión
Programas de televisión iniciados en 2013
Televisión de Chile en 2013
Bailando por un sueño | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,793 |
{"url":"https:\/\/juliabase.org\/programming\/remote_client_usage.html","text":"# Talking to JuliaBase\u00b6\n\nYou have a measurement or processing setup, and you have access to the program(s) that run this setup? This is the ideal situation to make a direct, bi-directional connection between your experimental setup and the central JuliaBase server. This has many benefits:\n\n\u2022 You can assure that the header data in your data files is correct, in particular operator name and sample name.\n\u2022 Each run is immediately available in the central database.\n\u2022 Operators needn\u2019t enter each run into the browser, which is inconvenient and error-prone.\n\u2022 Operators can pick the affected sample(s) comfortably from a list instead of typing the sample names explicitly.\n\u2022 If a run is connected with a task, this task can automatically updated.\n\nThis chapter is intended as a gentle introduction to how to realise this.\n\n## The big picture\u00b6\n\nJuliaBase is written in the Python programming language. This is also true for the Remote Client, which is a Python library that you can install on any lab or office computer and use it to talk to the central JuliaBase server.\n\nHowever, you needn\u2019t use Python to communicate with JuliaBase. This is just the most natural way. But JuliaBase\u2019s source code includes bindings to Delphi, Visual Basic, and LabVIEW. Further bindings can be added very easily once the demand is there.\n\nSuch a binding works by calling the Python interpreter in the background. This indirection causes a very slight performance loss. Moreover, you need to install a Python interpreter. However, the simplicity and maintainability of this approach make up for it.\n\n## Installation\u00b6\n\nFirst, make sure that Python is installed on your computer.\n\nThen, install the remote client package (which is an adaption of the original JuliaBase client to your institute or group). Ask your local JuliaBase guru for how to do this. Ideally, it is available in a shared directory, so that you don\u2019t have to do anything. You should make sure that the remote client package\u2019s directory is in the PYTHONPATH. In the following, I call the adapted module jb_institute_inm; your name with probably be different.\n\n## Basic usage\u00b6\n\nThe next steps differ depending on the programming language you use. The basic principle is always the same, though: You log in on server with user name and password, execute commands that read from or write to the database, and log out.\n\nIn all non-Python languages, however, you cannot give the commands directly. Instead, you build a string that contains the Python commands and pass it to a special function called execute_jb or similar.\n\n### Python\u00b6\n\nIn our example code, we read the data of sample \u201c14-JS-1\u201d and then change its current location:\n\nfrom jb_remote_inm import *\n\nsample = Sample(\"14-JS-1\")\nsample.current_location = \"main lab\"\nsample.edit_description = \"location changed\"\nsample.submit()\n\nlogout()\n\n\n### Visual Basic\u00b6\n\nThe Visual Basic binding in remote_client\/visual_basic\/juliabase.vb can be used like the following:\n\nImports System\nImports Juliabase\n\nPublic Module ModuleMain\nSub Main()\nJB_Module_Name = \"jb_remote_inm\"\n\nExecute_JB(\"juliabase\", \"12345\",\n\"sample = Sample('14-JS-1');\" &\n\"sample.current_location = 'main lab';\" &\n\"sample.edit_description = 'location changed';\" &\n\"sample.submit()\")\nEnd Sub\nEnd Module\n\n\n### Delphi\u00b6\n\nFor Delphi, in order to achieve the same as in the previous sections, you say\n\nprogram juliabase_example;\n\n{\\$APPTYPE CONSOLE}\n\nuses\nSysUtils, juliabase;\n\nbegin\njb_module_name := 'jb_remote_inm';\nexecute_jb('juliabase', '12345',\n'sample = Sample(\"14-JS-1\");' +\n'sample.current_location = \"main lab\";' +\n'sample.edit_description = \"location changed\";' +\n'sample.submit()');\nend.\n\n\nThe necessary unit can be found in remote_client\/delphi\/juliabase.pas.\n\n### LabVIEW\u00b6\n\nThe LabVIEW virtual instrument \u201cexecute jb.vi\u201d in remote_client\/labview\/juliabase.llb is very different from the other bindings for obvious reasons, but the general method is the same: You pass login, password, and the module name in a data structure called \u201csettings\u201d to the VI, and the result of the Python process is returned:\n\n### Getting data in non-Python languages\u00b6\n\nIn the non-Python languages, you don\u2019t have direct access to the results of the commands. Instead, you may use Python\u2019s print() to send data to the standard output, which in turn is the return value of the execute_jb function. Then, you can extract the original data from this value. For example in Delphi, you may write:\n\ntopic := execute_jb('juliabase', '12345', 'print(Sample(\"14-JS-1\").topic)');\n\n\nThen, topic contains the topic of the sample. Note that topic is a string. If you need other data types, you have to convert the result string yourself.\n\nFor more complex return values, this conversion can be cumbersome. In languages with JSON support, there is a convenience function defined in the remote client called as_json(). It can be used instead of print(). It prints its argument in JSON format to standard output. The LabVIEW example above demonstrates the usage of this function in the second VI call.\n\n### The test server\u00b6\n\nYour institution may provide a test server for easier developing. This way, you do not manipulate valuable data on the production server. You choose the test server by passing testserver=True to the login() function:\n\nlogin(\"juliabase\", \"12345\", testserver=True)\n\n\nIn non-Python languages, you pass the same parameter to the execute_jb function.\n\n## Error handling\u00b6\n\nIf something goes wrong while executing the commands, an exception is raised. If it is a JuliaBase-related error, this is a special exception class:\n\nlanguage exception class name error code attribute name\nPython JuliaBaseError error_code\nVisual Basic JuliabaseException code\nDelphi EJuliaBaseError ErrorCode\n\nMoreover, the error message is stored in the exception attribute typical of the respective language.\n\nIf the error is not JuliaBase-related (for example, a syntax error), the language-typical basic exception class is raised, containing a proper error message.\n\nAs usual, in LabVIEW, things are slightly different. If an error occurs, it is set in the error output of the VI. Error numbers greater than 6000 indicate JuliaBase errors. The error message contains the details.\n\n### Error pages in the browser\u00b6\n\nIn case of JuliaBase errors, non-Python languages may open a browser automatically showing a detailed problem description. You may turn off this behaviour by setting the global variable jb_open_error_page_in_browser to false.\n\nPasswords are sensitive data. Never store them on the disk. Assure that they never appear anywhere on the screen (use the \u2022\u2022\u2022\u2022 display). Let the user input their password, store it in a variable, and use it to login to JuliaBase \u2013 that\u2019s all.\n\n## How do I\u00a0\u2026\u00b6\n\n### \u2026 check whether the user is known to JuliaBase?\u00b6\n\nYou login the user with the user name and password they give and check whether this raises a JuliaBase exception with error code\u00a04. If it does, the user name and\/or the password is wrong.\n\nIn Python:\n\ntry:\nexcept JuliaBaseError as error:\nif error.error_code == 4:\n\n\nIn Visual Basic:\n\nTry\nCatch e As JuliabaseException:\nIf e.code = 4 Then\nEnd If\nEnd Try\n\n\n### \u2026 check whether the user is allowed to use my setup?\u00b6\n\nYou retrieve the permissions attribute of a User instance. Then, you check whether the \u201cadd\u201d permission occurs in this attribute.\n\nIn Python:\n\npermissions = User(username).permissions\nprint(\"You are not authorised to make PDS measurements!\")\n\n\nIn Visual Basic:\n\nDim result As String\nMessageBox.Show(\"You are not authorised to make PDS measurements!\")\nEnd If\n\n\n### \u2026 check whether a sample exists?\u00b6\n\nYou retrieve the sample and check whether this raises an exception with error code\u00a02. If it does, a sample with that name was not found.\n\nIn Python:\n\ntry:\nSample(sample_name)\nexcept JuliaBaseError as error:\nif error.error_code == 2:\nprint(\"A sample with this name does not exist!\")\n\n\nIn Visual Basic:\n\nTry\nCatch e As JuliabaseException:\nIf e.code = 2 Then\nMessageBox.Show(\"A sample with this name does not exist!\")\nEnd If\nEnd Try\n\n\n### \u2026 add a new process?\u00b6\n\nYou instantiate the process class, set sample ID, operator, timestamp, and the process-specific attributes, and call the submit() method of the process instance.\n\nIn Python:\n\npds_measurement = PDSMeasurement()\npds_measurement.sample_id = Sample(sample_name).id\npds_measurement.timestamp = datetime.datetime.now()\npds_measurement.number = next_number\npds_measurement.apparatus = \"pds1\"\npds_measurement.raw_datafile = filepath\npds_measurement.submit()\n\n\nIn Visual Basic:\n\nExecute_JB(login, password,\n\"pds_measurement = PDSMeasurement();\" &\n\"pds_measurement.sample_id = Sample('\" & sample_name & \"').id;\" &\n\"pds_measurement.operator = '\" & login & \"';\" &\n\"pds_measurement.timestamp = '\" & Format(Now, \"yyyy-MM-dd HH:mm:ss\") & \"';\" &\n\"pds_measurement.number = \" & next_number & \";\" &\n\"pds_measurement.apparatus = 'pds1';\" &\n\"pds_measurement.raw_datafile = '\" & filepath & \"';\" &\n\"pds_measurement.submit()\")\n\n\nIn order to know which instance attributes you need to set and how, look for documentation in the Python remote client module, or ask your local JuliaBase guru.","date":"2020-02-19 23:52:12","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\": 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\": 0.18671101331710815, \"perplexity\": 4351.912394490756}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875144429.5\/warc\/CC-MAIN-20200219214816-20200220004816-00120.warc.gz\"}"} | null | null |
Now most of you might think that this is the most boring thing to do but believe us, this will just pave the path for the whole of the assignment for you and you won't even have to look for any assignment help online.
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Just let your heart do the talking from here on in. All you have to do is write whatever you think is related to the topic and what isn't going to make the reader think that you are boring. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,450 |
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