text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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<?php
/**
* class NoCache
* No cache class
* @package Framework
* @subpackage Cache
*/
class NoCache extends Cache {
// instance of the class
static private $instance = null;
/**
* singleton function to return
* the instance of the class
*
* @return NoCache
*/
public static function singleton() {
if (!self::$instance) {
self::$instance = new NoCache();
}
return self::$instance;
}
/*****************************************************
*** Implement the abstract classes of parent ***
*****************************************************/
public function initCache($cfg) {
// do nothing
}
protected function setData($key, $data, $args=NULL) {
// do nothing
}
protected function getData($key) {
return false;
}
protected function deleteData($key) {
// do nothing
}
protected function setLifetime($lifetime) {
// do nothing
}
protected function setCompressed($compressed) {
// do nothing
}
}
?>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,286 |
Q: Batch file If statement - syntax error? I'm making a small batch file to look up the French perfect tense verbs that take etre, which comes from a text file.
The whole file works, except this segment:
echo Type Q and hit enter to quit, enter any other key to try another verb.
set /p option=Enter Option:
IF %option%==Q(
goto :quit
) else (
goto :1
)
Typing Q and entering it works fine, but anything else comes up with the error 'incorrect syntax'. I can't see what's incorrect here, can anyone help?
A: Just add a space.
IF %option%==Q (
| {
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} | 4,802 |
Q: Php API Nuki doorlock Post I am a very beginner at PHP and need to unlock a Nuki from a webpage.
I've generated an api, I have a bearer token etc and apache running on my computer.
When I run the command from my commandline everything is working but nothing is happening when I try it like this... Did I miss something?
<?php
$url = "https://api.nuki.io/smartlock/HIDDEN/action/unlock";
$curl = curl_init($url);
curl_setopt($curl, CURLOPT_URL, $url);
curl_setopt($curl, CURLOPT_RETURNTRANSFER, true);
$headers = array(
"Content-Type: application/json",
"Accept: application/json",
"Authorization: Bearer {HIDDEN}",
);
curl_setopt($curl, CURLOPT_HTTPHEADER, $headers);
$resp = curl_exec($curl);
curl_close($curl);
var_dump($resp);
?>
Btw: the working Curl (commandline) is this:
curl -X POST --header 'Content-Type: application/json' --header 'Accept: application/json' --header 'Authorization: Bearer HIDDEN' 'https://api.nuki.io/smartlock/HIDDEN/action/unlock'
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,831 |
{"url":"http:\/\/algebra2.thinkport.org\/module6\/prize-winner-part3-page-3.html","text":"# Prize Winner, Part 3\n\n### Resources for this lesson:\n\nKey Term\n\nTheoretical probability\n\nYou will use your Algebra II Journal on this page.\n\nAs you can see, having the letters in the game repeat makes for a much more challenging game. Let\u2019s keep the game the way it was originally designed, with three doors and the possibility of the letters repeating.\n\nWhat\u2019s Behind the Door?\n\nIf you can guess the letter behind the door, you win a prize!\n\nLarge Prize:\nGuess all three doors correctly\n\nMedium Prize:\nGuess two doors correctly\n\nSmall Prize:\nGuess one door correctly\n\nPrize\n\nProbability to Win\n\nSmall\n\n$\\frac{12}{27}$\n\nMedium\n\n$\\frac{6}{27}$\n\nLarge\n\n$\\frac{1}{27}$\n\nThese probabilities are the theoretical probabilities. Theoretically, a person should win a small prize twice as often as a medium prize, and approximately 50% of the time a contestant wins a small prize. Seldom should a contestant win a large prize.\n\nLet\u2019s conduct a probability simulation to see if these theoretical probabilities hold true!\n\n## Algebra II Journal: Reflection 1\n\nUse the following simulation of the Guess the Letter Behind the Door game. Run the simulation for at least 50 trials. Record the results in your Algebra II Journal and submit to your teacher.","date":"2018-01-22 14:28:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 3, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5828847289085388, \"perplexity\": 2481.9004411479955}, \"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-2018-05\/segments\/1516084891377.59\/warc\/CC-MAIN-20180122133636-20180122153636-00610.warc.gz\"}"} | null | null |
Summary: CERTAIN VEHICLES FAIL TO COMPLY WITH THE REQUIREMENTS OF FEDERAL MOTOR VEHICLE SAFETY STANDARD NO. 108, "LAMPS, REFLECTIVE DEVICES, AND ASSOCIATED EQUIPMENT." SOME VEHICLES MAY NOT HAVE HAD A REQUIRED CAP INSTALLED WHICH DISABLES THE HEADLIGHT HORIZONTAL AIM AND SOME VEHICLES MAY CONTAIN A CAP THAT DISABLES THE VERTICAL AIMING SCREW.
Consequence: INAPPROPRIATE, NON-STANDARD LIGHTING COULD REDUCE VISIBILITY OF VEHICLES AND INCREASE THE POTENTIAL FOR A CRASH.
Remedy: DEALERS WILL INSPECT FOR THE PRESENCE OF A CAP IN THE SOCKET OF THE LOW BEAM HORIZONTAL AIMING SCREW AND INSTALL ONE IF MISSING, AND INSPECT FOR THE PRESENCE OF A CAP IN THE SOCKET OF THE VERTICAL AIMING SCREW AND IF PRESENT, THE CAP WILL BE REMOVED. THE RECALL IS EXPECTED TO BEGIN ON OR ABOUT OCTOBER 31, 2007. OWNERS MAY CONTACT VW AT 1-800-822-8987.
Notes: VOLKSWAGEN RECALL NO. Q8. CUSTOMERS MAY CONTACT THE NATIONAL HIGHWAY TRAFFIC SAFETY ADMINISTRATION'S VEHICLE SAFETY HOTLINE AT 1-888-327-4236 (TTY: 1-800-424-9153); OR GO TO HTTP://WWW.SAFERCAR.GOV.
Volkswagen provides a basic warranty for the GTI that covers the first four years or 50,000 miles, and a five-year/60,000-mile limited powertrain warranty.
Calculate 2008 Volkswagen GTI Monthly Payment Which Cars You Can Afford? | {
"redpajama_set_name": "RedPajamaC4"
} | 789 |
Q: How to not crop the background when resizing the browser to smaller size? You have for example: stackoverflow.com when you resize the width of your browser to smaller size you will see a large padding from the right, same for this webpage http://www.hyper.no/ (bottom page footer)
I'm not sure if they are using Compass blueprint, but I am and I have this problem when adding the @include container it adds the following css:
width: 950px;
margin: 0 auto;
overflow: hidden;
*zoom: 1;
I would like that the width get 100% even resizing the browser to smaller size
A: I finally found the answer for this, it's quite simple for example you put a width of 960px and when you resize your windows to a smaller size, just put min-width:960px on your body and you won't see the gap. I spent 3 hours to find it...
| {
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} | 7,580 |
\section{Introduction}
\label{sec:intro}
The established paradigm for structure formation offers a clear road
map for galaxy formation. Primordial fluctuations in the dominant cold
dark matter (CDM) component of the Universe grow via gravitational
instability, sweeping baryons into an evolving hierarchy of dark
matter haloes that grow through mergers of preexisting units as well as
through the accretion of material from the intergalactic medium
\citep{White1978}. On galaxy mass scales, baryons caught in a halo are
able to radiate away the gravitational energy gained through the
collapse, sink to the center of the halo, and assemble into the dense
aggregations of gas and stars that we call galaxies \citep{Blumenthal1985}.
The structure and morphology of a galaxy results from the complex
interplay between the time of collapse, the mode of assembly, the
efficiency of cooling, and the rate of transformation of gas into
stars \citep[see, e.g.,][]{SteinmetzNavarro2002}. Where cooling
dominates and outpaces star formation, baryons collect into thin,
rotationally-supported disks \citep{Fall1980}. Stars formed in these
disks inherit these morphological features, but are vulnerable to
swift transformation into dispersion-supported spheroids during
subsequent merger events \citep{Toomre1977}. Disks may re-form if
mergers or accretion bring fresh supplies of cooled gas, making
morphology a constantly evolving rather than an abiding feature of a
galaxy \citep{Cole2000,Robertson2006}.
The galaxy formation scenario driven by gravitational collapse and
radiative losses outlined above is compelling, but incomplete.
Indeed, cooling is so effective at early times that, unless impeded
somehow, most baryons would be turned into stars in early-collapsing
protogalaxies, which would then merge away to form by the present time
a majority of spheroid-dominated remnants, in vehement disagreement
with observations \citep{White1978,Cole1991,White1991}. The problem is
compounded by the fact that, during mergers, cooled gas tends to
transfer its angular momentum to the surrounding dark matter halo. As
a result, even in cases where disks could form, their structural
properties would be at odds with those of spiral galaxies
\citep{Navarro1991,Navarro1995,NavarroSteinmetz1997}.
A gas heating mechanism that prevents runaway cooling and that
regulates the formation of stars in step with mergers and accretion is
widely believed to be the most likely solution to these problems. The
energetic output from evolving stars and supernovae is a natural
candidate. It scales directly with star formation and, in a typical
galaxy, the total energy released by supernovae can be comparable to
the binding energy of the baryons. Thus, if channeled properly,
feedback energy from supernovae may temper the gravitational
deposition of cooled gas into a galaxy and effectively self-regulate
its star formation history \citep{White1991}.
The even standing of gravity, feedback and cooling may thus help
reconcile the observed galaxy population with hierarchical clustering
models, but it comes at the price of complexity: the main structural
properties of a galaxy, such as stellar mass, rotation speed, and
morphology, are then expected to depend on details of its assembly
history, such as the exact timing, geometry and mass spectrum of
accretion events \citep[see,
e.g.,][]{Abadi2003a,Abadi2003b,Meza2003,Governato2007,Zavala2008,
Scannapieco2009,Governato2010}.
Such sensitivity to feedback has held back progress in direct
simulation of the process of galaxy formation. As recent work
demonstrates, different but plausible implementations of feedback
within the {\it same} dark halo lead to galaxies of very different
mass, morphology, dynamics, and star formation history
\citep[see, e.g.,][]{Okamoto2005}. Numerical parameters may thus be tuned to
reproduce some properties of individual galaxies, but at the expense
of wider predictability in the modeling.
These results suggest that further progress in the subject
requires the testing of different feedback schemes on a statistically
significant sample of dark haloes formed with representative assembly
histories. The viability of each feedback implementation may then be
assessed by contrasting the statistics of such samples with
observational constraints such as the stellar mass function,
clustering, color distribution, and scaling laws.
We take a step in this direction here by analyzing a subset of
cosmological N-body/gasdynamical simulations from the OverWhelmingly
Large Simulations ({\small OWLS}) project \citep{Schaye2010}. We present
results regarding the morphology, stellar mass, and angular
momentum content of galaxies assembled at $z=2$, and compare them with
the few observational constraints available at that epoch. We limit
our analysis to the $z=2$ galaxy population because most
high-resolution {\small OWLS} runs follow volumes too small to be
evolved until $z=0$. In future papers, we plan to extend this
analysis to the present-day galaxy population using samples drawn from
the closely-related {\small GIMIC} project, designed to follow a few
representative volumes selected from the Millennium Simulation
\citep{Crain2009}.
The paper is organized as follows. In Sec.~\ref{sec:numsim} we present a
short overview of the simulations and feedback models. We then present
our main numerical results in Sec.~\ref{sec:numres} and analyze them
in the context of available observational constraints in
Sec.~\ref{sec:obsdiag}. We end with a brief summary in
Sec.~\ref{sec:conc}.
\section{The Numerical Simulations}
\label{sec:numsim}
\subsection{The OWLS runs}
The {\small OWLS} project consists of a suite of $\sim 50$ different
cosmological N-body/SPH simulations that follow the evolution of dark
matter and baryons in boxes of 25 and 100 $h^{-1}$ Mpc (comoving). Each
box is run many times, varying the numerical implementation of
various aspects of the gas cooling, star formation and feedback
modules \citep[see ][ for further details]{Schaye2010}.
We have selected for our analysis nine $25 \, h^{-1}$ Mpc-box {\small
OWLS} runs, eight of which explore different feedback
implementations with $512^3$ dark matter and $512^3$ baryonic
particles whilst keeping other subgrid parameters constant, such as
the stellar initial mass function (IMF), the star formation threshold
and its efficiency. The ninth repeats one of the runs, at $8\times$
lower mass resolution (and $2\times$ lower spatial resolution), in
order to provide some guidance regarding the sensitivity of our results
to numerical resolution.
All simulations assume a standard WMAP-3 $\Lambda$CDM cosmogony, start
at $z_i=127$ and, because of their small box size, they have only been
carried out to $z=2$. We adopt this cosmology for all physical
quantities listed here. We make explicit the dependence on the Hubble
constant, $h$, for simulation parameters, but drop the $h$ dependence and
adopt $h=0.73$ when comparing with observations.
The high-resolution runs have a comoving gravitational softening lengthscale of
1/25 of the initial mean inter-particle spacing at high redshift. These are
switched later to a fixed physical value so that the softening never
exceeds $0.5 \, h^{-1}$ kpc (physical). The mass per baryonic particle is $\sim 1.4
\times 10^6 \, h^{-1} {\rm M}_\odot$ and $4.5$ times higher for the
dark matter component. All runs assume that the
Universe is reionized at $z=9$ (for H) and at $z=3.5$ (He) by a bath
of energetic photons whose properties evolve as proposed by \citet{Haardt2001}.
Table \ref{tab:simpar} summarizes the most important numerical
parameters of the simulations, as well as the cosmological parameters.
\begin{center}
\begin{table}
\caption{Simulation parameters}
\begin{tabular}{|c|c|}
\hline
\hline
$\Omega_{\rm M}$ & 0.238\\
$\Omega_{\rm CDM}$ & 0.1962\\
$\Omega_b$ & 0.0418\\
$\Omega_\Lambda$ & 0.762\\
$\sigma_8$ & 0.74\\
$h$ & 0.73 \\
$n$ & 0.951\\
Reionization redshift & 9 (H), 3.5 (He)\\
Mass per DM particle & $m_p=6.3 \times 10^6 h^{-1} \rm M_\odot$ \\
Mass per baryonic particle & $m_p = 1.4 \times 10^6 h^{-1} \rm M_\odot$\\
Number of particles & $2 \times 512^3$ \\
Box size & $25 \, h^{-1}$ Mpc\\
\hline
\end{tabular}
\label{tab:simpar}
\end{table}
\end{center}
\subsection{Subgrid gas physics}
Baryons are assumed to trace the dark matter distribution at the
initial redshift. Whilst in gaseous form, they are followed
hydrodynamically and are subject to pressure gradients and
shocks. Radiative cooling and heating is implemented following
Wiersma et al. (2009a), which also accounts for the photo-ionisation of
metals due to the UV background.\nocite{Wiersma2009a}
In collapsed structures, gas can cool and sink to the center of these
haloes, where it may reach high overdensities before turning into
stars. With limited numbers of particles, these regions are poorly
resolved and vulnerable to numerical instabilities, such as artificial
clumping and fragmentation. As discussed by \citet{Springel2003},
these shortcomings can be alleviated by adopting, in high-density
regions, a multi-phase description for the gas where the effective
equation of state differs from the simple ideal gas law. In practice,
we impose a polytropic equation of state (PEOS; $P \propto
\rho^{\gamma}$, with $\gamma=4/3$) on all gas particles whose density
exceeds a critical value of $n_c=0.1$ cm$^{-3}$, the density above
which the gas is expected to be multiphase and unstable to star
formation \citep{Schaye2004}. This choice ensures that the Jeans mass
in high-density regions is independent of $\rho$, effectively
suppressing artificial clumping and reducing the dependence of
star formation algorithms on numerical resolution \citep{Schaye2008}.
\subsection{Star formation algorithm}
Star formation is implemented as described in detail by
\citet{Schaye2008}. In brief, stars form out of PEOS gas particles
with pressure-dependent parameters chosen to reproduce a
Kennicutt-Schmidt law with index $1.4$ \citep{Kennicutt1998}. We
assume a Chabrier initial mass function \citep{Chabrier2003} in order
to take into account the enrichment and energy injected into the
surroundings of young star particles by the explosion of SNII and SNIa
supernovae. The energy per supernova explosion is chosen to be
$10^{51}$ ergs. These events, together with mass loss from
intermediate mass stars, pollute neighboring gas particles with
metals, as described in Wiersma et al. (2009b). We track 11 species
and include them in the computation of the cooling function following
(Wiersma et al 2009a) in all our runs, with the exception of the
``NoF'' model described below. For the latter case, chemical enrichment is
modelled in the same way but is not considered in the computation of
cooling, which instead assumes primordial abundances.
\nocite{Wiersma2009b}
\subsection{Feedback Models}
The runs we analyze here explore alternative feedback implementations
where the total amount of energy injected by supernovae into the
surrounding ISM is kept constant, but the numerical algorithm used to
inject this energy is varied. All runs that include feedback from core
collapse supernova feedback assume a total energy input of $10^{51}$
ergs per solar mass of stars formed, 40\% of which is invested into
driving outflowing winds. The remainder is assumed to be lost to
radiative processes.
\subsubsection{Thermal Feedback}
The simplest possibility, which we label "thermal feedback" (ThF), is
to use the supernova energy to raise the internal energy of the
surrounding gas particles. As reported in earlier work
\citep{Katz1992}, these regions typically have such short cooling
times that the injected energy is quickly radiated away, with little
hydrodynamical effect on the surrounding gas. As a result, thermal
feedback is rather inefficient, and has little effect in regulating
gas cooling and star formation, even though the implementation here follows the
stochastic heating method described in \citet{Schaye2010} and presented in more detail
in Dalla Vecchia et al. (in preparation), which is
more resilient to numerical resolution limitations than the
implementations adopted in earlier work \citep[see also][]{Kay2003}.
\subsubsection{Kinetic Feedback}
A second possibility is to invest part of the feedback energy directly
into gas bulk motions, with the aim of allowing gas to outflow from
regions of active star formation, thus increasing the overall
efficiency of feedback. These wind models are characterized by a
couple of parameters: a ``mass loading'' factor, $\eta$, specifying the
number of gas particles amongst which the injected energy is shared,
and a ``wind velocity'', $v_{\rm w}$, characterizing the kinetic energy of the
outflow. For given energy, $\eta$ and $v_{\rm w}$ are related by a constant
$\eta \, v_{\rm w}^2$.
For the reference model in OWLS (WF2 in our notation) the wind
velocity $v_{\rm w}=600$ km/s is chosen partly motivated by observation of
local starburst galaxies \citep{Veilleux2005}. The mass
loading $\eta$ is thus fixed to 2 particles as the integer number that
best reproduces the peak in the cosmic star formation history. This
combination of $v_{\rm w}$ and $\eta$ imply that 40\% of the total energy
liberated by supernovae impacts the kinematics of the surrounding
gaseous medium. Because all these parameters are highly uncertain, in the
OWLS runs we contrast the results obtained with three different values
of $\eta$=1,2,4 (we refer to these runs as WF1, WF2, and WF4, respectively).
The wind velocities in each model are adjusted so
that the same amount of energy (40\%) is input in every case (see Table~\ref{tab:fpar}).
Further
details can be found in Dalla Vecchia \& Schaye (2008) and Schaye et
al. (2010). WF2LR is equivalent to WF2
but run at $8 \times$ poorer mass resolution and $2\times$ poorer
spatial resolution.
As discussed by \citet{Springel2003}, a possible modification that can
enhance feedback efficiency is to temporarily "decouple" the wind
particle(s) hydrodynamically from the surrounding ISM. This
facilitates large-scale galactic outflows and regulates star formation
more effectively by enhancing the removal of gas from active
star-forming regions \citep[see e.g.,][]{DallaVecchia2008}. The
criterion for re-coupling particles to the gas is as in Springel \&
Hernquist, and occurs as soon as either: $a)$ the density has fallen
to 0.1 $n_c$, where $n_c$ is the density threshold for
star formation, or $b)$ 50 Myr have elapsed since decoupling. We
label this run WF2Dec.
A further run probes the possibility that the efficiency of feedback
should correlate with the local density of the gas. We explore a model
in which the wind velocity and mass loading are related to the gas
density by $v_{\rm w} \propto \rho^{1/6}$ and $\eta \propto
\rho^{-1/3}$. This guarantees that the wind velocity scales with the
local gas sound speed ($v_{\rm w} \propto c_s$) given the
aforementioned effective PEOS that holds in
star-forming regions. The $v_{\rm w}$ and $\eta$ relations are
normalized so that, at the gas density corresponding to the star
formation threshold ($n_c = 0.1$ cm$^{-3}$), they match $v_{\rm
w}=600$ km/s and $\eta=2$ particles, consistent with the WF2 run. We
will refer to this run as WDENS.
\subsubsection{AGN Feedback}
Our next model enhances feedback by adding to the WF2 feedback the
extra energetic input from AGN. This model, which we refer to as AGN,
for short, follows the numerical procedure introduced by
\citet{Booth2009} and summarized in \citet{Schaye2010}. Seed black
holes with mass $m_{\rm seed}=9 \times 10^{4} M_\odot$ are placed at
the center of all haloes that exceed a threshold virial mass of $4
\times 10^{10} h^{-1}M_\odot$. BHs can then grow by mass accretion and
mergers with other BHs. Gas accretion onto the BH is modelled
according to a modified version of the Bondi-Hoyle-Lyttleton
\citep{BondiHoyle1944,Hoyle1939} formula: $\dot{m_{\rm accr}} =\alpha
\, 4\pi \, G^2 \, m_{\rm BH}^2 \, \rho/(c_s^2+ v^2)^{3/2}$, where
$m_{\rm BH}$ is the mass of the BH, $v$ is the velocity of the BH with
respect to the ambient medium, $c_s$ is the local speed of sound and
$\rho$ the local gas density. $\alpha$ is an extra ``efficiency''
parameter that did not appear in the original versions of the
Bondi-Hoyle formula but was introduced by \citet{Springel2005c}, who
set it to alpha = 100, to account for the finite numerical resolution
and for the fact that the cold, interstellar phase is not explicitly
modeled. In our model this factor is set to unity in the regime that
the physics is modeled correctly, but increases with the ISM density
(i.e., for particles on the PEOS, see Booth \& Schaye (2009) for
further details and discussion).
A fraction $\epsilon_f$ of the total radiated energy due to the mass
accretion onto the BHs is assumed to couple to the surrounding
ISM. This efficiency is set to $\epsilon_f=0.15$ to match local
constraints on the number density \citep[see, e.g.,][]
{Shankar2004,Marconi2004} as well as relations between BHs and
host galaxy properties \citep{Haring2004,Tremaine2002},
both at redshift zero. AGN feedback is implemented as a {\it
thermal} injection of energy (as opposed to the kinetic prescription
used to model the stellar feedback), in the way described in
\citet{Booth2009}. Because it combines the supernova and AGN energetic
outputs, the AGN run is the most effective feedback model tried in our
series.
\subsubsection{No Feedback}
Finally, mainly for comparison purposes we also analyze a run that
follows star formation like in the other implementations but neglects
all energy injection into the ISM due to either supernovae or AGN. Gas
cooling in this ``no feedback'' model, NoF, adopts the cooling
function of a gas with primordial abundances, but in the absence of
feedback this is only a minor difference that has little impact on the
results. The NoF model stands at the opposite extreme as AGN, allowing
for unimpeded transformation of gas into stars in regions able to
collapse and condense into galaxies. Although unrealistic as a galaxy
formation model, it serves to provide a useful framework where the
relative importance of feedback effects may be gauged and understood.
Table~\ref{tab:fpar} summarizes the relevant parameters of each
feedback implementation. For ease of reference, we also quote in each
case, the name used to label each simulation by \citet{Schaye2010}.
\begin{center}
\begin{table}
\caption{Parameters of the different feedback models
probed in each run. First and second columns list the short name
(used throughout this paper) and the name originally used in
\citet{Schaye2010}, respectively. The third and fourth columns list
the mass loading ($\eta$) and wind velocity
($v_{\rm w}$) parameters of each model. The WF2Dec is the only
model where wind gas particles are temporarily kinematically
decoupled from the surrounding gas. This aids the removal of gas
from galaxies and results in increased feedback efficiency.
The characteristic density $n_{\rm w}$ used for scaling the
WDENS wind parameters is that corresponding to
the star formation threshold: $n_c = 0.1$ cm$^{-3}$.}
\begin{tabular}{|c|c|c|c|}
\hline
Short name & OWLS name & $\eta$ & $v_{\rm w}$\\
& & [particles] & [km/s]\\
\hline
\hline
NoF & NOSN$\_$NOZCOOL & -- & --\\
ThF & WTHERMAL & -- & -- \\
WF4 & WML4V424 & 4 & 424 \\
WF2 & REF & 2 & 600 \\
WF1 & WML1V848 & 1 & 848 \\
WF2Dec & WHYDRODEC & 2 & 600 \\
WDENS & WDENS & $ 2 (n/n_{\rm w})^{-1/3} $ & $ 600(n/n_{\rm w})^{1/6}$ \\
AGN & AGN & 2 & 600\\
\hline
\end{tabular}
\label{tab:fpar}
\end{table}
\end{center}
\begin{center}
\begin{figure*}
\includegraphics[width=0.75\linewidth,clip]{figs/fig_xyz_gas_rvir.ps}
\caption{
Gas particles within the virial radius of four WF2 haloes spanning the
mass range of systems selected for analysis. The virial mass and
radius are given in the label of each panel. Gas particles are colored
according to temperature: red, green, and blue correspond to particles
in the hot, warm, and cold phases, respectively. Hot particles are
those with $T> (1/4) T_{\rm vir}$, where $T_{\rm vir}=35.9 (V_{\rm
vir}/$ km s$^{-1})^2$ K is the virial temperature of the halo. Cold
particles are those with $T< 3 \times 10^4$ K. Warm are those with
intermediate temperatures. Cyan particles denote dense, star-forming
gas in the PEOS phase. Particles are plotted sequentially in order of descending
temperature, so colder particles may occult hotter ones in regions of
high density. Small circles show the radius, $r_{\rm gal}=0.15 \, r_{\rm
vir}$, used to define the central galaxy.
}
\label{fig:GasRvir}
\end{figure*}
\end{center}
\begin{center}
\begin{figure*}
\includegraphics[width=0.75\linewidth,clip]{figs/fig_xyz_gas_rglx.ps}
\caption{
Zoomed-in view of the galaxies at the centers of the haloes shown in Fig.~\ref{fig:GasRvir}. Colors
are as described in the caption of that figure, except that yellow now
denotes ``star'' particles. The circles show the galaxy ``radius'', $r_{\rm gal}=0.15 \,
r_{\rm vir}$. Each box has been rotated so that the spin axis of the
PEOS gas is aligned with the $z$ axis of each panel. This
``edge-on'' projection emphasizes the presence of disk-like structures
in all four haloes. Besides $r_{\rm gal}$, labels in each panel specify
the baryonic mass of the galaxy, $M_{\rm gal}$, and the spin parameter
of the surrounding halo, $\lambda$.
}
\label{fig:GasStarsRgal}
\end{figure*}
\end{center}
\begin{center}
\begin{figure*}
\includegraphics[width=0.475\linewidth,clip]{figs/fig_xyz_samehalo_box1_faceon.ps}
\hspace{0.4cm}
\includegraphics[width=0.475\linewidth,clip]{figs/fig_xyz_samehalo_box2_faceon.ps}
\caption{ Face-on view of the central galaxy formed at the center of
an $M_{\rm vir}=1.2 \times 10^{12} \, h^{-1} \, M_{\odot}$ halo. All
panels correspond to the {\it same} halo, but in runs with different
feedback implementations, as labelled in the bottom right of each panel. Only baryonic
particles are shown. Colors indicate gas temperature, classified as
hot (red), warm (green), cold (blue) and star-forming (cyan). See
the caption to Fig.~\ref{fig:GasRvir} for details. Yellow dots
correspond to ``star'' particles. The circle in each panel indicate
the radius used to define the central galaxy, $r_{\rm gal}$. Each
galaxy has been rotated so that it is seen ``face on''; i.e., the
angular momentum of the PEOS gas is aligned with the line of sight
of the projection. The mass in stars and gas within $r_{\rm gal}$ is
labelled in each panel (units are $10^{10} \, h^{-1} M_\odot$). }
\label{fig:FaceOnGx}
\end{figure*}
\end{center}
\begin{center}
\begin{figure*}
\includegraphics[width=0.475\linewidth,clip]{figs/fig_xyz_samehalo_box1_edgeon.ps}
\hspace{0.4cm}
\includegraphics[width=0.475\linewidth,clip]{figs/fig_xyz_samehalo_box2_edgeon.ps}
\caption{ Same as Fig.~\ref{fig:FaceOnGx}, but each galaxy has been
rotated so that it is seen ``edge-on''. Labels in each panel give the
fraction of kinetic energy of the stellar component in ordered
rotation. }
\label{fig:EdgeOnGx}
\end{figure*}
\end{center}
\section{Numerical Results}
\label{sec:numres}
\subsection{The halo sample}
Our sample consists of all galaxies at the centers of haloes with
virial mass\footnote{Virial values are measured at or within the virial radius,
$r_{\rm vir}$, of a halo, defined as the radius where the mean inner
density exceeds the critical density of the universe by a factor
$\Delta_{\rm vir}(z)=18\pi^2+82f(z)-39f(z)^2$. Here $f(z)=[\Omega_{\rm
M}(1+z)^3/(\Omega_{\rm M}(1+z)^3+\Omega_\Lambda))]-1$ and $\Omega_{\rm
M}=\Omega_{\rm CDM}+\Omega_{\rm
bar}$ \citep{bryanandnorman98}. $\Delta_{\rm vir}\sim 170$ at $z=2$ for
our choice of cosmological parameters.}
$M_{\rm vir} > 10^{11} \, h^{-1}\, M_\odot$. There are about
$150$ haloes at $z=2$ in each $25 \, h^{-1}$ Mpc-box
{\small OWLS} run with masses between $10^{11} \,
h^{-1} \, M_\odot< M_{\rm vir} < 3\times 10^{12} h^{-1} \,
M_\odot$. The median of the sample is $M_{\rm vir} \sim 1.8\times
10^{11} h^{-1} \, M_\odot$. Halos are identified by the
substructure finding algorithm {\small SUBFIND} \citep{Springel2001a,
Dolag2009} and centers are defined by the minimum of the potential.
All runs use the same initial conditions, and therefore the number
(and identity) of haloes selected for analysis is roughly the same
in each simulation.
We begin with an overview of the properties of the gaseous component
within the virial radius and its halo mass dependence
(Sec.~\ref{SecGasRvir}), and follow on with a description of the
properties of the stellar component of the central galaxy
(Sec.~\ref{SecCentrGx}). We discuss the link between feedback and
morphology in Sec.~\ref{SecFbMorph}, and compare the number of massive
galaxies in various runs in Sec.~\ref{SecMassGx}. We end this section
by discussing the mass and angular momentum of central galaxies, as
well as their dependence on feedback (Secs.~\ref{SecMGx} and
~\ref{SecJGx}), before proceeding to compare these results with
observations.
\subsection{Gas within the virial radius}
\label{SecGasRvir}
Fig.~\ref{fig:GasRvir} illustrates the distribution of gas within the
virial radius in four haloes selected from the WF2 run at
$z=2$. Each panel corresponds to haloes differing by consecutive
factors of two in virial mass. The box size in each panel has been
adjusted to the virial radius of each halo. Only gas particles within
the virial radius are shown, and have been colored according to their
density/temperature.
Red particles are those with temperatures exceeding $(1/4) \, T_{\rm
vir}$, where $T_{\rm vir}=35.9 \, (V_{\rm vir}/$km s$^{-1})^2$ K is
the virial temperature of a halo ($V_{\rm vir}$ is the circular
velocity at $r_{\rm vir}$). Gas particles in this ``hot phase'' are
all found in a low-density, largely pressure-supported atmosphere that
fills the halo out to the virial radius. The virial temperature is
$\sim 10^6$ K for haloes with $V_{\rm vir} \sim 170$ km/s, about the
median virial velocity range spanned by our sample.
The fraction of gas in the hot phase increases with halo mass; it
makes up $68\%$ of all the gas within $r_{\rm vir}$ in the most
massive halo but only $21\%$ in the least massive system shown in
Fig.~\ref{fig:GasRvir} . This is a result of the steady increase in
cooling time with increasing halo mass, which favors the formation of
a hot tenuous gas atmosphere in massive systems.
Particles in green are those in the ``warm'' phase, which we define as
those satisfying $3\times 10^4$K$<T< (1/4) \, T_{\rm vir}$. These are
particles at moderate overdensities, and make up a small fraction of
all the gas within $r_{\rm vir}$; from $\sim 7\%$ in the most massive halo
to $\sim 15\%$ in the least massive one. This gas typically traces
material accreted relatively recently, which has yet to be pressurized
by shocks, or material ejected during accretion events in "tidal
tails" that expand and cool as they recede from the center. Because
accretion occurs frequently through filaments, and tidal tails are
likewise highly asymmetric, the warm component distribution is
non-uniform, with discernible large-scale features suggestive of
recent mergers and accretion events.
Cold ($T<3\times 10^4$ K) gas of moderate density ($n < n_c =0.1$
cm$^{-3}$) is shown in blue, and is rather clumpy in
appearance. Large-scale features similar to those noted for the warm
component are also visible here, suggesting that this is also mostly
gas recently accreted or affected by accretion events. In terms of
mass, this component is negligible ($\sim 5\%$) in the $3.7 \times
10^{12} \, h^{-1} \, M_{\odot}$ halo but increases in importance with
decreasing halo mass. Indeed, it makes up $\sim 30\%$ of all the gas
in the $3.1\times 10^{11} \, M_\odot$ halo shown in
Fig.~\ref{fig:GasRvir}.
The star-forming gaseous component is, by definition, the densest
($n > n_c = 0.1$ cm$^{-3}$), and is shown in cyan in
Fig.~\ref{fig:GasRvir}. Most of this gas is at the bottom of the
potential well of the main halo and of its substructure haloes, and
makes up between $20$ and $30\%$ of the gas within $r_{\rm vir}$, with
little dependence on halo mass.
The generally strong halo mass dependence of the various gaseous phases
highlights the different modes of accretion that shape the evolution of
a central galaxy. In massive haloes galaxies grow by accreting cooled
material from the surrounding reservoir of hot gas, whereas in low
mass haloes the gas is likely to flow virtually unimpeded to the
central regions, where it may be swiftly accreted into the central
galaxy \citep{White1991,Keres2005,Dekel2006,Birnboim2007,Keres2009,Brooks2009}.
These different accretion modes highlight the complex assembly
history of a galaxy, a complexity that is further compounded by the
effects of feedback that we discuss below.
\begin{center}
\begin{figure}
\includegraphics[width=84mm]{figs/fig_numberdens.ps}
\caption{ Number density of galaxies with stellar mass exceeding $5
\times 10^9 \, M_{\odot}$, shown for the various feedback
implementations explored in this paper. Runs are labeled as in
Table~\ref{tab:fpar}, and are listed in the abscissa roughly in
order of increasing feedback efficiency. The open circle correspond
to the WF2 low resolution run WF2LR. ``Error bars'' denote $\sqrt N$
uncertainties corresponding to the number of systems in our
computational box. The shaded area outline observational constraints
from estimates of the galaxy stellar mass function at $z=2$, as
compiled by \citet{Marchesini2009}. Note the strong decline in the
number of massive galaxies as a function of increasing feedback
efficiency. }
\label{fig:NumDens}
\end{figure}
\end{center}
\subsection{Central galaxies}
\label{SecCentrGx}
Fig.~\ref{fig:GasStarsRgal} shows a zoomed-in view of the four WF2
haloes depicted in Fig.~\ref{fig:GasRvir}, including the stellar
component, which is shown in yellow. The circle centered on the main
galaxy indicates the radius, $r_{\rm gal}=0.15 \, r_{\rm vir}$, that
we use to define the central galaxy inhabiting each halo. As is clear
from the figure, this definition includes virtually all stars and
dense gas obviously associated with the galaxy.
It also emphasizes the halo mass dependence of the various phases in
which baryons may flow into the central galaxy. As discussed above,
whereas galaxies in low mass haloes grow through the smooth accretion
of cold gas, a fair fraction of the star forming gas in the most
massive systems include ``clouds'' that condense out of the hot and
warm phases. Little star formation happens in these clouds, however,
since their typical densities are well below those reached in the main
body of the galaxy.
Gas turns swiftly into stars once it settles into a dense, thin,
rotationally supported disk in the central galaxy. In systems that
avoid major mergers, the stellar component inherits the disk-like
structure of the gaseous component. All 4 galaxies shown in
Fig.~\ref{fig:GasStarsRgal} sport well-defined stellar disks, which
have been rotated to be seen ``edge-on'' in this figure. Disks of gas
and stars are indeed quite common in the WF2 run that we have chosen
to illustrate the main general features of our simulated galaxies.
\subsection{Feedback and morphology}
\label{SecFbMorph}
Varying the feedback implementation has a dramatic effect on the
properties of central galaxies. We illustrate this in
Figs.~\ref{fig:FaceOnGx} and ~\ref{fig:EdgeOnGx}, where we show, for
the {\it same} dark matter halo, how the appearance of its central
galaxy varies with feedback. Although the assembly history of the dark
halo is identical in all cases, differences in feedback lead to
drastic variations in the stellar mass, gaseous content, and
morphology of the central galaxy.
When feedback is inefficient, such as in the ThF and WF4 runs, a
stellar disk is clearly present, but its mass is small compared with
that of the spheroidal component. This is because most stars form in
early collapsing protogalaxies which are later stirred into a
spheroidal component when these subsystems coalesce to form the final
galaxy. The extreme case is NoF, where the absence of feedback allows
for early and highly efficient star formation that converts most of
the available gas into stars. The large number of satellites seen
around the NoF central galaxy is also a result of the lack of
feedback. This preserves star formation even in small subhaloes, where
modestly energetic feedback might lead to drastic changes in the
availability of star formation fuel and in the total mass of stars
formed.
When feedback effects are strong, such as in the WF2Dec, WDENS, and
AGN runs, fewer stars form since the gas is constantly pushed out of
star-forming galaxies by outflows. These outflows also disrupt the
smooth settling of gas into disks and its gradual transformation into
stars. In the most extreme case (AGN), the gas outflows are so violent
that there is little gas left in the central galaxy. In none of these
cases do central galaxies have an extended and easily recognizable
stellar disk component.
As may be seen from Fig.~\ref{fig:EdgeOnGx}, more moderate feedback
implementations, such as WF2 and WF1, yield systems with a
well-defined stellar disk, and a gas/stellar mass fraction of roughly
$1$:$1$.
This impression is corroborated quantitatively by the fraction of {\it
stellar} kinetic energy in ordered rotation:
\begin{equation}
{\kappa_{\rm rot}^{\rm Star}=K_{\rm rot}/K; \phantom{ii}\rm with\phantom{ii} K_{\rm rot}=\sum (1/2) m (j_z/R)^2}
\end{equation}
Here, $m$
is the mass of a star particle; $j_z$ is the z-component of the
specific angular momentum, assuming that the z-axis coincides with the
angular momentum vector of the galaxy, and $R$ is the (cylindrical)
distance to the z-axis. $\kappa_{\rm rot}$ is listed in each panel of
Fig.~\ref{fig:EdgeOnGx} for the stellar component: it is highest
for WF2, and minimum for AGN.
\subsection{Feedback and massive galaxies}
\label{SecMassGx}
A robust way of assessing the effectiveness of the various feedback
implementations explored in these runs is to compute the abundance of
massive galaxies that each predicts. Because the total amount of stars
formed decreases as the feedback becomes more effective, the abundance
of massive galaxies is expected to depend sensitively on
feedback. This is shown in Fig.~\ref{fig:NumDens}, where we plot, for
each implementation, the number of galaxies (per unit volume) with
stellar masses exceeding $5 \times 10^9 \, M_{\odot}$. This mass
threshold is chosen to roughly coincide with $10,000$ baryonic
particles
The runs in Fig.~\ref{fig:NumDens} are ranked, from left to right, in
order of decreasing number of massive galaxies (i.e., increasing
feedback efficiency). This figure confirms the strong effect of
feedback on the abundance of massive galaxies. For example, the AGN
run has $\sim 16$ times fewer such galaxies than the run with only
thermal feedback, ThF, and $\sim 60$ fewer than NoF, the model without
feedback energy injection.
Not only does the total amount of feedback energy matter, but also the
manner in which it is injected. Indeed, large differences are also
obtained for models with the {\it same} feedback strength (as measured by
the total feedback energy per unit stellar mass formed), but that
differ in the combinations of mass loading and wind velocities: ThF, WF4,
WF2, WF1, WF2Dec and WDENS all assume that $40\%$ of the available
supernova energy is invested into winds, yet their predictions for
the number of bright galaxies differ by up to a factor of $\sim 4$.
The results do not seem to depend dramatically on numerical
resolution, as shown by the good agreement between the WF2 and WF2LR
runs (the latter is shown with an open symbol in
Fig.~\ref{fig:NumDens}). Reducing the number of particles by a factor
of eight (as in WF2LR) brings down the number of $M_{\rm gal}>5\times 10^{9} \,
M_{\odot}$ galaxies in the box from $133$ to $123$. The trends shown
in Fig.~\ref{fig:NumDens} are therefore unlikely to be an artifact of
limited numerical resolution.
The shaded band in Fig.~\ref{fig:NumDens} indicates the expected
number of massive galaxies, taken from \citet{Marchesini2009}, after
interpolating their fits to $z=2$ and correcting volume elements to
account for the different cosmology assumed in their work. The band
aims to represent uncertainties due to photometric redshift
inaccuracies, cosmic variance and systematics in the modeling. Notice
that the bright end of the luminosity function traces the abundance of
the most massive objects present at $z=2$ and is, as such,
particularly sensitive to the adopted cosmological parameters.
Given these large uncertainties, it would be premature to use
Fig.~\ref{fig:NumDens} to rule in or out any particular
implementation of feedback but, as the data improve, it might be
useful to revisit this issue to learn which feedback modeling
procedure is favored or disfavored by the data. For clarity, many of the
plots in the analysis that follows will focus on 4 cases that
span the full range of feedback strength shown in
Fig.~\ref{fig:NumDens}: i.e., NoF, WF2, WF2Dec, and AGN.
We end by noting that different observational diagnostics, such as the
specific star formation rate or the gas content as a function of
galaxy luminosity, could be used to provide further constraints on the
viability of each feedback model. We plan to present a detailed
analysis along these lines in a future paper (see Haas et al., {\it in
preparation}).
\begin{center}
\begin{figure*}
\includegraphics[width=0.475\linewidth,clip]{figs/fig_barfrac2.ps}
\hspace{0.4cm}
\includegraphics[width=0.475\linewidth,clip]{figs/fig_barfrac_mvir_j_box1.ps}
\caption{{\it Left:} Central galaxy mass as a function of virial mass
for all haloes selected in the WF2 run. The top dashed line indicates
the mass in baryons in each halo corresponding to the universal
baryon fraction, $f_{\rm bar}=\Omega_b/\Omega_M=0.175$, adopted in
the simulations. Dots correspond to the baryon mass of the central
galaxy (i.e.; within $r_{\rm gal}$); the thick solid (blue) curve
tracks its median as a function of $M_{\rm vir}$. The two bottom
curves track the median for the stellar (red, dot-dashed) and
gaseous (green, dashed) mass within $r_{\rm gal}$. The thick dotted
(magenta) line shows the median of the total mass in baryons within
the virial radius, $r_{\rm vir}$. {\it Right:} Galaxy formation
``efficiency'', $\eta_{\rm gal}=M_{\rm gal}/(f_{\rm bar} M_{\rm
vir})$, as a function of halo virial mass for the various
runs. For clarity, only the results corresponding to 4 selected runs
are shown, spanning the effective range of feedback strength, from
the ``no feedback'' (NoF) case to the AGN case, where feedback
effects are maximal. Cases not shown fall between these two
extremes. The scatter around each curve is large, typically
$\sim 0.19$ dex rms. The thin line labelled WF2LR has the same physics
are WF2 but $8 \times$ poorer mass resolution and $2 \times$ lower
spatial resolution. Note that the galaxy formation efficiency is, on
average, very sensitive to feedback, but only weakly dependent on
halo mass, at least for the range of masses considered here.}
\label{fig:BarM}
\end{figure*}
\end{center}
\subsection{Galaxy masses}
\label{SecMGx}
The stellar and gaseous masses of galaxies assembled at the centers of
dark matter haloes are determined largely by the virial masses of the
systems, modulated by the efficiency of radiative cooling and the
regulating effects of feedback. We show this in the left panel of
Fig.~\ref{fig:BarM} for all galaxies selected from the WF2 run. The
dots in the figure correspond to $M_{\rm gal}$, the total baryon mass
within the radius, $r_{\rm gal}$, used to define the central galaxy;
the solid line traces the median as a function of $M_{\rm vir}$. As
expected, the central galaxy mass correlates well with $M_{\rm vir}$,
albeit with fairly large scatter (the global rms about the median trend is
$\sim 0.19$ dex).
The top dashed line in this panel indicates the mass, $f_{\rm
bar}M_{\rm vir}$, galaxies would have if all baryons in the halo
have assembled at the center (the universal baryon fraction is
$f_{\rm bar}=\Omega_b/\Omega_{\rm M}=0.175$). The thick dotted magenta line
shows the median baryon mass within $r_{\rm vir}$ as a function of
halo mass. This shows that massive systems have retained all baryons within the
virial radius, but also that the effects of feedback are clear at the low mass end:
$10^{11} \, h^{-1} \, M_\odot$ haloes have only retained about
half of their baryons within the virial radius. Of those, only one
third or so have collected in the central galaxy.
Thus, the ``efficiency'' of galaxy formation, as measured by the mass
of the galaxy expressed in units of the total baryon mass
corresponding to its halo, $\eta_{\rm gal}=M_{\rm gal}/(f_{\rm bar}\,
M_{\rm vir})$, increases steadily with halo mass, from $\sim 10\%$ in
$10^{11} \, h^{-1} M_{\odot}$ haloes to a maximum of roughly $40\%$
for $M_{\rm vir} \sim 5 \times 10^{11} \, h^{-1} M_\odot$. There is
also indication that the efficiency decreases in more massive
systems, to roughly $\sim 30\%$ in the most massive haloes.
These trends (i.e., low galaxy formation efficiency in low and high
mass haloes) are qualitatively in line with what is required to
reconcile the shape of the galaxy luminosity function with the dark
matter halo mass function \citep[see,
e.g.,][]{Yang2005,Conroy2009,Guo2010}. Feedback is the main mechanism
responsible for reducing efficiency in low-mass haloes. Together
with long cooling timescales, it also helps prevent the formation of too
massive galaxies in high-mass haloes.
Although the trends seem qualitatively correct, it remains to be seen
whether a model like WF2, evolved to $z=0$, is able to satisfy the
stringent constraints placed by the stellar mass function in the local
Universe. Indeed, the recent estimate of \citet{Li2009} suggest
that only $3.5\%$ of all baryons in the Universe are today locked up
in stars, and \citet{McCarthy2009} argue that supernova feedback alone
is not enough to ensure such a low efficiency of transformation of
baryons into stars. Since the runs we analyze here have only been evolved
to $z=2$, we are unable to address this issue in a conclusive manner,
but we plan to return to it when extending the present analysis to
the {\small GIMIC} simulations.
\subsubsection{Feedback dependence}
The right panel of Fig.~\ref{fig:BarM} shows that the overall
efficiency of galaxy formation is quite sensitive to feedback. Each
curve here tracks the median trend of $\eta_{\rm gal}$ with $M_{\rm
vir}$ for different runs. As expected from the discussion of
Fig.~\ref{fig:NumDens}, $\eta_{\rm gal}$ is highest for NoF and lowest
for AGN, with more moderate results for WF2 and WF2Dec, as well as the
other runs, which are omitted from this panel for clarity.
Clearly, not only the total feedback energy input, but also the
details of its implementation can affect dramatically the galaxy
formation efficiency. Central galaxies in the NoF case can be up to
$10\times$ more massive than in the AGN run. WF2 galaxies are a factor
of two to three more massive than those formed in WF2Dec, although the
only difference between these two runs is the choice to ``decouple''
hydrodynamically the supernova-driven winds in the latter. The
reasonable agreement (given the large scatter) between WF2 and WF2LR suggests that this result
is not unduly influenced by numerical resolution.
Although the average galaxy formation efficiency depends strongly on
feedback, its dependence on halo mass is weak: $\eta_{\rm gal}$ varies
by less than a factor of $2$ over the factor of $\sim 30$ range in
virial mass spanned by the simulations. In the absence of feedback
$\eta_{\rm gal}$ peaks at low masses: feedback is clearly needed to
counter the high efficiency of gas cooling in low mass haloes.
We highlight however that the dependence of $\eta_{\rm gal}$ on
halo mass must become significantly stronger for halo masses below
those studied here (i.e. $M_{\rm vir} < 10^{11} h^{-1}M_\odot$) in order to
successfully reproduce the measured faint end of the luminosity function.
\begin{center}
\begin{figure*}
\includegraphics[width=0.475\linewidth,clip]{figs/fig_mvir_j.ps}
\hspace{0.4cm}
\includegraphics[width=0.475\linewidth,clip]{figs/fig_barfrac_mvir_j_box2.ps}
\caption{ {\it Left:} Specific angular momentum, $j$, as a function of
virial mass. The black dashed line tracks the median $j$ of the
dark matter component as a function of $M_{\rm vir}$. This follows
closely the $j \propto M^{2/3}$ correlation expected for systems
with constant spin parameter, $\lambda$. The other symbols, colors,
and line types are the same as in Fig.~\ref{fig:BarM}. Note that the
specific angular momentum of {\it all} baryons within $r_{\rm vir}$
is quite similar to that of the halo as a whole (top dotted
curve). The specific angular momentum of the central galaxy is
typically lower than that of the halo; although it correlates well
with $M_{\rm vir}$, the scatter is large. {\it Right:} Angular
momentum ``efficiency'', $\eta_j=j_{\rm gal}/j_{\rm vir}$, as a
function of mass for various runs. For clarity, only the median of
galaxies in runs NoF, WF2, WF2Dec, and AGN, are shown as a function
of mass. Note that, unlike $\eta_{\rm gal}$, the angular momentum
efficiency, $\eta_j$, is a weak function of both mass and of
feedback. See text for further discussion. }
\label{fig:BarJ}
\end{figure*}
\end{center}
\subsection{Galaxy Angular Momentum}
\label{SecJGx}
The size and rotation speed of galaxy disks place powerful
observational constraints on galaxy formation models, and are directly
linked to the angular momentum acquired and retained by the baryons
that make up the galaxy. We explore this in Fig~\ref{fig:BarJ}, where
we show, in the left panel, the specific angular momentum of the
various galaxy components as a function of halo virial mass.
As in Fig.~\ref{fig:BarM}, dots correspond to the baryonic component
inside $r_{\rm gal}$ for individual systems in run WF2. Although the
scatter is large (an rms of $\sim 0.27$ dex), the solid curve, which
tracks the median $j$ as a function of $M_{\rm vir}$, shows that the
specific angular momentum scales roughly like $j \propto
M^{2/3}$. This is the same scaling found for the dark matter component
within $r_{\rm vir}$ (dashed black line), and is indeed the expected
scaling if the dimensionless halo spin parameter,
$\lambda=J|E|^{1/2}/GM^{5/2}$, is constant.
When all baryons within the virial radius are considered, their
specific angular momentum agrees well with that of the dark matter
(magenta dotted line). On the other hand, the specific angular
momentum of central galaxies is, on average, about $50\%$ that of its
surrounding halo. This fraction, which we refer to as the ``angular
momentum efficiency'', $\eta_j=j_{\rm gal}/j_{\rm vir}$, appears, on
average, to be roughly independent of halo mass for WF2 galaxies.
Interestingly, the gaseous and stellar components of WF2 galaxies have
distinctly different angular momenta. The gas has $2$ to $3$ times
larger specific angular momentum than the stars, implying that the
radial extent of gaseous disks in these galaxies is substantially
larger than that of the stellar component. This was already noted by
\citet{Sales2009} as a possible way to explain the large sizes of the
$z=2$ star-forming (gaseous) disks analyzed by
the SINS survey \citep{Forster2009}. We shall return to this issue in
Sec.~\ref{sec:obsdiag}.
\subsubsection{Feedback dependence}
The feedback dependence of the angular momentum efficiency, $\eta_j$,
is shown in the right panel of Fig.~\ref{fig:BarJ} as a function of
virial mass. Like the galaxy formation efficiency, $\eta_{\rm gal}$,
the halo mass dependence of $\eta_j$ is weak. Unlike $\eta_{\rm gal}$,
however, $\eta_j$ depends only weakly on feedback. In a given halo,
NoF galaxies have approximately the {\it same} angular momentum as
galaxies in the AGN run. This is striking, since their baryonic masses
differ on average by a factor of $\sim 10$. Feedback affects the
mass of a galaxy much more severely than its spin: 9 out of 10 baryons
in NoF galaxies are missing from AGN galaxies, but their specific
angular momenta are, on average, the same.
The thin line in the right panel of Fig.~\ref{fig:BarJ} shows the
results for WF2LR. Despite the large scatter, numerical resolution
effects are clearly noticeable below $\sim 4 \times 10^{11} h^{-1} \,
M_{\odot}$. This corresponds to $\sim 10^4$ particles per halo for WF2LR;
extrapolating this to WF2, it would mean that our results there should
be credible down to $\sim 5 \times 10^{10} h^{-1} \, M_{\odot}$. It
would therefore appear as if the main trends shown in
Fig.~\ref{fig:BarJ} are safe from resolution-induced numerical
artifacts.
The angular momentum efficiency peaks in moderate feedback runs (such
as WF2) at roughly $50\%$ and its dependence on feedback strength is
non-monotonic. Despite this apparent complexity, galaxy masses and
angular momenta are actually well correlated. Following
\citet{Sales2009}, we define the mass and angular momentum {\it
fractions} $m_d$ and $j_d$, as
\begin{equation}
m_d=\eta_{\rm gal} \, f_{\rm bar}=M_{\rm gal}/M_{\rm vir},
\label{eq:md}
\end{equation}
and
\begin{equation}
j_d={J_{\rm gal} \over J_{\rm vir}}={M_{\rm gal} \, j_{\rm gal} \over
M_{\rm vir} \, j_{\rm vir}} = \eta_{\rm gal}\eta_{\rm j} f_{\rm bar}
\label{eq:jd}
\end{equation}
These parameters were introduced by \citet{Mo1998}, and have become
standard fare in semianalytic models of disk galaxy formation.
\citet{Sales2009} noted that $j_d$ and $m_d$ correlate well, but in a
manner different from the typical $j_d=m_d$ assumption of
semianalytic models \citep[e.g.,][]{Cole2000} and, perhaps more
importantly, insensitive to feedback. These authors showed that the
simple expression
\begin{equation}
j_d=9.71 \, m_d^2\, (1-\exp[-1/(9.71\,m_d)])
\label{eq:jdmd}
\end{equation}
provides a good approximation to the results of four {\small OWLS} runs with
supernova-driven winds: WF1, WF2, WF4 and WF2Dec.
We revisit this result in Fig.~\ref{fig:JdMd}, where we show the
$j_d$-$m_d$ correlation for all the {\small OWLS} runs considered in this
paper. The dots show individual WF2 galaxies, and are meant to
illustrate the typical scatter in the relation; the curves trace the median
trend of $j_d$ with $m_d$ for the different runs while the black dotted curve outline
the relation in Eq.~\ref{eq:jdmd}. Although the AGN and NoF galaxies deviate
somewhat from the trend outlined in eq.~\ref{eq:jdmd} (indicated by
the dotted thick line), the departures are relatively small and the agreement
between runs seems remarkable given the extreme range in feedback models
explored here.
The bottom panel in Fig.~\ref{fig:JdMd} shows the {\it distribution}
of $m_d$ for four different feedback implementations. Clearly,
feedback, at least as implemented in our models, affects mostly the
baryonic mass of galaxies, but largely preserves the link between the spins of
haloes and galaxies. This link imprints correlations between galaxy mass,
size, and rotation speed that may be contrasted with observations. We
turn to this issue next.
\begin{center}
\begin{figure}
\includegraphics[width=84mm]{figs/figpaper_jdmd_allsphs.ps}
\caption{ {\it Top:} The angular momentum fraction $j_d=J_{\rm
gal}/J_{\rm vir}$ vs the galaxy mass fraction, $m_d=M_{\rm
gal}/M_{\rm vir}$. Dots correspond to individual galaxies in WF2
and are meant to illustrate the scatter;
the median trend is traced by the black solid line. Other curves are
analogous, but for each feedback model analyzed here. The black
dotted curve is the fit proposed by \citet{Sales2009}. The straight
line labeled $j_d=m_d$ corresponds to the commonly-adopted assumption
that the specific angular momentum of a galaxy equals that of its
surrounding halo. {\it Bottom:} Distribution of galaxy mass
fraction, $m_d$, for four different runs spanning the range of
feedback strengths of our simulations: NoF, WF2, WF2Dec, and AGN. }
\label{fig:JdMd}
\end{figure}
\end{center}
\section{Observational Diagnostics}
\label{sec:obsdiag}
\subsection{Size and Stellar Mass of $z=2$ Galaxies}
\label{ssec:rhmstr}
The feedback-driven trends of galaxy mass and angular momentum
efficiencies discussed above imprint different relations between
the stellar mass, $M_{\rm str}$, and the size of a galaxy. This is
shown in Fig.~\ref{fig:RhMstr}, where the various panels compare (for
runs NoF, WF2, WF2Dec, and AGN) the half-mass radius of the galaxy
versus $M_{\rm str}$. The panels on the left show the the half-mass
radius of the gas component whereas those on the right correspond to
the stars. The thick solid curve in each panel traces the median trend
as a function of $M_{\rm str}$. As noted above, simulated galaxies are
substantially more extended in gas than in stars.
The simulated galaxies are contrasted with data for the large
star-forming gas disks studied by the SINS survey
\citep[][hereafter FS09]{Forster2009}, as well as with the quiescent
compact red galaxies of \citet[][hereafter vD08]{vanDokkum2008}. These
two datasets probably bracket the extremes in the size distribution of
massive galaxies at $z=2$, from the most extended to the most compact.
When feedback is inefficient (e.g., the NoF run) most stars form in
dense, early-collapsing progenitors that merge later on to form the
final galaxies. During such mergers the baryonic component transfers
angular momentum to the surrounding halo, leading to the formation of
very compact massive galaxies
\citep{Navarro1991,Navarro1995,NavarroSteinmetz1997}. The galaxies
that result are therefore nearly as compact as the quiescent vD08
spheroids, although we note that many of those simulated galaxies have
half-mass radii even smaller than the gravitational softening of our
simulations, so their true sizes are actually uncertain. The gaseous
component in these simulations is also quite compact, with radii rarely
matching those of SINS disks.
Intermediate strength feedback (e.g., the WF2 run) has little effect on the most
massive galaxies, which are generally as compact as the vD08 spheroids. On
the other hand, feedback affects more strongly less massive systems,
leading to a correlation between the mass and size of the stellar
component where, at the massive end, size decreases with increasing
mass. This trend runs counter the well-established galaxy scaling laws
at z=0 (brighter galaxies tend to be bigger). The trend is reversed at
lower masses and results, overall, in systems whose gaseous disks
overlap in properties with those of galaxies in the SINS survey.
Increasing the effects of feedback (as in the WF2Dec and AGN runs)
continues this trend, gradually reducing the mass of galaxies and
increasing their size at given $M_{\rm str}$. This is because the more
efficient the feedback the more massive the halo inhabited by a galaxy
of given stellar mass. More massive haloes are larger and have higher
specific angular momenta. Since, as we saw above, galaxies generally
inherit the specific angular momenta of their surrounding haloes, it is
possible to have fairly large galaxies of modest stellar mass because
they actually inhabit large, massive haloes. Indeed, many gaseous disks
in the AGN run are even more extended than the rather extreme examples
surveyed by SINS.
\begin{center}
\begin{figure*}
\includegraphics[width=0.475\linewidth,clip]{figs/mstr_rh_sphs3_gas.ps}
\hspace{0.4cm}
\includegraphics[width=0.475\linewidth,clip]{figs/mstr_rh_sphs3_str.ps}
\caption{ {\it Left:} half-mass radius of the gas as a function of
stellar mass. Solid black dots in each panel show the results for
four of our simulations NoF, WF2, WF2Dec and AGN. The thick solid
line tracks the median as a function of mass. Open symbols with
error bars correspond to the extended star-forming disk galaxies
from the SINS survey \citep{Forster2009}, while the red shaded
ellipsoid indicates the area of the plot occupied by the sizes of
the compact quiescent galaxies from \citet{vanDokkum2008}. {\it
Right:} same as before, but for the half-mass radii of the stars. In
this case, open symbols and error bars are used to indicate the
sizes of the compact stellar spheroids from \citet{vanDokkum2008},
while the shaded blue area indicates the region of the plot occupied
by extended gaseous disks from the SINS sample. The extended disks
reported by SINS \citep{Forster2009} and the compact spheroidal
galaxies from \citet{vanDokkum2008} probably bracket the size
distribution of massive galaxies at $z=2$. The black shaded area
indicates the gravitational softening of the simulations. Half-mass
radii for the gaseous components of simulated galaxies are typically
larger than for the stars. Note that the size-stellar mass
correlation is heavily dependent on feedback. When feedback is very
efficient (e.g., WF2Dec/AGN) the size of the gaseous disks increase
with stellar mass, a correlation that is reversed when feedback
efficiency is low.}
\label{fig:RhMstr}
\end{figure*}
\end{center}
\subsection{The Tully-Fisher Relation at $z=2$}
\label{ssec:tf}
The structural diversity of $z=2$ galaxies discussed above should also
be manifest in their kinematics. We explore this in Fig.~\ref{fig:TF},
where we plot, as a function of stellar mass, the circular velocity
estimated for SINS galaxies and for the compact vD08 galaxies. For
SINS, we use the ``maximum'' gas rotation speed, as quoted by FS09,
whereas for vD08 we estimate the circular velocity at the
effective radius based only on the contribution of the stellar
component; i.e., $V_c^2=G(M_{\rm str}/2)/R_{\rm eff}$. This is clearly
a {\it lower limit} to the circular velocity at that radius, since it
neglects the possible contributions of dark matter and gas
components. We note this in Fig.~\ref{fig:TF} by small arrows on the
vD08 data points (open triangles).
It is clear from this rendition of the data that the two populations
of $z=2$ galaxies follow very different Tully-Fisher relations. At
given stellar mass, the compact galaxies are expected to have circular
velocities at least {\it twice} higher than SINS disks. Although
kinematic data for such galaxies is scarce, \citet{vanDokkum2009}
report a preliminary measurement of the velocity dispersion of one of
these galaxies. The high velocity dispersion reported, $\sim 510$
km/s, agrees with this interpretation.
The circular velocity of the simulated galaxies is measured at the
half-mass radius of the stellar (red solid curve) or the gaseous (blue
dashed curve) component, respectively. The comparison between
simulations and observations yields similar conclusions as in the
previous subsection.
Inefficient or absent feedback (e.g., NoF) yields galaxies that are
more concentrated than the SINS disks, and therefore have, at given
stellar mass, typically higher circular velocities. Forming large,
extended disks is difficult in the absence of efficient feedback. By
contrast, accounting for the compact spheroids studied by vD08 is
relatively easy.
In the case of AGN or WF2Dec, the most efficient feedback schemes
explored in Fig.~\ref{fig:TF}, many simulated galaxies are as
spatially extended as the SINS disks, and the good agreement extends
to the Tully-Fisher relation for those galaxies. Few very massive
galaxies form as a result of the efficient feedback, and very few of
those that form are as compact as those in the vD08 sample.
More moderate feedback choices give intermediate results. We consider
it encouraging that some galaxies in the WF2 runs overlap with both
SINS and vD08 in Figs.~\ref{fig:RhMstr} and ~\ref{fig:TF}. If these
models are correct, then there should be a sizable population of
galaxies at $z=2$ with properties intermediate to the SINS disks and
vD08.
To summarize, the results shown in Figs.~\ref{fig:RhMstr} and
~\ref{fig:TF} indicate that neither the extreme compact sizes of
massive spheroids nor the large spatial extent of star-forming disks
at $z=2$ pose insurmountable challenges to the standard
paradigm. Indeed, it is possible, with adjustments to the feedback
algorithm, to reproduce either population without resorting to unusual
halo spin or halo formation histories.
At the same time, reproducing the striking diversity in the observed
sizes and masses of $z=2$ galaxies with a single feedback recipe might be
challenging, but we are encouraged by the large scatter in the
properties of simulated galaxies at given stellar mass that arises
naturally in {\it any} feedback model. The relative
abundance of either population is still poorly constrained
observationally, and our small simulation box might not be adequate to
study or search for rare, extreme populations. Improved observational
constraints on the relative abundance of extended vs compact galaxies
and a better characterization of the ``average'' population of $z=2$
galaxies will certainly help to constrain which feedback
implementation gives results that agree best with observation.
\begin{center}
\begin{figure}
\includegraphics[width=\linewidth,clip]{figs/tf_v6.ps}
\caption{ The stellar mass-circular velocity (Tully-Fisher) relation
for $z=2$ galaxies identified in runs with four different feedback
implementations. Symbols are as in Fig.~\ref{fig:RhMstr}. The median
circular velocity measured at the stellar half-mass radius is shown
by the solid red line. Vertical lines show the 25-75 percentiles of
the distribution. We also show this relation when the circular
velocity is measured at the half-mass radius of the star forming gas
(dashed blue curve). Open circles and triangles show the
observational determinations for disks and compact galaxies at z=2
taken from \citet{Forster2009} and \citet{vanDokkum2008}. For the
latter we assign velocities by neglecting the dark matter
distribution; i.e., we assume $V_c^2=G(M_{\rm str}/2)/R_{\rm eff}$,
which constitutes a lower limit to the true circular velocity. This
is indicated by the horizontal arrows in each panel. The thick
dotted line is the \citet{Bell2001} relation for late-type galaxies
at z=0 corrected to a Chabrier IMF.}
\label{fig:TF}
\end{figure}
\end{center}
\subsection{Disks and Mergers at $z=2$}
\label{ssec:dskmrg}
Another interesting constraint is provided by the frequency of systems
actively forming stars in rotationally-supported disks. Before
surveys such as SINS and OSIRIS \citep[][ and references
therein]{Forster2009,Law2009} started to resolve the kinematics of
star-forming galaxies at high $z$, it had been commonplace to assume
that systems where star formation was progressing in earnest would
almost invariably be ongoing major mergers. It is now clear, however,
that at least about one third of the galaxies surveyed by SINS and
OSIRIS are forming stars in relatively quiescent disks
rather than ongoing mergers with disturbed and transient kinematics
\citep[for an alternative view, however,
see][]{Robertson2008}.
We use $\kappa_{\rm rot}$, the simple measure of the importance of
ordered rotation introduced in Sec.~\ref{SecFbMorph}, to explore this
issue in our simulations. When most of the gas is in a
rotationally-supported disk, the parameter $\kappa_{\rm rot}$ should
approach unity. Fig.~\ref{fig:RotMorph} enables a visual calibration
of this parameter by showing edge-on projections of 12 galaxies
arranged by the value of $\kappa_{\rm rot}$ of the central galaxy (in
this case only the star-forming gas is used to compute $\kappa_{\rm
rot}$). Figure~\ref{fig:RotMorph} shows an image of the projected gas
density within a sphere of radius $1.3 \, r_{\rm gal}$. Thin, extended
disks are the norm when $\kappa_{\rm rot} \lower .75ex \hbox{$\sim$} \llap{\raise .27ex \hbox{$>$}} 0.75$. Ongoing mergers
typically have $\kappa_{\rm rot} \lower .75ex \hbox{$\sim$} \llap{\raise .27ex \hbox{$<$}} 0.5$; those with intermediate
values of $\kappa_{\rm rot}$ have disturbed morphologies, and tend to
be late-stage mergers or systems where accretion is ongoing but minor.
Using this simple measure, the fraction of ongoing mergers vs
quiescent disks may be readily estimated, and is shown in the top
panel of Fig.~\ref{fig:HistErot} for the case of WF2. The
distribution of $\kappa_{\rm rot}$ for all WF2 galaxies is shown by
the top histogram; the shaded histogram is for the same run, but
reducing the sample of galaxies to one half by selecting only those in
haloes more massive than $2 \times 10^{11} h^{-1}
M_{\odot}$. Encouragingly, the shape of the two histograms is quite
similar. This is further confirmed by the distribution of $\kappa_{\rm
rot}$ in WF2LR galaxies (for $M_{\rm vir}>2\times 10^{11} h^{-1} \,
M_{\odot}$) which is shown as the thin solid line in the bottom panel of
Fig.~\ref{fig:HistErot} . The good agreement between WF2 and WF2LR
indicates that numerical resolution effects are unlikely to compromise
our conclusions.
According to the definition above, about $45\%$ of WF2 galaxies are
reasonably quiescent star-forming disks, and only about $20\%$ are
ongoing major mergers. These fractions are similar for WF2 and WF1,
and seem consistent with the observational data quoted above.
For the run without feedback, NoF, over $\sim 75\%$ of the galaxies
are classified as disks. This is because, in the absence of feedback,
the gas cools and flows unimpeded to the center, where it settles into
disks and forms stars profusely. These disks are, however, quite small
(see Fig.~\ref{fig:RhMstr}). The absence of effective feedback allows
the gas to remain undisturbed in such disks, which are quickly
reconstituted after mergers \citep[see, e.g.,][]{Springel2005d,Robertson2006}. At
the other extreme, only $5\%$ of all galaxies in the AGN run, and
$\sim 20\%$ of those in the WF2Dec run, would be classified as disks
according to this criterion.
Strong feedback-driven winds can clearly disturb quiescent disk
morphologies, and their kinematic effects may be difficult to
disentangle from those of ongoing mergers. It remains to be seen
whether a simple feedback model can account for both the observed
frequency of galaxies with disk-like kinematics as well as the
mounting evidence for large scale galactic outflows at $z\sim 2$
\citep[][]{Steidel2010}.
\begin{center}
\begin{figure*}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0001_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0019_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0055_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0085_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0048_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0066_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0046_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0072_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0047_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0056_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0011_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0057_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0032_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0034_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0069_axis2.ps}
\includegraphics[width=0.2375\linewidth,clip]{figs/fig11/loggasoverdens__halo0022_axis2.ps}
\caption{ Edge-on view of galaxies spanning a wide range in rotational
support taken from the WF2 run. Each panel is labelled by the value of $\kappa_{\rm rot}$
of the star-forming gas. Colors are assigned according to the (projected)
logarithmic densities of the gas. Well-defined disk systems are
apparent when $\kappa_{\rm rot} \lower .75ex \hbox{$\sim$} \llap{\raise .27ex \hbox{$>$}} 0.75$; lower values of this
parameter indicate ongoing mergers and/or systems with disturbed
morphology. Solid circles indicate the region selected as the galaxy radii:
$r_{\rm gal} = 0.15 \, r_{\rm vir}$.}
\label{fig:RotMorph}
\end{figure*}
\end{center}
\section{Summary and Conclusions}
\label{sec:conc}
We study the effects of various feedback implementations on the
structure and morphology of simulated galaxies at $z=2$. Our analysis
uses nine runs from the OverWhelmingly Large Simulations ({\small
OWLS}) project, and probe a variety of possible feedback
implementations, from ``no feedback'' to supernova-driven wind
feedback to strong outflows aided by the contribution from AGNs.
Except for the no-feedback and AGN-feedback cases, all other runs
assume that the {\it same amount} of feedback energy (per mass of
stars formed) is devolved by supernovae to the interstellar medium:
the main difference is {\it how} this energy is coupled to the medium,
which in turn determines the overall effectiveness of the feedback.
Each run follows the evolution of the {\it same} $25 \, h^{-1}$ Mpc
box up to $z=2$, with $512^3$ dark matter particles and $512^3$
particles for the baryonic component. All other simulation parameters
(star formation algorithm, stellar initial mass function, etc) are
kept constant, so any differences between runs may be traced solely to
feedback. In total, we analyze for each run $\sim 150$ galaxies formed
at the centers of haloes with virial mass in the range $10^{11} \,
h^{-1} \, M_\odot< M_{\rm vir} < 3\times 10^{12} h^{-1} \,
M_\odot$. Our main results may be summarized as follows.
\begin{itemize}
\item
Varying the feedback implementation can lead to dramatic differences in
the mass of galaxies formed in a given dark matter halo. The galaxy
formation efficiency, $\eta_{\rm gal}=M_{\rm gal}/(f_{\rm bar}
M_{\rm vir})$, varies by roughly an order of magnitude when
comparing the no-feedback run (NoF, where $\eta_{\rm gal}\sim 0.5$) to
the AGN+supernova feedback run (AGN, where $\eta_{\rm gal} \sim 0.05$), the
two extremes probed by our simulations.
\item The ability of feedback to regulate the efficiency of galaxy
formation in haloes of different mass varies according to the details
of the adopted numerical implementation of the feedback. Weak or
ineffective feedback leads to a decrease in galaxy formation
efficiency with mass, whereas strong feedback curtails
preferentially the formation of galaxies in low-mass haloes. The
mass dependence is, however, modest, with variations in $\eta_{\rm gal}$ of less than
a factor of $\sim 2$ over the (factor of $\sim 30$) mass range
spanned by haloes in our sample.
\item Feedback results in strong correlations between galaxy mass and
angular momentum. This leaves an imprint on galaxy
morphologies and on the scaling laws relating mass, size, and
circular velocity.
\item Weak feedback minimizes disturbances to the settling of gas in
rotationally-supported structures, and favors the formation and
survival of quiescent {\it gaseous} disks. However, weak feedback
also allows much of the gas to form stars early in dense
protogalactic clumps that are later disrupted in mergers as the
final galaxy assembles. Such mergers also transfer angular momentum
from the baryons to the halo. The net result is a predominance of
dense, spheroid-dominated stellar components and a scarcity of
spatially-extended star-forming disks.
\item Strong feedback, on the other hand, promotes the formation of
large, extended galaxies. Indeed, the more efficient the feedback the
more massive (and therefore, larger) the halo inhabited by a galaxy
of given stellar mass. It is thus possible to have fairly large
galaxies of modest stellar mass because, when feedback is strong,
they inhabit large, massive haloes. The size, mass, and rotation speeds of these extended galaxies
compare favorably with those reported by the SINS survey. This,
however, comes at the expense of inhibiting the survival of
rotationally-supported disks of quiescent kinematics and of
preventing the formation of compact stellar spheroids.
\item Moderate-feedback runs result in galaxies that follow scaling
laws that are intermediate between large star-forming disks, such as
those studied by the SINS collaboration \citep{Forster2009}, and the
compact, quiescent early-type systems analyzed by
\citet{vanDokkum2008}. Disk-like morphologies in both gas and stars
are common in these runs, in numbers that appear commensurate with
current constraints.
\end{itemize}
Although far from definitive, the results outlined above are encouraging.
Properly calibrated, simple feedback recipes such as the ones we explore
here seem able to produce galaxies with properties in broad agreement with
observation. One should be aware, however, of the numerical sensitivity of
the results to details of feedback implementation. Nevertheless, if
developed in step with observational progress in the characterization of
the high-redshift galaxy population, simulations are likely to become more
and more reliable tools, useful when trying to make sense of the striking
diversity of high-z galaxies in terms of the current paradigm of structure
formation.
\begin{center}
\begin{figure}
\includegraphics[width=84mm]{figs/fig_angmom_ekinet.ps}
\caption{ Upper panel shows the histogram of $\kappa_{\rm rot}=K_{\rm rot}/K$, the fraction
of kinetic energy of star-forming gas particles in ordered rotation
for all galaxies in the WF2 run. The shaded red histogram is the same,
but only for the half most massive, and therefore best numerically
resolved, systems. The similarity between the two suggests that
numerical resolution does not play a significant role in the
statistics. $\kappa_{\rm rot}$ should be approximately unity for a
disk where all particles are in circular orbits and much smaller for
systems where ordered rotation plays a less important role. The
large number of systems around $\kappa_{\rm rot} \sim 0.8$ indicates
that systems where star formation occurs in well-defined disks are
quite common in this run (see Fig.~\ref{fig:RotMorph} for
examples). The cumulative fraction of systems as a fraction of
$\kappa_{\rm rot}$ for the four different feedback implementations
are shown in the bottom panel. A trend for gaseous disks becoming more
prevalent as feedback efficiency decreases is clearly seen.}
\label{fig:HistErot}
\end{figure}
\end{center}
\section*{Acknowledgements}
\label{acknowledgements}
LVS thanks the hospitality of the University of Massachusetts and
Kavli Institute for Theoretical Physics, Santa Barbara, where part of
this work was completed. LVS is grateful to Amina Helmi, Marcel Haas,
Natasha F{\"o}rster Schreiber and Thiago Gon\c{c}alvez for useful
comments and discussions, as well as to Freeke van de Voort for help
with the plotting routine used in Figure 11. LVS also acknowledge
Amina Helmi, NWO and NOVA for financial support. This research was
also supported in part by the National Science Foundation under Grant
No. PHY05-51164. The simulations presented here were run on Stella,
the LOFAR BlueGene/L system in Groningen, on the Cosmology Machine at
the Institute for Computational Cosmology in Durham as part of the
Virgo Consortium research programme, and on Darwin in Cambridge. This
work was sponsored by National Computing Facilities Foundation (NCF)
for the use of supercomputer facilities, with financial support from
the Netherlands Organization for Scientific Research (NWO). This work
was supported by Marie Curie Excellence Grant MEXT-CT-2004-014112 and
by an NWO VIDI grant. We thank useful comments from the anonymous
referee that helped to improve the presentation and clarity of this
paper.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 335 |
Q: How do I set a start time for a zoompan filter that overlays a video background in ffmpeg The below code works fine but I cannot seem to succeed at starting this image zoom at a specific time. i.e I want the zoom to be effective at 5 seconds not immediately when the background video starts.
ffmpeg -i background.avi -i image.png \
-filter_complex "1:v]scale=8000x4000,setsar=1/1,zoompan=z='min(zoom+0.005,10)':d=125:s=530x680,trim=duration=3[v1];[0:v][v1]overlay=20:20" \
-c:v libx264 output.avi
A: Use
ffmpeg -i background.avi -loop 1 -i image.png \
-filter_complex
"[1:v]scale=8000x4000,setsar=1/1,zoompan=z='if(gte(in,125),min(pzoom+0.005,10),1)':d=1:s=530x680,
trim=duration=3[v1];[0:v][v1]overlay=20:20"
-c:v libx264 output.avi
Since you're applying this on a video, pzoom is the correct variable to use. d should be set to 1 since it represents the duration of the zoom effect interval for each individual frame. Be sure to use a recent ffmpeg version. There was a bug in earlier versions where pzoom did not function correctly.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,889 |
\section{Introduction}
Dwarf spheroidal (dSph) galaxies are believed to be the most common type of galaxies
in the Universe, and the building blocks of more massive galaxies in hierarchical
formation scenarios.The large mass-to-light ratios of these dSph galaxies suggest
that the dynamics of such galaxies are dominated by dark matter (DM). Therefore,
the study of dSphs is crucial for a better understanding of the nature of DM. In particular,
the dSph of the Local Group (LG) of galaxies satellites of the Milky Way (MW) and
Andromeda galaxies, are good targets for study, since individual stars can be resolved and
evolutionary histories can be derived in great detail.
\cite{walker:09} proposed that all dSphs follow a universal mass profile,
which means that all dSphs are embedded within a universal dark matter halo.
They found the best-fit for the relationship between velocity
dispersion and half-light radius for a maximum velocity $V_{max} = 13$~km/s,
and a scale radius, $r_0 = 150$~pc, for a cored DM halo. For the NFW profile, the best
fit is achieved for $V_{max} = 15$~km/s, and a scale radius $r_0 = 795$~pc.
More recently, \cite{collins:14} extended the previous work, incorporating kinematic information
of $25$ Andromeda dSph \citep{collins:13}. They found a discrepancy between the DM
density profile best-fit for the dSphs in the MW compared with that for Andromeda's dSphs (for
both, cored and NFW cases). A good agreement between both MW dSphs and Andromeda
dSphs samples is only reached, when three Andromeda outliers (And XIX, And XXI and
And XXV) and three MW outliers (Hercules, CVnI and Sagittarius) are removed from the
dwarf galaxy sample.
The discrepancy between the cuspy density profiles of DM halos predicted in simulations,
with the cored density profiles derived from observations of dSph galaxies and Low
Surface Brightness galaxies \citep{bosch:00,kleyna:03,blok:02,lora:09,walker:11,amorisco:12,
jardel:12} (the cusp/core problem), the overpopulation of dark substructure \citep{klypin99},
and the to-big-to-fail problem \citep{read:06,boylan:11,boylan:12}, have motivated alternative
DM candidates to the $\Lambda$CDM model.
An alternative model that lately has gained interest, is to consider that the DM is
made of bosons described by a real (or complex) scalar field $\Phi$: the Bose-Einstein condensate/scalar
field DM model (BEC/SFDM) \citep{sin:94, ji:94, jaeweon:96, peebles:99, matos:00,
guzman:00,matos:09,magana:12b}. The BEC/SFDM model is also known as Fuzzy DM \citep{hu:00} or
recently as Wave DM \citep{schive:2014}.
A very interesting feature of the BEC/SFDM model, is that it naturally
produces cored halos. The size of the cores of such halos (for a fixed
self-interacting parameter $\Lambda$) depends on the mass of the BEC/SFDM boson
and the mass of the SFDM halo ($M_{DM}$).
The SFDM model has been proved to be very successful \citep{magana:12a}. It is consistent with the
anisotropies of the cosmic microwave background radiation (CMB) \citep{rodriguez:10},
and it can also achieve a better fit to high-resolution rotation curves of
low-surface-brightness galaxies \citep{robles:12}, compared to the NFW \citep{nfw}
profile.
Lately, \cite{lora:12} and \cite{lora:14} used the internal stellar structures of
dSph galaxies to establish a preferred range for the mass $m_{\phi}$ of the
bosonic particle. They performed $N$-body simulations and explored how the
dissolution time-scale of the cold stellar clump in Ursa Minor (UMi) and Sextans depends on
$m_{\phi}$. They found that for a boson mass in the range of $(3<m_{\phi}<8)\times10^{-22}$~eV,
the BEC/SFDM model would have large enough cores to explain the stellar substructure
in dSph galaxies.
Moreover, \cite{diez-tejedor:14} find a preferred scale radius of $\sim0.5-1$~kpc,
from the kinematics of the eight brightest dSphs satellites of the MW.
In this work I investigate the idea of a universal mass profile for the dSph
population of the MW \citep{walker:09} and Andromeda (M31) under the SFDM model,
and compare it with a cored and NFW profile \citep{collins:14}.
The article is organized as follows: in \S \ref{sec:SFDM} the SFDM model is described
and the Schr\"{o}dinger-Poisson system is briefly reviewed.
In \S\ref{sec:dwarfs}, I discuss the universal NFW and cored
DM profile for the dSphs in the MW and M31. In \S\ref{sec:results}, I describe
our results, and finally, in section \S\ref{sec:conclusions} I discuss the results and
give our conclusions.
\section{The BEC/SFDM halos}
\label{sec:SFDM}
The DM halos can be interpreted as BEC/SFDM Newtonian gravitational configurations
in equilibrium, which can be described by the so called Schr\"odinger-Poisson system,
which is the Newtonian limit of the Einstein-Klein-Gordon equations
\begin{equation}
i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m_{\phi}} \nabla^2 \psi +
U m_{\phi} \psi
+ \frac{\lambda}{2m_{\phi}} |\psi|^2\psi \mbox{ , }
\label{schroedingerA}
\end{equation}
\begin{equation}
\nabla^2 U = 4\pi G m_{\phi}^{2} \psi \psi^\ast \mbox{ . }
\label{poissonA}
\end{equation}
The critical mass for such configurations is
$M_{crit}\sim 0.6 \left( \frac{ m_{P}}{m_{\phi}}\right)\sim10^{12}$~M$_{\odot}$
\citep{ruffini:69,seidel:91}.
In Equations~\ref{schroedingerA} and \ref{poissonA}, $m_{\phi}$ is the mass of the boson
associated with the wave function $\psi$. $U$ is the gravitational potential (produced by the
mass density source $\rho=m_{\phi}^2|\psi|^2$), and $\lambda$ is the self-interacting parameter. It has been
shown that the latter term determines the compactness of the structure \citep{guzman:06}.
It is convenient to work with the dimensionless Schr\"odinger-Poisson system, given as follows
\citep{ThesisArgelia}
\begin{equation}
i\frac{\partial \hat{\psi}}{\partial \hat{t}} = -\frac{1}{2} \hat{\nabla^2} \hat{\psi} +
\hat{U} \hat{\psi} + \hat{\Lambda} |\hat{\psi}|^2 \hat{\psi} \mbox{ , }
\label{schroedinger_nounits}
\end{equation}
\begin{equation}
\hat{\nabla}^2 \hat{U} = \hat{\psi} \hat{\psi}^\ast \mbox{ ,}
\label{poisson_nounits}
\end{equation}
where
\begin{equation}
\label{lambda}
\Lambda=\frac{\lambda}{4 \pi m_{\phi}^{2} G }
\end{equation}
The solutions to the Schr\"odinger-Poisson system relevant for this work
are those with spherical equilibrium \citep{ThesisKevin,ThesisArgelia}. Hence, it is
convenient to work with the Schr\"odinger-Poisson system in spherical coordinates
\begin{equation}
\label{schrodinger_spherical}
i\frac{\partial \hat{\psi}}{\partial \hat{t}} = -\frac{1}{2\hat{r}}
\frac {\partial^{2}}{\partial \hat{r}} (\hat{r} \hat{\psi}) +
\hat{U} \hat{\psi} + \hat{\Lambda} |\hat{\psi}|^2 \hat{\psi} \mbox{ , }
\end{equation}
\begin{equation}
\label{poisson_spherical}
\frac {\partial^{2}} { \partial \hat{r}} (\hat{r}\hat{U}) = \hat{r} \hat{\psi} \hat{\psi}^\ast \mbox{ .}
\end{equation}
The solutions of the Schr\"odinger-Poisson system are obtained assuming an harmonic
behavior of the scalar field, such that
\begin{equation}
\hat{\psi}(r,t)=e^{-(i \hat{\gamma} \hat{t})}\hat{\phi}(\hat{r}) \mbox{ ,}
\end{equation}
where $\hat{\gamma}$ is an adimensional frequency.
Substituting $\hat{\psi}$ in Equations~\ref{schrodinger_spherical} and \ref{poisson_spherical},
the Schr\"odinger-Poisson system now reads as
\begin{equation}
\frac{d^2}{d\hat{r}^{2}}(\hat{r}\hat{\phi})= 2 \hat{r} (\hat{U}-\hat{\gamma})
+ 2 \hat{r} \Lambda \hat{\phi}{^3} \mbox{ ,}
\label{S-icA}
\end{equation}
\begin{equation}
\frac{d^2}{d\hat{r}^{2}}(\hat{r}\hat{U})= \hat{r}\hat{\phi}^2 \mbox{ .}
\label{P-icA}
\end{equation}
The SFDM halos are constructed by obtaining ground state (stable) solutions of the
Schr\"odinger-Poisson system (\ref{S-icA}-\ref{P-icA}) \citep{guzman:04,ThesisArgelia,lora:12,lora:14}.
To guarantee regular solutions, the boundary conditions must satisfy that at $r=0$
$\partial_{r} U=0$, $\partial_{r} \phi=0$, and that $\phi(0)=\phi_c=$, where $\phi_c$
is an arbitrary value.
The mass of this BEC/SFDM halo can be estimated as
\begin{equation}
M= 4\pi \int^{\infty}_{0} {{\phi}}^2 {r}^2 d {r} \mbox{ ,}
\label{eq:masa_r}
\end{equation}
and must be finite number. The radius of this configuration is defined as
$r_{95}$, which is the radius containing $95\%$ of the mass. Note that both
properties, the mass and the radius of the SF halo depend on the boson
mass, and the self-interacting term.
Three parameters $\phi_c$, $m_{\phi}$ and $\Lambda$, define a model completely.
\begin{figure*}
\begin{center}
\includegraphics[width=.55\linewidth]{FIG1.eps}
\caption{Mass of the SFDM halo, as a function of the SFDM core radius for the $\Lambda=0$ case.
Each diagonal line corresponds to a different value of the SFDM boson $m_{\phi}$: $5\times10^{-23}$(green),
$10^{-22}$(blue), $2\times10^{-22}$(pink), $5\times10^{-22}$(red), and $10^{-21}$(yellow)~eV.
The shaded region shows the permitted core radius $r_{c}$, such that the stellar clumps in Ursa Minor
and Sextans are not destroyed
\citep{lora:12,lora:14}.}
\label{fig:FIG1}
\end{center}
\end{figure*}
\subsection{The $\Lambda\gg1$ case}
\label{sec:big_lambda}
When the self interaction between the bosons in a Bose-Einstein condensate is taken into
account, the Schr\"odinger equation can be interpreted as the mean-field approximation at
zero temperature of the Gross-Pitaevskii equation (see Equation ~\ref{schroedingerA}).
The limit where the number of particles in the BEC is very large, and thus the self
interacting term dominates, is called the Thomas-Fermi limit (TFL) \citep{pitaevskii:61,dalfovo:99}.
As the number of particles in the BEC becomes infinite, the TFL approximation becomes
exact \citep{barcelo:05}, giving as a result the classical limit of the theory.
Then, the equations describing the static BEC in a gravitational field with potential
$U$ take the following form \citep{boehmer:07}:
\begin{equation}
\nabla P\left(\frac{\rho}{m_{\phi}}\right)=-\rho \nabla \left(\frac{U}{m}\right) \mbox{ ,}\\
\end{equation}
\begin{equation}
\nabla^{2} U =4 \pi G \rho \mbox{ .}
\end{equation}
With appropriate boundary conditions, one can integrate the latter equations, assuming an
equation of state of the form $P=P(\rho)$.
Assuming the non-linearity in the Gross-Pitaevskii equation
of the form $g(\rho)=\alpha \rho^{\Gamma}$ (where $\alpha$ and $\Gamma$ are constants greater
than zero) \citep{boehmer:07}, then the BEC equation of state is given by
\begin{equation}
P=P(\rho)=\alpha(\Gamma -1) \rho^{\Gamma} \mbox{ .}
\end{equation}
If $\Gamma$ is represented by $\Gamma=1+\frac{1}{n}$ (where $n$ is the polytropic index),
then the equation of state of the static gravitationally bounded BEC is described by
the Lane-Emden equation
\begin{equation}
\frac{1}{\xi^{2}} \frac{d}{d\xi} \left(\xi^{2} \frac{d\theta}{d\xi}\right) + \theta^{n}=0 \mbox{ .}
\end{equation}
In particular, for a static BEC in the TFL with a politropic index $n=1$, the Gross-Pitaevskii
equation reduces to a Lane-Emden equation of the form
\begin{equation}
\frac{d^2\theta}{d\xi^{2}}+\frac{2}{\xi} \frac{d\theta}{d\xi}+\theta =0 \mbox{ ,}
\end{equation}
with an analytical solution
\begin{equation}
\theta(\xi) = \frac{sin \xi}{\xi} \mbox{ .}
\end{equation}
Using the solution given by \cite{boehmer:07} for the density profile, one has
\begin{equation}
\rho(r)= \left\{
\begin{array}{ll}
\rho_{0} \frac{sin (\pi r /R_{max})}{\pi r /R_{max}} & \mbox{if } r \leq R_{max} \\
0 & \mbox{if } r > R_{max} \mbox{ ,}
\end{array}
\right.
\label{eq:TFL}
\end{equation}
\textbf{where $R_{max}$ can be interpreted as the size of the core radius of the SFDM halo.}
Finally, from Equation~\ref{eq:masa_r} one can obtain the mass within a radius $r$
in the TFL
\begin{equation}
M(r)=\frac{4}{\pi}\frac{\rho_{0} R_{max}^{3}}{r} \left[ sin\left(\frac{\pi r}{R_{max}} \right) -
\frac{\pi r}{R_{max}} cos\left(\frac{\pi r}{R_{max}} \right)\right] \mbox{ .}
\end{equation}
Thus, the quantities that describe a system completely in the $\Lambda\gg1$ limit (TFL)
are the central mass density $\rho(c)$ and the size of the SFDM halo $R_{max}$.
As \cite{diez-tejedor:14} point out, the size of the DM halos in the TFL can be expressed as
\begin{equation}
R_{max}=48.93 \left( \frac{\lambda^{1/4}}{m_{\phi}} \right)^{2}
\end{equation}
showing the dependency of $R_{max}$ with $\left(\frac{m_{\phi}}{\lambda^{1/4}}\right)$.
\section{NFW and cored profiles for the Milky Way and Andromeda dSphs}
\label{sec:dwarfs}
The measurements of central velocity dispersions of dSph galaxies very well constrain
the dynamical mass within the deprojected half-light radii ($M_{1/2}$) \citep{walker:09,wolf:10},
then one can compare such dSph mass ($M_{1/2}$ Vs $r_{1/2}$), with a certain integrated DM mass
profile, given a total DM halo mass $M$.
\cite{walker:09} applied the Jeans equation to the velocity dispersion profiles of eight
of the brightest dSph galaxies of the MW and consider the hypothesis that all dSphs follow
a universal mass profile, i.e. that all dSphs are embedded within a universal dark matter
halo.
Using the density Equation,
\begin{equation}
\label{eq:core-nfw}
\rho(r)=\frac{\rho_{0}}{(r/r_{0})^{\alpha} (1+r/r_{0})^{\beta-\alpha}} \mbox{ ,}
\end{equation}
\cite{walker:09} fit a cored ($\alpha=0$, $\beta=3$) and an NFW \citep{nfw}
($\alpha=1$, $\beta=3$) DM profile.
For the cored DM halo, they found the best-fit for the relationship between velocity
dispersion and the half-light radius, for a maximum velocity $V_{max} = 13$~km/s,
and a scale radius, $R_S = 150$~pc. For the NFW profile, the best fit is achieved
for $V_{max} = 15$~km/s, and a scale radius $R_S = 795$~pc.
\cite{collins:14} extended the idea of a universal mass profile, now including the M31
objects into the analysis \citep{collins:13}.
\begin{table*}
\centering
\caption{Parameters of the best fit after a CMA optimization to the SFDM profile for three groups:
the MW dSph(19), the M31 dSph(22), and ALL the sample(41).}
\medskip
\begin{tabular}{@{}cccccccccc@{}}
\hline
Group&Number & M & m$_{\phi}$ & $\epsilon$ & $r_{95}$ & V$_{c}$ & $r_{core}$ \\
& of galaxies &[$10^{7}$~M$_{\odot}$] & [$10^{-22}$~eV]& $10^{-5}$& [kpc] & [km/s] & [kpc] \\
\hline
& & & & & & & \\
M31& 22 & $5.12$ & 3.80 &7.07 & 0.90 & 16.15 & 0.46 \\
MW & 19 & $4.44$ & 4.54 &7.31 & 0.73 & 16.71 & 0.37 \\
ALL& 41 & $4.80$ & 4.17 &7.28 & 0.80 & 16.63 & 0.41 \\
& & & & & & \\
\hline
& & & & & & \\
& & & & & & \\
\end{tabular}
\label{table:1}
\end{table*}
\cite{collins:14} also use the NFW and core mass density profiles.
They obtain the best fit for the NFW (cored) profile, to the whole population of dSphs, of
$V_{max}=14.7\pm0.5$~km/s ($V_{max}=14.0\pm0.4$~km/s) and a scale radius of
$R_S=876\pm284$~pc ($R_S=242\pm124$~pc). They concluded that neither value is a good
fit for many of the LG dSphs. They found that if they do not take into account five outliers,
three in M31 (And~XIX, And~XXI, and And~XXV), and two in the MW (Hercules and CVn~I), the fit is
significantly better for the NFW (core) mass profile, $V_{max}=16.2\substack{+2.6 \\ -1.7}$~km/s
($V_{max}=15.6\substack{+1.5 \\ -1.3}$~km/s) and $R_S=664\substack{+412 \\ -232}$~pc
($R_S=225\substack{+70 \\ -55}$~pc).
\section{A universal SFDM halo?}
\label{sec:results}
\subsection{The $\Lambda=0$ case}
One can consider the self-interacting term $\Lambda$ to be zero in Equation
\ref{schroedingerA} (i. e. the self-interaction is negligible), such a case is also
known as the fuzzy DM model \citep{hu:00}.
In Figure~\ref{fig:FIG1}, the mass of the SFDM halo is plotted as a function of the
core radius ($r_c$) for $\Lambda=0$. The core radius is defined as the radius at which
the initial density has dropped by a factor of two.
\textbf{When analyzing the behavior of the different models with fixed $M_{DM}$ and
$m_{\phi}$, I found out that there is a correlation between the mass of the SFDM halo
and the core radius, for a given $m_{\phi}$. The mass of the SFDM halo is inversely
proportional to the DM core radius independently of the value of $m_{\phi}$ (i. e.
$M_{DM} \propto r_{c}^{-1}$), but the intercept varies depending on the value of
$m_{\phi}$. Then, I found a fit over different values of $m_{\phi}$ and their corresponding
intercepts ($\alpha$), and found the relation between the intercept and the value of
$m_{\phi}$ of $\alpha \propto m_{\phi}^{8.7}$.}
The complete relation between the mass of the dark
matter halo and the dark matter halo's core radius for a given $m_{\phi}$, is given by
\begin{equation}
M_{DM}=10^{\alpha} r_{c}^{-1} \mbox{ ,}
\end{equation}
where $\alpha$ depends on the mass of the boson as
\begin{equation}
m_{\phi}=10^{-13.9} \alpha^{-8.7} \mbox{ .}
\end{equation}
For a given $M_{DM}$ and $m_{\phi}$, there is a slope $\alpha$, for which a unique
$r_{c}$ will be defined. With the latter relations, one can build a set of continuous
models for a given $M_{DM}$ and $m_{\phi}$. The continuous models are shown as diagonal
colored lines in Figure~\ref{fig:FIG1}. Each color represent a mass of the boson $m_{\phi}$
(green $\rightarrow 5\times10^{-23}$~eV, blue $\rightarrow 1\times10^{-22}$~eV, pink
$\rightarrow 2\times10^{-22}$~eV, red $\rightarrow 5\times10^{-22}$~eV, yellow
$\rightarrow 1\times10^{-21}$~eV).
It is clear to see that for a given halo mass, lower values of $m_{\phi}$'s have
larger core-radius, and higher values of $m_{\phi}$ have smaller core-radius. For example,
for a fixed SFDM halo mass of $10^8$~$M_{\odot}$, if $m_{\phi}=10^{-22}$~eV the corresponding
core radius is $\sim3.5$~kpc. But if $m_{\phi}=10^{-21}$~eV the corresponding core radius
is $\sim30$~pc.
I consider three different groups of dSph data; the dSphs in the MW ($19$ galaxies), the
dSph in the M31 ($22$ galaxies), and \scriptsize{ALL} \normalsize the galaxies ($41$ galaxies).
In order to compare with \cite{collins:14} results, I do not take into account five dSphs
(Sagittarius, Hercules, CvnI, AndXIX, AndXXI and And XXV).
In order to see how well these three groups of dSph galaxies can be fit with a single SFDM
profile (for a SFDM halo mass $M$ and a SFDM boson mass $m_{\phi}$), I use the following maximum
likelihood fitting routine
\begin{multline}
\label{eq:max_prob}
L_{SFDM} (\{r_{h,i},V_{c,i},\delta_{V_{c},i}\}|m_{\phi},M) = \\
\prod_{i=0}^{N} \frac{1}{\sqrt{2\pi} \delta_{V_{c},i}}
\times exp \Biggl[-\frac{(V_{c,SFDM}-V_{c,i})^{2}}{2\delta_{V_{c},i}^2}\Biggr] ,
\end{multline}
where $V_{c,SFDM}$ is the circular velocity predicted by the SFDM model, $r_{h,i}$ is the
half-light radius of the $i-$ dSph, $V_{c,i}$ is the measured circular velocity at the
half-light radius, and $\delta_{V_{c},i}$ is its uncertainty (the observational data are
taken from \citeauthor{mcconnachie:12} \citeyear{mcconnachie:12}, \citeauthor{collins:14}
\citeyear{collins:14} and \citeauthor{martin:15} \citeyear{martin:15}).
Equation~\ref{eq:max_prob} is a measure of how well a model with specific parameters
$m_{\phi}$ and $M$ represents a given group of galaxies. In order to find the best fit
for the three samples I use the Evolution Strategy with Covariance Matrix Adaptation
(CMA-ES) optimization. For details on the CMA-ES routine, see Appendix~\ref{appendix}.
The best fit for the 19 MW sample is $M=5.12\times10^7$~$M_{\odot}$ and a $m_{\phi}=3.8\times10^{-22}$~eV.
For the 22 M31 sample the best fit corresponds to $M=4.44\times10^7$~$M_{\odot}$ and
a $m_{\phi}=4.53\times10^{-22}$~eV. For the whole sample ($41$) the best fit corresponds
to $M=4.8\times10^7$~$M_{\odot}$ and $m_{\phi}=4.17\times10^{-22}$~eV (see Table~\ref{table:1}).
The MW fit gives as a result a slightly more massive DM halo, and a smaller
mass of the boson, compared with the M31 and the \scriptsize{ALL} \normalsize data sets. As a
consequence the core radius for the MW sample, is the largest ($r_c\sim0.45$~kpc). Such a size
of the DM core radius is large enough to explain the longevity of the old-cold stellar clump
in UMi \citep{lora:12}, and the two stellar substructures in Sextans \citep{lora:14}.
\begin{figure*}
\begin{center}
\includegraphics[width=.99\linewidth]{FIG2.eps}
\caption{Best fit to the NFW (dark blue), and the core (purple) DM density profile from \cite{collins:14},
leaving out the MW dSphs' outliers (Hercules, CVnI and Sagittarius, see white triangles), and three M31
outliers M31 (AndXIX, AndXXI, AndXXV, white circles).
The SFDM profile studied in this work, is shown in light-blue. The shaded regions correspond to a
$1\sigma$ deviation, from each DM model fit.
The upper panels show the circular velocity as a function of the half-light radius, for ALL, M31, and
MW samples, respectively.
The lower panels show the DM halo mass, as a function of the half-light radius, for ALL, M31, and
MW samples, respectively. The dSph galaxies used in this work are over plotted with their corresponding
uncertainties. The gray circles correspond to the M31 dwarfs, and the black triangles correspond to the
MW dSphs.}
\label{fig:FIG2}
\end{center}
\end{figure*}
In the upper panels of Figure~\ref{fig:FIG2}, I show the half-light radius circular velocity
($V_{c,1/2}=\sqrt{\frac{GM_{1/2}}{r_{1/2}}}$) as a function of the half-light radius of all
M31 and MW dSphs in the sample, along with the best fit to \scriptsize{ALL} \normalsize sample
for an NFW density mass profile (see dashed dark-blue line) and a cored density mass profile
(see dashed purple lines) \citep{collins:14}, and the SFDM profile (see light-blue dash line).
The shaded regions in Figure~\ref{fig:FIG2} indicate the $1\sigma$ error. The computed maximum
circular velocity for the SFDM model for the \scriptsize{ALL} \normalsize sample is
$V_{max}\approx16.63$~km/s. This value of the maximum circular velocity is in a very good
agreement with both computed values of the NFW profile
($V_{max}=16.2\substack{+2.6 \\ -1.7}$~km/s), and the core profile
($V_{max}=15.6\substack{+1.5 \\ -1.3}$~km/s).
The dSph galaxies with low $V_{c}$s and small half-light radii can be well reproduced
with the cored and the NFW density profile, but cannot be well reproduced with the SFDM
profile. On the other hand, the dSphs with $V_{c}$s between $6-10$~km/s, and high values
of the half-light radii ($\sim1000$~pc) can be reproduced with the SFDM model, but
cannot be well reproduced with either the NFW profile, or with the core profile.
In the lower panels of Figure~\ref{fig:FIG2}, I show the same fits (core, NFW and SFDM) for
the mass of the DM halo, as a function of the half-light radius. It is very clear from this
Figure, that the NFW fits the \scriptsize{ALL} \normalsize data better at
small-mass-small-half-light-radius, whereas the SFDM fits better the
\scriptsize{ALL} \normalsize data at high-mass-high-half-light-radius, where both core and
NFW profiles fail.
In order to look in more detail at the properties of the dSphs that are in better agreement with
the SFDM profile, I plot the mass inside the half-light radius as a function of the luminosity
at the same radius (see left panel of Figure~\ref{fig:FIG3}). The diagonal dashed lines in
Figure~\ref{fig:FIG3} correspond to four fixed mass-to-light ratios ($M/L=1000$, $100$, $15$
and $3$). The circles represent the M31 dSphs, and the triangles represent the MW dSphs.
For each dSph I computed the ratio $\zeta$ between the theoretical mass values from the
best DM fits of the \scriptsize{ALL} \normalsize data set, and the observed mass
values \citep{collins:14}, $\zeta=M_{DM}/M_{dSph}$.
I then define that a dSph is well reproduced by a certain DM model if $\zeta$ is at most 2.
The latter means that the dSph mass obtained from the best DM model
differs from the observed value by a maximum of a factor 2. A dSph galaxy is in agreement
with an specific DM model when $\zeta$ is minimized.
In the left panel of Figure~\ref{fig:FIG3}, the color code is as follows: the light-blue symbols
(both M31 circles and MW triangles) represent the dSphs which are in good agreement with the
SFDM profile. The purple symbols show the dSphs which are better reproduced with the cored DM
profile, and the dark-blue symbols represent the dSphs which are in better agreement with the
NFW profile. On the other hand, the yellow symbols correspond to the dSph that are well reproduced
with the three DM models, whereas the red symbols show the dwarf galaxies that cannot be reproduced
with any DM model studied in this work (i. e. $\zeta>2$).
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=80mm]{FIG3_A.eps}&
\includegraphics[width=80mm]{FIG3_B.eps}\\
\end{tabular}
\caption{Mass at the half-light radius as a function of the luminosity at the half-light radius.
The M31 dSphs are shown with circles and the MW dSphs are shown with triangles. The dark-blue(purple)
symbols show the dSphs that are in better agreement with an NFW (cored) DM profile. The light-blue symbols
show the dSphs which are in better agreement with the SFDM profile. The yellow symbols show the dwarfs
that are well reproduced with the three DM models, and the red symbols show the dwarfs that are not well
reproduced with any DM model. The white symbols represent the galaxies that were not taken into account
(Hercules, CVnI, AndXIX, AndXXI and AndXXV). The diagonal dashed lines show the mass-to-light ratios
$M/L=1000$, $100$, $15$, and $3$.
In the right panel, the dSph that are in a better agreement
with the SFDM in the TFL i. e. $\Lambda>>1$, are also plotted (see pink symbols).}
\label{fig:FIG3}
\end{figure*}
One can immediately see from Figure~\ref{fig:FIG3} that the dSphs which are in a better
agreement with an NFW DM profile are very dark matter dominated, with mass-to-light ratios
between $1600\gtrsim M/L \gtrsim 100$. There are only two exceptions LeoI and LeoII, which mass-to-light
ratios of $7.1$ and $12.9$, respectively.
\begin{table*}
\centering
\caption{Parameters of the best fit in the TFL ($\Lambda>>0$) for three groups:
the MW dSph(19), the M31 dSph(22), and ALL the sample(41).}
\medskip
\begin{tabular}{@{}cccccccccc@{}}
\hline
Group &Number &$\rho_{0}$ &$R_{max}$& M$_{max}$ & $\frac{m_{\phi}^4}{\lambda}$&$\frac{m_{\phi}}{\lambda^{1/4}}$&V$_{c}$ & $r_{core}$ \\
&of galaxies&[$10^7$~M$_{\odot}$~kpc$^{3}$] &[kpc]& [$10^{7}$~M$_{\odot}$] &[eV$^{4}$] &[eV] &[km/s] & [kpc] \\
\hline
& & & & & & & & \\
M31 & 22& $8.06$ & $0.86$ & $6.716$& $3180.96$ & $7.51$ & $18.24$ & $0.52$ \\
MW & 19& $17.96$& $0.59$ & $4.80$ & $6788.80$ & $9.08$ & $18.64$ & $0.36$ \\
ALL & 41& $9.74$ & $0.85$ & $7.60$ & $3325.20$ & $7.59$ & $19.61$ & $0.51$ \\
& & & & & & & & \\
\hline
& & & & & & \\
\end{tabular}
\label{table:2}
\end{table*}
The galaxies which are better reproduced with an NFW profile are also the less
luminous of the sample with $L \lesssim 10^{4}$~L$_{\odot}$. The only outlier is UMaI, which
is in a better agreement with the SFDM model. On the other hand, it seems that these galaxies
have a mass at the half-light ratio limit of $\sim10^7$~M$_{\odot}$.
The upper-right gray shaded region in Figure~\ref{fig:FIG3} contains a set of five M31 dSphs,
and two MW dSphs which are in good agreement with the SFDM profile (light-blue symbols). If one
also takes into account the yellow symbols, which are dSphs well reproduced with either DM model, then
four more can be added, resulting in a set of $11$ galaxies which are very well reproduced
with the SFDM model. These galaxies share the property of being less DM dominated than those
which are better reproduced with an NFW DM profile. The mass-luminosity ratios of these galaxies
range from $\sim100$ to $5$. On the other hand, these dSphs are the most luminous galaxies of the
sample, with a luminosity ranging from $\sim10^{5}$ to $5\times10^{6}$~L$_{\odot}$.
It is remarkable that there are no dSph in this region which are more in agreement with an NFW profile.
\begin{figure*}
\epsscale{1.2}
\plotone{FIG4.eps}
\caption{Same as Figure~\ref{fig:FIG2} but for the best fits of the SFDM model, and the SFDM
in the TFL.}
\label{fig:FIG4}
\end{figure*}
\subsection{The $\Lambda>>1$ case}
\label{sec:Big lambda}
In this subsection, the kinematics of the dSphs in the MW and M31, for large
values of the self-interacting parameter $\Lambda$ , are analyzed.
In the $\Lambda\gg1$ limit, as mentioned in section~\ref{sec:Big lambda}, the
SFDM density profile is described by Equation~\ref{eq:TFL}.
In order to build the DM halos the value of the central DM density $\rho_{0}$ and the
size of the SFDM halo $R_{max}$ must be fixed.
To study how well the circular velocity, and mass profile of the three
groups of dSphs (M31, MW and ALL) are fit with a single SFDM halo in $\Lambda>>1$
regime the maximum likelihood fitting routine is modified, so that $\rho_{0}$ and
$R_{max}$ are the free parameters. Then similarly to Equation~\ref{eq:max_prob} one
has
\begin{multline}
\label{eq:max_prob2}
L_{TF} (\{r_{h,i},V_{c,i},\delta_{V_{c},i}\}|\rho_{0},R_{max}) = \\
\prod_{i=0}^{N} \frac{1}{\sqrt{2\pi} \delta_{V_{c},i}}
\times exp \Biggl[-\frac{(V_{c,TF}-V_{c,i})^{2}}{2\delta_{V_{c},i}^2}\Biggr] ,
\end{multline}
where $V_{c,TF}$ is the circular velocity in the TFL.
Using the CMA-ES optimization routine described in Appendix~\ref{appendix},
I find that for the M31 dwarfs, the best fit is achieved for a value of $m_{\phi}^4/\lambda=3180$,
which corresponds to a $\rho_{0} \sim 8\times10^7$~M$_{\odot}$~kpc$^{-3}$, a $R_{max}=0.87$~kpc,
and at circular velocity value of $18.24$~km/s (see Table~\ref{table:2}). The latter value is
approximately $13$\% higher than the one obtained for the $\Lambda=0$ case ($16.1$~km/s).
The core radius (halo DM mass $M$) in the TFL for M31 dwarfs, is $0.52$~kpc ($6\times10^7$~M$_{\odot}$),
very similar to the $0.45$~kpc ($5\times10^7$~M$_{\odot}$) find for the $\Lambda=0$ case.
The results are summarized in Table~\ref{table:2}.
For the MW dwarfs the central density is the highest of the three groups,
$1.8\times10^8$~M$_{\odot}$~kpc$^{-3}$, whereas the maximum DM radius for the MW dSphs is the smallest
of the three groups, $R_{max}=0.59$~kpc (which corresponds to a core radius of $0.36$~kpc).
Such a core radius is very similar to the one obtained for the $\Lambda=0$ case, and thus, it is
large enough to explain the stellar substructures found within the UMi and the Sextans dSphs.
The maximum circular velocity is $18.6$~km/s, again approximately $11$\%
higher than that found in the $\Lambda=0$ case ($16.7$~km/s). For the MW dwarfs a
value of $m_{\phi}^4/\lambda=6788$ is obtained (i. e. $m_{\phi}/\lambda^{1/4}\approx9$).
The upper panels of Figure~\ref{fig:FIG4} show the circular velocity as a function of the half-light
radius, and DM halo mass as a function of the half-light radius (lower panels) for the
SFDM model for $\Lambda=0$ and $\Lambda>>1$.
The pink lines show the best fit for the SFDM model in the TFL ($\Lambda>>1$)
for each of the dSph data sets: ALL (left panels), M31 (central panels) and MW (right panels),
respectively. The shaded regions show the $1\sigma$ deviation.
The results obtained for the $\Lambda=0$ case (light-blue), are also plotted in
order to facilitate comparison between both SFDM profiles.
In the right panel of Figure~\ref{fig:FIG3} I show the seven dSphs which are in better agreement
with a SFDM model in the TFL (such galaxies are plotted with pink symbols) also including
the previous studied DM profiles. The galaxies that are better reproduced with the SFDM in the
TFL are also mostly contained in the upper-right shaded gray region. This means that the
SFDM model in the TFL better fit galaxies with low mass-to-light ratios.
\subsection{The universal SFDM profile compared}
\label{sec:The universal SFDM profile compared}
In \cite{lora:12} we performed $N$-body simulations of the UMi dSph, which contains a
cold-old stellar substructure \citep{kleyna:03}. We explored how the dissolution time-scale of such
stellar substructure depends on the mass of the boson $m_{\phi}$. The boson mass range obtained
from UMi's dynamics is $0.3\times10^{-22}<m_{\phi}<10^{-22}$~eV.
Similarly, in \cite{lora:14} we investigated the dSph Sextans, which has two different old-cold stellar
substructures \citep{battaglia11,walker06}. For Sextans we require a mass for the SFDM boson of
$0.12\times10^{-22}<m_{\phi}<8\times10^{-22}$~eV, in order to guarantee the survival of both stellar
substructures.
In Figure~\ref{fig:FIG1}, the corresponding mass of the boson is plotted (purple blue stars) for UMi
and for Sextans. The error-bars denote the $1\sigma$ deviation. The best fit of the MW(triangle),
M31(circle), and \scriptsize{ALL} \normalsize the sample (square) are also plotted.
For the Sextans dSph, all the values for the mass $M$ and the core radius $r_c$ in the light-gray
area in Figure~\ref{fig:FIG1} are permitted. In particular, those corresponding to the three best
SFDM fits (MW, M31 and \scriptsize{ALL}\normalsize). The latter means that the dynamics of the
Sextans dwarf, is in agreement with a unique SFDM mass profile. The UMi dSph is located above the
MW, and \scriptsize{ALL} \normalsize fits in Figure~\ref{fig:FIG1}, being the most MW dSph
restrictive case. However, it is still in agreement with the MW (circle) case at $1\sigma$ confidence.
My findings in this work in the TFL can be directly compared with the recent
results of \cite{diez-tejedor:14}. They constrained the parameters of the SFDM self-interacting
parameter with the kinematics of the eight brightest dSph of the MW.
They reported a preferred
value of $R_{max}\sim1$~kpc, which corresponds to $m_{\phi}/\lambda^{1/4}\sim7$~eV. In this work I
obtain for the MW sample a value of $R_{max}\sim0.6$~kpc, which corresponds to
$m_{\phi}/\lambda^{1/4}\sim 9$~eV. It has to be noted that \cite{diez-tejedor:14} analyzed the eight
brightest dSph in the MW; in this work the MW sample was comprised of 19 dwarf galaxies.
If the MW sample is restricted to the eight brightest dSph, I obtain a value $R_{max}=0.57$~kpc,
which corresponds to $m_{\phi}/\lambda^{1/4}=9.24$~eV. This result is still in agreement with the
results obtained from the 19 MW dwarf sample, but $m_{\phi}/\lambda^{1/4}$ (and $R_{max}$) is
somewhat higher (lower) that the $m_{\phi}/\lambda^{1/4}\sim7$~eV ($R_{max}\sim1$~kpc) values
reported by \cite{diez-tejedor:14}.
\textbf{It has to be noted that \cite{diez-tejedor:14} analyze the velocity dispersion of the
eight brightest dSph galaxies in the MW, arguing that the stellar component of each of the
eight galaxies are in dynamical equilibrium, and that the stellar distribution traces the DM
distribution.}
\textbf{They fit three free parameters in their models: $R_{max}$, $M_{max}$ and
the orbital anisotropy of the stellar component of each galaxy, which is a difference
in the approach of this work. It is encouraging that, even with these differences, when the uncertainty
averaged over the eight dSphs is taken into account in $Rmax$ (and thus $m_{\phi}/\lambda^{1/4}$)
one obtains $R_{max}=1.05^{+0.3}_{-0.22}$ ($m_{\phi}/\lambda^{1/4}=6.82^{+0.87}_{-0.81}$), which
is roughly in agreement with the findings reported in this work.}
\cite{strigari:08} suggested that dwarf galaxies satellites of the MW, have a common mass
within $300$~pc of $\sim10^7$~M$_{\odot}$. They suggest that such a finding could shed light
in a characteristic scale for the clustering of dark matter.
In Figure~\ref{fig:FIG5} the gray shaded region shows \cite{strigari:08}'s results. Overplotted
are the results of this work. The light blue symbols correspond to the $\Lambda=0$ case, and
the pink ones correspond to the $\lambda\gg1$ case. It is encouraging to find that for the best
fits of the SFDM model for the M31 and MW dSph galaxies, the mass within $300$~pc is in agreement
with \cite{strigari:08} findings.
It has to be noted that the central mass density of the MW's dSphs range from $\rho_{0}\approx 0.03$
to $0.3$~M$_{\odot}$~pc$^{-3}$ \citep{kormendy:14,burkert:14}. It is encouraging that the latter values
of the central density are not only in agreement with the central density obtained for the MW's sample
($\sim0.18$~M$_{\odot}$~kpc$^{-3}$), but also with the whole sample
($\sim0.1$~M$_{\odot}$~kpc$^{-3}$).
\section{Conclusions}
\label{sec:conclusions}
\begin{figure}
\epsscale{1.00}
\plotone{FIG5.eps}
\caption{The gray shaded region shows the results of \cite{strigari:08}. The resulting
mass within $300$~pc for $M31$ (circle), the $MW$ (triangle), and all the sample (square)
are shown, for the $\Lambda=0$ case (light blue symbols) and the $\Lambda\gg1$ case (pink
symbols).}
\label{fig:FIG5}
\end{figure}
The very high dark-to-stellar mass ratios of dSph galaxies suggest that they are the most
DM dominated objects in the universe, and therefore ideal laboratories to test any DM
alternative model. In this work I compare the kinematics of $22$ dSph galaxies satellites of M31,
and $19$ dSph galaxies of the MW. I study also a third group of galaxies containing the sum of
all M31 and MW dSphs (ALL).
I study the hypothesis that all dSph are embedded in dark matter halos with a same
mass \citep{walker:09,collins:14}, by fitting the dSph's half-light radius,
and the velocity at the half-light radius, to the SFDM model, in two different regimes:
when $\Lambda=0$ and when $\Lambda\gg1$.
There is a very good agreement between the velocity at the half-light radius for the best fit
for the ALL sample ($16.4$~km/s) from the $\Lambda=0$ case, with those obtained using an NFW ($16.2$~km/s),
and a cored ($15.6$~km/s) DM profile (see Equation~\ref{eq:core-nfw}). This corresponds to a mass
of the SFDM boson of $\sim4\times10^{-22}$~eV, which is in agreement with our previous findings
\citep{lora:12,lora:14}.
A higher value for the velocity at the half-light radius was found for the $\Lambda>>1$ case
($19.6$~km/s) resulting in a $m_{\phi}/\lambda^{1/4}\sim7.6$~eV.
The M31-dSph galaxies with high luminosities and mass-to-light ratios ranging from $27$ to
$78$, are better reproduced with the SFDM model than with an NFW or core DM model (see light-blue
symbols in Figure~\ref{fig:FIG3}). Hence, MW-dSph with very high mass-to-light ratios
(ranging from $1500$ to $\sim100$) and very low values of the luminosity are in better agreement
with a NFW/core DM profile than with a SFDM (see dark-blue and purple symbols in Figure~\ref{fig:FIG3}).
The mass within $300$~pc for the SFDM model (for both $\Lambda=0$ and $\gg1$) is in a very good
agreement with \cite{strigari:08}'s findings. The latter suggests a universal SFDM halo
mass for the dSph galaxies in the MW and M31. These results are encouraging, since also a unique
mass of the SFDM boson (for the $\Lambda=0$ case) of $\sim4.8\times10^{-22}$~eV is obtained. One
would expect a unique mass of the SFDM boson, and not to have a different SFDM boson mass for
different dSph galaxies.
\acknowledgments
I would like to thank Michelle L. M. Collins and Nicolas F. Martin, for making their
data available to me.
I also thank Steffen Brinkmann and Colin W. Glass from HLRS, for making their
CMA routine available to me.
I thank Andreas Just, Juan Maga\~na and Avon Huxor for very helpful comments and discussions, that gave
as a result an improved version of this paper.
I gratefully acknowledges support from the Heidelberg University Innovation Fund FRONTIER.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 782 |
\section{Introduction}
Throughout this paper,
we assume that
\begin{equation}
\text{$X$ is a real Hilbert space}
\end{equation}
with inner product $\scal{\cdot}{\cdot}$ and induced norm
$\|\cdot\|$.
We also assume that
$T \colon X\to X$ is a \emph{cutter},
i.e.,
$\Fix T := \menge{y\in X}{y=Ty}\neq\varnothing$ and
that furthermore
$(\forall x\in X)(\forall y\in \Fix T)$
$\scal{y-Tx}{x-Tx}\leq 0$;
equivalently,
\begin{equation}
(\forall x\in X)(\forall y\in \Fix T)\quad
\|Tx-y\|^2 + \|x-Tx\|^2 \leq \|x-y\|^2.
\end{equation}
Cutters are also known as \emph{quasi firmly nonexpansive operators}.
We also assume that
$C$ is a closed and convex subset of $X$
such that
$C \cap \Fix T \neq\varnothing$.
Our aim is to
\begin{equation}
\label{e:0424a}
\text{find a point in $C\cap\Fix T\neq\varnothing$.}
\end{equation}
Because $T$ can be a subgradient projector
(see Example~\ref{ex:0424b} below),
\eqref{e:0424a} is quite flexible and includes the problem
of solving convex inequalities.
For further information on cutters and subgradient projectors,
we refer the reader to
\cite{bb96,MOR,BWWX1,Cegielski,CL,CS,CenZen,Comb93,Comb97,CombLuo,Pauwels,Poljak,Polyakbook,PolyakHaifa,SY,YO1,YO2,YSY,YY}
and the references therein.
Given $r\geq 0$, we follow Crombez \cite{Crombez} and define the operator
$U_r\colon X\to X$ at $x\in X$ by
\begin{equation}
\label{e:Ur}
U_rx :=
\begin{cases}
\displaystyle x + \frac{r+\|Tx-x\|}{\|Tx-x\|}(Tx-x) = Tx +
\frac{r}{\|Tx-x\|}(Tx-x), &\text{if $x\neq Tx$;}\\
x, &\text{otherwise.}
\end{cases}
\end{equation}
When $T$ is a subgradient projector,
then $U_r$ was also studied by Polyak \cite{PolyakHaifa}.
Note that $\Fix U_r = \Fix T$.
Our goal is to solve \eqref{e:0424a} algorithmically via
sequence $(x_n)_\nnn$ generated by $x_0\in X$ and
\begin{equation}
(\forall \nnn)
\quad
x_{n+1} := P_{C}U_{r_n}x_n,
\end{equation}
where $P_C$ is the projector\footnote{$P_C$ is the unique
operator from $X$ to $C$ satisfying $(\forall x\in X)(\forall c\in C)$
$\|x-P_Cx\|\leq\|x-c\|$.} onto $C$
and the sequence of parameters $(r_n)_\nnn$ lying in $\RPP :=
\menge{\xi\in\RR}{\xi>0}$ satisfies a
divergent-series condition.
\emph{
We will obtain \emph{finite convergence results} for this and more general
algorithms provided some constraint
qualification is satisfied. In the present setting,
our results complement and extend
results by Crombez for cutters and by Polyak
for subgradient projectors.
}
The paper is organized as follows.
In Section~\ref{s:aux}, we collect various
auxiliary results, that will facilitate the presentation
of the main results in Section~\ref{s:main}.
Limiting examples are presented in Section~\ref{s:limex}.
In Section~\ref{s:compare}, we compare to existing results.
Future research directions are discussed in
Section~\ref{s:persp}. Finally, Section~\ref{s:conc} concludes the
paper.
Notation is standard and follows e.g., \cite{BC2011}.
\section{Auxiliary Results}
\label{s:aux}
\subsection{Cutters}
We start with the most important instance of a cutter,
namely Polyak's subgradient projector \cite{Poljak}.
\begin{example}[subgradient projector]
\label{ex:0424b}
Let $f\colon X\to\RR$ be convex and continuous such that $\menge{x\in
X}{f(x)\leq 0}\neq\varnothing$,
and let $s\colon X\to X$ be a selection of $\partial f$,
i.e., $(\forall x\in X)$ $s(x)\in\partial f(x)$.
Then the \emph{associated subgradient projector},
defined by
\begin{equation}
(\forall x\in X)\quad
G_fx :=
\begin{cases}
\displaystyle x - \frac{f(x)}{\|s(x)\|^2}s(x), &\text{if $f(x)>0$;}\\
x, &\text{otherwise,}
\end{cases}
\end{equation}
is a cutter.
\end{example}
We now collect some inequalities and identities that
will facilitate the proofs of the main results.
The inequality
$\|U_rx-y\|^2 \leq \|Tx-y\|^2 - r^2$,
which is a consequence of \ref{l:0424e2} in the next lemma,
was also observed by Crombez in \cite[Lemma~2.3]{Crombez}.
\begin{lemma}
\label{l:0424e}
Let $y\in \Fix T$, let $r\in\RPP$, and
suppose that $\ball(y;r)\subseteq \Fix T$ and that
$x\in X\smallsetminus \Fix T$.
Set
\begin{equation}
\tau_x := \scal{x-y}{(x-Tx)/\|x-Tx\|} - \big(r + \|x-Tx\|\big).
\end{equation}
Then the following hold:
\begin{enumerate}
\item
\label{l:0424e1}
$\tau_x \geq 0$.
\item
\label{l:0424e2}
$\|U_rx-y\|^2 = \|Tx-y\|^2 - r^2 -2r\tau_x \leq \|Tx-y\|^2 - r^2$.
\item
\label{l:0424e3}
$\|U_rx-y\|^2 = \|x-y\|^2 - (r+\|x-Tx\|)^2 -2\tau_x(r+\|x-Tx\|)
\leq \|x-y\|^2 - (r+\|x-Tx\|)^2 \leq \|x-y\|^2 - r^2 - \|x-Tx\|^2$.
\end{enumerate}
\end{lemma}
\begin{proof}
\ref{l:0424e1}:
Set $z := y + r(x-Tx)/\|x-Tx\|$.
Then $z\in \ball(y;r)\subseteq \Fix T$.
Since $T$ is a cutter, we obtain
\begin{subequations}
\begin{align}
0 &\geq \scal{z-Tx}{x-Tx}\\
&=\scal{y+r(x-Tx)/\|x-Tx\|-Tx}{x-Tx}\\
&=\scal{y-Tx}{x-Tx} + r\|x-Tx\|\\
&=\scal{y-x}{x-Tx} + \|x-Tx\|^2 + r\|x-Tx\|.
\end{align}
\end{subequations}
Rearranging and dividing by $\|x-Tx\|$ yields
$\scal{x-y}{(x-Tx)/\|x-Tx\|} \geq r+\|x-Tx\|$ and
hence $\tau_x\geq 0$.
\ref{l:0424e2}:
Using \eqref{e:Ur}, we derive the identity from
\begin{subequations}
\begin{align}
\|U_rx-y\|^2 &= \big\| x+ (\|x-Tx\|+r)/\|x-Tx\|(Tx-x)-y\big\|^2\\
&= \big\|(Tx-y) + r (Tx-x)/\|Tx-x\|\big\|^2\\
&=\|Tx-y\|^2 + r^2 + 2r\scal{(Tx-x)+(x-y)}{(Tx-x)/\|Tx-x\|}\\
&=\|Tx-y\|^2 + r^2 +2r\|x-Tx\| - 2r\scal{x-y}{(x-Tx)/\|x-Tx\|}\\
&=\|Tx-y\|^2 - r^2 - 2r\tau_x.
\end{align}
\end{subequations}
The inequality follows immediately from \ref{l:0424e1}.
\ref{l:0424e3}:
Using \ref{l:0424e2}, we obtain
\begin{subequations}
\begin{align}
\|U_rx-y\|^2 &= \|(x-y)+(Tx-x)\|^2 - r^2 - 2r\tau_x\\
&= \|x-y\|^2 + \|x-Tx\|^2 + 2\scal{x-y}{Tx-x} - r^2 - 2r\tau_x\\
&= \|x-y\|^2 - \|x-Tx\|^2 - 2(\tau_x+r)\|x-Tx\|-r^2-2r\tau_x\\
&= \|x-y\|^2 - (r+\|x-Tx\|)^2 -2\tau_x(r+\|x-Tx\|).
\end{align}
\end{subequations}
The inequalities now follow from \ref{l:0424e1}.
\qed
\end{proof}
We note in passing that $U_r$ itself is not necessarily a cutter:
\begin{example}[$U_r$ need not be a cutter]
\label{ex:0425a}
Suppose that $X=\RR$ and that
$T$ is the subgradient projector associated with the function
$f\colon\RR\to\RR\colon x\mapsto x^2-1$.
Then $\Fix T = [-1,1]$.
Let $r\in\RP := \menge{\xi\in\RR}{\xi\geq 0}$.
Then
\begin{equation}
(\forall x\in \RR\smallsetminus\Fix T)
\quad
U_rx = \frac{x}{2} + \frac{1}{2x} - r\sgn(x).
\end{equation}
Choosing $y:=1\in\Fix T$ and $x:=y+\varepsilon\notin\Fix T$, where
$\varepsilon\in\RPP$,
we may check that $U_r$ is not a cutter\footnote{In fact, $U_r$ is not even
a relaxed cutter in the sense of \cite[Definition~2.1.30]{Cegielski}.}
when $\varepsilon$ is sufficiently small and $r>0$.
\end{example}
We now obtain the following result
concerning a relaxed version\footnote{$U_{r,\eta}$ can also be called a generalized
relaxation of $T$ with relaxation parameter $\eta$; see \cite[Definition~2.4.1]{Cegielski} .} of $U_r$.
Item~\ref{c:0424f5} also follows from \cite[Corollary~2.4.3]{Cegielski}.
\begin{corollary}
\label{c:0424f}
Let $y\in \Fix T$, let $r\in\RPP$, let $\eta\in\RP$,
and suppose that $\ball(y;r)\subseteq \Fix T$ and that
$x\in X\smallsetminus \Fix T$.
Set
\begin{equation}
U_{r,\eta}x :=
x + \eta \frac{r+\|x-Tx\|}{\|Tx-x\|}(Tx-x).
\end{equation}
Then the following hold\footnote{We note that
item~\ref{c:0424f4} can also be deduced from \cite[(2.27)]{Cegielski} with $\lambda
= (r+\|x-Tx\|)/\|x-Tx\|$, $z=y$, and $\delta=r$ in
\cite[Proposition~2.1.41]{Cegielski}.
This observation, as well as a similar one for \ref{c:0424f5}, is due to a referee.}:
\begin{enumerate}
\item
\label{c:0424f1}
$U_{r,\eta}x = (1-\eta)x + \eta U_r x$.
\item
\label{c:0424f2}
$\|U_{r,\eta}x-y\|^2 =
\eta\|U_{r}x-y\|^2 + (1-\eta)\|x-y\|^2 - \eta(1-\eta)\|x-U_{r}x\|^2$.
\item
\label{c:0424f3}
$\|U_rx-x\| = r + \|x-Tx\|$.
\item
\label{c:0424f4}
$\|U_rx-y\|^2 \leq \|x-y\|^2 - (r + \|x-Tx\|)^2
= \|x-y\|^2 - \|x-U_{r}x\|^2$.
\item
\label{c:0424f5}
$\|U_{r,\eta}x-y\|^2 \leq \|x-y\|^2 - \eta(2-\eta)(r + \|x-Tx\|)^2
= \|x-y\|^2 - \eta^{-1}(2-\eta)\|x-U_{r,\eta}x\|^2$.
\end{enumerate}
\end{corollary}
\begin{proof}
\ref{c:0424f1}: This is a simple verification.
\ref{c:0424f2}:
Using \ref{c:0424f1}, we obtain
$\|U_{r,\eta}x-y\|^2 = \|(1-\eta)(x-y)+\eta(U_{r}x-y)\|^2$.
Now use \cite[Corollary~2.14]{BC2011} to obtain the identity.
\ref{c:0424f3}:
This is immediate from \eqref{e:Ur}.
\ref{c:0424f4}:
Combine \ref{c:0424f3} with Lemma~\ref{l:0424e}\ref{l:0424e3}.
\ref{c:0424f5}:
Combine \ref{c:0424f1}--\ref{c:0424f4}.
\qed
\end{proof}
\subsection{Quasi Projectors}
\begin{definition}[quasi projector]
$Q\colon X\to X$ is a
\emph{quasi projector} of $C$
if $\ran Q = \Fix Q = C$ and
$(\forall x\in X)(\forall c\in C)$
$\|Qx-c\|\leq\|x-c\|$.
\end{definition}
\begin{example}[projectors are quasi projectors]
$P_C$ is a quasi projector of $C$.
More generally\footnote{This observation is a due to a referee.},
if $R\colon X\to X$ is quasi nonexpansive, i.e.,
$(\forall x\in X)(\forall y\in \Fix R)$ $\|Rx-y\|\leq\|x-y\|$ and
$C\subseteq \Fix R$, then $P_C\circ R$ is a quasi projector of
$C$.
\end{example}
It can be shown (see \cite[Proposition~3.4.4]{Thesis})
that when $C$ is an affine subspace, then
the only quasi projector of $C$ is the projector.
However, we will now see that for certain cones there
are quasi projectors different from projectors.
\begin{proposition}[reflector of an obtuse cone]
\label{p:0424c}
{\rm (See \cite[Lemma~2.1]{BK}.)}
Suppose that
$C$ is an obtuse cone, i.e.,
$\RP C = C$ and
$C^\ominus := \menge{x\in X}{\sup\scal{C}{x}=0} \subseteq - C$.
Then the reflector $R_C := 2P_C-\Id$ is nonexpansive and
$\ran R_C = \Fix C = C$.
\end{proposition}
\begin{corollary}
\label{c:0424d}
Suppose that $C$ is an obtuse cone and let
$\lambda \colon X \to [1,2]$.
Then
\begin{equation}
Q\colon X\to X\colon x\mapsto \big(1-\lambda(x)\big)x + \lambda(x) P_Cx
\end{equation}
is a quasi projector of $C$.
\end{corollary}
\begin{proof}
Since, for every $x\in X$,
we have $Q(x)\in [P_Cx,R_Cx]$ and the result thus follows from
Proposition~\ref{p:0424c}.
\qed
\end{proof}
\begin{example}
Suppose $X=\RR^d$ and $C=\RP^d$.
Then $R_C$ is a quasi projector.
\end{example}
\begin{proof}
Because $C^\ominus = -C$,
this follows from Corollary~\ref{c:0424d} with $\lambda(x)\equiv 2$.
\qed
\end{proof}
\begin{remark}
A quasi projector need not be
continuous because we may choose $\lambda$ in Proposition~\ref{p:0424c}
discontinuously.
\end{remark}
\subsection{\fejer\ Monotone Sequences}
Recall that a sequence $(x_n)_\nnn$ in $X$ is \fejer\ monotone with
respect to a nonempty subset $S$ of $X$ if
\begin{equation}
(\forall s\in S)(\forall\nnn)\quad
\|x_{n+1}-s\|\leq\|x_n-s\|.
\end{equation}
Clearly, every \fejer\ monotone sequence is bounded.
We will require the following key result.
\begin{fact}[Raik]
\label{f:Raik}
Let $(x_n)_\nnn$ be a sequence in $X$ that
is \fejer\ monotone with respect to a
subset $S$ of $X$.
If $\inte S\neq\varnothing$, then
$(x_n)_\nnn$ converges strongly to some point in $X$ and
$\sum_\nnn\|x_n-x_{n+1}\| <\pinf$.
\end{fact}
\begin{proof}
See \cite{Raik} or e.g.\ \cite[Proposition~5.10]{BC2011}.
\qed
\end{proof}
\subsection{Differentiability}
\begin{lemma}
\label{l:0426a}
Suppose that $X$ is finite-dimensional,
let $f\colon X\to\RR$ be convex and \frechet\ differentiable such
that $\inf f(X)<0$.
Then for every $\rho\in\RPP$, we have
\begin{equation}
\inf \menge{\|\nabla f(x)\|}{x\in \ball(0;\rho)\cap
f^{-1}(\RPP)}>0.
\end{equation}
\end{lemma}
\begin{proof}
Let $\rho\in\RPP$ and assume to the contrary that the conclusion
fails.
Then there exists a sequence $(x_n)_\nnn$ in
$\ball(0;\rho)\cap f^{-1}(\RPP)$ and a point $x\in
\ball(0;\rho)$ such that $x_n\to x$
and $\nabla f(x_n)\to 0$.
It follows that $f(x)\geq 0$ and $\nabla f(x)=0$, which is
clearly
absurd.
\qed
\end{proof}
\section{Finitely Convergent Cutter Methods}
\label{s:main}
From now on, we assume that
\begin{subequations}
\begin{equation}
\label{e:onr}
\text{$(r_n)_\nnn$ is a sequence in $\RPP$ such that $r_n\to 0$,}
\end{equation}
that
\begin{equation}
\text{$(\eta_n)_\nnn$ is a sequence in $\left]0,2\right]$,}
\end{equation}
and that
\begin{equation}
\text{$Q_C$ is a quasi projector of $C$.}
\end{equation}
\end{subequations}
We further assume that
$x_0\in C$ and that
$(x_n)_\nnn$ is generated by
\begin{equation}
\label{e:seq}
(\forall\nnn)\quad
x_{n+1} :=
\begin{cases}
Q_C\big(x_n+\eta_n(U_{r_n}x_n-x_n)\big), &\text{if $x_n\notin \Fix T$;}\\
x_n, &\text{otherwise.}
\end{cases}
\end{equation}
Note that $(x_n)_\nnn$ lies in $C$.
Also observe that if $x_n$ lies in $\Fix T$, then so does $x_{n+1}$.
We are now ready for our first main result.
\begin{theorem}
\label{t:main1}
Suppose that $\inte(C\cap \Fix T)\neq\varnothing$ and
that $\sum_{\nnn}\eta_nr_n=\pinf$.
Then $(x_n)_\nnn$ lies eventually in $C\cap\Fix T$.
\end{theorem}
\begin{proof}
We argue by contradiction.
If the conclusion is false, then \emph{no} term of the sequence in $(x_n)_\nnn$
lies in $\Fix T$, i.e., $(x_n)_\nnn$ lies in $X\smallsetminus \Fix T$.
By assumption, there exist
$z\in C\cap\Fix T$ and
$r\in\RPP$ and such that
$\ball(z;2r)\subseteq C\cap\Fix T$.
Hence
\begin{equation}
\label{e:0424g}
\big(\forall y\in \ball(z;r)\big)\quad
\ball(y;r)\subseteq C\cap \Fix T.
\end{equation}
Since $r_n\to 0$, there exists $m\in\NN$ such
that $n\geq m$ implies $r_n\leq r$.
Now let $n\geq m$ and $y\in\ball(z;r)$.
Using the assumption that $Q_C$ is a quasi projector of $C$,
that $y\in C$, \eqref{e:0424g}, and Corollary~\ref{c:0424f}, we obtain
\begin{subequations}
\begin{align}
\|x_{n+1}-y\| &= \big\|Q_C\big(x_n+\eta_n(U_{r_n}x_n-x_n)\big)-y\big\|\\
&\leq \|x_n+\eta_n(U_{r_n}x_n-x_n)-y\|\\
&\leq \|x_n-y\|.
\end{align}
\end{subequations}
Hence the sequence
\begin{equation}
\big(x_m,x_m+\eta_m(U_{r_m}x_m-x_m),x_{m+1},x_{m+1}+\eta_{m+1}(U_{r_{m+1}}x_{m+1}-x_{m+1}),x_{m+2},\ldots\big)
\end{equation}
is \fejer\ monotone with respect to $\ball(z;r)$.
It follows from Fact~\ref{f:Raik} and Corollary~\ref{c:0424f}\ref{c:0424f3} that
\begin{equation}
\pinf>
\sum_{n\geq m} \eta_n\|x_n-U_{r_n}x_n\| =
\sum_{n\geq m} \eta_n\big(r_n+\|x_n-Tx_n\|\big) \geq
\sum_{n\geq m}\eta_nr_n,
\end{equation}
which is absurd because $\sum_\nnn \eta_nr_n=\pinf$.
\qed
\end{proof}
We now present our second main result.
Compared to Theorem~\ref{t:main1}, we have a less restrictive
assumption on $(\Fix T,C)$ but a more restrictive one
on the parameters $(r_n,\eta_n)$.
The proof of Theorem~\ref{t:main2} is more or less implicit in
the works by Crombez \cite{Crombez} and Polyak
\cite{PolyakHaifa}; see Remark~\ref{r:Polyak} and
Remark~\ref{r:Crombez}.
\begin{theorem}
\label{t:main2}
Suppose that $C\cap\inte \Fix T\neq\varnothing$ and that
$\sum_\nnn \eta_n(2-\eta_n)r_n^2=\pinf$.
Then $(x_n)_\nnn$ lies eventually in $C\cap\Fix T$.
\end{theorem}
\begin{proof}
Similarly to the proof of Theorem~\ref{t:main1},
we argue by contradiction and assume the conclusion is false.
Then $(x_n)_\nnn$ must lie in $X\smallsetminus \Fix T$.
By assumption, there exist
$y\in \Fix T$ and $r\in\RPP$ such that
$\ball(y;r)\subseteq \Fix T$.
Because $r_n\to 0$, there exists $m\in\NN$ such that
$n\geq m$ implies $r_n\leq r$.
Let $n\geq m$.
Using also the assumption that $Q_C$ is a quasi projector of $C$
and Corollary~\ref{c:0424f}\ref{c:0424f5},
we deduce that
\begin{subequations}
\begin{align}
\|x_{n+1}-y\|^2 &=
\big\|Q_C\big(x_n+\eta_n(U_{r_n}x_n-x_n)\big)-y\big\|^2\\
&\leq \|x_n+\eta_n(U_{r_n}x_n-x_n)-y\|^2\\
&\leq \|x_n-y\|^2 -
\eta_n(2-\eta_n)\big(r_n+\|x_n-Tx_n\|\big)^2\\
&\leq \|x_n-y\|^2 - \eta_n(2-\eta_n)r_n^2.
\end{align}
\end{subequations}
This implies
\begin{equation}
\|x_m-y\|^2 \geq \sum_{n\geq m}\big(\|x_n-y\|^2 -
\|x_{n+1}-y\|^2\big) \geq
\sum_{n\geq m} \eta_n(2-\eta_n)r_n^2 = \pinf,
\end{equation}
which contradicts our assumption on the parameters.
\qed
\end{proof}
Theorem~\ref{t:main1} and Theorem~\ref{t:main2} have various applications.
Since every resolvent of a maximally monotone operator is firmly
nonexpansive and hence a cutter, we obtain the following result.
\begin{corollary}
\label{c:Shawn}
Let $A\colon X\To X$ be maximally monotone, suppose that
$Q_C=P_C$, that $T= (\Id+A)^{-1}$, and
that one of following holds:
\begin{enumerate}
\item $\inte(C\cap A^{-1}0)\neq\varnothing$
and $\sum_\nnn \eta_nr_n=\pinf$.
\item $C\cap \inte A^{-1}0 \neq\varnothing$
and $\sum_\nnn \eta_n(2-\eta_n)r_n^2=\pinf$.
\end{enumerate}
Then $(x_n)_\nnn$ lies eventually in $C\cap A^{-1}0$.
\end{corollary}
Corollary~\ref{c:Shawn} applies in particular
to finding a constrained critical point of a
convex function. When specializing further
to a normal cone operator, we obtain the following result.
\begin{example}[convex feasibility]
Let $D$ be a nonempty closed convex subset of $X$,
and suppose that $Q_C = P_C$, that $T = P_D$, and that one of
the following holds:
\begin{enumerate}
\item
$\inte(C\cap D)\neq\varnothing$ and $\sum_\nnn r_n=\pinf$.
\item
$C\cap\inte D\neq\varnothing$ and $\sum_\nnn r_n^2 = \pinf$.
\end{enumerate}
Then the sequence $(x_n)_\nnn$, generated by
\begin{equation}
(\forall \nnn)\quad
x_{n+1} := P_C\bigg(P_Dx_n + r_n\frac{P_Dx_n-x_n}{\|P_Dx_n-x_n\|} \bigg)
\end{equation}
if $x_n\notin D$ and $x_{n+1}:=x_n$ if $x_n\in D$,
lies eventually in $C\cap D$.
\end{example}
\begin{remark}[relationship to Polyak's work]
\label{r:Polyak}
In \cite{PolyakHaifa},
B.T.\ Polyak considers random algorithms for solving constrained
systems of convex inequalities. Suppose that only one consistent
constrained convex inequality is
considered. Hence the cutters used are all subgradient
projectors (see Example~\ref{ex:0424b}).
Then his algorithm coincides with the one considered
in this section and thus is comparable.
We note that our Theorem~\ref{t:main1} is more flexible because
Polyak requires $\sum_\nnn r_n^2=\pinf$ (see
\cite[Theorem~1 and Section~4.2]{PolyakHaifa}) provided that
$0<\inf_\nnn \eta_n \leq \sup_\nnn \eta_n < 2$ while
we require only $\sum_\nnn r_n=\pinf$ in this case.
Regarding our Theorem~\ref{t:main2}, we note that
our proof essentially follows his proof which actually works
for cutters --- not just subgradient projectors --- and under a
less restrictive constraint qualification.
\end{remark}
\begin{remark}[relationship to Crombez's work]
\label{r:Crombez}
In \cite{Crombez}, G.\ Crombez considers asynchronous parallel
algorithms for finding a point in the intersection of the fixed
point sets of finitely
many cutters --- without the constraint set $C$.
Again, we consider the case when we are dealing with only one
cutter. Then Crombez's convergence result (see
\cite[Theorem~2.7]{Crombez}) is similar to Theorem~\ref{t:main2};
however, he requires that
the radius $r$ of some ball contained in $\Fix T$ be known which
may not always be realistic in practical applications.
\end{remark}
We will continue our comparison in
Section~\ref{s:compare}.
While it is not too difficult to extend Theorem~\ref{t:main1} and
Theorem~\ref{t:main2} to deal with finitely many cutters, we have
opted here for simplicity rather than maximal generality.
Instead, we focus in the next section
on limiting examples.
We conclude this section with a comment on the proximal point algorithm.
\begin{remark}[proximal point algorithm]
Suppose that $A$ is a maximally monotone operator on $X$
(see, e.g., \cite{BC2011} for relevant background information)
such that $Z := A^{-1}0\neq\varnothing$.
Then its resolvent $J_A := (\Id+A)^{-1}$ is firmly nonexpansive
--- hence a cutter --- with $\Fix J_A = Z$.
Let $y_0\in X$ and set $(\forall \nnn)$
$y_{n+1} := J_Ay_n$. Then $(y_n)_\nnn$,
the sequence generated by the proximal point algorithm,
converges weakly to a point in $Z$.
If
\begin{equation}
\label{e:0429a}
(\exi \bar{x}\in X)\quad
0\in\inte A\bar{x},
\end{equation}
then the convergence is finite
(see \cite[Theorem~3]{Rockprox}).
On the other hand, our algorithms impose that
$\inte \Fix T \neq\varnothing$, i.e.,
\begin{equation}
\label{e:0429b}
(\exi \bar{x}\in X)\quad
\bar{x}\in\inte A^{-1}0.
\end{equation}
(Note that \eqref{e:0429a} and \eqref{e:0429b} are
independent:
If $A$ is $\partial \|\cdot\|$, then
$0\in\inte A0$ yet $\inte A^{-1}0=\varnothing$.
And if $A = \nabla d^2_{\ball(0;1)}$,
then $0\in \inte A^{-1}0$ while $A = 2(\Id-P_{\ball(0;1)})$ is single-valued.)
\end{remark}
\section{Limiting Examples}
\label{s:limex}
In this section, we collect several examples that
illustrate the boundaries of the theory.
We start by showing that the conclusion of Theorem~\ref{t:main1} and
Theorem~\ref{t:main2} both may fail to hold if the divergent-series
condition is not satisfied.
\begin{example}[divergent-series condition is important]
Suppose that $X=C=\RR$,
that $f\colon \RR\to\RR\colon x\mapsto x^2-1$,
and that $T=G_f$
is the subgradient projector associated with $f$.
Suppose that $x_0>1$,
set $r_{-1} := x_0-1>0$ and
$(\forall\nnn)$
$r_{n} := r_{n-1}^2/(4(1+r_{n-1}))$.
Then $(r_n)_\nnn$ lies in $\RPP$,
$r_n\to 0$, and
$\sum_{\nnn} r_n <\pinf$ and
hence $\sum_\nnn r_n^2 <\pinf$.
However, the sequence $(x_n)_\nnn$
generated by \eqref{e:seq} lies in
$\left]1,\pinf\right[$ and hence
does not converge finitely to a point in $\Fix T = [-1,1]$.
Furthermore, the classical subgradient projector
iteration $(\forall\nnn)$ $y_{n+1}=Ty_n$ converges
to some point in $\Fix T$, but not finitely when $y_0\notin \Fix T$.
\end{example}
\begin{proof}
It is clear that $\Fix T= [-1,1]$.
Observe that
$(\forall\nnn)$
$0<r_n \leq ({1}/{4})r_{n-1}\leq
(1/4)^{n+1} r_{-1}$.
It follows that $r_n\to 0$ and that
$\sum_\nnn r_n$ and $\sum_\nnn r_n^2$ are
both convergent series.
Now suppose that $r_{n-1} = x_n-1 > 0$ for some
$\nnn$.
It then follows from
Example~\ref{ex:0425a} that
\begin{equation}
x_{n+1} = \frac{x_n}{2} + \frac{1}{2x_n} - r_n
= \frac{(x_n-1)^2}{2x_n} + 1 - r_n
= \frac{r_{n-1}^2}{2(1+r_{n-1})} + 1 - r_n
= r_n+1.
\end{equation}
Hence, by induction,
$(\forall\nnn)$ $x_{n} = 1+r_{n-1}$ and
therefore $x_n\to 1^+$.
As for the sequence $(y_n)_\nnn$, it is follows from
Polyak's seminal work (see \cite{Poljak}) that
$(y_n)_\nnn$ converges to some point in $\Fix T$.
However, by e.g.\ \cite[Proposition~9.9]{BWWX1},
$(y_n)_\nnn$ lies outside $\Fix T$ whenever $y_0$ does.
\qed
\end{proof}
The next example illustrates that we cannot
expect finite convergence if the interior
of $\Fix T$ is empty, in the context
of Theorem~\ref{t:main1} and Theorem~\ref{t:main2}.
\begin{example}[nonempty-interior condition is important]
Suppose that $X=C=\RR$,
that $f\colon \RR\to\RR\colon x\mapsto x^2$,
and that $T=G_f$ is the subgradient projector
associated with $f$.
Then $\Fix T=\{0\}$ and hence
$\inte\Fix T = \varnothing$.
Set $x_0:=1/2$,
and set $(\forall\nnn)$
$w_n := (n+1)^{-1/2}$
and
$r_n = w_n$ if $U_{w_n}x_n\neq 0$
and $r_n=2w_n$ if $U_{w_n}x_n=0$.
Then $r_n\to 0$ and $\sum_\nnn r_n^2=\pinf$.
The sequence $(x_n)_\nnn$ generated
by \eqref{e:seq} converges to $0$ but not finitely.
\end{example}
\begin{proof}
The statements concerning $(r_n)_\nnn$ are clear.
It follows readily from the definition that
$(\forall x\in\RR)(\forall r\in\RP)$
$Tx= x/2$ and $U_rx = x/2 - r\sgn(x)$.
Since $x_0=1/2$, $w_0=1$,
$U_1x_0 = -3/4\neq 0$, and $r_0=w_0=1$,
it follows that $0< |x_0/2| < r_0$.
We now show that for every $\nnn$,
\begin{equation}
\label{e:0425b}
0 < |x_n/2| < r_n.
\end{equation}
This is clear for $n=0$. Now assume
\eqref{e:0425b} holds for some $\nnn$.
\emph{Case~1:} $|x_n| = 2w_n$.\\
Then $U_{w_n}x_n = x_n/2 - \sgn(x_n)w_n = 0$.
Hence $r_n=2w_n$ and thus
$x_{n+1} = U_{r_n}x_n
= x_n/2 - 2w_n\sgn(x_n)
= \sgn(x_n)w_n - 2w_n\sgn(x_n)
= -\sgn(x_n)w_n$.
Thus
$0 < |x_{n+1}/2| = w_n/2 = 1/(2\sqrt{n+1}) < 1/\sqrt{n+2}=w_{n+1}\leq
r_{n+1}$, which yields \eqref{e:0425b} with $n$ replaced by $n+1$.
\emph{Case~2:} $|x_n| \neq 2w_n$.\\
Then $U_{w_n}x_n = x_n/2 - \sgn(x_n)w_n \neq 0$.
Hence $r_n = w_n$ and thus
$x_{n+1} = U_{r_n}x_n = x_n/2 - r_n\sgn(x_n)$.
It follows that
$|x_{n+1}| = r_n - |x_n/2|>0$.
Hence $0<|x_{n+1}/2|$ and
also
$|x_{n+1}| < r_n = w_n < 2w_{n+1}\leq 2r_{n+1}$.
Again, this is \eqref{e:0425b} with $n$ replaced by $n+1$.
It follows now by induction that \eqref{e:0425b} holds for every $\nnn$.
\qed
\end{proof}
We now illustrate that when $\Fix T=\varnothing$, then
$(x_n)_\nnn$ may fail to converge.
\begin{example}
Suppose that $X=C=\RR$,
that $f\colon \RR\to\RR\colon x\mapsto x^2+1$,
and that $T=G_f$ is the subgradient projector associated with $f$.
Let $y_0\in\RR$ and suppose that
$(\forall\nnn)$ $y_{n+1}:=Ty_n$.
Then $(y_n)_\nnn$ is either not well defined or it diverges.
Suppose that $x_0 > 1/\sqrt{3}$,
set $k_0 := x_0 - 1/\sqrt{3}>0$ and
$(\forall\nnn)$ $k_{n+1} := \sqrt{(n+1)/(n+2)}k_n$.
Suppose that
\begin{equation}
(\forall\nnn)\quad
r_n := \frac{1}{2}\bigg( \sqrt{3} + 2k_{n+1} + k_n -
\frac{1}{k_n+1/\sqrt{3}}\bigg).
\end{equation}
Then $r_n\to 0^+$ and $\sum_\nnn r_n^2 = \pinf$.
Moreover, the sequence $(x_n)_\nnn$ generated by
\eqref{e:seq} diverges.
\end{example}
\begin{proof}
Clearly, $\Fix T = \varnothing$ and
one checks that
\begin{equation}
\label{e:0425c}
(\forall r\in\RP)(\forall x\in\RR\smallsetminus\{0\})\quad
U_rx = \frac{x}{2} - \frac{1}{2x} - r\sgn(x).
\end{equation}
If some $y_n=0$, then the sequence $(y_n)_\nnn$ is not well defined.
\emph{Case~1}: $(\exi\nnn)$ $y_n=1/\sqrt{3}$.\\
Then $x_{n+1}= Tx_n = U_0x_n = x_n/2 - 1/(2x_n) = -1/\sqrt{3}
= -x_n$ and similarly $x_{n+2}=-x_{n+1}=x_n$.
Hence the sequence eventually oscillates between
$1/\sqrt{3}$ and $-1/\sqrt{3}$.
\emph{Case~2}: $(\exi\nnn)$ $|y_n|=1$.\\
Then $y_{n+1} = 0$ and the sequence is not well defined.
\emph{Case~3}: $(\forall\nnn)$ $|y_n|\notin\{1,1/\sqrt{3}\}$.\\
Using the Arithmetic Mean--Geometric Mean inequality, we obtain
\begin{equation}
|y_{n+1}-y_n| = \left|\frac{y_n}{2}-\frac{1}{2y_n}-y_n\right|
= \frac{1}{2}\left| y_n + \frac{1}{y_n}\right|
= \frac{1}{2}\left( |y_n| + \frac{1}{|y_n|}\right) \geq 1
\end{equation}
for every $\nnn$.
Therefore, $(y_n)_\nnn$ is divergent or not well defined.
We now turn to the sequence $(x_n)_\nnn$.
Observe that $0 < k_n = \sqrt{n/(n+1)}k_{n-1} = \cdots =
k_0/\sqrt{n+1} \to 0^+$ and hence $(k_n)_\nnn$ is strictly
decreasing.
It follows that $r_n\to 0^+$ and
that $r_n > (2k_{n+1}+k_n)/2 > 3k_{n+1}/2 = 3k_0/(2\sqrt{n+2})$.
Thus, $\sum_\nnn r_n^2 = \pinf$.
Next, \eqref{e:0425c} yields
\begin{subequations}
\begin{align}
x_1 &= \frac{x_0}{2} - \frac{1}{2x_0} - r_0 \\
&= \frac{k_0+1/\sqrt{3}}{2} - \frac{1}{2\big(k_0+1/\sqrt{3})} -
\frac{1}{2} \bigg( \sqrt{3} + 2k_{1} + k_0 -
\frac{1}{k_0+1/\sqrt{3}}\bigg)\\
&= -\frac{1}{\sqrt{3}} - k_1.
\end{align}
\end{subequations}
Hence $x_1<0$ and we then see analogously that $x_2 =
1/\sqrt{3}+k_2 > 0$.
We inductively obtain
\begin{equation}
(\forall\nnn)\quad
0<x_{2n} = \frac{1}{\sqrt{3}} + k_{2n}
\;\;\text{and}\;\;
0>x_{2n+1} = -\frac{1}{\sqrt{3}} - k_{2n+1}.
\end{equation}
It follows that $(-1)^nx_n \to 1/\sqrt{3}$;
therefore, $(x_n)_\nnn$ is divergent.
\qed
\end{proof}
\section{Comparison}
\label{s:compare}
In this section, we assume for notational simplicity\footnote{If we replace
\frechet\ differentiability by mere continuity, then we may
consider a selection of the subdifferential operator $\partial f$
instead.} that
\begin{equation}
\text{$f\colon X\to\RR$ is convex and \frechet\ differentiable
with $\menge{x\in X}{f(x)\leq 0}\neq\varnothing$}
\end{equation}
and that
\begin{equation}
T=G_f\colon X\to X\colon x\mapsto
\begin{cases}
\displaystyle x - \frac{f(x)}{\|\nabla f(x)\|^2}\nabla f(x), &\text{if $f(x)>0$;}\\
x, &\text{otherwise}
\end{cases}
\end{equation}
is the associated subgradient projector (see
Example~\ref{ex:0424b}).
Then \eqref{e:Ur} turns into
\begin{equation}
U_rx =
\begin{cases}
\displaystyle x - \frac{f(x)+r\|\nabla f(x)\|}{\|\nabla f(x)\|^2}
\nabla f(x), &\text{if $f(x)>0$;}\\
x, &\text{otherwise}
\end{cases}
\end{equation}
and \eqref{e:seq} into
\begin{equation}
\label{e:ours}
(\forall\nnn)\quad
x_{n+1} =
\begin{cases}
Q_C\bigg(x_n-\eta_n\displaystyle\frac{f(x_n)+r_n\|\nabla
f(x_n)\|}{\|\nabla f(x_n)\|^2}\nabla
f(x_n)\bigg), &\text{if $f(x_n)>0$;}\\
x_n, &\text{otherwise.}
\end{cases}
\end{equation}
In the algorithmic setting of Section~\ref{s:main},
Polyak uses $\eta \equiv\eta_n\in \left]0,2,\right[$
(e.g.\ $\eta=1.8$; see \cite[Section~4.3]{PolyakHaifa}).
In the present setting, his framework requires $\sum_\nnn
r_n^2=\pinf$.
When $C=X$, one also has
the following similar yet different update formula
\begin{equation}
\label{e:ccp}
(\forall\nnn)\quad
y_{n+1} =
\begin{cases}
y_n-\eta_n\displaystyle\frac{f(y_n)+\varepsilon_n}{\|\nabla f(y_n)\|^2}\nabla
f(y_n), &\text{if $f(y_n)>0$;}\\
y_n, &\text{otherwise,}
\end{cases}
\end{equation}
where $0<\inf_\nnn \eta_n \leq \sup_\nnn \eta_n<2$ and
$(\varepsilon_n)_\nnn$ is a strictly decreasing sequence in $\RPP$
with $\sum_\nnn\varepsilon_n=\pinf$.
In this setting, this is also known as the
\emph{Modified Cyclic Subgradient Projection Algorithm (MCSPA)},
which finds its historical roots in works by
Fukushima \cite{Fuku}, by De Pierro and Iusem \cite{DPI}, and
by Censor and Lent \cite{CL};
see also \cite{CCP,IusMol86,IusMol87,Pang} for related works.
Note that MCSPA requires the existence of a \emph{Slater point}, i.e.,
$\inf f(X)<0$, which is more restrictive than our assumptions
(consider, e.g., the squared distance to the unit ball).
Let us now link the assumption on the parameters of the MCSPA \eqref{e:ccp}
to \eqref{e:ours}.
\begin{proposition}
Suppose that $X=C$ is finite-dimensional,
that $\inf f(X)<0$,
that $\eta_n\equiv 1$,
that $\sum_\nnn r_n=\pinf$ (recall \eqref{e:onr}), and
that $(\forall\nnn)$ $\varepsilon_n = r_n\|\nabla
f(x_n)\| > 0$.
Then $\varepsilon_n \to 0$ and $\sum_\nnn\varepsilon_n=\pinf$.
\end{proposition}
\begin{proof}
Corollary~\ref{c:0424f}\ref{c:0424f4} implies
that $(x_n)_\nnn$ is bounded.
Because $\nabla f$ is continuous, we obtain
that $\sigma := \sup_\nnn \|\nabla f(x_n)\|<\pinf$.
By Lemma~\ref{l:0426a}, there
exists $\alpha\in\RPP$ such that
if $f(x_n)>0$, then $\|\nabla f(x_n)\|\geq \alpha$.
Hence
\begin{equation}
(\forall\nnn)\quad
f(x_n)>0
\;\;\Rightarrow\;\;
0<\alpha r_n \leq \|\nabla f(x_n)\|r_n = \varepsilon_n \leq
\sigma r_n,
\end{equation}
and therefore $\sum_\nnn \varepsilon_n=\pinf$.
\qed
\end{proof}
The next example shows that our assumptions are independent of
those on the MCSPA.
\begin{example}
Suppose that $X=C=\RR$, that
$f\colon \RR\to\RR\colon x\mapsto x^2-1$,
that $r_n = (n+1)^{-1}$ if $n$ is even and
$r_n = n^{-1/2}$ if $n$ is odd,
and that $\eta_n\equiv 1$.
Clearly, $r_n\to 0$ and $\sum_\nnn r_n^2=\pinf$.
However, $(\varepsilon_n)_\nnn := (r_n|f'(x_n)|)_\nnn$
is not strictly decreasing.
\end{example}
\begin{proof}
The sequence $(x_n)_\nnn$ is bounded.
Suppose that $f(x_n)>0$ for some $\nnn$.
By Example~\ref{ex:0425a},
\begin{equation}
\label{e:0426b}
x_{n+1} = U_{r_n}x_n = \frac{x_n}{2} + \frac{1}{2x_n} -
r_n\sgn(x_n).
\end{equation}
Assume that $n$ is even, say $n=2m$, where $m\geq 2$,
and that $1<x_{2m} < (2m+1)/2$.
Then $x_{2m} > 2x_{2m}/\sqrt{2m+1}$ and
\begin{equation}
\varepsilon_{2m} =
r_{2m}|f'(x_{2m})|
= 2r_{2m}x_{2m}
= \frac{2x_{2m}}{2m+1}.
\end{equation}
Hence, using \eqref{e:0426b},
\begin{equation}
x_{2m+1} = \frac{x_{2m}}{2} + \frac{1}{2x_{2m}} - r_{2m}
> \frac{x_{2m}}{2} + \frac{1}{2m+1} - \frac{1}{2m+1} =
\frac{x_{2m}}{2},
\end{equation}
and therefore
\begin{equation}
2x_{2m+1} > x_{2m} > \frac{2x_{2m}}{\sqrt{2m+1}}.
\end{equation}
Thus
$\varepsilon_{2m+1} = r_{2m+1}|f'(x_{2m+1})| = 2r_{2m+1}x_{2m+1}$.
It follows that
\begin{equation}
\varepsilon_{2m+1} = \frac{2x_{2m+1}}{\sqrt{2m+1}}
> \frac{2x_{2m}}{2m+1} = \varepsilon_{2m}
\end{equation}
and the proof is complete.
\qed
\end{proof}
\section{Perspectives}
\label{s:persp}
Suppose that $X=\RR$ and
that $f\colon X\to\RR\colon x\mapsto x^2-1$.
Let $T$ be the subgradient projector associated with $f$ and
assume that $C=X$.
We chose 100 randomly chosen starting points in
the interval $[1,10^6]$.
In the following table, we record the performance
of the algorithms;
here $(r_n,\eta_n)$ signals that \eqref{e:ours} was used,
while $\varepsilon_n$ points to \eqref{e:ccp} with
$\eta_n\equiv 1$. Mean and median refer to the number of
iterations until the current iterate was $10^{-6}$ feasible.
\begin{table}[h!] \centering
\begin{tabular}{@{}lrr@{}} \toprule
Algorithm for $x^2-1$ &Mean &Median\\ \midrule
$(r_n,\eta_n)=\big(1/(n+1),1\big)$ &$11.49$ & $13$ \\
$(r_n,\eta_n)=\big(1/(n+1),2\big)$ &$2$ & $2$ \\
$(r_n,\eta_n)=\big(1/\sqrt{n+1},1\big)$ &$10.83$ & $12$ \\
$(r_n,\eta_n)=\big(1/\sqrt{n+1},2\big)$ &$2$ & $2$ \\
$\varepsilon_n=1/(n+1)$ &$11.81$ & $13$ \\
$\varepsilon_n=1/\sqrt{n+1}$ &$12.19$ & $13$ \\
\bottomrule
\end{tabular}
\end{table}
\noindent
Now let us instead consider $f\colon X\to\RR\colon x\mapsto 100 x^2-1$.
The corresponding data are in the following table.
\begin{table}[h!] \centering
\begin{tabular}{@{}lrr@{}} \toprule
Algorithm for $100x^2-1$ &Mean &Median\\ \midrule
$(r_n,\eta_n)=\big(1/(n+1),1\big)$ &$13.29$ & $14$ \\
$(r_n,\eta_n)=\big(1/(n+1),2\big)$ &$12$ & $12$ \\
$(r_n,\eta_n)=\big(1/\sqrt{n+1},1\big)$ &$17.52$ & $19$ \\
$(r_n,\eta_n)=\big(1/\sqrt{n+1},2\big)$ &$105$ & $105$ \\
$\varepsilon_n=1/(n+1)$ &$15.27$ & $16$ \\
$\varepsilon_n=1/\sqrt{n+1}$ &$15.76$ & $17$ \\
\bottomrule
\end{tabular}
\end{table}
\noindent
We observe that the performance of the algorithms
clearly depends on the step lengths $r_n$ and $\varepsilon_n$, on
the relaxation parameter $\eta_n$, and on the underlying
objective function $f$; however, \emph{the precise nature of this
dependence is rather unclear}.
It would thus be
interesting to perform numerical experiments on a wide variety of
problems and parameter choices with the goal to \emph{obtain guidelines
in the choice of algorithms and parameters} for the user.
Another avenue for future research is to
\emph{construct a broad framework} that
encompasses the present as well as previous related
finite convergence results (see references in Section~\ref{s:compare}).
\section{Conclusions}
\label{s:conc}
We have obtained new and more general finite convergence results
for a class of algorithms based on cutters.
A key tool was Raik's result on
\fejer\ monotone sequences (Fact~\ref{f:Raik}).
\begin{acknowledgements}
The authors thank two anonymous referees for careful reading,
constructive comments, and for bringing additional references to
our attention.
The authors also thank
Jeffrey Pang for helpful discussions and for pointing
out additional references.
\end{acknowledgements}
| {
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} | 9,381 |
In the Nick is a 1960 British comedy film directed by Ken Hughes and starring Anthony Newley, Anne Aubrey, Bernie Winters, James Booth and Harry Andrews. In the film, a gang of incompetent criminals are placed in a special type of new prison. Featured song Must Be was written by Lionel Bart.
Plot
A progressive experimental prison without bars is run by young psychiatrist Dr. Newcombe (Anthony Newley) and harsh but fair Chief Officer Williams (Harry Andrews). Four hardened criminals, the Spider Gang, arrive at this minimum security prison, the leader of whom is Spider Kelly (James Booth). Dr. Newcombe has his work cut out trying to reform the boys and enlists the aid of Spider's girlfriend Doll (Anne Aubrey), who, to Spider's anger, is now working as a stripper in Soho. Newcombe seems to be straightening Spider out, while Spider is in turn sorting out a rival imprisoned gang, led by Ted Ross (Ian Hendry), who hold the monopoly in smuggled cigarettes.
Cast
Anthony Newley - Dr. Newcombe
Anne Aubrey - The Doll
Bernie Winters - Jinx Shortbottom
James Booth - Spider Kelly
Harry Andrews - Chief Officer Williams
Al Mulock - Dancer
Derren Nesbitt - Mick
Niall MacGinnis - Prison Governor
Victor Brooks - Screw Smith
Ian Hendry - Ted Ross
Kynaston Reeves - Judge
Barry Keegan - Screw Jenkins
Diana Chesney - Barmaid
Andria Lawrence (uncredited)
Sam Kydd - Inmate
Production
Many of the same team had just made Jazz Boat.
Critical reception
The Monthly Film Bulletin noted "A sequel to Jazz Boat, with the same leading characters and production team, In the Nick is cast very much in the same mould--easy-going mixture of farce and fantasy, loose and ingenuous scripting, excellent (if bizarre) team-playing. James Booth stands out for his genuinely observed portrait of Spider, Bernie Winters appears to be one of those rare comedians who can keep his moronic style of clowning free from offensiveness, and Niall MacGinnis (Governor), Harry Andrews (Chief Officer) and Ian Hendry (rival mobsters) all catch the eye. Anthony Newley is rather at sea as a psychiatrist, but plays with a likable modesty and warmth, and an improved Anne Aubrey discretely burlesques Jayne Mansfield. There is much in this film that is conventionally weak and structurally uneven, yet it gets closer to contemporary feeling than numerous more ambitious comedies. The dialogue, particularly, strikes an authentic note, and Ken Hughes' debt to Frank Norman, who wrote the original story, seems considerable." while TV Guide noted, "Though there are some genuinely funny moments in the film, Newley is miscast as the compassionate psychologist. Though relatively straightforward for its first half, the plot becomes convoluted and the motivations are twisted in the second half."
References
External links
1960 films
1960s English-language films
Films directed by Ken Hughes
1960 comedy films
British comedy films
1960s British films | {
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} | 5,081 |
Cătălin Gheorghiță Golofca ()
Echipe de club
Golofca a jucat timp de 6 ani între 2010 și 2016 în liga a II-a la Rapid CFR Suceava.
Botoșani
S-a transferat la clubul de Liga I FC Botoșani în ianuarie 2016. El a petrecut restul sezonului 2015-2016 revenit la Rapid CFR Suceava sub formă de împrumut.
În sezonul următor, noul său club l-a reținut în lot. Primul gol în prima ligă l-a înscris la 18 august 2016, într-o victorie cu 5-0 în deplasare cu ACS Poli Timișoara. Până la sfârșitul sezonului, Golofca a înscris în total 9 goluri în 34 de partide. În sezonul următor, după primele patru etape, FC Botoșani se afla pe primul loc în clasament, Golofca jucând în toate cele patru meciuri și marcând alte trei goluri.
FCSB
Pe 7 august 2017, FCSB a anunțat transferul lui Golofca pe un contract pe patru ani, în schimbul sumei de 400.000 de euro plus TVA pentru, FC Botoșani păstrând 60% din drepturile financiare în vederea unui viitor transfer.
Transferul nu a fost unul reușit, Golofca având prestații modeste, și fiind acuzat de indisciplină. La finalul sezonului de toamnă din 2017, Golofca a revenit la FC Botoșani, după ce a jucat în doar 8 meciuri de campionat pentru FCSB fără a înscrie vreun gol.
CFR Cluj
După 3 ani la FC Botoșani, Golofca a plecat la CFR Cluj în august 2019, pentru suma de 500.000 de euro.
Referințe
Legături externe
Profil și statistitici pe RomanianSoccer
Nașteri în 1990
Atacanți
Mijlocași
Fotbaliști ai FC Botoșani
Fotbaliști ai FC Steaua București
Oameni în viață
Suceveni
Fotbaliști români | {
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The Géant Glacier () is a large glacier on the French side of the Mont Blanc massif in the Alps. It is the main supplier of ice (via the Vallée Blanche) to the Mer de Glace which flows down towards Montenvers. It gets its name from the nearby Dent du Géant.
Access
It is possible to take the Vallée Blanche Cable Car which travels right over the Géant Glacier, from the Aiguille du Midi in France, to Pointe Helbronner on the Italian/French border. Experienced skiers also have an "outstanding attraction" in the challenging run from the Aiguille du Midi telepherique station, down the Vallée Blanche and onto the Géant Glacier then, avoiding the Seracs du Géant, merging onto the Mer du Glace to reach the resort of Montenvers
Early studies
In 1862, the physicist and mountaineer John Tyndall gave a series of lectures to the Royal Institution in which he reported on his studies into heat as a form of motion. By placing lines of stakes upon the ice, he made numerous measurements of ice flow from the Géant Glacier and surrounding glaciers into the Mer de Glace. He subsequently published further accounts in his 'Glaciers of the Alps.
Further reading
Tyndall, J. (1860?) Glaciers of the Alps
References
External links
Geant Glacier on French IGN mapping portal
Glaciers of Metropolitan France
Glaciers of the Alps | {
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The Diamond Dual Gate Opener (MMDIA30D) is designed for dual swing gates up to a maximum of 16 ft. Long or maximum weight of 500 lbs. per gate panel. Compatible with all gate types of gates including chain link, tube, panel, vinyl and wood. The Diamond series can be utilized for many applications including residential, agricultural, farm and ranch, etc. The Diamond series and all Mighty Mule automatic gate Openers are solar capable with the addition of a Mighty Mule 10-Watt solar panel (model # FM123) This opener comes equipped with Mighty Mule's exclusive dual sense technology, this meets UL 325s 6th edition standards for additional protection and safety. The Mighty Mule dual sense technology provides users with additional safety and precaution without the need of secondary entrapment device, such as photo eye sensors (model # R4222). Please note that all gate opener installations are unique and may require additional protection devices based on the type of installation. The Diamond series gate operators includes everything you need for easy DIY installation.
Designed for dual swing gates up to a maximum of 16 ft. long or maximum of weight of 500 lbs. per gate panel.
I bought these gate openers Feb. 2018 and have had nothing but problems out of them. I have called several times and had replacement parts sent to me and they still don't work right. Every time I get them to work right they will work for a couple of days and then they quit. I would not recommend buying any of these gate openers and I really wouldn't even give it a star but it wouldn't let me not select one! I made the mistake of buying them this time but it will not happen again!
Installed in less than 1/2 a day and works flawlessly.
Diamond dual openers. Installed about 5 days ago. Work well, installed on same hardware as MM352 that quit working with minor repositioning of gate brackets. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,588 |
SLEEPY HOLLOW, NY (Sept. 8, 2010) — Historic Hudson Valley, creator of the Great Jack O'Lantern Blaze, is expanding its portfolio of Halloween inspired experiences to include a professionally produced haunted attraction: Horseman's Hollow. It joins another new experience, Jonathan Kruk's 'Legend,' and two returning classics: Blaze and Legend Celebration, to round out a full complement of classic fall events in Sleepy Hollow Country.
Washington Irving's macabre The Legend of Sleepy Hollow inspires the brand new Horseman's Hollow, an interactive haunted attraction at Philipsburg Manor recommended for ages 14 and up. Stocked with professional actors and state-of-the-art special effects, Horseman's Hollow has a very high fear factor and is not for the faint of heart.
Jonathan Kruk's 'Legend' brings the master storyteller into the historic, candlelit interior of the circa-1685 Old Dutch Church, where he will offer a dramatic re-telling of The Legend of Sleepy Hollow accompanied by live organ music.
" with crows are some of the new jack o'lantern installations included in this massive display of Halloween-inspired creativity. A team of artists come together to carve more than 4,000 jacks, many of them fused together in enormous and elaborate constructions, all lit up throughout the wooded walkways, orchards, and gardens of historic Van Cortlandt Manor in Croton-on-Hudson, N.Y.
Also returning is Legend Celebration at Washington Irving's Sunnyside, a daytime event perennially popular with young children, where visitors are encouraged to come in costume.
All events are held rain or shine. All proceeds support Historic Hudson Valley, the Tarrytown-based non-profit educational organization which owns and operates the historic sites that host these events. Historic Hudson Valley is using Twitter, Facebook, Flickr (HHValley) and YouTube (InTheValley1), as well as its own HVBlog to connect with fans and offer behind the scenes info on event preparation.
For Blaze, Horseman's Hollow, and Jonathan Kruk's 'Legend,' all admissions are by timed ticket, which must be purchased in advance.
Blaze dates are Oct. 2-3, 8-11, 15-17, 21-24, 28-31, and Nov. 5-7. The first reservation is 7 p.m. on Oct. 2-3 and 8-11, and 6:30 p.m. for the other dates. Tickets are $16 for adults, $12 for children 5-17, free for children under 5.
Horseman's Hollow dates are Oct. 15-16, 22-24, and 28-30, with the first reservation at 7 p.m. Tickets are $20.
Jonathan Kruk's 'Legend' dates are Oct. 15-16 and 29-30. Seating is limited, and there are four performances each evening on the hour, beginning at 6pm. Tickets are $16 for adults, $12 for children under 18.
Legend Celebration dates are Oct. 23-24 and 30-31, from 10-4 p.m. Tickets are $12 for adults, $6 for children 5-17, and free for those under 5.
Buy tickets online at www.hudsonvalley.org or by calling 914-631-8200 ($2 per ticket surcharge for phone orders).
Historic Hudson Valley members receive discounted admission to Jonathan Kruk's 'Legend' and free admission to the other events. Details on becoming a member are online.
Horseman's Hollow, a brand new haunted experience right in the heart of Sleepy Hollow, takes the tale of The Legend of Sleepy Hollow to its darkest extremes. For eight October evenings, historic Philipsburg Manor will transform into a terrifying landscape ruled by vampires, witches, undead soldiers, ghouls, and ghosts, all serving the Headless Horseman himself.
Historic Hudson Valley is working with haunted house professional Lance Hallowell to create an immersive, interactive, pleasantly terrifying experience stocked with professional actors and state-of-the-art special effects. Visitors begin walking a haunted trail, stumbling upon scary scenes of a town driven mad by the Headless Horseman. The Hollow's unfortunate inhabitants are all too ready to keep visitors from ever leaving.
Creatures, some human and some not, lurk in the shadows, ready to terrify the unsuspecting, while state-of-the-art special effects disorient and unsettle. Lighting is by Emmy award-winning designer Deke Hazirjian of New York City Lites.
Custom built set pieces and period-correct costumes help orient the experience in Philipsburg Manor's traditional time period of the mid-1700s.
Philipsburg Manor is at 381 North Broadway (Route 9) in Sleepy Hollow.
The Great Jack O' Lantern Blaze, which drew 68,000 visitors last year, is a Halloween spectacle integrating thousands of hand-carved pumpkins — everything from your standard jack o' lantern to extremely elaborate abstract designs — lit up throughout the landscape of Van Cortlandt Manor in various thematic and conceptual arrangements such as Egypt, the Undersea Aquarium, Pirates' Cove, life-sized dinosaurs in Jurassic Park, "Pumpkinhenge," (a riff on Stonehenge), a giant "cornfield," an enormous spider web, Celtic knots, a circus arena with clown carvings, and more.
Professional lighting by designer Jay Woods and a spooky aural soundscape of original music and noises help create a complete all-senses immersion.
Scores of Blaze videos uploaded by fans can be found on YouTube.
Describing the event as an art installation, Creative Director Michael Natiello, who also coordinates the music, said Van Cortlandt Manor itself is the ultimate inspiration for Blaze. The jack o'lantern arrangements are meant to complement and draw attention to the site's architecture, Revolutionary-era history, and landscape.
Natiello leads a team of Historic Hudson Valley staff and local artists who carve the pumpkins. In addition, more than 1,000 volunteers help scoop and light the pumpkins, including teams from Westchester and Putnam ARC, NY Life, dozens of Girl Scout and Boy Scout troops, Pace University students and faculty, and many schoolchildren.
Blaze artists will be carving on site during the event, with their finished products available for purchase. In addition, the Great Jack O' Lantern Blaze Shop will offer a full bounty of Blaze-specific merchandise including hats, notepads, games, T-shirts, magnets, caps, mugs, and jewelry. Café Blazé, by Geordane's of Irvington, will offer culinary treats including soup, veggie chili, muffins, pumpkin cookies, and cider.
Blaze Title Sponsor is Entergy; Blaze Media Sponsor is 100.7 WHUD.
Van Cortlandt Manor is at 525 South Riverside Avenue, just off Route 9 in Croton-on-Hudson.
Master storyteller Jonathan Kruk offers a dramatic re-telling of Washington Irving's classic tale, The Legend of Sleepy Hollow, featuring the Headless Horseman, Ichabod Crane, Brom Bones, and Katrina Van Tassel. Flavored with live spooky organ music by Jim Keyes, Kruk's storytelling will take place in the historic, candlelit setting of the Old Dutch Church. The 1685 stone church is across the street from Philipsburg Manor, where visitors will park.
Ideal for the youngest Halloween fans, Legend Celebration offers kids and their families a chance to come in costume to Washington Irving's Sunnyside for Halloween fun.
Visitors can get their head examined by a phrenologist and enjoy a Punch Van Winkle puppet show as well as games, magic, sing-a-longs, Irish ghost stories, and other kid friendly Halloween experiences. Sunnyside also offers spooky woodland walks complete with ghost stories, which require online advance reservations. Sunnyside is at 89 West Sunnyside Lane, off Route 9 in Tarrytown.
Video-savvy visitors may enter the Blaze YouTube Video Contest by filming the event, editing their work, and uploading their video (three minutes or less) to YouTube.
Historic Hudson Valley will award an Apple iPad prize provided by 100.7 WHUD for the video that best achieves the contest goal of capturing the creativity and excitement of Blaze. Full contest details and requirements are online atwww.hudsonvalley.org. | {
"redpajama_set_name": "RedPajamaC4"
} | 3 |
using System.Linq;
using System.Reflection;
using System.Text.RegularExpressions;
using Baseline;
using Weasel.Postgresql;
using Weasel.Postgresql.Tables;
namespace Marten.Schema
{
public class FullTextIndex: IndexDefinition
{
public const string DefaultRegConfig = "english";
public const string DefaultDataConfig = "data";
private string _regConfig;
private string _dataConfig;
private readonly DbObjectName _table;
private string _indexName;
public FullTextIndex(DocumentMapping mapping, string regConfig = null, string dataConfig = null, string indexName = null)
{
_table = mapping.TableName;
RegConfig = regConfig;
DataConfig = dataConfig;
_indexName = indexName;
Method = IndexMethod.gin;
}
public FullTextIndex(DocumentMapping mapping, string regConfig, MemberInfo[][] members)
: this(mapping, regConfig, GetDataConfig(mapping, members))
{
}
protected override string deriveIndexName()
{
var lowerValue = _indexName?.ToLowerInvariant();
if (lowerValue?.StartsWith(SchemaConstants.MartenPrefix) == true)
return lowerValue.ToLowerInvariant();
else if (lowerValue?.IsNotEmpty() == true)
return SchemaConstants.MartenPrefix + lowerValue.ToLowerInvariant();
else if (_regConfig != DefaultRegConfig)
return $"{_table.Name}_{_regConfig}_idx_fts";
else
return $"{_table.Name}_idx_fts";
}
public string RegConfig
{
get => _regConfig;
set => _regConfig = value ?? DefaultRegConfig;
}
public string DataConfig
{
get => _dataConfig;
set => _dataConfig = value ?? DefaultDataConfig;
}
public override string[] Columns
{
get
{
return new string[] { $"( to_tsvector('{_regConfig}', {_dataConfig}) )"};
}
set
{
// nothing
}
}
// TODO -- keep this just in case?
// public bool Matches(ActualIndex index)
// {
// var ddl = index?.DDL.ToLowerInvariant();
//
// // To omit the null conditional operators that were following here before
// if (ddl == null)
// {
// return false;
// }
//
// var regexStripType = new Regex(@"('.+?')::text");
// var regexStripParentheses = new Regex("[()]");
//
// // Check for the existence of the 'to_tsvector' function, the correct table name, and the use of the data column
// return ddl.Contains("to_tsvector") == true
// && ddl.Contains(IndexName) == true
// && ddl.Contains(_table.QualifiedName) == true
// && ddl.Contains(_regConfig.ToLowerInvariant()) == true
// // For comparison, strip out types (generated by pg_get_indexdef, but not by Marten) and parentheses (again, pg_get_indexdef produces a bit different output to Marten).
// && regexStripParentheses.Replace(regexStripType.Replace(ddl, "$1"), string.Empty).Contains(regexStripParentheses.Replace(_dataConfig.ToLowerInvariant(), string.Empty));
// }
private static string GetDataConfig(DocumentMapping mapping, MemberInfo[][] members)
{
var dataConfig = members
.Select(m => $"({mapping.FieldFor(m).TypedLocator.Replace("d.", "")})")
.Join(" || ' ' || ");
return $"({dataConfig})";
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,235 |
Q: How to use .format() string in a MySQL statement? I'm trying to format a MySQL statement with today's data, as well as 7 days back. I'm pretty sure the date is in the correct format, so I don't think that's the issue.
The error report is:
Warning: (1292, "Incorrect datetime value: '{} 16:00:00' for column 'run_start_date' at row 1")
result = self._query(query)
Traceback (most recent call last):
AttributeError: 'int' object has no attribute 'format'
e.g.
today = DT.date.today()
week_ago = today - DT.timedelta(days=7)
print(today.strftime('%Y-%m-%d'))
print(week_ago.strftime('%Y-%m-%d'))
cursor.execute(SELECT * FROM db WHERE run_start_date BETWEEN '{} 16:00:00' AND '{} 16:00:00'format(week_ago, today)
A: Using prepared statements (safe way):
qry = """
SELECT *
FROM db
WHERE run_start_date BETWEEN '%s 16:00:00' AND '%s 16:00:00'
""" #
today = DT.date.today()
week_ago = today - DT.timedelta(days=7)
today = str(today.strftime('%Y-%m-%d')) # Convert to string
week_ago = str(week_ago.strftime('%Y-%m-%d')) # Convert to string
cursor.execute(qry, [today, week_ago])
Using .format() which leaves you at risk for sql injections (if you pass user input to .format() e.g.)
qry = """
SELECT *
FROM db
WHERE run_start_date BETWEEN '{today} 16:00:00' AND '{week_ago} 16:00:00'
""" # Use named placeholders, nicer to read, prevents you having to repeat variables multiple time when calling .format()
today = DT.date.today()
week_ago = today - DT.timedelta(days=7)
today = str(today.strftime('%Y-%m-%d')) # Convert to string
week_ago = str(week_ago.strftime('%Y-%m-%d')) # Convert to string
qry = qry.format(today=today, week_ago=week_ago)
cursor.execute(qry)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,137 |
The Venice State Archive, or State Archive of Venice, () is located at Campo dei Frari, San Polo Venice.
Significance
The archive contains most of the historical sources that the Republic of Venice has had since the city fire of 976 (cf. Pietro IV. Candiano). up to 1797, as well as the holdings of Italian state offices based in Venice from 1866, which were ceded to the archive. The locally created archival materials from the French and Austrian periods between 1797 and 1866 are also located there. There are also isolated documents from before 976.
The more recent municipal holdings are in the Archivio storico del Comune di Venezia, while those of the parishes, the defunct dioceses and the patriarchate are in the Archivio storico del Patriarcato di Venezia.
Also in the state archive are numerous holdings from monasteries and churches, professional associations and families, the six Scuole Grandi and the numerous Scuole piccole and the brotherhoods, notaries, etc. There is also a library with a stock of around 59,000 volumes.
Headquarters is a former Franciscan monastery at the Frari Church in the Sestiere of San Polo.
The house is not only the most important archive for Venetian history and that of Veneto, but also of the greatest importance for the entire area of the former colonial empire, i.e., the area between the upper Adriatic and Cyprus. The same applies to the history of the Mediterranean area touched by Venice's foreign relations, the Black Sea, but also the North Sea area and the southern German cities. It is also the most important archive for the history of the Balkans until well into the Ottoman period.
History
The Franciscan Monastery
The Franciscan monastery goes back to a donation from Doge Jacopo Tiepolo in 1246. The Order was permitted to drain the lacus Badovarius or Badovariorum. The small lake was named after the neighboring palace of the Badoer family. The number of monks and the amount of donations grew rapidly, so that on April 28, 1250 the foundation stone could be laid. One of the largest Franciscan churches in Europe was built by the end of the 15th century. Therefore, the double convent building was also called domus magna or cà granda. The outer, larger monastery was dedicated to the Trinity, the inner, smaller monastery to Sant'Antonio. The order was dissolved in 1810.
Beginnings
Jacopo Chiodo, who preferred the baptismal name Giacomo Chiodo, worked as an archivist both before and after 1797. He tried to initiate the establishment of a central archive, to which both Vienna and Paris agreed. However, it was not set up initially, instead there were still three departments in different locations: the "political" archives were kept in San Teodoro, a branch managed by Carlo Antonio Marin, while the court files were moved to San Giovanni and Archival documents of the Treasury and the Domains stored in San Provolo.
In 1815 Venice was returned to Austria, and in the same year the decision was made to set up a central archive. The state archive was created from 1817 under the name Archivio generale veneto, its first director was Chiodo, who retired in 1840. Actually, all archival material should be transferred to the current archive of the Austrians, but Chiodo managed to avert this. Between 1817 and 1822 the state files from the time of the Republic were brought there. In Napoleon's time, if they had not been taken to Paris and later to Vienna, they were spread over three locations. The state holdings from the period up to 1797 were originally located in the Doge's Palace, in the Procuratie or in the institutions at the Rialto Bridge. The files of the political organs were now in the Scuola grande di S. Teodoro, the court files were in San Zanipolo, while the economic files, especially those of the tax authorities, were in a palazzo near San Provolo. The notarial files were initially located at Rialto, but were relocated several times.
From 1797, French and Austrian authorities were in the city, and their holdings were transferred to the State Archives. The state institutions that arose in the city from 1866, when the city became part of Italy, also bequeathed their holdings to the archive, which was now the state archive. In 1875 the archive expanded and incorporated the former monastery of Ss. Trinità and S. Antonio beyond S. Nicoletto ai Frari. Therefore, the cul-de-sac behind the monastery is now called Calle dietro l'archivio, ().
The archive was by no means open to the public for a long time. In 1825 Emmanuele Antonio Cicogna and in 1829 Leopold von Ranke had to ask the Emperor in Vienna for permission. At the same time, numerous documents and entire holdings migrated to Vienna or Milan. In 1805, a total of 44 crates first went across the Alps, only to be brought to Milan in 1815. They were not brought back to Venice until 1837 and 1842. The diplomatic dispute over the stocks continued for a long time.
Expansion after 1866
In 1866 Venice became part of Italy. In 1876 the State Archives received part of the Palace of the Dieci savi alle decime in Rialto together with the adjoining Scuola dei Orefici. Directors were Girolamo Dandolo (1796–1867, director from 1860 to 1867), Tommaso Gar, Teodoro Toderini (until 1876) and Bartolomeo Cecchetti (1838–89, director from 1876 to 1889), Luigi Lanfranchi. Carlo Malagola was director of the Bologna State Archive, before took the director position in Venice Archive.
On November 4, 1966, the city experienced extreme flooding, which also endangered the archive holdings, which initially had to be placed on higher shelves. In the years that followed, protective measures were taken against future floods, and the convent of San Nicoletto was restructured. An enlarged reading room was created. In addition, the former summer refectory, which overlooks one of the two cloisters, was enlarged. That's where the financial files had been until then. Elements that had been installed at the beginning of the 20th century for structural reasons were removed during the conversion work, so that the room regained its original dimensions. The new reading room was opened in August 1989. At the same time, the main entrance was moved to Campo dei Frari. At the end of 2008, another flood threatened the security of the stocks.
The branch on the Giudecca (Fondamenta della Croce, 17) was originally a Benedictine convent. This became state property in 1806, used as a prison from 1811, then as a tobacco warehouse. In 1925 the State Archives exchanged some buildings on the Giudecca of the Magistrato alle Acque for the Palace of the Dieci savi alle decime in Rialto, which had served as a subsidiary until then. However, these spaces proved to be unsuitable, and so the archive acquired the former Benedictine building in the 1960s. Some items from the main building, above all the items concerning the Rialto Bridge, were recorded, subjected to conservation measures and transferred to the Giudecca at the end of the 1970s. In the 1980s, the state archives occupied the Benedictine church as well as the monastery. The dependance mainly contains court files, but also police, prefectural and financial files from the 19th century.
Principle of provenance, inventory records
Teodoro Toderini, director of the archive who fell ill at the end of 1875 and died in 1876, was an advocate of the principle of provenance, which ultimately prevailed, while his successor Bartolomeo Cecchetti took a different view.
Andrea Da Mosto (1937–1940), also director of the archive, published the first overview of the holdings, which is still useful today[5], and was followed by Raimondo Morozzo della Rocca (1905–1980, director from 1952 to 1968).
In 1994 a Guida generale was published, In 1997 the digital recording of the holdings began, which are to be gradually made available to the public via the Internet, albeit only partially. In December 2006, the Repertorio dei fondi e degli strumenti di ricerca was completed, which provides a general overview of the holdings. This is still not available online.
From 1977 to 1990 Maria Francesca Tiepolo was director of the archive, she was succeeded by Paolo Selmi († August 28, 2010) until 2003, followed by Raffaele Santoro until 2018, who in turn was followed by Stefania Piersanti.
See also
Ottoman Archive
Dubrovnik Archive
References
Literature
Guida generale degli Archivi di Stato Italiani. Band 4: S – Z. Ministero per i beni culturali e ambientali – Ufficio centrale per i beni archivistici, Rom 1994, ISBN 88-7125-080-X, S. 869–881.
Rawdon Brown: L'archivio di Venezia con riguardo speciale alla storia inglese (= Nuova Collezione di Opere Storiche. Band 4, ). G. Antonelli u. a., Venedig u. a. 1865, (Digitalisat).
Bartolomeo Cecchetti: L'archivio di stato in Venezia negli anni 1876–1880. Naratovich, Venedig 1881. (Digitalisat)
Maria Pia Pedani Fabris, Alessio Bombaci (Hrsg.): I "documenti turchi" dell'Archivio di Stato di Venezia (= Pubblicazioni degli Archivi di Stato. Strumenti. Band 122). Ministero per i beni culturali e ambientali – Ufficio centrale per i beni archivistici, Rom 1994, ISBN 88-7125-090-7.
Daniele Ceschin: L'Archivio dei Frari. In: Daniele Ceschin, Anna Scannapieco: L'Archivio dei Frari. La casa di Goldoni (= Novecento a Venezia. Band 5). Il poligrafo, Padua 2005, ISBN 88-7115-472-X, S. 11–48.
External links
Internetseite des Staatsarchivs
Überblick über die Bestände
Hans-Jürgen Hübner: Quellen und Editionen zur Geschichte Venedigs
Archives in Italy
Education in Venice
State archives | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,055 |
\section{Introduction}
A theoretical understanding of the behavior of many-body systems
is a great challenge and provides fundamental insights into quantum
mechanical studies, as well
as offering potential areas of applications.
However, apart from some few analytically solvable problems,
the typical absence of an exactly solvable contribution to the
many-particle
Hamiltonian
means that we need reliable numerical many-body methods.
These methods should allow for controlled approximations
and provide a computational scheme which accounts for successive
many-body corrections in a systematic way.
Typical examples of
popular many-body methods are coupled-cluster methods
\cite{bartlett81,helgaker,Dean2004},
various types of
Monte Carlo methods \cite{Pudliner1997,kdl97,ceperley1995},
perturbative expansions \cite{lindgren,mhj95},
Green's function methods \cite{dickhoff,blaizot},
the density-matrix renormalization group \cite{white1992,schollwock2005},
ab initio density functional theory
\cite{bartlett2005,peirs2003,vanneck2006}
and large-scale diagonalization methods
\cite{Whitehead1977,caurier2005,navratil2004,horoi2006}.
However, all these methods have to face in some form or the other the problem of
an exponential growth in dimensionality. For a system of $P$ fermions
which
can be placed
into $N$ levels, the total number of basis states are given by
$\left(\begin{array}{c}N\\P\end{array}\right)$.
The dimensional curse means that most quantum
mechanical calculations on classical computers have exponential
complexity and therefore are very hard to solve for larger systems. On
the other hand, a so-called
quantum computer, a particularly dedicated computer,
can improve greatly on the size of systems that can be simulated, as
foreseen by Feynman \cite{feynman1982,feynman1986}. A quantum computer
does not need
an exponential amount of memory to represent a quantum state.
The basic unit of information for a quantum computer
is the so-called qubit or quantum bit. Any
suitable
two-level quantum system can be a qubit, but the
standard model of quantum computation is a model where two-level
quantum systems are located at different points in
space, and are manipulated by a small universal set of operations.
These operations are called gates in the same fashion as operations on
bits in classical computers are called gates.
For the example of $P$ spin $1/2$ particles,
a classical computer needs $2^P$ bits to represent all possible states, while
a quantum computer needs only $P$ qubits. The complexity in
number of qubits is thus linear. Based on these ideas, several groups have proposed
various algorithms for simulating quantal many-body systems on quantum computers.
Abrams
and Lloyd, see for example Refs.~\cite{lloyd1997,lloyd1999a}, introduced a
quantum algorithm that uses the quantum fast Fourier transform to find
eigenvalues and eigenvectors of a given Hamiltonian,
illustrating how one could solve classically intractable problems with
less than 100 qubits.
Achieving a polynomial complexity in the number of operations needed
to simulate a quantum system is not that straightforward however.
To get efficient simulations in time one
needs to transform the many-body Hamiltonian into a sum of operations
on qubits, the building blocks of the quantum simulator and computer,
so that the time evolution operator can be implemented in polynomial
time.
In Refs.~\cite{somma2002,somma2005,ortiz2002} it was shown how the
Jordan-Wigner
transformation in principle
does this for all Hamiltonians acting on fermionic many-body states.
Based on this approach, recently two groups, see Refs.~\cite{krbrown2006,yang2006}, published
results where
they used Nuclear Magnetic Resonance (NMR) qubits to simulate the
pairing Hamiltonian.
The aim of this work is to develop an algorithm than allows one to perform
a quantum computer simulation (or simply quantum simulation hereafter) of any many-body
fermionic Hamiltonian. We show how to generate, via various Jordan-Wigner
transformations, all qubit operations
needed to simulate the time evolution operator of a given Hamiltonian.
We also show that for a given term in an $m$-body fermionic
Hamiltonian, the number of operations needed to simulate it is linear
in the number of qubits or energy-levels of the system. The number of
terms in the Hamiltonian is of the order of $m^2$ for a
general $m$-body interaction, making the simulation increasingly
harder with higher order interactions. We specialize our examples to a two-body Hamiltonian,
since this is also the most general type of Hamiltonian encountered in many-body physics.
Besides fields like nuclear physics, where three-body forces play a non-neglible role,
a two-body Hamiltonian captures most of the relevant physics. The various transformations are
detailed in the next section. In Sec.~\ref{sec:details}
we show in detail how to simulate a quantum computer finding the
eigenvalues of any two-body Hamiltonian, with all available particle
numbers,
using the simulated time
evolution operator. In that section we describe also the techniques which are necessary for the
extraction of information using a phase-estimation algorithm.
To demonstrate the feasibility of our algorithm,
we present in Sec.~\ref{sec:results} selected results from applications of
our algorithm to two simple model-Hamiltonians, a pairing Hamiltonian and the Hubbard model.
We summarize our results and present future perspectives in
Sec.~\ref{sec:conclusion}.
\section{Algorithm for quantum computations of fermionic systems}
\label{sec:algo}
\subsection{Hamiltonians}
A general two-body Hamiltonian for fermionic system
can be written as
\begin{equation}
\label{eq:twobodyH}
H = E_0 + \sum_{ij=1} E_{ij} a^\dag_i a_j
+\sum_{ijkl = 1} V_{ijkl} a^\dag_i a^\dag_j a_l a_k,
\end{equation}
where $E_0$ is a constant energy term, $E_{ij}$ represent all the
one-particle terms, allowing for non-diagonal terms as well. The
one-body term can represent
a chosen single-particle potential, the kinetic energy or other more
specialized terms such as
those discussed in connection with the Hubbard model
\cite{hubbardmodel} or the pairing Hamiltonian
discussed below.
The two-body interaction part is given by $V_{ijkl}$ and can be any
two-body interaction, from
Coulomb interaction to the interaction between nucleons.
The sums run over all possible single-particle levels $N$.
Note that
this model includes particle numbers from zero to the number of
available quantum levels, $n$. To simulate states with fixed numbers
of fermions one would have to either rewrite the Hamiltonian or
generate specialized input states in the simulation.
The algorithm which we will develop in this section and in
Sec.~\ref{sec:details} can treat any
two-body Hamiltonian. However,
in our demonstrations of the quantum computing algorithm, we will limit ourselves to
two simple models, which however capture much of the important physics
in quantum mechanical
many-body systems. We will also limit ourselves to spin $j=1/2$
systems, although our algorithm
can also simulate higher $j$-values, such as those which occur in nuclear, atomic and
molecular physics, it simply uses one qubit for every available
quantum state.
These simple models are the Hubbard model and a pairing
Hamiltonian.
We start with the spin $1/2$ Hubbard model, described by the following Hamiltonian
\begin{eqnarray}
H_H &&= \epsilon \sum_{i, \sigma} a_{i\sigma}^\dag a_{i\sigma}
-t \sum_{i, \sigma} \left(a^\dag_{i+1, \sigma}a_{i, \sigma}
+a^\dag_{i, \sigma}a_{i+1, \sigma} \right) \notag \\
&& + U \sum_{i=1} a_{i+}^\dag a_{i-}^\dag a_{i-}a_{i+},
\label{eq:hubbard}
\end{eqnarray}
where $a^{\dagger}$ and $a$ are fermion creation and annihilation operators, respectively.
This is a chain of sites where each site has room for one spin up
fermion and one spin down fermion.
The number of sites is $N$, and the sums over $\sigma$ are sums over
spin up and down only.
Each site has a single-particle
energy $\epsilon$. There is a repulsive term $U$ if there is a pair
of particles at the same site. It is energetically favourable to tunnel to
neighbouring sites, described by
the hopping terms with coupling constant $-t$.
The second model-Hamiltonian is the simple pairing Hamiltonian
\begin{equation}
H_P=\sum_i \varepsilon_i a^{\dagger}_i a_i -\frac{1}{2} g\sum_{ij>0}
a^{\dagger}_{i}
a^{\dagger}_{\bar{\imath}}a_{\bar{\jmath}}a_{j},
\label{eq:pairing}
\end{equation}
The indices $i$ and $j$ run over the number of levels $N$, and the label $\bar{\imath}$
stands for a time-reversed state. The parameter $g$ is the strength of the pairing force
while $\varepsilon_i$ is the single-particle energy of level $i$.
In our case
we assume that the single-particle levels are equidistant (or
degenerate) with a fixed spacing $d$.
Moreover, in our simple model, the degeneracy of the single-particle
levels is set to $2j+1=2$, with $j=1/2$
being the spin of the particle. This gives a set of single-particle
states with the same spin projections as
for the Hubbard model. Whereas in the Hubbard model we operate with
different sites with
spin up or spin down particles, our pairing models deals thus with
levels with double degeneracy.
Introducing the pair-creation operator
$S^+_i=a^{\dagger}_{im}a^{\dagger}_{i-m}$,
one can rewrite the Hamiltonian in
Eq.\ (\ref{eq:pairing}) as
\[
H_P=d\sum_iiN_i+
\frac{1}{2} G\sum_{ij>0}S^+_iS^-_j,
\]
where $N_i=a^{\dagger}_i a_i$
is the number operator, and
$\varepsilon_i = id$ so that the single-particle orbitals
are equally spaced at intervals $d$. The latter commutes with the
Hamiltonian $H$. In this model, quantum numbers like seniority
$\cal{S}$ are good quantum numbers, and the eigenvalue problem
can be rewritten in terms of blocks with good seniority.
Loosely
speaking, the seniority quantum number $\cal{S}$ is equal to
the number of unpaired particles; see \cite{Talmi1993} for
further details.
Furthermore, in a series of papers, Richardson, see for example
Refs.~\cite{richardson1,richardson2,richardson3},
obtained the exact solution of the pairing Hamiltonian, with
semi-analytic (since there is still the need for a numerical solution)
expressions for the eigenvalues and eigenvectors. The exact solutions
have had important consequences for several fields, from Bose condensates to
nuclear superconductivity and is currently a very active field of studies, see for example
Refs.~\cite{dukelsky2004,rmp75mhj}.
Finally, for particle numbers up to $P \sim 20$, the above model can be
solved exactly through numerical diagonalization and one can obtain all eigenvalues.
It serves therefore also as an excellent ground for comparison with our algorithm based
on models from quantum computing.
\subsection{Basic quantum gates}
Benioff showed that one could make a quantum mechanical Turing machine
by using various unitary operations on a quantum system, see Ref.~\cite{benioff}.
Benioff demonstrated
that a quantum computer can calculate anything a
classical computer can. To do this one needs a quantum system and
basic operations that can approximate all unitary operations
on the chosen many-body system. We describe in this subsection the basic ingredients entering
our algorithms.
\subsubsection{Qubits, gates and circuits}
\label{sec:gates}
In this article we will use the standard model of quantum information,
where
the basic unit of information is the qubit, the quantum bit.
As mentioned in the introduction, any
suitable
two-level quantum system can be a qubit,
it is the smallest system there is with the
least complex dynamics.
Qubits are both abstract measures of information and physical objects.
Actual physical qubits can be ions trapped in magnetic fields where
lasers can access only two energy levels or the nuclear spins of some of
the atoms in molecules accessed and manipulated by an NMR machine.
Several other ideas have been proposed and some tested, see
\cite{nielsen2000}.
The computational basis for one qubit is ${\ensuremath{|0\rangle}}$ (representing for example bit $0$)
for the first state
and ${\ensuremath{|1\rangle}}$ (representing bit $1$) for the second, and for a set of qubits
the tensor products of
these basis states for each qubit form a product basis. Below we write out the different
basis states for a system of $n$ qubits.
\begin{eqnarray}
\label{eq:compBasis}
&{\ensuremath{|0\rangle}} \equiv {\ensuremath{|00\cdots 0\rangle}} =
{\ensuremath{|0\rangle}} \otimes {\ensuremath{| 0\rangle}} \otimes
\cdots
\otimes {\ensuremath{|0\rangle}}
\notag \\
&{\ensuremath{|1\rangle}} \equiv {\ensuremath{|00\cdots 1\rangle}} =
{\ensuremath{|0\rangle}} \otimes {\ensuremath{| 0\rangle}} \otimes
\cdots
\otimes {\ensuremath{|1\rangle}}
\notag \\
&\vdots \notag \\
&{\ensuremath{|2^n-1\rangle}} \equiv {\ensuremath{|11\cdots 1\rangle}} =
{\ensuremath{|1\rangle}} \otimes {\ensuremath{| 1\rangle}} \otimes
\cdots \otimes {\ensuremath{|1 \rangle}}.
\notag \\
\end{eqnarray}
This is a $2^n$-dimensional system and we number the different basis
states using binary numbers corresponding to the order in which they appear in the
tensor product.
Quantum computing means to manipulate and measure qubits in such a
way that the results from a measurement yield the solutions to a given problem.
The quantum operations we need to be able to perform our simulations are
a small set of elementary single-qubit
operations, or single-qubit gates, and one universal two-qubit gate,
in our case the so-called CNOT gate defined below.
To represent quantum computer algorithms graphically we use circuit
diagrams. In a circuit diagram each qubit is represented by a line,
and operations on the different qubits are represented by boxes.
In fig.~\ref{fig:circ} we show an example of a quantum circuit, with the arrow
indicating the time evolution,
\begin{figure}[h]
\begin{picture}(0,65)(160,55)
\put(60,87){\makebox(0,0){${\ensuremath{| a\rangle}}$}}
\put(60,63){\makebox(0,0){${\ensuremath{| b\rangle}}$}}
\put(90,50){\input{A.pstex_t}}
\put(240, 87){\makebox(0,0){${\ensuremath{|{a^\prime\rangle}}}$}}
\put(240, 63){\makebox(0,0){${\ensuremath{| {b^\prime\rangle}}}$}}
\end{picture}
\caption{A quantum circuit showing a single-qubit gate $A$ and a
two-qubit gate acting on a pair of qubits, represented by the horizontal lines.}
\label{fig:circ}
\end{figure}
The states ${\ensuremath{| a\rangle}}$ and ${\ensuremath{| b\rangle}}$ in the figure represent qubit
states. In general, the total state will be a
superposition of different qubit states.
A single-qubit gate is an operation that only affects one physical
qubit, for example one ion or one nuclear spin in a molecule. It is represented by
a box on the line corresponding to the qubit in question. A
single-qubit
gate operates on one qubit and is therefore represented
mathematically by a $2\times2$ matrix while a two-qubit gate is
represented by a $4\times4$ matrix. As an example we can portray the so-called CNOT
gate as matrix,
\begin{equation}
\begin{pmatrix}
1 & 0 & 0 & 0\\
0&1&0&0\\
0&0&0&1\\
0&0&1&0
\end{pmatrix}.
\label{eq:CNOT}
\end{equation}
This is a very important gate, since one can show that it behaves as a universal two-qubit
gate, and that we only need this two-qubit gate and a small set of single-qubit
gates to be able to approximate any multi-qubit operation. One example of a
universal set of single-qubit gates is given in
Fig.~\ref{fig:elementarySingleQubitGates}. The products of these three
operations on one qubit can approximate to an arbitrary precision any
unitary operation on that qubit.
\begin{figure}[h]
\begin{picture}(200,20)
\put(25,-10){\makebox(0,0){Hadamard}}
\put(50,-22){\input{fig2.latex}}
\put(150, -10){\makebox(0,0){\ensuremath{
\frac{1}{\sqrt2}
\begin{pmatrix}
1 & 1 \\
1 &-1 \end{pmatrix}
}}}
\end{picture}
\end{figure}
\begin{figure}[h]
\begin{picture}(200,10)
\scalebox{1.5}{\put(10,-10){\makebox(0,0){$\frac\pi8$}}}
\put(50,-22){\input{fig3.latex}}
\put(150, -10){\makebox(0,0){\ensuremath{
\begin{pmatrix}
1 & 0 \\
0 &e^{i\pi/4} \end{pmatrix}
}}}
\end{picture}
\end{figure}
\begin{figure}[h!]
\begin{picture}(200,30)
\scalebox{1.1}{\put(10,10){\makebox(0,0){phase}}}
\put(50,-2){\input{fig4.latex}}
\put(150, 10){\makebox(0,0){\ensuremath{
\begin{pmatrix}
1 & 0 \\
0 & i \end{pmatrix}
}}}
\end{picture}
\caption{Set of three elementary single-qubit gates and their matrix representations.
The products of these three
operations on one qubit can approximate to an arbitrary precision any
unitary operation on that qubit.}
\label{fig:elementarySingleQubitGates}
\end{figure}
\subsubsection{Decomposing unitary operations into gates}
The next step is to find elementary operations
on a set of qubits that can be put together in order to approximate any unitary
operation on the qubits. In this way we can perform computations on
a quantum computer by performing many of these elementary operations
in the correct order.
There are three steps in finding the elementary operations needed to
simulate any unitary operation.
First, any
$d\times d$ unitary matrix can be factorized into a product of at most
$d(d-1)/2$ two-level unitary matrices, see for example Ref.~\cite{nielsen2000} for details.
A two-level unitary matrix is a
matrix that only acts non-trivially on two vector components when
multiplied with a vector. For all other vector components it acts as
the identity operation.
The next step is to prove that any two-level unitary matrix can be
implemented by one kind of two-qubit gate, for example the CNOT gate
in Eq.~(\ref{eq:CNOT}),
and single-qubit gates only.
This simplifies the making of actual quantum
computers as we only need one type of interaction between pairs of
qubits. All other operations are operations on one qubit at the time.
Finally, these single-qubit operations can be approximated to an
arbitrary precision by a finite set of single-qubit gates. Such a set
is called a universal set and one example is the phase gate, the
so-called Hadamard gate and
the $\pi/8$ gate. Fig.~\ref{fig:elementarySingleQubitGates} shows these gates.
By combining these properly
with the CNOT gate one can approximate any unitary operation on a set
of qubits.
\subsubsection{Quantum calculations}
The aspect of quantum computers we are focusing on in this article is
their use in computing properties of quantum systems.
When we want to use a quantum computer to
find the energy levels of a quantum system
or simulate it's dynamics, we need to simulate
the time evolution operator of the Hamiltonian, $U=\exp(-iH\Delta t)$.
To do that on a quantum computer we must find a set of single- and
two-qubit gates that would implement the time evolution on a set of
qubits. For example, if we have one qubit in the state ${\ensuremath{|
a\rangle}}$, we must find the single-qubit gates that when applied
results in the qubit being in the state
$\exp(-iH\Delta t)|a\rangle$.
From what we have written so far
the naive way of simulating $U$ would be to calculate it directly as
a matrix in an appropriate basis, factorize it into two-level unitary
matrices and then implement these by a set of universal gates.
In a many-body fermion system for example, one could use the Fock basis to
calculate $U$ as a matrix. A fermion system with $n$ different
quantum levels can have from zero to $n$ particles in each Fock basis
state. A
two-level system has four different basis states, $|00\rangle$,
$|01\rangle$, $|10\rangle$ and $|11\rangle$, where $|0\rangle$ corresponds to an
occupied quantum level. The time evolution matrix is then a $2^n\times
2^n$ matrix. This matrix is then factorized into at most
$2^n(2^n-1)/2$ two-level unitary matrices. An exponential
amount of operations, in terms of the number of quantum levels, is
needed to simulate $U$; by definition not an effective simulation.
This shows that quantum computers performing quantum
simulations not necessarily fulfill their promise. For each physical
system to be simulated one has to find representations of the
Hamiltonian that leads to polynomial complexity in the number of
operations. After one has found a proper representation of the
Hamiltonian, the time evolution operator $\exp(-iH\Delta t)$ is found by using a
Trotter approximation, for example
\begin{equation}
\label{eq:Trotter1}
U=e^{-iH\Delta t}=e^{-i(\sum_k H_k) \Delta t} = \prod_k e^{-iH_k\Delta
t} + {\cal O}(\Delta t^2).
\end{equation}
There are different ways to approximate $U$ by products of
exponentials of the different terms of the Hamiltonian, see
Ref.~\cite{nielsen2000} and Eq.~(\ref{eq:Trotter2}). The essential
idea is to find a form of the Hamiltonian where these factors in the
approximated time evolution operator can be further
factorized into single- and two-qubit operations effectively.
In
Refs.~\cite{ortiz2001,ortiz2002}
it was shown how to do this in principle for
any many-body fermion system using the Jordan-Wigner transformation.
In this article we
show how to create a quantum compiler that takes
any many-body fermion Hamiltonian and outputs the quantum gates needed
to simulate the time evolution operator. We implement it for the case
of two-body fermion Hamiltonians and show results from numerical
calculations finding the energylevels of the well known pairing and
Hubbard models.
\subsection{The Jordan-Wigner transformation}
For a spin-$1/2$ one-dimensional quantum spin-chain a fermionization
procedure exists
which allows the mapping between spin operators and fermionic
creation-annihilation operators.
The algebra governing the spin chain is the $SU(2)$ algebra, represented by
the $\sigma$-matrices. The Jordan-Wigner transformation is a
transformation from fermionic annihilation and creation operators to
the $\sigma$-matrices of a spin-$1/2$ chain, see
for example Ref.~\cite{dargis1998} for more details on the Jordan-Wigner
transformation.
There is an isomorphism
between the two systems, meaning that any $a$ or $a^\dag$ operator can be
transformed into a tensor product of $\sigma$-matrices operating on a
set of qubits. This was explored by Somma {\em et al.} in Refs.~\cite{somma2002,ortiz2002}.
The authors demonstrated, with an emphasis on single-particle fermionic operators,
that the Jordan-Wigner transformation
ensures efficient, i.e., not exponential complexity, simulations of a
fermionic system on a quantum computer.
Similar transformations must be found for other systems, in order to
efficiently simulate many-body systems. This was the main
point in Ref.~\cite{somma2002}.
We present here the various ingredients needed in order to transform a given
Hamiltonian into a practical form suitable for quantum mechanical simulations.
We begin with the fermionic creation and annihilation operators, which satisfy the following
anticommutation relations
\begin{equation}
\label{eq:anticommutationrelations}
\{a_k, a_l\}=\{a_k^\dag, a_l^\dag\}= 0, \quad
\{a_k^\dag, a_l\} = \delta_{kl}.
\end{equation}
Thereafter we define the three traceless and Hermitian generators of the $SU(2)$ group, the
$\sigma$-matrices $\sigma_x$, $\sigma_y$ and $\sigma_z$. Together
with the
identity matrix ${\bf 1}$ they form a complete basis for all Hermitian $2\times2$
matrices. They can be used to write all Hamiltonians on a spin $1/2$
chain when taking sums of tensor products of these, in other words
they form a product basis for the operators on the qubits.
The three $\sigma$-matrices are
\begin{equation}
\sigma_x = \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix},
\sigma_y = \begin{pmatrix}
0 & -i\\
i & 0
\end{pmatrix}, \quad
\sigma_z = \begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}.
\end{equation}
We define the raising and lowering matrices as
\[
\sigma_+ = \frac{1}{2}(\sigma_x + i\sigma_y)=
\begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix},
\]
\begin{equation}
\label{eq:raisingAndLowerin}
\sigma_- = \frac{1}{2}(\sigma_x - i\sigma_y)
=\begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix}.
\end{equation}
The transformation is based on the fact that for each possible
quantum state of the fermion system, there can be either one or zero
fermions. Therefore we need $n$ qubits for a system with $n$ possible
fermion states. A qubit in state $ |0\rangle ^i=a^\dag_i|vacuum\rangle$
represents a state with a
fermion, while $ |1\rangle ^i=|vacuum\rangle$ represents no fermions. Then the raising
operator $\sigma_+$ changes $ |1\rangle $ into $ |0\rangle $ when
\begin{equation}
|0\rangle \equiv \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad
|1\rangle \equiv \begin{pmatrix} 0 \\ 1 \end{pmatrix}.
\end{equation}
This means that $\sigma_+$ acts as a creation operator, and $\sigma_-$
acts as an annihilation operator. In addition, because of the
anticommutation of creation(annihilation) operators for different states we have
$a_1^\dag a_2^\dag
|vacuum\rangle = - a_2^\dag a_1^\dag |vacuum\rangle$, meaning that for creation and
annihilation operators for states higher than the state corresponding
to the first qubit, we need to multiply with a $\sigma_z$-matrix on
all the qubits leading up to the one in question, in order to get the correct
sign in the final operation. This leads us to the Jordan-Wigner
transformation \cite{somma2002,ortiz2002}
\begin{equation}
\label{eq:JWtransformation}
a^\dag_n = \left(\prod_{k=1}^{n-1} \sigma_z^k\right) \sigma_+^n, \quad
a_n = \left(\prod_{k=1}^{n-1} \sigma_z^k\right) \sigma_-^n.
\end{equation}
The notation $\sigma_z^i\sigma_+^j$ means a tensor product of the
identity matrix on all qubits other than $i$ and $j$, ${\bf 1}\otimes
\sigma_z \otimes {\bf 1} \otimes \sigma_+\otimes{\bf 1}$, if $i<j$, with ${\bf 1}$
being the identity matrices of appropriate dimension.
\subsection{Single-particle Hamiltonian}
\label{sec:1partH}
What we must do now is to apply the Jordan-Wigner transformation to a
general fermionic Hamiltonian composed of creation and annihilation
operators, so we can write it as a sum of products of $\sigma$
matrices. The matrix $\sigma^k$ is then an operation on the $k^{th}$
qubit representing the $k^{th}$ quantum level of the fermion system.
When we have expressed the Hamiltonian as a sum of products of
operations on the qubits representing the system, we must find a
representation of the time evolution operator as products of two-qubit
operations. These operations can be further decomposed into elementary
operations, see subsection \ref{sec:gates} for further discussion.
\subsubsection{Jordan-Wigner transformation of the one-body part}
We first describe the procedure for the
simplest case of a general single-particle Hamiltonian,
\begin{equation}
H_1=\sum_{i} E_{ii} a^\dag_i a_i +
\sum_{i<j} E_{ij} (a^\dag_i a_j + a^\dag_j a_i).
\end{equation}
We now use the transformation of
Eq.~(\ref{eq:JWtransformation}) on the terms $a^\dag_i a_j$.
The diagonal terms of the one-particle Hamiltonian,
that is the case where $i=j$, can be rewritten as
\[
a^\dag_i a_i = \left(\prod_{k=1}^{i-1} \sigma_z^k\right) \sigma_+^i
\left(\prod_{k=1}^{i-1} \sigma_z^k \right)\sigma_-^i
\]
\begin{equation}
\label{eq:H_1Sigmas}
= \sigma_+^i \sigma_-^i = \frac{1}{2}\left({\bf 1}^i + \sigma_z^i\right),
\end{equation}
since $(\sigma_z)^2= {\bf 1}$ which is the number operator. It counts whether or not a fermion is in
state $i$. In the case of qubits counting whether or not the qubit
is in state $ |0\rangle $, we have eigenvalue one for $ |0\rangle $ and eigenvalue
zero for $ |1\rangle $. The action of this Hamiltonian on qubit $i$ can be simulated using
the single-qubit operation
\begin{equation}
U=e^{-i({\bf 1} + \sigma_z) E_{ii}\Delta t} = \begin{pmatrix}
e^{-iE_{ii}\Delta t} & 0 \\
0 & 1
\end{pmatrix},
\end{equation}
see subsection \ref{sec:gates} for other examples of single-qubit gates.
For the non-diagonal elements, $i<j$, not all of the $\sigma_z$ matrices multiply with each
other and end up in the identity operation.
As an example we will consider a five level system, $n=5$, and look
at the transformation of the term $a^\dag_ia_j$ whith $i=2$ and $j=4$,
\begin{eqnarray}
\label{eq:fiveQubitSystem}
a_2^\dag& = &\sigma_z\otimes\sigma_+\otimes {\bf 1} \otimes {\bf 1}\otimes {\bf 1}, \notag\\
a_4 & = & \sigma_z\otimes\sigma_z\otimes\sigma_z\otimes\sigma_-\otimes {\bf 1}, \notag\\
& \Downarrow & \notag\\
a_2^\dag a_4 & = & {\bf 1}\otimes (\sigma_+\sigma_z)\otimes \sigma_z \otimes \sigma_-\otimes {\bf 1}.
\end{eqnarray}
The operation on all qubits
before $i$ and after $j$ is identity, on qubits $i+1$ through
$j-1$ it is $\sigma_z$.
We can then write the non-diagonal one-body operators as
\begin{widetext}
\begin{eqnarray}
\label{eq:singleParticle}
a^\dag_i a_j + a^\dag_j a_i &&= (\sigma_+^i\sigma_z^i)
\left(\prod_{k=i+1}^{j-1}
\sigma_z^k \right)\sigma_-^j
+
(\sigma_z^i\sigma_-^i) \left(\prod_{k=i+1}^{j-1}
\sigma_z^k \right)\sigma_+^j \notag\\
&&= - \sigma_+^i \left(\prod_{k=i+1}^{j-1}\sigma_z^k \right)\sigma_-^j
- \sigma_-^i \left(\prod_{k=i+1}^{j-1}
\sigma_z^k \right)\sigma_+^j \notag\\
&&= -\frac{1}{2} \left\{
\sigma_x^i \left(\prod_{k=i+1}^{j-1}\sigma_z^k \right)\sigma_x^j
+ \sigma_y^i \left(\prod_{k=i+1}^{j-1}\sigma_z^k \right)\sigma_y^j
\right\}.
\end{eqnarray}
\end{widetext}
Using Eqs.~(\ref{eq:H_1Sigmas}) and (\ref{eq:singleParticle})
the total single-particle fermionic Hamiltonian of $n$ quantum levels,
transformed using the Jordan-Wigner transformation of
Eq.~(\ref{eq:JWtransformation}), is written as
\begin{eqnarray}
\label{eq:singleParticleTotal}
H_1 &=& \sum_{i} E_{ii} a^\dag_i a_i +
\sum_{i<j} E_{ij} (a^\dag_i a_j + a^\dag_j a_i) \notag\\
&=&
\frac{1}{2}\sum_{i} E_{ii}\left({\bf 1}^i + \sigma_z^i\right)
\notag\\
&-& \frac{1}{2}\sum_{i<j} E_{ij}
\left\{
\sigma_x^i \left(\prod_{k=i+1}^{j-1}\sigma_z^k \right)\sigma_x^j \right.\notag\\
&+& \left.\sigma_y^i \left(\prod_{k=i+1}^{j-1}\sigma_z^k \right)\sigma_y^j
\right\}.
\end{eqnarray}
\subsubsection{Transformation into two-qubit operations}
The Hamiltonian is now transformed into a sum of many-qubit
operations, $H=\sum_l H_l$. The $a_2^\dag a_4$ term in Eq.~(\ref{eq:fiveQubitSystem})
for example, is transformed into a three-qubit operation. The next
step is to factorize these many-qubit operations $H_l$ into products of
two-qubit operations, so that we in the end can get a product of
two-qubit operations $U_{kl}$, that when performed in order give us the time
evolution operator corresponding to each term in the Hamiltonian,
$\exp(-iH_l\Delta t) = \prod_k U_{kl}$.
The first thing we do is to find a set of two-qubit operations that
together give us the Hamiltonian, and later we will see that to
find the time evolution from there is straightforward.
The many-qubit terms in Eq.~(\ref{eq:singleParticleTotal}) are
products of the type $\sigma_x \sigma_z \cdots \sigma_z \sigma_x$
with $\sigma_x$ or $\sigma_y$ at either end. These products have to be
factorized into a series of two-qubit operations.
What we wish to do is successively build up the operator using
different unitary transformations.
This can be achieved with successive operations with the $\sigma$-matrices, starting with for example
$\sigma_z^i$, which can be transformed
into $\sigma_x^i$, then transformed into $\sigma_y^i\sigma_z^{i+1}$
and so forth.
Our goal now is to express each term in the Hamiltonian
Eq.~(\ref{eq:singleParticleTotal}) as a product of the type
$\sigma_x^i \sigma_z \cdots \sigma_z
\sigma_x^j=
(\prod_k U_k^\dag ) \sigma_z^i
(\prod_{k^\prime} U_{k^\prime})$, with a different form in the case
where the Hamiltonian term starts and ends with a $\sigma_y$ matrix.
To achieve this we need the transformations in
Eqs.~(\ref{eq:rotations1})-(\ref{eq:rotations4}).
We will use this to find the time-evolution operator for each Hamiltonian, see
Eq.~(\ref{eq:Us}) below.
To understand how we factorize the Hamiltonian terms into single- and
two-qubit operations we follow a bottom up procedure.
First,
if we have a two qubit system, with the operator $\sigma_z \otimes
{\bf 1}$, we see that the unitary operation $\exp(i\pi/4 \sigma_z\otimes
\sigma_z)$ transforms it into
\begin{equation}
e^{-i\pi/4 \sigma_z\otimes\sigma_z}\left( \sigma_z\otimes{\bf 1} \right)
e^{i\pi/4 \sigma_z\otimes\sigma_z} = \sigma_z\otimes\sigma_z.
\end{equation}
In addition, if we start out with the operator $\sigma_z^i$ we can transform it
into $\sigma_x^i$ or $\sigma_y^i$ using the operators
$\exp(i\pi/4\sigma_y)$ or $\exp(-i\pi/4\sigma_x)$ accordingly.
We can then generate the $\prod_k \sigma_z^k$ part of the terms in
Eq.~(\ref{eq:singleParticleTotal}) by succesively
applying the operator $\exp(i\pi/4\sigma_z^i \sigma_z^l)$ for $l=2$
through $l=j$. Yielding the operator $\sigma_a^i \prod_{k=i+1}^{j}
\sigma_z^k$ with a phase of $\pm1$, because of the sign change in
Eqs.~(\ref{eq:rotations3}) and (\ref{eq:rotations4}). We write
$\sigma_a$ to show that we can start with both a $\sigma_x$ and
a $\sigma_y$ matrix. To avoid the sign
change we can simply use the operator $\exp(-i\pi/4\sigma_z^i
\sigma_z^l)$ instead for those cases where we have $\sigma_y^i$ on site
$i$
instead of $\sigma_x^i$. This way we always keep the same phase.
Finally, we use the operator $\exp(i\pi/4\sigma_y)$ if we want the
string of operators to end with $\sigma_x$, or $\exp(-i\pi/4\sigma_x)$
if we want it to end with $\sigma_y$.
The string of operators starts with either $\sigma_x$ or $\sigma_y$. For an odd
number of $\exp(\pm i\pi/4\sigma_z^i \sigma_z^l)$ operations, the
operations that add a $\sigma_z$ to the string, the first
operator has changed from what we started with. In other words we have
$\sigma_x$ instead of $\sigma_y$ at the start of the string or vice
versa, see Eqs.~(\ref{eq:rotations3}) and (\ref{eq:rotations4}).
By counting, we see that we do
$j-i$ of the $\exp(\pm i\pi/4\sigma_z^i \sigma_z^l)$
operations to get the string
$\sigma_a^i\sigma_z^{i+1}\cdots\sigma_z^j$.
and therefore if $j-i$ is odd, the first matrix is the opposite of
what we want in the final string.
The following simple example can serve to clarify.
We want the Hamiltonian
$\sigma_x^1\sigma_z^2\sigma_x^3= \sigma_x\otimes \sigma_z \otimes
\sigma_x$, and by using the transformations in
Eqs.~(\ref{eq:rotations1})-(\ref{eq:rotations4}) we can construct it
as a product of single- and two-qubit operations in the following way,
\begin{eqnarray}
(e^{-\pi/4\sigma_y^1}) \sigma_z^1 (e^{\pi/4\sigma_y^1})&=&\sigma_x^1
\notag\\
(e^{-i\pi/4\sigma_z^1\sigma_z^2}) \sigma_x^1
(e^{i\pi/4\sigma_z^1\sigma_z^2}) & = & \sigma_y^1\sigma_z^2 \notag \\
(e^{i\pi/4\sigma_z^1\sigma_z^3}) \sigma_y^1\sigma_z^2
(e^{-i\pi/4\sigma_z^1\sigma_z^3}) & = & \sigma_x^1\sigma_z^2 \sigma_z^3
\notag\\
(e^{-i\pi/4\sigma_y^3}) \sigma_x^1\sigma_z^2 \sigma_z^3
(e^{i\pi/4\sigma_y^3}) & =& \sigma_x^1\sigma_z^2\sigma_x^3.
\end{eqnarray}
We see that we have factorized $\sigma_x^1\sigma_z^2\sigma_x^3$ into
$U_4^\dag U_3^\dag U_2^\dag U_1^\dag \sigma_z^1 U_1 U_2 U_3U_4$.
Now we can find
the time-evolution operator $\exp(-iH\Delta t)$ corresponding to each
term of the Hamiltonian, which is the quantity of interest.
Instead of starting
with the operator $\sigma_z^i$ we start with the corresponding
evolution operator and observe that
\begin{eqnarray}
U^\dag e^{-i\sigma_z a} U &&= U^\dag \left( \cos(a){\bf 1}
-i\sin(a) \sigma_z \right) U\notag \\
&&=\cos(a){\bf 1} -i \sin(a) U^\dag \sigma_zU \notag \\
&&=e^{-i U^\dag\sigma_zU a},
\end{eqnarray}
where $a$ is a scalar.
This means that we have a series of unitary transformations on this
operator yielding the final evolution, namely
\begin{equation}
\label{eq:Us}
e^{-i \sigma_x^i \sigma_z \cdots \sigma_z \sigma_x^j a}
= \left(\prod_k U_k^\dag \right) e^{-i\sigma_z^ia}
\left(\prod_{k^\prime}
U_{k^\prime}\right),
\end{equation}
with the exact same unitary operations $U_k$ as we find when we
factorize the Hamiltonian. These are now the single- and two-qubit operations we
were looking for, first we apply the operations $U_k$ to the
appropriate qubits, then $\exp(-i\sigma_z^ia)$ to qubit $i$, and then
the $U_k^\dag$ operations, all in usual matrix multiplication order.
\subsection{Two-body Hamiltonian}
\label{sec:2bH}
In this section we will do the same for the general two-body fermionic
Hamiltonian.
The two-body part of the Hamiltonian can be classified into
diagonal elements and non-diagonal elements. Because of the Pauli
principle and the anti-commutation relations for the creation and
annihilation operators, some combinations of indices are not allowed.
The two-body part of our Hamiltonian is
\begin{equation}
H_2 = \sum_{ijkl} V_{ijkl} a_i^\dag a_j^\dag a_l a_k,
\end{equation}
where the indices run over all possible states and $n$ is the total number
of available quantum states.
The single-particle labels $ijkl$ refer to their
corresponding sets of quantum numbers, such as
projection of total spin, number of nodes in the single-particle wave function etc. Since every state
$ijkl$ is uniquely defined,
we cannot have two equal
creation or annihilation operators and therefore $i\neq j$ and $k\neq
l$.
When $i=l$ and $j=k$, or $i=k$ and $j=l$, we have a
diagonal element in the Hamiltonian matrix, and the output state
is the same as the input state.
The operator term corresponding to $V_{ijji}$ has these equalities due
to the anti-commutation relations
\begin{eqnarray}
a^\dag_i a^\dag_j a_i a_j &&= a^\dag_j a^\dag_i a_j a_i \notag\\
&&= -a^\dag_i a^\dag_j a_j a_i \notag\\
&&= -a^\dag_j a^\dag_i a_i a_j,
\end{eqnarray}
which means that
\begin{equation}
\label{eq:Vdiags}
V_{ijji} = V_{jiij} = - V_{ijij} = - V_{jiji}.
\end{equation}
The term $a^\dag_i a^\dag_j a_i a_j$
with $i<j$ is described using the Pauli matrices
\begin{eqnarray}
&&a^\dag_i a^\dag_j a_i a_j \\ &&=
\left(\prod_{s=1}^{i-1} \sigma_z \right) \sigma_+^i
\left(\prod_{t=1}^{j-i} \sigma_z \right)
\sigma_+^j \notag \\
&&\times\left(\prod_{t=1}^{j-i} \sigma_z \right) \sigma_-^j
\left(\prod_{s=1}^{i-1} \sigma_z \right) \sigma_-^i \notag \\
&&=\left(\prod_{s=1}^{i-1} (\sigma_z)^4 \right) \left( \sigma_+^i \sigma_z^i
\sigma_z^i \sigma_-^i\right) \left(\prod_{t=i+1}^{j-1} (\sigma_z)^2 \right) \left(
\sigma_+^j
\sigma_-^j\right) \notag \\
&&=\sigma_+^i\sigma_-^i\sigma_+^j\sigma_-^j \notag\\
&&=\frac{1}{16} \left( {\bf 1} + \sigma_z^i\right) \left( {\bf 1} + \sigma_z^j
\right).
\end{eqnarray}
When we add all four different permutations of
$i$ and $j$ this is the number operator on qubit $i$ multiplied with
the number operator on qubit $j$.
The eigenvalue is one if both qubits are in
the state $ |0\rangle $, that is the corresponding quantum states are both
populated, and zero otherwise.
We can in turn rewrite the sets of creation and annihilations in terms of the
$\sigma$-matrices as
\begin{eqnarray}
a^\dag_i a^\dag_j a_i a_j + a^\dag_j a^\dag_i a_j a_i
-a^\dag_i a^\dag_j a_j a_i-a^\dag_j a^\dag_i a_i a_j \notag \\
= \frac{1}{4} \left( {\bf 1}
+\sigma_z^i + \sigma_z^j +\sigma_z^i\sigma_z^j \right).
\end{eqnarray}
In the general case we can have three different sets of non-equal
indices. Firstly, we see that $a^\dag_i a^\dag_j a_l a_k = a^\dag_k
a^\dag_l a_j a_i$, meaning that the exchange of $i$ with $k$ and $j$ with $l$
gives the same operator $\rightarrow V_{ijkl}=V_{klij}$. This results in
a two-body Hamiltonian with no equal indices
\begin{equation}
\label{eq:H2nonequalSymmetric}
H_{ijkl} =
\sum_{i < k}\sum_{ j < l} V_{ijkl} (a_i^\dag a_j^\dag a_l a_k + a_k^\dag
a_l^\dag a_j a_i).
\end{equation}
Choosing to order the
indices from lowest to highest gives us the position
where there will be $\sigma_z$-matrices to multiply with the
different raising and lowering operators, when we perform the
Jordan-Wigner transformation Eq.~(\ref{eq:JWtransformation}). The
order of matrix
multiplications is fixed once and for all, resulting in three
different groups into which these terms fall, namely
\begin{equation}
\label{eq:threeGroups}
\begin{array}{ccccc}
I & i<j<l<k, & i \leftrightarrow j, & k\leftrightarrow l, \\
II & i<l<j<k, \quad& i \leftrightarrow j, & k\leftrightarrow l, \\
III & i<l<k<j, \quad& i \leftrightarrow j, & k\leftrightarrow l. \\
\end{array}
\end{equation}
These $12$ possibilities for $a^\dag_i a^\dag_j a_l a_k$
are mirrored in the symmetric term
in Eq.~(\ref{eq:H2nonequalSymmetric}) giving us the $24$ different
possibilities when permuting four indices.
The $ijkl$ term of Eq.~(\ref{eq:H2nonequalSymmetric}) is
\begin{eqnarray}
a^\dag_i a^\dag_j a_l a_k + a_k^\dag
a_l^\dag a_j a_i=
\notag\\
\left(\prod \sigma_z \right) \sigma_+^i \left(\prod \sigma_z \right)
\sigma_+^j \notag \\
\times\left(\prod \sigma_z \right) \sigma_-^l
\left(\prod \sigma_z \right) \sigma_-^k \notag \\
+\left(\prod \sigma_z \right) \sigma_+^k \left(\prod \sigma_z \right)
\sigma_+^l \notag \\
\times
\left(\prod \sigma_z \right) \sigma_-^j
\left(\prod \sigma_z \right) \sigma_-^i .
\end{eqnarray}
In the case of $i<j<l<k$ we have
\begin{eqnarray}
\label{eq:VijklPhi}
a^\dag_i a^\dag_j a_l a_k+ a_k^\dag a_l^\dag a_j a_i=\notag\\
\left(\prod (\sigma_z)^4\right) \left(\sigma_+^i \sigma_z^i\right)
\left(\prod (\sigma_z)^3\right) \sigma_+^j \notag \\
\times
\left(\prod (\sigma_z)^2\right) \left(\sigma_-^l \sigma_z^l \right)
\left(\prod \sigma_z\right) \sigma_-^k \notag \\
+\left(\prod (\sigma_z)^4\right) \left(\sigma_z^i\sigma_-^i\right)
\left(\prod (\sigma_z)^3\right) \sigma_-^j \notag \\
\times
\left(\prod (\sigma_z)^2\right) \left(\sigma_z^l\sigma_+^l \right)
\left(\prod \sigma_z\right) \sigma_+^k .
\end{eqnarray}
Using Eq.~(\ref{eq:pmzs}), where we have the rules for sign changes when
multiplying the raising and lowering operators with the $\sigma_z$
matrices, gives us
\begin{eqnarray}
-\left(\sigma_+^i \sigma_z^{i+1}\cdots \sigma_z^{j-1} \sigma_+^j
\sigma_-^l \sigma_z^{l+1}\cdots \sigma_z^{k-1}\sigma_-^k\right.
\notag\\
+ \left.\sigma_-^i \sigma_z^{i+1}\cdots \sigma_z^{j-1} \sigma_-^j
\sigma_+^l \sigma_z^{l+1}\cdots \sigma_z^{k-1}\sigma_+^k\right).
\end{eqnarray}
If we switch the order of $i$ and $j$ so that $j<i<l<k$, we change the
order in which the $\sigma_z$-matrix is multiplied with the first
raising and lowering matrices, resulting in a sign change.
\begin{eqnarray}
a^\dag_i a^\dag_j a_l a_k + a_k^\dag a_l^\dag a_j a_i
=\notag\\
\left(\prod (\sigma_z)^4\right) \left(\sigma_z^j \sigma_+^j\right)
\left(\prod (\sigma_z)^3\right) \sigma_+^i \notag \\
\times
\left(\prod (\sigma_z)^2\right) \left(\sigma_-^l \sigma_z^l \right)
\left(\prod \sigma_z\right) \sigma_-^k \notag \\
+\left(\prod (\sigma_z)^4\right) \left(\sigma_-^j\sigma_z^j\right)
\left(\prod (\sigma_z)^3\right) \sigma_-^i \notag \\
\times
\left(\prod (\sigma_z)^2\right) \left(\sigma_z^l\sigma_+^l \right)
\left(\prod \sigma_z\right) \sigma_+^k \notag \\
=+\left(\sigma_+^j \sigma_z^{j+1}\cdots \sigma_z^{i-1} \sigma_+^i
\sigma_-^l \sigma_z^{l+1}\cdots \sigma_z^{k-1}\sigma_-^k\right.
\notag\\
\left.+ \sigma_-^j \sigma_z^{j+1}\cdots \sigma_z^{i-1} \sigma_-^i
\sigma_+^l \sigma_z^{l+1}\cdots \sigma_z^{k-1}\sigma_+^k\right).
\end{eqnarray}
We get a change in sign for every permutation of the ordering
of the indices from lowest to highest because of the matrix
multiplication ordering. The ordering is described by another set of
indices \newline $\{s_\alpha, s_\beta, s_\gamma, s_\delta\} \in \{i, j, k, l\}$
where
$s_\alpha< s_\beta< s_\gamma< s_\delta$. We assign a number to each of
the four indices, $i\leftrightarrow 1$, $j\leftrightarrow 2$,
$l\leftrightarrow 3$ and $k\leftrightarrow 4$. If $i<j<l<k$ we say the
ordering is $\alpha =1$, $\beta =2$, $\gamma=3$ and $\delta=4$, where
$\alpha$ is a number from one to four indicating which of the indices $i$, $j$,
$l$ and $k$ is the smallest. If $i$ is the smallest, $\alpha=1$ and
$s_\alpha= i$. This allows us to give the sign of a
given $(a^\dag_i a^\dag_j a_l a_k + a_k^\dag a_l^\dag a_j a_i)$
term using the totally anti-symmetric tensor with
four indices, which is $+1$ for even permutations, and $-1$ for odd
permutations. For each of the three groups in Eq.~(\ref{eq:threeGroups})
we get a different set of raising and lowering operators on the
lowest, next lowest and so on, indices, while the sign for the whole
set is given by $-\varepsilon^{\alpha\beta\gamma\delta}$.
We are in the position where we can use the relation in
Eq.~(\ref{eq:raisingAndLowerin}) to express
the Hamiltonian in terms of the $\sigma$-matrices.
We get 16 terms with products of four
$\sigma_x$ and or $\sigma_y$ matrices in the first part of
Eq.~(\ref{eq:VijklPhi}),
then when we add the Hermitian conjugate we get another 16 terms.
The terms with an odd number of $\sigma_y$ matrices have an imaginary
phase and are therefore cancelled out
when adding the conjugates in Eq.~(\ref{eq:H2nonequalSymmetric}).
This leaves us with just the terms with four $\sigma_x$
matrices, four $\sigma_y$ matrices and two of each in different
orderings.
The final result is given as an array with a global sign and factor
given by the permutation of the ordering, and eight terms with
different signs depending on which of the three groups,
Eq.~(\ref{eq:threeGroups}), the set of indices belong to.
These differing rules are due to the rules for $\sigma_z$
multiplication with the raising and lowering operators, resulting in
\begin{eqnarray}
\label{eq:Phiijkl}
&&a^\dag_i a^\dag_j a_l a_k + a_k^\dag a_l^\dag a_j a_i= \notag \\
&&-\frac{\varepsilon^{\alpha\beta\gamma\delta}}{8} \left\{
\begin{array}{cccc}
I & II & III & \\
+& + & + & \sigma_x^{s_\alpha} \sigma_z \cdots \sigma_z
\sigma_x^{s_\beta} \sigma_x^{s_\gamma}\sigma_z\cdots \sigma_z
\sigma_x^{s_\delta}\\
-& +& +& \sigma_x \cdots \sigma_x \quad \sigma_y \cdots\sigma_y \\
+& -& +& \sigma_x \cdots \sigma_y \quad \sigma_x \cdots\sigma_y \\
+& +& -& \sigma_x \cdots \sigma_y \quad \sigma_y \cdots\sigma_x \\
+& +& -& \sigma_y \cdots \sigma_x \quad \sigma_x \cdots\sigma_y \\
+& -& +& \sigma_y \cdots \sigma_x \quad \sigma_y \cdots\sigma_x \\
-& +& +& \sigma_y \cdots \sigma_y \quad \sigma_x \cdots\sigma_x \\
+& +& +& \sigma_y \cdots \sigma_y \quad \sigma_y \cdots\sigma_y \\
\end{array}
\right.
\end{eqnarray}
where the letters $I$, $II$ and $III$ refer to the subgroups defined
in Eq.~(\ref{eq:threeGroups}).
As for the single-particle operators in subsection \ref{sec:1partH}
we now need to factorize these multi-qubit
terms in the Hamiltonian to
products of two-qubit and single-qubit operators.
Instead of transforming a product of the form $a z\cdots z b$, we now
need to transform a product of the form $a z\cdots zb cz\cdots zd$,
where $a$, $b$, $c$ and $d$ are short for either $\sigma_x$ or
$\sigma_y$ while $z$ is short for $\sigma_z$.
The generalization is quite straightforward, as we see that if the
initial operator is $\sigma_z^{s_\alpha} \sigma_z^{s_\gamma}$ instead
of just $\sigma_z^{s_\alpha}$, we can use the same set of
transformations as for the single-particle case,
\begin{eqnarray}
&&U_k^\dag \cdots U_1^\dag \sigma_z^{s_\alpha} U_1 \cdots U_k
\notag \\ &&=
\sigma_a^{s_\alpha}
\sigma_z \cdots \sigma_z \sigma_b^{\beta}
\notag \\
\Rightarrow&&
U_k^\dag \cdots U_1^\dag \sigma_z^{s_\alpha} \sigma_z^{s_\gamma}
U_1 \cdots U_k \notag \\
&&= \sigma_a^{s_\alpha}
\sigma_z \cdots \sigma_z \sigma_b^{s_\beta} \sigma_z^{s_\gamma}.
\end{eqnarray}
Using the same unitary two-qubit transformations, which we now call
$V$,
that take $\sigma_z^{s_\gamma}$ to $\sigma_c^{s_\gamma} \sigma_z
\cdots \sigma_z \sigma_d^{s_\delta}$,
we find
\begin{eqnarray}
&&V_s^\dag\cdots V_1^\dag
U_k^\dag \cdots U_1^\dag \sigma_z^{s_\alpha}\sigma_z^{s_\gamma}
U_1 \cdots U_k
V_1\cdots V_s \notag \\
&&=
\sigma_a^{s_\alpha}
\sigma_z \cdots \sigma_z \sigma_b^{\beta}
\sigma_c^{s_\gamma}
\sigma_z \cdots \sigma_z \sigma_d^{\delta}.
\end{eqnarray}
This straightforward generalization of the procedure from the single-particle
Hamiltonian case is possible because the operations commute when performed
on different qubits.
With the above expressions, we can start with
the unitary operator $\exp(-ia \sigma_z^{s_\alpha}\sigma_z^{s_\gamma})$
and have two different series of unitary operators that give us the
evolution operator of the desired Hamiltonian.
The $U$ operators are defined as in
Eq.~(\ref{eq:Us}),
\begin{equation}
e^{-i\sigma^{s_\alpha} \sigma_z \cdots \sigma_z
\sigma^{s_\beta}a}
= \left(\prod_k U_k^\dag \right) e^{-i\sigma_z^{s_\alpha}a}
\left(\prod_{k^\prime}
U_{k^\prime}\right),
\end{equation}
while the
$V$ operators are defined in a similar way
\begin{equation}
e^{-i\sigma^{s_\gamma} \sigma_z \cdots \sigma_z
\sigma^{s_\delta}a}
= \left(\prod_s V_s^\dag \right) e^{-i\sigma_z^{s_\gamma}a}
\left(\prod_{s^\prime}
V_{s^\prime}\right),
\end{equation}
where the $\sigma$-matrices without subscripts represent that we can
have $\sigma_x$ or $\sigma_y$ in each position.
This gives us the total evolution operator for each term in
Eq.~(\ref{eq:Phiijkl})
\begin{eqnarray}
&&e^{-i\sigma^{s_\alpha} \sigma_z \cdots \sigma_z
\sigma^{s_\beta} \sigma^{s_\gamma} \sigma_z \cdots \sigma_z
\sigma^{s_\delta} a} \notag \\
&&=\left(\prod_s V_s^\dag \right)\left(\prod_k U_k^\dag \right)
e^{-i \sigma_z^{s_\alpha}\sigma_z^{s_\gamma}a} \notag\\
&&\times\left(\prod_{k^\prime}
U_{k^\prime}\right)
\left(\prod_{s^\prime}
V_{s^\prime}\right).
\end{eqnarray}
Here we have all the single- and two-qubit operations we need to
perform on our set of qubits, that were initially in the state
$|\psi\rangle$, to simulate the time evolution $\exp(-iH_k\Delta
t)|\psi\rangle$ of the Hamiltonian term $H_k= \sigma^{s_\alpha} \sigma_z \cdots \sigma_z
\sigma^{s_\beta} \sigma^{s_\gamma} \sigma_z \cdots \sigma_z
\sigma^{s_\delta}$. Every factor in the above equation is a single- or
two-qubit operation that must be performed on the qubits in proper
matrix multiplication order.
When using the Jordan-Wigner transformation of
Eq.~(\ref{eq:JWtransformation}) applied to our two model Hamiltonians of
Eqs.~(\ref{eq:hubbard}) and (\ref{eq:pairing}), we choose a representation with two qubits at each site. These
correspond to fermions with spin up and down, respectively.
The number of qubits, $n$, is always the total number of available
quantum states and therefore
it is straightforward to use this model on systems with higher degeneracy, such
as those encountered in quantum chemistry \cite{helgaker} or nuclear physics \cite{caurier2005}.
Site one spin up is qubit one, site one spin
down is qubit two and site two spin up is qubit three and so on.
To get all the quantum gates one needs to simulate a given Hamiltonian
one needs to input the correct $E_{ij}$ and $V_{ijkl}$ values.
\subsection{Complexity of the quantum computing algorithm}
\label{sec:complexityOfFermionicSimulator}
In order to test the efficiency of a quantum algorithm, one needs to know how many
qubits, and how many operations on these, are needed to implement the
algorithm. Usually this is a
function of the dimension of the Hilbert space on which the
Hamiltonian acts. The natural input scale
in the fermionic simulator is the number of quantum states, $n$, that are
available to the fermions.
In our simulations of the Hubbard and the pairing
models of Eqs.~(\ref{eq:hubbard}) and (\ref{eq:pairing}), respectively,
the number of qubits is $n=2N$ since we have chosen systems with
double-degeneracy for every single-particle state, where $N$ is the
number of energy-levels in the model.
We use one qubit to represent each possible fermion state,
on a real quantum computer, however, one should implement some
error-correction procedure using several qubits for each state, see
Ref.~\cite{nielsen2000}.
The complexity in number of qubits remains linear, however, since ${\cal O} (n)$
qubits are needed for error correction.
The single-particle Hamiltonian has potentially ${\cal O} (n^2)$
different $E_{ij}$ terms. The two-particle Hamiltonian has up to
${\cal O} (n^4)$ $V_{ijkl}$ terms. A general $m$-body interaction has
in the worst case ${\cal O} (n^{2m})$ terms. It is straightforward to
convince oneself that the
pairing model has ${\cal O} (n^2)$ terms while
in the Hubbard model we end up with ${\cal O}
(n)$ terms. Not all models have maximum complexity in the different
$m$-body interactions.
How many two-qubit operations do each of these terms need to be
simulated? First of all a two-qubit operation will in general have to
be decomposed into a series of universal single- and two-qubit
operations, depending entirely on the given quantum simulator. A particular
physical realization might have a natural implementation of the
$\sigma_z^i\otimes \sigma_z^j$ gate and save a lot of intermediary
operations.
Others will have to use a fixed number of operations in order to apply the
operation on any two qubits. A system with only nearest neighbor
interactions would have to use ${\cal O}(n)$ operations for each
$\sigma_z^i\otimes \sigma_z^j$ gate, and thereby increase the
polynomial complexity by one degree.
In our discussion on the one-body part of the Hamiltonian,
we saw that for each $E_{ij}$ we obtained the
$a^\dag_ia_j + a^\dag_j a_i$ operator which is transformed into the
two terms in Eq.~(\ref{eq:singleParticle}),
$\sigma_x\sigma_z\cdots\sigma_z \sigma_x$ and
$\sigma_y\sigma_z\cdots\sigma_z \sigma_y$.
We showed how these terms are
decomposed into $j-i+2$ operations, leading to twice as many
unitary transformations on an operator, $VAV^\dag$ for the time evolution.
The average of $j-i$ is $n/2$ in this case and in total we need to
perform $2\times 2\times n/2 = 2n$ two-qubit operations per
single-particle term in the Hamiltonian, a linear complexity.
In the two-particle case each term $V_{ijkl}(a^\dag_i a^\dag_j a_l a_k
+ a^\dag_k a^\dag_l a_j a_i)$ is transformed into a sum of eight
operators of the form $\sigma^{s_\alpha} \sigma_z \cdots \sigma_z
\sigma^{s_\beta} \sigma^{s_\gamma} \sigma_z \cdots \sigma_z
\sigma^{s_\delta}$, Eq.~(\ref{eq:Phiijkl}). The two parts of these
operators are implemented in the same way as the $\sigma^i
\sigma_z\cdots \sigma_z \sigma^j$ term of the single-particle
Hamiltonian, which means they require $s_\beta - s_\alpha$ and
$s_\delta -s_\gamma$ operations, since
$s_\alpha<s_\beta<s_\gamma<s_\delta$ the average is $n/4$.
For both of these parts we need to perform both the unitary operation
$V$ and it's
Hermitian conjugate $V^\dag$. In the end we need $2\times 2
\times 8 \times n/4=8n$ two-qubit operations per two-particle term in
the Hamiltonian, the complexity is linear.
A term of an $m$-body Hamiltonian will be transformed into $2^{2m}$
operators since each annihilation and creation operator is transformed
into a sum of $\sigma_x$ and $\sigma_y$ matrices.
All the imaginary terms cancel out and we are left with $2^{2m-1}$
terms.
Each of these terms
will include $2m$ $\sigma$ matrices, in products of the form
$\prod_{k=1}^m \sigma^i \sigma_z \cdots \sigma_z \sigma^j$,
and we use the same procedure as discussed above
to decompose these $m$ factors into
unitary
transformations. In this case each factor will require an average of
$n/2m$ operations for the same reasons as in the two-body case.
All in all, each $m$-body term in the Hamiltonian requires
$2^{2m-1}\times 2\times m \times n/2m = 2^{2m-1}n$ operations.
Thus, the complexity for simulating one $m$-body term of a fermionic
many-body Hamiltonian is linear in the number of two-qubit
operations, but the number of terms is not. For a
full-fledged simulation of general three-body forces, in common use in
nuclear physics \cite{Pieper2001,navratil2002,ccsdt03},
the total complexity of the simulation is ${\cal O}
(n^7)$. A complete two-particle Hamiltonian would be ${\cal O}
(n^5)$.The bottleneck in these simulations is the number of terms in the
Hamiltonian, and for systems with less than the full number of terms
the simulation will be faster.
This is much better than the exponential complexity of
most simulations on classical computers
\section{Algorithmic details}
\label{sec:details}
Having detailed how a general Hamiltonian, of two-body nature in our case,
can be decomposed in terms of various quantum gates,we present here details of the
implementation of our algorithm for finding eigenvalues and eigenvectors of
a many-fermion system.
For our tests of the fermionic simulation algorithm we have
implemented the phase-estimation algorithm from \cite{nielsen2000}
which finds the eigenvalues of an Hamiltonian operating on a set of
simulation qubits. There are also other quantum computer algorithms for
finding expectation values and correlation functions, as discussed by Somma {\em et al.} in
Refs.~\cite{somma2002,somma2005}.
In the following we first describe the phase-estimation algorithm, and
then describe its implementation and methods we have developed
in using this algorithm. A much more thorough description of quantum
computers and the phase-estimation algorithm can be found in
\cite{ovrum2003}.
\subsection{Phase-estimation algorithm}
To find the eigenvalues of the Hamiltonian we use
the unitary time evolution operator we get from the Hamiltonian.
We have a set of simulation qubits representing the system governed by
the Hamiltonian, and a set of auxiliary qubits, called work qubits \cite{lloyd1997,lloyd1999a}, in
which we will store the eigenvalues of the time evolution operator.
The procedure is to perform several controlled time evolutions with
work
qubits as control qubits and the simulation qubits as targets,
see for example Ref.~\cite{nielsen2000} for information on controlled qubit
operations.
For
each work qubit we perform the controlled operation on the simulation
qubits with a different time parameter, giving all the work qubits
different phases. The information stored in their phases is extracted
using first an inverse Fourier transform on the work qubits alone, and
then performing a measurement on them. The values of the
measurements give us directly the eigenvalues of the Hamiltonian after
the algorithm has been performed a number of times.
The input state of the simulation qubits is a random state in our
implementation, which is
also a random superposition of the eigenvectors of the Hamiltonian
$|\psi\rangle=\sum_k c_k | k\rangle$. It does not have to be a random state,
and in \cite{lawu2002} the authors describe a quasi-adiabatic
approach, where the initial state is created by starting in the ground
state for the non-interacting Hamiltonian,
a qubit basis state, e.g. $|0101\cdots101\rangle$,
and
then slowly the interacting part of the Hamiltonian is turned on.
This gives us an initial state mostly comprising the true ground state,
but it can also have parts of the lower excited states if the
interacting Hamiltonian is turned on a bit faster.
In for example nuclear physics it is common to use a starting state for large-scale
diagonalizations that reflects some of the features of the states one wishes to study.
A typical example is to include pairing correlations in the trial wave function, see
for example Refs.~\cite{caurier2005,rmp75mhj}.
Iterative methods such as the Lanczo's diagonalization technique \cite{Whitehead1977,golub1996}
converge much faster if such starting vectors are used. However, although more iterations are needed,
even a random starting vector converges to the wanted states.
The final state of all the qubits
after an inverse Fourier transform on the work qubits is
\begin{equation}
\label{eq:FinalState}
\sum_{k}c_k |\phi^{[k]} 2^t\rangle \otimes |k\rangle.
\end{equation}
If the
algorithm works perfectly, $|k\rangle$ should be an exact eigenstate of
$U$, with an exact eigenvalue $\phi^{[k]}$.
When we have the
eigenvalues of the time evolution operator we easily find
the eigenvalues of the
Hamiltonian.
We can summarize schematically the phase-estimation algorithm as follows:
\begin{enumerate}
\item Intialize each of the work qubits to $1/\sqrt2 ( |0\rangle + |1\rangle )$
by initializing to $ |0\rangle $ and applying the Hadamard gate, H,
see Fig.~\ref{fig:elementarySingleQubitGates}.
\item Initialize the simulation qubits to a random or specified
state, depending on the whether one wants the whole eigenvalue
spectrum.
\item Perform conditional time evolutions on the simulation qubits, with
different timesteps $\Delta t$ and different work qubits as the
control qubits.
\item Perform an inverse Fourier transform on the work qubits.
\item Measure the work qubits to extract the phase.
\item Repeat steps 1-6 until the probability distribution gathered
from the measurement results is good enough to read out the wanted
eigenvalues.
\end{enumerate}
\begin{widetext}
\onecolumngrid
\begin{figure
\begin{picture}(250,200)
\put(15,0){\input{fig5.latex}}
\put(0,15){\makebox(0,0){\ensuremath{|i\rangle}}}
\put(0,67){\makebox(0,0){\ensuremath{|0\rangle}}}
\put(0,95){\makebox(0,0){\ensuremath{|0\rangle}}}
\put(0,123){\makebox(0,0){\ensuremath{|0\rangle}}}
\put(0,163){\makebox(0,0){\ensuremath{| 0\rangle}}}
\put(240,15){\makebox(0,0)[l]{\ensuremath{|i\rangle}}}
\put(240,67){\makebox(0,0)[l]{\ensuremath{ |0\rangle + e^{2\pi
i(2^0\phi)} |1\rangle = |0\rangle + e^{2\pi i0.\phi_1\cdots\phi_w} |1\rangle }}}
\put(240,95){\makebox(0,0)[l]{\ensuremath{ |0\rangle + e^{2\pi i(2^1\phi)} |1\rangle
= |0\rangle + e^{2\pi i0.\phi_2\cdots\phi_w} |1\rangle }}}
\put(240,123){\makebox(0,0)[l]{\ensuremath{ |0\rangle + e^{2\pi i(2^2\phi)} |1\rangle =
|0\rangle + e^{2\pi i 0.\phi_3\cdots\phi_w} |1\rangle }}}
\put(240,163){\makebox(0,0)[l]{\ensuremath{ |0\rangle + e^{2\pi
i(2^{w-1}\phi)} |1\rangle
= |0\rangle + e^{2\pi i0.\phi_w} |1\rangle }}}
\end{picture}
\caption{Phase estimation circuit showing all the different qubit
lines schematically with operations represented by boxes. The boxes
connected by vertical lines to other qubit lines are controlled
operations, with the qubit with the black dot as the control qubit.}
\label{fig:phaseEstCircuit}
\end{figure}
\end{widetext}
\twocolumngrid
As discussed above a set of two-qubit
operations can be simulated by the CNOT two-qubit operation and a
universal set of single-qubit operations.
We will not use or discuss any such
implementation in this article, as one will have to use a different
set for each physical realization of a quantum computer. When
simulating a fermion system with a given
quantum computer, our algorithm will first take the
fermionic many-body evolution operator to a series of two-qubit and
single-qubit operations, and then one will have to have a system
dependent setup that takes these operations to the basic building
blocks that form the appropriate universal set.
In subsection \ref{sec:2bH} we showed how to take any two-particle
fermionic Hamiltonian to a set of two-qubit operations that
approximate
the evolution operator.
In addition we must use one of the Trotter approximations \cite{trotter1959,suzukitrotter,suzuki1985}
Eqs.~(\ref{eq:Trotter1}) and (\ref{eq:Trotter2}) that take
the evolution operator of a sum of terms to the product of the
evolution operator of the individual terms,
see for example Ref.~\cite{nielsen2000} for details. To order ${\cal O}(\Delta
t^2)$ in the error we use Eq.~(\ref{eq:Trotter1})
while to order ${\cal O} (\Delta t^3)$ we have
\begin{equation}
\label{eq:Trotter2}
e^{-i(A+B)\Delta t}=e^{-iA\Delta
t/2}e^{-iB\Delta t}e^{-iA\Delta t/2} + {\cal O} (\Delta t^3).
\end{equation}
\subsection{Output of the phase-estimation algorithm}
\label{sec:output}
The output of the phase-estimation algorithm is a series of
measurements of the $w$ number of work qubits. Putting them all together
we get a
probability distribution that estimates the amplitudes $|c_k|^2$ for
each eigenvalue $\phi^{[k]}$.
The $\phi^{[k]}2^w$
values we measure from the work qubits, see Eq.~(\ref{eq:FinalState}),
are binary numbers from zero to $2^w-1$,
where each one translates to a given eigenvalue of the Hamiltonian depending
on the parameters we have used in our simulation.
When accurate, a set of simulated
measurements will give a distribution with peaks around the true
eigenvalues.
The probability
distribution is calculated by applying non-normalized projection
operators to the qubit state,
\[
\left(| \phi^{[k]}2^t\rangle \langle\phi^{[k]}2^t | \otimes {\bf 1} \right)
\left( \sum_{i}c_i |\phi_i 2^t\rangle \otimes |i\rangle \right)
= c_k|\phi^{[k]} 2^t\rangle \otimes |k\rangle.
\]
The length of this vector squared
gives us the probability,
\begin{equation}
\left|c_k |\phi^{[k]}\rangle 2^t \otimes |k\rangle \right|^2=|c_k|^2
\langle \phi^{[k]} 2^t|\phi^{[k]} 2^t\rangle\langle k |k \rangle= |c_k|^2.
\end{equation}
Since we do not employ the exact evolution due to different
approximations, we can have non-zero
probabilities for all values of $\phi$, yielding a distribution
without sharp peaks for the correct eigenvalues and possibly peaks in
the wrong places. If we use different random input states for every
run through the quantum computer and gather
all the measurements in
one probability distribution, all the eigenvectors in the input state
$|\psi\rangle=\sum_k c_k | k\rangle$ should average out to the same amplitude.
This means that eigenvalues with higher multiplicity, i.e., higher
degeneracy, will show up as taller peaks in the probability distribution,
while non-degenerate eigenvalues might be difficult to find.
To properly estimate the eigenvalues
$E_k$ of the Hamiltonian from this distribution, one must take into account the
periodicity of $e^{2\pi i\phi}$. If $0 < \phi^\prime < 1$ and $\phi =
\phi^\prime +s$, where $s$ is an integer, then
$e^{2\pi i\phi}=e^{2\pi i\phi^\prime}$. This means that to get all
the eigenvalues correctly $\phi$ must be positive and less than one.
Since $\phi = -E_k \Delta t/2\pi$ this means all the eigenvalues $E_k$
must be negative, this merely means subtracting a constant we denote
$E_{max}$ from the Hamiltonian, $H^\prime = H -E_{max}$, where
$E_{max}$ is greater than the largest eigenvalue of $H$.
The values we read out from the work qubits are integers from zero to
$2^w-1$. In other words, we have
$\phi^{[k]}2^w\in [0, 2^w-1]$, with $\phi=0$ for $\Delta t=0$.
The value $\phi=0$ corresponds to the lowest eigenvalue possible to measure,
$E_{min}$, while $\phi=1$ corresponds to $E_{max}$. The interval of
possible values is then $E_{max}-E_{min} = 2\pi/\Delta t$. If we want to
have all possible eigenvalues in the interval the largest value
$\Delta t$ can have is
\begin{equation}
\mathrm{max}(\Delta t) = \frac{2\pi}{E_{max}-E_{min}}
\end{equation}
\subsubsection{Spectrum analysis}
In the general case one does not know the upper and lower bounds on
the eigenvalues beforehand, and therefore for a given $E_{max}$ and
$\Delta t$ one does not know if the $\phi^{[k]}$ values are the correct
ones, or if an integer has been lost in the exponential function.
When $\phi=\phi^\prime +s$ for one
$\Delta t$, and we slightly change $\Delta t$, $\phi^\prime$ will
change if $s\neq 0$
as the period of the exponential function is a function of
$\Delta t$.
To find out which of $\phi^{[k]}$s are greater than
one, we perform the phase-estimation algorithm with different values for $\Delta
t$ and
see which eigenvalues shift.
If we measure the same $\phi$ after adding $\delta t$ to
the time step, and $(\Delta t + \delta t)/\Delta t$ is not a rational
number, we know that $\phi <1$. In practice it does not have to be an
irrational number, but only some unlikely fraction.
There are at least two
methods for finding the eigenvalues. One can start with a
large positive $E_{max}$ and a small $\Delta t$, hoping to find that
the whole spectrum falls within the range $[E_{min}, E_{max}]$, and
from there zoom in until the maximal eigenvalue is slightly less than
$E_{max}$ and the groundstate energy is slightly larger than
$E_{min}$. This way the whole spectrum is covered at once.
From there we can also zoom in on specific areas of the spectrum,
searching the location of the true eigenvalues by shifting $\Delta t$.
The number of measurements needed will depend entirely on the
statistics of the probability distribution. The number of eigenvalues
within the given energy range determines the resolution needed.
That said, the number of
measurements is not a bottleneck in quantum computer calculations. The
quantum computer will prepare the states, apply all the operations
in the circuit and measure. Then it will do it all again. Each
measurement will be independent of the others as the system is
restarted each time. This way the serious problem of decoherence only
apply within each run, and the number of measurements is only
limited by the patience of the scientists operating the quantum
computer.
\section{Results and discussion}
\label{sec:results}
In this section we present the results for the Hubbard model and the
pairing model of Eqs.~(\ref{eq:hubbard}) and (\ref{eq:pairing}), respectively, and
compare the simulations to exact diagonalization results.
In Fig.~\ref{fig:P24-13-15IM} we see the resulting probability
distribution from the simulated measurements, giving us the eigenvalues
of the pairing model with six degenerate energy levels and from zero
to 12 particles.
The pairing strength was set to $g=1$.
The eigenvalues from the exact solutions of these many-particle
problems are $0$, $-1$,
$-2$, $-3$, $-4$, $-5$, $-6$, $-8$, $ -9$, $-12$.
All the eigenvalues are not seen as this is the probability
distribution resulting from one random input state. A
different random input state in each run could be implemented on
an actual quantum computer. These are results for the degenerate
model, where the single-particle energies of the doubly degenerate levels are set to
zero for illustrate purposes only, since analytic formula are available for the
exact eigenvalues. The block diagonal structure of the pairing
Hamiltonian has not been used to our advantage in this straightforward
simulation as the qubit basis includes all particle numbers.
\begin{figure}[h!]
\begin{center}
\scalebox{0.5}{
\psfig{file=fig6.eps}
}
\caption{Resulting probability
distribution from the simulated measurements, giving us the eigenvalues
of the pairing model with six degenerate energy levels with a total
possibility of 12 particles and pairing strength $g=1$.
The correct eigenvalues are $0$, $-1$,
$-2$, $-3$, $-4$, $-5$, $-6$, $-8$, $ -9$, $-12$.
All the eigenvalues are not seen as this is the probability
distribution resulting from one random input state. A
different random input state in each run could be implemented on
an actual quantum computer and would eventually yield peaks
of height corresponding to the degeneracy of each eigenvalue. }
\label{fig:P24-13-15IM}
\end{center}
\end{figure}
We have also performed tests of the algorithm for
the non-degenerate case, with excellent agreement with our diagonalization codes,
see discussion in Ref.~\cite{rmp75mhj}.
This is seen in Fig.~\ref{fig:P24-17-e3IMd} where we have simulated the pairing model
with four energy levels with a total possibility of eight fermions.
We have chosen $g=1$ and $d=0.5$, so this is a model with low degeneray
and since
the dimension of the system is $2^8= 256$ there is a lot of different
eigenvalues. To find the whole spectrum one would have to employ
some of the techniques discussed in subsection~\ref{sec:output}.
\begin{figure}[h!]
\begin{center}
\scalebox{0.5}{
\psfig{file=fig7.eps}
}
\caption{The eigenvalues
of the non-degenerate pairing model with four energy levels with a total
possibility of 8 particles, the level spacing $d$ is $0.5$
and the pairing strength $g$ is $1$.
The correct eigenvalues are obtained from exact
diagonalization, but in this case there is a multitude of
eigenvalues and only some eigenvalues are found from this
first simulation.
}
\label{fig:P24-17-e3IMd}
\end{center}
\end{figure}
\subsection{Number of work qubits versus number of simulation qubits}
The largest possible amount of different eigenvalues is $2^s$, where
$s$ is the number of simulation qubits. The resolution in the energy
spectrum we get from measuring upon the work qubits is $2^w$, with $w$ the number of
work qubits.
Therefore the resolution per eigenvalue in a non-degenerate system is
$2^{w-s}$. The higher the degeneracy the less work qubits are needed.
In Fig.~\ref{fig:24-17-1T0} we see the results for the Hubbard model
Eq.~(\ref{eq:hubbard})
with $\epsilon=1$, $t=0$ and $U=1$. The reason we chose $t=0$ was just
because of the higher degeneracy and therefore fewer eigenvalues.
The number of work qubits is $16$
and the number of simulation qubits is eight for a total of $24$
qubits. The difference between work qubits and simulation qubits is
eight which means there are $2^8$ possible energy values for each
eigenvalue. Combining that with the high degeneracy we get a very sharp
resolution. The correct eigenvalues with degeneracies are obtained from exact
diagonalization of the Hamiltonian, the degeneracy follows the
eigenvalue in paranthesis: 0(1), 1(8), 2(24), 3(36), 4(40), 5(48), 6(38),
7(24), 8(24), 9(4), 10(8), 12(1). We can clearly see that even though
we have a random input state, with a random superposition of the
eigenvectors, there is a correspondence between the height of the
peaks in the plot and the degeneracy of the eigenvalues they represent.
\begin{figure}[h!]
\begin{center}
\scalebox{0.5}{
\psfig{file=fig8.eps}
}
\caption{The energy levels of the Hubbard model of Eq.~(\ref{eq:hubbard}),
simulated with a total of $24$ qubits, of which eight were
simulation qubits and $16$ were work qubits.
In this run we chose
$\epsilon=1$, $t=0$ and $U=1$. The reason we chose $t=0$ was just
because of the higher degeneracy and therefore fewer
eigenvalues. The correct eigenvalues are obtained from exact
diagonalization, with the level of degeneracy following in
paranthesis: 0(1), 1(8), 2(24), 3(36), 4(40), 5(48), 6(38),
7(24), 8(24), 9(4), 10(8), 12(1).
}
\label{fig:24-17-1T0}
\end{center}
\end{figure}
\subsection{Number of time intervals}
The number of time intervals, $I$, is the number of times we must
implement the time evolution operator in order to reduce the error in the
Trotter approximation \cite{trotter1959,suzukitrotter,suzuki1985},
see Eq.~(\ref{eq:Trotter1}).
In our program we have only implemented the simplest Trotter
approximation and in our case we find that we do not need a large $I$
before the error is small enough. In Fig.~\ref{fig:24-17-1T0}
$I$ is only one, but here we have a large number of work qubits.
For other or larger systems it
might pay off to use a higher order Trotter approximation. The total
number of operations that have to be done is a multiple of $I$, but
this number
also increases for higher order Trotter approximations, so for each
case there is an optimal choice of approximation.
In Figs.~\ref{fig:P24-15-1IM} and \ref{fig:P24-15-e1IM} we
see the errors deriving from the Trotter approximation, and how they
are reduced by increasing the number of time intervals. The results in this figure are
for the degenerate pairing model with 24 qubits in total, and ten
simulation qubits with $d=0$ and $g=1$. In Fig.~\ref{fig:P24-15-1IM}
we had $I=1$ while in Fig.~\ref{fig:P24-15-e1IM} $I$ was set to ten. Both
simulations used the same starting state. The errors are seen as the
small spikes around the large ones which represent some of the
eigenvalues of the system. The exact eigenvalues are $0$, $-1$,
$-2$, $-3$, $-4$, $-5$, $-6$, $-8$, $ -9$.
\begin{figure}[h!]
\begin{center}
\scalebox{0.5}{
\psfig{file=fig9.eps}
}
\caption{Pairing model simulated with $24$ qubits, where $14$
were simulation qubits, i.e. there are $14$ available
quantum levels, and
$10$ were work qubits. The correct eigenvalues are $0$, $-1$,
$-2$, $-3$, $-4$, $-5$, $-6$, $-8$, $ -9$. In this run we
did not divide up the time interval to reduce the error in
the Trotter approximation, i.e., $I=1$.}
\label{fig:P24-15-1IM}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\scalebox{0.5}{
\psfig{file=fig10.eps}
}
\caption{Pairing model simulated with $24$ qubits, where $14$
were simulation qubits, i.e. there are $14$ available
quantum levels, and
$10$ were work qubits.
The correct eigenvalues are $0$, $-1$,
$-2$, $-3$, $-4$, $-5$, $-6$, $-8$, $ -9$. In this run we
divided the time interval into $10$ equally space parts
in order to reduce the error in
the Trotter approximation, i.e., $I=10$.}
\label{fig:P24-15-e1IM}
\end{center}
\end{figure}
\subsection{Number of operations}
Counting the number of single-qubit and $\sigma_z\sigma_z$ operations
for different sizes of systems simulated gives us an indication of the
decoherence time needed for different physical realizations of a
quantum simulator or computer. The decoherence time is an average time
in which the state of the qubits will be destroyed by noise, also called
decoherence, while the operation time is the average time an operation takes
to perform on the given system. Their fraction is the number of
operations possible to perform before decoherence destroys the
computation. In table~\ref{fig:noOfGates} we have listed the number of
gates used for the pairing model, $H_P$, and the Hubbard model, $H_H$,
for different number of simulation qubits.
\begin{table}[h!]
\begin{center}
\begin{tabular}{l|cccccc}\hline\hline
& $s=2$&$s=4$&$s=6$&$s=8$&$s=10$ & $s=12$ \\ \hline
$H_P$ & 9 & 119 & 333 & 651 & 1073 & 1598 \\ \hline
$H_H$ & 9 & 51 & 93 & 135 & 177 & 219 \\ \hline
\end{tabular}
\end{center}
\caption{Number of two-qubit gates used in simulating the time
evolution operator of the pairing model, $H_P$, and the Hubbard
model, $H_H$, for different number of simulation qubits $s$.}
\label{fig:noOfGates}
\end{table}
\section{Conclusion}
\label{sec:conclusion}
In this article we have shown explicitly how the Jordan-Wigner
transformation is used to simulate any many-body fermionic Hamiltonian
by two-qubit operations. We have shown how the simulation of such
Hamiltonian terms of products of creation and annihilation operators
are represented by a number of operations linear in the number of
qubits.
To perform efficient quantum simulations on quantum computers one
needs transformations that take the Hamiltonian in question to a set
of operations on the qubits simulating the physical system. An example of
such a transformation employed in ths work, is
the Jordan-Wigner transformation. With the appropriate transformation and
relevant gates or quantum circuits, one can taylor an actual quantum computer to
simulate and solve the eigenvalue and eigenvector problems for different quantum systems.
Specialized quantum simulators might be more efficient in solving some
problems than others because of similarities in algebras between physical system
of qubits and the physical system simulated.
We have limited the applications to two simple and well-studied models that provide,
via exact eigenvalues, a good testing ground for our quantum computing based
algorithm. For both the pairing model and the Hubbard model we obtain an excellent agreement.
We plan to extend the area of application to quantum mechanical studies of systems in nuclear physics,
such as a comparison of shell-model studies of oxygen or calcium isotopes where the nucleons are active
in a given number of single-particle orbits \cite{mhj95,caurier2005}. These single-particle orbits have normally a higher
degeneracy than $2$, a degeneracy studied here. However, the algorithm we have developed allows
for the inclusion of any degeneracy, meaning in turn that with a given interaction $V_{ijkl}$
and single-particle energies, we can compare the nuclear shell-model (configuration interaction) calculations
with our algorithm.
\section*{Acknowledgment}
This work has received support from the Research Council of Norway through the center of
excellence program.
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\section{Introduction}
Relativistic jets in blazars transport energy in the form of bulk motion of protons,
leptons and magnetic field.
When part of this power is dissipated, the particles emit the beamed radiation we
observe, consisting of two broad humps.
The origin of the low frequency hump is well established, believed to be
synchrotron radiation from relativistic (in the comoving frame) leptons.
The nature of the high energy hump is a controversial issue.
In general, we can group the existing models into three families:
i) the high energy radiation is generated by the same leptons producing the
synchrotron, through the inverse Compton process
(Maraschi, Ghisellini \& Celotti 1992;
Dermer \& Schlickeiser 1993;
Sikora, Begelman \& Rees 1994;
Ghisellini \& Madau 1996;
Bloom \& Marscher 1996;
Celotti \& Ghisellini 2008);
ii) There are two populations of leptons, one accelerated directly by the
acceleration mechanism (i.e. shocks), and a second one resulting from
cascades initiated by ultra--relativistic protons
(Mannheim 1993;
M\"ucke et al. 2003;
B\"ottcher 2007);
iii) Ultrarelativistic protons emit by the proton--synchrotron process at
high energies (Aharonian 2000; M\"ucke \& Protheroe 2001).
Another issue of debate is the role of electron--positron pairs.
Sikora \& Madejski (2000) discussed this problem concluding that,
though they can exist, their amount is limited to a few pairs per proton,
and a similar conclusion was reached by Celotti \& Ghisellini (2008)
analysing a large number spectral energy distributions (SEDs) for deriving
the jet powers of blazars.
This contrasts the idea, originally put forward by Blandford \& Levinson (1995)
that the $\gamma$--rays spectrum is the superposition of the spectra
originating at different distances from the black hole, each one cutted--off
at a different $\gamma$--ray energy as a result of photon--photon absorption.
In this scheme a large amount of pairs is produced, since the high energy hump carries
most of the emitted power: if this is partly absorbed, we transform most of the
total power into (energetic) pairs.
Ghisellini \& Madau (1996) argued that the Blandford \& Levinson idea
has one important observational consequence in powerful blazars
with a standard accretion disk: since the pairs are born relativistic,
they contribute to the emission mainly by inverse Compton scattering
the dense UV radiation field coming from the accretion disk.
In fact, if we want the emission region be compact, it is reasonable to locate
it close to the accretion disk and its X--ray corona.
This implies that the coronal X--rays are targets for the
$\gamma$--$\gamma \to e^\pm$ process, and that the accretion disk UV photons
become seeds for the scattering made by the newly born pairs.
The resulting radiation is mainly in the X--ray band, that should have a
power comparable to the power absorbed in the $\gamma$--ray band, contrary
to what observed.
This led Ghisellini \& Madau (1996) to conclude that the dissipation region
in blazar jets cannot be very close to the accretion disk.
On the other hand the observed fast variability argues for not too large distances.
Taken together, these two limits strongly suggest that there is a preferred distance
at which {\it the $\gamma$--ray radiation is produced}, at some hundreds of Schwarzschild\ radii.
Since this argument makes use of the reprocessed radiation produced by the pairs,
it cannot be applied when most of the radiation is produced below the pair
production energy threshold and/or if the accretion disk is
radiatively inefficient: in both cases very few pairs are created.
Indeed, Katarzynski \& Ghisellini (2007) proposed a jet model in which
a dissipation close to the accretion disk resulted in a low level
$\gamma$--ray emission, with no contradiction with the existing X--ray data.
These arguments, although correct, are qualitative, and
in our opinion a detailed computation of the reprocessing due to pairs
is not yet present in the literature.
Therefore one of the aims of the present paper is to derive
some limits on the location of the dissipation region in
the jet of powerful blazars.
We will do it in the framework of the ``leptonic" class of models
[family i) mentioned above], and we limit our analysis to blazars
having a ``standard" (Shakura \& Sunyaev 1973) accretion disk.
This implies to consider Flat Spectrum Radio Quasars (FSRQs, and their likely
parent population, FR II radio--galaxies) and not BL Lac objects, that are
likely to have radiatively inefficient accretion flows (see
Celotti \& Ghisellini 2008 and Ghisellini \& Tavecchio 2008 for more discussion
about this point).
While doing this, we will study the relative importance of
different sources of seed photons as a function of the distance $R_{\rm diss}$
of the dissipation region from the black hole, and their spectrum
as seen in the comoving frame.
This is the second aim of the paper.
Besides the radiation coming directly from the accretion disk and
its X--ray corona, we will consider the radiation produced in the
Broad Line Region (BLR) and in a relatively more distant dusty torus, intercepting
a fraction of the disk radiation and re--emitting it in the infrared.
Finally, we also include the contribution of the Cosmic Microwave Background
(CMB), important for very large (beyond 1 kpc) $R_{\rm diss}$.
The term ``canonical high power blazars'' refers to the rather standard
choice for both the environment of these sources and their jets:
\begin{enumerate}
\item
The accretion disk is a standard, ``Shakura \& Syunyaev" (1973) disk.
\item
Above this accretion disk, there is an X--ray
corona, emitting a luminosity less than, but comparable to, the luminosity
emitted by the accretion disk.
\item
the BLR and the IR torus are located at distances
that scale as the square root of the disk luminosity
(Bentz et al. 2006;
Kaspi et al. 2007;
Bentz et al. 2008;
see the discussion about this point made in Ghisellini \& Tavecchio 2008).
\item
The jet is assumed to dissipate only a fraction of its
total power, which is then conserved.
After the acceleration phase, possibly magnetic in origin,
we assume that also the Poynting flux
is conserved (see e.g. Celotti \& Ghisellini 2008).
\end{enumerate}
Our study is not completely new, since several of its ``ingredients"
have already been discussed in the literature.
The paper more german to our is Dermer et al. (2009), but we include
some new ingredients.
The novel features of our investigation concern mainly: i) the inclusion of the
X--ray corona as an important producer of target photons for the
$\gamma$--$\gamma \to e^\pm$ process; ii) the calculation of the emitting
particle distribution, including pair creation;
iii) the effects of jet acceleration at small distances from the black hole;
iv) the strict link between the properties of the accretion disk and the
amount of
the external radiation and v) the overall scenario allowing to
describe in a more general way (than done before)
the SED properties of high power jets at all scales.
The paper is divided into four parts.
In the first we study the different sources of external radiation,
and the corresponding energy densities, as seen in the comoving frame,
as a function of distance from the black hole and as a function of the
bulk Lorentz factor of the jet.
In the second part of the paper we investigate the role of pair production
processes when the dissipation region is close to the black hole, with the aim
to find quantitative constraints.
In the third part we construct the expected SED as a function of distance,
highlightening what are the relevant external seed photons.
Finally, in the fourth part, we apply some of the above results
and considerations when modelling the SED of some blazars, used
as illustrative examples, and we check if our scenario can
reproduce the phenomenological blazar sequence (Fossati et al. 1998).
We use a cosmology with $h_0=\Omega_\Lambda=0.7$ and $\Omega_{\rm M}=0.3$.
We also use the notation $Q=10^xQ_X$ in cgs units, unless noted otherwise.
\section{Setup of the model}
Our model is characterised by the following setup.
The accretion disk extends from $R_{\rm in}=3 R_{\rm S}$ to
$R_{\rm out}=500 R_{\rm S}$ ($R_{\rm S}$ is the Schwarzschild\ radius)
and is producing a total luminosity
$L_{\rm d}=\eta \dot M c^2$, where $\dot M$ is the accretion rate and $\eta$
is the accretion efficiency.
Locally, its emission is black--body, with a temperature
\begin{equation}
T^4 \, =\, { 3 R_{\rm S} L_{\rm d } \over 16 \pi\eta\sigma_{\rm MB} R^3 }
\left[ 1- \left( {3 R_{\rm S} \over R}\right)^{1/2} \right]
\end{equation}
Below and above the accretion disk there is a hot corona, emitting UV
and X--rays with a luminosity $L_X=f_X L_{\rm d}$.
For simplicity, the corona is assumed to be homogeneous between 3 and 30 Schwarzschild\ radii.
The spectrum is assumed to be a cut--off power law:
$L_X(\nu)\propto \nu^{-\alpha_X}\exp(-\nu/\nu_{\rm c})$.
The broad line region (BLR) is assumed to be a shell located at a distance
\begin{equation}
R_{\rm BLR} \, =\, 10^{17} \, L_{\rm d, 45}^{1/2}\,\, {\rm cm}
\end{equation}
It reprocesses a fraction $f_{\rm BLR}$ of $L_{\rm d}$ in lines, especially
the hydrogen Lyman--$\alpha$ line, and continuum.
Following Tavecchio \& Ghisellini (2008), we assume that
the spectral shape of the BLR observed in the comoving frame is a black--body
peaking at a factor $\Gamma$ times the (rest frame) frequency of the Lyman--$\alpha$ line.
We also assume the presence of a torus (see B{\l}azejowski et al. 2000;
Sikora et al. 2002), at a distance
\begin{equation}
R_{\rm IR} \, =\, 2.5\times 10^{18} \, L_{\rm d, 45}^{1/2}\,\, {\rm cm}
\end{equation}
reprocessing a fraction $f_{\rm IR}$ of the disk radiation in the infrared.
Note that both $R_{\rm BLR}$ and $R_{\rm IR}$ scale as the square root
of $L_{\rm d}$: this implies that, in the lab frame, the radiation energy densities
of these two components are constant, as long as $R_{\rm diss}$ is smaller than
these two radii.
We emphasise that our treatment of the torus emission is approximate:
it is likely that the torus itself is a complex structure, possibly clumpy
(Nenkova et al. 2008) with a range of radii, extending also quite close to
the black hole, where the temperature is just below dust sublimation
(i.e. $\sim 1500$ K).
Our approach follows in part the results of Cleary et al. (2007),
finding weak signs of hot dust emission in the studied spectra, and
partly is dictated by simplicity.
The emitting region is moving with a velocity $\beta c$ corresponding to a bulk
Lorentz factor $\Gamma$.
We call $R_{\rm diss}$ the distance of the dissipation region from the black hole.
We consider either a constant $\Gamma$ or include an acceleration phase of the kind
(see e.g. Komissarov et al. 2007; Vlahakis \& K\"onigl 2004):
\begin{equation}
\Gamma \, =\, \min\left[\Gamma_{\rm max},
\left({R\over 3R_{\rm S}}\right)^{1/2}\right]
\end{equation}
When the acceleration phase is taken into account, the jet is assumed
to be parabolic in shape, becoming conical when $\Gamma$ reaches
its maximum value (see e.g. Vlahakis \& K\"onigl 2004).
Calling $r$ the cross sectional radius of the jet, and $R$ the distance
from the black hole, we have
\begin{eqnarray}
r \, &=&\, \phi\, R^{1/2}, \quad \Gamma \le \Gamma_{\rm max} \nonumber \\
r \, &=&\, \psi \, R, \quad\quad\, \Gamma \ge \Gamma_{\rm max}
\end{eqnarray}
where $\psi$ is the semi--aperture angle of the jet in its conical part,
and the constant $\phi$ is fixed by assuming that at the start of the jet
we have $r_o=R_0=3R_{\rm S}$.
The parabolic and conical parts of the jet connect at
$R=3R_{\rm S}\Gamma^2_{\rm max}$.
The total power carried by the jet, $P_{\rm j}$, was assumed
in Ghisellini \& Tavecchio (2008) to be related to
the mass accretion rate, i.e. $P_{\rm j}=\eta_{\rm j} \dot M c^2$.
Since Celotti \& Ghisellini (2008)
found that $P_{\rm j}$ is greater than
the disk accretion luminosity, i.e. $P_{\rm j} \lower .75ex \hbox{$\sim$} \llap{\raise .27ex \hbox{$>$}} L_{\rm d}$,
we assumed (in Ghisellini \& Tavecchio 2008)
that $\eta_{\rm j}$ was greater than the
corresponding efficiency of transforming the accretion
rate $\dot M$ in disk luminosity.
When fitting the data of specific sources,
we do not specify a priori the total jet luminosity,
which is instead a result of the modelling
(once we assume how many protons there are for each emitting
lepton).
Therefore $P_{\rm j}$ is not an input parameter in our scheme,
but it is a quantity derived a posteriori.
What we will specify, instead, is the power injected
in the dissipation region, that is related to the power carried
by the jet in relativistic electrons (and by cold protons,
once we specify how many protons there are per emitting electron).
We assume a value of the magnetic field in the dissipation region,
that corresponds to a Poynting flux $P_{\rm B}$.
If the jet is magnetically accelerated, then the initial (i.e. close to $R_0$)
$P_{\rm B}$ should be of the same order of $P_{\rm j}$, becoming
$P_{\rm B}=\epsilon_{\rm B}P_{\rm j}=$const when $\Gamma$ reaches its maximum value.
To describe the profile of $P_{\rm B}$ we assume the
following prescription:
\begin{equation}
P_{\rm B} \, =\, \pi r^2 \Gamma^2 c U_{\rm B} \,
=\, P_{\rm j} \left[ 1- {\Gamma\beta \over \Gamma_{\rm max}\beta_{\rm max} }
\left(1-\ \epsilon_{\rm B}\right)\right]
\end{equation}
In this way $P_{\rm B}=P_{\rm j}$ initially, becoming a
constant fraction
$\epsilon_{\rm B}$ of $P_{\rm j}$ when the jet is conical.
Here $U_{\rm B}=B^2/(8\pi)$ is the magnetic energy density.
The energy distribution of the particles responsible for the emission
is derived through the continuity equation, assuming
a continuous injection of particles throughout the source lasting
for a finite time. This time is the light crossing time
$t_{\rm cross}=r_{\rm diss}/c$, where $r_{\rm diss}$ is the
size of the emitting blob, located at the distance $R_{\rm diss}$ from
black hole.
We always calculate the particle distribution at this time.
The reason for this approach is suggested by the fast variability
shown by blazars, indicating that the release of energy
is short and intermittent.
Besides, we believe that this approach is the simplest that can
nevertheless describe in some detail the particle distribution.
In fact, it allows to neglect: i) adiabatic losses
(important after $r_{\rm diss}/c$, which is also the time needed
to double the radius);
ii) particle escape (again important for times longer than $r_{\rm diss}/c$) and
iii) the changed conditions in the emitting region (since the
source is travelling and expanding, the magnetic field changes).
High energy particles can radiatively cool in a time shorter than
$t_{\rm cross}$. Let us call $\gamma_{\rm cool}$ the energy
of those particles halving their energy in a time $t_{\rm cross}$.
Above $\gamma_{\rm cool}$, and at $t=t_{\rm cross}$,
the particle energy distribution $N(\gamma, t_{\rm cross})$
can be found by solving
\begin{equation}
{\partial \over \partial \gamma }\left[ \dot \gamma N(\gamma, t_{\rm cross}) \right] + Q(\gamma)
+P(\gamma)\, =\, 0
\label{cont}
\end{equation}
where $\dot\gamma$ is the
cooling rate of a particle of energy $\gamma mc^2$, $Q(\gamma)$ is the source
term (i.e. the injection of primary particles) assumed constant in time, and
$P(\gamma)$ is the term corresponding to the electron--positron pairs that
are produced in photon--photon collisions.
The formal solution of Eq. \ref{cont} is
\begin{equation}
N(\gamma) \, =\, { \int_{\gamma}^{\gamma_{\rm max}} [Q(\gamma)+P(\gamma)] d\gamma
\over \dot \gamma }, \quad \gamma>\gamma_{\rm cool}
\label{ngamma}
\end{equation}
When electrons with $1<\gamma<\gamma_{\rm cool}$ do not cool in $t_{\rm cross}$
we approximate the low energy part of $N(\gamma)$ with
\begin{equation}
N(\gamma) \, \sim \, t_{\rm cross} [Q(\gamma)+P(\gamma)],
\quad \gamma<\gamma_{\rm cool}
\label{ngamma2}
\end{equation}
Note that, within our assumptions, the particle distribution of
Eq. \ref{ngamma} and Eq. \ref{ngamma2}
correspond to the maximum $N(\gamma)$.
The injection of primary particles $Q(\gamma)$ is a smoothly joining
broken power law:
\begin{equation}
Q(\gamma) \, = \, Q_0\, { (\gamma/\gamma_{\rm b})^{-s_1} \over 1+
(\gamma/\gamma_{\rm b})^{-s_1+s_2} }
\label{qgamma}
\end{equation}
where $\gamma_{\rm b}$ is a break energy.
The pair injection term $P(\gamma)$ corresponds to the $\gamma$--$\gamma \to e^\pm$
process only, and it is calculated with
the prescriptions given by Svensson (1987) and Ghisellini (1989).
The total power injected in the form of relativistic electrons,
calculated in the comoving frame, is
\begin{equation}
P^\prime_{\rm i} \, = \, m_{\rm e}c^2 V
\int_1^{\gamma_{\rm max}} \gamma Q(\gamma)d\gamma
\label{leprime}
\end{equation}
where $V=(4\pi/3)r_{\rm diss}^3$ is the emitting volume.
Note that this is not equivalent to the power that the jet transports
in the form of relativistic particles (as measured in the comoving frame),
since $P^\prime_{\rm i}$ includes also the energy that will be
emitted, and possibly transformed into pairs.
\section{Energy densities}
\begin{figure}
\psfig{figure=cartoon3.ps,width=9cm,height=9cm}
\caption{Cartoon illustrating the accretion disk, its X--ray corona,
the broad line region and
a schematic representation of the IR torus.
At the distance $R_{\rm diss}$ the jet is assumed to dissipate.
At this distance, here assumed to be outside the BLR, we
label the relevant angles for calculating the contribution of the BLR
radiation to the corresponding energy density.
}
\label{cartoon}
\end{figure}
\subsection{Direct disk radiation}
Each annulus of the accretion disk
is characterised by a different temperature and it is seen under
a different angle $\xi$ (with respect to the jet axis),
thus its radiation is boosted in a different way.
A stationary observer with respect to the black hole (lab frame)
will see a flux, integrated over all annuli, given by
\begin{equation}
F_{\rm d}(\nu) \,=\, 2\pi \int^1_{\mu_{\rm d}} I(\nu)d\mu \, = \,
2\pi\int^1_{\mu_{\rm d}} { 2h\nu^3/c^2 \over \exp[ h\nu /(kT)] -1} d\mu
\end{equation}
where $\mu=\cos\xi$, and $\mu_{\rm d}$ is given by
\begin{equation}
\mu_{\rm d} \, =\, [1+R^2_{\rm out} /R^2_{\rm diss} ]^{-1/2}
\end{equation}
In the comoving frame of the blob, frequency are transformed as:
\begin{equation}
\nu^\prime =b\nu, \qquad b\, \equiv\, \, \Gamma(1-\beta\mu)
\end{equation}
and solid angles transform as:
\begin{equation}
d\Omega^\prime\, =\, {d\Omega \over b^2}\, =\, 2\pi {d\mu \over b^2}
\end{equation}
where primed quantities are in the comoving frame.
The intensities as seen in the comoving frame transform as:
\begin{equation}
I^\prime_{\rm d} (\nu^\prime) \, =\, b^3 I_{\rm d}(\nu) \, =\, b^3 I_{\rm d}(\nu^\prime/b)
\end{equation}
The specific radiation energy density
seen in the comoving frame is
\begin{equation}
U^\prime_{\rm d}(\nu^\prime) \, =\, {1\over c}
\int I^\prime_{\rm d} (\nu^\prime) d\Omega^\prime \, =
{2\pi\over c}
\int^1_{{\mu_{\rm d}}} {I^\prime_{\rm d}(\nu^\prime) \over b^2} d\mu
\end{equation}
\subsection{Radiation from the X--ray corona}
According to our assumptions, the
total radiation energy density $U^\prime_X$ of this component is
(see e.g. Ghisellini \& Madau 1996):
\begin{eqnarray}
U^\prime_X &=&
{f_X L_{\rm d} \Gamma^2 \over \pi R_X^2 c}
\left[ 1-\mu_X-\beta(1-\mu_X^2)+{\beta^2\over 3}(1-\mu_X^3) \right] \nonumber \\
\mu_X &=& [1+R^2_X/R^2_{\rm diss}]^{-1/2}
\end{eqnarray}
where $R_X$ is the extension of the X--ray corona.
\subsection{BLR radiation}
Within $R_{\rm BLR}$ the corresponding energy density seen in the comoving frame
can be approximated as (Ghisellini \& Madau 1996):
\begin{equation}
U^\prime_{\rm BLR} \, \sim \, { 17 \Gamma^2 \over 12} \, { f_{\rm BLR} L_{\rm d}
\over 4\pi R_{\rm BLR}^2 c}
\,\, \quad R_{\rm diss}< R_{\rm BLR}
\label{ublr1}
\end{equation}
At distances much larger than $R_{\rm BLR}$, and calling $\mu=\cos\alpha$,
we have
(see the cartoon in Fig. \ref{cartoon})
\begin{eqnarray}
U^\prime_{\rm BLR} \, &\sim& \, \,
{ f_{\rm BLR} L_{\rm d} \over 4\pi R_{\rm BLR}^2 c} \, {\Gamma^2 \over 3\beta}
\, [ 2(1-\beta\mu_1)^3- (1-\beta\mu_2)^3 \nonumber \\
&-&(1-\beta)^3)]
\,\, \qquad R_{\rm diss}\gg R_{\rm BLR} \nonumber \\
\mu_1 \, &=&\, [1+R^2_{\rm BLR}/R^2_{\rm diss}]^{-1/2} \nonumber \\
\mu_2 \, &=&\, [1 - R^2_{\rm BLR}/R^2_{\rm diss}]^{1/2}
\label{ublr2}
\end{eqnarray}
For $R_{\rm diss} \lower .75ex \hbox{$\sim$} \llap{\raise .27ex \hbox{$>$}} R_{\rm BLR}$ the exact value of $U^\prime_{\rm BLR}$
depends on the width of the BLR, which is poorly known.
For this reason, in the range $R_{\rm BLR}<R_{\rm diss}<3R_{\rm BLR}$ we
use a simple (power--law) interpolation.
The BLR is assumed to ``reflect" (Compton scatter) a fraction $f_{\rm BLR,X}$
(of the order of 1 per cent) of the corona emission.
The existence of this diffuse X--ray radiation is a natural outcome
of photo--ionisation models for the BLR (see e.g. Tavecchio \& Ghisellini 2008,
Tavecchio \& Mazin 2009).
The assumed value, $f_{BLR, X}\sim 0.01$,
is the average value found for typical parameters of the clouds.
\subsection{Radiation from the IR torus}
This component scales as $U_{\rm BLR}$, but substituting $R_{\rm BLR}$
with $R_{\rm IR}$. We have
\begin{equation}
U^\prime_{\rm IR} \, \sim \, { f_{\rm IR} L_{\rm d}\, \Gamma^2
\over 4\pi R_{\rm IR}^2 c}
\,\, \quad R_{\rm diss}< R_{\rm IR}
\end{equation}
For $R_{\rm diss}>R_{\rm IR}$ we have the same behaviour as in Eq. \ref{ublr2},
but with $R_{\rm IR}$ replacing $R_{\rm BLR}$.
In a $\nu F_\nu$ plot the (lab frame) peak frequency of this component is assumed
to be at $\nu_{\rm IR} = 3\times 10^{13}$ Hz (see Cleary et al. 2007),
independent of the disk luminosity, since $R_{\rm IR}$ scales as $L_{\rm d}^{1/2}$.
The corresponding temperature is $T_{\rm IR} = h\nu_{\rm IR}/(3.93 k)$
(we must use the factor 3.93, instead of the usual 2.82, because we are
using the peak frequency in $\nu F_\nu$).
In the comoving frame this corresponds to
\begin{equation}
T^\prime_{\rm IR} \, \sim\, 370\, b\, \,\, {\rm K}
\end{equation}
\subsection{Radiation from the host galaxy bulge}
The bulge of the galaxy hosting the blazar can be a non--negligible
emitter of ambient optical radiation (see e.g.
(Stawarz, Sikora \& Ostrowski 2003).
Within the bulge radius $R_{\rm star}$ emitting a luminosity $L_{\rm star}$
we have
\begin{equation}
U^\prime_{\rm star} \, = \, \Gamma^2 \, {L_{\rm star} \over 4\pi R_{\rm star}^2 c}
\end{equation}
As an order of magnitude estimate, we have
$U^\prime_{\rm star} \sim 10^{-10} \Gamma^2 $ erg cm$^{-3}$
using $L_{\rm star} = 3 \times 10^{44}$ erg s$^{-1}$
produced within a bulge radius $R_{\rm star}\sim $1 kpc.
When $R_{\rm diss}>R_{\rm star}$, $U^\prime_{\rm star}$ decreases
in an analogous way as $U^\prime_{\rm IR}$ and $U^\prime_{\rm BLR}$
(once we substitute $R_{\rm star}$ to $R_{\rm IR}$ or $R_{\rm BLR}$).
\subsection{Radiation from the cosmic background}
The energy density of the Cosmic Background Radiation (CMB),
as seen in the comoving frame, is
\begin{equation}
U^\prime_{\rm CMB} \, =\, a T_0^4 \Gamma^2 (1+z)^4
\end{equation}
where $a=7.65\times 10^{-15}$ erg cm$^{-3}$ deg$^{-4}$ and
$T_0=2.7$ K is the temperature of the CMB now (i.e. at $z=0$).
\subsection{Magnetic field}
Following our prescriptions concerning the power carried in the form
of Poynting flux, we have
\begin{equation}
U^\prime_{\rm B} \, = \, {P_{\rm B} \over \pi r^2_{\rm diss} \Gamma^2 c } \, =\,
{P_{\rm j} \over \pi r^2_{\rm diss} \Gamma^2 c }
\left[ 1- {\Gamma\beta \over \Gamma_{\rm max}\beta_{\rm max} }
\left(1-\ \epsilon_{\rm B}\right)\right]
\end{equation}
The magnetic field $B$ scales as $1/R_{\rm diss}$
both in the acceleration and in the coasting (i.e. $\Gamma=$const)
phases, but with a different normalisation.
\subsection{Internal radiation}
This component corresponds mainly to the radiation produced by the blob
that can be efficiently scattered through the inverse Compton process.
This cannot be calculated without specifying the relativistic particle
distribution.
If the emitting volume $V$ is a sphere,
the synchrotron radiation energy density formally is:
\begin{eqnarray}
U^\prime_{\rm syn} \, &=& \, { V \over 4\pi r_{\rm diss}^2 c} m_{\rm e} c^2
\int N(\gamma) \dot \gamma_{\rm syn} d\gamma
\nonumber \\
&=& { 4 \over 9 } [n r_{\rm diss} \sigma_{\rm T}] U^\prime_{\rm B}
{ \int N(\gamma) \gamma^2 d\gamma \over \int N(\gamma) d\gamma }
\nonumber \\
&=& { 4 \over 9 } [n r_{\rm diss} \sigma_{\rm T} \langle \gamma^2 \rangle ]
U^\prime_{\rm B}
\, = \, { 4 \over 9 }\, y \, U^\prime_{\rm B}
\end{eqnarray}
where $\sigma_{\rm T}$ is the Thomson cross section, $n$ is the number
density of the emitting particles and
$y\equiv \sigma_{\rm T} n r_{\rm diss} \langle\gamma^2 \rangle$ is the relativistic
Comptonization parameter.
This parametrisation follows the one in Ghisellini \& Tavecchio (2008).
From the fits to a sample of blazars performed in Celotti \& Ghisellini (2008)
we have that $y$ is in the range 0.1--10.
Since the found distribution of $y$ values is rather narrow, one
has a first order estimate of what is the internal radiation through the
value and profile of the magnetic energy density $U_{\rm B}$.
\begin{figure}
\hskip -0.2 cm
\psfig{figure=u_r2.ps,width=9cm,height=9.7cm}
\vskip -1 cm
\psfig{figure=u_r3.ps,width=9cm,height=9.7cm}
\vskip -0.5 cm
\caption{
Top panel: Comparison of different energy densities
as measured in the comoving frame.
The moving blob is assumed to have a bulk Lorentz factor
$\Gamma =\min[15, (R_{\rm diss}/3R_{\rm S})^{1/2}]$.
The black hole has a mass $M=10^9 M_\odot$.
The different contributions are labelled.
The disk emits as a blackbody, and extends from 3 to 500 Schwarzschild\ radii.
The two sets of lines correspond to two disk luminosities:
$10^{45}$ and $10^{47}$ erg s$^{-1}$.
The radius of the broad line region is assumed to scale
with the disk luminosity as
$R_{\rm BLR}=10^{17}L_{\rm d, 45}^{1/2}$ cm.
The radius of the IR torus scales as
$R_{\rm IR} = 2.5\times 10^{18}L_{\rm d, 45}^{1/2}$ cm.
The X--ray corona is assumed to be homogeneous,
to extend up to 30 Schwarzschild\ radii and to emit 10 per cent of the
disk luminosity.
The contribution of the BLR between 1 and 3 $R_{\rm BLR}$
depends on the unknown width of the BLR itself (dotted line).
The magnetic energy density (long dashed lines) is calculated
assuming $P_{\rm j}=L_{\rm d}$ and $\epsilon_{\rm B}=0.1$.
Bottom panel: as above, but assuming a constant $\Gamma=15$
all along the jet.
}
\label{ur}
\end{figure}
\begin{figure}
\hskip -0.2 cm
\psfig{figure=u_rout.ps,width=9cm,height=9.7cm}
\vskip -0.5 cm
\caption{
The energy density as seen in the comoving frame of the
blob, moving with a bulk Lorentz factor
$\Gamma =\min[15, (R_{\rm diss}/3R_{\rm S})^{1/2}]$.
The black hole has a mass $M=10^9 M_\odot$, and the disk
emit $L_{\rm disk}=10^{46}$ erg s$^{-1}$.
The different contributions are labelled.
This figure shows the effect of changing the outer radius
of the accretion disk, from 150 to 5000 $R_{\rm S}$,
as indicated by the labels and arrows.
As the outer radius increases, $U^\prime_{\rm d}$ increases
at distances $R_{\rm diss}\sim R_{\rm out}$, but there the
disk contribution is overtaken by $U^\prime_{\rm BLR}$
and $U^\prime_{\rm IR}$.
}
\label{urout}
\end{figure}
\subsection{Comparing the different components}
We have calculated the contributions to the energy density given by the
different components as a function of $R_{\rm diss}$,
plotting them in Fig. \ref{ur}.
For these cases, we have chosen a black hole
mass $M=10^9 M_\odot$, an accretion disk extending from 3$R_{\rm s}$
to 500$R_{\rm S}$, an accretion efficiency
$\eta=0.08$, $f_{\rm BLR} = f_X=0.1$, $f_{\rm IR}=0.5$.
We plot the results for two values of $L_{\rm d}$: $10^{45}$ and $10^{47}$
erg s$^{-1}$.
In the top panel we show the case of an accelerating jet, whose
bulk Lorentz factor is $\Gamma=\min[15, (R_{\rm diss}/3R_{\rm S})^{1/2}]$.
In the bottom panel we assume a constant $\Gamma=15$ along the entire jet.
As can be seen, the dominant energy density is different at different
$R_{\rm diss}$.
As a rule, all the external radiation energy densities drop when
$R_{\rm diss}$ is greater than the corresponding typical size of
the structure producing the seed photons.
Particularly interesting is the comparison between
$U^\prime_{\rm d}$ and $U^\prime_{\rm BLR}$.
The distance above which $U^\prime_{\rm BLR}>U^\prime_{\rm d}$
depends upon the disk luminosity, since the BLR adjusts its radius so
to give a constant (in the lab frame) energy density.
So for $L_{\rm d}=10^{47}$ erg s$^{-1}$ the energy density from the BLR dominates
only above $10^{17}$ cm, equivalent to $\sim 300 R_{\rm S}$,
while for $L_{\rm d}=10^{45}$ erg s$^{-1}$ it starts to dominate about
three times closer, when $\Gamma$ has not reached yet its maximum value.
Note that, for the shown cases, $U^\prime_{\rm B}$ dominates
over the external radiation energy density only
at the start of the jet and up to 100--300 $R_{\rm S}$,
where $U^\prime_{\rm BLR}$ takes over.
In this particular examples, we assumed $P_{\rm B}=0.1 L_{\rm d}$.
Fig. \ref{urout} illustrates the effects of changing the outer radius of
the accretion disk, from $R_{\rm out}=150$ to $5000$ Schwarzschild\ radii.
As expected, $U^\prime_{\rm d}$ remains the same at small and very high
distances, and increases with $R_{\rm out}$ in between, at distances
comparable with $R_{\rm out}$.
The increase of $U^\prime_{\rm d}$ is modest, and occurs when the
external radiation energy density is dominated by the BLR and torus
components.
\begin{figure}
\vskip -0.5cm
\hskip -0.2cm
\psfig{figure=udisk.ps,width=9cm,height=10cm}
\vskip -1 cm
\caption{
The radiation energy density $U^\prime_{\rm d}$
(as seen in the comoving frame) produced by the direct radiation
of the accretion disk as a function of $\Gamma$, for different
distances $R_{\rm diss}$, for a total disk luminosity $L_{\rm d}=10^{46}$
erg s$^{-1}$ and a black hole mass $M=10^9 M_\odot$.
For comparison we show the radiation energy density observed {\it within} the BLR.
It can be directly compared with $U^\prime_{\rm d}$
only for $R_{\rm diss}<R_{\rm BLR}$.
}
\label{ug}
\end{figure}
In Fig. \ref{ug} we show $U^\prime_{\rm d}$ as a function
of $\Gamma$ for different locations of the dissipation region.
Of course the largest $U^\prime_{\rm d}$ are obtained when
the source is very close to the accretion disk.
For each $R_{\rm diss}$ there is a value of $\Gamma$
that minimises $U^\prime_{\rm d}$.
To understand this (somewhat anti--intuitive) behaviour
we must recall that the boosting factor $b=\Gamma(1-\beta\mu)$
has a minimum for $\beta=\mu$.
This corresponds, in a comoving frame, to $\mu^\prime=0$, i.e. an angle
of $90^\circ$.
Therefore, when the external radiation is all coming from a ring,
we expect that in the comoving frame, the radiation energy density
initially decreases increasing $\beta$, but when the photons
are starting to come from the forward hemisphere (i.e. $\mu^\prime<0$,
or, equivalently, $\mu<\beta$), $U^\prime_{\rm d}$ increases increasing $\beta$.
This explains the minima shown in Fig. \ref{ug}, once we
weight the above effect with the intensity produced at each ring.
\begin{figure}
\vskip -0.5cm
\hskip -0.2cm
\psfig{figure=urad_z.ps,width=9cm,height=10cm}
\vskip -1 cm
\caption{
The spectra of the radiation energy density (as seen in the
comoving frame) produced by the direct disk plus the
X--ray corona ($U^\prime_{\rm d}$, solid line),
the BLR ($U^\prime_{\rm BLR}$, including the ``reflected'' X--rays,
dashed line) and the IR torus ($U^\prime_{\rm IR}$, dotted line).
We have assumed a constant $\Gamma=15$, $M=10^9 M_\odot$ and
$L_{\rm d}=10^{45}$ erg s$^{-1}$.
The spectra are shown for 5 different $R_{\rm diss}$.
For $R_{\rm diss}=10$ and 100 Schwarzschild\ radii $U^\prime_{\rm BLR}(\nu^\prime)$
is the same, since for these cases $R_{\rm diss}<R_{\rm BLR}$.
The contribution of the IR emitting torus is the same up
to $R_{\rm diss}=10^3 R_{\rm S}$, and starts to decline
afterwards, when $R_{\rm diss}>R_{\rm IR}$.
}
\label{uradz}
\end{figure}
\subsection{Spectra of the external radiation}
Besides comparing the frequency integrated components
forming the radiation energy density in the comoving frame, it is
instructive to compare also their different spectra.
This is because the Doppler boost appropriate for each
component is not the same, and it changes for different
distances from the black hole.
In Fig. \ref{uradz} we show one illustrative example, calculated
for a blob moving with a constant $\Gamma=15$ and located at different $R_{\rm diss}$.
The disk is assumed to emit $L_{\rm d}=10^{45}$ erg s$^{-1}$ and the black hole
mass is $M=10^9 M_\odot$, corresponding to $R_{\rm S}=3\times 10^{14}$ cm.
With this disk luminosity, the BLR is located at $R_{\rm BLR}=10^{17}$ cm,
implying that for $R_{\rm diss}< R_{\rm BLR}$ the radiation energy density
from the BLR is the same.
For larger $R_{\rm diss}$ the total $U^\prime_{\rm BLR}$ decreases
(see Eq. \ref{ublr2}).
The radiation energy density from the disk and its corona
always decreases increasing $R_{\rm diss}$, being larger than
$U^\prime_{\rm BLR}$ for $R_{\rm diss}< 100R_{\rm S}\sim 3\times 10^{16}$ cm
(see also Fig. \ref{ur}).
The disk radiation dominates again for $R_{\rm diss} \lower .75ex \hbox{$\sim$} \llap{\raise .27ex \hbox{$>$}} \times 10^{18}$ cm
(see Fig. \ref{ur}).
For small $R_{\rm diss}$, the high energy X--ray photons coming
from the corona will be responsible of a large optical depth
for the $\gamma$--$\gamma$ process, as discussed below.
The IR energy density coming form the torus is constant
for all $R_{\rm diss}$ but for the largest value
(since $R_{\rm IR}=8.3\times 10^3
R_{\rm S}=2.5\times 10^{18}$ cm).
This component will be larger than $U^\prime_{\rm d}$ and
$U^\prime_{\rm BLR}$ for $R_{\rm diss}$ greater than $\sim 500R_{\rm S}
\sim 2\times 10^{17}$ cm.
\subsection{Input parameters}
Having included many components of external radiation,
and having tried to link the jet emission to the
accretion disk luminosity and the black hole mass,
we necessarily have a large numbers of parameters
for a specific model.
It can be useful to summarise them here, dividing them into
two separated lists.
In the latter we have those parameters that we fix for
all models, based on physical considerations or
a priori knowledge, or that do not influence the model SED
(apart from pathological cases); in the former we list
the important input parameters.
\begin{itemize}
\item
$M$: the mass of the black hole;
\item
$L_{\rm d}$: the luminosity of the accretion disk.
\item
$R_{\rm diss}$: the distance from the black hole where the
jet dissipates.
\item
$P^\prime_{\rm i}$: the power, calculated in the jet rest frame,
injected into the source in the form of relativistic electrons.
\item
$B$: the magnetic field on the dissipation region;
\item
$\Gamma_{\rm max}$: the value of bulk Lorentz factor after the acceleration
phase, in which the bulk Lorentz factor increases as
$[R/(3R_{\rm S})]^{1/2}$.
\item
$\theta_{\rm v}$: the viewing angle.
\item
$\gamma_{\rm b}$: value of the random Lorentz factor at which
the injected particle distribution $Q(\gamma)$ changes slope.
\item
$\gamma_{\rm max}$: value of maximum random Lorentz factor of
the injected electrons.
\item
$s_2$: slope of $Q(\gamma)$ above $\gamma_{\rm b}$.
\end{itemize}
These amount to 10 parameters.
If we use $\theta_{\rm v}\sim 1/\Gamma$, as discussed in \S \ref{caveats},
then the number of relevant parameters decreases to 9.
Note that we do not consider $P_{\rm j}$ as an input parameter,
since it is derived once calculating the total electron energy density
in the dissipation region, and assuming a given number of (cold) proton
per emitting electron.
The following parameters are either unimportant or are well constrained:
\begin{itemize}
\item
$R_X$: the extension of the X--ray corona.
We fix it to $30R_{\rm S}$.
\item $R_{\rm out}$: the outer radius of the accretion disk.
For the shown illustrative examples, we fixed it at 500 Schwarzschild\ radii,
and we have shown that changing it has a modest influence on $U^\prime_{\rm d}$.
\item
$s_1$: slope of $Q(\gamma)$ below $\gamma_{\rm b}$.
It is bound to be very flat (i.e. $-1<s_1<1$).
\item
$\alpha_X$: the spectral shape of the coronal X--ray flux.
We fix it to $\alpha_X=1$.
\item
$h\nu_{\rm c}$: the high energy cut--off of the X--ray coronal flux.
We fix it at 150 keV.
\item
$L_X$: total X--ray luminosity of the corona.
We fix it to $L_X=0.3L_{\rm d}$.
\item
$f_{\rm BLR}$: fraction of $L_{\rm d}$ intercepted by the BLR and
re--emitted in broad lines.
We fix it to $f_{\rm BLR}=0.1$
\item
$f_{\rm BLR,X}$: fraction of $L_X$ of the corona scattered by the BLR.
We fix it to $f_{\rm BLR,X}=0.01$
\item
$f_{\rm IR}$: fraction of $L_{\rm d}$ intercepted by the torus
and re--emitted in IR.
We fix it to $f_{\rm IR}=0.5$.
\item
$\psi$: semi--aperture angle of the jet.
We fix it to $\psi=0.1$.
\item $U_{\rm star}$: this is the radiation energy density within the bulge of the
host galaxy. We approximated it with the constant value $U_{\rm star}=10^{-10}$ erg cm$^{-3}.$
\end{itemize}
\section{The role of pairs}
At large distances from the black hole, the density
of the photon targets for the $\gamma$--$\gamma \to e^\pm$ process
is small, with few pairs being produced inside the emitting region.
At small distances, instead, the presence of the X--ray corona
makes pairs easy to create.
With our assumptions, we can see the effects of the reprocessing of the
spectrum due to pairs.
Whenever pairs are produced
inside the emitting region at small $R_{\rm diss}$,
the radiative cooling rates are large,
implying that all particles cool in one light crossing time.
This in turn implies a particle distribution of the form given
by Eq. \ref{ngamma}, with $\gamma_{\rm cool}\sim 1$.
One necessary ingredient for making pairs important is the relative
amount of power emitted above the energy threshold $m_ec^2$ (in the comoving frame).
If this power is a small fraction of the total, the
reprocessing will be modest, even if all the $\gamma$--rays get absorbed.
There is then the possibility to have a large jet power dissipated close
to the accretion disk, but with electrons of lower energies, whose emission
is mostly below threshold.
Furthermore, even if the electron energies are large, the bulk Lorentz
factor could be small, especially very close to the accretion disk,
where presumably the jet just starts to accelerate.
In this case the magnetic energy density becomes more dominant
with respect to the disk radiation, implying a relatively modest high energy
emission, and furthermore the whole spectrum is much less boosted.
In the following we will consider the case of two jets
with the same parameters, except that the first is moving with
a large $\Gamma$ already at small $R_{\rm diss}$,
the other, instead, is accelerating.
\begin{figure}
\vskip -0.5cm
\psfig{figure=repro_pairs.ps,width=9cm,height=8cm}
\vskip -0.5cm
\psfig{figure=comov_pairs.ps,width=9.cm,height=10.5cm}
\vskip -0.5cm
\caption{
Top panel:
example of how reprocessing due to electron--positron pairs
can be important, if the dissipation region is located too close
to the accretion disk and its X--ray corona, {\it and} if this
region produces $\gamma$--rays above the pair--production threshold.
The two humps at low energies correspond to the flux produced by the
accretion disk and the IR flux from the torus.
See Tab. \ref{para} for the parameters used to construct the shown SED.
Mid panel: the SED as observed in the comoving frame.
Bottom panel:
the particle distribution $\gamma^3\tau(\gamma)$.
For this example, the maximum particle Lorentz factor is
$\gamma_{\rm max}=10^4$ and $\gamma_{\rm b}=10^3$.
A redshift $z=3$ has been assumed.
}
\label{pairs}
\end{figure}
Consider Fig. \ref{pairs}: it
shows the effects of including the $\gamma$--$\gamma \to e^\pm$
on the final spectrum.
In the middle panel of Fig. \ref{pairs} the same spectra are shown as seen in
the comoving frame.
In this case on the y--axis we plot the specific compactness, defined as
$\ell^\prime(\nu^\prime) \equiv
\sigma_{\rm T} L^\prime(\nu^\prime)/ (r_{\rm diss} m_e c^3)$.
The bottom panel of this figure shows the particle distribution
(considering or not the produced pairs) in the form $\gamma^3\tau(\gamma)\equiv
\gamma^3 \sigma_{\rm T} r_{\rm diss} N(\gamma)$.
The $\gamma^3$ factor makes the $\gamma^3\tau(\gamma)$ distribution to have
a peak: electrons at this peak are the ones producing the peaks in the
$\nu L_\nu$ synchrotron and inverse Compton spectra.
The parameters used for this example are listed in Tab. \ref{para}.
One can see that pairs redistribute the power from high to low frequencies.
The $N(\gamma)$ distribution steepens at low energies (where
pairs contribute the most), and this softens the emitted spectrum
especially in the X--ray range.
The reprocessing of the X--ray spectrum
is stronger than at lower frequencies because the low energy
electrons emit, by synchrotron, self absorbed radiation, whose
shape and normalisation are largely independent of the slope of
$N(\gamma)$.
Instead, in the X--ray range, the presence of the intense optical--UV
radiation field coming from the accretion disk implies that the dominant
process is the external Compton (EC) scattering with these photons.
In turn, this implies that the radiation we see in the soft and hard
X--ray range is due to relatively low energy electrons.
In fact, the scattered frequency $\nu \sim \gamma^2\Gamma^2\nu_0$,
where $\nu_0$ is the (observed) frequency of the seed photon.
Therefore, at 10 keV, we are observing the radiation produced by electrons
with $\gamma \sim 50 \, \nu_{0,15}^{-1/2}\Gamma^{-1} \sim$ a few.
\begin{figure}
\hskip -0.2 cm
\vskip -0.3 cm
\psfig{figure=elle.ps,width=9cm,height=9.7cm}
\vskip -0.5cm
\caption{
The compactness
$\ell^\prime \equiv 4\pi U^\prime \sigma_{\rm T} r_{\rm diss}/ (m_{\rm e}c^2)$
as measured in the comoving frame of a source of size
$r_{\rm diss}$ as a function of $R_{\rm diss}$.
We use the same parameters used to construct the top panel
of Fig. \ref{ur}.
}
\label{elle}
\end{figure}
In the above example we have, on purpose, assumed that
the dissipation region, although very close to the
black hole, is already moving at large speeds,
corresponding to a bulk Lorentz factor of 10.
We may ask, instead, what happens if the same kind of
dissipation occurs while the jet is still accelerating,
and has therefore a relatively small bulk velocity when it is
close to the black hole.
To this aim, Fig. \ref{elle} shows the compactness $\ell^\prime$,
as measured in the comoving frame, as a function of $R_{\rm diss}$,
for an accelerating jet.
We define this comoving compactness as:
\begin{equation}
\ell^\prime\, \equiv \, {4\pi U^\prime \sigma_{\rm T} r_{\rm diss}
\over m_{\rm e}c^2 }
\end{equation}
where $r_{\rm diss}$ is the size of the blob.
The term $U^\prime$ is the external radiation energy densities.
The compactness is directly associated to the optical depth of the
$\gamma$--$\gamma \to e^\pm$ process, and it describes the absorption
probability of a $\gamma$--ray photon while it is inside the blob.
In other words, this definition does not account for the absorption
a photon can suffer while it has already escaped the blob.
It also neglects the contribution of the internal radiation energy
density due to the synchrotron and the synchrotron self Compton (SSC)
emission.
Fig. \ref{elle} shows that there are three distances where the
compactness is large:
\begin{enumerate}
\item
At the base of the jet the main targets
for the $\gamma$--$\gamma$ process are the X--ray photons
produced by the corona.
They can interact with $\gamma$--ray photons just above threshold,
implying that the absorption is important. However, at these distances,
the magnetic energy density is larger than the external radiation energy
density (dominated by the disk radiation), and this implies a modest
external Compton scattering, and a modest production of $\gamma$--rays.
\item
The compactness becomes large again for $R_{\rm diss}\sim R_{\rm BLR}$.
Since the main contribution to $U^\prime$ is due to the BLR photons,
(seen at UV frequencies in the comoving frame), the absorbed $\gamma$--rays
will have GeV energies (see next subsection for more details).
\item
The third relevant range of distances corresponds to
$R_{\rm diss}\sim R_{\rm IR}$.
The main targets for the absorption process are the IR photons produces
by the torus, absorbing TeV $\gamma$--rays.
\end{enumerate}
Fig. \ref{rpairs} shows three SED corresponding to
$R_{\rm diss}=10$, 100 and $10^3R_{\rm S}$, with an
electron injection function equal to the one used in Fig. \ref{pairs},
but assuming $\Gamma=\min[15, (R_{\rm diss}/3R_{\rm S})^{1/2}]$.
Tab. \ref{para} lists the parameters.
We can see the dramatic changes with respect to Fig. \ref{pairs},
and also among the three shown models.
Pair production, although present (compare solid and dashed lines),
is marginal and there is no noticeable reprocessing of the primary spectrum.
This is due to the much reduced importance of the external radiation
when $\Gamma$ is small.
As a consequence, the spectrum is dominated by the synchrotron emission,
with a small fraction of the power being emitted above
the pair production energy threshold.
This in turn implies that early (i.e. close to the black hole)
dissipation is not an efficient mechanism to produce
electron--positron pairs.
Since a small $\Gamma$ means small Doppler boosting, the
produced spectra at small distances are undetectable,
overwhelmed by the emission of more distant and fastly
moving components.
When $\Gamma$ is large, therefore for the $R_{\rm diss}=10^3R_{\rm S}$ case,
the relevant absorbing photons are those of the broad line region,
with a very modest contribution from the coronal X--rays reflected by the BLR
material.
The overall effect is marginal.
We can conclude that a ``canonical" jet can dissipate
part of its kinetic energy even at small distances.
One way to avoid strong pair production is to have
small electron energies, emitting a small power above
the pair production threshold.
Alternatively, if the jet is accelerating, and close to
the black hole it has a modest $\Gamma$--factor,
the produced radiation is hardly observable because
the Doppler boost is limited.
In relative terms, the synchrotron luminosity should be
dominant, with a small fraction of power being emitted
at high energies.
\begin{figure}
\vskip -0.5cm
\psfig{figure=rpairs.ps,width=9cm,height=8cm}
\vskip -0.5cm
\caption{
Same as Fig. \ref{pairs}, but for an accelerating jet and
for different $R_{\rm diss}$.
In this case, in the comoving frame of the jet, the magnetic energy
density is dominant for $R_{\rm diss}=$10 and 100$R_{\rm S}$.
Furthermore, the much reduced Doppler boosting for small values
of $R_{\rm diss}$ implies a very small received flux.
There is some $\gamma$--$\gamma$ absorption (compare solid and
dashed lines, the latter corresponding to switching off the
$\gamma$--$\gamma$ process), but involving a very small amount of power.
Consequently, pairs reprocessing is unimportant.
Light grey lines (green in the electronic version) correspond to the
non--thermal spectrum, black lines include the contribution from the
accretion disk, its X--ray corona and the emission from the IR torus.
}
\label{rpairs}
\end{figure}
\subsection{Pair production versus Klein--Nishina effects}
In powerful blazars, the high energy flux, at $\sim$GeV energies, is
dominated by the inverse Compton process between high energy electrons
and external radiation.
In Tavecchio \& Ghisellini (2008) we pointed out that when the dominant
contribution to the seed photon for scattering is given by the BLR, the
hydrogen Lyman--$\alpha$ photons are the most prominent ones.
The fact that there is a characteristic frequency of the seed photons
allows to easily calculate when the Klein--Nishina effects
(decrease of the scattering cross section with energy) are important.
For completeness, we briefly repeat here the argument, with the
aim to extend it to cases in which $R_{\rm diss}$ is beyond the BLR, and
as a consequence, the relevant seed photons becomes the IR ones.
In the comoving frame, and as long as we are within the BLR,
The observed seed Lyman--$\alpha$ frequency is observed at
$\nu_{\rm L\alpha}^\prime=2\Gamma \nu_{\rm L\alpha}$.
To be in the Thomson scattering regime we require that
\begin{equation}
2\Gamma h\nu_{\rm L\alpha}\, < \, { m_{\rm e}c^2\over \gamma}
\end{equation}
If the random Lorentz factor $\gamma$ of the electron satisfies this
condition, the energy of the scattered photon is
\begin{equation}
\nu_{\rm KN} \, =\, {4\over 3}\gamma^2 \nu_{\rm L\alpha}
{2 \Gamma \delta \over 1+z} \, =\,
15\, {\delta \over \Gamma (1+z)} \,\, {\rm GeV}
\label{kn}
\end{equation}
Above this energy, the spectrum steepens due to the
decreased efficiency of the scattering process.
This may well mimic the effect of photon--photon absorption,
making it difficult to discriminate between the two effects.
Consider also that the Lyman--$\alpha$ photons are the
best targets for the photon--photon process at these energies
and in these conditions (namely, relatively far from the accretion disk,
but still within the BLR).
We can repeat the very same argument assuming that the relevant seed
photons are the ones produced by the IR torus. This occurs when the
dissipation region is beyond the BLR but within $R_{\rm IR}$.
In this case the relevant frequency is $\nu_{\rm IR}=3\times 10^{13}$ Hz
and we can rewrite Eq. \ref{kn} as
\begin{equation}
\nu_{\rm KN} \, =\, {4\over 3}\gamma^2 \nu_{\rm IR}{2 \Gamma \delta \over 1+z} \, =\,
1.2\, {\delta \over \Gamma (1+z)} \,\, {\rm TeV}
\label{kn2}
\end{equation}
This implies that, if we do see a spectrum which is unbroken even above $15/(1+z)$ GeV,
we can conclude that the dissipation region is beyond the BLR.
Also in this case, the ``intrinsic" absorption due to the $\gamma$--$\gamma$ process
is expected to become important at $\sim \nu_{\rm KN}$.
There is therefore a range, between $15/(1+z)$ and $\sim 1000/(1+z)$ GeV,
in which the intrinsic spectrum should not suffer from the Klein--Nishina
nor from the $\gamma$--$\gamma$ process due to ``internal" absorption.
These sources are thus the best candidates for studying the
photon--photon absorption due to the cosmic optical--UV background radiation.
\section{Dissipation at large distances}
Although there are strong indications that {\it most} of the
observed radiation originates in a well localised region of the
jet, there is also compelling evidence that the jet dissipates
also in other regions, larger and more distant from its base.
In fact, in the radio band, we do see bright knots at different
distances contributing to form the characteristic flat total
radio spectrum of the entire jet, and we do observed optical
and X--ray knots also at hundreds of kpc from the jet apex.
The power associated to these, often resolved, features is
a small fraction of the total observed jet bolometric luminosity,
but in specific bands the flux produced at larger distances can
be comparable to the flux produced in the most active region.
It is therefore instructive to calculate the predicted spectrum
as a function of distance, extending our analysis well beyond
the region of influence of the IR flux produced by the torus,
and therefore analysing the effects of the radiation energy
density of the cosmic background radiation (CMB).
This has already been pointed out as an important source of
seed photons for enhancing the inverse Compton flux of the
bright X--ray knots detected by the Chandra satellite at tens and
hundreds of kpc from the core of the jet (Tavecchio et al. 2000;
Celotti Ghisellini \& Chiaberge 2001).
Fig. \ref{cmb} shows the different contributions to the radiation energy
densities as seen in the comoving frame, compared to $U^\prime_{\rm B}$,
and including $U^\prime_{\rm CMB}$, important for large distances.
To construct this figure we have assumed $M=10^9 M_\odot$,
$L_{\rm d}=10^{47}$erg s$^{-1}$,
$P^\prime_{\rm i}=10^{44}$ erg s$^{-1}$, $P_{\rm B}=10^{46}$ erg s$^{-1}$,
$z=3$ and
$\Gamma =\min[15, (R_{\rm diss}/3R_{\rm S})^{1/2}]$.
This figures shows that $U^\prime_{\rm B}$ dominates only in two
regions of the jet: in the vicinity of the black hole and
at $R_{\rm diss}\sim 10^{20}$--$10^{21}$ cm.
Within these two distances, $U^\prime_{\rm BLR}$ and
$U^\prime_{\rm IR}$ dominate, and beyond $10^{21}$ cm
$U^\prime_{\rm CMB}$ takes over.
This has two important and immediate consequences:
\begin{enumerate}
\item The external Compton radiation will be more important than
the synchrotron (and the SSC, if $y\sim 1$)
radiation at all distances, except the (relatively narrow)
distance intervals where $U^\prime_{\rm B}$ dominates.
\item
At large distances the jet is conical,
and the blob size scales linearly with $R_{\rm diss}$
(namely the jet is a cone with the same aperture angle $\psi$).
Then the light crossing time $t_{\rm cross}=\psi R_{\rm diss}/c$.
This hypothesis means that $\gamma_{\rm cool}$ reaches a
maximum where $U^\prime_{\rm B}=U^\prime_{\rm CMB}$.
\end{enumerate}
To understand the second issue, consider
the value of $\gamma_{\rm cool}$ after a time $t_{\rm cross}$.
\begin{equation}
\gamma_{\rm cool} \, =
\, {3 m_{\rm e} c^2 \over 4\sigma_{\rm T} r_{\rm diss} U^\prime}
\label{gcool1}
\end{equation}
In the regions of interest (i.e. above 1 kpc),
$U^\prime_{\rm CMB}$ is constant, while
$U^\prime_{\rm B}\propto r^{-2}_{\rm diss}$.
Therefore the maximum $\gamma_{\rm cool} $ occur when
$U^\prime_{\rm CMB}=U^\prime_{\rm B}$, i.e. at
\begin{equation}
\psi R_{\rm eq} \, = \, \left[ { P_{\rm B} \over
\pi a c}\right]^{1/2} \, { 1\over T_0^2 (1+z)^2 \Gamma^2}
\end{equation}
Inserting this in Eq. \ref{gcool1} and setting
$U^\prime=U^\prime_{\rm CMB}+U^\prime_{\rm B}
=2 U^\prime_{\rm CMB}$, we obtain:
\begin{eqnarray}
\gamma^{\rm max}_{\rm cool} \, &=& \,
{3 m_{\rm e} c^2 \over 4\sigma_{\rm T} T_0^2(1+z)^2}
\,\left( { \pi c \over a P_{\rm B}}\right)^{1/2} \nonumber \\
& =& \, { 3.08\times 10^6 \over (1+z)^2 \, P_{\rm B,46}^{1/2}}
\label{gcool2}
\end{eqnarray}
Independent of $\Gamma$.
Electrons with this energy emit an observed synchrotron frequency
$\nu_{\rm cool}$ given by:
\begin{eqnarray}
\nu^{\rm syn}_{\rm cool} \, &\sim& \, 3.6 \times 10^6
B\gamma_{\rm cool}^2 \delta \nonumber \\
&=&\, 1.09 \times 10^{14}\, { \Gamma\delta \over
(1+z)^3 L_{\rm B,46} }\quad {\rm Hz}
\label{vcoolsyn}
\end{eqnarray}
Analogously, the observed frequency emitted by these electrons scattering the peak
of the CMB radiation is
\begin{eqnarray}
\nu^{\rm IC}_{\rm cool} \, &\sim& \, {4 \over 3} {3.93 kT_0 \over h} \gamma_{\rm cool}^2
\Gamma\delta \nonumber \\
&=&\, 2.8 \times 10^{24}\, { \Gamma\delta \over
(1+z)^4 P_{\rm B,46} }\quad {\rm Hz}
\label{vcoolic}
\end{eqnarray}
We thus expect that the synchrotron spectrum produced in
powerful jets, at large distances, cuts--off at a frequency
given by Eq. \ref{vcoolsyn}, or somewhat larger if, at these
scales, the active region has a size smaller than $\psi R$
(implying a smaller cooling time, and thus electron energies
greater than the ones given by Eq. \ref{gcool2}).
\begin{figure}
\vskip -0.7cm
\hskip -1.6cm
\psfig{figure=u_cmb_gc.ps,width=12cm,height=12cm}
\vskip -0.5 cm
\caption{
Top panel: the contributions to the radiation energy density as
seen in the comoving frame of the emitting blob, as labelled, and the
magnetic energy density as a function of the distance of the blob
to the black hole.
We assumed: $M=10^9 M_\odot$,
$L_{\rm d}=10^{47}$ erg s$^{-1}$, $z=3$
and $\Gamma =\min[15, (R_{\rm diss}/3R_{\rm S})^{1/2}]$.
The Poynting flux is $P_{\rm B}=10^{46}$ erg s$^{-1}$ once the jet
has reached its maximum bulk Lorentz factor.
The grey vertical lines indicates the distances assumed to construct
the SEDs in Fig. \ref{rdiss}, the different numbers help to identify
the corresponding SED.
The bottom bottom panel shows $\gamma_{\rm cool}$ (after one light crossing
time since the start of the injection) as a function of $R_{\rm diss}$.
}
\label{cmb}
\end{figure}
\begin{figure}
\vskip -0.5cm
\psfig{figure=rdiss.ps,width=9cm,height=9cm}
\vskip -0.5cm
\caption{
Sequence of SEDs calculated for different $R_{\rm diss}$
from $10^3 R_{\rm S}$ to $10^9 R_{\rm S}$ (one per decade).
The injected electron luminosity is $P^\prime_{\rm i}=10^{44}$ erg s$^{-1}$
for $R_{\rm diss}=10^3 R_{\rm S}$ and is reduced by a factor 3 each decade.
The particle distribution has always the same $\gamma_{\rm b}=100$, while
$\gamma_{\rm max}$ increases by a factor 3 each decade starting
from $\gamma_{\rm max}=10^4 $ for $R_{\rm diss}=10^3 R_S$.
All other parameters are the same as in Fig. \ref{cmb}.
What shown are the observed spectra neglecting the absorption of the
high energy flux due to the IR--opt--UV cosmic background.
The numbers correspond to the same numbers in Fig. \ref{cmb} and
correspond to the SED at different $R_{\rm diss}$.
The thicker black line is the sum of all the SED.
The grey line at radio frequencies indicates $F(\nu) \propto \nu^0$.
The received flux is calculated assuming that the source is at $z=3$.
}
\label{rdiss}
\end{figure}
Fig. \ref{rdiss} shows the predicted SEDs corresponding to
different $R_{\rm diss}$ (indicated by the vertical grey lines
in Fig. \ref{cmb} with the corresponding numbers);
all the relevant input parameters for the different SEDs
are reported in Tab. \ref{para}.
The profile of the external radiation and the value of the magnetic
field correspond to what shown in Fig. \ref{cmb}.
We have also assumed that, increasing $R_{\rm diss}$, the
injected power in relativistic electrons decreases
by a factor 3 increasing $R_{\rm diss}$ by a factor 10.
At the same time, we assumed that the maximum energy
of the injected electrons increases
by a factor 3 for a tenfold increase of $R_{\rm diss}$.
These choices are arbitrary, but reflect the observational evidence
that the bolometric radiative output of jets decreases with distance.
Furthermore the existence of optical jets, whose emission is due to
the synchrotron process, ensures that at large jet scales there
are very energetic electrons.
The first three SEDs (number 1, 2 and 3, corresponding to
$R_{\rm diss}=10^3$, $10^4$ and $10^5 R_{\rm S}$, respectively)
have a large Compton dominance,
corresponding to the large ratio between the external radiation
and the magnetic energy densities.
Note that only the first uses the BLR photons as the main seeds
for the Compton process (and thus it has the largest peak frequency),
while the SED 2 and SED 3 use the IR photons as seeds.
SED 1 is dominating in the $\gamma$--ray band (above $\sim$1 MeV), while
in the far IR to UV bands there is the contribution of the
thermal radiation (accretion disk and IR torus).
Remarkably, the soft X--rays are produced almost equally
by these three jet dissipation sites (see also below).
SED 4 (i.e. $R_{\rm diss}=10^6 R_{\rm S}$)
is the only one where the magnetic energy density dominates.
Correspondingly, the synchrotron flux dominates the bolometric output.
Comparing SED 3 and SED 4, we note that they have total
luminosities that differ by more than a factor 3.
This is due to the incomplete electron cooling, and implies that
not all the power injected in random energy of the electrons can
be radiated in one crossing time.
For the same reason, there is a rather large jump in $\gamma_{\rm cool}$
between $R_{\rm diss}=10^5$ and $10^6R_{\rm S}$
(see the bottom panel of Fig. \ref{cmb}), corresponding to the
fast drop of the energy density of the external radiation between
these two distances.
SED 5 ($R_{\rm diss}=10^7 R_{\rm S}$, or 1 kpc)
has again a relatively large Compton dominance,
due to the prevailing of the CMB
and the starlight (from the galaxy bulge)
energy densities over the magnetic one.
This SED has almost the largest possible $\gamma_{\rm cool}$
(see Fig. \ref{cmb}) and this is reflected in the calculated
SED, having the largest peak frequency of the high energy component.
SED 6 and 7 ($R_{\rm diss}=10^8$ and $10^9 R_{\rm S}$, or 10 and 100 kpc,
respectively) have a very large Compton dominance, due to the fact
that $U^\prime_{\rm CMB}$ is constant, while $U^\prime_{\rm B}$
decreases with distance.
As discussed previously, the constancy of $U^\prime_{\rm CMB}$
makes $\gamma_{\rm cool}$ to decrease with $R_{\rm diss}$,
since the cooling time at which $\gamma_{\rm cool}$ is calculated increases.
This also implies that, despite the decrease of $P^\prime_{\rm i}$,
the SED 5, 6 and 7 have the same bolometric luminosity,
because a larger portion of the electron population can cool
in one $t_{\rm cross}$.
\vskip 0.3 cm
The sum of all SEDs is shown by the black thick line in Fig. \ref{rdiss}.
Note that, in the radio band, the total flux $F_\nu\propto \nu^0$
(or slightly harder), as observed.
At frequencies greater than the radio ones,
the total flux is dominated by SED 1, but the flux originating at relatively
large scales can be important in some frequency bands, as the
soft X--ray one, with important consequences on the observed
variability and the correlation of variability in different bands.
According to Fig. \ref{rdiss} (which is, we re--iterate, only one
possible example, shown for illustration) the correlation
between the $\gamma$--ray and the IR flux should be tighter
than the correlation between the $\gamma$--ray and the soft X--ray flux,
diluted by the the flux originating at larger scales
(that can vary on longer timescales).
Furthermore, if the soft X--ray flux is produced by external Compton,
the energies of the electron emitting it are relatively modest
(namely $\gamma\sim$1--10)
while, in the $\gamma$--ray band, we see the emission of the
electrons with the highest energies (which emit, by synchrotron,
at IR--optical frequencies).
The cooling timescales are therefore different, producing
different variability if $t_{\rm cool}$ at low energies is longer
than the light crossing time.
If not, then we expect the same variability pattern for
emission produced in the same zone.
One example of different variability behaviour is displayed
by the blazar 3C 454.3, as recently studied by Bonning et
al. (2008), and has been
interpreted by these authors as a consequence of the different energies
of the electron contributing in the X--ray and $\gamma$--ray bands.
The contribution to the X--ray flux by other zones of the jet
can be an alternative possibility.
\begin{table*}
\centering
\begin{tabular}{lllllllllllllll}
\hline
\hline
Fig. &$R_{\rm diss}$ &$M$ &$R_{\rm BLR}$ &$P^\prime_{\rm i}$ &$L_{\rm d}$ &$B$ &$\Gamma$ &$\theta_{\rm v}$
&$\gamma_{\rm b}$ &$\gamma_{\rm max}$ &$s_1$ &$s_2$&$z$ &Notes \\
~[1] &[2] &[3] &[4] &[5] &[6] &[7] &[8] &[9] &[10] &[11] &[12] &[13] &[14] &[15] \\
\hline
\ref{pairs} &6 (20) &1e9 &1e3 &0.1 &100 (0.67) &200 &10 &3 &1e3 &1e4 &0 &2.5 &3 &Pairs--no pairs \\
\ref{rpairs} &3 (10) &1e9 &1e3 &0.1 &100 (0.67) &1.6e3 &1.8 &3 &1e3 &1e4 &0 &2.5 &3 &Pairs--no pairs \\
\ref{rpairs} &30 (100) &1e9 &1e3 &0.1 &100 (0.67) &120 &5.8 &3 &1e3 &1e4 &0 &2.5 &3 &Pairs--no pairs \\
\ref{rpairs} &300 (1e3) &1e9 &1e3 &0.1 &100 (0.67) &5.6 &10 &3 &1e3 &1e4 &0 &2.5 &3 &Pairs--no pairs \\
\hline
\ref{rdiss} &3e2 (1e3) &1e9 &1e3 &7.3e--2 &100 (0.67) &3.6 &15 &3 &100 &1e4 &1 &2.5 &3 &$R_{\rm diss}$ seq. \\
\ref{rdiss} &3e3 (1e4) &1e9 &1e3 &2.4e--2 &100 (0.67) &0.36 &15 &3 &100 &3e4 &1 &2.5 &3 &$R_{\rm diss}$ seq. \\
\ref{rdiss} &3e4 (1e5) &1e9 &1e3 &8.1e--3 &100 (0.67) &3.6e--2 &15 &3 &100 &9e4 &1 &2.5 &3 &$R_{\rm diss}$ seq. \\
\ref{rdiss} &3e5 (1e6) &1e9 &1e3 &2.7e--3 &100 (0.67) &3.6e--3 &15 &3 &100 &2.7e5 &1 &2.5 &3 &$R_{\rm diss}$ seq. \\
\ref{rdiss} &3e6 (1e7) &1e9 &1e3 &9e--4 &100 (0.67) &3.6e--4 &15 &3 &100 &8.1e5 &1 &2.5 &3 &$R_{\rm diss}$ seq. \\
\ref{rdiss} &3e7 (1e8) &1e9 &1e3 &3e--4 &100 (0.67) &3.6e--5 &15 &3 &100 &2.4e6 &1 &2.5 &3 &$R_{\rm diss}$ seq. \\
\ref{rdiss} &3e8 (1e9) &1e9 &1e3 &1e--4 &100 (0.67) &3.6e--6 &15 &3 &100 &7.3e6 &1 &2.5 &3 &$R_{\rm diss}$ seq. \\
\hline
\ref{m} &4.5 (500) &3e7 &55 &3e--3 &0.3 (0.067) &13.3 &15 &3 &100 &1e4 &1 &2.5 &3 &$M$ seq. \\
\ref{m} &15 (500) &1e8 &100 &0.01 &1 (0.067) &7.3 &15 &3 &100 &1e4 &1 &2.5 &3 &$M$ seq. \\
\ref{m} &45 (500) &3e8 &173 &0.03 &3 (0.067) &4.2 &15 &3 &100 &1e4 &1 &2.5 &3 &$M$ seq. \\
\ref{m} &150 (500) &1e9 &317 &0.1 &10 (0.067) &2.3 &15 &3 &100 &1e4 &1 &2.5 &3 &$M$ seq. \\
\ref{m} &450 (500) &3e9 &549 &0.3 &30 (0.067) &1.3 &15 &3 &100 &1e4 &1 &2.5 &3 &$M$ seq. \\
\hline
\ref{1253}: 3C 279 &114 (380) &1e9 &173 &0.06 &3 (0.02) &3.1 &11.3 &3 &250 &4e3 &1 &2.2 &0.536 &high EGRET \\
\ref{1253}: 3C 279 &300 (1e3) &1e9 &173 &0.04 &3 (0.02) &0.42 &16 &3 &3e3 &3e5 &0 &2.7 &0.536 &TeV \\
\ref{1253}: 3C 279 &72 (240) &1e9 &173 &9e--3 &3 (0.02) &7.7 &8.9 &3 &160 &1.2e3 &0 &2.7 &0.536 &low EGRET \\
\hline
\ref{1428}: 1428 &680 (1.5e3) &1.5e9 &1.2e3 &0.1 &135 (0.6) &1.2 &15 &2.5 &30 &3e3 &0 &2.4 &4.72 &{\it Beppo}SAX \\
\ref{1428}: 1428 &6.8e3 (1.5e4) &1.5e9 &1.2e3 &0.1 &135 (0.6) &0.12 &15 &2.5 &30 &3e3 &0 &2.4 &4.72 &$R_{\rm diss}\times 10$ \\
\ref{1428}: 1428 &68 (150) &1.5e9 &1.2e3 &0.1 &135 (0.6) &44 &7.1 &2.5 &30 &3e3 &0 &2.4 &4.72 &$R_{\rm diss}\times 0.1$ \\
\hline
\ref{2149}: 2149 &960 (800) &4e9 &1.2e3 &0.1 &150 (0.25) &1.7 &12 &3 &10 &1e3 &--1 &2.6 &2.345 &{\it Swift} data \\
\ref{2149}: 2149 &9.6e3 (8e3) &4e9 &1.2e3 &0.1 &150 (0.25) &0.17 &12 &3 &10 &1e3 &--1 &2.6 &2.345 &$R_{\rm diss}\times 10$ \\
\ref{2149}: 2149 &96 (80) &4e9 &1.2e3 &0.1 &150 (0.25) &43 &5.2 &3 &10 &1e3 &--1 &2.6 &2.345 &$R_{\rm diss}\times 0.1$ \\
\hline
\ref{sequence} &810 (900) &3e9 &1.7e3 &0.4 &302 (0.67) &2.4 &13 &3 &100 &3e3 &0 &2.5 &--- &Blazar seq. \\
\ref{sequence} &810 (900) &3e9 &636 &0.04 &41 (0.09) &0.9 &13 &3 &200 &3e4 &0 &2.5 &--- &Blazar seq. \\
\ref{sequence} &120 (400) &1e9 &387 &4e-3 &15 (0.1) &5.5 &11.5 &3 &200 &1.5e4 &0 &2.5 &--- &Blazar seq. \\
\ref{sequence} &210 (700) &1e9 &--- &1e-3 &--- &0.2 &15 &3 &600 &2e5 &0 &2.5 &--- &Blazar seq. \\
\ref{sequence} &210 (700) &1e9 &--- &5e-4 &--- &0.1 &15 &3 &3e3 &7e5 &0 &2.5 &--- &Blazar seq. \\
\hline
\end{tabular}
\vskip 0.4 true cm
\caption{List of parameters used to construct the SED shown in Figg. 6--13.
Col. [1]: figure number where the model is shown;
Col. [2]: dissipation radius in units of $10^{15}$ cm and (in parenthesis) in units of $R_{\rm S}$;
Col. [3]: black hole mass in solar masses;
Col. [4]: size of the BLR in units of $10^{15}$ cm;
Col. [5]: power injected in the blob calculated in the comoving frame, in units of $10^{45}$ erg s$^{-1}$;
Col. [6]: accretion disk luminosity in units of $10^{45}$ erg s$^{-1}$ and
(in parenthesis) in units of $L_{\rm Edd}$;
Col. [7]: magnetic field in Gauss;
Col. [8]: bulk Lorentz factor at $R_{\rm diss}$;
Col. [9]: viewing angle in degrees;
Col. [10] and [11]: break and maximum random Lorentz factors of the injected electrons;
Col. [12] and [13]: slopes of the injected electron distribution [$Q(\gamma)$] below and above $\gamma_{\rm b}$;
Col. [14]: redshift;
Col. [15]: some notes.
For all cases the X--ray corona luminosity $L_X=0.3 L_{\rm d}$.
Its spectral shape is assumed to be $\propto \nu^{-1} \exp(-h\nu/150~{\rm keV})$.
}
\label{para}
\end{table*}
\section{Changing the black hole mass}
It is instructive to study the SED produced by the jet
in FSRQs of different black hole masses, scaling the relevant
quantities with the Schwarzschild\ radius and the Eddington luminosity.
To this aim we show in Fig. \ref{m}
a sequence of SED with the black hole mass ranging from
$3\times 10^7$ to $3\times 10^9 M_\odot$.
We assume that $R_{\rm diss}$ is always at 500 Schwarzschild\ radii and
that $L_{\rm d}=0.067 L_{\rm Edd}$, with a corona with
an X--ray luminosity equal to one third that of the disk.
We assume that the power in relativistic electrons injected into
the dissipation region of the jet scales with the black hole mass as
$P^\prime_{\rm i}=10^{44}M_9$ erg s$^{-1}$.
Electrons are injected
between $\gamma_1=10^2$ and $\gamma_2=10^4$ in all cases.
The bulk Lorentz factor is $\Gamma=15$ for all cases and the redshift is $z=3$.
At the assumed $R_{\rm diss}$ the jet has already reached its
maximum $\Gamma$, and at these distances the Poynting flux
is assumed to scale as
$P_{\rm B}=6.7\times 10^{-3} L_{\rm Edd}$.
We then have
\begin{equation}
U_{\rm B} \, \propto { L_{\rm Edd}
\over \Gamma^2 R^2_{\rm diss} } \,
\propto \, {M \over M^2 } \, \propto {1\over M}
\end{equation}
which follows from the assumption of $R_{\rm diss}/R_{\rm S}=$const
and $\Gamma=$const.
For all our cases, the dissipation occurs within the BLR, which yields a constant
radiation energy density $U^\prime_{\rm BLR}$ since $\Gamma$ is the same.
As a consequence, the ratio
$L_{\rm EC}/L_{\rm syn} \sim U^\prime_{\rm BLR}/U_{\rm B} \propto M$.
This is the reason of the increasing dominance of the inverse Compton
emission increasing the black hole mass.
This implies that blazars with large black hole masses
should preferentially be more Compton dominated, and therefore
more easily detected by the Fermi satellite.
In fact, in Fig. \ref{m} one can see the 5$\sigma$ sensitivity
of Fermi for 1 year of operation (grey line), suggesting that, at high redshifts,
the detected blazars will preferentially have large black hole masses.
The importance of the EC relative to the SSC emission increases
with the black hole mass, hardening the X--ray spectral shape.
The SSC and EC components are shown separately
for the SED corresponding to $M=3\times 10^7M_\odot$,
to illustrate the importance of the SSC flux.
For larger masses the SSC components becomes
relatively less important than the EC one.
For the SED with $M=3\times 10^9M_\odot$ we show the effects
of neglecting the $\gamma$--$\gamma$ absorption and the
consequent reprocessing (dashed line).
One can see that the primary (i.e. neglecting pairs)
spectrum as a rather sharp
cut--off due to the Klein--Nishina limit given by
Eq. \ref{kn}.
\begin{figure}
\vskip -0.5cm
\psfig{figure=m.ps,width=9cm,height=9cm}
\vskip -0.5cm
\caption{
The observed SED for different black hole masses
(from $3\times 10^7$ to $3\times 10^9M_\odot$, as labelled)
assuming that the dissipation takes place at
500$R_{\rm S}$, that the accretion disk luminosity is
$L_{\rm d} =0.067 L_{\rm Edd}$, and that the
power injected in the jet dissipation region scales as
$P^\prime_{\rm i} =10^{44}M_9$ erg s$^{-1}$.
The bulk Lorentz factor is kept fixed at $\Gamma=15$.
The particles are always injected between $\gamma_1=10^2$ and
$\gamma_2=10^4$.
The grey line is the 5$\sigma$ detection sensitivity of Fermi,
after 1 year of operation.
For the $M=3\times 10^7M_\odot$ case we show, besides the total spectrum,
the SSC and the EC components separately.
This illustrates the importance of the SSC process for
low values of the black hole masses.
The dashed line (for the case with $M=3\times 10^9 M_\odot$)
shows the spectrum neglecting photon--photon absorption and pair
reprocessing.
The received flux is calculated assuming that all sources are
at $z=3$.
}
\label{m}
\end{figure}
\section{Some illustrative examples}
\begin{figure}
\vskip -0.5cm
\psfig{figure=1253f_tev_test.ps,width=9cm,height=9cm}
\vskip -0.5cm
\caption{
SED of 3C 279 in different states together with the corresponding
models, whose input parameters are listed in Tab. \ref{para}.
See Ballo et al. (2002) and references therein for the sources
of data points.
Note that the high energy data--points (above 100 GeV) have been
de--absorbed according to the Primack, Bullock \& Somerville (2005)
model used in Tavecchio \& Mazin (2009).
According to Eq. \ref{kn2}, the flux at these energies can
be produced at relatively large distances from the black hole, beyond
the BLR. In this case the main contribution to the seed photons
is coming from the IR torus (solid light grey line,
orange in the electronic version).
In the low state (blue triangles) the SSC process
is dominating the 2--10 keV X--ray flux.
For this state only we show the SSC (long dashed line) and the EC
flux (dot--dashed line) separately.
}
\label{1253}
\end{figure}
\subsection{TeV FSRQs: the case of 3C 279}
3C 279 has been recently detected in the TeV band (Albert et al. 2008),
although its redshift, $z=0.536$, implies a strong absorption
of high energy $\gamma$--rays by the IR cosmic background.
This demonstrates that also powerful blazars emit at large
energies, up to the TeV band, even if the peak of their
high energy hump may lye in the MeV--GeV band.
Besides the consequences that this result has on the
cosmic background, discussed in Sitarek \& Bednarek (2008);
Tavecchio \& Mazin (2009); Liu, Bai \& Ma (2008),
we would like to discuss here another consequence,
which concerns the primary spectrum of the source.
We have discussed previously that if the bulk of the
inverse Compton spectrum uses BLR photons as seeds, then
we expect a steepening of the intrinsic
spectrum at $\sim 15/(1+z)\sim 10$ GeV due to Klein--Nishina effects.
This is shown in Fig. \ref{1253} as the darker line passing through
an archival EGRET spectrum of the source (squares).
For this model, in fact, the dissipation region is within the BLR,
which is then giving most of the seed photons used for the scattering
(see Tab. \ref{para} for the all the input parameters).
The rather abrupt cut--off seen at $\sim 2.5\times 10^{24}$ Hz $\sim 10$ GeV
is due this effect, and not to $\gamma$--$\gamma$ internal absorption,
which, in this particular example, is unnoticeable.
To produce photons of larger energies and evade the Klein--Nishina limit
we are forced to locate $R_{\rm diss}$ beyond the BLR, and then
use the IR photons produced by the torus.
This is what the model does (lighter grey line, orange in
the electronic version).
Note that the presence of a very high energy component in 3C 279
was predicted (before detection) by B{\l}azejowski et al. (2000), who
included the presence of a IR emitting torus (see their Fig. 4).
The main difference of our modelling with respect to B{\l}azejowski et al.
(2000) is in the assumed temperature of the torus, assumed to be larger
in that paper, implying a smaller size and an enhanced $U^\prime_{\rm IR}$.
If the jet of 3C 279 is ``canonical" in the sense described in this paper,
then a larger $R_{\rm diss}$ means a smaller magnetic field, and a larger
bulk Lorentz factor (if it has not yet reached its maximum value).
Then, according to these ideas, we show the entire modelled SED to compare it
with the ``high EGRET state" (squares) and the ``low EGRET state" (triangles).
For the latter model we have assumed a smaller $R_{\rm diss}$
and $P^\prime_{\rm i}$.
This implies a small bulk Lorentz factor (hence a decreased importance
of the external seed photons) and a large magnetic field, resulting in
a SED of equal synchrotron and inverse Compton power.
3C 279 is one of the best studied $\gamma$--ray blazars, partly
because it was very active during the observations of EGRET.
It should not be taken, however, as the prototypical high power FSRQ,
since its disk emission is very modest, as also directly
suggested by its SED in the low state (Pian et al. 1999;
see the triangles in Fig. \ref{1253}, see Ballo et al. 2002
and references therein for the data): for our models we have assumed
a black hole mass $M=10^9M_\odot$ and $L_{\rm d}=0.02 L_{\rm Edd}$.
It is therefore instructive to
show example of more powerful blazars, with accretion disk
emitting close to the Eddington limit.
\begin{figure}
\vskip -0.5cm
\psfig{figure=1428f4.ps,width=9cm,height=9cm}
\vskip -0.5cm
\psfig{figure=u1428.ps,width=8.5cm,height=7cm}
\vskip -0.5cm
\caption{
Top panel: the SED of one of the most distant blazars,
together to 3 different models with different $R_{\rm diss}$
(see Tab. \ref{para} for the set of parameters) to illustrate
possible different states of the source.
See Fabian et al. (2001) and references therein for the sources of data,
and Celotti et al. (2007) for further discussion about this source.
}
\label{1428}
\end{figure}
\begin{figure}
\vskip -0.5cm
\psfig{figure=2149_307f.ps,width=9cm,height=9cm}
\vskip -0.5cm
\psfig{figure=u2149.ps,width=8.5cm,height=7cm}
\vskip -0.5cm
\caption{
Top panel: the SED PKS 2149--307 together to 3 different
models (see Tab. \ref{para} for the set of parameters)
to illustrate possible different states of the source.
These relatively distant, close to Eddington, high black hole
mass blazars should be at the extreme of the blazar sequence,
showing a high energy peak in the 100 keV--1 MeV energy band.
Note that if the dissipation takes place very close to the black hole,
when the jet is still accelerating and with a strong magnetic field,
the resulting spectrum becomes unconspicuous at high energies, even if
the intrinsic dissipated power is the same of the higher states.
}
\label{2149}
\end{figure}
\subsection{MeV blazars at large redshifts}
GB 1428+4217, at $z=4.73$, is the second most distant blazar known
(the most distant is Q0906--6930, with $z=5.47$).
Its SED is shown in the top panel of Fig. \ref{1428}, together with
the fitting model (black line).
The figure shows that this source has a prominent UV--bump,
with a luminosity around $10^{47}$ erg s$^{-1}$ (see also Tab. \ref{para}).
According to our model, the peak of the high energy hump of the SED
during the {\it Beppo}SAX observations (Fabian et al. 2001)
is predicted to lye at $\sim 1$ MeV, locating this source at
one extreme of the blazar sequence.
The model assumes a black hole mass $M=1.5\times 10^9 M_\odot$, an
accretion disk emitting at $\sim 2/3$ of the Eddington ratio (the other third
is emitted by the X--ray corona).
The dissipation region is within the BLR, at 1500 Schwarzschild\ radii from the centre.
The fit requires $B=1.2$ G, corresponding to $P_{\rm B}=5.4\times10^{45}$
erg s$^{-1}$, and $L_{\rm d}=1.35\times 10^{47}$ erg s$^{-1}$.
The bottom panel of the same figures shows the radiation and magnetic
energy density profiles for the parameters used to model this blazars.
The mid grey vertical line corresponds to the used value of $R_{\rm diss}$.
This figure shows that $U^\prime_{\rm BLR}$ is about two orders of magnitude
larger than $U^\prime_{\rm B}$, corresponding to the ratio of the external
Compton to synchrotron luminosities.
For illustration purposes, we also show the SED corresponding to increase
(decrease) $R_{\rm diss}$ tenfold (see the vertical grey lines in the bottom
panel, and the corresponding SED in the top panel).
Shifting $R_{\rm diss}$ to $1.5\times 10^4 R_{\rm S}$ the high energy peak
moves to lower frequencies, as the relevant seed photons are softer,
and at the same time the X--ray spectrum hardens.
This is obtained even if the particle injection function is unchanged,
and is due to the incomplete cooling occurring for the larger $R_{\rm diss}$ case.
Since $\gamma_{\rm cool}$ shifts from 1 to $\sim$34, in the
X--ray band we see the emission from the uncooled electron populations,
that retains the original, ``injection", slope $s=0$.
Note that the high energy peak is now slightly below 100 keV, and this would
make the source even more extreme in terms of the blazar sequence.
The EC to synchrotron luminosity ratio is still $\sim 100$, corresponding
now to the ratio between $U^\prime_{\rm IR}$ and $U^\prime_{\rm B}$,
as can be seen in the bottom panel.
Instead, if $R_{\rm diss}$ is at 150 Schwarzschild\ radii, the SED
changes more dramatically.
Since in this case the bulk Lorentz factor is smaller and the magnetic field
larger, the inverse Compton and synchrotron powers become comparable
(see the bottom panel, showing that for this $R_{\rm diss}$ we have
$U^\prime_{\rm B}\sim U^\prime_{\rm d}$).
The X--ray spectrum brightens and softens considerably, in the IR--UV
band the flux increases (because of the increased magnetic field)
even if the bolometric observed luminosity decreases because of the decreased
Doppler boosting.
The fact that we did not see this kind of SED in GB 1428+4217 suggests that
this state rarely occurs.
This is particularly true considering that this blazars
was not discovered because it was particularly bright in hard X--rays or
in the $\gamma$--ray band, so there was no bias against a soft X--ray spectrum.
\vskip 0.3 cm
We lastly consider PKS 2149--307 at $z=2.345$, as observed by the XRT and BAT
instruments onboard {\it Swift} during its first 9 months of observations
(Sambruna et al. 2007).
Fig. \ref{2149} shows its SED (top panel) and the profiles of the radiation
and magnetic energy densities for the considered models (bottom panels).
As done for GB 1428+4217, we show what we considered the best fitting model
(black line in the top panel) corresponding to $R_{\rm diss}=9.6\times 10^{17}$ cm,
(corresponding to 800 $R_{\rm S}$ for the assumed black hole mass of
$M=4\times 10^9 M_\odot$), and also the models corresponding to a tenfold
increase (decrease) of $R_{\rm diss}$ (light grey lines in the top panel and
vertical lines in the bottom panel).
The BAT data have large error bars, precluding the possibility to firmly claim that
its high energy peak is within the BAT energy range (namely, at $\sim 100$ keV),
but this possibility is indeed suggested by the present data.
The SED of this source can be explained by a set of parameters
similar to the ones chosen for GB 1428+4217, and similar considerations apply.
Increasing $R_{\rm diss}$ tenfold makes the IR radiation from the torus to dominate,
leaving almost unchanged the ratio between the radiation and the magnetic energy
densities (and thus the corresponding EC to synchrotron luminosity ratio).
Again a decrease in $R_{\rm diss}$ by a factor 10 has a dramatic impact on the
predicted SED, dominated in this case by the synchrotron luminosity (see
also the bottom panel).
However, having only data up to the medium--soft X--ray energy range,
it would be almost impossible to tell what is the real appearance of the
bolometric SED of the source, as almost all the changes occur above
10 keV.
\begin{figure}
\vskip -0.5cm
\hskip -0.5 cm
\psfig{figure=repro_gfos.ps,width=9.5cm,height=9.5cm}
\vskip -0.5cm
\caption{
The blazar sequence (Fossati et al. 1998; Donato et al. 2001)
interpreted in the framework of our canonical jet
scenario. Parameters are listed in Tab. \ref{para}.
The SED changes according to changing the accretion rate
and the power of the jet, and assuming that below
some critical accretion rate the accretion regime
changes regime, becoming very radiatively inefficient.
In out case, this occurs for the two least powerful SED,
that should corresponds to low power, line--less BL Lacs.
}
\label{sequence}
\end{figure}
\section{Canonical jets and the blazar sequence}
We can ask if our ``canonical" jet can reproduce the
phenomenological blazar sequence as proposed by Fossati et al. (1998)
(see also Donato et al. 2001 and Ghisellini et al. 1998).
This sequence was constructed by taking flux limited samples
of blazars (in the radio and X--ray bands),
dividing the sources in radio luminosity bins, and
averaging the flux of the sources in each (radio luminosity)
bin at selected frequencies.
The resulting data are shown in Fig. \ref{sequence}.
As discussed in Ghisellini \& Tavecchio (2008)
and in Maraschi et al. (2008), this sequence does
not pretend to describe ``the average blazar",
since it likely represents the most beamed sources,
successfully entering in the flux limited sample
they belong to.
Nevertheless it is appropriate to ask if our ``canonical" jet can
reproduce this sequence without strong modifications of the
basic assumptions.
Fig. \ref{sequence} then shows the SED resulting from
our modelling, whose parameters are listed in Tab. \ref{para}.
As can be seen the agreement is quite good, and it is achieved
by assuming (as in Ghisellini \& Tavecchio 2008), that the
accretion disk becomes radiatively inefficient below
some critical accretion rate (thus around a critical
luminosity in units of the Eddington one, that we take
as a few $\times10^{-3}$).
The first three, more powerful, SED correspond to sources
with a standard disk, while for the two SED at lower luminosities
we have ``switched--off" all external radiation.
The black hole mass is $3\times 10^9M_\odot$ for the
two most powerful SED, and is $10^9M_\odot$ for the
other three.
The luminosity, in the comoving frame, injected in relativistic electrons
is monotonically decreasing, as the value of the magnetic field.
The dissipation radius $R_{\rm diss}$, in units of the Schwarzschild\ radius,
changes slightly, but less than a factor 2.5.
The energy of the injected particles need to increase as the
cooling decreases.
The decreased cooling also makes $\gamma_{\rm cool}$ to increase.
These effects are very important for the two ``BL Lac"
SED, and much less for the other three most luminous SEDs.
We can conclude that the blazar sequence, as illustrated in
Fig. \ref{sequence}, can be interpreted as a sequence of jet powers,
in sources having more or less the same (large) black hole mass.
The jet power correlates with the accretion disk luminosity
and below some critical value the disk changes
accretion regime, becoming radiatively inefficient and
unable to photo--ionise the broad line clouds.
A intriguing manifestation of this behavior is shown
by the bright blazars recently detected by the {\it Fermi} satellite
(Abdo et al. 2009) discussed in Ghisellini, Maraschi \& Tavecchio
(2009), and mirrors what has been suggested to occur for the
FRI--FRII radio--galaxy divide (Ghisellini \& Celotti 2001).
\section{Caveats}
\label{caveats}
Our aim was to describe the main properties of the produced
spectrum in a ``canonical" jet.
Specific sources can of course deviate somewhat by our description.
For instance:
\begin{itemize}
\item
Assuming $R_{\rm BLR}\propto L_{\rm d}^{1/2}$,
may be approximately obeyed on average,
but specific sources might behave differently.
The assumed jet geometric profile
and the assumption that the jet, at its start,
is magnetically dominated could be two oversimplifications.
We hope to test
these assumptions
by direct observations, since the combined efforts
of the {\it Fermi} and {\it Swift} satellites
can give us really simultaneous spectra on a large
frequency band.
\item
We have assumed that the ``dissipation"
mechanism, converting some fraction of the jet power into
radiation, has the form of an injection of primary leptons
lasting for a light crossing time.
With this assumption, we have derived the particle distribution
at this time, which corresponds to the maximum produced flux.
However, the injection can last for a longer or shorter time.
Our framework is appropriate for describing a ``snapshot"
of the SED, and not for a time--dependent analysis.
\item
We assume that the magnetic field in the dissipation region
is the same as the magnetic field transported by the jet.
It is not amplified by e.g. shocks, nor it is reduced
by e.g. reconnection events (see e.g. Giannios, Uzdensky \& Begelman 2009).
Although we can assume any value of the magnetic field
in the dissipation region (i.e. we can fix any value
in the numerical code calculating the SED), we prefer this choice
for all the examples shown, because of simplicity.
\item
We focused on high power blazars, thought to have ``standard"
accretion disks and broad line regions, and therefore we did
not discuss in detail the expected SED from low power
(and TeV emitting) BL Lacs.
However, we could simply extend our study to these sources
if we ``switch--off" the external radiation components,
as expected if the accretion disk becomes radiatively inefficient,
making very weak or absent broad emission lines.
In Fig. \ref{sequence} we indeed show two examples of
the expected spectra in this case.
\item
All the shown SED do not take into account the
absorption of $\gamma$--rays due to the IR, optical
and UV background, but only the ``internal" absorption
caused by the radiation fields existing close to the
emitting blob.
\item
We have always assumed a relatively small viewing
angle ($\theta_{\rm v}\sim 3^\circ$), of the same order of $1/\Gamma$.
Since $\Gamma$ and $\theta_{\rm v}$ are two separated input parameters,
we could change $\theta_{\rm v}$ (and thus the Doppler factor $\delta$)
for a fixed $\Gamma$.
However, this would introduce a delicate issue, since the external
radiation field, in the comoving frame, is not isotropic.
If the emitting blob is inside the BLR, for instance, then
all external photons appear to come from one direction, and
we could apply the formalism of Dermer (1995) to describe
the pattern of the scattered radiation.
Note that, in this case, if $\theta_{\rm v}=1/\Gamma$
(and thus $\delta=\Gamma$) one has the observed luminhosity
as for radiation isotropic in the rest frame.
On the other hand, if the emitting region is beyond the BLR,
then the arrival directions of the BLR external photons are
spread, and it is not trivial to reconstruct the exact pattern.
But in this cases the BLR component is hardly dominant with respect to
the IR radiation from the torus.
We have decided, for simplicity, to use $\theta_{\rm v}\sim 1/\Gamma$,
and to treat the emitted radiation pattern is the same way as for an
isotropic seed photon distribution.
\item
We neglected, for simplicity, the possibility of a jet
composed by a fast spine surrounded by a slower layer.
This structured jet have been proposed for low power
BL Lacs and FR I radio--galaxies (Ghisellini, Tavecchio
\& Chiaberge 2005), and can ease some problems in explaining the
SED properties in these sources.
While in high power blazars there is no compelling need, yet,
for such a structure, its inclusion in the present study
would make our description much more complex and model--dependent,
given the freedom to choice the parameters for the layer emission.
\item
For simplicity, we neglected the possibility that the BLR and
the associated inter--cloud material
can absorb and re-emit, or scatter, part of the synchrotron radiation
produced by the jet (i.e. the ``mirror model", see Ghisellini \& Madau 1996,
and some criticism by Bednarek 1998; B\"ottcher \& Dermer 1998).
\item
The jet, even if cold (i.e. before dissipation),
is relativistic, and the carried leptons could scatter
the external radiation by bulk Comptonization.
This process, proposed by Begelman \& Sikora (1987) and
Sikora et al. (1994) and studied
in detail in Celotti, Ghisellini \& Fabian (2007), could produce
a black--body like component in the X--ray band, whose level
depends on how the jet accelerates and on the amount of
leptons carried by the jet.
We have neglected this component for simplicity, although
it can be a very important diagnostic for deriving, at the same
time, the bulk Lorentz factor of the jet and its matter content.
\end{itemize}
\section{Conclusions}
The relatively recent observations of blazars, especially
in the $\gamma$--ray band, have disclosed some of the
crucial ingredient for our understanding of jets, namely
the bolometric power output and the real shape of
the produced SED.
Although there is still discussion about the origin
of the high energy emission of blazars, the relatively simple
one--zone leptonic model was rather successful to
interpret the existing data of specific sources.
In this paper, rather than modelling single sources,
we tried to propose a more general description of the emission
produced by high power jets.
For this aim we have first summarised in a simple
way the expected radiation fields as a function of the
distance from the black hole, and then we have studied
how the expected SED changes if the main dissipation
occurs at different locations in the jet.
We have done this by assuming rather reasonable prescription for
the accretion disk and its X--ray corona emission, and
using reasonable prescriptions for the radiation fields
corresponding to the broad line region and the infrared torus.
We have also assumed that the jet conserves its Poynting flux
at all distances, thus fixing the profile of the magnetic field.
Since all these prescriptions correspond to rather uncontroversial
assumptions, we believe that they should well characterise the
properties of an average, canonical, large power jet.
The main conclusions of our work are:
\begin{itemize}
\item
The magnetic energy density is
always smaller than the external radiation energy densities, except
at the very beginning of the jet and at the kpc scale.
Beyond these scales, the contribution of the cosmic microwave background
becomes dominant.
\item
The inclusion of the X--ray radiation from the disk corona
allowed to quantify the effect of pair production and reprocessing
for small $R_{\rm diss}$.
This is found to be severe if the primary spectrum emits
most of its power at energy above the pair production threshold.
This requires that the bulk Lorentz factor is large
even in the very vicinity of the black hole.
In this case
there is a transfer of power from the $\gamma$--ray to the X--ray band.
Our quantitative analysis confirms earlier, more qualitative, statements
concerning the non--dissipative nature of the inner (i.e. up to a few hundreds
Schwarzschild\ radii) jet.
\item
If instead the jet accelerates gradually (i.e. as $\Gamma\propto R^{1/2}$
up to a maximum value), the magnetic energy density dominates
the cooling at the start of the jet,
resulting in a synchrotron dominated SED,
with a very small fraction of power being emitted above the pair
production energy threshold.
These SED would hardly be detectable, since the small $\Gamma$
means a much reduced Doppler enhancement of the flux, and thus
these components are easily overtaken by the emission at larger distances
(that have a larger $\Gamma$).
In any case, early dissipation (i.e. small $R_{\rm diss}$)
is not an efficient process to create electron--positron pairs.
\item
Besides the inner part of the jet, the ``internal" pair absorption
(i.e. calculated within the emitting blob) can be particularly important
also if dissipation takes place close to (but within) the BLR and
close to (but within) $R_{\rm diss}\sim R_{\rm IR}$.
In these cases, however, a small fraction of the
emitted power can be absorbed, and the reprocessing is modest.
\item
The Klein--Nishina decrease (with energy) of the scattering cross section
imprints a characteristic steepening in the spectrum above $\sim 10$ GeV
if the dissipation takes place within the BLR, and at $\sim 1$ TeV if
the dissipation occurs beyond the BLR but within $R_{\rm IR}$.
\item
The above point has important consequences
on the possible use of the high energy (GeV) data of blazars to
study the optical--UV background.
In fact the signature of the $\gamma$--$\gamma$ absorption
due to this component would be a rather sharp steepening
of the received spectrum, that would be very similar to the
intrinsic steepening of the primary spectrum due to
Klein--Nishina effects {\it if most of the dissipation
takes place within the BLR}. In this case also the
``internal" $\gamma$--$\gamma$ absorption due to BLR photons
will be relevant.
It will be very difficult to disentangle these effects.
On the other hand, if the observed spectrum continues unbroken above
$15(1+z)$ GeV, we can infer that most of the dissipation
has occurred beyond $R_{\rm BLR}$.
In this case both Klein--Nishina and ``internal" absorption effects are
unimportant up to the TeV band.
These therefore would be the best candidates to study the
opt--UV cosmic backgrounds.
\item
The magnetic field decreases with distance, while
the $U^\prime_{\rm BLR}$ is approximately constant up to $R_{\rm BLR}$,
and $U^\prime_{\rm IR}$ is approximately constant up to $R_{\rm IR}$.
At these distances (i.e. $R_{\rm diss}\sim R_{\rm BLR}$ and
$R_{\rm diss}\sim R_{\rm IR}$) the Compton dominance
is very large.
Existing observations of large power blazars already indicate that
these blazars are characterised by a weak synchrotron component,
a strong thermal, disk component (unhidden by the synchrotron flux)
and a very large high energy flux.
These are the sources with the largest Compton dominance,
strongly suggesting that $R_{\rm diss}$ is relatively large
(i.e. close to the $R_{\rm BLR}$ or $R_{\rm IR}$).
This helps to understand why in these sources the soft X--rays
are relatively weak and very hard, demanding a very weak SSC flux:
since the magnetic field is small, not only the synchrotron, but also
the SSC flux is small with respect to the EC radiation.
\item
Concerning the above point, the fact that the high power blazars
we already know of have the mentioned properties does not imply that
they preferentially dissipate at large distances with respect to less
powerful sources, since we may have selected them on the basis of
their hard X--ray or $\gamma$--ray flux (therefore taking advantage of
the large Compton dominance they have when $R_{\rm diss}$ is large).
In other words: they may dissipate at different $R_{\rm diss}$ at
different times, but we select them only when $R_{\rm diss}$ is large.
\item
Although most of the dissipation should take place in
one zone of the jet (with $R_{\rm diss}\sim$500--1000 $R_{\rm S}$),
it is very likely that dissipation occurs also at larger
distances, albeit with less power.
The flux originating in these zones can be important in
some frequency bands, as the soft X--ray one, with
important consequences on the observed variability.
Instead, in the $\gamma$--ray energy band (at least when the $\gamma$--ray
dominates the bolometric output) most of the flux should originate
in one zone.
\item
Blazars with large masses should be characterised by the largest
inverse Compton to synchrotron ratios, and thus they will be
well represented in $\gamma$--ray all sky surveys, as
performed by the {\it Fermi} satellite.
\item
The most powerful blazars, with the most powerful associated accretion disk,
should have a high energy hump peaking in the 100 keV--1 MeV energy range.
The exact location of the peak frequency depends somewhat on $R_{\rm diss}$,
being smaller if dissipation occurs beyond the BLR.
These blazars should form the high power extension of the blazar sequence.
They {\it are not} easily detectable by the {\it Fermi} satellite, since their MeV
peak implies relatively small fluxes in the GeV band.
The brightest of them have already been revealed by the serendipitous
survey of the BAT instrument onboard {\it Swift}.
\item
The phenomenological blazar sequence as presented in Fossati et al. (1998)
can be interpreted as a sequence of ``canonical jets" in sources with
more or less the same (large) black hole mass, but
with different accretion rates and different jet powers.
\end{itemize}
\section*{Acknowledgments}
We sincerely thank the referee for criticism that helped to
substantially improve the paper.
We thank fruitful discussions with Laura Maraschi and Annalisa Celotti.
This work was partly financially supported by a 2007 COFIN-MIUR grant.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,822 |
\section{\@startsection {section}{1}{\z@}%
{-3.5ex \@plus -1ex \@minus -.2ex}%
{2.3ex \@plus.2ex}%
{\normalfont\large\bfseries}}
\newcommand{\bBigg@{3}}{\bBigg@{3}}
\newcommand{\bBigg@{8}}{\bBigg@{8}}
\makeatother
\numberwithin{equation}{section}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\begin{document}
\thispagestyle{empty}
\begin{center}
{\huge Yang-Mills theory for semidirect products }\\\vskip 6pt %
{\huge ${\rm G}\ltimes\mathfrak{g}^*$ and its instantons} %
\vskip 45pt {F.~Ruiz~Ruiz} \vskip 3pt
\emph{Departamento de F\'{\i}sica Te\'orica I, Universidad Complutense de
Madrid\\ 28040 Madrid, Spain}
\bigskip\bigskip\bigskip
{{Dedicated to Ram\'on~F.~Alvarez-Estrada on occasion of his 70th birthday}}
\vskip 55pt
\end{center}
{\leftskip=30pt\rightskip=30pt
\noindent Yang-Mills theory with a symmetry algebra that is the semidirect
product $\mathfrak{h}\ltimes\mathfrak{h}^*$ defined by the coadjoint action
of a Lie algebra $\mathfrak{h}$ on its dual $\mathfrak{h}^*$ is studied.
The gauge group is the semidirect product ${\rm
G}_{\mathfrak{h}}\ltimes{\mathfrak{h}^*}$, a noncompact group given by the
coadjoint action on $\mathfrak{h}^*$ of the Lie group ${\rm
G}_{\mathfrak{h}}$ of $\mathfrak{h}$. For $\mathfrak{h}$ simple, a method
to construct the self-antiself dual instantons of the theory and their gauge
non\-equivalent deformations is presented. Every ${\rm
G}_{\mathfrak{h}}\ltimes{\mathfrak{h}^*}$ instanton has an embedded ${\rm
G}_{\mathfrak{h}}$ instanton with the same instanton charge, in terms of
which the construction is realized. As an example,
$\mathfrak{h}=\mathfrak{s}\mathfrak{u}(2)$ and instanton charge one is
considered. The gauge group is in this case $SU(2)\ltimes{\bf
R}^3$. Explicit expressions for the selfdual connection, the zero modes
and the metric and complex structures of the moduli space are
given. \\[15pt]%
{\sc keywords:} Gauge theory, classical double, semidirect product,
self-antiself dual instanton, moduli space
\par}
\vspace{30pt}
\section{Introduction}
Motivated by an interest in finding new gauge configurations, we consider
Yang-Mills theory with a symmetry algebra that is the classical double of a real Lie
algebra and study its self-antiself dual solutions. By the classical double of
a real Lie algebra $\mathfrak{h}$, we understand in this paper the semidirect product
$\mathfrak{h}\ltimes\mathfrak{h}^*$ defined by the action of $\mathfrak{h}$ on its dual $\mathfrak{h}^*$ via the
coadjoint representation. Our concern here is Yang-Mills theory with gauge
group the simply connected Lie group ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ obtained from
$\mathfrak{h}\ltimes\mathfrak{h}^*$ by exponentiation.
The group ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ admits several descriptions. From a
geometric point of view, it is the cotangent bundle of the Lie group ${\rm
G}_\mathfrak{h}$ of $\mathfrak{h}$. Algebraically, it can be regarded as the
semidirect product \hbox{${\rm G}_\mathfrak{h}\ltimes{\rm G}_{\mathfrak{h}^*}$} of ${\rm G}_\mathfrak{h}$
with the Lie group ${\rm G}_{\mathfrak{h}^*}$ of~$\mathfrak{h}^*\!$. The cotangent bundle
construction is standard in symplectic mechanics.
The semidirect product approach is not new either in the physics literature.
The Chern-Simons formulation of three-dimensional
gravity~\hbox{\cite{Achucarro-Townsend,Witten-three}} is probably the most
celebrated example of a gauge theory with a gauge group of this type. In that
case, $\mathfrak{h}$ is the Lorentz algebra in three dimensions, $\mathfrak{h}^*$ is the algebra
of three-dimensional translations, \hbox{$\mathfrak{h}\ltimes\mathfrak{h}^*$} is the algebra of
isometries $\mathfrak{i}\mathfrak{s}\mathfrak{o}(1,2)$, and\, ${\rm G}_\mathfrak{h}\ltimes{\rm G}_{\mathfrak{h}^*}$ \,is
the isometry group \hbox{$\textnormal{ISO}(1,2)$}. Other forms of semidirect
products, some involving finite groups, have been employed in various
scenarios, including quantization of monopoles with nonabelian magnetic
charges~\cite{Bais}, neutrino mixing~\cite{Altarelli-Feruglio,King} and
hypercharge quantization~\cite{Hattori,Hashimoto}.
An important property of $\mathfrak{h}\ltimes\mathfrak{h}^*$ is that it is a metric Lie
algebra. This means that it admits an invariant, nondegenerate, symmetric,
bilinear form, called metric, that takes values in~${\bf R}$. The relevance
of this property comes from the observation that if $\mathfrak{g}$ is a metric Lie
algebra and ${\Omega}$ is a metric on it, it is possible to formulate Yang-Mills
theory with gauge group the Lie group ${\rm G}_\mathfrak{g}$ of $\mathfrak{g}$. To do this on a
\hbox{\emph{d}-dimensional} spacetime manifold, introduce a one-form gauge
field ${\kappa}$ and its two-form field strength $K=\textnormal{d}{\kappa}+ {\kappa}\wedge{\kappa}$, both
valued in $\mathfrak{g}$, and consider the Yang-Mills \hbox{\emph{d}-form}\,
\hbox{${\cal L}_{\textnormal{\sc
ym}}={\Omega}\hspace{0.5pt}(K,\star\/K)$}. Nondegeneracy of ${\Omega}$ ensures
that\, ${\cal L}_{\textnormal{\sc ym}}$ \,contains a kinetic term for the
gauge field ${\kappa}$, while invariance of ${\Omega}$ guarantees that\, ${\cal
L}_{\textnormal{\sc ym}}$ is invariant under ${\rm G}_\mathfrak{g}$ gauge
transformations. By considering the classical double $\mathfrak{h}\ltimes\mathfrak{h}^*$, it is
thus possible to define a Yang-Mills theory even if $\mathfrak{h}$ is not
metric. Similarly, four-dimensional topological field theory and
three-dimensional Chern-Simons theory can be considered, with Lagrangians
given by \hbox{${\Omega}\hspace{0.5pt}(K,K)$} \,and\,
\hbox{${\Omega}\hspace{0.5pt}({\kappa},\textnormal{d}{\kappa}+\tfrac{2}{3}{\kappa}\wedge{\kappa})$}.
In view of this, it seems natural to ask how many different real metric Lie
algebras there are. The list of them is exhausted by (i) reductive algebras,
(ii) classical doubles and (iii) double extensions. Reductive algebras are
direct sums of semisimple Lie algebras and the Abelian algebra. They are the
Lie algebras of the compact Lie groups, and their gauge theories have been the
subject of continuous study over the last forty years. Less is known about the
gauge theories for algebras of type (ii) and (iii). Yang-Mills theory for
classical doubles is the object of this paper. As regards double extensions,
they are obtained by a nontrivial generalization~\cite{Medina-Revoy} due to
Medina and Revoy of the semidirect product that defines the classical double.
In fact, a classical double can be regarded as a double extension of the
trivial algebra. These Authors proved a structure theorem that states (a) that
every real metric Lie algebra is an orthogonal sum of indecomposable real
metric Lie algebras, and (b) that every indecomposable real metric Lie algebra
is simple, one-dimensional or the double extension of a metric Lie algebra by
either a simple or a one-dimensional Lie algebra. A discussion of the
theorem can be found in Ref.~\cite{FO-Stanciu-double}. Some Wess-Zumino-Witten
models and gauge theories for double extensions have been considered in
Refs.~\cite{FO-Stanciu-double,Sfetsos,Tseytlin,FO-Stanciu-nonreductive}.
Let us center on the case of interest here, gauge theories with symmetry
algebra $\mathfrak{h}\ltimes\mathfrak{h}^*$. In these theories, the gauge field ${\kappa}$ takes
values in $\mathfrak{h}\ltimes\mathfrak{h}^*$ and has nonzero projections onto~$\mathfrak{h}$
and~$\mathfrak{h}^{*\!}$. New degrees of freedom are thus introduced when $\mathfrak{h}$ is
replaced with $\mathfrak{h}\ltimes\mathfrak{h}^*$. In Section~2, it s shown however that the
homology and homotopy invariants for the group ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ are
the same as for ${\rm G}_\mathfrak{h}$. This has two implications. Homotopically
nontrivial solutions for ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ gauge theory exist if
they do for ${\rm G}_\mathfrak{h}$ gauge theory, and the $\mathfrak{h}^*$-component of the gauge
field ${\kappa}$ does not contribute to the theory's invariants. Here we study
these questions. It will be shown that~${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ instantons
indeed have the same instanton charge as their embedded~${\rm G}_{\mathfrak{h}}$
instantons, but larger moduli spaces. A method to construct~${\rm
G}_{\mathfrak{h}\ltimes\mathfrak{h}^*\!}\cong\/T^{\hspace{1pt}*}{\rm G}_\mathfrak{h}\cong{\rm
G}_\mathfrak{h}\ltimes{\rm G}_{\mathfrak{h}^*}$ instantons and their moduli spaces from those
of~${\rm G}_\mathfrak{h}$ instantons will be presented.
This paper is organized as follows. Section 2 is dedicated to review the
definition and basic properties of $\mathfrak{h}\ltimes\mathfrak{h}^*$ and its Lie group ${\rm
G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$. The Lagrangian and field content of ${\rm
G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ Yang-Mills theory are discussed in Section~3. The
construction of self-antiself dual ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ instantons in
terms of the embedded ${\rm G}_{\mathfrak{h}}$ instantons is presented in Section~4.
This construction is explicitly realized for $\mathfrak{h}=\mathfrak{s}\mathfrak{u}(2)$ and instanton
charge one in Section 5, where expressions for the gauge field, the zero modes
and the metric and complex structures of the moduli space are presented. In
Section~6 we collect our final comments.
\section{The classical double of a Lie algebra and its Lie group}
Let us start by reviewing the construction of the classical double as a
semidirect product. Assume that~$\mathfrak{h}$ is a real Lie algebra of dimension $n$
with basis $\{T_i\}$ satisfying $[T_i,T_j]=f_{ij}{}^kT_k$. Denote by $\mathfrak{h}^*$
its dual vector space, and take for $\mathfrak{h}^*$ the canonical dual basis\,
$\{Z^i\}$, defined by\, \hbox{$Z^i(T_j)={\delta}^i{}_j$}. Form the vector space\,
$\mathfrak{h}\oplus\mathfrak{h}^*$.
Its elements are pairs $(T,Z)$, with $T$ in $\mathfrak{h}$ and $Z$ in $\mathfrak{h}^*$, and as a
basis on it one may take \hbox{$\{(0,T_i),(0,Z^j)\}$}. Consider the semidirect
product $\mathfrak{h}\ltimes\mathfrak{h}^*$ that results from acting with $\mathfrak{h}$ on $\mathfrak{h}^*$ via
the coadjoint representation. For $T$ in $\mathfrak{h}$, the coadjoint representation\,
\hbox{$\textnormal{ad}_T^*\!:\mathfrak{h}^*\to\mathfrak{h}^*$} \,associates\,
$Z\mapsto\textnormal{ad}^*_T Z$, with action on $T^{\hspace{1pt}\prime}$ in
$\mathfrak{h}$ given by\, $\textnormal{ad}^*_T Z(T^{\hspace{1pt}\prime}) =
Z(\textnormal{ad}_T T^{\hspace{1pt}\prime}) =
Z([T,T^{\hspace{1pt}\prime}])$. This results in a Lie algebra of dimension
$2n$ with Lie bracket
\begin{equation}
[(T,Z), (T^{\hspace{1pt}\prime},Z^{\hspace{1pt}\prime})] =
\big(\,[T,T^{\hspace{1pt}\prime}]\,,
- \,\textnormal{ad}^\star_{T}Z^{\hspace{1pt}\prime}\!
+ \textnormal{ad}^\star_{T^{\hspace{1pt}\prime}}Z\,\big)\,.
\label{Lie-bracket}
\end{equation}
For the bases $\{T_i\}$ and $\{Z^i\}$, one has\, $\textnormal{ad}^\star_{T_i}
Z^j(T_k) = f_{ik}{}^j$, so the Lie bracket becomes
\begin{equation}
[T_i,T_j]=f_{ij}{}^kT_k\,,\quad
[T_i,Z^j]=-f_{ik}{}^j\,{Z}^k\,, \quad
[Z^i,Z^j]=0\,.
\label{SDalgebra}
\end{equation}
Here we have introduced the notation, which we will often use,
$T_i+Z^j\!:=(T_i,Z^j)$, so that $T_i\!:=(T_i,0)$ and $Z^i\!:=(0,Z^i)$. The
semidirect product $\mathfrak{h}\ltimes\mathfrak{h}^*$ is a paticular type of Drinfeld
double~\cite{Drinfeld}, namely the one specified by the trivial bialgebra
structure on $\mathfrak{h}$.
Let us also recall that a bilinear symmetric form ${\Omega}$ on a Lie algebra is
invariant if, for all $A,\,B$ and $C$ in the algebra, it satisfies
\begin{equation}
{\Omega}\,(A\,,[B,C])\,={\Omega}\,(\,[A,B]\,,C) \,.
\label{invariance}
\end{equation}
This in turn implies invariance under the a group adjoint action, or more
precisely
\begin{equation}
{\Omega}\,(e^{-C}A\,e^{C},\,e^{-C}\/B\,e^{C}) = {\Omega}\,(A,B)\,.
\label{adjointinvariance}
\end{equation}
Coming back to $\mathfrak{h}\ltimes\mathfrak{h}^*$, it is very easy to see that
\begin{equation}
\begin{tabular}{cccccc}
& & & $T_j$ & $Z^j$ & \\[3pt]
\multirow{2}{*}{${\Omega}=$}
& $T_i$ & \multirow{2}{*}{$\Bigg(\!\!\!\!$} & ${\omega}_{ij}$
&${\delta}_i{}^j$ &\multirow{2}{*}{$\!\!\!\!\Bigg)$} \\[4.5pt]
& $Z^i$ & &${\delta}_i{}^j$ & 0 &
\end{tabular}
\label{SDmetric}
\end{equation}
is nondegenerate and solves condition~(\ref{invariance}) for the
commutators~(\ref{SDalgebra}), where\, ${\omega}_{ij}={\omega}(T_i,T_j)$ \,are the
components of an arbitrary symmetric, \emph{possibly degenerate}, invariant,
bilinear form~${\omega}$ on~$\mathfrak{h}$. Hence $\mathfrak{h}\ltimes\mathfrak{h}^*$ is a real metric Lie
algebra, even if~$\mathfrak{h}$ is not, and ${\Omega}$ is a metric on it.
The algebras $\mathfrak{h}$, $\mathfrak{h}^*$ and $\mathfrak{h}\ltimes\mathfrak{h}^*$ define through
exponentiation simply connected Lie groups that we denote by ${\rm G}_\mathfrak{h},\,
{\rm G}_{\mathfrak{h}^*}$ and ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$. From a geometric point of
view, ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ is the cotangent bundle $T^*{\rm G}_\mathfrak{h}$ of
${\rm G}_\mathfrak{h}$, a standard construction in geometry. $T^*{\rm G}_\mathfrak{h}$ is in
turn isomorphic to the semidirect product ${\rm G}_\mathfrak{h}\ltimes \mathfrak{h}^*$, where
${\rm G}_\mathfrak{h}$ acts on $\mathfrak{h}^*$ by the coadjoint action. For $h$ in ${\rm
G}_\mathfrak{h}$, the coadjoint representation\, ${\rm Ad}^*_h\!:\mathfrak{h}^*\to\mathfrak{h}^*$
\,maps\, $Z$ to ${\rm Ad}_h^*Z$, whose action on $T'$ in $\mathfrak{h}$ \,is given by\,
${\rm Ad}_h^*Z(T^\prime)=Z({\rm Ad}_hT^\prime) = Z(h^{-1}T^\prime\/h)$. The
elements of ${\rm G}_\mathfrak{h}\ltimes\mathfrak{h}^*$ are pairs $(h,Z)$ with product law
$(h_1,Z_1)\,(h_2,Z_2)=(h_1h_2\,, {\rm Ad}^*_{h_2\!}Z_1\!+Z_2)$. Since $h$ in
${\rm G}_\mathfrak{h}$ can be uniquely written as\, $h=e^T\!$, with $T$ in $\mathfrak{h}$, the
derivative of\, ${\rm Ad}_h^*$ \,is the coadjoint action\, ${\rm ad}_T^*$
\,used to construct the semidirect product $\mathfrak{h}\ltimes\mathfrak{h}^*$. As a group,
$\mathfrak{h}^*$ is Abelian, noncompact and homeomorphic to ${\bf R}^{n}$, and
$\{0\}\times\mathfrak{h}^*$ is a normal subgroup. For example, for $\mathfrak{h}=\mathfrak{s}\mathfrak{u}(2)$,
this gives ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}\cong\/SU(2)\ltimes{\bf R}^3$.
One may also adopt the following approach to ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$.
Consider the Cartesian product \hbox{${\rm G}_{\mathfrak{h}}\times{\rm G}_{\mathfrak{h}^*}$},
whose elements are pairs $(h,n)$ that can be uniquely written as
$(e^T\!,e^Z)$, for some $T$ in $\mathfrak{h}$ and some $Z$ in $\mathfrak{h}^*\!$. The
homomorphism\, $\varphi\!: {\rm G}_{\mathfrak{h}}\!\to{\rm Aut}({\rm G}_{\mathfrak{h}^*})$,
where\, \hbox{$\varphi(h)=\varphi_h$} \,acts on\, ${\rm G}_{\mathfrak{h}^*}$ by
conjugation, \hbox{$\varphi_h(n)=h^{-1}nh$}, defines a group structure on
${\rm G}_{\mathfrak{h}}\times{\rm G}_{\mathfrak{h}^*}$. This results in the semidirect product
${\rm G}_{\mathfrak{h}}\ltimes{\rm G}_{\mathfrak{h}^*}$, with group law\, $(h_1,n_1)\,(h_2,n_2)=
\big(h_1h_2,\,(h_2^{-1}n_1\,h_2)\,n_2\big)$ \,and Lie algebra
$\mathfrak{h}\ltimes\mathfrak{h}^*\!$. As a group, ${\rm G}_{\mathfrak{h}^*}$ is Abelian, noncompact and
homeomorphic to~${\bf R}^n_+$. The map\, $[0,1]\times({\rm G}_{\mathfrak{h}}\ltimes{\rm
G}_{\mathfrak{h}^*}) \to {\rm G}_{\mathfrak{h}}\times\{0\}$, given by
\hbox{$\big(t,(h,n)\big)\mapsto(h,tn)$}, is then a homotopy. This means that
${\rm G}_{\mathfrak{h}}\ltimes{\rm G}_{\mathfrak{h}^*}$ and ${\rm G}_{\mathfrak{h}}\times\{0\}$ are
homotopically equivalent, hence have the same homology and homotopy
invariants. In particular, they have the same third homotopy group. For the
elements of ${\rm G}_{\mathfrak{h}}\ltimes{\rm G}_{\mathfrak{h}^*}$ we will use the notation
$g=hn=(h,n)$. It is clear that ${\rm G}_{\mathfrak{h}}\ltimes{\rm G}_{\mathfrak{h}^*}$ and ${\rm
G}_{\mathfrak{h}}\ltimes\mathfrak{h}^*$ are isomorphic.
We finish this section with two comments, one on representations and one on
deformations.
\medskip {\bf Comment 1}.
Given any \hbox{\emph{p}-dimensional} matrix representation of $\mathfrak{h}$ that
associates to its basis $\{T_i\}$ matrices\, $\{{\bf M}_i\}$ \,with\, $[{\bf
M}_i,{\bf M}_j]=f_{ij}{}^k\,{\bf M}_k$, it is very easy to see that
\begin{equation}
\renewcommand*{\arraystretch}{1}
\rho(T_i,0)= \left(\begin{array}{c|c}
{\bf M}_i & 0\\\hline 0 & {\bf M}_i \end{array} \right),\quad
\rho(0,Z_i)= \left(\begin{array}{c|c}
0 & 0 \\\hline {\bf M}_i& 0 \end{array} \right)
\label{2p-representation}
\end{equation}
is a \hbox{$2p$-dimensional} matrix representation of $\mathfrak{h}\ltimes\mathfrak{h}^*$. In
the adjoint representation of $\mathfrak{h}$, the matrices $\{{\bf M}_i\}$ are\,
$n{\scriptstyle\times}n$ and have entries\, $({\bf M}_i^{\rm
ad})_j{}^k\!=\!-f_{ij}{}^k$. It is straightforward to check that $\rho$
above is then the adjoint representation of $\mathfrak{h}\ltimes\mathfrak{h}^*$. Representations
other than~(\ref{2p-representation}) are possible. An example is the
following. Let~${\bf e}_i$ be the unit column vector in ${\bf R}^n$, with
components $({\bf e}_i)_j\!={\delta}_{ij}$. Some simple algebra shows that the
matrices
\begin{equation}
\renewcommand*{\arraystretch}{1}
\rho^{\,\prime}(T_i,0) = \left(\begin{array}{c|c}
{\bf M}_{i}^{\rm ad} & 0\\\hline 0 & 0 \end{array} \right),\quad
\rho^{\,\prime}(0,Z_i)= \left(\begin{array}{c|c}
0 & {\bf e}_i\\\hline 0 & 0 \end{array} \right)
\label{adjoint-representation}
\end{equation}
form a $(n\!+\!1)$-dimensional representation of $\mathfrak{h}\ltimes\mathfrak{h}^*$. Note
finally that every matrix representation of $\mathfrak{h}\ltimes\mathfrak{h}^*$ induces a matrix
representation of ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ via matrix exponentiation.
\medskip{\bf Comment 2}. Assume that the algebra $\mathfrak{h}$ is metric, so that
${\omega}_{ij}$ in eq.~(\ref{SDmetric}) can be taken as the components of a metric.
One may use ${\omega}_{ij}$ and its inverse ${\omega}^{ij}$, given by
\hbox{${\omega}^{ik}{\omega}_{kj\!}={\delta}^i{\!}_j$}, to lower and raise indices in the
structure constants $f_{ij}{}^k$. This yields completely antisymmetric
structure constants
\begin{equation}
f_{ijk\!} =f_{ij}{}^l{\omega}_{lk}\,={\omega}([T_i,T_j],T_k)\,,\qquad f_{ijk}=-f_{jik}=f_{kji}\,.
\label{lowemetric}
\end{equation}
Perform in $\mathfrak{h}^*$ the change of generators $\{Z^i\}\! \to\! \{Z_i\}$,
with\, $Z_i={\omega}_{ik}Z^j$. This gives
\begin{equation}
[T_i,T_j]=f_{ij}{}^kT_k\,,\quad
[T_i,Z_j]=f_{ij}{}^k\,{Z}_k\,, \quad
[Z_i,Z_j]=0\,.
\label{undeformedSDalgebra}
\end{equation}
Consider the commutators
\begin{equation}
[T_i,T_j]=f_{ij}{}^kT_k\,,\quad
[T_i,Z_j]=f_{ij}{}^k\,{Z}_k\,, \quad
[Z_i,Z_j]=s^2 f_{ij}{}^k T_k\,,
\label{deformedSDalgebra}
\end{equation}
where $s$ in $[Z_i,Z_j]$ is an arbitrary real parameter.
These commutators satisfy the Jacobi identity for all $s$ and reduce to the
Lie bracket~(\ref{undeformedSDalgebra}) of the classical double when
$s\to\/0$. The vector space $\mathfrak{h}\oplus\mathfrak{h}^*$ with the Lie
bracket~(\ref{deformedSDalgebra}) is thus a Lie algebra, call it
$\mathfrak{h}\ltimes_{\!s}\mathfrak{h}^{*\!}$, and a deformation of $\mathfrak{h}\ltimes\mathfrak{h}^*$ with
deformation parameter $s$. The algebra $\mathfrak{h}\ltimes_{\!s}\mathfrak{h}^*$ is metric
since it admits the metric %
\begin{equation}
\begin{tabular}{cccccc}
& & & $T_j$ & $Z_j$ & \\[3pt]
\multirow{2}{*}{${\Omega}_s=$}
& $T_i$ & \multirow{2}{*}{$\Bigg(\!\!\!\!$} & ${\omega}_{ij}$
&${\omega}_{ij}$ &\multirow{2}{*}{$\!\!\!\!\Bigg)$\,.} \\[4.5pt]
& $Z_i$ & &${\omega}_{ij}$ & $s^2 {\omega}_{ij}$ &
\end{tabular}
\label{deformedSDmetric}
\end{equation}
In $\mathfrak{h}\ltimes_{\!s}\mathfrak{h}^*$ introduce generators $\{X_i,Y_j\}$ given by
\begin{equation}
X_i = \frac{1}{2}\,\Big( T_i+\frac{1}{s}\,Z_i\Big)\,,\qquad
Y_i = \frac{1}{2}\,\Big( T_i-\frac{1}{s}\,Z_i\Big)\,.
\label{generatorsXY}
\end{equation}
In the new basis, the Lie bracket~(\ref{deformedSDalgebra}) becomes
\begin{equation}
[X_i,X_j]=f_{ij}{}^kX_k\,,\quad
[X_i,Y_j]=0\,, \quad
[Y_i,Y_j]= f_{ij}{}^k Y_k\,,
\label{deformedSDalgebraXY}
\end{equation}
and the metric ${\Omega}_s$ takes the diagonal form%
\begin{equation}
\begin{tabular}{cccccc}
& & & $X_j$ & $Y_j$ & \\[3pt]
\multirow{2}{*}{${\Omega}_s=$}
& $X_i$ & \multirow{2}{*}{$\bBigg@{3}(\!\!\!\!$}
& $\frac{1}{2}\,\big(1+\frac{1}{s}\big)\,{\omega}_{ij}$
& 0 &\multirow{2}{*}{$\!\!\!\!\bBigg@{3})\,.$} \\[4.5pt]
& $Y_j$ & & 0 & $\frac{1}{2}\,\big(1-\frac{1}{s}\big)\,{\omega}_{ij}$ & \\[3pt]
\end{tabular}
\label{deformedSDmetricXY}
\end{equation}
The deformed algebra $\mathfrak{h}\ltimes_{\!s}\mathfrak{h}^*$ is thus the direct sum
$\mathfrak{h}\oplus\mathfrak{h}$ and its simply connected Lie group ${\rm
G}_{\,\mathfrak{h}\ltimes_{\scriptstyle s}\mathfrak{h}^*}$ becomes the direct product ${\rm
G}_{\mathfrak{h}\!}\times{\rm G}_\mathfrak{h}$.
\section{The gauge theory and its field content}
Our interest here is Yang-Mills theory with gauge group $G_{\mathfrak{h}\ltimes\mathfrak{h}^*}$.
Consider a spacetime manifold~$M_d$ of dimension~$d$ equipped with a metric
${\gamma}$. Greek letters ${\mu},{\nu},\ldots$ will label coordinate indices\, $1,2,
\ldots, d$ \,in a local chart $\{x^{\mu}\}$. In such a chart, ${\gamma}_{{\mu}{\nu}}$ will
denote the metric components and ${\gamma}^{{\mu}{\nu}}$ the components of the
inverse metrif. For an \hbox{\emph{r}-form} $\zeta$ we will adopt the normalization\,
\hbox{$\zeta=\frac{1}{r!}\,\zeta_{{\mu}_1\cdots{\mu}_r}\,\textnormal{d}\/x^{{\mu}_1\!}
\wedge\cdots\wedge\/\textnormal{d}\/x^{{\mu}_r}$}. Indices will be raised and lowered
using ${\gamma}^{{\mu}{\nu}}$ and ${\gamma}_{{\mu}{\nu}}$. For the commutator of an
\hbox{\emph{r}-form} $\zeta$ with an \hbox{\emph{s}-form}~$\xi$, both taking
values in $\mathfrak{h}\ltimes\mathfrak{h}^*$, we will use\,
\hbox{$[\,\zeta,\xi\,]=\zeta\wedge\xi-(-)^{rs}\,\xi\wedge\zeta$}.
The gauge filed is a connection one-form ${\kappa}$ on $M_d$ that takes values in
$\mathfrak{h}\ltimes\mathfrak{h}^*$. The connection defines a covariant
derivative~$\textnormal{d}_{\displaystyle {\kappa}}$, whose action on an
\hbox{($\mathfrak{h}\ltimes\mathfrak{h}^*$)}-valued \hbox{\emph{r}-form}~$\zeta$ \,is given by\,
$\textnormal{d}_{\displaystyle {\kappa}}\,\zeta=\textnormal{d}\zeta+[\,{\kappa},\zeta]$, and a curvature
two-form or field strength
\begin{equation}
K=\textnormal{d}\/{\kappa} + \tfrac{1}{2}\,[{\kappa},{\kappa}] \,.
\label{fieldstrength}
\end{equation}
The curvature takes values in $\mathfrak{h}\ltimes\mathfrak{h}^*$ and satisfies the Bianchi
identity\, $\textnormal{d}_{\displaystyle {\kappa}}\/K=0$. Gauge transformations
\begin{equation}
{\kappa} \to {\kappa}'\! = g^{-1}\,\textnormal{d}\/g + g^{-1}\,{\kappa}\,g\,,
\end{equation}
are implemented by\, ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*\!}$ valued functions
$g(x)$. Under such transformations, the curvature changes as
\begin{equation}
K \to K'\!=g^{-1}Kg\,.
\label{GT-K}
\end{equation}
As usual, infinitesimal gauge transformation are obtained by expanding
$g=e^Te^Z$ in powers of $T$ and $Z$ and keeping terms up to order one. With\,
$\Lambda\!:=T+Z$, they read
\begin{align}
{\kappa} \to {\kappa}'&={\kappa} + \textnormal{d}_{\displaystyle {\kappa}}\,\Lambda \,,\\
K \to K'&= K+[\,K,\,\Lambda\,]\,.
\end{align}
Consider the \hbox{$d$-form}\, \hbox{${\Omega}\hspace{0.5pt}(K,\star\/K)$}, where
$\star\/K$ is the Hodge dual of~$K$ and~${\Omega}$ is an invariant metric on
$\mathfrak{h}\ltimes\mathfrak{h}^*$. The transformation law~(\ref{GT-K}) for $K$, the
observation that any $g$ can be written as $g=e^Te^Z$, and the invariance
condition~(\ref{adjointinvariance}) imply that
\hbox{${\Omega}\hspace{0.5pt}(K,\star\/K)$} \,remains unchanged under gauge
transformations. The functional
\begin{equation}
S_{\textnormal{\sc ym}}= \frac{1}{8\pi^2}
\int_{M_d} {\Omega}\hspace{0.5pt}(K,\star\/K)
= \frac{1}{16\pi^2}\int_{M_d}\!\!\sqrt{{\gamma}\,}~\textnormal{d}\/^d\hspace{-0.5pt}x ~
{\Omega}\hspace{0.5pt}\big(K^{{\mu}{\nu}\!},K_{{\mu}{\nu}}\big)
\label{YMaction}
\end{equation}
is thus gauge invariant and can be taken as the classical action of ${\rm
G}_{\mathfrak{h}\ltimes\mathfrak{h}^*\!}$ Yang-Mills theory. Variation of $S_{\textnormal{\sc
ym}}$ with respect to ${\kappa}$ gives for the field equation
\begin{equation}
\textnormal{d}_{\kappa}\!\/\star\!K=0\,.
\end{equation}
For $d\geq\/4$, it is also possible to consider the gauge invariant
four-form\, \hbox{${\Omega}\hspace{0.5pt}(K,K)$}. Since
\hbox{${\Omega}\hspace{0.5pt}(K,K)$} does not require a metric, it can be regarded
as the Lagrangian of a topological field theory in four dimensions, the
classical action being
\begin{equation}
S_{\textnormal{\sc p}}= \frac{1}{8\pi^2}
\int_{M_4} {\Omega}\hspace{0.5pt}(K,K)\,.
\end{equation}
The form \hbox{${\Omega}\hspace{0.5pt}(K,K)$} is the first Pontrjagin class of
the principal bundle over $M_d$ with structure group ${\rm
G}_{\mathfrak{h}\ltimes\mathfrak{h}^*\!}$\,, and the exterior derivative of a Chern-Simons
three-form. That is, ${\Omega}\hspace{0.5pt}(K,K) =\textnormal{d}\/{\cal L}_{\textnormal{\sc
cs}}({\kappa})$ \,with
\begin{equation}
{\cal L}_{\textnormal{\sc cs}}({\kappa})
={\Omega}\hspace{0.5pt}\big({\kappa}\,,\,
\textnormal{d}{\kappa} + \tfrac{2}{3}\>{\kappa}\wedge\/{\kappa}\big)\,.
\end{equation}
In analogy with the case of semisimple Lie algebras, one may formulate
Chern-Simons field theory on a three-dimensional manifold $M_3$ with the
classical action
\begin{equation}
S_{\textnormal{\sc cs}} = \frac{1}{8\pi^2} \int_{M_3}
{\cal L}_{\textnormal{\sc cs}}({\kappa})\,.
\end{equation}
The connection ${\kappa}$ and the curvature $K$ can be expanded in the Lie algebra
basis $\{T_i,Z^j\}$ as
\begin{alignat}{6}
{\kappa} & = \a + \b\,, &\quad &\a:=\a^iT_i, &\quad &\b:= \b_iZ^i,\\
K &= F + B\,, &\quad& F:= F^iT_i, &\quad& B:=B_iZ^i,
\end{alignat}
where $\a^i$ and $\b_i$ are one-forms on $M_d$, and $F^i$ and $B_i$ are
two-forms. Substitution in eq.~(\ref{fieldstrength}) gives
\begin{alignat}{6}
&F = \textnormal{d}\a + \tfrac{1}{2}\,[\a,\a]
& &\Leftrightarrow& &
F^i =\textnormal{d}\a^i + \tfrac{1}{2}\,f_{jk}{}^i\, \a^j\wedge\a^k ,
\label{curvature1-SD} \\[2pt]
&B = \textnormal{d}\b + [\,\a,\b\,] &\quad&\Leftrightarrow&\quad&
B_i= \textnormal{d}\b_i + f_{ij}{}^k\, \a^j\wedge\b_k\,.
\label{curvature2-SD}
\end{alignat}
In infinitesimal form, gauge transformations read
\begin{align}
\a\to\a' & =\a+\textnormal{d}\/T +[\a, T]\,,\label{GT-infinitesimal-alpha} \\
\b\to\b'&=\b+\textnormal{d}\/Z+[\a,Z] + [\b,T]\,,
\label{GT-infinitesimal-beta}
\end{align}
whereas for the field strength they become
\begin{align}
F\to\/F' &=F+[F, T]\,,\\
B\to\/B' &=B+[B,T] + [F,Z]\,.
\end{align}
The Bianchi identity\, $\textnormal{d}_{\kappa}\/K=0$ \,unfolds in two identities
\begin{align}
& \textnormal{d}\/F + [\,\a, F\,] = 0 \,,
\label{BI1-SD}\\
&\textnormal{d}\/B + [\,\a,B\,] + [\,\b,F\,] =0\,,
\label{BI2-SD}
\end{align}
and the field equation $\textnormal{d}_{\kappa}\!\star\/K=0$ splits in
\begin{align}
& \textnormal{d}\star\/F + [\,\a, \star\/F\,] = 0 \,,
\label{FE1-SD} \\
& \textnormal{d}\star\/B + [\,\a,\star\/B\,] + [\,\b,\star\/F\,]=0 \,.
\label{FE2-SD}
\end{align}
There are a few observations that, despite their simplicity, are worth making.
Firstly, the curvature $F$ has the same dependence on $\a$ that results from
gauging the algebra~$\mathfrak{h}$. It is $B$ that mixes $\a$ with $\b$. Secondly, the
Lagrangian ${\Omega}\hspace{-0.5pt}(K,\star\/K)$ has a kinetic term for all the
field components $\a^i$ and $\b_i$ of the gauge field ${\kappa}$. Note in this
regard that, for ${\omega}$ degenerate, ${\omega}(F,\star\/F)$ does not define a
Yang-Mills Lagrangian since it does not contain a kinetic term for all the
$\a^i$. Thirdly, the field strength $B$, its Bianchi identity~(\ref{BI2-SD})
and its field equation~(\ref{FE2-SD}) are linear in $\b$. And lastly, the
field equations~(\ref{FE1-SD}) and~(\ref{FE2-SD}) do not depend on~${\omega}$.
The Pontrjagin and Chern-Simons forms read
\begin{equation}
{\Omega}\hspace{0.5pt}(K,K) = {\omega}(F,F) + 2\,{\Omega}(F,B)
\label{Pon-SD}
\end{equation}
and
\begin{equation}
{\cal L}_{\textnormal{\sc cs}}\hspace{0.5pt}({\kappa}) =
{\cal L}_{\textnormal{\sc cs}}\hspace{0.5pt}(\a) +
2\,{\Omega}(\b,F) + \textnormal{d}\hspace{0.5pt}{\Omega}\hspace{0.5pt}(\b,\a)\,.
\label{simplified-CS}
\end{equation}
The first term on the right hand side in eq.~(\ref{simplified-CS}) is the
Chern-Simons three-form for $\a$ computed with the invariant bilinear
form~${\omega}$,
\begin{equation}
{\cal L}_{\textnormal{\sc cs}}\hspace{0.5pt}(\a)
={\omega}\hspace{1pt}
\big(\a,\textnormal{d}\a+\tfrac{2}{3}\,\a\wedge\a\big)\,.
\label{CS-ah}
\end{equation}
For~$\mathfrak{h}$ the Lorentz algebra in three dimensions, the metric ${\Omega}$ has the
form in eq.~(\ref{SDmetric}) and ${\cal L}_{\textnormal{\sc cs}}({\kappa})$ in
eq.~(\ref{simplified-CS}) gives, for ${\omega}_{ij\!}=0$, the Chern-Simons
Lagrangian of three-dimensional gravity~\cite{Achucarro-Townsend,Witten-three}
modulo an exact form.
\section{Semidirect instantons: general analysis}
Let us turn our attention to self-antiself dual instantons on ${\bf
R}^4$. They are described by connections~${\kappa}_{\textnormal{\sc s}}$ that
solve equation\, $\star\/K\!=\!\pm\/K$, where the positive sign corresponds to
selfduality and the negative sign to anti-selfduality. For such connections,
the field equation reduces to the Bianchi identity, thus is trivially
satisfied, and $S_{\textnormal{\sc ym}}[{\kappa}_{\textnormal{\sc
s}}]=S_{\textnormal{\sc p}}[{\kappa}_{\textnormal{\sc s}}]$. Since the
Pontrjagin index $S_{\textnormal{\sc p}}[{\kappa}_{\textnormal{\sc s}}]$ is a
homotopy invariant and homotopy invariants are the same as for ${\rm G}_\mathfrak{h}$
gauge theory, one has
\begin{equation}
S_{\textnormal{\sc ym}}\big[{\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*} \hspace{0.5pt};
{\kappa}_{\textnormal{\sc s}}\big] = \pm
S_{\textnormal{\sc p}}\big[{\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*} \hspace{0.5pt};
{\kappa}_{\textnormal{\sc s}}\big] = \pm
S_{\textnormal{\sc p}}\big[{\rm G}_\mathfrak{h}\hspace{0.5pt};
\a_{\textnormal{\sc s}}\big] =
S_{\textnormal{\sc ym}}\big[{\rm G}_\mathfrak{h}\hspace{0.5pt};
\a_{\textnormal{\sc s}}\big]\,.
\end{equation}
Finiteness of the Yang-Mills action on the rightmost side of this equation
requires the curvature $\mathfrak{h}$-component $F_{\textnormal{\sc s}}$ to approach
zero at the three-sphere $\textnormal{S}^3_\infty$ at infinity. This in turn
demands $\a_{\textnormal{\sc s}}$ to approach a pure gauge configuration.
That is, $\a_{\textnormal{\sc s}}\to\/h^{-1}\textnormal{d}\/h$ \,at\,
$\textnormal{S}^3_\infty$ \,for some $h$ in ${\rm G}_\mathfrak{h}$. Note that no boundary
condition for $\b_{\textnormal{\sc s}}$ is needed. These arguments can be made
more explicit by noting that\, $S_{\textnormal{\sc p}}[{\kappa}]$ \,is the integral
over $\textnormal{S}^3_\infty$ of the Chern-Simons three-form ${\cal
L}_{\textnormal{\sc cs}}({\kappa})$ in eq.~(\ref{simplified-CS}). For a
connection\, \hbox{${\kappa}=(\a,\b)$} \,that approaches\,
$(\a_\infty\!=h^{-1}\textnormal{d}\/h\,,\,\b_{\infty})$ \,at\, $\textnormal{S}^3_\infty$,
with\, $\b_{\infty}$ arbitrary, eq.~(\ref{simplified-CS}) and $F_{\infty}\!=0$
imply that\, $S_{\textnormal{\sc p}}[{\rm
G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}\hspace{0.5pt};{\kappa}] = S_{\textnormal{\sc p}}[{\rm
G}_\mathfrak{h}\hspace{0.5pt}; \a]$.
All in all, the instanton charge, call it $N$, and the boundary conditions for
a self-antiself dual ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ instanton
${\kappa}_{\textnormal{\sc s}}=({\a}_{\textnormal{\sc s}},{\b}_{\textnormal{\sc
s}})$ are specified by those of the embedded ${\rm G}_\mathfrak{h}$ instanton,
\begin{equation}
N = S_{\textnormal{\sc p}}\big[{\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}\hspace{0.5pt};
{\a}_{\textnormal{\sc s}},{\b}_{\textnormal{\sc s}}\big] = \frac{1}{8\pi^2}
\int_{{\bf R}^4} {\omega}\hspace{1pt}({F}_{\textnormal{\sc s}},
{F}_{\textnormal{\sc s}})\,.
\label{Pon-reduced}
\end{equation}
This implies in particular that $\b_{\textnormal{\sc s}}$ does not contribute
to the instanton charge,
\begin{equation}
\frac{1}{8\pi^2}\int_{{\bf R}^4} {\Omega}\hspace{1pt}({F}_{\textnormal{\sc s}},
{B}_{\textnormal{\sc s}})=0\, .
\label{Pon-check}
\end{equation}
The self-antiself duality equation\, $\star\/K\!=\!\pm\/K$ \,splits in
\begin{alignat}{4}
\star\/F=\pm\/F &~~\Leftrightarrow~~&
\star\,\big(\, \textnormal{d}\a + \tfrac{1}{2}\;[\a,\a]\, \big)
& = \pm\, \big(\, \textnormal{d}\a + \tfrac{1}{2}\;[\a,\a]\, \big)\,,
\label{selfdualF} \\[6pt]
\star\/B=\pm\/B &~~\Leftrightarrow~~&
\star\,\big(\, \textnormal{d}\b + [\a,\b]\, \big)
& = \pm\, \big(\, \textnormal{d}\b + [\a,\b]\, \big) \, .
\label{selfdualB}
\end{alignat}
Equation~(\ref{selfdualF}) and the boundary condition\, $\a\to\/h^{-1}\textnormal{d}\/h$
set a differential problem for~$\a$, whose solutions are the self-antiself
dual ${\rm G}_\mathfrak{h}$ instantons. For every solution ${\a}_{\textnormal{\sc
s}}$, equation~(\ref{selfdualB}) becomes an homogeneous linear
differential problem for $\b$, with solution ${\b}_{\textnormal{\sc s}}$. In
what follows we present a method to find the most general solution
$\b_{\textnormal{\sc s}}$ for a given ${\a}_{\textnormal{\sc s}}$.
Take $\mathfrak{h}$ to be simple and ${\omega}_{ij}$ in eq.~(\ref{SDmetric}) a metric on
$\mathfrak{h}$. This is the case of all self-antiself dual ${\rm G}_\mathfrak{h}$ instantons
known to date~\cite{BPST, 'tHooft-unpublished, JNR, WittenSU2, ADHM, CWS,
CFGT, Schwarz, Jackiw-Rebbi, AHS, BCL}. Introduce generators
$Z_i={\omega}_{ij}Z^j$. The commutation relations for $\{T_i,Z_j\}$ and the metric
${\Omega}$ take the form~(\ref{undeformedSDalgebra}) and~(\ref{deformedSDmetric}).
Since any gauge field ${\kappa}^{\,\prime\!}=(\a^{\,\prime\!},\b^{\,\prime})$
obtained from a solution ${\kappa}_{\textnormal{\sc s}}=(\a_{\textnormal{\sc
s}},\b_{\textnormal{\sc s}})$ by a ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ gauge
transformation is trivially a solution, we restrict our attention to gauge
nonequivalent solutions. The space of all such solutions with instanton charge
$N$ is the moduli space ${\cal M}_N({\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*})$.
Standard arguments~\cite{Tong, Weinberg-book} show that if
${\kappa}_{\textnormal{\sc s}}$ is a solution to the self-antiself duality
equation, \hbox{${\kappa}^{\,\prime\!}={\kappa}_{\textnormal{\sc s}}+{\delta}{\kappa}$} \,is a
gauge nonequivalent solution if ${\delta}{\kappa}$ satisfies the equation
\begin{equation}
\textnormal{d}_{\displaystyle{{\kappa}}_{\textnormal{\sc s}}}{\delta}{\kappa}
=\star\,\textnormal{d}_{\displaystyle{{\kappa}}_{\textnormal{\sc s}}}{\delta}{\kappa}
\label{pert-kappa}
\end{equation}
and the gauge fixing condition
\begin{equation}
\textnormal{d}_{\displaystyle{{\kappa}}_\textnormal{\sc s}}\star{\delta}{\kappa} = 0 \,.
\label{GFkappa}
\end{equation}
Any infinitesimal local gauge transformation
$\textnormal{d}_{{\displaystyle{{\kappa}}_{\textnormal{\sc s}}}}\Lambda=\textnormal{d}\Lambda
+[{\kappa}_{\textnormal{\sc s}},\Lambda]$, with $\Lambda$ in $\mathfrak{h}\ltimes\mathfrak{h}^*$,
solves equation~(\ref{pert-kappa}). The solutions ${\delta}{\kappa}$
to~(\ref{pert-kappa}) may then include a
transformation of this type. The point is that for ${\kappa}'$ and
${\kappa}_{\textnormal{\sc s}}$ to be gauge nonequivalent, ${\delta}{\kappa}$ cannot just be
an infinitesimal gauge transformation, and this is what eq.~(\ref{GFkappa})
takes care of.
Expand ${\delta}{\kappa}$ in the basis $\{T_i,Z_j\}$~as
\begin{equation}
{\delta}{\kappa}= {\delta}\a + {\delta}\b, \quad {\delta}\a :={\delta}\a^i\,T_i, \quad
{\delta}\b:= {\delta}\b^{\,i}\,Z_i,
\label{pert-expansion}
\end{equation}
and substitute these expansions in eqs.~(\ref{pert-kappa})
and~(\ref{GFkappa}). This gives for~${\delta}\a$ and ${\delta}\b$ the equations
\begin{eqnarray}
& \star\, \big(\, \textnormal{d}\,{\delta}\a + [\a_{\textnormal{\sc s}},{\delta}\a]\, \big)
= \pm\, \big( \, \textnormal{d}\,{\delta}\a + [\a_{\textnormal{\sc s}},{\delta}\a]\, \big)\,, &
\label{pert-alpha} \\[3pt]
& \textnormal{d}\star{\delta}\a +\, [\hspace{1pt}\a_{\textnormal{\sc s}}\hspace{1pt},
\star\,{\delta}\a\hspace{1pt}] = 0\,, &
\label{GFalpha}
\end{eqnarray}
and
\begin{eqnarray}
& \star \big(\, \textnormal{d} \,{\delta}\b + [\a_{\textnormal{\sc s}}, {\delta}\b]
+ [\b_{\textnormal{\sc s}},{\delta}\a] \,\big) =
\pm\, \big(\, \textnormal{d} \,{\delta}\b + [\a_{\textnormal{\sc s}}, {\delta}\b]
+ [\b_{\textnormal{\sc s}},{\delta}\a]\, \big) \,,
& \label{pert-beta} \\[3pt]
& \textnormal{d}\star{\delta}\b
+ [\hspace{1pt}\a_{\textnormal{\sc s}}\hspace{1pt},
\star\,{\delta}\b\hspace{1pt} ]
+ [\hspace{1pt}\b_{\textnormal{\sc s}}\hspace{1pt},
\star\,{\delta}\a\hspace{1pt}] = 0\,.
& \label{GFbeta}
\end{eqnarray}
The solutions ${\delta}{\kappa}_{\textnormal{\sc s}}=({\delta}\a_{\textnormal{\sc
s}},{\delta}\b_{\textnormal{\sc s}})$ to these equations describe gauge
nonequivalent displacements in the moduli space ${\cal M}_N({\rm
G}_{\mathfrak{h}\ltimes\mathfrak{h}^*})$. We will use standard terminology and refer to them
as zero modes (since they are the zero modes of a linear differential
operator).
In eqs.~(\ref{selfdualF}), (\ref{pert-alpha}) and (\ref{GFalpha}) one
recognizes the problem of charge $N$ self-antiself dual~${\rm G}_\mathfrak{h}$
instantons and their zero modes. Given its solution $\{\a_{\textnormal{\sc
s}}, {\delta}\a_{\textnormal{\sc s}}\}$, we want to solve
eqs.~(\ref{selfdualB}), (\ref{pert-beta}) and~(\ref{GFbeta}) for $\b$ and
${\delta}\b$. Let us first understand the solution to the ${\rm G}_\mathfrak{h}$ problem. A
solution~$\a_{\textnormal{\sc s}}$ to eq.~(\ref{selfdualF}) depends on a set
of free parameters~$\{u^a\}$ that describe instanton degrees of freedom and
that occur in the differential problem as integration constants~\cite{CWS,
CFGT, Schwarz, Jackiw-Rebbi, AHS, BCL, Tong, Weinberg-book}. In the ADHM
approach, $\{u^a\}$ appear as free parameters in the quaternion matrices in
terms of which $\a_{\textnormal{\sc s}}$ is constructed. Using that partial
derivatives ${\partial}/{\partial}\/u^a$ commute with the exterior differential~$\textnormal{d}$ and
noting the Jacobi identity for the generators $\{T_i\}$ of $\mathfrak{h}$, it is
trivial to check that (i)~derivatives ${\partial}\a_{\textnormal{\sc s}}/{\partial}\/u^a$ of
$\a_{\textnormal{\sc s}}$ along $u^a$ and (ii)~rotations $[\a_{\textnormal{\sc
s}},T_i]$ of $\a_{\textnormal{\sc s}}$ about $T_i$ solve the moduli
equation~(\ref{pert-alpha}). The problem is that they may not satisfy the
gauge fixing condition~(\ref{GFalpha}). To correct this, one includes
infinitesimal local ${\rm G}_\mathfrak{h}$ transformations and writes for the zero
modes
\begin{align}
{\delta}_{(a)}\a_{\textnormal{\sc s}} & =
\frac{{\partial}\a_{\textnormal{\sc s}}}{{\partial}\/u^a} + \textnormal{d}\hspace{1pt}t_{(a)}
+ [\a_{\textnormal{\sc s}}\hspace{1pt},t_{(a)}] \,,\label{solalpha-1}\\[4.5pt]
{\delta}_{(i)}\a_{\textnormal{\sc s}} & =
[\a_{\textnormal{\sc s}},T_i] + \textnormal{d}\hspace{1pt}t_{(i)}
+ [\a_{\textnormal{\sc s}}\hspace{1pt},t_{(i)}] \,, \label{solalpha-2}
\end{align}
where $t_{(a)\!}=t_{(a)}^{\,j} T_j$ \,and\, $t_{(i)\!}=t_{(i)}^{\,j} T_j$ are
$\mathfrak{h}$-valued functions that must be chosen so that eq.~(\ref{GFalpha}) holds.
The zero modes\, ${\delta}_{(a)}\a_{\textnormal{\sc s}} $ \,and\,
${\delta}_{(i)}\a_{\textnormal{\sc s}} $ \,give the gauge nonequivalent
deformations of $\a_{\textnormal{\sc s}}$. Introducing angles~$\tau^i$ for
the rotations around $T_i$, one may take $\{u^a,\tau^i\}$ as local coordinates
on the moduli space of charge $N$ self-antiself dual ${\rm G}_\mathfrak{h}$ instantons
${\cal M}_N({\rm G}_\mathfrak{h})$.
\medskip{\bf The connection}. We now turn to equation~(\ref{selfdualB}).
Writing\, $\b=\b^{\,i}Z_i$ \,and noting the commutation relations\,
\hbox{$[T_i,T_j]=f_{ij}{}^kT_k$} \,and\, \hbox{$[T_i,Z_j]=f_{ij}{}^kZ_k$},
eq.~(\ref{selfdualB}) gives for~$\b^{\,i}$ the same equation as the moduli
equation~(\ref{pert-alpha}) gives for the components ${\delta}\a^i$ of ${\delta}\a$. The
latter is solved by derivatives ${\partial}\a_{\textnormal{\sc s}}/{\partial}\/u^a$ and
rotations $[\a_{\textnormal{\sc s}},T_i]$. Hence, modulo gauge
transformations, the most general solution for $\b$ is a linear combination
\begin{equation}
\b _{\textnormal{\sc s}}= \sum_{a}\,\tilde{u}^a\,
\frac{{\partial}\a^i_{\textnormal{\sc s}}}{{\partial}\/u^a}\;Z_i
+ \tilde{\tau}^i\, [\a_{\textnormal{\sc s}}\hspace{1pt},Z_{i}]
\label{betageneral}
\end{equation}
with arbitrary coefficients $\tilde{u}^a$ and $\tilde{\tau}^i$. Upon
substitution in eq.~(\ref{curvature2-SD}), the $\mathfrak{h}^*$-component of the
curvature becomes
\begin{equation}
B_{\textnormal{\sc s}} = \sum_{a}\,
\tilde{u}^a\,\frac{{\partial}\/F^i_{\textnormal{\sc s}}}{{\partial}\/u^a}\;Z_i
+ \tilde{\tau}^i\, [F_{\textnormal{\sc s}},Z_i]\,.
\label{Bgeneral}
\end{equation}
This is trivially self-antiself dual and does not contribute to the instanton
charge. To check the latter, use that\, ${\Omega}\hspace{0.5pt}\big(F,
[F,Z_i\,]\big)=0$ \,for any two form $F$, so that
\begin{equation}
\int_{{\bf R}^4} {\Omega}\hspace{1pt}(F_{\textnormal{\sc s}},
B_{\textnormal{\sc s}})=
\frac{1}{2}\; \sum_{a}\,\tilde{u}^a\,\frac{{\partial}}{{\partial}\/u^a}
\int_{{\bf R}^4} {\omega}\hspace{1pt}(F_{\textnormal{\sc s}},
F_{\textnormal{\sc s}})\,.
\label{Pon-check-2}
\end{equation}
Since $\int\!{\omega}\hspace{1pt}(F_{\textnormal{\sc s}}, F_{\textnormal{\sc s}})$
is a constant, equal to $8\pi^2N$, with $N$ the charge of the~${\rm G}_\mathfrak{h}$
instanton specified by $\a_{\textnormal{\sc s}}$, the derivatives on right
hand side vanish and eq.~(\ref{Pon-check}) is reproduced.
Once we have $(\a_{\textnormal{\sc s}},\b_{\textnormal{\sc s}})$, we look for
the solutions ${\delta}\b$ to equations~(\ref{pert-beta}) and~(\ref{GFbeta}). There
are two types of solutions. Those with ${\delta}\a={\delta}\a_{\textnormal{\sc
s}}\neq\/0$, and those with ${\delta}\a=0$.
\medskip {\bf Zero modes with $\boldsymbol{{\delta}\a\neq\/0}$}. A perturbation
$\a_{\textnormal{\sc s}}\to\a_{\textnormal{\sc s}}+{\delta}\a_{\textnormal{\sc s}}$
\,produces a change\, $\b_{\textnormal{\sc s}}\to\b_{\textnormal{\sc
s}}+{\delta}\b_{\textnormal{\sc s}}$ \,given by
\begin{equation}
{\delta}\b _{\textnormal{\sc s}}= \sum_{b}\,\tilde{u}^b\,
\frac{{\partial}\,{\delta}\a^j_{\textnormal{\sc s}}}{{\partial}\/u^b}\;Z_j
+ \tilde{\tau}^j\, [\,{\delta}\a_{\textnormal{\sc s}}\hspace{1pt},Z_{j}]\,.
\label{deltabeta}
\end{equation}
Employing that ${\delta}\a_{\textnormal{\sc s}}$ satisfies eqs.~(\ref{pert-alpha})
and~(\ref{GFalpha}), it is a matter of simple algebra to check that
${\delta}\b_{\textnormal{\sc s}}$ solves the moduli equation~(\ref{pert-beta}) and
the gauge fixing condition~(\ref{GFalpha}). Hence, to every ${\rm G}_\mathfrak{h}$
zero mode ${\delta}\a_{\textnormal{\sc s}}$ there corresponds a ${\rm
G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ zero mode $({\delta}\a_{\textnormal{\sc
s}},{\delta}\b_{\textnormal{\sc s}})$.
Using the expressions for ${\delta}\a_{\textnormal{\sc s}}$ in
eqs.~(\ref{solalpha-1}) and~(\ref{solalpha-2}), ${\delta}\b_{\textnormal{\sc s}}$
can be recast as
\begin{align}
{\delta}_{(a)}\b_{\textnormal{\sc s}} &= \frac{{\partial}\b^{\,i}_{\textnormal{\sc s}}}{{\partial}\/u^a}\>Z_i
+ \textnormal{d}\hspace{1pt}z_{(a)} + [\a_{\textnormal{\sc s}}, z_{(a)}]
+ [\b_{\textnormal{\sc s}}, t_{(a)}] \,, \label{deltabeta-a} \\[4.5pt]
{\delta}_{(i)}\b_{\textnormal{\sc s}} &= [\b_{\textnormal{\sc s}}, T_i] + \textnormal{d}\hspace{1pt}z_{(i)}
+ [\a_{\textnormal{\sc s}},z_{(i)}]
+ [\b_{\textnormal{\sc s}},t_{(i)}] \,.\label{deltabeta-i}
\end{align}
Here $z_{(a)}$ and $z_{(i)}$ are the $\mathfrak{h}^*$-valued functions
\begin{align}
z_{(a)} &= \sum_{b}\,\tilde{u}^b\, \frac{{\partial}\,t^j_{(a)}}{{\partial}\/u^b}\;Z_j
+ \tilde{\tau}^j\, [\,t_{(a)},Z_{j}] \,,
\label{z(a)}\\
z_{(i)} &= \sum_{b}\,\tilde{u}^b\, \frac{{\partial}}{{\partial}\/u^b}\;
\big(\, T_i + t^j_{(i)} Z_j\,\big)
+ \tilde{\tau}^j \,\big[\,t_{(i)\!}+T_i\, , \,Z_j\big]\,,
\label{z(i)}
\end{align}
and\, $t_{(a)\!}$ \,and\, $t_{(i)\!}$ \,are the same functions that occur in
the zero modes ${\delta}_{(a)}\a_{\textnormal{\sc s}}$ and
${\delta}_{(i)}\a_{\textnormal{\sc s}}$. The deformations
${\delta}_{(a)}\b_{\textnormal{\sc s}}$, ${\delta}_{(i)}\b_{\textnormal{\sc s}}$ in
eqs.~(\ref{deltabeta-a}),~(\ref{deltabeta-i}) exhibit the pattern of a
parametric derivative ${\partial}\b_{\textnormal{\sc s}}/{\partial}\/u^a$, rotation
$[\b_{\textnormal{\sc s}},T_i]$, followed by an infinitesimal gauge
transformation. Furthermore, ${\delta}\a_{\textnormal{\sc s}}$ and
${\delta}\b_{\textnormal{\sc s}}$ can be combined in
\begin{align}
{\delta}_{(a)}{\kappa}_{\textnormal{\sc s}} & =
\frac{{\partial}{\kappa}_{\textnormal{\sc s}}}{{\partial}\/u^a}
+ \textnormal{d}\hspace{1pt}\Lambda_{(a)}
+ \big[{\kappa}_{\textnormal{\sc s}}, \Lambda_{(a)}\big]
\,, \label{deltakappa-a} \\[4.5pt]
{\delta}_{(i)}{\kappa}_{\textnormal{\sc s}} &= [{\kappa}_{\textnormal{\sc s}}, T_i]
+ \textnormal{d}\hspace{1pt}\Lambda_{(i)}
+ [{\kappa}_{\textnormal{\sc s}},\Lambda_{(i)}]
\,.\label{deltakappa-i}
\end{align}
where $\Lambda_{(a)}=t_{(a)}+z_{(a)}$ and $\Lambda_{(i)} = t_{(i)}+z_{(i)}$.
\medskip {\bf Zero modes with $\boldsymbol{{\delta}\a=0}$}. For ${\delta}\a=0$, the
moduli equation~(\ref{pert-beta}) and the gauge fixing
condition~(\ref{GFbeta}) for ${\delta}\b^{\,i}$ reduce to those for the zero modes
${\delta}\a^i$ of the self-antiself dual ${\rm G}_\mathfrak{h}$ instanton
$\a_{\textnormal{\sc s}}$. It then trivially follows that there are
\hbox{$\textnormal{dim}_N({\rm G}_\mathfrak{h})$} additional zero modes
${\delta}{\kappa}_{\textnormal{\sc s}}=({\delta}\a_{\textnormal{\sc
s}},{\delta}\b_{\textnormal{\sc s}})$ with
\begin{alignat}{4}
{\delta}_{(\tilde{a})}\a_{\textnormal{\sc s}} =0\,,
&\qquad &{\delta}_{(\tilde{a})}\b _{\textnormal{\sc s}}& =
{\delta}_{(a)}\a_{\textnormal{\sc s}}^{\,j}\, Z_j
= \frac{{\partial}\b_{\textnormal{\sc s}}}{{\partial}\tilde{u}^a}
+ \textnormal{d}\hspace{1pt}t_{(a)}^jZ_j
+ [\,\a_{\textnormal{\sc s}}\,,\, t_{(a)}^jZ_j\,] \, ,
\label{solbeta-tilde-a}\\[4.5pt]
{\delta}_{(\tilde{i})}\a_{\textnormal{\sc s}} =0\,,
&\qquad & {\delta}_{(\tilde{j})}\b_{\textnormal{\sc s}} & =
{\delta}_{(i)}\a_{\textnormal{\sc s}}^{\,j} \,Z_j
= \frac{{\partial}\b_{\textnormal{\sc s}}}{{\partial}\/\tilde{\tau}^i}
+ \textnormal{d}\hspace{1pt}t_{(i)}^jZ_j
+ [\,\a_{\textnormal{\sc s}}\,,\, t_{(i)}^jZ_j\,] \,.
\label{solbeta-tilde-i}
\end{alignat}
These have the same structure of all zero modes, partial derivatives with
respect to moduli parameters, $\tilde{u}^a$ and $\tilde{\tau}^i$ in this case,
followed by infinitesimal gauge transformations.
To summarize, the gauge field $(\a_{\textnormal{\sc s}},\b_{\textnormal{\sc
s}})$, with $\a_{\textnormal{\sc s}}$ the connection of a charge $N$
self-antiself dual ${\rm G}_\mathfrak{h}$ instanton and $\b_{\textnormal{\sc s}}$ as in
eq.~(\ref{betageneral}), specifies a self-antiself dual ${\rm
G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ instanton with the same charge. The dimension of its
moduli space ${\cal M}_N({\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*\!})$ is twice the dimension
of ${\cal M}_N({\rm G}_\mathfrak{h})$. As local coordinates on ${\cal M}_N({\rm
G}_{\mathfrak{h}\ltimes\mathfrak{h}^*})$, one may take
$\{u^a,\tau^j,\tilde{u}^a,\tilde{\tau}^j\}$, where $u^a$ and $\tau^i$ are
local coordinates on ${\cal M}_N({\rm G}_\mathfrak{h})$, and $\tilde{u}^a$ and
$\tilde{\tau}^i$ are kind of dual coordinates. If the zero modes of the ${\rm
G}_\mathfrak{h}$ instanton $\a_{\textnormal{\sc s}}$ are given by
eqs.~(\ref{solalpha-1}) and (\ref{solalpha-2}), the zero modes of the
$(\a_{\textnormal{\sc s}},\b_{\textnormal{\sc s}})$ instanton take the form in
eqs.~(\ref{solalpha-1})-(\ref{solalpha-2}),
(\ref{deltabeta-a})-(\ref{deltabeta-i}) and
(\ref{solbeta-tilde-a})-(\ref{solbeta-tilde-i}). We may call these instantons
cotangent~\hbox{$T^*{\rm G}_\mathfrak{h}$}, or semidirect ${\rm G}_\mathfrak{h}\ltimes{\rm
G}_{\mathfrak{h}^*}$, instantons.
The moduli space ${\cal M}_N({\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*\!})$ inherits a natural
metric from the field theory defined by the overlap of deformations
${\delta}{\kappa}=({\delta}\a,{\delta}\b)$. If $U$ and $V$ stand for two arbitrary moduli
coordinates, the moduli space metric coefficients are given by
\begin{equation}
G_{UV} = \frac{1}{8\pi^2} \int_{{\bf R}^4} {\Omega}\,\big( {\delta}_{(U)}{\kappa}\,,
\,\star\,{\delta}_{(V)}{\kappa}\big)\,.
\label{modulimetricIJ}
\end{equation}
Denote by $H$ the metric on ${\cal M}_N({\rm G}_\mathfrak{h})$, with components
\begin{equation}
H_{pq} = \frac{1}{8\pi^2} \int_{{\bf R}^4} {\omega} \big( {\delta}_{(p)}\a\,,
\,\star\,{\delta}_{(q)}\a\big)\,. \qquad\qquad
\label{moduli-metric-usual}
\end{equation}
Using that ${\Omega}(T_i,T_j)={\Omega}(T_i,Z_j)={\omega}(T_i,T_j)$ and the results in this
Section for the zero modes, one has
\begin{equation}
\begin{tabular}{cccccc}
& & & $q$ & $ \tilde{q}$ & \\[3pt]
\multirow{2}{*}{$G_{UV} =$}
& $p$ & \multirow{2}{*}{$\Bigg(\!\!\!\!$} & $H_{pq}+\Delta_{pq}$
&$H_{pq}$ &\multirow{2}{*}{$\!\!\!\!\Bigg)$~,} \\[4.5pt]
& $\tilde{p}$ & &$H_{pq}$ & 0 &
\end{tabular}
\label{modulimetric}
\end{equation}
where $\Delta_{pq}$ stands for
\begin{equation}
\Delta_{pq} = \frac{1}{8\pi^2} \int_{{\bf R}^4} \big[\,
{\omega} \big( {\delta}_{(p)}\a\,, \,\star\,{\delta}_{(q)}\b\big)
+ (p\leftrightarrow q)\big]\,.
\label{moduli-metric-correction}
\end{equation}
The $\mathfrak{h}\ah$-coefficient $G_{pq}$ is the sum of $H_{pq}$ and a contribution
$\Delta_{pq}$ that arises from the \hbox{$\mathfrak{h}^*$-components} ${\delta}_{(p,q)}\b$
of the deformations along the moduli space directions $p$ and~$q$.
In the next section we explicitly realize this construction for
$\mathfrak{h}=\mathfrak{s}\mathfrak{u}(2)$ and instanton charge one.
\section{The semidirect extension BPST instanton and its moduli}
On ${\bf R}^4$ take coordinates $x^{\mu}\!=(x^1,x^2,x^3,x^4)$ and Euclidean
metric ${\delta}_{{\mu}{\nu}}$. Set \hbox{$\mathfrak{h}=\mathfrak{s}\mathfrak{u}(2)$}, with basis\,
$[T_i,T_j]={\epsilon}_{ijk}\,T_k$. The most general invariant bilinear form ${\omega}$
that can be defined on \hbox{$\mathfrak{s}\mathfrak{u}(2)$} is\,
\,\hbox{${\omega}_{ij}\!={\omega}_0{\delta}_{ij}$}, with ${\omega}_0$ an arbitrary constant that
is conventionally set equal to ${1/2g^2}$.
The classical double \hbox{$\mathfrak{s}\mathfrak{u}(2)\!\ltimes\!\mathfrak{s}\mathfrak{u}(2)^*$} has commutators
\begin{equation}
[T_i,T_j]={\epsilon}_{ijk}\,T_k\,,\quad
[T_i,{Z}_j]={\epsilon}_{ijk}\,{Z}_k\,, \quad
[{Z}_i,{Z}_j]=0\,, \quad i=1,2,3,
\end{equation}
and the most general metric ${\Omega}$ on it reads
\begin{equation}
\begin{tabular}{cccccc}
& & & $T_j$ & $Z_j$ & \\[3pt]
\multirow{2}{*}{${\Omega}=$}
& $T_i$ & \multirow{2}{*}%
{$\displaystyle{\frac{1}{2g^2}}\,\Bigg(\!\!\!\!$} & ${\delta}_{ij}$
&${\delta}_{ij}$ & \multirow{2}{*}{$\!\!\!\!\Bigg)\,.$} \\[4.5pt]
& $Z_i$ & &${\delta}_{ij}$ & 0 &
\end{tabular}
\label{SDsu(2)}
\end{equation}
This is of the form (\ref{SDmetric}), or more precisely, of the
form~(\ref{deformedSDmetric}) with $s=0$. In the basis $\{T_i,Z_j\}$ the
connection ${\kappa}$ has components $\a^i$ and $\b^{j}$, and the curvature $K$ has
components $F^{\,i}$ and $B^j$, given by
\begin{equation}
F^{\,i} = \textnormal{d}\a^i + \frac{1}{2}\,{\epsilon}^{ijk}\,\a^j\!\wedge\a^k,\quad
B^{\,i} = \textnormal{d}\b^i\!+{\epsilon}^{ijk}\,\a^j\!\wedge\/\b^k.
\label{FBsu(2)}
\end{equation}
In what follows we restrict ourselves to the positive sign in equation\,
\hbox{$\star\/K=\pm\/K$}. This corresponds to selfdual instantons and, with
the metric convention~(\ref{SDsu(2)}), positive instanton charge. The
negative sign, antiself dual instantons with negative instanton charge, is
analogously treated. The group ${\rm G}_{\mathfrak{s}\mathfrak{u}(2)\ltimes\mathfrak{s}\mathfrak{u}(2)^*}$ is the
cotangent bundle $T^{*\!}SU(2)$, isomorphic to the semidirect product\,
$SU(2)\ltimes{\bf R}^3$.
Equation\, \hbox{$\star\/F^{\,i\!}=F^{\,i}$}, with $F^{\,i}$ as in
eq.~(\ref{FBsu(2)}), is solved by $SU(2)$ selfdual instantons. Take as
solution\,the BPST instanton~\cite{BPST}, whose connection
$\a^{\,i}_{\textnormal{{\sc s}}}$ and curvature $F^{\,i}_{\textnormal{{\sc
s}}}$ are given in singular gauge by
\begin{equation}
\a^{\,i}_{\textnormal{{\sc s}}\!}=
\frac{2\rho^2}{\,r_a^2\,(r_a^2+\rho^2)\,}\>
\bar{\eta}^i{}_{{\mu}{\nu}}\,(x-a)^{\nu}\,\textnormal{d}\/x^{\mu}
\label{BPST-connection}
\end{equation}
and
\begin{equation}
F^{\,i}_{\textnormal{{\sc s}}}\! =
\frac{2\rho^2}{r_a^2{(r_a^2+\rho^2)}^2}~
\big[ 4 \,\bar{\eta}^{\,i\!}{}_{{\mu}{\gamma}}\, (x-a)^{\gamma}\, (x-a)_{\nu}
- \bar{\eta}^{\,i\!}{\!}_{{\mu}{\nu}}\,r_a^2\,\big]\,\textnormal{d}\/x^{\mu}\wedge\textnormal{d}\/x^{\nu}\,.
\label{BPST-curvature}
\end{equation}
Here $\rho$ is an arbitrary constant, $r_a$ is the radius of the three-sphere
\begin{equation}
r_a^2= (x-a)^{\mu} (x-a)_{\mu}
\end{equation}
centered at any point $a^{\mu}$ on ${\bf R}^4$, and $\bar{\eta}^i{}_{{\mu}{\nu}}$ are
the 't Hooft symbols~\cite{'tHooft}
\begin{equation}
\bar{\eta}^{\,i\!}{\!}_{{\mu}{\nu}}\!=-\,\bar{\eta}^{\,i\!}{\!}_{{\nu}{\mu}}\,,\quad
\bar{\eta}^{\,i\!}{\!}_{4j}\!={\delta}_{ij}\,,\quad
\bar{\eta}^{\,i\!}{\!}_{jk}\!={\epsilon}_{ijk}\,,
\label{thooft}
\end{equation}
whose properties are collected in the Appendix.
The BPST connection has instanton charge one in units of $1/g^2$,
\begin{equation}
S_{\textnormal{\sc p}}[SU(2); \a_{\textnormal{\sc s}}\,]
=\frac{1}{16\pi^2g^2} \int_{{\bf R}^4}
F^{\,i}_\textnormal{\sc s}\wedge F^{\,i}_\textnormal{\sc s}
= \frac{1}{g^2}\,.
\end{equation}
The moduli space of the BPST instanton
~\cite{Schwarz,Jackiw-Rebbi,AHS,BCL,Weinberg-book,Tong} is an eight
dimensional manifold on which one may take as global coordinates the instanton
size $\rho$, the four coordinates $a^{\mu}$ of the instanton center, and three
angles $\tau^i$ that account for rotations about the generators $\{T_i\}$ of
$\mathfrak{s}\mathfrak{u}(2)$. The deformations along these moduli directions are~\cite{Tong}
\begin{align}
{\delta}_{(\rho)}\a_{\textnormal{\sc s}} &
= \frac{{\partial}\a_{\textnormal{\sc s}}}{{\partial}\rho}\;,
\label{varalpharho}\\[3pt]
{\delta}_{(a^{\mu})}\a_{\textnormal{\sc s}} & =
\frac{{\partial}\a_{\textnormal{\sc s}}}{{\partial}\/a^{\mu}\,}
+\, \textnormal{d} \a_{{\mu}\,\textnormal{\sc s}} + [\,\a_{\textnormal{\sc s}},
\a_{{\mu}\,\textnormal{\sc s}}\,] =
-\,F_{{\mu}{\nu}\,\textnormal{\sc s}}\,\textnormal{d}\/x^{\nu}\,,
\label{varalphamu}\\[3pt]
{\delta}_{(\tau^i)} \a_{\textnormal{\sc s}} &= [\a_{\textnormal{\sc s}},T_i\,]
+ \textnormal{d}\hspace{1pt}t\,T_i
+ [\,\a_{\textnormal{\sc s}}\,,\hspace{1pt}t\,T_i\,]\,, \label{varalphaTi}
\end{align}
where $t$ is the function
\begin{equation}
t(r_a) = -\,\frac{\rho^2}{r_a^2+\rho^2}\;.
\label{t(ra)}
\end{equation}
\medskip{\bf The semidirect BPST instanton and its zero modes}. The results in
Section 4 imply that, for $\a=\a_{\textnormal{\sc s}}$, the most general
solution to equation\, \hbox{$\star\/B^{\,i\!}=B^{\,i}$} is, modulo gauge
transformations,
\begin{equation}
\b^{\,i}_{\textnormal{\sc s}} =
\tilde{\rho}\>\frac{{\partial}\a^i_{\textnormal{\sc s}}}{{\partial}\rho}
\,+ \,\tilde{a}^{\mu}\>\frac{{\partial}\a^i_{\textnormal{\sc s}}}{{\partial}\/a^{\mu}}
+ {\epsilon}^{ikj}\,\a^k_{\textnormal{\sc s}}\, \tilde{\tau}^j \,,
\label{betasu(2)}
\end{equation}
where $\tilde{\rho},\,\tilde{a}^{\mu}$ and $\tilde{\tau}^j$ are free parameters.
The curvature $B^{\,i}$ then becomes
\begin{equation}
B^{\,i}_{\textnormal{\sc s}} = \Big(
\tilde{\rho}\>\frac{{\partial}}{{\partial}\rho}\,
+ \,\tilde{a}^{\mu}\>\frac{{\partial}}{{\partial}\/a^{\mu}}\,\Big)\,
F^{\,i}_{\textnormal{\sc s}}
+ {\epsilon}^{ikj}\,F^{\,k}_{\textnormal{\sc s}}\,\tilde{\tau}^j \,.
\label{Bsu(2)}
\end{equation}
The \hbox{$\mathfrak{s}\mathfrak{u}(2)\ltimes\mathfrak{s}\mathfrak{u}(2)^*$} connection $(\a_{\textnormal{\sc s}},
\b_{\textnormal{\sc s}})$ \,specifies a charge one\, \hbox{$SU(2)\ltimes{\bf
R}^3$} \,instanton that we call semidirect or cotangent BPST instanton. It
depends on 16 moduli parameters, $\rho$, $a^{\mu}$, $\tau^j$, $\tilde{\rho}$,
$\tilde{a}^{\mu}$ and $\tilde{\tau}^j$. The derivatives entering
$\b^i_{\textnormal{\sc s}}$ and $B^i_{\textnormal{\sc s}}$ are trivially
calculated from the expression of $\a_{\textnormal{\sc s}}$.
The \hbox{$\mathfrak{s}\mathfrak{u}(2)$-components} of the zero modes along the moduli
directions $\rho$, $a^{\mu}$ and $\tau^i$ are those in eqs.~(\ref{varalpharho}),
~(\ref{varalphamu}) and~~(\ref{varalphaTi}). Upon substitution in
eqs.~(\ref{deltabeta-a}) and ~(\ref{deltabeta-i}), we obtain for their
\hbox{$\mathfrak{s}\mathfrak{u}(2)^*$-companions}
\begin{align}
{\delta}_{(\rho)}\b & = \frac{{\partial}\b_{\textnormal{\sc s}}}{{\partial}\rho} \,,
\label{varbetarho} \\
{\delta}_{(a^{\mu})}\b & = \frac{{\partial}\b_{\textnormal{\sc s}}}{{\partial}\/a^{\mu}}
+ \textnormal{d} \hspace{1pt}\b_{{\mu}\hspace{1pt}\textnormal{\sc s}}
+ [\a_{\textnormal{\sc s}}, \b_{{\mu} \hspace{1pt}\textnormal{\sc s}}]
+ [\b_{\textnormal{\sc s}}, \a_{{\mu}\hspace{1pt}\textnormal{\sc s}}]
=-\,B_{{\mu}{\nu}\hspace{1pt}\textnormal{\sc s}}\,\textnormal{d}\/x^{\nu}\,,
\label{varbetamu} \\[4.5pt]
{\delta}_{(\tau^i)}\b & = [\b_{\textnormal{\sc s}},T_i]
+ \textnormal{d}\hspace{1pt}z_{(\tau^i)} + [\a_{\textnormal{\sc s}},z_{(\tau^i)}]
+ [\b_{\textnormal{\sc s}}, t\hspace{1pt}T_i] \,,
\end{align}
where $z_{(\tau^i)}$ is a function of $x^{\mu}$ given by
\begin{equation}
z_{(\tau^i)}(x) =-\,\frac{2\rho}{\,(r_a^2+\rho^2)^2\,} \;
\big[ \tilde{\rho}\,r_a^2 + \rho\;\tilde{a}^{\lambda} (x-a)_{\lambda}\big]\, Z_i
+ \frac{r_a^2}{\,r_a^2+\rho^2\,}\; \tilde{\tau}^j\, [T_i,Z_j]\,.
\label{zi}
\end{equation}
As a cross check, one may directly verify, after a long but simple
calculation, that ${\delta}_{(\rho,a^{\mu},\tau^i)}\b_{\textnormal{\sc s}}$ \,indeed
satisfy the moduli equation~(\ref{pert-beta}) and the gauge-fixing
condition~(\ref{GFbeta}). We remark that
${\delta}_{(\rho,a^{\mu},\tau^i)}\b_{\textnormal{\sc s}}$ follow from
eqs.~(\ref{deltabeta-a}) and~(\ref{deltabeta-i}) and that no additional
gauge transformation has been fintrodued so as to ensure that the gauge fixing
condition holds.
The zero modes associated to the moduli coordinates $\tilde{\rho}$,
$\tilde{a}^{\mu}$ and $\tilde{\tau}^i$ are given by
eqs.~(\ref{solbeta-tilde-a})-(\ref{solbeta-tilde-i}), which in our case take
the form
\begin{alignat}{4}
&{\delta}_{(\tilde{\rho})}\,\a =0 \,, &\qquad &{\delta}_{(\tilde{\rho})}\b = \frac{{\partial}\a_{\textnormal{\sc s}}^{\,i}}
{{\partial}\rho} \, Z_i\,, \label{varbeta-tilderho} \\[1.5pt]
&{\delta}_{(\tilde{a}^{\mu})}\a=0\,, &\qquad & {\delta}_{(\tilde{a}^{\mu})}\b =
- F_{{\mu}{\nu}\hspace{1pt}\textnormal{\sc s}}^{\,i}\, Z_i\, \textnormal{d}\/x^{\nu} \,,
\label{varbeta-tildea} \\[4.5pt]
&{\delta}_{(\tilde{\tau}^i)}\a=0\,, &\qquad & {\delta}_{(\tilde{\tau}^i)}\b = \textnormal{d}\hspace{1pt}t\,Z_i
+ [\,\a_{\textnormal{\sc s}},(1+t)\hspace{1pt} Z_i\,]\,,
\label{varbeta-tildej}
\end{alignat}
with $t$ as in eq.~(\ref{t(ra)}).
\medskip{\bf The moduli space metric}. The expressions for the zero modes
above and some calculations lead to the moduli space metric
\begin{equation}
\begin{tabular}{cccccccccc}
& & & $\rho$ & $a^{\nu}$ & $\tau^j$ & $\tilde{\rho}$
& $\tilde{a}^{\nu}$ & $\tilde{\tau}^j$ & \\[9pt]
\multirow{7}{*}{$G_{UV}~=$} & $\rho$
& \multirow{6}{*}{$\displaystyle{\frac{1}{2g^2}}\>\bBigg@{8}(\!\!\!\!$}
& 2 & 0 & 0 & 2& 0 & 0 & \multirow{6}{*}{$\!\!\!\!\bBigg@{8})$\,.} \\[4.5pt]
& $a^{\mu}$ & & 0 & ${\delta}_{{\mu}{\nu}}$ & 0 & 0 & ${\delta}_{{\mu}{\nu}}$ & 0 & \\[4.5pt]
& $\tau^i$ & & 0 & 0
& $\frac{1}{2} \,\rho\, (\rho+2\tilde{\rho})\,{\delta}_{ij}$
& 0 & 0 & $\frac{1}{2}\,\rho^2\,{\delta}_{ij}$ & \\[4.5pt]
& $\tilde{\rho}$ & & 2 & 0 & 0 & 0 & 0 & 0 & \\[4.5pt]
& $\tilde{a}^{\mu}$ & & 0 & ${\delta}_{{\mu}{\nu}}$ & 0 & 0 & 0 & 0 & \\[4.5pt]
& $\tilde{\tau}^i$ & & 0 & 0
& $\frac{1}{2}\,\rho^2\, {\delta}_{ij}$ & 0 & 0 & 0 & \\
\end{tabular}
\label{modulimetric}
\end{equation}
The change of coordinates
\begin{equation}
\begin{array}{l}
{\sigma} = \tilde{\rho} -\rho\, r_-\,, \\[2pt]
\tilde{{\sigma}} = \tilde{\rho} -\rho\, r_+\,,
\end{array} \qquad
\begin{array}{l}
b^{\mu}=\, \tilde{a}^{\mu} -a^{\mu}\, r_-\,, \\[2pt]
\tilde{b}^{\mu} = \,\tilde{a}^{\mu} -a^{\mu}\, r_+\,,
\end{array} \qquad
\begin{array}{l}
\th^i = \tau^i -\tilde{\tau}^i \, s_+\,, \\[2pt]
\tilde{\th}^i = \tau^i -\tilde{\tau}^i\, s_-\,,
\end{array}
\label{CHANGE}
\end{equation}
with $r_\pm$ and $s_\pm$ given by
\begin{equation}
r_\pm = \frac{2}{1 \pm \sqrt{5}} \, \qquad
s_\pm = \frac{2\rho}{ \rho+2\tilde{\rho} \pm
\sqrt{4\rho^2 + (\rho+2\tilde{\rho})^2}\,}\,,
\label{CHANGE-2}
\end{equation}
brings the metric to the diagonal form
\begin{equation}
dL^2 = \frac{1}{2g^2}\> \big[\,d{\sigma}^{2\!} + db^{\mu} \, db^{\mu}
+ f\,d\th^i\,d\th^i - d\tilde{{\sigma}}^2
- d\tilde{b}^{\mu}\, d\tilde{b}^{\mu}
- \tilde{f}\,d\tilde{\th}^i\,d\tilde{\th}^i \,\big]\,,
\label{eigenvectors}
\end{equation}
where $f$ and $\tilde{f}$ are positive functions of ${\sigma}$ and $\tilde{{\sigma}}$.
This shows that the moduli metric has signature (8,8).
The field theory is invariant under translations and $SO(4)$ rotations in
${\bf R}^4$, and under $SU(2)\ltimes{\bf R}^3$ gauge transformations. These
symmetries go into isometries of the moduli metric. Indeed, ${\bf R}^4$
translations give rise to translations in $b^{\mu}$ and $\tilde{b}^{\mu}$, generated
by ${\partial}/{\partial}\/b^{\mu}$ and ${\partial}/{\partial}\tilde{b}^{\mu}$. Rotations become\,
$SO(4)\cong\/SU(2)_{+\!}\times\/SU(2)_-$ \,rotations in $b^{\mu}$ and
$\tilde{b}^{\mu}$, generated by
\begin{equation}
\chi_\pm^{\,i}\! = \frac{1}{2}\,\Big[
{\epsilon}^{ijk}\,b^j\,\frac{{\partial}}{{\partial}\/b^k} \pm \Big(
b^i\frac{{\partial}}{{\partial}\/b^4} - b^4\frac{{\partial}}{{\partial}\/b^i}\Big) \Big]
\label{SO(4)generators}
\end{equation}
and $\tilde{\chi}_\pm^{\,i}$, obtained from eq.~(\ref{SO(4)generators}) by
replacing $b^{\mu}$ with $\tilde{b}^{\mu}$. Finally gauge transformations become
translations in $\tau^i$ and $\tilde{\tau}^i$ generated by ${\partial}/{\partial}\tau^i$ and
${\partial}/{\partial}\tilde{\tau}^i$. Note that in the conventional BPST instanton, one has
translational and rotational invariance in $a^{\mu}$. The first one is an
isometry here, but the second one is not, due to the occurrence of the term
$da^{\mu}\/d\tilde{a}^{\mu}$ in the moduli metric.
\medskip{\bf Complex structures}. Let us show that the moduli
space\,\hbox{${\cal M}_1\big(SU(2)\ltimes{\bf R}^3\big)$} \,is a
hyper-K\"ahler manifold. We do this by finding three complex structures
$J^{i\!}=\tfrac{1}{2}\,(J^i)_{UV}\,\textnormal{d}\/U\!\wedge\textnormal{d}\/V$, with components\,
$(J^i)_{UV}$, such that
\begin{equation}
(J^i)^U{}_W\;(J^j)^W{}_V = -\,{\delta}^{ij}\,{\delta}^U{\!}_V
+ {\epsilon}^{ijk}\,(J^k)^U{}_V \,.
\label{hyperkahler}
\end{equation}
As in the BPST case, one expects the moduli space to inherit its complex
structures from those of ${\bf R}^4$, which can be written as
$-\bar{\eta}^i{\!}_{{\mu}{\nu}\,}\textnormal{d}\/x^{\mu}\wedge \textnormal{d}\/x^n$. This suggests the ansatz
\begin{equation}
\big(J^i\big)_{UV} = -\,\frac{1}{8\pi^2} \int_{{\bf R}^4}\textnormal{d}^4\/x~
\bar{\eta}^i{}_{{\mu}{\nu}}~ {\Omega}\hspace{0.5pt}\big({\delta}_{(U)}{\kappa}^{\mu}\,,\,
{\delta}_{(V)}{\kappa}^{\nu}\big)\,.
\label{ansatz-complex}
\end{equation}
Using the expressions for the zero modes and some algebra and integration, one
has
\begin{equation}
\begin{tabular}{cccccccccc}
& & & $\rho$ & $a^{\nu}$ & $\tau^k$ & $\tilde{\rho}$
& $\tilde{a}^{\nu}$ & $\tilde{\tau}^k$ & \\[9pt]
\multirow{7}{*}{$(J^i)_{UV}~=$} & $\rho$
& \multirow{6}{*}{$\!\!\displaystyle{-\,\frac{1}{2g^2}}\>\bBigg@{8}(\!\!\!\!$}
& 0 & 0 & $(\rho+\tilde{\rho})\,{\delta}^i{\!}_k$
& 0 & 0 & $\rho\,{\delta}^i{\!}_k$
& \multirow{6}{*}{$\!\!\!\!\bBigg@{8})$\,.} \\[4.5pt]
& $a^{\mu}$ & & 0 & $\bar{\eta}^i{\!}_{{\mu}{\nu}}$ & 0
& 0 & $\bar{\eta}^i{\!}_{{\mu}{\nu}}$ & 0 & \\[4.5pt]
& $\tau^j$ & & $-(\rho+\tilde{\rho})\,{\delta}^i{\!}_j$ & 0
& $\frac{1}{2} \rho\, (\rho+2\tilde{\rho})\,{\epsilon}^i{\!}_{jk}$
& $-\rho\,{\delta}^i{\!}_j$ & 0
& $\frac{1}{2} \rho^2\,{\epsilon}^i{\!}_{jk}$& \\[4.5pt]
&$ \tilde{\rho}$ & & 0 & 0 & $\rho\,{\delta}^i{\!}_k$
& 0 & 0 & 0 & \\[4.5pt]
& $a^{\mu}$ & & 0 & $\bar{\eta}^i{\!}_{{\mu}{\nu}}$ & 0
& 0 & 0 & 0 & \\[4.5pt]
& $\tau^j$ & & $-\rho\,{\delta}^i{\!}_j$ & 0
& $\frac{1}{2}\,\rho^2\,{\epsilon}^i{\!}_{jk}$
& 0 & 0 & 0 & \\
\end{tabular}
\label{complex-structure}
\end{equation}
Noting that $(J^i)^U{}_V=G^{UW}(J^i)_{WV}$, with $G^{UV}$ the inverse of
$G_{UV}$ in~(\ref{modulimetric}), it is straightforward to check that the
two-forms~$J^i$ in eq.~(\ref{complex-structure}) indeed satisfy the
relations~(\ref{hyperkahler}), hence are complex structures. It is worth
remarking that the moduli space is hyper-K\"ahler, despite not being a
Riemannian manifold. It looks like hyper-K\"ahlerity is ``transmitted'' to
${\cal M}_1(SU(2)\ltimes{\bf R}^3)$ via its Riemannian submanifolds.
We finish by studying the compatibility of the isometries of the moduli metric
with the complex structures. Recall that for an isometry generated by a
Killing vector $\xi$ to be compatible with a tensor $A$, the Lie derivative
${\cal L}_\xi\/A$ of $A$ along $\xi$ must vanish. For an isometry given in a
chart $\{u^a\}$ by $u^{\,a\!}\to\/u^{\prime\,\/a}\!=u^a
+\varepsilon\,\xi^a(u)$, we use for the Lie derivative the convention\, ${\cal
L}_{\xi\!}{A} = \lim_{\varepsilon\to\/0} \frac{1}{\varepsilon}\,
\big[A^{\,\prime}(u)-A(u)\big]$. With this convention, one may check that the
isometries generated by\, $\xi={\partial}_{b^{{\mu}}},\, {\partial}_{\,\tilde{b}^{\mu}},
\chi^{i}_+,\,{\partial}_{\tau^i}$ and ${\partial}_{\tilde{\tau}^i}$ \,are compatible with the complex
structures~$J^i$. However, for $\xi=\chi^{i}_-$, one has\, ${\cal
L}_{\chi^{i}_-\!}J^j\!={\epsilon}^{ijk}J^k$. The complex structures are thus
rotated by $SU(2)_-$ rotations, but they remain unchanged by the other isometries.
\section{Outlook}
In this paper we have proposed a method to obtain the self-antiself dual
solutions for a gauge group ${\rm G}_{\mathfrak{h}\ltimes\mathfrak{h}^*}$ from those for ${\rm
G}_{\mathfrak{h}}$. This hints to using Medina and Revoy's
theorem~\cite{Medina-Revoy} to find structure results for the self-antiself
dual instantons of the Lie groups with metric Lie algebras. One may advance a
few ideas on the subject. According to the theorem, it would suffice to
consider three cases: (1) simple Lie algebras, (2) Abelian algebras, and (3)
double extensions of a metric Lie algebra by a either a simple or a
one-dimensional Lie algebra.
Simple real Lie algebras are the Lie algebras of simple real Lie
groups, whose instantons would be regarded as the basic objects in terms of
which to state structure results. Next on the list is the Abelian Lie
algebra. This case is trivial, since on ${\bf R}^4$ there are no Abelian
instantons. One is left with the Lie groups of double extensions.
The double extension $\mathfrak{d}(\mathfrak{m},\mathfrak{h})$ of a metric Lie algebra $\mathfrak{m}$ by a Lie
algebra $\mathfrak{h}$ is obtained~\cite{Medina-Revoy,FO-Stanciu-double} by forming the
classical double $\mathfrak{h}\ltimes\mathfrak{h}^*$ and, then, by acting with $\mathfrak{h}$ on $\mathfrak{m}$
via antisymmetric derivations. Since $\mathfrak{m}$ needs to be metric, three
possibilities must be considered for $\mathfrak{m}$. The first one is $\mathfrak{m}$ a simple
real Lie algebra. In this case~\cite{FO-Stanciu-double}, the algebra of
antisymmetric derivations of $\mathfrak{m}$ is $\mathfrak{m}$ itself and the double extension is
isomorphic to the direct product $\mathfrak{m}\times(\mathfrak{m}\ltimes\mathfrak{m}^*)$. The
corresponding Lie group is then the direct product ${\rm G}_\mathfrak{m}\times\/{\rm
G}_{\mathfrak{m}\ltimes\mathfrak{m}^*}$ and its instantons are determined in terms of the
${\rm G}_\mathfrak{m}$ instantons using the construction presented here. The second
possibility is $\mathfrak{m}$ Abelian, of dimension $m$. Being Abelian, any
nondegenerate, symmetric bilinear form on $\mathfrak{m}$ is a metric, and this can
always be brought to a diagonal form with all its eigenvalues equal to either
$+1$ or $-1$. If there are $p$ positive and $q$ negative eigenvalues, the
algebra $\mathfrak{h}$ of antisymmetric derivations is any subalgebra of
$\mathfrak{s}\mathfrak{o}(p,q)$~\cite{FO-Stanciu-double}. In this case, by extending the
arguments at the beginning of Section~4, it can be shown that the third
homotopy group of ${\rm G}_{\mathfrak{d}(\mathfrak{m},\mathfrak{h})}$ is equal to the third homotopy
group of ${\rm G}_\mathfrak{h}$. This motivates studying the self-antiself dual
solutions of such theories in detail. The third option, $\mathfrak{m}$ a double
extension, takes us back to the starting point.
One would also like to include matter fields in the analysis. Their coupling
to an $\mathfrak{h}\ltimes\mathfrak{h}^*$ gauge field requires additional matter field
components, which introduce additional field equations that may lead to new
nontrivial configurations.
\section*{Appendix}
\renewcommand{\theequation}{A.\arabic{equation}}
The 't Hooft symbols, defined in eq.~(\ref{thooft}), satisfy the
algebraic identities~\cite{'tHooft}
\begin{align}
\bar{\eta}^i{}_{{\mu}{\nu}}\; \bar{\eta}^i{}_{{\gamma}\tau} & =
{\delta}_{{\mu}{\gamma}} \,{\delta}_{{\nu}\tau} - {\delta}_{{\mu}\tau} \,{\delta}_{{\nu}{\gamma}}
- {\epsilon}_{{\mu}{\nu}{\gamma}\tau}\,,\\
\bar{\eta}^i{}_{{\mu}{\nu}} \;\bar{\eta}^j{}_{{\mu}\tau} &
= {\delta}^{ij}\,{\delta}_{{\nu}\tau} + {\epsilon}^{ijk}\,\bar{\eta}^k{}_{{\nu}\tau}\,,\\
{\epsilon}_{{\mu}{\nu}{\sigma}\tau}\,\bar{\eta}^i{}_{\tau{\gamma}} & =
\bar{\eta}^i{}_{{\mu}{\nu}} \,{\delta}_{{\sigma}{\gamma}}
+ \bar{\eta}^i{}_{{\nu}{\sigma}}\, {\delta}_{{\mu}{\gamma}}
+ \bar{\eta}^i{}_{{\sigma}{\mu}}\, {\delta}_{{\nu}{\gamma}}\,,\\
{\epsilon}^{ijk}\,\bar{\eta}^j{}_{{\mu}{\nu}}\,\bar{\eta}^k{}_{{\gamma}\tau} & =
{\delta}_{{\mu}{\gamma}}\,\bar{\eta}^i{}_{{\nu}\tau}
- {\delta}_{{\mu}\tau}\,\bar{\eta}^i{}_{{\nu}{\gamma}}
- {\delta}_{{\nu}{\gamma}}\,\bar{\eta}^i{}_{{\mu}\tau}
+ {\delta}_{{\nu}\tau}\,\bar{\eta}^i{}_{{\mu}{\gamma}} \,.
\end{align}
These have been widely used in the computations of Section~5. The one-forms
\begin{equation}
\bar{\chi}^i=
\frac{1}{r^2_a}\>\bar{\eta}^i{}_{{\mu}{\nu}}\,(x-a)^{\nu}\textnormal{d}\/x^{\mu}\
\end{equation}
are Maurer-Cartan forms for $SU(2)\cong \/S_3$. Letting the radius $r_a$
vary, one obtains the frame\, $\bar{\cal F}=\{\bar{e}^{\,i}\!
=r_a\bar{\chi}^i,\,\bar{e}^{\,4}\!=\!-\textnormal{d}\/r_a\}$, which has the same
orientation as $\{\textnormal{d}\/x^{\mu}\}$. We could have worked in regular gauge, in
which the BPST connection reads
\begin{equation}
\a^{\,i}_{\textnormal{{\sc s},reg}} = \frac{2}{\,r_a^2+\rho^2\,}\>
\eta^i{}_{{\mu}{\nu}}\,(x-a)^{\nu}\,\textnormal{d}\/x^{\mu}\,,
\label{BPST-regular}
\end{equation}
with the 't~Hooft symbols $\eta^i{}_{{\mu}{\nu}}$ given in terms of
$\bar{\eta}^i{}_{{\mu}{\nu}}$ by
\begin{equation}
\eta^i{}_{j4}\!=-\bar{\eta}^i{}_{j4}\!={\delta}_{ij}\,,\quad
\eta^i{}_{jk}\!=\bar{\eta}^i{}_{jk}\!={\epsilon}_{ijk}\,.
\end{equation}
Maurer-Cartan one-forms can also be defined now,
\begin{equation}
{\chi}^i= \frac{1}{r^2_a}~{\eta}^i{}_{{\mu}{\nu}}\,(x-a)^{\nu}\textnormal{d}\/x^{\mu}\,.
\end{equation}
Together with $\textnormal{d}\/r_a$, they form a frame ${\cal
F}=\{{e}^{\,i}\!=r_a{\chi}^i, \,{e}^{\,4}\!=\textnormal{d}\/r_a\}$ with the same
orientation as $\{\textnormal{d}\/x^{\mu}\}$. All the calculations in Section~5 can be
analogously performed in this gauge.
\section*{Acknowledgment}
This work was partially funded by the Spanish Ministry of Education and
Science through grant FPA2011-24568.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,226 |
@interface FBWebDialogs : NSObject
{
}
+ (void)presentDialogModallyWithSession:(id)arg1 dialog:(id)arg2 parameters:(id)arg3 handler:(CDUnknownBlockType)arg4;
+ (void)presentDialogModallyWithSession:(id)arg1 dialog:(id)arg2 parameters:(id)arg3 handler:(CDUnknownBlockType)arg4 delegate:(id)arg5;
+ (void)presentFeedDialogModallyWithSession:(id)arg1 parameters:(id)arg2 handler:(CDUnknownBlockType)arg3;
+ (void)presentRequestsDialogModallyWithSession:(id)arg1 message:(id)arg2 title:(id)arg3 parameters:(id)arg4 handler:(CDUnknownBlockType)arg5;
+ (void)presentRequestsDialogModallyWithSession:(id)arg1 message:(id)arg2 title:(id)arg3 parameters:(id)arg4 handler:(CDUnknownBlockType)arg5 friendCache:(id)arg6;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,913 |
Le test d'émulsion est une méthode de révélation de la présence de lipides en utilisant la chimie par voie humide.
Principe
De l'éthanol, alcool qui dissout les lipides potentiellement présents, est ajouté à un échantillon et le mélange ainsi formé est agité, avant que de l'eau soit ajoutée et que le mélange soit agité à nouveau : si une certaine quantité de lipide est présente, le mélange devient opaque, blanc laiteux, car les lipides ne se dissolvant pas dans l'eau, il se forme une émulsion qui, par nature, est moins transparente qu'une solution.
Références
Test chimique
Analyse des lipides
Émulsion | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,181 |
Busco novia es una película de comedia peruana de 2022 dirigida por Daniel Vega y escrita por Renato Cisneros. Basado en el exitoso blog homónimo de El Comercio escrito por Renato Cisneros. Está protagonizada por César Ritter, Magdyel Ugaz, Vadhir Derbez, Fiorella Pennano, Gustavo Bueno y Grapa Paola.
Sinopsis
Cuenta la historia de Renzo Collazos, un periodista treintañero que busca novia y que, por obligación, debe empezar a escribir un blog sobre su fallida vida amorosa.
Reparto
Los actores que participaron en esta película son:
César Ritter como Renzo Collazos
Magdyel Ugaz como Lucía
Fiorella Pennano como Mariana
Vadhir Derbez como Robot
Gustavo Bueno como Ventoso
Grapa Paola como Dora
Producción
La película comenzó a rodarse a mediados de febrero de 2019 con una duración de 5 semanas.
Lanzamiento
La película iba a estrenarse el 7 de mayo de 2020 en los cines peruanos, pero el estreno fue cancelado debido al cierre de salas por la pandemia de COVID-19. Finalmente, la película se estrenó internacionalmente el 18 de noviembre de 2022 en Amazon Prime Video.
Referencias
Enlaces externos
Películas de 2022
Películas de Perú
Películas en español
Películas cómicas
Películas cómicas de Perú
Películas cómicas de los años 2020
Películas rodadas en Perú
Películas ambientadas en Perú
Películas sobre periodismo
Películas pospuestas debido a la pandemia de COVID-19
Películas no lanzadas en cines por la pandemia de COVID-19 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,475 |
Italian Ambassador Gaiani
presents Mattarella's credentials
(8 february 2019) The Italian Ambassador in Turkey, Massimo Gaiani, presented yesterday to the President of the Republic of Turkey, Recep Tayyip Erdoğan, the credential letters signed by President Sergio Mattarella. At the ceremony, held at the Presidential Palace, the Vice-Head of Mission, the Min. Cons. Filippo Andrea Colombo, and the Defense Attaché, Ship Captain Attilio Gattia. Ambassador Gaiani then tribute to the Mausoleum of Ataturk, where he deposited a wreath of flowers and signed the Book of Honour. Finally, he took part in a private visit to the museum. Ambassador Gaiani, recently established in Ankara, said: "Turkey is a strategic player in the Mediterranean region and a major economic partner for Italy. We must strive to further strengthen the already excellent relations in the political, economic, and cultural fields ". | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,816 |
Q: AFRAME: Event on completion of dynamic adding of component My use-case is as follows:
In a loop, entities are being created and components are being set up. This is via a json-object that is being passed to the function. The question I have is how best to get an event that the whole set of entities and their components are being initialised. The code is something like this
var parent = document.querySelector('#parent');
var ent = document.createElement('a-entity');
parent.appendChild(ent);
for(var i =0; i = components.length; i++) {
var arr = components[i];
var cl = arr[0]; // class name
var attr = arr[2]; // component name
var attrV = arr[3]; // component data
ent.setAttribute('class', cl);
AFRAME.utils.entity.setComponentProperty(ent, attr, attrV);
//ent.setAttribute(attr, attrV); tried with this too
}
console.log('loop completed')
The loop completed gets logged before the completion of the loading of some of the components. I would like to have some sort of a call back to know that all the components have been completed loaded.
There seems to be an event componentinitialized but it sends a return for only 1 component. My real requirement (not reflected in above code) is that an entity can have multiple components added.
To use the above, I may have to set this event for every component and keep track of whether it has been completed or not. Just wondering if there is a more elegant way to do it. Thanks
A: Entities emit the "loaded" event. It should be easier than listening for each component initialization within the entity.
Try out:
entity.addEventListener("loaded", (e) => {
console.log(e)
})
like i did here.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 169 |
Durness 4-H Club swept the competition by winning multiple prizes in this year's Co-op's 4-H essay competition. Federated Co-op Limited held an essay competition from January 1 to April 2 and received entries from 4-H clubs in Alberta, Manitoba and Saskatchewan.
The Durness Club took home the grand prize, as well as several second and third place prizes. Feed Sales and Marketing Manager Shawn McGuire said in a statement that the club's quality of applications were stellar.
The grand prize includes a set of clippers, backpack and a $1,000 donation to a non-profit or charity of the winner's choice. This year's winner, Kate Peregrym, is a member of the Durness 4-H Cloverbud program that focuses on hands-on, activity-based programming for youth aged six to eight. | {
"redpajama_set_name": "RedPajamaC4"
} | 88 |
\section{Outline}\label{sec:intro}
\section{Introduction}\label{sec:intro}
When developing improved medical imaging technologies, such as methods for image reconstruction, restoration, and analysis, it is crucial to objectively evaluate them via a diagnostic clinical task \cite{barrett, jha_objective_eval, sr_eval, eval_denoising}. Because a full-fledged clinical trial of rapidly developing imaging technologies often is infeasible \cite{hmi, fuli}, computer simulation studies \cite{vit} have been proposed as an alternative. In order to refine and assess any medical imaging technology via computer simulation, the nature and variability of the objects to-be-imaged must be accurately characterized. To this end, a variety of stochastic object models (SOMs) have been developed \cite{hmi, vit}; these enable simulation of random, and sufficiently realistic, digital medical objects.
{A generative model is a statistical model of an unknown data distribution that enables sampling from the data distribution via a learned representation of it.} The model is trained directly on a large sample drawn from the data distribution \cite{gdl_book}. Modern generative models learn a neural network-based mapping from a tractable distribution, such as a multivariate, independent, and identically distributed (i.i.d.) Gaussian distribution, to the intractable, high-dimensional object distribution of interest. This enables sampling from the unknown distribution, and provides the ability to perform inference. Therefore, generative models such as generative adversarial networks (GANs) are being actively investigated for applications in medical imaging, such as: image restoration \cite{gan_denoising, gancircle}, image reconstruction \cite{picgm, cs_sgm, inn_mri, iagan_sayantan}, image analysis \cite{madgan, abnormality_detection}, image-to-image translation \cite{cyclegan_medical}, data sharing \cite{gan_data_sharing} and objective image quality assessment \cite{gan_mcmc}.
Modern generative models, such as the StyleGAN and its successors \cite{stylegan, stylegan2, stylegan3}, represent a tremendous improvement in terms of the stability, controllability, diversity, and visual quality of generated images. However, state-of-the-art GANs trained on medical image datasets have been shown to produce images that look realistic, but nevertheless contain medically impactful errors \cite{gan_data_sharing, image_translation_hallucinations, hallucinations}. Therefore, in order for GANs to be safely used in medical imaging, they must first be objectively evaluated \cite{weimin_ambientgan}, for instance, with the help of a relevant diagnostic task.
Despite tremendous improvements in the quality of the images generated by a GAN, the question of whether or not a GAN correctly approximates the statistical features important to a medical imaging application remains largely unanswered. Although mathematical summaries, such as the Wasserstein metric \cite{wgan} and negative log-likelihood \cite{gan_eval_review} are correlated with the fidelity of the trained GAN, there is no guarantee that a favorable value achieved by these measures also translates to usefulness of the trained GAN for medical imaging applications. Although perceptual measures such as the Frechet Inception distance (FID) have grown to be immensely popular, they are agnostic to the downstream task a medical image GAN may be used for \cite{fid}. Furthermore, the above mentioned measures are ensemble measures. It has been shown that individual samples drawn from the GAN may contain impactful errors despite giving satisfactory ensemble measures \cite{rucha}. Lastly, medical image distributions typically consist of multiple classes or modes, and it has been shown that may produce critical errors while producing images from a mode that is rarely seen during training \cite{gan_data_sharing}.
The objective of this study is to assess the ability of a state-of-the-art GAN to learn the statistics of a canonical stochastic image model (SIM) that are relevant to the objective assessment of image quality (IQ), and to study how the performance assessment of GANs by task-agnostic measures such as FID score compares with the performance assessed by the medically meaningful measures identified for the canonical SIM under consideration. To this end, three canonical SIMs were identified, namely the modified clustered lumpy background model \cite{clb2}, the B-mode ultrasound speckle model \cite{insana} and the stylized two-dimensional (2D) VICTRE (S2V) model \cite{victre}. A state-of-the-art GAN architecture, namely StyleGAN2, was trained on images generated from these canonical SIMs. Statistical quantities that are meaningful and relevant to the above SIMs were computed from images from the canonical SIM as well as the images generated by the GAN. Summary measures computed from these identified statistical quantities were compared against the FID for the purpose of assessing the fidelity of the trained GAN. This work is an extension of a preliminary study conducted using an angiographic SIM \cite{gan_eval_spie}.
The remainder of this paper is organized as follows. Section \ref{sec:bkd} describes the relevant background on GANs and their evaluation, as well as the background on the SIMs used in this study. Section \ref{sec:num_methods} describes the setup for the specific numerical studies and the identification of the SIM-pertinent evaluation measures. Section \ref{sec:results} presents the results. A summary of the salient findings of this work is presented in Section \ref{sec:discussion}.
\section{Background}\label{sec:bkd}
\subsection{Generative adversarial networks (GANs)}
Generative adversarial networks (GANs) are a popular class of generative models that are aimed to approximate a data distribution by learning to map a sample $\mathbf{z} \in \mathbb{R}^k$ from a lower dimensional, tractable data distribution $p_{\mathbf{z}}$, such as the i.i.d. standard normal distribution, to a sample $\mathbf{f}$ from the high dimensional data distribution $p_{\mathbf{f}}$.
In GANs, two networks, namely a \textit{generator network} $G : \mathbb{R}^k \rightarrow \mathbb{R}^n$ with parameters $\Theta_G$ and a \textit{discriminator network} $D : \mathbb{R}^n \rightarrow \mathbb{R}$ with parameters $\Theta_D$ are jointly trained by approximately solving the following min-max optimization problem:
\begin{align*}\label{eqn:gan_loss}
\min_{\Theta_G} \max_{\Theta_D} ~ \mathbb{E}&_{\mathbf{f} \sim p_\mathbf{f}} \left[ \ell(D_{\Theta_D} (\mathbf{f})) \right]
+ \mathbb{E}_{\mathbf{z} \sim p_\mathbf{z}} \left[ \ell (1 - D_{\Theta_D}(G_{\Theta_G}(\mathbf{z}))) \right],\numberthis{}
\end{align*}
\noindent where $\ell(\cdot)$ is a utility function used to define the objective; for instance, a popular choice being $\ell(\mathbf{x}) = \log(\mathbf{x})$ \cite{gan_goodfellow}. The promise of a generative model such as a GAN comes from the fact that once trained, samples from the otherwise inaccessible high dimensional distribution $p_\mathbf{f}$ can be obtained by sampling low dimensional vectors, known as latent vectors $\mathbf{z}$ from $p_\mathbf{z}$ and computing $G(\mathbf{z})$. Thus, the GAN provides a tractable representation of $p_\mathbf{f}$ that may find use in downstream applications in imaging science, such as image reconstruction \cite{iagan_sayantan, picgm} and image quality assessment \cite{gan_mcmc}.
\subsection{Advanced GAN training strategies}
Under prescribed theoretical conditions, minimizing the GAN training loss described in \autoref{eqn:gan_loss} is equivalent to minimizing the empirical Jensen-Shannon (JS) divergence between the true and the estimated probability distribution functions (PDFs) of the data \cite{gan_goodfellow}. However, in practice, GAN training is known to be unstable \cite{gan_improved_training, gan_principled_training} and several strategies have been proposed to improve stability. For example, the use of different learning rates and update frequencies for the generator and discriminator weights aids in avoiding the vanishing gradients problem for the generator and premature overfitting of the discriminator \cite{gan_goodfellow, gan_review}. Novel loss functions, such as in so-called Wasserstein GANs \cite{wgan}, also help in improving the training stability. Karras, \textit{et al.} \cite{progan} proposed a strategy for scaling GANs by use of \textit{progressive training}, where both the generator and discriminator are trained on lower resolution images and are progressively grown to enable training on higher and higher resolution images. StyleGAN and its successor, StyleGAN2, introduce blocks of transformed latent vectors as inputs to different layers of the network at different resolutions, thus controlling features at different scales \cite{stylegan, stylegan2}. Although these improvements to the GAN architecture and training have cumulatively led to state-of-the-art performance in terms of diversity, controllability and realism of images generated, they are largely heuristic, and are not designed specifically to learn task-pertinent statistics of medical image distributions.
\subsection{Evaluation of generative adversarial networks}
Modern GANs, such as the StyleGAN2 \cite{stylegan2} have shown impressive performance in terms of the perceptual quality of the generated images, invertibility, and meaningful control over image semantics. However, evaluating the quality of the distribution learned by a generative model is an open problem \cite{gan_eval_review_new}. Some measures directly estimate analytical quantities and distance metrics related to the image probability density function (PDF), such as the negative log-likelihood \cite{gan_eval_review} or the Wasserstein metric \cite{wgan}. Other measures such as the perceptual path-length \cite{stylegan} analyze the nature of the manifold learned by the GAN. Motivated by subjective perceptual assessment by humans \cite{is}, perceptual evaluation measures such as the Inception score (IS) and more commonly, the Fr\'echet Inception distance (FID) score have become immensely popular \cite{is, fid}. In order to compute these scores, image features are first extracted using a pre-trained Inception network \cite{inception} and distance metrics on the extracted features are computed.
Although the FID score has shown excellent agreement with subjective visual assessments by humans \cite{fid}, it is agnostic to the downstream task a medical image GAN may be used for.
Additionally, it is an ensemble statistic, and hence could be blind to specific errors in high-order statistics of individual images \cite{rucha}.
The studies described below seek to assess the ability of medical image GANs to reproduce image statistics that are meaningful and pertinent to the medical stochastic image model under consideration, and to see how well traditional measures such as the FID correlate with these pertinent statistics. In order to do so, the data distributions used to train the GAN needs to be carefully chosen as follows. First, realistic canonical SIMs that are associated with a mathematical procedure for generating images need to be identified, because it allows for direct control over image properties of interest. For these canonical SIMs, statistical quantities that are medically meaningful for the particular canonical SIM need to be identified. These tasks are described next.
\begin{figure*}
\includegraphics[width=\linewidth]{images/real_and_fake.pdf}
\caption{Images simulated from the canonical CLB and USS SIMs and images generated by the GANs trained on images from the SIMs.}
\label{fig:real_and_fake_clb}
\end{figure*}
\begin{figure}
\includegraphics[width=\linewidth]{images/victre_real_and_fake.pdf}
\caption{Images simulated from the canonical VICTRE SIM and images generated by the GAN trained on images from the SIM.}
\label{fig:real_and_fake_victre}
\end{figure}
\subsection{Canonical stochastic image models}
Stochastic models of simulated medical images have been developed in order to approximately capture the variability in medical image distributions \cite{barrett, hmi, lumpy}. Traditionally, such stochastic image models (SIMs) have been established by developing a mathematical procedure for generating images that possess certain prescribed statistical properties. Examples of such SIMs include the lumpy background model \cite{lumpy}, the clustered lumpy background (CLB) model \cite{clb}, B-mode ultrasound speckle model \cite{insana}, among others. {Once a SIM is established, it can be used to model image statistics in virtual imaging trials \cite{vit}.} Here, the three canonical SIMs identified for the purpose of evaluating a GAN-based SIM are briefly reviewed. These SIMs are the modified clustered lumpy background model \cite{clb2}, the B-mode ultrasound speckle model \cite{insana} and the stylized 2D VICTRE breast phantom model \cite{victre}.
As compared to real medical images, simulated images from these SIMs provide the ability to examine the behavior of the GAN under a controlled setting, with several different parameter configurations of the canonical SIM.
\subsubsection{Modified clustered lumpy background (CLB) model}\label{sec:clb}
The CLB model was developed by Bochud et al. \cite{clb} for generating random backgrounds that resemble the image textures seen in mammography.
In 2008, Castella, \textit{et. al} proposed variations to the original CLB model so that the images from the model better resemble realistic mammographic textures as judged by human experts \cite{clb2}.
In addition to introducing oriented structures and long-range correlations, the authors
proposed to adjust the parameters of the CLB model in order to improve the realism of the images generated. This was done by computing 17 different texture features on both the real mammographic regions of interest (ROIs) as well as images generated from the CLB model. These were used to formulate a loss function that was minimized by tuning the parameters of the CLB model.
\subsubsection{B-Mode Ultrasound Speckle (USS)}
B-mode ultrasound speckle (USS) can be viewed as a random phasor sum of complex signals \cite{insana}. The received complex signal $E$ is a radio frequency voltage output from an ultrasound transducer and can be modeled as the sum of $N$ complex signals with phases statistically independent uniformly distributed on $[0,2\pi]$ \cite{insana}. The quantity $N$ is the number of scatterers per resolution cell or equivalently the scatterers per number density (SND) times the resolution cell size. The resolution cell size is defined as the axial resolution ($\rm AR$) times the lateral resolution ($\rm LR$), given in Ref. \cite{iaea_handbook},
where the parameters are the frequency of the carrier $f_c$, the wave velocity $v$, the ratio between the focal distance and the length of the aperture (called the $f$-number) and the number of cycles within the full width half maximum in the spatial direction (FWHM) $N_c$ . The USS SIM is modeled using the method proposed in Ref. \cite{insana} where the standard deviations of the 2-D Gaussian PSF are determined by the AR and LR.
If $N$ is large, the resulting USS follows Gaussian statistics and is called fully developed speckle. In this case, the envelope $|E|$ follows a Rayleigh distribution and thus the intensity $I = |E|^2$ follows an exponential distribution.
If $N$ is small then the resulting USS is called non-Gaussian speckle and its statistical properties are determined by $N$ \cite{insana}.
\subsubsection{The stylized 2D VICTRE (S2V) breast phantom model}
The US Food and Drug Administration's (FDA) Virtual Imaging Clinical Trials for Regulatory Evaluation (VICTRE) initiative has produced a set of software tools for simulating random anthropomorphic phantoms of the human female breast \cite{victre}.
These numerical breast phantoms (NBPs) are three dimensional (3D) voxelized maps, where a voxel value denotes the tissue type from one out of the following 10 tissues: fat, glandular tissue, skin, artery, vein, muscle, ligament, nipple and terminal duct lobular unit. Controlling the patient-specific input parameters such as breast type, size, shape, granularity and density, and setting the random seed number enables the generation of large ensembles of stochastic NBPs with realistic variation in breast anatomy, shape and fat-to-glandular tissue ratio. The VICTRE model is thus a general stochastic object model (SOM) that can be specialized to different imaging modalities by assigning the appropriate physical coefficients. In particular, by assigning X-ray linear attenuation coefficients to the various tissues in the NBPs and extracting 2D slices from the 3D phantom, a SIM can be obtained.
The VICTRE software creates NBPs that correspond to four breast types identified by the American College of Radiology's (ACR) Breast Imaging Reporting and Data System (BI-RADS) \cite{birads} and are distinguished by the amounts of fat and glandular tissue.
\section{Numerical studies}\label{sec:num_methods}
\subsection{SIM training data and GAN training}
\subsubsection{The CLB model}
The following four parameter configurations of the modified CLB model that were shown to produce realistic simulated mammographic images under radiologists' assessment \cite{clb2} were used in this study -- (1) \textit{doubiso}, a double-layered CLB model with isotropically oriented clusters, (2) \textit{simpiso}, a single-layered CLB model with isotropically oriented clusters, (3) \textit{doubori}, a double-layered CLB model with anisotropically oriented clusters, and (4) \textit{simpori}, a single-layered CLB model with anisotropically oriented clusters. Additionally, images from the original CLB model \textit{opex99}, proposed by Bochud \textit{et al.} \cite{clb} were employed. The gray levels and pixel value range were set in accordance with Castella \textit{et al.} \cite{clb2}. For each of the five canonical SIMs, a GAN was trained on a dataset of 100,000 256$\times$256 images from the SIM.
As discussed in the Introduction, medical image distributions are typically mixed distributions consisting of multiple classes or modes. In order to illustrate the effect of mixing distributions on the identified SIM-pertinent measures,
a stylized emulation of data coming from two different imaging sites or clinical systems having different resolution properties was constructed. Accordingly, one of the classes consisted of \textit{doubiso} images as are described above. The other class consisted of \textit{doubiso} images that were first degraded by use of a Gaussian blur followed by low-pass filter $\mathcal{H}_{\rm LPF}(\cdot)$ with cutoff at half the image bandwidth. Two such multi-class datasets were constructed, one having a 50\textbackslash50\% split and the other having a 95\textbackslash5\% split between the regular and degraded image classes. These two datasets will henceforth be referred to as the \textit{doubiso} 50-50 and \textit{doubiso} 95-5 datasets respectively.
\subsubsection{B-mode Ultrasound Speckle Model}
The parameter configurations chosen for the USS SIMs are follows. All images were $256 \times 256$ pixels in size {with each pixel corresponding to a $100 \mu \text{m} \times 100 \mu \text{m}$ square.} The velocity of the wave was set to $v = 1556 \text{m/s}$, the frequency $f_c$ was set to 3.5 MHz, the number of cycles within the FWHM was set to $N_c=2$, the $f$-number for the $y$ direction was set to $2$ and the $f$-number in the $z$ direction was set to $3$. The ultrasound wave was assumed to be propagating in the $x$ direction.
The SND parameter was varied to create four canonical USS SIM datasets, corresponding to SND values of 1, 2, 3 and 30 $\text{mm}^{-3}$ respectively. The first three values were chosen because they fall in the range of SND values that can be accurately estimated from the image \cite{insana}, which is not the case for the \textit{SND-30} SIM that represents a fully developed speckle \cite{iaea_handbook}. These four SIMs will henceforth be called (1) \textit{SND-1}, (2) \textit{SND-2}, (3) \textit{SND-3} and (4) \textit{SND-30} respectively.
Additionally, similar to the CLB case, two multi-class datasets were considered, where \textit{SND-2} and \textit{SND-3} were (1) distributed with a 50\% - 50\% split and (2) were distributed with a 95\% - 5\% split. Henceforth these datasets will be called \textit{USS Mixed 50-50} and \textit{USS Mixed 95-5}.
For each of the above described SIMs, a GAN was trained using 100,000 images from the SIM. Before training, each ensemble of training images was converted to an unsigned, 8-bit grayscale where 255 corresponds to the top 1\% pixel value in the ensemble.
\subsubsection{The S2V model}
The S2V was obtained from the 3D VICTRE NBP SOM described in \autoref{sec:bkd} as follows. First, a dataset of 1000 3D NBPs was generated using the VICTRE tool \cite{victre}. Next, linear attenuation coefficients in $\rm cm ^{-1}$ for X-rays of energy 30 keV were assigned to the pixels corresponding to each of the tissue types. These values were either directly obtained from literature, or calculated using the mass attenuation coefficient and material density values obtained from literature \cite{chen, nist, tomal}. Coronal slices were extracted from a central region of an NBP that ranges from 40\% through 70\% of the distance from the outermost coronal plane to the innermost coronal plane. This was done to avoid extracting slices too close to the chest wall or the nipple. A spacing of 50 pixels was maintained between two slices consecutively extracted from the same NBP. The extracted slices were then downsampled to an image dimension of 512$\times$512, which corresponds to the length scale of 0.4 $\mu$m per pixel.
The described procedure generated a 2D dataset of 130,000 slices, which was used for training a GAN.
{StyleGAN2, proposed by Karras \textit{et al.} \cite{stylegan2} was employed as the GAN in all the studies described in this work. All the default parameters and configurations of the StyleGAN2 architecture including the latent space dimensionality were kept the same as the the original code base, except for the number of channels in the output image, which was set to 1. The networks were trained using Tensorflow 1.14/Python \cite{tensorflow} on an Intel Xeon Gold 5218 CPU and two Nvidia Quadro RTX 8000 GPUs.}
\subsection{Identification and computation of SIM-pertinent evaluation measures}
A GAN may learn different types of image statistics to different levels of correctness. Hence, it is important to evaluate GANs using measures based on those statistics that are meaningful and pertinent to the SIM considered. In this study, such SIM-pertinent evaluation measures are based on statistics that either have been deemed important for assessing the realism of the canonical SIM images by human experts, or are known to be related to biomarkers important for a particular diagnostic task. These statistics are computed from both the ``direct-simulated" images, i.e. images directly simulated from the canonical SIM, as well as the GAN-generated images.
\subsubsection{The CLB model}
The 17 different texture features identified by Castella, \textit{et al.} mentioned in Section \ref{sec:bkd} have been demonstrated to be useful for improving the medical realism of CLB images under objective and psychophysical experiments involving the judgement of radiologists \cite{clb2}. Therefore, these statistics were chosen as the statistics meaningful for assessing a GAN trained on the CLB SIMs.
These texture features include those derived from the per-image, gray-level intensity distribution, gray-level co-occurrence matrices (GLCMs) \cite{haralick}, primitives matrices (GLRM), and the neighborhood gray tone difference matrix (NGTDM) \cite{ngdtm}.
For each of the five CLB model types in Section \ref{sec:num_methods}A, as well as the two multi-class CLB SIMs, the following 17 texture features described by Castella, \textit{et al.} \cite{clb2} were computed from each image of the evaluation datasets.
Mean, standard deviation, skewness and kurtosis were derived from the per-image gray-level intensity distribution. The texture features energy, entropy, maximum, contrast and homogeneity were computed from the GLCMs.
Four features were derived from the primitives matrices (GLRMs), namely, the short primitive emphasis (SPE), long primitive emphasis (LPE), gray level uniformity (GLU), and primitive length uniformity (PLU). The four features derived from the NGTDM \cite{ngdtm} were coarseness, contrast, complexity and strength. Various parameter values required for the computation of the texture features, such as the number of gray levels, two-point distances and angles were fixed to the values used in Castella, \textit{et al.} \cite{clb2}. The resulting feature data were then used for further analysis in order to summarize trends. Two types of analyses were conducted on the feature data. The first computed an empirical estimation of the JS divergence between the joint texture-feature distributions by utilizing the feature data \cite{empirical_kl}. The second plotted the joint empirical PDF over the first two principal components of texture features. The texture features used for this computation were selected as follows. First, principal component analysis (PCA) was conducted for each of the three spatial texture feature families, namely -- the GLCM, GLRM and NGTDM feature families. Next, the first two principal components were selected, and an empirical PDF over these two components was computed. The empirical PDFs that give the highest discrepancy between the direct-simulated and GAN-generated distributions in terms of the total variation (TV) distance were plotted.
\subsubsection{B-mode Ultrasound Speckle Model}
Previous studies have shown that the intensity signal-to-noise ratio (SNR) of USS images is associated with the envelope statistics \cite{oelze}. In regions of the body such as the liver and the breast, the envelope statistics have previously been successfully used for tissue characterization \cite{oelze}. Therefore, it was chosen as the SIM-pertinent statistic, though this preliminary study does not associate a given speckle model with a tissue type.
The PDF of the $\rm SNR^2$ estimate of USS speckle can be modeled as a Gaussian distribution centered around the true $\rm SNR^2$. If the scatterers per resolution cell $N$ follows a Poisson distribution, then one can estimate $N$ using $\rm SNR^2$. The SNR and $N$ estimate called $\hat{N}$ are defined as:
\begin{align}\label{eqn:SNR_2}
{\rm SNR} = \frac{\mu_I}{\sigma_I}, \text{ } \hat{N} = \frac{\rm SNR^2}{1-\rm SNR^2},
\end{align}
where $\mu_I$ and $\sigma_I$ are the mean and standard deviation of the intensity. The SNR and $\hat{N}$ were computed on a per-image basis for both the direct-simulated and GAN-generated images using empirically estimated $\mu_I$ and $\sigma_I$ from each image in the test dataset. The JS divergence was used as a measure to summarize the discrepancy between the $\rm SNR^2$ PDFs of the direct-simulated and GAN-generated images.
\subsubsection{The S2V model}
Human female breasts can be categorized into four different types based on the relative amount of fat and glandular tissue \cite{birads}. It is known that the amount of fat compared to the glandular tissue is an important factor impacting the risk of developing breast cancer, and the effectiveness of screening tests such as mammography in detecting breast masses \cite{wolfe1, fgr, birads}. Fat and glandular tissue have different linear attenuation coefficients \cite{chen, nist, tomal}. Therefore, the ratio of fat-to-glandular tissue was chosen as the SIM-pertinent measure for evaluating the GAN trained on the S2V SIM. For the idealized S2V SIM described in \autoref{sec:num_methods}A, the ratio $\rho_{F:G}$ of the amount of fat-to-glandular tissue in a thin coronal slice of an NBP can be computed by first calculating the number of pixels $F$ and $G$ relative to the total image pixels having linear attenuation coefficient values close to that of fat and glandular tissue respectively, and then computing their ratio $\rho_{F:G} = F / G$. Because the linear attenuation coefficient value of fat and glandular tissue are far enough to not confound a simple thresholding-based segmentation scheme, the values of $F, G$ and $\rho_{F:G}$ can be estimated accurately both for the direct-simulated and GAN-generated images.
Using this procedure, $\rho_{F:G}$ was estimated on a per-image basis for both the direct-simulated and GAN-generated images. The empirical PDFs of $\log \rho_{F:G}$ computed from both the direct-simulated and GAN-generated images were plotted, and the JS divergence between the two PDFs was computed.
Apart from the above-described SIM-pertinent measures, basic ensemble statistics, such as the histogram of gray level values and the empirical image autocorrelation were computed from direct-simulated and GAN-generated images from all the SIMs in order to assess the ability of the GAN to learn these statistics accurately. As described in Bochud \textit{et al.} \cite{clb}, a Papoulis window was used in order to overcome boundary artifacts in the computation of the autocorrelation.
The FID score between a direct-simulated and a GAN-generated test dataset, as well as two i.i.d. direct-simulated datasets was computed. The latter serves as a heuristic noise floor for the FID score for the particular SIM. A pre-trained InceptionV3 network \cite{inception} was employed for this purpose. All the evaluation measures were computed using 10,000 direct-simulated and GAN-generated images. Other test dataset sizes were examined, and the computed metrics were found to be qualitatively no different.
\begin{figure}
\includegraphics[width=\linewidth]{images/gray_histogram.pdf}
\caption{Sample empirical gray level PDFs of direct simulated and GAN-generated images for the three types of SIMs.}
\label{fig:intensity_histogram}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{images/autocorrelation.pdf}
\vspace{1pt}
\caption{Sample radial profiles of autocorrelation of direct simulated and GAN-generated images for the three types of SIMs.}
\label{fig:autocorrelation}
\vspace{-10pt}
\end{figure}
\section{Results}\label{sec:results}
This section is organized as follows. Section \ref{sec:results}A qualitatively describes the images generated by the GAN. Section \ref{sec:results}B describes the basic ensemble statistics learned by the GAN, such as the intensity histogram and the image autocorrelation. Section \ref{sec:results}C describes and compares the FID score and the identified meaningful measures based on their ability to assess the fidelity of the trained GAN. Finally, Section \ref{sec:results}D compares the ability of the FID score and the identified measures to assess multi-modal SIMs.
\begin{figure}
\includegraphics[width=\linewidth]{images/fid_js_singleclass.pdf}
\caption{FID and empirical feature JS divergence measures between the real and GAN-generated distrbutions for \textit{opex99}, \textit{simpiso}, and \textit{doubiso} models. The dotted lines represent the value of the measures between two direct-simulated datasets.}
\label{fig:fid_js_singleclass}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{images/fid_js_multiclass.pdf}
\caption{FID and empirical feature JS divergence between the real and GAN-generated distrbutions for \textit{opex99}, \textit{doubiso}, and the two \textit{doubiso-mixed} models. The dotted lines represent the value of the measures between two direct-simulated datasets.}
\vspace{-10pt}
\label{fig:fid_js_multiclass}
\end{figure}
\begin{figure*}
\includegraphics[width=\linewidth]{images/clb_pca2.pdf}
\caption{Empirical PDF over the first two principal components of the CLB feature data. The selected texture feature family for each of the models is shown below each plot. The blue and the orange contour plots denote the direct-simulated and GAN-generated distributions respectively.}
\label{fig:clb_worst_components}
\end{figure*}
\begin{figure}
\includegraphics[width=\linewidth]{images/clb_worst_components_individual.pdf}
\caption{Distributions of per-image NGTDM Complexity and Coarseness features learned by the GAN for the \textit{doubiso} and \textit{doubiso mixed 50-50} SIMs. The red arrows point to the parts of the distribution corresponding to the degraded class that are completely ignored by the GAN.}
\label{fig:clb_worst_components_individual}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{images/US-FID-JS-plot-new.pdf}
\caption{FID and $\rm SNR^2$-JS divergence between the real and GAN-generated distributions for \textit{SND-1}, \textit{SND-2}, \textit{SND-3}, \textit{SND-30}, \textit{USS Mixed 50-50} and \textit{USS Mixed 95-5}. The dotted lines represent the value of the measures between two direct-simulated datasets.}
\label{fig:US_FID_KL}
\vspace{-10pt}
\end{figure}
\begin{figure*}
\includegraphics[width=\linewidth]{images/snr_2_us_new3.pdf}
\caption{Estimated $\rm SNR^2$ PDFs of both direct-simulated and GAN-generated images for \textit{SND-1}, \textit{SND-2}, \textit{SND-3}, \textit{SND-30}, \textit{USS Mixed 50-50} and \textit{USS Mixed 95-5}. Although the direct-simulated and GAN-generated distributions tend to match well, occasionally this is not the case as can be seen in for \textit{SND-30} and \textit{USS Mixed 50-50}. Note that the \textit{USS Mixed 95-5} $\rm SNR^2$ PDF has the density in log scale with the red arrow pointing to the distribution of the \textit{SND-3} class having 5\% prevalence.}
\label{fig:SNR_2_us}
\end{figure*}
\subsection{Qualitative assessment of images generated by the GAN}
Figures \ref{fig:real_and_fake_clb} and \ref{fig:real_and_fake_victre} show the images generated by the trained GANs alongside direct-simulated images from the training dataset for the single-class CLB, USS and S2V models. It can be seen that there is obvious visual similarity between the direct-simulated and the GAN-generated images. Note that this is even true for the zoomed-in images of the S2V model shown in Fig. \ref{fig:real_and_fake_victre}. One important thing to note, however, is that some of the ligaments in the GAN-generated images appear broken at certain locations, which is not the case for the direct-simulated images.
\subsection{Basic ensemble statistics learned by GANs}
Figure \ref{fig:intensity_histogram} shows the ensemble empirical PDF of pixel gray levels for the CLB \textit{doubiso} SIM, the USS \textit{SND-1} and the \textit{SND-30} SIMs, and the S2V SIM, computed from both the direct-simulated and GAN-generated images. A close match between these empirical PDFs indicates that the GAN is able to reproduce first-order statistics. The GAN performs similarly for the other CLB SIMs, which have gray-level distributions similar to the ones shown in Fig. \ref{fig:intensity_histogram}a. It can be seen that for USS \textit{SND-30} SIM, which represents a fully developed speckle, the GAN reliably reproduces the expected Rayleigh distribution of grayscale values. For the USS \textit{SND-1} SIM, this distribution is far from Rayleigh both for the direct-simulated and GAN-generated images, yet the GAN recovers this distribution successfully. The pixel-value distributions corresponding to USS \textit{SND-2} and \textit{SND-3} SIMs appear intermediate between the ones shown in Fig. \ref{fig:intensity_histogram}b and c.
Fig. \ref{fig:autocorrelation} shows the radial profile of the image autocorrelation computed using the direct-simulated and GAN-generated images for the CLB \textit{doubiso}, USS \textit{SND-1} and S2V SIMs. It can be seen that the GAN was successful in recovering this particular second-order statistic. Similar results were obtained for the other CLB and USS SIMs considered.
\subsection{SIM-pertinent measures learned by GANs}
\subsubsection{CLB Model}
Figure \ref{fig:fid_js_singleclass} shows the FID as well as the texture feature JS divergence between the direct-simulated and GAN-generated distributions as a function of training iteration. In Fig. \ref{fig:fid_js_multiclass}, the FID scores and the feature JS divergences for the \textit{doubiso mixed 50-50} and \textit{doubiso mixed 95-5} datasets are shown along with those for the single class \textit{doubiso} and \textit{opex99} models. It can be seen that as the training progressed, both the FID as well as the empirical feature JS divergence converged for most of the SIMs considered. However, in some cases, these measures either diverged or varied erratically as the training progressed.
Furthermore, the high value of the feature JS divergence for the GAN trained on the \textit{doubiso mixed 50-50} model suggests that the GAN was not able to reproduce the meaningful feature statistics as well as the GAN trained on the single class dataset. On the other hand, the FID plot in Fig. \ref{fig:fid_js_multiclass} shows comparable FID scores for the various SIMs and does not predict the same trend as the feature JS divergence plots. This suggests that for this specific example, the FID score could be blind to telling if multiple modes in the distribution are learned correctly.
{These findings were further investigated using the principal components of the data from the texture feature family that was learnt the least accurately by the GAN. The procedure for computing these components was described earlier in Section \ref{sec:num_methods}B.} Figure \ref{fig:clb_worst_components} plots this joint empirical PDF for the direct-simulated and GAN-generated images. Note that these texture features are computed on a per-image basis. For most of the CLB SIMs, obvious dissimilarities between the original and learned distributions can be seen. These dissimilarities correlate well with the feature JS divergence values shown in Figures \ref{fig:fid_js_singleclass} and \ref{fig:fid_js_multiclass}, but not with the corresponding FID values. For the \textit{doubiso mixed 50-50} SIM, it can be seen that the GAN failed to correctly learn the distribution of principal NGTDM and GLRM texture components for one of the classes. On further investigation and comparison with the individual texture distributions for the \textit{doubiso} SIM, it was revealed that the GAN failed to learn the per-image NGTDM coarseness and complexity distributions of the images from the degraded class, as shown in Fig. \ref{fig:clb_worst_components_individual}. This was despite the GAN being able to learn ensemble measures such as the FID and basic first- and second-order statistics well.
\begin{table}[h!]
\centering
\resizebox{0.5\textwidth}{!}{
\begin{tabular}{lcccccccc}
\toprule
& \multicolumn{2}{c}{\textit{SND-1}} & \multicolumn{2}{c}{\textit{SND-2}} & \multicolumn{2}{c}{\textit{SND-3}} & \multicolumn{2}{c}{\textit{SND-30}}\\
\cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9}
& D.S. & G.G. & D.S. & G.G. & D.S. & G.G. & D.S. & G.G.\\
\midrule
$\mu$ & 0.41 & 0.41 & 0.48 & 0.49 & 0.58 & 0.58 & 0.89 & 0.89 \\
\midrule
$\sigma$ & 0.024 & 0.024 & 0.027 & 0.028 & 0.03 & 0.03 & 0.032 & 0.047 \\
\midrule
\textit{MSE} ($10^{-4}$) & 2.9 & 20.9 & 3.0 & 24.4 & 2.6 & 3.3 & 1.3 & 2.6 \\
\midrule
$\hat{N}$ & 0.71 & 0.70 & 0.94 & 0.97 & 1.37 & 1.39 & 7.54 & 9.78 \\
\bottomrule
\end{tabular}}
\caption{A table showing the mean $\mu$ and standard deviation $\sigma$ of the gaussian fit curve for both direct-simulated and GAN-generated $\rm SNR^2$ distributions, the mean squared error (MSE) between the gaussian fit and their respective $\rm SNR^2$ distributions and the mean scatterers per resolution cell estimate $\hat{N}$ of both direct-simulated (D.S.) and GAN-generated (G.G.) images.}
\label{tab:us_gauss_fit}
\vspace{-10pt}
\end{table}
\subsubsection{B-Mode Ultrasound Speckle Model}
The empirical JS divergence between the estimated $\rm SNR^2$ PDFs computed from the direct-simulated and GAN-generated USS images (henceforth refered to as the $\rm SNR^2$-JS divergence) is shown in Fig. \ref{fig:US_FID_KL} alongside the FID score computed between the direct-simulated and the GAN-generated images.
Although the $\rm SNR^2$-JS divergence approaches the noise floor and converges for most SIMs, it behaves erratically for a few SIMs, even as the FID score for the corresponding SIM converges.
In Fig. \ref{fig:SNR_2_us} the estimated $\rm SNR^2$ PDFs are plotted for both direct-simulated and GAN generated USS images. As can be seen the GAN generated images tend to give $\rm SNR^2$ distributions that somewhat match those of the direct-simulated images for the \textit{SND-1}, \textit{SND-2} and \textit{SND-3} SIMs. Since the $\rm SNR^2$ is theoretically expected to be distributed as a Gaussian for these SIMs \cite{wagner_auto},
each distribution of direct-simulated and GAN-generated images was fit to a Gaussian. In Table \ref{tab:us_gauss_fit} the mean and standard deviation of the best fit Gaussian distribution are shown in the first two rows while the third row shows the mean squared error between a given $\rm SNR^2$ distribution and its Gaussian fit.
The results for the mean and standard deviation of the Gaussian fit distributions confirm our visual inspection. The mean values were near perfect matches and so are the standard deviations with the exception of \textit{SND-30}.
However, the \textit{MSE} between the GAN-generated empirical $\rm SNR^2$ PDFs and their Gaussian fits was larger than the \textit{MSE} between the direct-simulated empirical $\rm SNR^2$ PDFs and their Gaussian fits.
Finally, the mean estimate of scatterers per resolution cell $\hat{N}$ computed from GAN-generated images was close to that computed from the direct-simulated images for all USS SIMs except for \textit{SND-30}. This is expected since the $\rm SNR^2$ distributions do not match well for the \textit{SND-30} SIM.
In Fig. \ref{fig:US_FID_KL}, the FID scores and the $\rm SNR^2$-JS divergences can be seen for \textit{USS Mixed 50-50} and \textit{USS Mixed 95-5} SIMs. As the training progresses, both the measures seem to converge in a similar fashion to most of the single class SIMs. Interestingly, the \textit{USS Mixed 95-5} SIM has one of the higher FID scores while also having the lowest $\rm SNR^2$-JS divergences over training. This could be because even if the $\rm SNR^2$ distribution over the class having 5\% prevalence was not learnt well, it may not significantly impact the JS divergence \cite{teaching}.
Finally, it can be seen that the the GAN struggles in this case to properly reproduce the direct-simulated $\rm SNR^2$ distributions.
In the case of \textit{USS Mixed 50-50}, the $\rm SNR^2$ distributions of the two classes have greater variance for the GAN-generated images. This results in the GAN producing more images having a value of $\rm SNR^2$ intermediate between the two classes. For the \textit{USS Mixed 95-5} SIM, the GAN was not able to reproduce the mode corresponding to the class having 5\% prevalence in the dataset, as seen in \autoref{fig:SNR_2_us}.
\begin{figure}
\includegraphics[width=\linewidth]{images/fid_js_victre.pdf}
\caption{FID and empirical ratio-JS divergence between real and GAN-generated distributions for the S2V dataset. The dotted line represents the value of the measures between two direct-simulated datasets.}
\label{fig:fid_js_victre}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{images/fat_and_glandular_kde.pdf}
\caption{(a-b) The estimated PDF over the per-image number of pixels corresponding to fat and glandular tissue respectively, as a fraction of the total image pixels (denoted by $F$ and $G$ respectively). (c) The estimated PDF over $\log(F/G)$.}
\vspace{-10pt}
\label{fig:fat_and_glandular_kde}
\end{figure}
\subsubsection{The S2V SIM}
Figure \ref{fig:fid_js_victre} shows the empirical JS divergence between the empirical PDFs of $\log\rho_{F:G}$ computed from the direct-simulated and GAN-generated images (henceforth refered to as the ratio-JS divergence) as a function of the training iteration. This is displayed alongside the plot of FID as a function of the training iteration. It can be seen that although the FID predictably converged as the training progresses, the ratio-JS divergence was erratic and did not converge the same way as FID. Figure \ref{fig:fat_and_glandular_kde} shows the empirical PDFs of $\log\rho_{F:G}$ computed on a per-image basis from the direct-simulated and GAN-generated images. The direct-simulated distribution clearly shows the four different breast types based on the $F:G$ ratio in their correct clinical prevalence. However, the GAN-generated distribution completely ignored or incorrectly represented many of the breast type modes. This was despite the GAN giving visually appealing images and accurate FID and other basic ensemble metrics.
\section{Summary}\label{sec:discussion}
Generative adversarial networks (GANs) could potentially be employed as stochastic image models for use in several tasks in medical imaging. However, GANs have traditionally been evaluated using mathematical or perceptual measures that may not correlate with those statistics that are important with respect to a downstream task. The objective of this work was to study the ability of GANs to reproduce medical image statistics that are meaningful and pertinent to the SIM under consideration, and to see how well traditional measures such as FID correlate with these pertinent statistics.
The GANs employed consistently produced images that visually appeared realistic, and were able to accurately and consistently reproduce basic statistics such as the intensity histograms and image autocorrelation. It was also observed that although most of the evaluation measures used in this paper converged, they did not necessarily converge at the same rate, and some of them diverged as the training progressed. This indicates that the convergence of a commonly used measure such as the FID score to a low value does not guarantee the correct convergence of those statistics that are meaningful to the particular medical SIM under consideration. Since the FID score measures the Fr\'echet distance in the feature space of an Inception network trained on the ImageNet dataset, it is not tailored to the specific medical image distribution considered. Additionally, the GAN may learn the distribution of different features to different degrees of fidelity, resulting in different performance rankings when examined by different measures.
We note that for all the SIMs considered in this paper, the GAN-generated images retained potentially impactful per-realization errors in some of the meaningful features identified. These errors manifested themselves in the empirical distributions of these meaningful features learned by the GAN, where among others, critical inaccuracies such as mode-dropping and merging of multiple classes or modes was observed. This was despite the GAN producing excellent agreement with the direct-simulated distribution in terms of ensemble measures, such as the FID and basic first- and second-order statistics.
These observations point to the need for choosing evaluation measures that are meaningful and pertinent to the SIM considered, are motivated by a downstream task, and are sensitive to the important aspects of a medical image distribution, such as multiple modes. While formulating such evaluation measures requires significant effort, it opens up the possibility of evaluating GANs in terms of those statistics that influence task-performance.
This study employed the StyleGAN2 architecture, since it has been shown to consistently produce realistic images when trained on a wide variety of datasets. However, the proposed analysis does not depend upon the GAN architecture employed, and could easily be performed on other GAN architectures. Canonical SIMs that produce simulated medical images provided the ability to examine the behavior of the GAN under a controlled setting with different parameter configurations. Nevertheless, evaluating GANs trained on real medical images remains a topic of future investigation. Lastly, although careful identification of meaningful evaluation measures is a key aspect of this study, it falls short of performing a task-based assessment of GANs. This will be the topic of a follow-up study.
\section{Results - Ultrasound frequency sweep} \label{sec:freq_sweep}
Figure \ref{fig:real_and_fake_freq_sweep} shows GAN generated images as well as images from the training set each at different carrier frequencies. In Figure \ref{fig:freq_sweep_US} results for the intensity $SNR^2$, $\mu_I$ and $\sigma_I$ are plotted versus frequency for both GAN generated and training set images. The line indicates the mean of each of the statistics over the 20,000 images used for both `real' and `fake' images, while the bars indicate plus or minus one standard deviation for each statistics' histogram. In Fig. \ref{fig:SNR_2_freq_US} histograms of $SNR^2$, $\mu_I$ and $\sigma_I$ for all six carrier frequencies are shown.
\begin{figure}[h]
\centering
\includegraphics[width=0.97\linewidth]{images/Frequency_Sweep.pdf}
\caption{$SNR^2$, $\mu_I$ and $\sigma_I$ values for real and GAN generated images versus the carrier frequency.}
\label{fig:freq_sweep_US}
\end{figure}
Visually, both the GAN generated images and the training set images look very similar. However, this is not the case when evaluating the similarity with regards to the three statistics mentioned. Interestingly, the GAN generated images have a lower $SNR^2$ on average than the `real' images for frequencies less than $4$ MHz, but have larger $SNR^2$ values on average for frequencies greater than or equal to $4$MHz. A deeper investigation into the $\mu_I$ and $\sigma_I$ statistics shows that the GAN generated images severely underestimate both statistics with respect to the training set images. In general, the larger the frequency the more the `real' and `fake' distributions diverge to the point where at $6$ and $7$ MHz, these two statistic distributions no longer have any overlap between the `reals' and `fakes'.
\begin{figure}
\includegraphics[width=\linewidth]{images/freq_sweep_imgs.pdf}
\caption{Real images from the US SIMs and fake images generated by the GANs for the SND-1-$f$ models.}
\label{fig:real_and_fake_freq_sweep}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{images/freq_histograms.pdf}
\caption{$SNR^2$, $\mu_I$ and $\sigma_I$ histograms for real and GAN generated ultrasound images with carrier frequencies 2,3,4,5,6 and 7 MHz.}
\label{fig:SNR_2_freq_US}
\end{figure*}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,414 |
Easy to use lightweight dialog library for JavaFX applications.
### Contents
* <a href='#features'>Features</a>
* <a href='#dialogs'>Dialogs</a>
* <a href='#colorthemes'>Color Themes</a>
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***
### <a name='features'></a>Features
<sup><a href='#home'>[back to top]</a></sup>
- Minimal design
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### <a name='dialogs'></a>Dialogs
<sup><a href='#home'>[back to top]</a></sup>
<p align="center">
<img src ="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/InfoDialog.png?token=AGk1WtIl0yQai-c3MiXwwyPwtbakmtY4ks5UmUB6wA%3D%3D" alt="Information Dialog" />
<br /><br />
<img src ="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/ConfirmDialog.png" alt="Confirmation Dialog" />
<br /><br />
<img src ="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/ConfirmAlt1Dialog.png" alt="Confirmation Alternative 1 Dialog" />
<br /><br />
<img src ="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/ConfirmAlt2Dialog.png" alt="Confirmation Alternative 2 Dialog" />
<br /><br />
<img src ="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/WarningDialog.png?token=AGk1WkOat2dl0G-gUxoeE8ockiVejWY-ks5UmUC6wA%3D%3D" alt="Warning Dialog" />
<br /><br />
<img src ="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/ErrorDialog.png?token=AGk1Why_GDl4ELib3F_X8rq1NS2chp1kks5UmUDQwA%3D%3D" alt="Error Dialog" />
<br /><br />
<img src="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/InputTextDialog.png" alt="Input Text Dialog" />
<br /><br />
<img src="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/ExceptionDialog.png" alt="Exception Dialog" />
</p>
### <a name='colorthemes'></a>Color Themes
<sup><a href='#home'>[back to top]</a></sup>
If none of those colors hook you, try mix and match various color styles.
##### Gloss Series
<p align="center">
<img src="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/HeaderColors/GlossSeries.png" alt="Gloss Series" />
</p>
##### Linear Fade Left Series
<p align="center">
<img src="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/HeaderColors/LinearFadeLeftSeries.png" alt="Linear Fade Left Series" />
</p>
##### Linear Fade Right Series
<p align="center">
<img src="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/HeaderColors/LinearFadeRightSeries.png" alt="Linear Fade Right Series" />
</p>
##### Opaque Series
<p align="center">
<img src="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/HeaderColors/OpaqueSeries.png" alt="Opaque Series" />
</p>
##### Generic Style
<p align="center">
<img src="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/HeaderColors/GenericSeries.png" alt="Generic Style" />
</p>
### <a name='installation'></a>Installation
<sup><a href='#home'>[back to top]</a></sup>
SimpleDialogFX is available in Maven Central. To start using, simply add the following elements to your pom.xml file:
```xml
<dependency>
<groupId>com.github.daytron</groupId>
<artifactId>SimpleDialogFX</artifactId>
<version>2.2.0</version>
</dependency>
```
### <a name='usage'></a>Usage
<sup><a href='#home'>[back to top]</a></sup>
#####<a name='overview'></a>Overview
<sup><a href='#home'>[back to top]</a></sup>
A dialog consists of the following areas shown in the figure below:
- Title
- Header
- Details
- and Buttons
<p align="center">
<img src="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/OverviewUsage.png" alt="Dialog overview" />
</p>
#####<a name='construction'></a>Construction
<sup><a href='#home'>[back to top]</a></sup>
To create a dialog, you only have to create a new `Dialog` object. For example, a confirmation dialog would look like this:
```java
Dialog dialog = new Dialog(
DialogType.CONFIRMATION,
"Confirm Action",
"Are you sure?");
dialog.showAndWait();
```
For an exception dialog:
```java
Dialog dialog = new Dialog(exception);
dialog.showAndWait();
```
Retrieving a response::
```java
DialogResponse response = dialog.getResponse();
```
Another example:
```java
Dialog dialog = new Dialog(
DialogType.CONFIRMATION,
"This is a sample title",
"Confirm Action",
"Are you sure?");
dialog.showAndWait();
if (dialog.getResponse() == DialogResponse.YES) {
// Rest of the code
}
```
Result:
<p align="center">
<img src ="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/Example1.png" alt="Result Example Dialog" />
</p>
<br />
For the complete list of constructors, see [Javadoc].
##### <a name='undecorated'></a>Undecorated
<sup><a href='#home'>[back to top]</a></sup>
For an undecorated window style approach, simply use the `DialogStyle` option, `UNDECORATED` in the constructor.
```java
Dialog dialog = new Dialog(
DialogType.CONFIRMATION,
DialogStyle.UNDECORATED,
"Confirm Action",
"Are you sure?");
dialog.showAndWait();
```
<p align="center">
<img src ="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/UndecoratedVsNative.png" alt="Undecorated style Dialog" />
</p>
<br />
##### <a name='headless'></a>Headless
<sup><a href='#home'>[back to top]</a></sup>
For a more simplistic approach, you may remove the header completely and show only the details section of the dialog. To choose a headless approach, simply use the `DialogStyle` option, `HEADLESS` in the constructor.
<p align="center">
<img src ="https://raw.githubusercontent.com/Daytron/SimpleDialogFX/master/Screenshots/HeadlessVsHead.png" alt="Headless style Dialog" />
</p>
<br />
##### <a name='colorstyle'></a>Color Style
<sup><a href='#home'>[back to top]</a></sup>
You can set the color style with `HeaderColorStyle` enum either via the constructor or through a method.
###### Via constructor:
`Dialog(DialogType dialogType, HeaderColorStyle headerColorStyle, String header, String details)` and
`Dialog(DialogType dialogType, DialogStyle dialogStyle, String title, String header, HeaderColorStyle headerColorStyle, String details, Exception exception)`
###### Via method:
`setHeaderColorStyle(HeaderColorStyle headerColorStyle)`
##### <a name='font'></a>Font
<sup><a href='#home'>[back to top]</a></sup>
Apply any style fonts using these methods:
```java
setFontSize(int font_size)
setFontSize(int header_font_size, int details_font_size)
setFontFamily(String font_family)
setFontFamily(String header_font_family, String details_font_family)
setFont(String font_family, int font_size)
setFont(String header_font_family, int header_font_size, String details_font_family, int details_font_size)
```
```java
setHeaderFontSize(int font_size)
setDetailsFontSize(int font_size)
setHeaderFontFamily(String font_family)
setDetailsFontFamily(String font_family)
setHeaderFont(String font_family, int font_size)
setDetailsFont(String font_family, int font_size)
```
##### <a name='responses'></a>Responses
<sup><a href='#home'>[back to top]</a></sup>
The list of all available dialog responses:
- `OK`
- `CANCEL`
- `YES`
- `NO`
- `CLOSE` (When user clicks dialog's close button instead)
- `SEND`
- `NO_RESPONSE` (Default value until the user interacts with it)
##### <a name='misc'></a>Misc
<sup><a href='#home'>[back to top]</a></sup>
UI components can be extracted, allowing you to customize the dialog as you see fit.
```java
getHeaderLabel() // The colored head label
getDetailsLabel() // The label text below header
getTextField() // For Input dialog's textfield
getExceptionArea() // For Exception dialog's textarea
```
In addition, the Dialog class itself is a subclass of the Stage class, so you can further customize the look and style of your dialogs.
### <a name='documentation'></a>Documentation
<sup><a href='#home'>[back to top]</a></sup>
See [Javadoc] for more information.
### <a name='development'></a>Development
<sup><a href='#home'>[back to top]</a></sup>
Want to contribute? Please do open up an issue for any bug reports, recommendation or feedback.
### <a name='license'></a>License
<sup><a href='#home'>[back to top]</a></sup>
MIT
[Javadoc]:https://daytron.github.io/SimpleDialogFX/apidocs/
| {
"redpajama_set_name": "RedPajamaGithub"
} | 108 |
George M. Wertz (* 19. Juli 1856 bei Johnstown, Cambria County, Pennsylvania; † 19. November 1928 ebenda) war ein US-amerikanischer Politiker. Zwischen 1923 und 1925 vertrat er den Bundesstaat Pennsylvania im US-Repräsentantenhaus.
Werdegang
George Wertz besuchte die öffentlichen Schulen seiner Heimat, die Ebensburg Academy und die National Normal School in Lebanon (Ohio). Zwischen 1876 und 1884 arbeitete er als Lehrer und danach bis 1894 als Schuldirektor. Von 1893 bis 1896 war er Bezirksrat und von 1897 bis 1901 Sheriff im Cambria County. Politisch schloss er sich der Republikanischen Partei an. Von 1908 bis 1912 saß Wertz im Senat von Pennsylvania, dessen Präsident er seit 1911 war. Er wurde außerdem im Zeitungsgeschäft tätig und leitete zwischen 1911 und 1917 die von ihm mitgegründete Zeitung Johnstown Daily Leader. In den Jahren 1914 bis 1916 übte er das Amt des Comptroller im Cambria County aus.
Bei den Kongresswahlen des Jahres 1922 wurde Wertz im 20. Wahlbezirk von Pennsylvania in das US-Repräsentantenhaus in Washington, D.C. gewählt, wo er am 4. März 1923 die Nachfolge von Edward Schroeder Brooks antrat. Da er im Jahr 1924 von seiner Partei nicht mehr zur Wiederwahl nominiert wurde, konnte er bis zum 3. März 1925 nur eine Legislaturperiode im Kongress absolvieren. Nach seiner Zeit im US-Repräsentantenhaus betätigte sich George Wertz in der Immobilienbranche. Er starb am 19. November 1928 in seiner Heimatstadt Johnstown.
Weblinks
Mitglied des Repräsentantenhauses der Vereinigten Staaten für Pennsylvania
Mitglied des Senats von Pennsylvania
Mitglied der Republikanischen Partei
US-Amerikaner
Geboren 1856
Gestorben 1928
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 713 |
\section{Introduction}
\begin{figure}[hb!]
\begin{center}
\includegraphics[width=0.8\linewidth]{general_idea_in_loss_space.pdf}
\caption{ {With BPN, one can switch at runtime the network parameters that are global optimal for each task. Training trajectories are illustrated in loss and parameter space. The green curve shows loss as a function of network parameters for a first task A, with optimal parameters shown by the green circle. The purple curve and circle correspond to a second task B. Training first task A then task B with stochastic gradient descend (SGD, without any constraints on parameters, gray) leads to optimal parameters for task B (purple circle), but those are destructive for task A. When, instead, learning task B using EWC or PSP (have some constraints on parameters, yellow), the solution is a compromise that can be sub-optimal for both tasks (black circle). Beneficial perturbations (blue curve for task A, red curve for task B) push the representation learned by EWC or PSP back to their task-optimal states.}}
\label{fig:concept_in_loss_space}
\end{center}
\end{figure}
{The human brain is the benchmark of adaptive learning. While interacting with new environments that are not fully known to an individual, it is able to quickly learn and adapt its behavior to achieve goals as well as possible, in a wide range of environments, situations, tasks, and problems. In contrast, deep neural networks only learn one sophisticated but fixed mapping between inputs and outputs, thereby limiting their application in more complex and dynamic situations in which the mapping rules are not kept the same but change according to different tasks or contexts. One of the failed situations is continual learning - learning new independent tasks sequentially without forgetting previous tasks. In the domain of image classification, for example, each task may consist of learning to recognize a small set of new objects. A standard neural network only learns a fixed mapping rule between inputs and outputs after training on each task. Training the same neural network on a new task would destroy the learned fixed mapping of an old task. Thus, current deep learning models based on stochastic gradient descent suffer from so-called "catastrophic forgetting" \citep{mccloskey1989catastrophic,french1994dynamically,sloman1992episodic}, in that they forget all previous tasks after training each new one.}
{Here, we propose a new biological plausible (Discussion) method~--- Beneficial Perturbation Network (BPN)~--- to accommodate these dynamic situations. The key new idea is to allow one neural network to learn potentially \textit{unlimited} task-dependent mappings and to switch between them at runtime. To achieve this, we first leverage existing lifelong learning methods to reduce interference between successive tasks (Elastic Weight Consolidation, EWC \citep{kirkpatrick2017overcoming}, or parameter superposition, PSP \cite{cheung2019superposition}). We then add out-of-network, task-dependent bias units, to provide per-task correction for any remaining parameter drifts due to the learning of a sequences of tasks. We compute the most beneficial biases~---~beneficial perturbations~---~for each task in a manner inspired by recent work on adversarial examples. The central difference is that, instead of adding adversarial perturbations that can force the network into misclassification, beneficial perturbations can push the drifted representations of old tasks back to their initial task-optimal working states (Fig.~\ref{fig:concept_in_loss_space}).}
\begin{figure*}[htb]
\begin{center}
\includegraphics[height = 15cm]{different_types_publication_ready.pdf}
\caption{{\bf Concept:} Type 1 - constrain the network weights while training the new task: (a) Retraining models such as elastic weight consolidation \citep{kirkpatrick2017overcoming}: retrains the entire network learned on previous tasks while using a regularizer to prevent drastic changes in the original model. Type 2 - expanding and retraining methods (b-c); (b) Expanding models such as progressive neural networks \citep{rusu2016progressive} expand the network for new task \textit{$t$} without any modifications to the network weights for previous tasks. (c) Expanding model with partial retraining such as dynamically expandable networks \citep{yoon2018lifelong} expand the network for new task t with partial retraining on the network weights for previous tasks. Type 3 - episodic memory methods (d): Methods such as Gradient Episodic Memory \citep{lopez2017gradient} store a subset of the original dataset from previous tasks into the episodic memory and replays them with new data during the training of new tasks. Type 4 - Partition network (e): these use context or mask matrices to partition the core network into several sub-networks for different tasks \citep{cheung2019superposition,zeng2019continual,mallya2018piggyback,du2019single,yoon2019oracle}. Type 5 - beneficial perturbation methods (f): Beneficial perturbation networks create beneficial perturbations which are stored in bias units for each task. Beneficial perturbations bias the network toward that task and thus allow the network to switch into different modes to process different independent tasks. It retrains the normal weights learned from previous tasks using elastic weight consolidation \citep{kirkpatrick2017overcoming} or parameter superposition \citep{cheung2019superposition}. (g) Strengths and weaknesses for each type of method.}
\label{fig:concept}
\end{center}
\end{figure*}
There are three major benefits of BPN: {\bf{1)}} BPN is memory and parameter efficient: to demonstrate it, we validate our BPN for continual learning on incremental tasks. We test it on multiple public datasets (incremental MNIST \citep{lecun1998gradient}, incremental CIFAR-10 and incremental CIFAR-100 \citep{krizhevsky2009learning}), on which it achieves better performance than the state-of-the-art. For each task, by adding bias units that store beneficial perturbations to every layer of a 5-layer fully connected network, we only introduce a 0.3\% increase in parameters, compared to a 100\% parameter increase for models that train a separate network, and 11.9\% - 60.3\% for dynamically expandable networks \citep{yoon2018lifelong}. Our model does not need any episodic memory to store data from the previous tasks and does not need to replay them during the training of new tasks, compared to episodic memory methods \citep{rebuffi2017icarl,lopez2017gradient,rannen2017encoder,rios2018closed}. Our model does not need large context matrices, compared to partition methods \citep{cheung2019superposition,zeng2019continual,yoon2019oracle,mallya2018piggyback,du2019single,farajtabar2020orthogonal,srivastava2013compete,masse2018alleviating}. {\bf{2)}} BPN achieves state-of-the-art performance across different datasets and domains: to demonstrate it, we consider a sequence of eight unrelated object recognition datasets (Experiments). After training on the eight complex datasets sequentially, the average test accuracy of BPN is better than the state-of-the-art. {\bf{3)}} BPN has capacity to accommodate a large number of tasks: to demonstrate it, we test a sequence of 100 permuted MNIST tasks (Experiments). A variant of BPN that uses PSP to constrain the normal network achieves 30.14\% better performance than the second best, the original PSP \citep{cheung2019superposition}, a partition method which performs well in incremental tasks and eight object recognition tasks. Thus, BPN has a promising future to solve continual learning compared to the other types of methods.
{To lay out the foundation of our approach we start by introducing the following key concepts: Sec.~\ref{sectypes}: Different types of methods for enabling lifelong learning; Sec.~\ref{adp}: Adversarial directions and perturbations; Sec.~\ref{bdp}: Beneficial directions and perturbations, and the effects of beneficial perturbations in sequential learning scenarios; Sec.~\ref{bpn}: Structure and updating rules for BPN.}
{We then present experiments (Sec.~\ref{experiments}), results (Sec.~\ref{results}) and discussion (Sec.~\ref{discussion}).}
\section{Types of methods for enabling lifelong learning}
\label{sectypes}
Four major types of methods have been proposed to alleviate catastrophic forgetting. Type 1: constrain the network weights to preserve performance on old tasks while training the new task \citep{kirkpatrick2017overcoming,lee2017overcoming,
aljundi2018memory} (Fig.~\ref{fig:concept}a); A famous example of type 1 methods is EWC \citep{kirkpatrick2017overcoming}. EWC constrains certain parameters based on how important they are to previously seen tasks. {The importance is calculated from their task-specific Fisher information matrix. However, solely relying on constraining the parameters of the core network eventually exhausts the core network's capacity to accommodate new tasks. After learning many tasks, EWC cannot learn anymore because the parameters become too constrained (see Results).} Type 2: dynamic network expansion \citep{li2017learning,lee2017overcoming,rusu2016progressive,yoon2018lifelong} creates new capacity for the new task, which can often be combined with constrained network weights for previous tasks (Fig.~\ref{fig:concept}b-c); {However, this type is not scalable because it is not parameter efficient (e.g., 11.9\% - 60.3\% additional parameters per task for dynamically expandable networks \citep{yoon2018lifelong})}. Type 3: using an episodic memory \citep{rebuffi2017icarl,lopez2017gradient,rannen2017encoder} to store a subset of the original dataset from previous tasks, then rehearsing it while learning new tasks to maintain accuracy on the old tasks (Fig.~\ref{fig:concept}d). { However, this type is not scalable because it is neither memory nor parameter efficient.} All three approaches attempt to shift the network's single fixed mapping initially obtained by learning the first task to a new one that satisfies both old and new tasks. They create a new, but still fixed mapping from inputs to outputs across all tasks so far, combined. Type 4: Partition Network: using task-dependent context \citep{cheung2019superposition,zeng2019continual,yoon2019oracle,masse2018alleviating} or mask matrices \citep{mallya2018piggyback,du2019single,du2019single,farajtabar2020orthogonal,srivastava2013compete} to partition the original network into several small sub-networks (Fig.~\ref{fig:concept}e, flow chart - Fig.~\ref{fig:Flow_charts}a). Zeng {\em et al.} \cite{zeng2019continual} used context matrices to partition the network into independent subspaces spanned by rows in the weight matrices to avoid interference between tasks. However, context matrices introduce as many additional parameters as training a separate neural network for each new task (additional 100\% parameters per task). To reduce parameter costs, Cheung {\em et al.} proposed binary context matrices \citep{cheung2019superposition}, further restricted to diagonal matrices with -1 and 1 values. The restricted context matrices \citep{zeng2019continual} (1 and -1 values) behave similarly to mask matrices \citep{mallya2018piggyback} (0 and 1 values) that split the core network into several sub-networks for different tasks. With too many tasks, the core network would eventually run out of capacity to accommodate any new task, because there is no vacant route or subspace left. Although type 4 methods create multiple input to output mappings for different tasks, many of these methods are too expensive in terms of parameters, and none of them has enough capacity to accommodate numerous tasks because methods such as PSP run out of unrealized capacity of the core network.
In marked contrast to the above artificial neural network methods, here, we propose a fundamentally new fifth type (Fig.~\ref{fig:concept}f, flow chart - Fig.~\ref{fig:Flow_charts} b): We add out-of-network, task-dependent bias units to neural network. Bias units enable a neural network to switch into different modes to process different independent tasks through beneficial perturbations (the memory storage cost of these new bias units is actually lower than the cost of adding a new mask or context matrix). With only an additional 0.3\% of parameters per mode \footnote{ {Check supplementary discussion for more information about additional parameter costs}}, this structure allows BPN to learn potentially unlimited task-dependent mappings from inputs to outputs for different tasks. The strengths and weaknesses of each type are in Fig.~\ref{fig:concept}g.
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=0.9\linewidth]{beneficial_perturbation_story_publication_ready.pdf}
\caption{Defining adversarial perturbations in input space vs. beneficial perturbations in activation space. We consider two digits recognition tasks; Task A (recognizing 1s and 2s) and Task B (recognizing 3s and 4s). (a) {\bf Adversarial directions (AD)}. Adding adversarial perturbations (calculated from digits 1) to input digits 2 can be viewed as adding an adversarial direction vector (gray arrow) to the clear input image of digit 2 in the input space. Thus, the network misclassifies the clear input image of digit 2 as digit 1. Beneficial directions are not operated as adding beneficial perturbations to the clear input image of digit 2 in the input space to assist the correct classification (orange arrow). (b) {\bf Beneficial directions (class specific) for each class of task A .} $R_1$ ($R_2$) is the classification region (region of constant estimated label) of digit 1 (digit 2) from the MNIST dataset. Subregion $R_{1\_high}$ ($R_{1\_low}$) is the high (low) confidence classification region of digit 1, and likewise for $R_{2\_high}$ ($R_{2\_low}$) for digit 2. The point $A_1$ is the activations of normal neurons of each layer from an input image of task A. It lies in the intersection of $R_{1\_low}$ and $R_{2\_low}$. $BD_1$ ($BD_2$) are beneficial directions for class digit 1 (digit 2). $A_1 + BD_1$, blue arrows,($A_+BD_2$, red arrows) pushes the activation $A_1$ across the decision boundary of $R_2$ ($R_1$) and towards $R_{1\_high}$ ($R_{2\_high}$). Thus, the network classifies $A_1 + BD_1$ ($A_1 + BD_2$) as digit 1 (digit 2) with high confidence. (c) {{\bf After training task B, beneficial perturbations (task specific) for task A push the drifted representation of inputs from task A back to its initial optimal working region of task A.}} $R_3$ ($R_4$ ) is the classification region (region of constant estimated label) of digit 3 (digit 4) from the MNIST dataset. $BD_1$ ($BD_2$) is a beneficial direction for digit 1 (digit 2). During the training of task A, the network has been trained on two images from digit 1 ($1^a$ and $1^b$) and two images from digit 2 ($2^a$ and $2^b$). Thus, the beneficial perturbations for task A are the vector ($BD_2^{a} + BD_1^{a} + BD_2^{b} + BD_1^{b}$). After training task B, with gradient descent, point $A_1$ in b) is drifted to the $A'_1$ which lies inside of the classification regions of task B ($R_2$ or $R_3$). The drifted point $A'_1$ alone cannot be correctly classified as digit 1 or 2 because it lies outside of the classification region of task A ($R_1$ or $R_2$). At test time, adding beneficial perturbations for task A to the activations of $A'_1$, can drag it back the correct classification regions for task A (intersection of $R_1$ and $R_2$). Thus, it biases the network's outputs toward the correct classification region and push task representations back to { their initial task-optimal working region.}
}\label{fig:beneficial_perturbations}
\end{center}
\end{figure*}
\section{Adversarial directions and perturbations}
\label{adp}
{Three spaces of a neural network are important for this and the following sections: The {\em input space} is the space of input data (e.g., pixels of an image); the {\em parameter space} is the space of all the weights and biases of the network; the {\em activation space} is the space of all outputs of all neurons in all layers in the network.}
By adding a carefully computed "noise" (adversarial perturbations) to the input space of a picture, without changing the neural network, one can force the network into misclassification. The noise is usually computed by backpropagating the gradient in a so-called "adversarial direction" such as by using the fast gradient sign method (FGSD) \citep{tramer2017space}. For example, consider a task of recognizing handwritten digits "1" versus "2". Adversarial perturbations aimed at misclassifying an image of digit 2 as digit 1 may be obtained by backpropagating from the class digit 1 to the input space, following any of the available adversarial directions. In Fig.~\ref{fig:beneficial_perturbations}a, adding adversarial perturbations to the input image can be viewed as adding an adversarial direction vector (gray arrows $AD$) to the clear (non-perturbated) input image of digit 2. The resulting vector crosses the decision boundary. Thus, adversarial perturbations can force the neural network into misclassification, here from digit 2 to digit 1. Because the dimensionality of adversarial directions is around 25 for MNIST \citep{tramer2017space}, when we project them into a 2D space, we use the fan-shaped gray arrows to depict those dimensions.
\section{Beneficial directions and perturbations, \& The effects of beneficial perturbations in multitask sequential learning scenario}
\label{bdp}
{In this section, we first introduce the definition of beneficial directions and beneficial perturbations. Then, we explain why beneficial perturbations can help a network recover from a parameter drifting of old tasks after learning new tasks and can push task representations back to their initial task-optimal working region.}
We consider two incremental digits recognition tasks; Task A (recognizing 1s and 2s) and Task B (recognizing 3s and 4s). Attack and defense researchers usually view adversarial examples as a curse of neural networks, but we view it as a gift to solve continual learning. Instead of adding input "noise" (adversarial perturbations) to the {\em input space} calculated from other classes to force the network into misclassification, {we add "noise" to the {\em activation space}, using {\em beneficial perturbations} stored in bias units added to the {\em parameter space} (Supplementary Fig.~\ref{fig:Flow_charts}b) calculated by the input's own correct class to assist in correct classification.} To understand beneficial perturbations, we first explain beneficial directions. Beneficial directions are vectors that point toward the direction of high confidence classification region for each class (Fig.~\ref{fig:beneficial_perturbations}b); ${BD}_1$ (${BD}_2$) are the beneficial directions that point to the high confidence classification region of digit 1 (digit 2). The point $A_1$ represents the activation of the normal neurons of each layer generated from an input image of task A. $A_1 + {BD}_1$ ($A_1 + {BD}_2$) pushes the activation $A_1$ across the decision boundary of $R_2$ ($R_1$) and toward $R_{1\_high}$ ($R_{2\_high}$). Thus, the network would classify the $A_1 + {BD}_1$ ($A_1 + {BD}_2$) as digit 1 (2) with high confidence.
To overcome catastrophic forgetting, we create some beneficial perturbations for each task and store them in task-dependent bias units (Fig.~\ref{fig:explanation_structure}, Supplementary Fig.~\ref{fig:Flow_charts}b). Beneficial perturbations allow a neural network to operate in different modes by biasing the network toward that particular task, even though the shared normal weights become contaminated by other tasks. The beneficial perturbations for each task are created by aggregating the beneficial direction vectors sequentially for each class through mini-batch backpropagation. For example, during the training of task A, the network has been trained on two images from digit 1 ($1^a$ and $1^b$) and two images from digit 2 ($2^a$ and $2^b$). The beneficial perturbations for task A are the summation of the beneficial directions calculated from each image ($BD_2^{a} + BD_1^{a} + BD_2^{b} + BD_1^{b}$ in Fig. ~\ref{fig:beneficial_perturbations}c, { $BD_i^j$ is the beneficial direction for sample $j$ in class $i$}). During the training of task B, with gradient descent, the point $A_1$ (Fig.~\ref{fig:beneficial_perturbations}b) is drifted to $A'_1$ which lies inside the classification regions for task B ($R_3\bigcup R_4$). The drifted $A'_1$ alone cannot be classified as digit 1 or 2 since it lies outside of the classification regions of task A ($R_1 \bigcup R_2$). However, during testing of task A, after training task B, adding beneficial perturbations for task A to the drifted activation ($A'_1$) drags it back to the correct classification regions for task A ( $R_1$ $\bigcup$ $R_2$ in Fig.~\ref{fig:beneficial_perturbations}c). Thus, beneficial perturbations bias the neural network toward that task and {push task representations back to their initial task-optimal working region. Note that in this work we focus on adding more compact beneficial perturbations to the activation space, as adding perturbations to the input space has already been explored in adversarial attack methods, and adding perturbations to the parameter space is unlikely to be scalable due to the very large number of parameters in a typical neural network.}
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=0.8\linewidth]{structure_publication_ready.pdf}
\caption{ {\bf{ Beneficial perturbation network (BD + EWC or BD + PSP variant) with two tasks.}} (a) Structure of beneficial perturbation network. (b) Train on task A. Backpropagating through the network to bias units for tasks A in beneficial direction (FGSD) using input's own correct class (digits label 1 and 2), normal weights (gradient descent). (c) Test on task A. Feed the input images to the network. Activating bias units for task A and adding the stored beneficial perturbations to the activations. The beneficial perturbations bias the network to mode on classifying digits 1, 2 task. (d) Train on task B. Backpropagating through the network to bias units for tasks B in beneficial direction (FGSD) using input's own correct class (digits label 3 and 4), normal weights (constrained by EWC or PSP). (e) Test on task B. Feed the input images to the network. Activating bias units for task B and adding the stored beneficial perturbations to the activations. The beneficial perturbations bias the network to mode on classifying digits 3, 4 task.}
\label{fig:explanation_structure}
\end{center}
\end{figure*}
\section{Beneficial Perturbation Network}
\label{bpn}
We implemented two variants of BPN: BD + EWC and BD + PSP (Experiments). The backbone - BD (updating extra out-of-network bias units in beneficial directions to create beneficial perturbations) is the same for both methods. The only difference is BD + EWC (BD + PSP) uses EWC (PSP) method to retrain the normal weights while attempting to minimize disruption of old tasks. Here, we choose BD + EWC to explain our method (for BD + PSP, see Supplementary). We use a scenario with two tasks for illustration; task A - recognizing MNIST digit 1s, 2s, task B - recognizing MNIST digit 3s, 4s. BPN has task-dependent bias units ($\mathbf{BIAS}_{t}^{i}\in R^{1{\times}K}$, K is the number of normal neurons in each layer, $i$ is the layer number, and $t$ is the task number) in each layer to store the beneficial perturbations. The beneficial perturbations are formulated as an additive contribution to each layer's weighted activations. Unlike most adversarial perturbations, beneficial perturbations are not specific to each example, but are applied to all examples in each task (Fig.~\ref{fig:beneficial_perturbations} c, d). We define beneficial perturbations as a task-dependent bias term:
\begin{equation} {\bm{V}}^{i+1} = \sigma({\bm{W}}^{i}{\bm{V}}^{i} +b^i +\mathbf{BIAS}_{t}^{i} ) \ \ \ \mathbf{\forall} \ \ i \in [1,n] \label{Eqn:activations_rules} \end{equation}
\noindent where $V^{i}$ is the activations at layer $i$, $W^{i}$ is the normal weights at layer $i$, $BIAS_{t}^{i}$ is the task dependent bias units at layer $i$ for task $t$, $\sigma(\cdot)$ is the nonlinear activation function at each layer, $b^i$ is the normal bias term at layer $i$, $n$ is the number of layers.
For a simple fully connected network (Fig.~\ref{fig:explanation_structure} a), the forward functions are:
\begin{equation}
{\bm{V}}^{1} = \sigma({\bm{W}}^{1}{\bm{X}}_{t}+ b^1 +\mathbf{BIAS}_{t}^{1} )
\end{equation}
\begin{equation}
{\bm{V}}^{2} = \sigma({\bm{W}}^{2}{\bm{V}}^1+b^2 +\mathbf{BIAS}_{t}^{2})
\end{equation}
\begin{equation}
\mathbf{y} = Softmax({\bm{W}}^{3}{\bm{V}}^2+ b^3 +\mathbf{BIAS}^{3}_{t})
\end{equation}
\noindent where $\mathbf{y}$ is the output logits, ${\bm{X}}_{t}$ is the input data for task t, $Softmax$ is the normalization function, other notations are the same as in Eqn.~\ref{Eqn:activations_rules}. During the training of a specific task, the bias units are the product of two terms\footnote{the factorization provides more degrees of freedom to better learn the beneficial perturbations \citep{haeffele2017global,du2019gradient}}: ${\bm{M}}_{t}^{i}\in R^{1{\times}H}$ and ${\bm{W}}_{t}^{i}\in R^{H{\times}K}$ (H is the hidden dimension (a hyper-parameter), K is the number of normal neurons in each layer, and $t$ is the task number). After training a specific task, we discard both ${\bm{M}}_{t}^{i}$ and ${\bm{W}}_{t}^{i}$, and only keep their product $\mathbf{BIAS}_{t}^{i}$, reducing memory and parameter costs to a negligible amount (0.3$\%$ increase for parameters per task, and 4*K Bytes increase per layer per task, it is just a bias term). After training on different sequential tasks, at test time, the stored beneficial perturbations from the specific bias units can bias the neural network outputs to each task. Thus, these allow the BPN to switch into different modes to process different tasks. We use the forward and backward rules (Alg.~\ref{alg:FORTA}, Alg.~\ref{alg:BACTA}) to update the BPN.
{{\bf{For training}}}, first, during the training of task A, our goal is to maximize the probability $P(\mathbf{y} = \mathbf{y}_{A}|{\bm{X}}_{A},{\bm{W}}^{i},\mathbf{BIAS}_{A}^{i}) \ \ \ \mathbf{\forall} \ \ i \in [1,n]$ by selecting the bias units corresponding to tasks A . Thus, we set up our optimization function as:
\begin{equation}
\small
\begin{aligned}
{\bm{W}}^{i},&\,\mathbf{BIAS}_{A}^{i} =
{ \mathop{\arg\min}_{ {\bm{W}}^{i},\,\mathbf{BIAS}_{A}^{i}}} \\
& { -\, log\,[\,P(\mathbf{y} = \mathbf{y}_{A}|{\bm{X}}_{A},\,{\bm{W}}^{i},\,\mathbf{BIAS}_{A}^{i})\,]}
\ \ \ \ \mathbf{\forall} \ \ i \in [1,n]
\end{aligned}
\label{Eqn:training_A} \end{equation}
\noindent where $\mathbf{y}_{A}$ is the true label for data in task A (MNIST input images 1, 2), $X_{A}$ is the data for task A, other notations are the same as notations in Eqn.~\ref{Eqn:activations_rules}. We update ${\bm{M}}_{A}^{i}$ in the beneficial direction (FGSD) as $\epsilon sign(\nabla_{{\bm{M}}_{A}^{i}} L({\bm{M}}_{A}^{i},{\bm{y}}_{A}))$ to generate beneficial perturbations for task A, where ${\bm{M}}_{A}^{i}$ are the first term of bias units for task A. We update ${\bm{W}}_{A}^{i}$ (the second term of bias units for task A) in the gradient direction. The factorization allows the bias units for task A to better learn the beneficial perturbations for task A (a vector towards the work space of task A that has non-negligible network response for MNIST digits 1, 2, similar to Fig.~\ref{fig:beneficial_perturbations}b, c ). {We use a softmax cross entropy loss to optimize Eqn.~\ref{Eqn:activations_rules}.} After training task A, the bias units for task A ($\mathbf{BIAS}_{A}^{i}$) are the product of ${\bm{M}}_{A}^{i}$ and ${\bm{W}}_{A}^{i}$. We discard ${\bm{M}}_{A}^{i}$ and ${\bm{W}}_{A}^{i}$ to reduce the memory storage and parameter costs and freeze the $\mathbf{BIAS}_{A}^{i}$ to ensure that the beneficial perturbations are not being corrupted by other tasks (Task B). Then, we discard all of the MNIST input images 1, 2 because all of the information is stored inside the bias units for task A and we do not need to replay these images when we train on the following sequential tasks.
After training task A, during the training of task B (Fig.~\ref{fig:explanation_structure} d), our goal is to maximize the
probability ${P(\mathbf{y} = \mathbf{y}_{B}|{\bm{X}}_{B},{\bm{W}}^{i},\mathbf{BIAS}_{B}^{i})}$ ${ \mathbf{\forall} \ i \in [1,n]}$ by selecting the bias units corresponding to tasks B. To minimize the disruption for task A, we apply EWC or PSP constraints on normal weights. We set up our optimization function as
\begin{equation}
\small
\begin{aligned}
{\bm{W}}^{i},&\,\mathbf{BIAS}_{B}^{i} = \mathop{\arg\min}_{ {\bm{W}}^{i},\,\mathbf{BIAS}_{B}^{i}} \\
& -\, log\,[\,P(\mathbf{y} = \mathbf{y}_{B}|{\bm{X}}_{B},\,{\bm{W}}^{i},\,\mathbf{BIAS}_{B}^{i})\,]
+ EWC({\bm{W}}^{i}) \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{\forall} \ \ i \in [1,n]
\end{aligned}
\label{Eqn:training_EWC_constrained}\end{equation}
where $\mathbf{y}_{B}$ is the true label for data in task B (MNIST input images 3, 4), $X_{B}$ is the data for task B, $EWC(\cdot)$ is the EWC constraint \citep{kirkpatrick2017overcoming} on normal weights, other notations are the same as in Eqn.~\ref{Eqn:activations_rules}. In the loss function of Alg.~\ref{alg:BACTA}, $\lambda F_j(W_j-W^{A*}_{j})^2$ is the EWC constraint on the normal weights, where $j$ labels each parameter, $F_j$ is the Fisher information matrix for each parameter $j$ {(determine which parameters are most important for a task \citep{kirkpatrick2017overcoming})}, $\lambda$ sets how important the old task is compared to the new one, $W_j$ is normal weight $j$, and $W_{j}^{A*}$ is the optimal normal weight $j$ after training on task A. Apart from the additional EWC constraint, training task B and all subsequent tasks then simply proceeds in the same manner as for task A above.
{{{\bf For testing}}}, after training task B, we test the accuracy for task A on a test set by manually activating the bias units corresponding to task A (Fig.~\ref{fig:explanation_structure} c, Alg.~\ref{alg:FORTA}). Although the shared normal weights have been contaminated by task B, the integrity of bias units for task A that store the beneficial perturbations still can bias the network outputs to task A (set the network into a mode to process input from task A, see Results). In another word, the task-dependent bias units can still maintain a high probability ~--~ $P(\mathbf{y} = \mathbf{y}_{A}|{\bm{X}}_{A},{\bm{W}}^{i},\mathbf{BIAS}_{A}^{i})$ for task A. During testing of task B, we test the accuracy for task B on a test set by manually activating the bias units corresponding to task B (Fig.~\ref{fig:explanation_structure} e, Alg.~\ref{alg:FORTA}). The bias units for task B can bias the network outputs to task B and maintain a high probability ~--~ $P(\mathbf{y} = \mathbf{y}_{B}|{\bm{X}}_{B},{\bm{W}}^{i},\mathbf{BIAS}_{B}^{i})$ for task B, in case the shared normal weights are further modified by later tasks. In scenarios with more than two tasks, the forward and backward algorithms for later tasks are the same as for task B, except that they will select and update their own bias units.
In sum, beneficial perturbations act upon the network not by adding biases to the input data (like adversarial examples do, Fig.~\ref{fig:beneficial_perturbations}a), but instead by dragging the drifted activations back to the correct working region in activation space for the current task ( Fig.~\ref{fig:concept_in_loss_space} and Fig. ~\ref{fig:beneficial_perturbations} c).
The intriguing properties of task-dependent beneficial perturbations on maintaining high probabilities for different tasks can further be explained in two ways. The beneficial perturbations from the bias units can be viewed as features that capture how "furry" the images are for task A (or B). Olshausen {\em et al.} \citep{cheung2019superposition} showed that training a neural network only on these features is sufficient to make correct classification on the dataset that generates these features. They argued that these features have sufficient information for a neural network to make correct classification. In our continual learning scenarios, although the shared normal weights (${\bm{W}}^{i}$) have been contaminated after the sequential training of all tasks, by activating corresponding bias units, the task-dependent bias units still have sufficient information to bias the network toward that task. In other words, the task-dependent bias units can maintain high probabilities ~--~ $P(\mathbf{y} = \mathbf{y}_{A}|{\bm{X}}_{A},{\bm{W}}^{i},\mathbf{BIAS}_{A}^{i})$ for task A or $P(\mathbf{y} = \mathbf{y}_{B}|{\bm{X}}_{B},{\bm{W}}^{i},\mathbf{BIAS}_{B}^{i})$ for task B . Thus, bias units can assist the network to make correct classification. In addition, Elsayed {\em et al.} \cite{elsayed2018adversarial} showed how a carefully computed adversarial perturbations for each new task embedded in the input space can repurpose machine learning models to perform a new task. Here, these beneficial perturbations can be viewed as task-dependent beneficial "programs"\cite{elsayed2018adversarial} in the parameter space. Once activated, these task-dependent "programs" can maximize the probability for corresponding tasks.
\begin{algorithm*}[h]
\small
\caption{BD + EWC : forward rules for task t }
\label{alg:FORTA}
\begin{algorithmic}
\State {\quad For each fully connected layer i}
\State {\quad Select bias units (task t): $\mathbf{BIAS}_{t}^{i}$) for the current task }
\State{\quad{\bfseries Input:}\hspace{0.15cm} $\mathbf{BIAS}_{t}^{i}$ \textemdash
\hspace{0.08cm} Bias units for task t} \hfill {\color{green}// provide beneficial perturbations to bias the neural network}
\State{\quad\quad\quad\quad\hspace{0.33cm}${\bm{V}}^{i-1}$\textemdash
\hspace{0.08cm} Activations from the last layer}
\State{\quad{\bfseries Output:} ${\bm{V}}^{i} = \sigma({\bm{W}}^{i} \cdot {\bm{V}}^{i-1} + \mathbf{b}^{i} +\mathbf{BIAS}_{t}^{i}) \ \ \ \mathbf{\forall} \ i \in [1,n]$ \hfill {\color{green}// activations for the next layer}}
\State{\quad\quad\quad\qquad \hspace{0.13cm} where: ${\bm{W}}^{i}$\textemdash
\hspace{0.08cm} normal neuron weights at layer $i$. $\mathbf{b}^{i}$\textemdash \hspace{0.08cm} normal bias term at layer $i$}
\State{\quad\quad\quad\qquad \hspace{1.23cm} $n$ \textemdash \hspace{0.08cm} the number of FC layers. \hspace{1.1cm} $\sigma(\cdot)$ \textemdash \hspace{0.08cm} the nonlinear activation function at each layer}
\end{algorithmic}
\end{algorithm*}
\begin{algorithm*}[h]
\small
\caption{BD + EWC : backward rules for task t }
\label{alg:BACTA}
\begin{algorithmic}
\State {{\underline {For the first task A ($t = 1$):}}}
\State{{\quad Minimizing loss function: $L({\bm{X}}_{A},{\bm{W}}^{i},\mathbf{BIAS}_{A}^{i}) \ \ \ \mathbf{\forall} \ i \in [1,n]$}}
\State{{\quad \quad where: ${\bm{X}}_{A}$\textemdash
\hspace{0.08cm} data for task One.\quad ${\bm{W}}^{i}$\textemdash
\hspace{0.08cm} normal neuron weights at layer $i$. }}
\State{{ \quad\quad\quad\quad \hspace{0.16cm} $\mathbf{BIAS}_{A}^{i}$ \textemdash
\hspace{0.08cm} bias units for task One from FC layers $i$, which is the product of $({\bm{M}}_{A}^{i}, {\bm{W}}_{A}^{i})$}}
\State{{\hspace{0.16cm}\quad\quad\quad\quad\quad $n$\textemdash
\hspace{0.08cm} the number of FC layers.}}
\State{}
\State {\underline {For task B ($t > 1$):}}
\State{\quad Minimizing loss function: $L({\bm{X}}_{t},{\bm{W}}^{i}, \mathbf{BIAS}_{B}^{i})+ \sum_{j} \lambda F_j(W_j-W^{A*}_{j})^2 \ \ \ \mathbf{\forall} \ i \in [1,n]$}
\State{{\quad \quad where: ${\bm{X}}_{B}$\textemdash
\hspace{0.08cm} data for task B.\quad ${\bm{W}}^{i}$\textemdash
\hspace{0.08cm} normal neuron weights from FC at layers $i$}}
\State{{\hspace{0.1cm}\quad\quad\quad\quad\quad $j$\textemdash
\hspace{0.08cm} labels each parameter.\quad $F_{j}$ \textemdash
\hspace{0.08cm} Fisher information matrix for parameter j.}}
\State{{\hspace{0.1cm}\quad\quad\quad\quad\quad $W_j$\textemdash
\hspace{0.08cm} normal weight j.\hfill $W^{A*}_{j}$ \textemdash
\hspace{0.08cm} optimal normal weight j after training on task A.}}
\State{{ \quad\quad\quad\quad \hspace{0.16cm} $\mathbf{BIAS}_{B}^{i}$ \textemdash
\hspace{0.08cm} bias units for task B at FC layers $i$, which is the product of $({\bm{M}}_{B}^{i}, {\bm{W}}_{B}^{i})$}}
\State{{\hspace{0.1cm}\quad\quad\quad\quad\quad $n$\textemdash
\hspace{0.08cm} the number of FC layers.}}
\State{}
\State{\bf \underline{For each fully connected layer i:}}
\State{}
\State{\quad \underline {During the training of task t}}
\State {\quad \hspace{0.07cm} Select bias units for the current task t ($\mathbf{BIAS}_{t}^{i}$)}
\State{\hspace{0.07cm} \quad{\bfseries Input:}\hspace{0.15cm} $\mathbf{Grad}$ \textemdash
\hspace{0.08cm} Gradients from the next layer}
\State{\hspace{0.07cm} \quad{\bfseries output:} $\mathbf{dW_{t}^{i}} = \mathbf{Grad}\cdot(({\bm{M}}_{t}^{i})^T)$ {\color{green}// gradients for the second term of bias units for task t at layer i } }
\State{\hspace{0.05cm} \quad $\hspace{33pt}$ $\mathbf{dM_{t}^{i}} = \epsilon\; sign\;((\mathbf{W}_{t}^{{i}})^T\cdot(\mathbf{Grad}))$ }
\State{\hspace{0.07cm} \hfill {\color{green}// gradients for the first term of bias units for task t at layer i using FGSD method}}
\State{\hspace{0.08cm} \quad $\hspace{33pt}$ $\mathbf{dW^i} = \mathbf{Grad}\cdot((\mathbf{V}^{i})^T)$\hfill {\color{green}// gradients for normal weights at layer i}}
\State{\hspace{0.08cm} \quad $\hspace{33pt}$ $\mathbf{dV^{i}} = (\mathbf{W}^{i})^T \cdot (\mathbf{Grad})$ \hfill {\color{green}// gradients for activations at layer i to last layer i -1}}
\State{\hspace{0.07cm} \quad $\hspace{33pt}$ $\mathbf{db^i} = \sum_{j} {Grad}_j $ \hfill {\color{green}// gradients for normal bias at layer i, j is iterator over the first dimension of {\bf{Grad}} }}
\State{}
\State {\quad \underline {After training of task t}}
\State{\quad \hspace{0.07cm} Freeze the $\mathbf{BIAS}_{t}^{i}$}
\State{\quad \hspace{0.07cm} Delete the $\mathbf{W_{t}^{i}}$ and $\mathbf{M_{t}^{i}}$ to reduce parameter and memory storage cost}
\end{algorithmic}
\end{algorithm*}
\section{Experiments}
\label{experiments}
\subsection{Experimental Setup For Incremental Tasks}
To demonstrate that BPN is very parameter efficient and can learn different tasks in an online and continual manner, we used a fully-connected neural network with 5 hidden layers of 300 ReLU units. We tested it on three public computer vision datasets with "single-head evaluation", where the output space consists of all the classes from all tasks learned so far.
{\bf 1. Incremental MNIST.} A variant of the MNIST dataset \citep{lecun1998gradient} of handwritten digits with 10 classes, where each task introduces a new set of classes. We consider 5 tasks; each new task concerns examples from a disjoint subset of 2 classes.
{\bf 2. Incremental CIFAR-10.} A variant of the CIFAR object recognition dataset \citep{krizhevsky2009learning} with 10 classes. We consider 5 tasks; each new task has 2 classes.
{\bf 3. Incremental CIFAR-100.} A variant of the CIFAR object recognition dataset \citep{krizhevsky2009learning} with 100 classes. We consider 10 tasks; each new task has 2 classes. We use 20 classes for CIFAR-100 experiment.
\subsection{Experimental Setup For Eight Sequential Object Recognition Tasks}
To demonstrate the superior performance of BPN across different datasets and domains, we consider a sequence of eight object recognition datasets: {\bf 1.} Oxford \textit{Flowers} \citep{nilsback2008automated} for fine-grained flower classification (8,189 images in 102 categories); {\bf 2.} MIT \textit{Scenes} \citep{quattoni2009recognizing} for indoor scene classification (15,620 images in 67 categories); {\bf 3.} Caltech-UCSD \textit{Birds} \citep{wah2011caltech} for fine-grained bird classification (11,788 images in 200 categories); {\bf 4.} Stanford \textit{Cars} \citep{krause20133d} for fine-grained car classification (16,185 images of 196 categories); {\bf 5.} FGVC-\textit{Aircraft} \citep{maji2013fine} for fined-grained aircraft classification (10,200 images in 70 categories); {\bf 6.} VOC \textit{actions} \citep{everingham2015pascal}, the human action classification subset of the VOC challenge 2012 (3,334 images in 10 categories); {\bf 7.} \textit{Letters}, the Chars74K datasets \citep{de2009character} for character recognition in natural images (62,992 images in 62 categories); and {\bf 8.} the Google Street View House Number \textit{SVHN} dataset \citep{netzer2011reading} for digit recognition (99,289 images in 10 categories). To have a fair comparison, we use the same AlexNet \citep{krizhevsky2012imagenet} architecture pretrained on ImageNet \citep{russakovsky2015imagenet} as Aljundi {\em et al.} \cite{aljundi2018selfless,aljundi2018memory}, and tested on 8 sequential tasks with a "multi-head evaluation", where each task has its own classification layer (introduce same parameter costs for every method) and output space. All methods have a task oracle at test time to decide which classification layer to use. We run the different methods on the following sequence: Flower -> Scenes -> Birds -> Cars -> Aircraft -> Action -> Letters -> SVHM.
\subsection{Experimental Setup for 100 permuted MNIST dataset}
To demonstrate that BPN has capacity to accommodate a large number of tasks, we tested it on 100 permuted MNIST datasets generated from randomly permuted handwritten MNIST digits. We consider 100 tasks; each new task has 10 classes. We used a fully-connected neural network with 4 hidden layers of 128 ReLu Units (a core network with small capacity) to compare the performances of different methods. The type 4 methods, such as the Parameter Superposition (PSP \citep{cheung2019superposition}) would exhaust the unrealized capacity and inevitably dilute the capacity of the core network under a large number of tasks: in their Fig. 2, with a network that has 128 hidden units (leftmost panel), the average task performance for all tasks trained so far, is 95\% after training one task, but decreases to 50\% after training fifty tasks. While a larger network with 2048 hidden units shows much smaller decrease from 96\% to about 90\% (see Fig. 2 in their paper, rightmost panel). The reason is that this method generates a random diagonal binary matrix for each task, which in essence is a key or selector for that task. As more and more tasks are learned, those keys start to overlap more, causing interference among tasks. In comparison, BPN can counteract the dilution, hence it can accommodate a large number of tasks.
\subsection{Our model and baselines} We compared the proposed Beneficial Perturbation Network ( Beneficial Perturbation + Elastic Weight Consolidation (the eleventh model), BD + EWC (variant 1) and Beneficial Perturbation + Parameter Superposition (the twelfth model), BD + PSP (variant 2)) to 11 alternatives to demonstrate its superior performance.
{\bf 1. Single Task Learning (STL).} We consider several 5-layer fully-connected neural networks. Each network is trained for each task separately. Thus, STL does not suffer from catastrophic forgetting at all. It is used as an upper bound.
{\bf 2. Elastic Weight Consolidation (EWC) \citep{kirkpatrick2017overcoming}.} The loss is regularized to avoid catastrophic forgetting.
{\bf 3. Gradient Episodic Memory with task oracle (GEM (*)) \citep{lopez2017gradient},} GEM uses a task oracle to build a final linear classifier (FLC) per task. The final linear classifier adapts the output distributions to the subset of classes for each task. GEM uses an episodic memory to store a subset of the observed examples from previous tasks, which are interleaved with new data from the latest task to produce a new classifier for all tasks so far. We use notation GEM (*) for the rest of the paper, where * is the size of episodic memory (number of training images stored) for each class.
{\bf 4. Incremental Moment Matching \citep{lee2017overcoming} (IMM)} IMM incrementally matches the moment of the posterior distribution of the neural network with a L2 penalty and equally applies it to changes to the shared parameters.
{\bf 5. Learning without forgetting \citep{li2017learning} (LwF)} First, LwF freezes the shared parameters while learning a new task. Then, LwF trains all the parameters until convergence.
{\bf 6. Encoder based lifelong learning \citep{rannen2017encoder} (EBLL)} Based on LwF, using an autoencoder to capture the features that are crucial for each task.
{\bf 7. Synaptic Intelligence \citep{zenke2017continual} (SI)} While training on new task, SI estimates the importance weights in an online manner. Parameters important for previous tasks are penalized during the training of new task.
{\bf 8. Memory Aware Synapses \citep{aljundi2018memory} (MAS)} Similar to SI method, MAS estimates the importance weights through the sensitivity of the learned function on training data. Parameters important for previous tasks are penalized during the training of new task.
{\bf 9. Sparse coding through Local Neural Inhibition and Discounting \citep{aljundi2018selfless} (SLNID)} SLNID proposed a new regularizer that penalizes neurons that are active at the same time to create sparse and decorrelated representations for different tasks.
{\bf 10. Parameter Superposition \citep{cheung2019superposition} (PSP)} PSP used task-specific context matrices to map different inputs from different tasks to different subspaces spanned by rows in the weight matrices to avoid interference between tasks. We use the binary superposition model of PSP throughout the paper, because it is not only more memory efficient, but also, in our testing, it performed better than other PSP variants (e.g., complex superposition).
{\bf 11. BD + EWC (ours):} Beneficial Perturbation Network (variant 1). The first term (${\bm{M}}_{t}$) of the bias units is updated in the beneficial direction (BD) using FGSD method. The second term (${\bm{W}}_{t}$) of the bias units is updated in the gradient direction. The normal weights are updated with EWC constraints.
{\bf 12. BD + PSP (ours):} Beneficial Perturbation Network (variant 2). The first term (${\bm{M}}_{t}$) of the bias units is updated in the beneficial direction (BD) using FGSD method. The second term (${\bm{W}}_{t}$) of the bias units is updated in the gradient direction. The normal weights are updated using PSP (binary superposition model, Supplementary).
\begin{figure}[]
\begin{center}
\includegraphics[width=0.85\linewidth]{visualization_vertical_publication_ready.pdf}
\caption{{\bf Visualization of classification regions:} classify 3 randomly generated normal distributed clusters. Task 1: separate black from red clusters. Task 2: separate black from light blue clusters. The yellower (bluer) the heatmap, the higher (lower) the chance the neural network classifies a location as the black cluster. After training task 2, only BD + EWC remembers task 1 by maintaining its decision boundary between the black and red clusters. Both plain gradient descent and GD + EWC forget task 1 entirely.}
\label{fig:visualization}
\end{center}
\end{figure}
{\bf 13. GD + EWC:} The update rules and network structure are the same as BD + EWC, except the first term (${\bm{M}}_{t}$) of the bias units is updated in the Gradient direction (GD). This method has the same parameter costs as BD + EWC . The failure of GD + EWC suggests that the good performance of BD + EWC is not from the additional dimensions provided by bias units.
\section{ Results:}
\label{results}
\subsection{The beneficial perturbations can bias the network and maintain the decision boundary} To show the advantages of our method are really from the beneficial perturbations and not just from additional dimensions to the neural network, we compare between updating the first term of the bias units in the beneficial direction (BD + EWC which comes from beneficial perturbations) and in the gradient direction (GD + EWC, which just comes from the additional dimensions that our bias units provide). We use a toy example (classifying 3 groups of Normal distributed clusters) to demonstrate it and to visualize the decision boundary (Fig.~\ref{fig:visualization}). We randomly generate 3 normal distributed clusters different locations. We have two tasks - Task 1: separate the black cluster from the red cluster. Task 2: separate the black cluster from the light blue cluster. The yellower (bluer) the heatmap, the higher (lower) the confidence that the neural network classifies a location into the black cluster. After training task 2, both plain gradient descent and GD + EWC forget task 1 (dark blue boundary around the red cluster disappeared). However, BD + EWC not only learns how to classify task 2 (clear decision boundary between light blue and black clusters), but also remembers how to classify the old task 1 (clear decision boundary between red and black clusters). Thus, the beneficial perturbations are what can bias the network outputs and maintain the decision boundary for each task, not just adding more dimensions.
\begin{figure*}[h]
\begin{center}
\includegraphics[height=10.8cm]{mnist_5_tasks_pnas_with_psp_publication_ready.pdf}
\caption{ Results for a fully-connected network with 5 hidden layers of 300 ReLU units. (a) Incremental MNIST tasks (5 tasks, 2 classes per task). (b) Incremental CIFAR-10 tasks (5 tasks, 2 classes per task). For a and b, the dashed line indicates the start of a new task. The vertical axis is the accuracy for each task. The horizontal axis is the number of epochs. (c) Incremental CIFAR-100 tasks (10 tasks, 2 classes per task). The vertical axis is the accuracy for task 1. The horizontal axis is the number of tasks.}
\label{fig:quantative_results}
\end{center}
\end{figure*}
\begin{table*}[h!]
\caption{Task 1 performance with "single-head" evaluation after training all sequential tasks on incremental MNIST, CIFAR-10 and CIFAR-100 Dataset. We include additional memory storage costs per task (extra components that are necessary to be stored onto the disks after training each task, Supplementary) of GEM , BD+EWC, BD + PSP and PSP method.}
\label{tab:memory_performance}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{cccl}
\toprule
Dataset & Method & \makecell{Task 1 performance after\\ training all sequential tasks} & \makecell{additional memory storage \\ costs per task (Bytes)}\\
\midrule
\makecell{Incremental MNIST\\ (5 tasks, 2 classes per task)} & \makecell{GEM(10)\\BD+EWC}& \makecell{0.980 \\\bf{0.980}}& \makecell[r]{\ \ \ \ \ 47,040 \\\bf{4,808} }\\\hline
\makecell{Incremental CIFAR-10\\ (5 tasks, 2 classes per task)} & \makecell{GEM(256)\\GEM(150)\\BD+EWC}& \makecell{\bf{0.800} \\0.698\\0.795}& \makecell[r]{4,718,592 \\2,764,800 \\\bf{4,808} }\\\hline
\makecell{Incremental CIFAR-100\\ (10 tasks, 2 classes per task)} & \makecell{GEM(256)\\GEM(209)\\BD+PSP\\PSP\\BD+EWC}& \makecell{0.790\\0.775 \\ \bf{0.850}\\0.830\\0.845}& \makecell[r]{4,718,592 \\3,852,288 \\20,776\\15,968\\\bf{4,808} }\\ \hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table*}
\subsection{Quantitative analysis for incremental tasks} Our BPN achieves a comparable or better performance than PSP, GEM, EWC, GD + EWC in "single-head" evaluations, where the output space consists of all the classes from all tasks learned so far. In addition, it introduces negligible parameter and memory storage costs per task. Fig.~\ref{fig:quantative_results} and Tab.~\ref{tab:memory_performance} summarize performance for all datasets and methods. STL has the best performance since it trained for each task separately and did not suffer from catastrophic forgetting at all. Thus, STL is the upper bound. BD + EWC performed slightly worse than STL (1\%,4\%,1\% worse for incremental MNIST, CIFAR-10, CIFAR-100 datasets). BD + EWC achieved comparable or better performance than GEM. On incremental CIFAR-100 (10 tasks, 2 classes per task), BD + EWC outperformed PSP, GEM (256) and GEM (10) by 1.80\%, 6.96\%, and 22.4\%. BD + PSP outperformed PSP, GEM (256) and GEM (10) by 2.40\%, 7.59\%, and 23.1\%. By comparing the memory storage costs (Tab.~\ref{tab:memory_performance}, Supplementary), to achieve similar performance, BD + EWC only introduces an additional 4,808 Bytes memory per task, which is only 0.1\% of the memory storage cost required by GEM (256). BD + PSP only introduces 20,776 Bytes, or 0.44\% of the memory storage cost required by GEM (256).
The memory storage costs of BD + EWC is 30\% of that of PSP. The memory storage costs of BD + PSP is of the same order of magnitude as PSP. EWC alone rapidly decreased to 0\% accuracy. This confirms similar results on EWC performance on incremental datasets \citep{rios2018closed, kemker2017fearnet, parisi2019continual, kemker2018measuring} in "single-head" evaluations although EWC generally performs well in "multi-head" tasks. GD + EWC has the same additional dimensions as BD + EWC, but GD + EWC failed in the continual learning scenario. This result suggests that it is not the additional dimensions of the bias units, but the beneficial perturbations, which help overcome catastrophic forgetting.
\begin{table*}[h]
\caption{Test accuracy (in percent correct) achieved by each method with "multi-head" evaluation for each dataset after training on the 8 sequential object recognition datasets. (Dash (--) means that the results are not available in their papers. Star (*) means that we didn't reproduce the methods and the results were taken from SLNID \citep{aljundi2018selfless} and MAS \citep{aljundi2018memory}. Thus, we keep the same percentage table format as theirs).}
\label{tab:eighttasks}
\begin{center}
\includegraphics[width=0.97\linewidth]{eight_datasets_table_IEEE_TNNLS.pdf}
\end{center}
\end{table*}
\subsection{Quantitative analysis for eight sequential object recognition tasks} The eight sequential object recognition tasks demonstrate the superior performance of BPN (BD + PSP or BD + EWC) compared to the state-of-the-art and the ability to learn sequential tasks across different datasets and different domains. Our BPN achieves much better performance than IMM \citep{lee2017overcoming}, LwF\citep{li2017learning}, EWC \citep{kirkpatrick2017overcoming}, EBLL \citep{rannen2017encoder}, SI \citep{zenke2017continual}, MAS \citep{aljundi2018memory}, SLNID \citep{aljundi2018selfless}, PSP \citep{cheung2019superposition} in "multi-head" evaluations, where each task has its own classification layer and output space. After training on the 8 sequential object recognition datasets, we measured the test accuracy for each dataset and calculated their average performance (Tab.~\ref{tab:eighttasks}). On average, BD + PSP (ours) outperforms all other methods: PSP (7.52\% better), SLNID (8.02\% better), MAS (11.73\% better), SI (16.60\% better), EBLL (17.07\% better), EWC (17.75\% better), LwF (18.96\% better) and IMM (35.62\% better). Although MAS, SI and EBLL performed better than EWC alone, with the help of our beneficial perturbations (BD), BD + EWC can achieve a better performance than these methods: MAS (0.34\% better), SI (4.71\% better), EBLL (5.13\% better) and EWC (5.74\% better). By including the BD (BD + PSP and BD + EWC), we can significantly boost performance when compared to using PSP or EWC alone (black arrows in the Tab.~\ref{tab:eighttasks}).
\begin{figure}[h]
\includegraphics[width=\columnwidth]{permuted_MNIST_100_tasks_ewc_publication_ready.pdf}
\caption{100 permuted MNIST datasets results for a fully-connected network with 4 hidden layers of 128 ReLU units. This network is relatively small for these tasks and hence does not offer much available redundancy or unrealized capacity. (a) The average task accuracy of all tasks trained so far as the number of tasks increases. (b) After training 100 tasks, the average task accuracy for a group 10 tasks. We use t-test to validate the results. \label{fig:100_permuted_MNIST_tasks}}
\end{figure}
\subsection{Quantitative analysis for 100 permuted MNIST dataset} 100 permuted MNIST dataset demonstrates that our BPN has capacity to accommodate a large number of tasks. After training 100 permuted MNIST tasks, the average task performance of BD + PSP is 30.14\% better than PSP. The average task performance of BD + EWC is 35.47\% higher than EWC (Fig.~\ref{fig:100_permuted_MNIST_tasks}.a). As the number of tasks increases (Fig.~\ref{fig:100_permuted_MNIST_tasks}.a), the average task performance of BD + PSP becomes increasingly better than PSP. The reason is that adding new tasks significantly dilutes the capacity of the original network in Type 4 methods (e.g., PSP) as there are limited routes or subspaces to form sub-networks. In this case, even though the core network can no longer fully separate each task, the Beneficial perturbations (BD) can drag the misrepresented activations back to their correct work space of each task and recover their separation (as demonstrated in Fig.~\ref{fig:beneficial_perturbations}). Thus, the BD of BD + PSP can still increase the capacity of the network and boost the performance. Similarly, BD components in BD + EWC can boost performance, increasing the capacity of the network to accommodate more tasks than EWC alone (Fig.~\ref{fig:100_permuted_MNIST_tasks}.b). In addition, after training 100 tasks (Fig.~\ref{fig:100_permuted_MNIST_tasks}. b), the accuracy of BD + EWC for the first 50 tasks is higher than PSP, likely because BD+EWC did not severely dilute the core network's capacity while PSP did. This means BD + EWC has a larger capacity than PSP. In contrast, the lower performance of the last 50 tasks for BD + EWC comes from the constraints of EWC (do not allow the parameters of the network learned from the new tasks to have large deviations from the parameters trained from old tasks). Although the performance of PSP is much better than EWC, with the help of BD, BD + EWC still reaches a similar performance as PSP.
\section{Discussion}
\label{discussion}
We proposed a fundamentally new biologically plausible type of method - beneficial perturbation network (BPN), a neural network that can switch into different modes to process independent tasks, allowing the network to create potentially unlimited mappings between inputs and outputs. We successfully demonstrated this in the continual learning scenario. Our experiments demonstrate the performance of BPN is better than the state-of-the-art. 1) BPN is more parameter efficient (0.3\% increase per task) than the various network expansion and network partition methods. it does not need a large episodic memory to store any data from previous tasks, compared to episodic memory methods, or large context matrices, compared to partition methods. 2) BPN achieves state-of-the-art performance across different datasets and domains. 3) BPN has a larger capacity to accommodate a higher number of tasks than the partition networks. Through visualization of classification regions and quantitative results, we validate that beneficial perturbations can bias the network towards a task, allowing the network to switch into different modes. Thus, BPN significantly contributes to alleviating catastrophic forgetting and achieves much better performance than other types of methods.
Elsayed {\em et al.} \cite{elsayed2018adversarial} showed how carefully computed adversarial perturbations embedded in the input space can repurpose machine learning models to perform a new task without changing the parameters of the models. This attack finds a single adversarial perturbation for each task, to cause the model to perform a task chosen by the adversary.
This adversarial perturbation can thus be considered as a program to execute each task. Here, we leverage similar ideas. But, in sharp contrast, instead of using malicious programs embedded in the input space to attack a system, we embedded beneficial perturbations ('beneficial programs') into the network's parameter space (the bias terms), enabling the network to switch into different modes to process different tasks. The goal of both approaches is similar - maximizing the probability ($P(current \ task|image\ input,\ program)$) of the current task given the image input and the corresponding program for the current task. This can be achieved by either forcing the network to perform an attack task in Elsayed {\em et al.}, or assisting it to perform a beneficial task in our method. The addition of programs to either input space (Elsayed {\em et al.}'s method) or the network's activation space (our method) helps the network maximize this probability for a specific task.
We suggest that the intriguing property of the beneficial perturbations that can bias the network toward a task might come from the property of adversarial subspaces. Following the adversarial direction, such as by using the fast gradient sign method (FGSD) \cite{goodfellow6572explaining}, can help in generating adversarial examples that span a continuous subspace of large dimensionality (adversarial subspace). Because of "excessive linearity" in many neural networks \cite{tramer2017space} \cite{goodfellow2016}, due to features including Rectified linear units and Maxout, the adversarial subspace often takes a large portion of the total input space. Once an adversarial input lies in the adversarial subspace, nearby inputs also tend to lie in it. Interestingly, this corroborates recent findings by Ilyas {\em et al.} \citep{ilyas2019adversarial} that imperceptible adversarial noise can not only be used for adversarial attacks on an already-trained network, but also as features during training. For instance, after training a network on dog images perturbed with adversarial perturbation calculated from cat images, the network can achieve a good classification accuracy on the test set of cat images. This result shows that those features (adversarial perturbations) calculated from the cat training sets, contain sufficient information for a machine learning system to make correct classification on the test set of cat images. In our method, we calculate those features for each task, and store them into the bias units. In this case, although the normal weights have been modified (information from old tasks are corrupted), the stored beneficial features for each task have sufficient information to bias the network and enable the network to make correct predictions.
{BPN is loosely inspired by its counterpart in the human brain: having task-dependent modules such as bias units in our Beneficial Perturbation Network, and long-term memories in hippocampus (HPC, \cite{bakker2008pattern}) in a brain network, are crucial for a system to switch into different modes to process different tasks. During weight consolidation, the HPC \citep{lesburgueres2011early,squire1995retrograde,frankland2005organization,helfrich2019bidirectional} fuses features from different tasks into coherent memory traces. Over days to weeks, as memories mature, the HPC progressively stores permanent abstract high-level long-term memories to remote memory storage areas (neocortical regions). The HPC can then maintain and mediate their retrieval independently when a specific memory is in need.
We suggest that when a specific memory is retrieved, it helps the HPC switch into distinct modes to process different tasks.
Thus, our analogy between HPC and BPN can be formulated as: during the training of BPN, updating the shared normal weights using EWC or PSP in theory leads to distinct task-dependent representations (similar to the coherent memory traces in HPC). However, some overlap between these representations is inevitable because model parameters become too constrained for EWC, or PSP runs out of unrealized capacity of the core network. To circumvent this effect, Bias units (akin to the long-term memories in the neocortical areas) are trained independently for each task. At test time, bias units for a given task are activated to push representations of old tasks back to their initial task-optimal working regions in an analogous manner to maintaining and mediating the retrieval of Long-term memories independently in HPC.}
{An alternative biological explanation evokes the concept of factorized codes. In biological neuronal populations, neurons can be active for one task or, in many cases, for more than one tasks. At the population level, different tasks are encoded by different neuronal ensembles which can overlap. In our model, the PSP component deploys binary keys to activate task-specific readouts in hidden layers, in an analogy to neuronal task ensembles. When activating a BD component for a task, we would be further disambiguating a task-specific ensemble, particularly across neurons which are active for more than one task. The reason for this is that adding task-specific beneficial perturbations to activations of hidden layers can shift the distribution of the net activation (akin to a DC offset or carrier frequency). Evidence from nonhuman primate experiments \citep{roy2010prefrontal,cromer2010representation} and human behavioral results \citep{flesch2018comparing} support this factorized code theory. Electrophysiological experiments using monkeys demonstrated that neurons in prefrontal cortex are either representing competing categories independently \citep{roy2010prefrontal} or could represent multiple categories \citep{cromer2010representation}. In human behavior experiments, "humans tend to form factorized representation that optimally segregated the tasks \citep{flesch2018comparing}". In addition, recent neural network simulations \citep{yang2019task} demonstrated that "network developed mixed task selectivity similar to recorded prefrontal neurons after learning multiple tasks sequentially with a continual learning technique". Thus, having factorized representations for different tasks is important for enabling life-long learning and designing a general adaptive artificial intelligence system. }
\section*{Acknowledgment}
This work was supported by the National Science Foundation (grant number CCF-1317433), C-BRIC (one of six centers in JUMP, a Semiconductor Research Corporation (SRC) program sponsored by DARPA), and the Intel Corporation. The authors affirm that the views expressed herein are solely their own, and do not represent the views of the United States government or any agency thereof.
\bibliographystyle{abbrv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,168 |
Q: Stacked column chart in Google Sheets taking data from multiple columns I have a bunch of book club "star" votes in a table, broken into columns by participant. Some cells are blank where a participant didn't assign a rating.
A B C
Title 1 3 4 2
Title 2 5 5 4
Title 3 2 2
Title 4 1 4 3
I want a stacked column chart (histogram?) with 5 bars, one for each possible star rating subdivided by the person making the rating. So for the above, it would look something like:
C C
C C B B
A A A B A
-------------
1 2 3 4 5
I cannot for the life of me figure out how to do this with Google Sheets. Both the standard Setup and "switch rows/columns" are wrong. I think maybe I need to pre-analyse the data somehow?
A: try:
=ARRAYFORMULA(QUERY(TRIM(SPLIT(TRANSPOSE(SPLIT(QUERY(TRANSPOSE(QUERY(
IF(B2:D="",,"♠"&B2:D&"♦"&B1:D1),,999^99)),,999^99), "♠")), "♦")),
"select Col1,count(Col1) group by Col1 pivot Col2"))
A: I think you'll need to count the quantity of each star type broken down by person, and organize your data like this for the table:
Star Amt A B C
1 Star 1 0 0
2 Star 1 0 2
3 Star 1 0 1
4 Star 0 2 1
5 Star 1 1 0
The numbers in the table represent the number of times the user (column) voted a particular star amount (row).
Maybe this video can help too.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,330 |
package com.insightml.models.optimization;
import com.insightml.data.samples.Sample;
import com.insightml.data.samples.ISamples;
import com.insightml.evaluation.functions.ObjectiveFunction;
import com.insightml.math.Vectors;
import com.insightml.math.optimization.AbstractOptimizable;
public final class ObjectiveFunc<I extends Sample, E, T, C> extends AbstractOptimizable {
private final ISamples<I, E> instances;
private final ParameterModel<I, T, C> model;
private final C[] cachable;
private final int labelIndex;
private final ObjectiveFunction<? super E, ? super Double> objective;
public ObjectiveFunc(final ParameterModel<I, T, C> model, final ISamples<I, E> instances, final C[] cachable,
final double[][] initial, final ObjectiveFunction<? super E, ? super Double> objective,
final int labelIndex) {
super(10000, 0.000001, Vectors.fill(-10, initial[labelIndex].length), Vectors.fill(10,
initial[labelIndex].length), null, true);
this.instances = instances;
this.model = model;
this.cachable = cachable;
this.labelIndex = labelIndex;
this.objective = objective;
}
@Override
public double value(final double[] point) {
model.params = point;
return objective.normalize(objective.label(model.run(instances, cachable),
instances.expected(labelIndex),
instances.weights(labelIndex),
instances,
labelIndex).getMean());
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 29 |
Ilumetsa este un crater situat în sud-estul Estoniei. S-a format la impactul unui meteorit cu Pământul.
Date generale
Craterul are 80 de metri în diametru și are vârsta estimată la mai mult de 6600 ani (Holocen). Craterul este expus la suprafață.
Descoperit în 1938, este unul dintre cele șase situri dovedite de impact al meteoriților din Estonia.
Vezi și
Lista craterelor de impact de pe Pământ
Lista craterelor de impact din Europa
Referințe
Legături externe
Cele mai spectaculoase cratere din lume
Obiecte astronomice
Estonia
Cratere de impact | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,144 |
Q: How do I convert temp from Celsius to Fahrenheit? Almost done with my weather app just having trouble converting my temp from Celsius to Fahrenheit. I made a button and used an event handler to see if it works but when I fetch a city's api it shows both conversion. Also when I press the conversion button it makes the Fahrenheit disappear (honestly would like this to happen but in reverse, so it pops up on click). This is honestly the last thing I need done.
App.js:
import React, { Component } from "react";
import Greeting from "./Greeting";
const API_key = "dsdfghjkjhgfd";
class App extends Component {
constructor(props) {
super(props);
this.state = {
city: "",
country: undefined,
weather: "",
temp: "",
fahr: true,
};
this.handleChange = this.handleChange.bind(this);
this.handleConvert = this.handleConvert.bind(this);
this.newTemp = this.newTemp.bind(this);
}
newTemp(temp) {
let cell = Math.floor((temp * 9) / 5) + 32;
return cell;
}
calCelsius(temp) {
let cell = Math.floor(temp);
return cell;
}
/*shouldComponentUpdate() {
this.handleConvert();
}*/
handleConvert() {
const { fahr } = this.state;
this.setState({
fahr: true,
});
}
componentDidMount() {
this.handleConvert();
}
handleChange(event) {
const { name, value } = event.target;
this.setState({ [name]: value });
this.handleSubmit = async (e) => {
e.preventDefault();
const city = event.target.value;
fetch(
`http://api.openweathermap.org/data/2.5/weather?q=${city}&units=metric&appid=${API_key}`
)
.then((response) => response.json())
.then((response) => {
console.log(response);
this.setState({
city: `${response.name}`,
country: `${response.sys.country}`,
temp: this.calCelsius(`${response.main.temp}`),
fahr: this.newTemp(`${response.main.temp}`),
weather: `${response.weather[0].main}`,
});
});
};
}
render() {
let date = String(new window.Date());
date = date.slice(0, 15);
return (
<div
className={
typeof this.state.weather != "undefined"
? this.state.temp > 16
? "app hot"
: "app"
: "app"
}
>
<main>
<div className="search-box">
<form onSubmit={this.handleSubmit}>
<input
type="text"
className="search-bar"
placeholder="Enter a city"
name="city"
onChange={this.handleChange}
value={this.state.value}
/>
</form>
</div>
<div>
<div className="location-box">
<div className="location">
{this.state.city},{this.state.country}{" "}
</div>
<div className="date">{date}</div>
<div className="greet">
<Greeting />
</div>
<div className="weather-box">
<div className="temp">{this.state.temp}°C</div>
<div className="fahr">{this.state.fahr}°F</div>
<div className="weather">{this.state.weather}</div>
<button onClick={this.handleConvert}>Convert Temp</button>
</div>
</div>
</div>
</main>
</div>
);
}
}
export default App;
A: You can convert any celsius value to fahrenheit by multiplying it with 9/5 and adding 32.
Example:
function convertToF(celsius) {
return celsius * 9/5 + 32
}
convertToF(30); // 86
A: 0C is 32F, and the scale is 5/9th, so it goes Celsius = (Fahrenheit - 32) * 5/9
const celsius = -23;
const fahrenheit = 113;
const celsiusDegree = (fahrenheit - 32) * 5 / 9;
console.log(celsiusDegree); //-9.4
const fahrenheitDegree = (celsius * 9 / 5) + 32;
console.log(fahrenheitDegree); //45
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,154 |
Les Serres és una serra situada al municipi de Palamós a la comarca del Baix Empordà, amb una elevació màxima de 51 metres.
Referències
Serres del Baix Empordà
Geografia de Palamós | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,715 |
<?php
namespace Emonkak\Di\Benchmarks\Fixtures;
interface BarInterface
{
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,783 |
#include "test_coap_security_handler.h"
#include "coap_security_handler.h"
#include <string.h>
#include "nsdynmemLIB_stub.h"
#include "mbedtls_stub.h"
#include "mbedtls/ssl.h"
static int send_to_socket(int8_t socket_id, void *handle, const unsigned char *buf, size_t len)
{
}
static int receive_from_socket(int8_t socket_id, unsigned char *buf, size_t len)
{
}
static void start_timer_callback(int8_t timer_id, uint32_t int_ms, uint32_t fin_ms)
{
}
static int timer_status_callback(int8_t timer_id)
{
}
bool test_thread_security_create()
{
if( NULL != coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, NULL) )
return false;
if( NULL != coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, &timer_status_callback) )
return false;
nsdynmemlib_stub.returnCounter = 1;
mbedtls_stub.expected_int = -1;
if( NULL != coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, &timer_status_callback) )
return false;
mbedtls_stub.expected_int = 0;
nsdynmemlib_stub.returnCounter = 2;
mbedtls_stub.crt_expected_int = -1;
if( NULL != coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, &timer_status_callback) )
return false;
nsdynmemlib_stub.returnCounter = 2;
mbedtls_stub.crt_expected_int = 0;
coap_security_t *handle = coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, &timer_status_callback);
if( NULL == handle )
return false;
coap_security_destroy(handle);
return true;
}
bool test_thread_security_destroy()
{
nsdynmemlib_stub.returnCounter = 2;
mbedtls_stub.crt_expected_int = 0;
coap_security_t *handle = coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, &timer_status_callback);
if( NULL == handle )
return false;
coap_security_destroy(handle);
return true;
}
bool test_coap_security_handler_connect()
{
nsdynmemlib_stub.returnCounter = 2;
mbedtls_stub.crt_expected_int = 0;
coap_security_t *handle = coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, &timer_status_callback);
if( NULL == handle )
return false;
unsigned char pw = "pwd";
coap_security_keys_t keys;
keys._priv = &pw;
keys._priv_len = 3;
if( -1 != coap_security_handler_connect_non_blocking(NULL, true, DTLS, keys, 0, 1) )
return false;
mbedtls_stub.useCounter = true;
mbedtls_stub.counter = 0;
mbedtls_stub.retArray[0] = -1;
mbedtls_stub.retArray[1] = -1;
mbedtls_stub.retArray[2] = -1;
mbedtls_stub.retArray[3] = -1;
mbedtls_stub.retArray[4] = -1;
mbedtls_stub.retArray[5] = MBEDTLS_ERR_SSL_HELLO_VERIFY_REQUIRED;
mbedtls_stub.retArray[6] = -1;
mbedtls_stub.retArray[7] = -1;
if( -1 != coap_security_handler_connect_non_blocking(handle, true, DTLS, keys, 0, 1) )
return false;
mbedtls_stub.counter = 0;
mbedtls_stub.retArray[0] = 0;
if( -1 != coap_security_handler_connect_non_blocking(handle, true, DTLS, keys, 0, 1) )
return false;
mbedtls_stub.counter = 0;
// mbedtls_stub.retArray[0] = 0;
mbedtls_stub.retArray[1] = 0;
if( -1 != coap_security_handler_connect_non_blocking(handle, true, DTLS, keys, 0, 1) )
return false;
simple_cookie_t c;
memset(&c, 0, sizeof(simple_cookie_t));
mbedtls_stub.cookie_obj = &c;
memset(&mbedtls_stub.cookie_value, 1, 8);
mbedtls_stub.cookie_len = 2;
mbedtls_stub.counter = 0;
// mbedtls_stub.retArray[0] = 0;
// mbedtls_stub.retArray[1] = 0;
mbedtls_stub.retArray[2] = 0;
if( -1 != coap_security_handler_connect_non_blocking(handle, true, DTLS, keys, 0, 1) )
return false;
c.len = 8;
memset(&c.value, 1, 8);
mbedtls_stub.cookie_obj = &c;
memset(&mbedtls_stub.cookie_value, 1, 8);
mbedtls_stub.cookie_len = 8;
mbedtls_stub.counter = 0;
// mbedtls_stub.retArray[0] = 0;
// mbedtls_stub.retArray[1] = 0;
// mbedtls_stub.retArray[2] = 0;
mbedtls_stub.retArray[3] = 0;
if( -1 != coap_security_handler_connect_non_blocking(handle, true, DTLS, keys, 0, 1) )
return false;
mbedtls_stub.counter = 0;
// mbedtls_stub.retArray[0] = 0;
// mbedtls_stub.retArray[1] = 0;
// mbedtls_stub.retArray[2] = 0;
// mbedtls_stub.retArray[3] = 0;
mbedtls_stub.retArray[4] = 0;
if( -1 != coap_security_handler_connect_non_blocking(handle, true, DTLS, keys, 0, 1) )
return false;
mbedtls_stub.counter = 0;
// mbedtls_stub.retArray[0] = 0;
// mbedtls_stub.retArray[1] = 0;
// mbedtls_stub.retArray[2] = 0;
// mbedtls_stub.retArray[3] = 0;
// mbedtls_stub.retArray[4] = 0;
mbedtls_stub.retArray[6] = 0;
mbedtls_stub.retArray[7] = 0;
if( 1 != coap_security_handler_connect_non_blocking(handle, true, DTLS, keys, 0, 1) )
return false;
mbedtls_stub.counter = 0;
mbedtls_stub.retArray[5] = MBEDTLS_ERR_SSL_BAD_HS_FINISHED;
if( -1 != coap_security_handler_connect_non_blocking(handle, true, DTLS, keys, 0, 1) )
return false;
mbedtls_stub.counter = 0;
mbedtls_stub.retArray[5] = HANDSHAKE_FINISHED_VALUE;
if( 1 != coap_security_handler_connect_non_blocking(handle, true, DTLS, keys, 0, 1) )
return false;
coap_security_destroy(handle);
return true;
}
bool test_coap_security_handler_continue_connecting()
{
nsdynmemlib_stub.returnCounter = 2;
mbedtls_stub.crt_expected_int = 0;
coap_security_t *handle = coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, &timer_status_callback);
if( NULL == handle )
return false;
mbedtls_stub.useCounter = true;
mbedtls_stub.counter = 0;
mbedtls_stub.retArray[0] = MBEDTLS_ERR_SSL_HELLO_VERIFY_REQUIRED;
mbedtls_stub.retArray[1] = -1;
mbedtls_stub.retArray[2] = -1;
if( -1 != coap_security_handler_continue_connecting(handle) )
return false;
mbedtls_stub.counter = 0;
mbedtls_stub.retArray[0] = MBEDTLS_ERR_SSL_HELLO_VERIFY_REQUIRED;
mbedtls_stub.retArray[1] = 0;
mbedtls_stub.retArray[2] = 0;
if( 1 != coap_security_handler_continue_connecting(handle) )
return false;
mbedtls_stub.counter = 0;
mbedtls_stub.retArray[0] = MBEDTLS_ERR_SSL_BAD_HS_FINISHED;
if( MBEDTLS_ERR_SSL_BAD_HS_FINISHED != coap_security_handler_continue_connecting(handle) )
return false;
mbedtls_stub.counter = 0;
mbedtls_stub.retArray[0] = MBEDTLS_ERR_SSL_WANT_READ;
if( 1 != coap_security_handler_continue_connecting(handle) )
return false;
mbedtls_stub.counter = 0;
mbedtls_stub.retArray[0] = HANDSHAKE_FINISHED_VALUE_RETURN_ZERO;
if( 0 != coap_security_handler_continue_connecting(handle) )
return false;
coap_security_destroy(handle);
return true;
}
bool test_coap_security_handler_send_message()
{
nsdynmemlib_stub.returnCounter = 2;
mbedtls_stub.crt_expected_int = 0;
coap_security_t *handle = coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, &timer_status_callback);
if( NULL == handle )
return false;
if( -1 != coap_security_handler_send_message(NULL, NULL, 0))
return false;
mbedtls_stub.expected_int = 6;
unsigned char cbuf[6];
if( 6 != coap_security_handler_send_message(handle, &cbuf, 6))
return false;
coap_security_destroy(handle);
return true;
}
bool test_thread_security_send_close_alert()
{
nsdynmemlib_stub.returnCounter = 2;
mbedtls_stub.crt_expected_int = 0;
coap_security_t *handle = coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, &timer_status_callback);
if( NULL == handle )
return false;
if( -1 != coap_security_send_close_alert(NULL))
return false;
mbedtls_stub.expected_int = 0;
if( 0 != coap_security_send_close_alert(handle))
return false;
coap_security_destroy(handle);
return true;
}
bool test_coap_security_handler_read()
{
nsdynmemlib_stub.returnCounter = 2;
mbedtls_stub.crt_expected_int = 0;
coap_security_t *handle = coap_security_create(1,2,NULL,ECJPAKE,&send_to_socket, &receive_from_socket, &start_timer_callback, &timer_status_callback);
if( NULL == handle )
return false;
if( -1 != coap_security_handler_read(NULL, NULL, 0))
return false;
mbedtls_stub.expected_int = 6;
unsigned char cbuf[6];
if( 6 != coap_security_handler_read(handle, &cbuf, 6))
return false;
coap_security_destroy(handle);
return true;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,578 |
WinQueueStream::WinQueueStream():base_(0){
}
bool
WinQueueStream::initialize(){
base_ = SHCreateMemStream(NULL, 0);
if (base_ == 0)
return false;
base_->AddRef();
return true;
}
IStream*
WinQueueStream::getBaseStream(){
return base_;
}
void
WinQueueStream::startWritingPhase(){
LARGE_INTEGER zero = { 0 };
base_->Seek(zero, STREAM_SEEK_SET, NULL);
base_->SetSize(*reinterpret_cast<ULARGE_INTEGER*>(&zero));
}
void
WinQueueStream::endWritingPhase(){
LARGE_INTEGER zero = { 0 };
base_->Seek(zero, STREAM_SEEK_SET, NULL);
}
int WinQueueStream::in_avail(){
if (base_ == 0)
return 0;
STATSTG stats;
base_->Stat(&stats, STATFLAG_NONAME);
return (stats.cbSize.HighPart == 0 ? stats.cbSize.LowPart : stats.cbSize.HighPart); //agnostic to endianness
}
const char*
WinQueueStream::nextChunk(int* size){
ULONG read = 0;
base_->Read(audio_, AUDIO_CHUNK_SIZE, &read);
if (read == 0)
return 0;
*size = (int) read;
return audio_;
}
void WinQueueStream::dispose(){
if (base_ != 0){
base_->Release();
base_ = 0;
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,286 |
\section{Introduction}
\label{sec:introducao}
Innovative computing techniques are deployed each year in order to increase the efficiency and responsiveness of the user needs in data networks, enhancing it both in scale and complexity. The need to constantly evaluate the status of these networks in this context is imminent, driving the development of monitoring systems with a wide range of features/functionalities. Network management has become an indispensable task from the network's correct functioning, an issue that is essential to maintaining the quality demanded by the market \citep{Stallings:1998}.
The continuous growth in the number and diversity of computer-network equipment and, consequently, in the volume of information signals coming from these, has turned network management into a complex task. This also applies to the field of automation, where automation manufacturers, with the popularisation of ethernet networks, started to provide communication modules with Simple Network Management Protocol (SNMP). The Industrial Automation Open Network (IAONA) protocol specification reinforces this trend by publishing a Management Information Base (MIB) for SNMP categorisation of industrial applications \citep{fonseca2006}.
As a result, manufacturers now provide Management Information Bases (MIBs) for their products, which add specific Object Identifiers (OIDs) for monitoring automation devices. Some examples of data about OIDs found in automation equipment in the MIBs and in the IAONA are:
\begin{itemize}
\item Application information (name, {\em in operation});
\item Protocol data (Ethernet / IP, ModbusTCP);
\item Device name, serial number, vendor name, device version
firmware.
\end{itemize}
According to \citep{fonseca2006,Villela19,flores09,freitas14}, in contrast to the current fast development of MIBs for data network automation, these devices usually end up not monitored or monitored in isolation from the rest of the plant. In this context, concentrating the monitoring data on a centralised data centre becomes a suitable solution. Supervisory Control and Data Acquisition (SCADA) systems at the operation centres provide some of the desirable features for monitoring data network devices involved in the automation of critical processes.
Specifically in electrical systems, the need for monitoring is often more pronounced because there are completely isolated facilities and devices, without any monitoring events, not even local. This is particularly relevant in the context of the Brazilian power grids. In addition, there is an increasing amount of data to be analysed that are related to fault classification of network equipment, where contradictory and inaccurate information often occur. Therefore, in addition to more detailed monitoring, it is necessary to use algorithms that address this information gap. The application of algorithms with non-classical logic is an alternative to alleviate this problem, as they allow for a more refined, explicit, treatment of information than black-box systems, or traditional logic-based systems.
In this paper, Fuzzy logic and a Paraconsistent Annotated Logic with Annotation of Two Values (PAL2v) are combined to analyse and classify the operating conditions of equipment in data networks within two SCADA-monitored electrical installations of a Regional Operations Centre (ROC) in Brazil. In this analysis, information (variables) are collected from each equipment in the power grid using two distinct sources (used as evidences), the degree of contradiction between these variables is calculated and then, through a hybrid Fuzzy-paraconsistent inference tool (PAL2v lattice), the condition of the equipment operation ({\em normal, failure, unstable or undetermined / inconsistent}) is inferred.
The next section presents the background upon which this work was developed. The method for the analysis of data networks proposed in this paper is described in Section \ref{parafuzzy} and the related tests are described in Section \ref{exp}. Sections \ref{res} and \ref{disc} present the results and discussions, Section \ref{related} presents the related works, and Section \ref{conclu} close this paper with conclusions and future works.
\section{Background}
This section presents some fundamental concepts of automation, remote control of data network equipment, monitoring of electrical systems (Section \ref{2.1}), and also about Fuzzy logic and the Paraconsistent Annotated Logic with Annotation of Two Values (PAL2v) used in this work (Section \ref{fp-intro}).
\subsection{Concepts of automation and monitoring systems for the remote control of data network equipment of electrical systems}\label{2.1}
This section describes some concepts of the electrical substation automation systems, hydroelectric plants and operation centres, including Regional Operation Centres (ROC) and System Operations Centre (SOC), according to \citep{Jardini97,Paredes02,PE2003} that are in agreement with the technical procedure for the Brazilian electric system (Sub-Module 2.7), prepared by the National Operator of the Brazilian Electric System (ONS).
A substation automation system aims to provide the means for its operation and maintenance. It is characterised by two hierarchical levels: the interface level, with the process and data acquisition, and the level of command and supervision, also called central system.
The interface level hosts the Acquisition and Control Units (ACUs), and some dedicated equipment such as protective relays, oscillography equipment, interlocking units, among others. The central system level is typically comprised of multiple workstations connected to a local area network that is connected to a process-level digital equipment. The central system has some functions such as monitoring the equipment status, measurement display, data monitoring protection, equipment control, alarm controls, and the indication of sequences of events. Figure \ref{1a} presents the typical architecture of a substation automation system, according to \citep{Jardini97,Paredes02,PE2003}.
The automation system of a hydroelectric plant also aims to provide the means for its operation and maintenance; it has a hierarchical-level configuration similar to a substation automation system, as shown in Figure \ref{1b}, and can display a dual local area network (LAN).
In this system, the process interface units can be composed of several modules, including a generator ACU, auxiliary services ACU, spillway ACU etc. These ACU units are integrated with speed regulators, voltage regulators, local controllers (in floodgates, pumps, compressors and others). Generally in large plants there are independent subsystems built for specific activities such as machine supervision (obtaining data about vibration, temperature, partial discharges etc.) and dam supervision.
In Regional Operation Centres (ROC) and System Operation Centre (SOC), substation and the hydroelectric plant automation are organised into a larger hierarchy of supervision and control subsystems:
\begin{itemize}
\item The Regional Operation Centre (ROC) concentrates the operation and service of substations and hydroelectric plants in a region. From the ROC, for example, remote control signals can be issued to start and stop a given generator in a plant, given the data collected from the ACUs. In short, ROC concentrates the Supervisory Control and Data Acquisition (SCADA) systems.
\item The System Operation Centre (SOC) provides facilities for centralised system operation, generation and load coordination. SOC has a hardware and software structure with high-level functions, from which the information necessary for the proper and safe operation of the system is obtained. SOCs are often directly linked to ACUs, and a SCADA function is also included in each SOC.
\end{itemize}
The plant-automation system architecture can be represented as the hierarchy shown in Figure \ref{1c}.
\begin{figure}[ht!]
\centering
\begin{subfigure}[b]{1\textwidth}
\centering
\includegraphics[scale=0.074]{fig1a.png}
\caption{Substation}\label{1a}
\end{subfigure}
\hfill
\begin{subfigure}[b]{1\textwidth}
\centering
\includegraphics[scale=0.074]{fig1b.png}
\caption{Hydroelectric plant}\label{1b}
\end{subfigure}
\hfill
\begin{subfigure}[b]{1\textwidth}
\centering
\includegraphics[scale=0.074]{fig1c.png}
\caption{SOC, ROC, substations and hydroelectric plants}\label{1c}
\end{subfigure}
\caption{Typical architectures}\label{fig1}
\end{figure}
The most relevant points in a data network are usually monitored to ensure the reliability of SCADA supervision and the control of facilities. Some basic equipment, and related data, that are monitored are the following:
\begin{itemize}
\item ACUs (substations and hydroelectric plants): processing, memory, file allocation, protocol connections with ROCs SCADA systems and states of additional resources (communication cards, power supplies etc.).
\item Switches and routers: status of communication {\em ports} (traffic, latency, errors).
\item Servers (SOC / ROC): processing, memory, file allocation (disk units) data.
\end{itemize}
In this work, we considered the following sources of data: ACUs (information of protocol links between ACUs and SCADAs of the ROCs), routers (monitoring the outgoing traffic of substations and hydroelectric plants) and servers (providing data about the processing, memory and disk allocation).
\subsection{Fuzzy and Paraconsistent Annotated Logic}\label{fp-intro}
This section presents a brief introduction to Fuzzy set theory \citep{pedrycz1993fuzzy,ZADEH1965338} and to the Paraconsistent Annotated Logic with Annotation of Two Values (PAL2v) \citep{dacosta1974,Abe06} used in this work.
\subsection*{Fuzzy Sets}
Let $U$ be an universe of discourse, a collection of objects $\{u\}$. A Fuzzy set $A\subset U$ is defined by a membership function $\mu_A$ that assumes values within the range [0,1]: \[\mu_A: U\rightarrow [0,1].\]
The subset of points $u\in U$ such that $\mu_{A}\left ( u \right ) > 0$ is called the {\em support set of $A$}.
Let $A$ and $B$ be two distinct Fuzzy sets in $U$ with respective membership functions $\mu_{A}$ and $\mu_{B}$. The Fuzzy set operations between $A$ and $B$ are defined as:
\begin{itemize}
\item Union: $\mu_{A \cup B}(u)$ = $\mu_{A}(u)\; \delta\; \mu_{B}(u)$, where $\delta$ is a triangular co-norm (called t-conorm);
\item Intersection: $\mu_{A \cap B}(u)$ = $\mu_{A}(u) \;\tau\; \mu_{B}(u)$, where $\tau$ is a triangular norm (called t-norm); and,
\item Complement: $\mu_{\neg A}(u)$ = 1 - $\mu_{A}(u)$.
\end{itemize}
A t-norm is a function $\tau:[0,1]\times [0,1]\rightarrow [0,1]$ that satisfies the properties of commutativity, monotonicity, associativity and has 1 as a neutral element. The following functions are t-norms,for instance: $TM(x,y)=min(x,y)$ (minimum or Gödel t-norm), $TP(x,y)=x.y$ (product t-norm)
$TL(x,y)=max(x+y-1,0)$ (Lukasiewicz t-norm) \citep{Navara:2007}. A triangular co-norm ($t-conorm$) is a commutative, associative and monotonic function that has 0 as the neutral element. Examples of $t-conorm$s are: $SM(x,y)=max(x,y)$ (maximum or Gödel t-conorm), $SP(x,y)=x+y-x.y$ (product t-conorm, or probabilistic sum), $SL(x,y)=min(x+y,1)$ (Lukasiewicz t-conorm, or bounded sum) \citep{Navara:2007}. T-norms and t-conorms are generalisations of the classical conjunction and disjunction, respectively \citep{JAYARAM20092063}.
Given Fuzzy sets $A_{1}, A_{2}, ...,A_{n}$ defined respectively over the universes of discourse $U_{1}, U_{2}, ...,U_{n}$, an n-ary Fuzzy relation $R$ is a Fuzzy set in $U_{1}, U_{2}, ...,U_{n}$, expressed as: \[R = {[(u_{1}, u_{2}, ...,u_{n}), \mu _{R}(u_{1}, u_{2}, ...,u_{n})]|(u_{1}, u_{2}, ...,u_{n}) \in U_{1}, U_{2}, ...,U_{n}}\]
A Fuzzy knowledge base is represented by means of sets of Fuzzy conditional statements (Fuzzy rules). Let $A_{i}, B_{i}$ and $C_{i}$ be Fuzzy sets respectively defined over the universes of discourse $U, V$ and $W$. A Fuzzy rule such as: {\em if (x is $A_{i}$ and y is $B_{i}$) then (z is $C_{i}$)} can be interpreted as a the following Fuzzy $R_{i}$ relation:
\[\mu_{R_{i}} = [\mu_{A_i}(u)\;\; and\;\; \mu_{B_i}(v)]\rightarrow \mu_{C_i}(w),\;\; for \;u\in U, v \in V,\; w \in W,\]
where the symbol ``$\rightarrow$" is the Fuzzy implication. There are various types of Fuzzy implications \citep{JAYARAM20092063,4374117}. Assuming $x$ and $y$ such that $x, y\in [0,1]$, where $x$ is the antecedent and $y$ as the consequent of an implication ($x\rightarrow y$), the most usual interpretations of the Fuzzy implication are the following \citep{RUAN199323}:
\begin{itemize}
\item Zadeh implication \citep{zadeh75}: $\mathscr{I}(x,y) = max(1-x, min(x,y))$
\item {\L}ukasiewicz implication \citep{zadeh75}: $\mathscr{I}(x,y) = min(1, 1 - x + y)$
\item Mamdani implication \citep{1674779}: $\mathscr{I}(x,y) = min(x,y)$
\end{itemize}
In a Fuzzy system, each Fuzzy rule is represented by a Fuzzy relation. A set of Fuzzy relations describes the behaviour of the domain modelled by them. Therefore, a domain can be represented by a single Fuzzy relation, that is the combination of the set of relations describing it \citep{Gomide}. This combination can be obtained by applying the aggregation connective $aggreg$: {\em R = aggreg($R_{1}, R_{2}, ...,R_{n}$ )}. Usually, $aggreg$ is interpreted as set union ($max$ operator), however there are various possible aggregation connectives \citep{DUBOIS198585}.
In Fuzzy set theory, an important inference rule is the Generalised Modus Ponens (GMP) \citep{5408575,Cornelis2000} that is illustrated in the following example:
\begin{quote}
$x$ is $A$ \hspace{1.23in} (e.g., ``John is very tall"),\\
if ($x$ is $A$) then ($y$ is $B$)\hspace{0.3in} (e.g. ``If John is tall, then he is heavy")\\
$\therefore$ $y$ is $B$ \hspace{1.15in}(e.g. ``John is considerably heavy").
\end{quote}
Commonly this rule is interpreted by the law of compositional inference, suggested by \cite{5408575}. In this context, the rule {\em if $x$ is A then $y$ is B}, written as $A \rightarrow B$, is first transformed into a Fuzzy relation $R_{A \rightarrow B}$ that, assuming Mamdani's implication \citep{1674779}, becomes:
\[\mu_{A \rightarrow B} = min(\mu_A(u), \mu_B(v)); u\in U, v\in V\]
Given the fact that {\em if x is A} (or, simply $A$) and $R_{A \rightarrow B}$, Zadeh's compositional inference law states that:\[B = A \circ R_{A \rightarrow B},\]
where the composition of $A$ with $R_{A \rightarrow B}$ can be given by the max-min inference rule: \[\mu_{B}(v) = \underset{u}max({min(\mu_A(u), \mu_{R_{A \rightarrow B}}(u,v))})\]
The procedure above can be easily extended for any finite number of Fuzzy rules representing a domain. Figure \ref{2} \citep{Gomide} illustrates the $max-min$ inference process for two rules, $A_{i} \rightarrow B_{i}$ and $A_{j} \rightarrow B_{j}$, where $A'$ is the input fact and $B'$ is the output (the result of the Fuzzy inference procedure).
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.1]{fig2.png}
\caption{Fuzzy inference engine \citep{Gomide}.}\label{2}
\end{figure}
The Fuzzy modelling of a control system is based on translating the expert's knowledge into Fuzzy rules (as above), whereby the sensor data (which is the input to the system) is mapped to appropriate membership functions, this information is used to evaluate each of the Fuzzy rules representing the domain. The contributions of each of these rules is combined and then converted back into specific control output values. This conversion to control output value is called {\em defuzzification}, that extracts a {\em crisp} value from the results of the Fuzzy inference (for instance, extracting a single value form the Fuzzy set $B'$ in Figure \ref{2}). There are multiple defuzzification strategies available, and a description of them is outside the scope of this paper \citep{LEEKWIJCK1999159} .
In the present work, this process of Fuzzy modelling of control systems is used to bridge sensor data from the electrical data network to logical states in a paraconsistent expert system.
\subsection*{Paraconsistent Annotated Logic}
Paraconsistent Logic (PL) \citep{dacosta1974,Abe15} is a non-classical logic that opposes the principle of non-contradiction and can be applied for the rigorous treatment of contradictory sensor information. \cite{Abe15ch} provides a clear description of the principles of Paraconsistent Logic. The present paper is based on a subset of PL, called Paraconsistent Annotated Logic.
Paraconsistent Annotated Logic (PAL) embraces concepts of uncertainty, inconsistency and incompleteness in its semantic structure, allowing reasoning with and about these concepts.
With this in mind, PAL has four truth values: $\tau$ = \{ $\top$ (inconsistency), t (truth), F (falsehood), and $\perp$ (paracomplete or indeterminate) \}. The set $\tau$ forms a complete lattice, characterised by the Hasse diagram shown in Figure \ref{3} \citep{Abe15ch}.
In this logic, the negation $\sim$ is considered in the following way: $\sim$(1)=0, $\sim$(0)=1, $\sim$($\top$)=$\top$ and $\sim$($\perp$)=$\perp$.
A proposition in PAL is accompanied by annotations ($\mu$), or degrees of evidence, that ascribe values (or {\em logical connotations} \citep{filho12}) to the proposition. For instance, given a proposition $P$ and a related annotation $\mu$, $P\mu $ can be interpreted as: "I believe the proposition $P$ with a maximum of $\mu$ degrees of evidence".
Annotations can be made by $1$ or $n$ values, in order to acquire the correct intensity of the representations regarding how much the annotations expose the knowledge about a proposition $P$.
It is common to use a lattice in the real plane where the annotations take the form of two numeric values ($\mu $,$\lambda $). In this case, the logic is referred to as Paraconsistent Annotated logic with Annotation of Two Values (PAL2v).
The annotation of two values in the lattice $\tau$ is represented by: $\tau = \{ ( \mu ,\lambda) |\,\mu ,\lambda \in [ 0,1 ] \subset \mathds{R}\}$. The operator "$\thicksim$" can now be defined as: \[\thicksim~[( \mu ,\lambda)] = (\lambda, \mu )\]
Thus, the association of an annotation $( \mu ,\lambda)$ with a proposition $P$ means that the {\em favourable} evidence degree about $P$ is $\mu $, while its {\em unfavourable} evidence degree is $\lambda $. PAL2v defines the lattice shown in Figure \ref{3}, representing the following:
\begin{itemize}
\item $P_t$ = $P\left( \mu, \lambda \right )$ = $P\left( 1,0 \right )$: indicating a total favourable evidence and null unfavourable evidence for $P$, assigning a logical connotation of {\em True}.
\item $P_F$ = $P\left( \mu, \lambda \right )$ = $P\left( 0,1 \right )$: indicating a null favourable evidence and total unfavourable evidence for $P$, assigning a logical connotation of {\em False}.
\item $P_\top $ = $P\left( \mu, \lambda \right )$ = $P\left( 1,1 \right )$: indicating a total favourable evidence and a total unfavourable evidence for $P$, thus assigning a connotation of {\em Inconsistency}.
\item $P_\perp $ = $P\left( \mu, \lambda \right )$ = $P\left( 0,0 \right )$: indicating a null favourable evidence and null unfavourable evidence for $P$, assigning a logical connotation of {\em Paracompleteness}.
\end{itemize}
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.10]{fig3.png}
\caption{PAL2v lattice.}\label{3}
\end{figure}
The collection of values for the degrees of favourable evidence ($\mu $) and unfavourable evidence ($\lambda $) aim to solve the problem of contradictory signals, facilitating actions to modify the behaviour of a system in order to decrease the intensity (value) of the contradictory signals.
The PAL2v also considers the definitions for the {\em degree of certainty} ($D_c$) and a {\em degree of contradiction} ($D_{ct}$) as given by the Equations \ref{eq10} and \ref{eq11} below:
\begin{align}
&D_c = \mu - \lambda \label{eq10}\\
&D_{ct} = \left (\mu + \lambda \right ) -1 \label{eq11}
\end{align}
The degree of certainty ($D_{c}$) and contradiction ($D_{ct}$) are defined on the interval $[-1,1]\in \mathds{R}$.
The meeting point between the degree of certainty and the degree of contradiction (as shown in Figure \ref{4}) is considered a paraconsistent logical state $\varepsilon_{\tau}$ = $( D_c,D_{ct}) $. As $\tau$ assumes values closer to the point {\bf D}, paraconsistent logical states $\varepsilon_{\tau}$ with $D_c$ approximately 1 and with $D_{ct}$ approximately 0 indicate proximity to the truth ($P_t$). In turn, as $\tau$ assumes values closer to the point {\bf B}, the paraconsistent logic states $\varepsilon_{\tau}$ with values $D_c$ approximately -1 and $D_{ct}$ approximately 0 indicate a tendency towards falsehood ($P_F$). Considering that the point of intersection between the degrees of certainty and contradiction axes is the origin of these values, that is, at this point, $D_c$ = 0 and $D_{ct}$ = 0, when propositions present a high contradiction value, for $D_{ct}$ tending to +1 and $D_c$ to 0 (point {\bf A} in Figure \ref{4}), this characterises the logical $\varepsilon_{\tau}$ states that are close to inconsistency ($P_\top $), and the opposite, for $D_{ct}$ approximately -1 and $D_c$ approximately 0 (point {\bf C} in Figure \ref{4}), characterises logical $\varepsilon_{\tau}$ states with proximity to indeterminacy ($P_\perp $).
In the case of a paraconsistent verification system, when a logical state of a proposition remains close to the line segment defined by the {\bf BD} points (Figure \ref{4}), the degree of contradiction in the evidence is small and tends to be null. Similarly, logical whose degree of contradiction is $D_{ct}$ = +1 is an inconsistent state (analogously to the indeterminate state, $D_{ct}$ = -1).
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.08]{fig4.png}
\caption{Representation of $\mu $, $\lambda $, $D_{c}$ and $D_{ct}$ in the lattice.}\label{4}
\end{figure}
With the degrees of certainty and uncertainty, the following 12 output logical states can be defined (as shown in Figure \ref{5}):
\begin{itemize}
\item the {\em extreme logical states}: True (t), False (F), Inconsistent($\top $) and Paracomplete($\perp $);
\item the {\em Near True tending to Inconsistent} (Qt-$\top $);
\item the {\em Near True tending to Paracomplete} (Qt-$\perp $);
\item the {\em Near False tending to Inconsistent} (QF-$\top $);
\item the {\em Near False tending to Paracomplete} (QF-$\perp $);
\item the {\em Near Inconsistent tending to True} (Q$\top $-t);
\item the {\em Near Inconsistent tending to False} (Q$\top $-F);
\item the {\em Nearly Full Tending to True} (Q$\perp $-t); and,
\item the {\em Nearly Full Tending to False} (Q$\perp $-F).
\end{itemize}
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.07]{fig5.png}
\caption{Representation of the 12-state lattice subdivision (Para-Analyser)}\label{5}
\end{figure}
Figure \ref{5} also shows the boundaries of the regions that define the transition between extreme and non-extreme logical states as: $V_{cic}$ representing the {\em Maximum Uncertainty Control Value}; $V_{cve}$ standing for the {\em Maximum Certainty Control Value}; the $V_{cpa}$, representing the {\em Minimum Uncertainty Control Value}; and, the $V_{cfa}$ representing the {\em Minimum Certainty Control Value}.
Based on the 12-state lattice (Figure \ref{5}), an algorithm called Para-Analyser was created to calculate an output considering the 12 logical states (extremes and non-extremes) as inputs. The pseudocode for this process is shown in Algorithm \ref{alg1} \citep{Abe15ch}.
\begin{algorithm}
\caption{Para-Analyser} \label{alg1}
\begin{algorithmic}[1]
\Procedure{para-analyser}{$mi$ , $la$} \Comment{Reading of $\mu $ and $\lambda $ through $mi$ and $la$}
\State $c1 \gets 0.5$; \Comment{Maximum Certainty Control Value ($Vcve$)}
\State $c2 \gets -0.5$; \Comment{Minimum Certainty Control Value ($Vcfa$)}
\State $c3 \gets 0.5$; \Comment{Maximum Uncertainty Control Value ($Vcic$)}
\State $c4 \gets -0.5$; \Comment{Minimum Uncertainty control Value ($Vcpa$)}
\State
\State $dc \gets mi - la$; \Comment{$D_{c}$ calculation}
\State $dct \gets (mi + la) - 1$; \Comment{$D_{ct}$ calculation}
\State
\If{($dc \geq c1$)} $s1 \gets t$; \Comment{Section of extreme logical states} \EndIf
\If{($dc \leq c2$)} $s1 \gets F$; \EndIf
\If{($dct \geq c3$)} $s1 \gets \top $; \EndIf
\If{($dct \leq c4$)} $s1 \gets \perp $; \EndIf
\State
\If{($0 \leq dc < c1$) \& ($0 \leq dct < c3 $)} \Comment{Section of non-extreme logical states} \If {($dc \geq dct$)} $s1 \gets Qt-\top$; \Else \State $s1 \gets Q\top -t$; \EndIf \EndIf
\If{($0 \leq dc < c1$) \& ($c4 < dct \leq 0 $)} \If {($dc \geq dct$)} $s1 \gets Qt-\perp$; \Else \State $s1 \gets Q\perp -t$; \EndIf \EndIf
\If{($c2 < dc \leq 0$) \& ($c4 < dct \leq 0 $)} \If {($dc \geq dct$)} $s1 \gets QF-\perp$; \Else \State $s1 \gets Q\perp -F$; \EndIf \EndIf
\newpage
\If{($c2 < dc \leq 0$) \& ($0 \leq dct < c3 $)} \If {($dc \geq dct$)} $s1 \gets QF-\top$; \Else \State $s1 \gets Q\top -F$; \EndIf \EndIf
\State
\State $s2a \gets dc$;
\State $s2b \gets dct$;
\State
\State \textbf{return}(s1, s2a, s2b); \Comment{Return of current logical state, $D_{c}$ and $D_{ct}$ through $s1$, $s2a$ and $s2b$}
\EndProcedure
\end{algorithmic}
\end{algorithm}
\subsection{Using Fuzzy Sets to obtain paraconsistent logical states}\label{fuzzysets}
This section presents an introduction to the combination of Fuzzy set theory with Paraconsistent Logics for control systems \citep{Cortes17}.
Degrees of evidence are processed by the Para-Analyser algorithm described above \citep{Abe15ch}, which transforms them into degrees of certainty $D_{c}$ and contradiction $D_{ct}$. These two signals have their values bounded to a normalised interval $[-1,+1]$. This facilitates the application of Fuzzy set strategies for turning each obtained value of $D_{c}$ and $D_{ct}$ into a single output signal that falls in one of the 12 PAL2v states (Figure \ref{5}). This facilitates a finer control of the system, taking into account various degrees of evidence and a rich set of logical states, fine-tuning the decision process of the situation at hand.
The steps of applying Fuzzy set methods to paraconsistent logical states are summarised in Figure \ref{6}, representing (a) the fuzzification of the degrees of certainty and contradiction; (b) the elaboration of the inference rule table; (c) the application of inference methods; and, (d) defuzzification.
In step (a) (Figure \ref{6}) the axis representing the degrees of contradiction ($D_{ct}$) and the axis representing the degrees of certainty ($D_{c}$) were fuzzified (as shown in Figures \ref{7} and \ref{8}), using triangular membership functions. These functions were defined in the contradiction axis as: $\perp$ (Indeterminacy), Q$\perp$ (Almost Indeterminate), Ri$\perp$ (Common Region Tending to Undefined) and Ri$\top$ (Common Region Tending to Inconsistent), Q$\top$ (Quasi-Inconsistency) and $\top$ (Inconsistency), and on the certainty axis such as: F (Falsehood), QF (Quasi-Falsehood), RCF (Common Region of Falsehood), RCt (Common Region of Truth) and t (Truth).
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.08]{fig6.png}
\caption{Para-fuzzy stages.}\label{6}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.1]{fig7.png}
\caption{Fuzzy membership functions for the contradiction axis.}\label{7}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.1]{fig8.png}
\caption{Fuzzy membership functions for the certainty axis.}\label{8}
\end{figure}
Thereafter, by means of correlation between $D_{c}$ and $D_{ct}$, the 12 states of PAL2v (Figure \ref{5}), the pertinence functions (Figures \ref{7} and \ref{8}), and associating the corresponding PAL2v state, the inference table of step (b) was generated. This inference table is shown in Figure \ref{9}a, where the possible combinations of $D_{c}$ and $D_{ct}$, generate PAL2v states, whereas non-possible combinations are represented by ``*" in the table.
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.08]{fig9.png}
\caption{Inference rule table, outputs membership functions and crisp.}\label{9}
\end{figure}
The inference rules, from step (c), were performed considering the inference table in Figure \ref{9}a. For step (d), that corresponds to defuzzification, two Fuzzy sets of outputs were used (called output Fuzzy 1 and output Fuzzy 2, in Figure \ref{9}b), to achieve a single output value (crisp) ranging from -1 to +1, that is the result of the combination of paraconsistent logic and Fuzzy logic.
With this single output value (crisp), it becomes possible to indicate whether a proposition is in one of the 12 regions of the PAL2v lattice (Figure \ref{5}) in a refined way.
The next section presents how this combined Paraconsistent-Fuzzy System was used in the implementation of an expert system for monitoring equipment of a data network in a Brazilian electrical substation.
\section{Paraconsistent-Fuzzy (Para-Fuzzy) Expert System for Data Network Equipment Monitoring of electrical systems}\label{parafuzzy}
This section presents the expert system and subsystems that compose the data network equipment monitoring system developed in this work.
This research was developed assuming the following hardware and software\footnote{That was available in the power systems laboratory at Santa Cecília University - UNISANTA, where this work was developed.}: HP PROLIANT 360 G8 server, HP WORKSTATION Z240 server and CISCO 2911 router, and Scatex, FE500 5.4.0, CLP500 5.4.0, Wireshark, Matlab R2008a and VMware. With these tools, two test environments (denoted {\em installation 1} and {\em installation 2} below) were constructed by simulating the monitoring of a Regional Operation Centre with the basic Scatex software feature and two remote-controlled facilities. {\em Installation 1} is a Generic Hydroelectric Plant with two Acquisition and Control Units (ACU), three corporate stations and two routers (through which the outgoing network interfaces provide access to the Regional Operation Centre). {\em Installation 2} is a Generic Substation with an ACU, two corporate stations and a router. The Regional Operation Centre was composed of a Scatex Server (with its two aggregated Frontend and Watchdog servers), two UC500 / CLP500 (among them the UC500 / CLP500 Para-Fuzzy Server, where the Para-Fuzzy Expert System was installed), four corporate stations, one firewall, and two routers allowing access to the external facilities to be controlled. This setup is schematised in Figure \ref{10}.
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.06]{fig10.png}
\caption{Network architecture.}\label{10}
\end{figure}
The subnetwork B of the Regional Operation Centre (Figure \ref{10}) includes the SCADA environment behind the firewall, whereas subnetworks A, C, D and E are connected directly to the routers. Subnetworks B, C, D, G, H and J communicate in an operative segment (logical segmentation). Subnetworks A, E, F, I, and K communicate and interact with other elements in the data network cloud, and form the enterprise segment. The elements identified by the numbers 1, 2, 3, 4, and 5 are the outbound interfaces of routers that connect locations to the data network cloud, that is, all the information flow {\em in and from} a location passes through one of these interfaces. Data were collected from the network using SNMP and IEC104 channels (shown in Figure \ref{11}) that served as input to the Para-Fuzzy Expert System for data network equipment monitoring of electrical systems developed in this work. A scheme of how the expert system is integrated in the data network is shown in Figure \ref{12}.
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.055]{fig11.png}
\caption{Architecture of the expert system for data network equipment monitoring of electrical systems.}\label{11}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.06]{fig12.png}
\caption{General scheme of the Para-Fuzzy Expert System for data network equipment monitoring of electrical systems.}\label{12}
\end{figure}
The input of the Para-Fuzzy Expert System (step 1 in Figure \ref{12}) are variables representing the equipment status via communication protocols (as represented in Figure \ref{11}). The values obtained were transformed into degrees evidence by means of a Fuzzy procedure (step 2). With these degrees, the parameters $D_{c}$ and $D_{ct}$ (degrees of certainty and contradiction) are calculated (step 3), and they are used by another function (step 4), that combines paraconsistent and Fuzzy procedures, as described in Section \ref{fuzzysets}, that allows the assignment of one of the 12 PAL2v logical states for each equipment analysed. This is then used to infer the operating condition of the equipment, classified into {\em failure, unstable, normal, inconsistent} or {\em undetermined} (step 5). Information about the entire process is finally sent to the server with Scatex software, so that the automation specialist and the system operators can monitor the network (step 6).
The scheme shown in Figure \ref{12} was used in three subsystems, each of which dedicated to monitoring the: (A) servers; (B) routers' communication; and, (C) ACU connection.
Subsystem (A) receives as input the processing information, as well as the memory status and file allocation, and outputs the operating condition of the analysed server. Subsystem (B) uses the data-rate information and communication-interface errors to assign the operating condition of the analysed router. Similarly, subsystem (C) analyses a particular protocol link from an ACU, using as input the data-rate information and communication interface errors, and infers the operating condition of the analysed ACU.
Steps 2 to 4 (Figure \ref{12}) were summarised into a structure called the {\em analysis node}, this was done for code optimisation and also due to the fact that the paraconsistent-fuzzy combination analyses at most two variables at a time. Thus, in the subsystem (A) two nodes were used and in the subsystems (B) and (C) only one node was used. In the subsystem (A), one node analyses the aggregation of processing and memory, while the other analyses processing and file allocation. In the subsystem (B), the analysis was conducted with the aggregation of the data rate and errors in the router network interface, and in the subsystem (C) the analysis was done with data rate messages and the errors in the protocol connections, given in industrial format via the ACUs.
The expert subsystems run in parallel on a 5-minute loop on the UC500 / CLP500 Para-Fuzzy Server (from Figure \ref{10}). There can be a number of these subsystems for analysing a data network, one for each equipment to be monitored.
The degrees of evidence (favourable and unfavourable) were obtained from raw values using Fuzzy trapezoidal membership functions with three categories representing three levels of failures: {\em not failure}, {\em undefined}, {\em failure}. This allowed the projection of the degrees of evidence in the lattice of the PAL2v, as described in Section \ref{fp-intro}, accommodating the operating condition of the equipment into the states Q$\perp $-F, QF-$\perp $, QF-$\top $, Q$\top$-F, Q$\top $-t, Qt-$\top $, Qt-$\perp $ and Q$\perp $-t representing conditions of instability (without sufficient evidence to infer failure), t representing a condition of failure (with evidence of failure), F is a condition of normality (with evidence of non-failure), and $\perp $ and $\top $ are conditions of indeterminacy / inconsistency (evidences with a high level of contradiction).
In the experimental evaluation of the expert system (presented later in this article), the values for the trapezoidal functions and the rules of the Fuzzy procedure of step 2 were defined individually for each expert subsystem (corresponding to the input and output variables) and for each equipment. These values and rules were defined manually by the electrical-system specialist and are shown in Tables \ref{long}, \ref{long_1} and \ref{long_2} in the appendix. The other Fuzzy procedures in step 4 are those described in Section \ref{fuzzysets}, and also in \citep{inacio99}.
\section{Experimental Setup}\label{exp}
In order to evaluate the proposed expert system, simulations were carried out in MATLAB and in the software CLP500 and Scatex. In the MATLAB simulations, the subsystems described in the previous section were modelled as functions whose inputs were numerical values representing the information obtained by monitoring the various equipment. The outputs of these functions were matrices of values, one matrix for each subsystem, representing the individual operating conditions. For subsystem (A), the matrix represented values for processing, memory, file allocation, $\mu $, $\lambda $, $D_{c}$, $D_{ct}$, $crisp$ and equipment operating condition; for subsystem (B) the output matrix had rate, errors, $\mu $, $\lambda $, $D_{c}$, $D_{ct}$, $crisp$ and equipment operating condition; and for subsystem (C): rate, errors, $\mu $, $\lambda $, $D_{c}$, $D_{ct}$, $crisp$ and equipment operating condition. The value $crisp$ is the control action obtained after the defuzzification procedure (as described in Section \ref{fp-intro}).
In order to conduct the system evaluation using the CLP500 and Scatex software, a local test network was created simulating the following equipment of the architecture shown in Figure \ref{10}: the subnet B servers, the router 1 of the Generic Hydroelectric Plant and the UAC 1 of the Generic Hydroelectric Plant, which used for this test procedure the hardware and software presented at the beginning of Section \ref{parafuzzy}. The simulation of this local network was sufficient to test the three distinct expert subsystems. In this simulation, functions corresponding to the specialist subsystems were programmed and, on the Scatex Server, information displays were created. Using the connection to the local network, equipment data were collected by protocols and, then, these functions analysed the data and generated the results in tables for the UC500 / CLP500 Para-Fuzzy Server itself and on the displays of the Scatex Server. It is important to highlight that the output tables of the UC500 / CLP500 Para-Fuzzy Server had the same data pattern and they were subjected to the same operating conditions as the MATLAB simulations, allowing a consistent comparison of the results.
\section{Results}\label{res}
The results of the MATLAB simulation are described in Section \ref{comb_results}. Tests were conducted by using as input the whole range of $D_{c}$ and $D_{ct}$ values along the lattice. Two lattices were plotted as results, one representing the crisp values obtained and another highlighting the PAL2v logical state assigned. A PAL2v state assigned to the system was that associated to the highest value of the Fuzzy membership.
Section \ref{esp_ml_results} presents the MATLAB simulations of subsystems (A), (B) and (C). Similar tests with these subsystems were also performed using the CLP500 and the Scatex Server (instead of MATLAB), with the router 1 and the ACU 1 of the Generic Hydroelectric Plant (as described in Section \ref{esp_sf_results}). Each of these devices were subjected to operating conditions of normality, instability and failure.
\subsection{MATLAB simulations of the Para-Fuzzy Expert System}\label{comb_results}
Figures \ref{13} and \ref{14} show the results of the paraconsistent and Fuzzy combination presented in Section \ref{fuzzysets}. In Figure \ref{13} , for better representation of the results, the lattice was divided as follows: on the horizontal axis, $D_{c}$ assuming values from $[-1.0$ $to$ $+1.0]$, represented by $[-1.0, -0.9,$ $..., -0.1, -0.05 , +0.05, +0.1, ..., +0.9, +1.0]$, and, in the vertical axis, $D_{ct}$ the values from $[-1.0$ $to$ $+1.0]$, represented by $[-1.0, -0.9, ..., -0.1, +0.1, ..., +0.9, +1.0]$. Still, in Figure \ref{13} the values were represented with different levels of precision, with $10^{-2}$, $10^{-3}$ and $10^{-4}$, and highlighted in different colours. In Figure \ref{14} the 12 PAL2v logical states were represented with numbers 1 to 12, which represent respectively: $\perp $, Q$\perp $-F, QF-$\perp $, F, QF-$\top $, Q$\top$-F, $\top $, Q$\top $-t, Qt-$\top $, t, Qt-$\perp $ and Q$\perp $-t.
Figure \ref{13} shows the crisp values\footnote{The unique values obtained as a result of the defuzzification procedure of the paraconsistent-fuzzy combination.} for the different inputs of $D_{c}$ and $D_{ct}$. In these results, it can be seen that when achieving a value of Truth (t), at $D_{c}$ = + 1 and $D_{ct}$ = 0, the crisp values tend to +0.5. In contrast, when tending to False (F), at $D_{c}$ = -1 and $D_{ct}$ = 0, the crisp values tend to -0.5. For high contradictions, high divergence in the measured values for the same situation, Figure \ref{13} shows Inconsistency ($\top $), at $D_{c}$ = 0 and $D_{ct}$ = + 1, and the crisp tending to 0. In the case of Indeterminate ($\perp $), at $D_{c}$ = 0 and $D_{ct}$ = -1, the crisp values tend to -1 or +1, depending on whether the values come from the right (from the truth) or from the left (from the falsehood).
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.055]{fig13.png}
\caption{Crisp values in the PAL2v lattice.}\label{13}
\end{figure}
Figure \ref{14} shows the PAL2v logical states as numbers 1 to 12, for different input values of $D_{c}$ and $D_{ct}$ used in the simulation. The PAL2v logical states are decided according to highest values of the Fuzzy membership function of the combination of inputs, considering a processing interval of 5 minutes. In this figure we can observe that, for a combination of the extreme values for $D_{c}$ (1 or -1), and $D_{ct}$ (+1 or -1) we have true (t) (states with number 10 assigned, in Figure \ref{14}), false (F) (number 4), Inconsistent ($\top $) (number 7) and Indeterminate ($\perp $) (number 1).
\vspace{5mm}
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.055]{fig14.png}
\caption{Logical states (higher Fuzzy membership value) in the PAL2v lattice.}\label{14}
\end{figure}
\subsection{Results of the MATLAB simulations for the expert subsystems}\label{esp_ml_results}
Table \ref{long_paper_1} shows the test results of the MATLAB simulation for the expert subsystem (A), representing the Scatex Server (Figure \ref{11}). Three tests were performed, with the equipment operating in normal condition (test 1), unstable condition (test 2) and failure (test 3). The results for this subsystem were presented for each node (according to the structure presented in Section 3), for better understanding.
In test 1, for node 1 (label (1,1) in Table \ref{long_paper_1}) and node 2 (label (1,2) in Table \ref{long_paper_1}), a negative certainty value ($D_{c}$) was obtained (-0.8400), whereas the unfavourable evidence level ($\lambda $) was positive (+0.9200) and the crisp value was close to -0.50 (-0.499983866). These values triggered the false logical state (F) of the PAL2v lattice. In this case, the expert system considers as false the statement: ``There is a failure". Thus, confirming that the equipment is in normal working condition.
Test 2 shows the activation of the false logical state (F) for node 2 (label (2,2) in Table \ref{long_paper_1}), for values of $D_c, \lambda$ and $crisp$ similar to those obtained in test 1. However, in this case, node 1 (label (2,1)) shows values for $\mu $ and $\lambda$ that are under 0.5 and, thus, below the threshold to allow the attribution of a categorical truth or false value for the confirming, or not, the statement: ``There is a failure". Thus, with a low $D_{c}$ value (-0.1218), the triggered states were QF-$\top $ and QF-$\perp $, that correspond to the situations of ``the Near False tending to Inconsistent" and ``the Near False tending to Paracomplete". This situation was, then, considered to be between normal and failure (with crisp values not so close to -0.5 and +0.5), that is, the equipment analysed is in an unstable state.
For test 3, node 1 ((3,1) in Table \ref{long_paper_1}) shows a large positive $D_{c}$ value of +0.8400, and also a large $\mu$ (+0.9200); the crisp value was close to +0.5 (+0.499999996). This situation triggered the truth value (t), according to the PAL2v lattice. Therefore, the statement "There is a failure" is considered true, regardless of the results of node 2 ((3,2) in Table \ref{long_paper_1}).
\vspace{7.5mm}
\label{my-label-table1}
{\footnotesize
\begin{longtable}[h]{ c c c c c c}
\caption{Tests of the subsystem (A) in the analysis of the Scatex Server (MATLAB simulations)}\label{long_paper_1}\\
\textbf{\shortstack{EQUIP-\\AMENT}} & \textbf{\shortstack{TEST/ \\NODE}} & \textbf{\shortstack{ENGINEERING\\ VARIABLES \\READ}} & \textbf{\shortstack{DEGREES}} & \textbf{\shortstack{CRISP AND \\LOGICAL \\ STATE}} & \textbf{\shortstack{OPERATION \\CONDITION}} \\
\shortstack{(description)} & \shortstack{(test number,\\node \\identify)} & \shortstack{(P1,P2;M1,M2;\\A1,A2)*} & \shortstack{($\mu $, $\lambda $;\\$D_{c}$, $D_{ct}$)} & \shortstack{(crisp;\\logical state)} & \shortstack{(equipment \\condition)} \\
& & & & & \\
\hline
\endfirsthead
\multicolumn{6}{c}{Continuation of Table \ref{long_paper_1}}\\
\hline
\textbf{\shortstack{EQUIP-\\AMENT}} & \textbf{\shortstack{TEST/ \\NODE}} & \textbf{\shortstack{ENGINEERING\\ VARIABLES \\READ}} & \textbf{\shortstack{DEGREES}} & \textbf{\shortstack{CRISP AND \\logical \\ STATE}} & \textbf{\shortstack{OPERATION \\CONDITION}} \\
\shortstack{(description)} & \shortstack{(test numer,\\node \\identify)} & \shortstack{(P1,P2;M1,M2;\\A1,A2)*} & \shortstack{($\mu $, $\lambda $;\\$D_{c}$, $D_{ct}$)} & \shortstack{(crisp;\\logical state)} & \shortstack{(equipment \\condition)} \\
& & & & & \\
\hline
\endhead
\hline
\endfoot
\hline
\endlastfoot
\multirow{4}{*}{\shortstack{SCATEX \\SERVER}} & & & & & \\
& (1,1) & \shortstack{(1\%,0.8\%;20\%\\22\%;60\%,60\%)} & \shortstack{(0.0800,0.9200;\\-0.8400,0.0000)} & (-0,499983866;F) & \multirow{1}{*}{\shortstack{NORMAL}} \\ \cline{2-5}
& (1,2) & \shortstack{(1\%,0.8\%;20\%\\22\%;60\%,60\%)} & \shortstack{(0.0800,0.9200;\\-0.8400,0.0000)} & (-0,499983866;F) & \\
& & & & & \\ \cline{2-6}
& & & & & \\
\multirow{8}{*}{\shortstack{SCATEX \\SERVER}} & (2,1) & \shortstack{(28\%,28\%;85\%\\85\%;60\%,60\%)} & \shortstack{(0.4392,0.5609;\\-0.1218,0.0000)} & \shortstack{(-0,499976939;\\QF-$\perp $ / $\top $)} & \multirow{2}{*}{\shortstack{UNSTABLE}} \\ \cline{2-5}
& (2,2) & \shortstack{(28\%,28\%;85\%\\85\%;60\%,60\%)} & \shortstack{(0.0890,0.9110;\\-0.8220,0.0000)} & (-0,499979480;F) & \\
& & & & & \\ \cline{2-6}
& & & & & \\
& (3,1) & \shortstack{(10\%,10\%;95\%\\95.5\%;70\%,70\%)} & \shortstack{(0.9200,0.0800;\\0.8400,0.0000)} & (0,499999996;V) & \multirow{2}{*}{\shortstack{FAIL}} \\ \cline{2-5}
& (3,2) & \shortstack{(28\%,28\%;95\%\\95.5\%;60\%,60\%)} & \shortstack{(0.0800,0.9200;\\-0.8400,0.0000)} & (-0,499983866;F) & \\
& & & & & \\ \cline{2-6}
\end{longtable}}
* P1 - processing reading 1; P2 - processing reading 2; M1 - memory reading 1; M2 - memory reading 2; A1 - reading 1 of file allocation; A2 - reading 2 of file allocation.
Table \ref{long_paper_2} shows the test results of the expert subsystems (B) and (C) applied, respectively, to the equipment analysis of router 1 and ACU 1 from the Generic Hydroelectric Plant shown in Figure 10. In this case, three tests were also performed for each equipment representing: normal operation, instability and failure. These expert subsystems were composed of only one node, since there is only one decision, with two variables, for each equipment: rate and error. It is possible to observe in Table \ref{long_paper_2} that, similarly to subsystem (A), in the case of normal operation, $D_{c}$ had a large negative value ($< 0.75$), $\lambda$ had a large positive value ($> 0.75$), and crisp was close to -0.5. This case triggered the false (F) logical state, falsifying the proposition. In the case of failure, the opposite occurred: $D_{c}> 0.5$, $\mu > 0.5$, and crisp was close to +0.5, which triggered the True (t) logical state for the statement. Under the conditions of instability, $\mu$ and $\lambda$ were between $-0.5$ and $0.5$ that is, in these cases, it was insufficient to assign an absolute truth or false value to the proposition.
\label{my-label-table2}
{\footnotesize
\begin{longtable}[h]{ c c c c c c}
\caption{Tests for subsystems (B) and (C) in the analysis of router 1 and UAC 1 of the Generic Hydroelectric Plant (MATLAB simulations)}\label{long_paper_2}\\
\textbf{\shortstack{EQUIP-\\AMENT}} & \textbf{\shortstack{TEST/ \\NODE}} & \textbf{\shortstack{ENGINEERING\\ VARIABLES \\READ}} & \textbf{\shortstack{DEGREES}} & \textbf{\shortstack{CRISP AND \\LOGICAL \\ STATE}} & \textbf{\shortstack{OPERATION \\CONDITION}} \\
\shortstack{(description)} & \shortstack{(test number,\\node \\identify)} & \shortstack{(T1,T2;\\E1,E2)} & \shortstack{($\mu $, $\lambda $;\\$D_{c}$, $D_{ct}$)} & \shortstack{(crisp;\\logical state)} & \shortstack{(equipament \\condition)} \\
& & & & & \\
\hline
\endfirsthead
\multicolumn{6}{c}{Continuation of Table \ref{long_paper_2}}\\
\hline
\textbf{\shortstack{EQUIP-\\AMENT}} & \textbf{\shortstack{TEST/ \\NODE}} & \textbf{\shortstack{ENGINEERING\\ VARIABLES \\READ}} & \textbf{\shortstack{DEGREES}} & \textbf{\shortstack{CRISP AND \\logical \\ STATE}} & \textbf{\shortstack{OPERATION \\CONDITION}} \\
\shortstack{(description)} & \shortstack{(test numer,\\node \\identify)} & \shortstack{(T1,T2;\\E1,E2)} & \shortstack{($\mu $, $\lambda $;\\$D_{c}$, $D_{ct}$)} & \shortstack{(crisp;\\logical state)} & \shortstack{(equipament \\condition)} \\
& & & & & \\
\hline
\endhead
\hline
\endfoot
\hline
\endlastfoot
\multirow{3}{*}{\shortstack{ROUTER 1\\HYDROELEC.\\PLANT}} & (1,1) & \shortstack{(900k**,1110k;\\1,0)} & \shortstack{(0.0800,0.9200;\\-0.8400,0.0000)} & (-0,499983866;F) & \shortstack{NORMAL} \\ \cline{2-6} \vspace{5pt}
& (2,1) & \shortstack{(1300k,1300k;\\9,9)} & \shortstack{(0.3941,0.6059;\\-0.2118,0.0000)} & (-0,499978294;F) & \shortstack{UNSTABLE} \\ \cline{2-6} \vspace{5pt}
& (3,1) & \shortstack{(0k,0k;\\9,9)} & \shortstack{(0.9200,0.0800;\\0.8400,0.0000)} & (0,499999996;V) & \shortstack{FAIL} \\ \cline{1-6} \vspace{5pt}
\multirow{3}{*}{\shortstack{ACU 1\\HYDROELEC.\\PLANT}} & (1,1) & \shortstack{(1000msg***,990msg;\\0,0)} & \shortstack{(0.0800,0.9200;\\-0.8400,0.0000)} & (-0,499983866;F) & \shortstack{NORMAL} \\ \cline{2-6} \vspace{5pt}
& (2,1) & \shortstack{(1000msg,990msg;\\0,0)} & \shortstack{(0.5331,0.4669;\\0.0662,0.0000)} & (-0,499999994;F) & \shortstack{UNSTABLE} \\ \cline{2-6} \vspace{5pt}
& (3,1) & \shortstack{(31msg,30msg;\\0,0)} & \shortstack{(0.9200,0.0800;\\0.8400,0.0000)} & (0,499999996;V) & \shortstack{FAIL} \\
\end{longtable}}
** number of kbits / 300s; *** number of messages / 300s; T1 - data rate reading 1; T2 - data rate reading 2; E1 - error rate reading 1; E2 - error rate reading 2.
\vspace{2mm}
It is worth noting that all the tests presented in Tables \ref{long_paper_1} and \ref{long_paper_2} were made with readings of variables (processing, memory, etc) that did not cause high values for $D_{ct}$. If this had happened, we would have the Inconsistent condition (for $D_{ct}$ $> 0.5$) and Undetermined ( for $D_{ct}$ $<-0.5$).
\subsection{Results of of the Scatex and CLP500 implementation of the expert subsystems}\label{esp_sf_results}
Tables \ref{long_paper_3} and \ref{long_paper_4} present analogous results to the test sequence presented in the previous section, but obtained using the Scatex and CLP500 software. These systems are responsible for monitoring and analysing the equipment. They are allocated, respectively, to the servers .244 and .158, of the architecture shown in Figure \ref{10} in a real electrical installation. The results obtained in these simulations were analogous to those presented in the previous section and the system's behaviours received the same classification for similar cases, as shown in Tables \ref{long_paper_3}, \ref{long_paper_4}.
\label{my-label-table3}
{\footnotesize
\begin{longtable}[h]{ c c c c c c}
\caption{Tests of the subsystem (A) in the analysis of the Scatex Server (in Scatex and CLP500 software)}\label{long_paper_3}\\
\textbf{\shortstack{EQUIP-\\AMENT}} & \textbf{\shortstack{TEST/ \\NODE}} & \textbf{\shortstack{ENGINEERING\\ VARIABLES \\READ}} & \textbf{\shortstack{DEGREES}} & \textbf{\shortstack{CRISP AND \\LOGICAL \\ STATE}} & \textbf{\shortstack{OPERATION \\CONDITION}} \\
\shortstack{(description)} & \shortstack{(test numer,\\node \\identify)} & \shortstack{(P1,P2;M1,M2;\\A1,A2)} & \shortstack{($\mu $, $\lambda $;\\$D_{c}$, $D_{ct}$)} & \shortstack{(crisp;\\logical state)} & \shortstack{(equipament \\condition)} \\
& & & & & \\
\hline
\endfirsthead
\multicolumn{6}{c}{Continuation of Table \ref{long_paper_3}}\\
\hline
\textbf{\shortstack{EQUIP-\\AMENT}} & \textbf{\shortstack{TEST/ \\NODE}} & \textbf{\shortstack{ENGINEERING\\ VARIABLES \\READ}} & \textbf{\shortstack{DEGREES}} & \textbf{\shortstack{CRISP AND \\LOGICAL \\ STATE}} & \textbf{\shortstack{OPERATION \\CONDITION}} \\
\shortstack{(description)} & \shortstack{(test number,\\node \\identify)} & \shortstack{(P1,P2;M1,M2;\\A1,A2)*} & \shortstack{($\mu $, $\lambda $;\\$D_{c}$, $D_{ct}$)} & \shortstack{(crisp;\\logical state)} & \shortstack{(equipment \\condition)} \\
& & & & & \\
\hline
\endhead
\hline
\endfoot
\hline
\endlastfoot
\multirow{4}{*}{\shortstack{SCATEX \\SERVER}} & & & & & \\
& (1,1) & \shortstack{(1\%,0.8\%;20\%\\22\%;60\%,60\%)} & \shortstack{(0.0833,0.9166;\\-0.8333,-2.23E-08)} & (-0,499982;F) & \multirow{2}{*}{\shortstack{NORMAL}} \\ \cline{2-5}
& (1,2) & \shortstack{(1\%,0.8\%;20\%\\22\%;60\%,60\%)} & \shortstack{(0.0833,0.9166;\\-0.8333,-2.23E-08)} & (-0,499982;F) & \\
& & & & & \\ \cline{2-6}
\multirow{8}{*}{\shortstack{SCATEX \\SERVER}} & & & & & \\
& (2,1) & \shortstack{(28\%,28\%;85\%\\85\%;60\%,60\%)} & \shortstack{(0.4391,0.5609;\\-0.1218,0.0000)} & \shortstack{(-0,499979939;\\QF-$\perp $ / $\top $)} & \multirow{2}{*}{\shortstack{UNSTABLE}} \\ \cline{2-5}
& (2,2) & \shortstack{(28\%,28\%;85\%\\85\%;60\%,60\%)} & \shortstack{(0.0890,0.9110;\\-0.8220,0.0000)} & (-0,499979480;F) & \\
& & & & & \\ \cline{2-6}
& & & & & \\
& (3,1) & \shortstack{(10\%,10\%;95\%\\95.5\%;70\%,70\%)} & \shortstack{(0.9200,0.0800;\\0.8400,0.0000)} & (0,499999996;V) & \multirow{2}{*}{\shortstack{FAIL}} \\ \cline{2-5}
& (3,2) & \shortstack{(28\%,28\%;95\%\\95.5\%;60\%,60\%)} & \shortstack{(0.0800,0.9200;\\-0.8400,0.0000)} & (-0,499983866;F) & \\
& & & & & \\ \cline{2-6}
\end{longtable}}
\label{my-label-table4}
{\footnotesize
\begin{longtable}[h]{ c c c c c c}
\caption{Tests of subsystems (B) and (C) in the analysis of router 1 and UAC 1 of the Generic Hydroelectric Plant (in Scatex and CLP500 software)}\label{long_paper_4}\\
\textbf{\shortstack{EQUIP-\\AMENT}} & \textbf{\shortstack{TEST/ \\NODE}} & \textbf{\shortstack{ENGINEERING\\ VARIABLES \\READ}} & \textbf{\shortstack{DEGREES}} & \textbf{\shortstack{CRISP AND \\LOGICAL \\ STATE}} & \textbf{\shortstack{OPERATION \\CONDITION}} \\
\shortstack{(description)} & \shortstack{(test number,\\node \\identify)} & \shortstack{(T1,T2;\\E1,E2)} & \shortstack{($\mu $, $\lambda $;\\$D_{c}$, $D_{ct}$)} & \shortstack{(crisp;\\logical state)} & \shortstack{(equipment \\condition)} \\
& & & & & \\
\hline
\endfirsthead
\multicolumn{6}{c}{Continuation of Table \ref{long_paper_4}}\\
\hline
\textbf{\shortstack{EQUIP-\\AMENT}} & \textbf{\shortstack{TEST/ \\NODE}} & \textbf{\shortstack{ENGINEERING\\ VARIABLES \\READ}} & \textbf{\shortstack{DEGREES}} & \textbf{\shortstack{CRISP AND \\LOGICAL \\ STATE}} & \textbf{\shortstack{OPERATION \\CONDITION}} \\
\shortstack{(description)} & \shortstack{(test number,\\node \\identify)} & \shortstack{(T1,T2;\\E1,E2)*} & \shortstack{($\mu $, $\lambda $;\\$D_{c}$, $D_{ct}$)} & \shortstack{(crisp;\\logical state)} & \shortstack{(equipment \\condition)} \\
& & & & & \\
\hline
\endhead
\hline
\endfoot
\hline
\endlastfoot
\multirow{3}{*}{\shortstack{ROUTER 1\\HYDROELEC.\\PLANT}} & (1,1) & \shortstack{(900k,1110k;\\1,0)} & \shortstack{(0.0833,0.9166;\\-0.833333,2.23E-08)} & (-0,499982;F) & \shortstack{NORMAL} \\ \cline{2-6} \vspace{5pt}
& (2,1) & \shortstack{(1300k,1300k;\\9,9)} & \shortstack{(0.3625,0.5988;\\-0.236325,-0.03867)} & (-0,556207;F) & \shortstack{UNSTABLE} \\ \cline{2-6} \vspace{5pt}
& (3,1) & \shortstack{(0k,0k;\\9,9)} & \shortstack{(0.9200,0.0800;\\0.8400,0.0000)} & (0,499999996;V) & \shortstack{FAIL} \\ \cline{1-6} \vspace{5pt}
\multirow{3}{*}{\shortstack{ACU 1\\HYDROELEC.\\PLANT}} & (1,1) & \shortstack{(1000msg**,990msg;\\0,0)} & \shortstack{(0.0833,0.9166;\\-0.833333,2.23E-08)} & (-0,499982;F) & \shortstack{NORMAL} \\ \cline{2-6} \vspace{5pt}
& (2,1) & \shortstack{(1000msg,990msg;\\0,0)} & \shortstack{(0.5328,0.5321;\\0.0007332,0.06495)} & (-0,494548;F) & \shortstack{UNSTABLE} \\ \cline{2-6} \vspace{5pt}
& (3,1) & \shortstack{(31msg,30msg;\\0,0)} & \shortstack{(0.9166,0.0833;\\0.833333,2.23E-08)} & (0,500;V) & \shortstack{FAIL} \\
\end{longtable}}
\section{Discussion}\label{disc}
Figures \ref{13} and \ref{14} show that, as $D_{c}$ $> 0.75$, there is a gradual increase of trust in assigning the logical state true (t) to the system, with the crisp value tending to +0.5. In contrast, as $D_{c}$ $< -0.75$ the false logical state (F) is inferred, with the crisp value tending to -0.5 . Furthermore, for $D_{ct}$ $> 0.75$ or $D_{ct}$ $< -0.75$ there is the gradual acceptance of $\top $ or $\perp $, and the approximation of crisp to 0, or -1 (on the left-hand side of the diagram) and +1 (on the right-hand side of the diagram).
For $D_{c} > 0$ and $D_{c} < 0.75$, and also for $D_{ct} < 0$ and $D_{ct} > -0.75$, if $D_{c} > D_{ct}$, the logical state Qt-$\perp $ is assigned. As for the opposite, for $D_{c}>D_{ct}$, $Q\perp$-t is assigned to the system, with the crisp assuming values in the interval [0.50, 0.80]. For $D_{c} > 0$ and $D_{c} < 0.75$, and $D_{ct} > 0$ and $D_{ct} < 0.75$, with $D_{c} > D_{ct}$, the system is classified as Qt-$\top $; Q$\top $-t (with the crisp assuming values in the interval [0.20,0.50]). This occurs also when $D_{c} > 0$ and $D_{c} < 0.75$ and $D_{ct} < 0.75$, but with $D_{c} < D_{ct}$. For $D_{c} < 0$ and $D_{c} > -0.75$ (on the left side of the lattice), the opposite occurs, with the assignment of the respective logical states and crisp.
These results (in numbers and logical states) prove the correct implementation of the paraconsistent-fuzzy combination, since the values obtained in the simulations agree with whose of the general theory presented in Section \ref{fuzzysets}.
With the association of Fuzzy to the determination of logical states t, F, $\perp $ and $\top $, the affirmation or not of a proposition occurs in a two-tier way: observing the pertinence functions of the competing logical states, and defining the output based on the highest value of pertinence. This contrasts with the previous work presented in \citep{inacio99,Abe15ch} that use the Para-Analyser algorithm without the application of Fuzzy sets.
In relation to the tests of the expert subsystems in the classification of the operating condition of the different equipment tested, it was possible to see that: although there were differences in the values obtained in the MATLAB with respect to the CLP500 e Scatex implementation, the decision obtained by the expert system was consistent in both cases, for the same situations.
In this work, just a limited number of variables were used, but that is not a physical limitation, as more nodes could be implemented in the expert system. The set of variables used was selected based on the minimum number of nodes necessary for the remote supervision of a large electrical installation.
Finally, it is important to highlight that the underlying formalism assumed from the start the existence of contradictions between the readings of the engineering variables. This work has shown that, even in stated with contradictory signs, the classification was made in a consistent way. Moreover, no significant distinction was observed between the MATLAB simulation and the equivalent implementation in the softwares CLP500 and Scatex.
\section{Related Works}\label{related}
\cite{pimenta15} describes a way of analysing a communication network using PAL2v from device-network request logs. These logs were composed of four factors, which are: average response time, standard deviation, average packet size and total transactions, which were collected for five days over three periods (morning, afternoon and evening). These factors were modelled in paraconsistent logic, individually, through degrees of evidence favourable and unfavourable. As the interpretation of these factors in isolation did not lead to a satisfactory conclusion, a global analysis, using paraconsistent logic, was conducted considering the favourable and unfavourable evidence of each factor, multiplied by respective weights to finally diagnose and analyse the behaviour of the network.
\cite{pimenta16} showed a methodology with PAL2v for analysing and detecting problems in a computer network of a public university with 200 computers. In this methodology, from a proxy, parameters of the network were collected and the response time, data volume, bandwidth, number of requests and re-transmissions were calculated. These parameters were then transformed into favourable and unfavourable evidence related to anomaly propositions, which are then dealt appropriately by assigning states in a lattice, interpreting them as the behaviours of the analysed network.
\cite{pimenta18} presented a network anomaly detection methodology, which used as a source of information for analysis, logs from a proxy for a custom router, called a Squid proxy. In the information acquired by the logs, PAL2v concepts were applied to each equipment analysed, with favourable and unfavourable evidence determined according to its parameters. With the aid of a traffic analyser, it was possible to determine the behaviour of the hosts within a specific interval, daily. Considering various logical states, it was possible to modulate a general analysis corresponding to the detection of network anomalies, considering the behaviour of hosts outside the previously stipulated profile.
Similar to that carried out in \citep{pimenta15}, \citep{pimenta16} and \citep{pimenta18}, in the present paper, the measured parameters of network equipment (servers, routers and ACUs) are associated with favourable and unfavourable evidence, and the PAL2v is used for characterisation of failures in data networks. However, in addition to PAL2v, Fuzzy logic is used which provides the identification network states in a more refined way.
There is also work on the applications of PAL2v based on the traffic-profile analysis of Digital Signatures of Network Segment using Flow Analysis (DSNSF). \cite{pena14} presented a network-anomaly detection approach employing DSNSF, generated with an Autoregressive Integrated Moving Average (ARIMA) model. In addition, a functional algorithm based on PAL2v was proposed, with the objective of avoiding high false alarm rates, due to traffic variations beyond the expected profile, but identifying the behaviour of the traffic patterns that really harm the network services.
In \cite{pena17}, a methodology for detecting anomalies in computer networks by means of DSNSF was proposed with the help of PAL2v. In this study, DSNSFs are organised under two distinct models, ARIMA and Ant Colony Optimisation for Digital Signature (ACODS). Through the analysis of traffic records, each structure of a DSNSF was used as a standard for the observed traffic resources. A Correlational Paraconsistent Machine (CPM) was then created, based on LPA2v, with the objective of assimilating the DSNSFs of both models and the traffic disturbances caused by network anomalies, which by comparing them with the use of paraconsistent concepts determines the anomaly in a more adequate way, with the consideration of uncertainties in the information. The experimental results of a real assessment of traffic monitoring suggested that CPM responses indicated the possibility of improvement in anomaly detection rates.
In contrast to the work presented in \citep{pena14} and \citep{pena17}, in which the behaviour of the network traffic modelled over time is considered as the default pattern that defines normal functioning, in the present paper the network behaviour is defined within fixed ranges for all the measured parameters of each equipment in the network, taking into account the conditions of normality, instability and failure of each one, regardless of specific antecedent temporal associations and computational resources of historical data. These value ranges are defined by a specialist.
Related work on Fuzzy logic is described in \citep{olajubu13}, where a scheme for the predictive maintenance in communication networks is proposed using Fuzzy logic with real-time data monitoring. The model combines efficiency with reasonable cost, and works well in modern networks with heterogeneous elements, including: memory, processing cores and hard disk. The collected information is sent to a Data Analysis (DA) system, which analyses the collected information and advises the network administrator on the condition of the network element. The DA is modelled using the Matlab Fuzzy-logic toolbox, and the data used for the Fuzzy-logic model was collected from data sheets from different component / equipment suppliers. To verify the feasibility of the proposed model, an {\em ad-hoc} network of ten computers was created and evaluated.
In \cite{kotenco17}, a new approach was proposed for the monitoring of Network Elements (NE), in a Multi-Service Network (MSN), based on the application of diffuse logical inference, in which the need to use Fuzzy methods was due to three factors: (1) uncertainty on the causes that can result in node failures in the communication channels; (2) incomplete information about the status of NEs and MSN as a whole, which are subject to processing; (3) delay in the transmission of NE state data to the processing nodes.
A combination of Fuzzy logic with neural networks in presented in \citep{rudrusamy13}, where a Fuzzy-based diagnostic system was proposed to recognise and identify network operation anomalies, using neural network as a tuner. The focus was on building a Fuzzy system for handling decoded data packets to identify anomalies. Takagi Sugeno's Fuzzy model was used in the implementation of the system, which allowed the detection of network operation anomalies by their intensity, and the ability to choose the appropriate type of alerts. This combination of the diffuse model with the neural network minimised the number of interruptions of the network operators, relieving them from analysing possible false problems.
It is observed that Fuzzy logic has been used in the treatment and analysis of the operation and maintenance of equipment and systems in electrical network systems for a long time, as can be seen in \citep{tomsovic99} and \citep{haiwen06}. In these cases, the expert knowledge is used to create Fuzzy rules for efficient inferences, facilitating the equipment control.
Analogous to the work presented in \citep{olajubu13} and \citep{kotenco17}, the present paper used Fuzzy logic to monitor and identify network equipment failures by analysing the intrinsic resources of network elements (processing, memory and allocation of disk), basically using rules defined by specialists. Additionally, this paper uses the logic PAL2v to represent the contradiction between network parameter measurements and to conduct a real-time diagnosis of failures.
Similar to the work presented in \citep{tomsovic99} and \citep{haiwen06}, this paper used the knowledge from specialists to model critical networks of SCADA systems in the context of the remote control of electrical systems, creating Fuzzy association rules and parameterisation of membership functions, modelling equipment failures of these types of data networks.
In a more general way, the work reported in \citep{pimenta15,pimenta16,pimenta18}, \citep{pena14, pena17} and \citep{rudrusamy13} model the network traffic, and its aggregated subparameters (such as errors, latency, etc.), in a way to indicate anomalies in data networks; PAL2v is used to represent contradiction, as an additional resource for the correct indication of failure. In \citep{olajubu13,kotenco17}, it is observed that the specific resources of network equipment, such as: memory, processing, hard disk, temperature, etc., were modelled according to specialist knowledge as Fuzzy expert systems but without considering contradictions.
\section{Conclusion}\label{conclu}
\label{conclusion}
In this article, a prototype of an expert system was developed dedicated to the monitoring of data network equipment in the context of the remote control of electrical systems. This expert system treats inconsistencies explicitly, that could arise from different measurements related to data network monitoring equipment parameters. In a context where there are several monitored equipment, contradictions are common occurrences.
The expert system developed in this work is based on Fuzzy logic and Paraconsistent Annotated Logic with Annotation of Two Values and it was verified to be capable of analysing uncertainties in the readings of real engineering quantities, and generating the operating conditions for the equipment of data networks.
It is also important to note that the case studied in this article was applied to electrical energy systems. However, the techniques presented using the combination of paraconsistent annotated and Fuzzy sets can be applied to other industrial installations, in which the remote monitoring is carried out through one or more SCADA systems.
Future work should consider the monitoring of a larger set of variables in each expert subsystem than that considered in the present work, allowing the monitoring of more equipment. This would imply on the implementation of additional expert subsystems and also the use of more sources of information for the engineering quantities (processing, memory, data rate, etc). The monitoring of more variables by the expert subsystems would allow a more complete analysis of each equipment, generating a more accurate classification of the installation operating condition.
As a suggestion for a larger set of monitoring variables, one can, for example, assume for subsystem (A) the monitoring of the data rate and errors on network interfaces on servers; for the subsystem (B), the number of ICMP packets delivered and not delivered on the network interfaces, and use of memory and processing in routers; and, for the subsystem (C), the quantity of TCP messages with re-transmission and discard, and also the use of memory, processing and allocation of files in the ACUs.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,923 |
package cn.org.eshow.service;
import cn.org.eshow.bean.query.CategoryQuery;
import cn.org.eshow.common.page.Page;
import cn.org.eshow.model.Category;
import javax.jws.WebService;
import java.util.List;
@WebService
public interface CategoryManager extends GenericManager<Category, Integer> {
/**
*
* @param query
* @return
*/
List<Category> list(CategoryQuery query);
/**
*
* @param query
* @return
*/
Page<Category> search(CategoryQuery query);
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,715 |
Hanroth''' est une municipalité du Verbandsgemeinde'' Puderbach, dans l'arrondissement de Neuwied, en Rhénanie-Palatinat, dans l'ouest de l'Allemagne.
Références
Site de la municipalité de Hanroth
Commune en Rhénanie-Palatinat | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,760 |
{"url":"https:\/\/cs.stackexchange.com\/questions\/91853\/does-the-working-set-paging-algorithm-use-a-separate-page-table","text":"# Does the Working Set Paging Algorithm use a separate page table?\n\nI am doing research on paging algorithms. While learning about the working set algorithm from several scientific sources I was not really able to figure out where exactly the working set is defined or realized.\n\nI can only imagine two ways of realising this. The first is to put it into the page table file with indication bits; the second is to have another page table for the working set.\n\nTo be more specific: is the Working Set a completely different, separate list derived from the page table entries, or is it defined in the page table entries?\n\n\u2022 Are you asking about the definition of a working set itself? Can you provide a quote or two from these scientific sources to clarify the question? May 14 '18 at 1:19\n\u2022 The sources are: [Peter J. Denning - The Working Set Model for Program Behavior (1968)] (denninginstitute.com\/pjd\/PUBS\/WSModel_1968.pdf) and Andrew S. Tanenbaum - Modern Operating Systems (any edition is fine to use) May 14 '18 at 9:08\n\nDenning provides a rough definition for a working set in Section 2:\n\nRoughly speaking, a working set of pages is the minimum collection of pages that must be loaded in main memory for a process to operate efficiently, without \"unnecessary\" page faults.\n\nAnd provides the precise definition in Section 3:\n\nWe define the working set of information W(t, r) of a process at time t to be the collection of information referenced by the process during the process time interval (t - r, t).\n\nThus, the information a process has referenced during the last r seconds of its execution constitutes its working set (Figure 2).\n\nThe term \"page\" in these definitions refers to physical pages, not virtual pages. That's because only those physical pages that have resided in main memory at least once during that period of time are part of the working set of a process.\n\nis the Working Set a completely different, separate list derived from the page table entries, or is it defined in the page table entries?\n\nThe working set is not defined by the page table entries. These entries define the whole virtual address space of the process. Also, by only looking at the page table entires, it's not possible to determine or derive the working set. The pages that are resident in main memory in a particular period of time and that have been accessed by the process in that period constitute the working set for that period.\n\nEven if a page table entry includes an accessed bit (like the x86 page table entries) and even if the OS supports the accessed bits, it's still not possible to determine the working set just by looking at the page table entries. That's because the same page table entry may point to multiple physical pages during a period of time. At the end of that period, there is no way to determine all of the physical pages that any page table entry has defined during that period. So some additional data structures need to be used.\n\n\u2022 Hey, thanks for the effort for an answer. What you provided in your information was kind of clear to me already, nevertheless thank you for your information. There is actually another Bit used for the Virtual\/CPU Time, giving the OS an idea if a page belongs to the recent working set. I am just not sure if this analysis of the virtual time is enough for the OS to know if such pages belong to the working set, or if it keeps track of all the working sets in a seperate list? May 14 '18 at 17:52\n\u2022 @ThomasChristopherDavies What other bit? What do you mean by the virtual time? May 14 '18 at 17:56\n\u2022 According to Carr and Hennesy[81] WSClock \u00b4The virtual time VT of a task is the number of references that have been completed for that task\u00b4 And \u00b4Typically, we require (1) a task's virtual time VT, (2) the last reference time LR(p) for each page, and (3) a procedure to detect pages for which VT-LR(p) >= 0\u00b4. May 15 '18 at 9:51\n\nI spoke to my professor in operating systems about this matter. He told me, that it is up to the designer of the operating system how he realises the concept of the working set.\n\nThe working set is defined by the page table entries. The page table consists of the part for adressing the virtual and the possibly according physical adress and some extra bits, which could differ on different operating systems. Those bits like the \"dirty\"-bit, the valid-bit or the read-bit help to implement the paging algorithm.\n\nIf the working set is used in an operating system, there is an actual need of the virtual time in the page table entry. How this entry in the page table is used is up to the designer of the operating system. The working set could be realised by scanning the page table and finding the entries which by comparing the virtual time of use of the page to the virtual time of the process. Another approach can be to create lists for the complete working set or for different parts of the working set (like read-entries, modified-entries).","date":"2021-11-27 16:40:54","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\": 0.6293306946754456, \"perplexity\": 841.9239417759787}, \"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-2021-49\/segments\/1637964358208.31\/warc\/CC-MAIN-20211127163427-20211127193427-00174.warc.gz\"}"} | null | null |
Q: javascript code written in separate to the html markup what's wrong with this code, i have included the file but when i run it doesn't give the result i intended(customized design of video player),any one who is aware about javascript can help.......
function dofirst() {
//get values of buttons
var movie = document.getElementById("movie");
var playOrpause = document.getElementById("playOrpause");
var mute = document.getElementById("mute");
var fullscreen = document.getElementById("fullscreen");
//get values of sliders
var seekbar = document.getElementById("seekbar");
var myvolume = document.getElementById("volume");
//add the event listeners to buttons
playOrpause.addEventListener('click', playme, false);
mute.addEventListener('click', mute_me, false);
fullscreen.addEventListener('click', scree_full, false);
//add the event listener to sliders
seekbar.addEventListener('change', change_me, false);
seekbar.addEventListener('timeupdate', update_me, false);
seekbar.addEventListener('mousedown', mous_down, false);
seekbar.addEventListener('mouseup', mous_up, false);
myvolume.addEventListener('change', volume_up, false);
}
//the functions of play button
function playme() {
if (movie.paused == true) {
movie.play();
//update button status
playOrpause.innerHTML = 'pause';
else {
movie.pause();
//update button status
playOrpause.innerHTML = 'play';
}
}
}
//the functions of mute button
function mute_me() {
if (movie.muted == false) {
movie.muted = true;
//update button status
mute.innerHTML = 'unmute';
} else {
movie.muted = false;
//update button status
mute.innerHTML = 'mute';
}
}
//the functions of fullscreen button
function scree_full() {
if (movie.requestFullscreen) {
movie.requestFullscreen();
}
//for mozilla firefox browser
else if (movie.mozRequestFullscreen) {
movie.mozRequestFullscreen();
}
//for google chrome browsers
else if (movie.webkitRequestFullscreen) {
movie.webkitRequestFullscreen();
}
}
//the functions for seekbar
function change_me() {
//calculate current time of the video
var time = movie.duration * (seekbar.value / 100);
//update the current time of the video
movie.currentTime = time;
}
//update the seekbar when the video plays
function update_me() {
var size = movie.currentTime * (100 / movie.duration);
//update the size of the seekbar
seekbar.value = size;
}
//pause the video when the seekbar is dragged
function mous_down() {
movie.pause();
}
//play the video when the seekbar is dropped
function mous_up() {
movie.play();
}
//the function for the volume bar
function volume_up() {
movie.volume = myvolume.value;
}
window.addEventListener("load", dofirst, false);
A: Make sure that there are no syntax errors in your code....For instance, you seem to be using 'mous_up()' instead of 'mouse_up()'. From my own experience, javascript code doesn't run when there are any syntax errors.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 971 |
For most drivers that have their privileges suspended to operate a motor vehicle pending prosecution of their case, they can apply to the court for a hardship license.
In many instances the judge will grant a hardship license. However, a hardship license will not be valid for the operation of a commercial motor vehicle.
PreviousPrevious post:What Do New York Police Officers Look For After Pulling Someone Over For Drunk Driving?NextNext post:Can I Still Win My Case If I Refused The Breath Test? | {
"redpajama_set_name": "RedPajamaC4"
} | 3,021 |
{"url":"http:\/\/ia.cr\/cryptodb\/data\/author.php?authorkey=154","text":"CryptoDB\n\nPhong Q. Nguyen\n\nPublications\n\nYear\nVenue\nTitle\n2020\nCRYPTO\nWe show how to generalize Gama and Nguyen's slide reduction algorithm [STOC '08] for solving the approximate Shortest Vector Problem over lattices (SVP) to allow for arbitrary block sizes, rather than just block sizes that divide the rank n of the lattice. This leads to significantly better running times for most approximation factors. We accomplish this by combining slide reduction with the DBKZ algorithm of Micciancio and Walter [Eurocrypt '16]. We also show a different algorithm that works when the block size is quite large---at least half the total rank. This yields the first non-trivial algorithm for sublinear approximation factors. Together with some additional optimizations, these results yield significantly faster provably correct algorithms for \\delta-approximate SVP for all approximation factors n^{1\/2+\\eps} \\leq \\delta \\leq n^{O(1)}, which is the regime most relevant for cryptography. For the specific values of \\delta = n^{1-\\eps} and \\delta = n^{2-\\eps}, we improve the exponent in the running time by a factor of 2 and a factor of 1.5 respectively.\n2018\nCRYPTO\nAt Eurocrypt \u201910, Gama, Nguyen and Regev introduced lattice enumeration with extreme pruning: this algorithm is implemented in state-of-the-art lattice reduction software and used in challenge records. They showed that extreme pruning provided an exponential speed-up over full enumeration. However, no limit on its efficiency was known, which was problematic for long-term security estimates of lattice-based cryptosystems. We prove the first lower bounds on lattice enumeration with extreme pruning: if the success probability is lower bounded, we can lower bound the global running time taken by extreme pruning. Our results are based on geometric properties of cylinder intersections and some form of isoperimetry. We discuss their impact on lattice security estimates.\n2018\nASIACRYPT\nEnumeration is a fundamental lattice algorithm. We show how to speed up enumeration on a quantum computer, which affects the security estimates of several lattice-based submissions to NIST: if T is the number of operations of enumeration, our quantum enumeration runs in roughly $\\sqrt{T}$ operations. This applies to the two most efficient forms of enumeration known in the extreme pruning setting: cylinder pruning but also discrete pruning introduced at Eurocrypt \u201917. Our results are based on recent quantum tree algorithms by Montanaro and Ambainis-Kokainis. The discrete pruning case requires a crucial tweak: we modify the preprocessing so that the running time can be rigorously proved to be essentially optimal, which was the main open problem in discrete pruning. We also introduce another tweak to solve the more general problem of finding close lattice vectors.\n2017\nEUROCRYPT\n2016\nEUROCRYPT\n2015\nPKC\n2014\nPKC\n2012\nEUROCRYPT\n2012\nASIACRYPT\n2012\nASIACRYPT\n2011\nEUROCRYPT\n2011\nCHES\n2011\nASIACRYPT\n2010\nEUROCRYPT\n2009\nASIACRYPT\n2009\nJOFC\n2009\nCRYPTO\n2008\nEUROCRYPT\n2007\nCRYPTO\n2007\nPKC\n2006\nCRYPTO\n2006\nEUROCRYPT\n2006\nEUROCRYPT\n2005\nASIACRYPT\n2005\nEUROCRYPT\n2005\nFSE\n2005\nPKC\n2004\nEUROCRYPT\n2003\nCRYPTO\n2002\nASIACRYPT\n2002\nCRYPTO\n2002\nJOFC\n2001\nASIACRYPT\n2000\nASIACRYPT\n2000\nASIACRYPT\n2000\nEUROCRYPT\n1999\nCRYPTO\n1999\nCRYPTO\n1999\nPKC\n1998\nASIACRYPT\n1998\nCRYPTO\n1997\nCRYPTO\n\nProgram Committees\n\nPKC 2016\nCrypto 2016\nEurocrypt 2014 (Program chair)\nAsiacrypt 2013\nEurocrypt 2013 (Program chair)\nAsiacrypt 2012\nAsiacrypt 2011\nCrypto 2011\nAsiacrypt 2010\nPKC 2010 (Program chair)\nTCC 2010\nCrypto 2009\nAsiacrypt 2009\nEurocrypt 2008\nEurocrypt 2007\nCrypto 2006\nAsiacrypt 2005\nEurocrypt 2005\nPKC 2004\nAsiacrypt 2003\nAsiacrypt 2002\nEurocrypt 2002","date":"2022-05-20 01:38:29","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\": 0.9148560762405396, \"perplexity\": 2119.341096719096}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662530553.34\/warc\/CC-MAIN-20220519235259-20220520025259-00749.warc.gz\"}"} | null | null |
Vesislav Ilchev (born 27 May 1977) is a Bulgarian football defender, who currently plays for Master Burgas in the Bulgarian V Football Group.There's a wife and child Rosica and Simona.
External links
Player Profile at Burgas24
Bulgarian footballers
1977 births
Living people
Association football defenders
FC Chernomorets Burgas players
First Professional Football League (Bulgaria) players | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,030 |
'use strict';
define(['app'], function (app) {
app.factory('Auth', [
'$location',
'$rootScope',
'Session',
'UsersFactory',
'$cookieStore',
function ($location, $rootScope, Session, UsersFactory, $cookieStore) {
/* Get currentUser from cookie
=================================================================== */
$rootScope.currentUser = $cookieStore.get('user') || null;
//$cookieStore.remove('user');
return {
/* Authenticate user
=================================================================== */
login: function (user, callback) {
var cb = callback || angular.noop;
return Session.save({
username: user.username,
password: user.password
}, function (user) {
$rootScope.currentUser = user;
return cb();
}, function (err) {
return cb(err);
}).$promise;
},
/* Unauthenticate user
=================================================================== */
logout: function (callback) {
var cb = callback || angular.noop;
return Session.delete(
function () {
$cookieStore.remove('user');
$rootScope.currentUser = null;
return cb();
},
function (err) {
return cb(err);
}
).$promise;
},
/* Create a new user
=================================================================== */
createUser: function (user, callback) {
var cb = callback || angular.noop;
return UsersFactory.save(user,
function (user) {
$rootScope.currentUser = user;
return cb(user);
},
function (err) {
return cb(err);
}).$promise;
},
/* Change password
=================================================================== */
changePassword: function (oldPassword, newPassword, callback) {
var cb = callback || angular.noop;
return UsersFactory.update({
oldPassword: oldPassword,
newPassword: newPassword
}, function (user) {
return cb(user);
}, function (err) {
return cb(err);
}).$promise;
},
/* Gets all available info on authenticated user
=================================================================== */
currentUser: function () {
return UsersFactory.get();
},
/* Simple check to see if a user is logged in
=================================================================== */
isLoggedIn: function () {
var user = $rootScope.currentUser;
return !!user;
}
};
}
]);
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 1,084 |
\section{Introduction}
A general result on the existence and uniqueness of the solution for multidimensional, time dependent,
stochastic differential equations (SDEs) driven by a fractional Brownian motion (fBm) with Hurst parameter $H>\frac{1}{2}$
has been given by Nualart and R\u{a}\c{s}canu in~\cite{pl} using a techniques of the classical fractional calculus.
The notion of viable trajectories, used in the theory
of deterministic and stochastic differential equations, refers to those trajectories which
remain at any time in a fixed subset of the state space. The viability is to find necessary and sufficient conditions such that
a fixed subset is viable for the differential equation.
In the theory of viable solutions the concept of the tangent sets and contingent sets play
a fundamental role. In fact, the pioneering theorem, proved in 1942 by
Nagumo, gives a criterion of the viability in terms
of contingent sets. Namely, the Nagumo theorem states that if $f$ is a
bounded, continuous map from a closed subset $K$ of $\mathbb{R}^m$ to $\mathbb{R}^m$, then a
necessary and sufficient condition such that $K$ is viable for the differential equation
\[x^{\prime}(t) = f(x(t)), \quad x(0) = x_{0}\in K.
\]
is that
\[\langle f(x),p\rangle\leq0,\quad\forall\text{~}x\in K\text{ and
\forall\text{~}p~\;\text{a normal vector at}\;K\text{ in }x.
\]
Various generalizations of the Nagumo theorem provide viability conditions
in terms of contingent cones (see for instance ~\cite{pa} Th. 1, p. 191).
Viability and invariance with respect to It\^{o} equations have been investigated
first by J.-P. Aubin and G. Da Prato in~\cite{pc}.
Criterions for the viability and invariance of closed and convex subset of $\mathbb{R}^m$, given
in~\cite{pc}, are expressed in terms of stochastic contingent sets. Their results were generalized to arbitrary subsets
(which can also be time-dependent and random) in~\cite{pj}.
Another approach has been developed by Buckdahn, Peng, Quincampoix, Rainer and R\u{a}\c{s}canu in~\cite{pe},~\cite{pf},~\cite{pg},~\cite{ph}.
The main point of their work consist in proving that the viability property for SDE and
also for backward SDE holds true if and only if the square of the distance to the constraint sets is a viscosity supersolution(subsolution)
of the Hamilton-Jacobi-Bellman equation associated.
With respect to the SDE driven by fBm, I. Ciotir and {A.R\u{a}\c{s}canu} proved a type of Nagumo Theorem on viability properties of close bounded
subsets with respect to a stochastic differential equation driven by fractional Brownian motion in~\cite{pi}.
Conditions expressed by stochastic contingent sets which are given in~\cite{pi} are general but unfortunately
not easy to check and the aim of the present paper is to give
checkable conditions for general stochastic differential equation driven by the fractional Brownian
motion and some particular sets $K$.
Studying from~\cite{pi}, we find the deterministic necessary and sufficient conditions that guarantee
that the solution of a stochastic differential equation driven by the fractional Brownian motion $B^{H}$ with Hurst parameter $\frac{1}{2}<H<1$
(in short: f-SDE), $\mathbb{P}$-$a.s. \omega\in\Omega$
\[
X_{s}^{t,x}=x+{\displaystyle\int_{t}^{s}}b(r,X_{r}^{t,x})dr+{\displaystyle\int_{t}^{s}}\sigma(r,X_{r}^{t,x})dB_{r}^{H
,\quad s\in[t,T],
\]
$(t,x)\in[0,T]\times\mathbb{R}^d$ envolves in some particular sets $K$ i.e. under which it holds that for all $t\in[0,T]$
and for all $x\in K$:
\[
X_{s}^{t,x}\in K \quad a.s. \omega\in\Omega, \quad\forall s\in[t,T] .
\]
Here
\begin{itemize}
\item $B^{H}=\left\{ B^{H}_{t},t\geq0\right\} $ is a
fractional Brownian motion with Hurst parameter $\frac{1}{2}<H<1$, and the integral with respect to $B^{H}$ is a pathwise Riemann-Stieltjes integral;
\item
$b(t,x) :\left[ 0,T\right] \times
\mathbb{R}^{d}\mathbb{\rightarrow}\mathbb{R}^{d}$ and
$\sigma(t,x) :\left[ 0,T\right] \times\mathbb{R
^{d}\mathbb{\rightarrow R}^{d}$ are continuous functions.
\end{itemize}
The characterization of viability of $K$ is obtained through the study of the direct and inverse images for fractional stochastic tangent sets.
This idea comes from~\cite{pc}. In fact we extend the direct and inverse images of stochastic tangent sets to the fractional form and
using our main theorem \ref{th3.2},
we character the viability of some particular sets $K$ with the conditions on $b$ and $\sigma$ and we also obtain a comparison theorem.
We now explain how the paper is organized. In the second section,
we recall some classical definitions and the assumptions on the coefficients supposed to hold.
we also recall the main result in~\cite{pi}, which we will use later.
In section 3 we state our main result and some applications are given.
The section 4 is devoted to the proof of the main result and section 5 is for the proof of a general comparison theorem.
\section{Preliminaries}
Consider the equation on $\mathbb{R}^{d}$
\begin{equation}\label{equ2.2}
X_{s}=X_{0}+{\displaystyle\int_{0}^{s}}b(r,X_{r})dr+{\displaystyle\int_{0}^{s}}\sigma(r,X_{r})dB_{r}^{H}
,\quad s\in[0,T],
\end{equation}
\begin{itemize}
\item $B^{H}=\left\{ B^{H}_{t},t\geq0\right\} $ is a
fractional Brownian motion defined on a complete probability space $(\Omega
,\mathcal{F},\mathbb{P)};$ with Hurst parameter $\frac{1
{2}<H<1$, and the integral with respect to $B^{H}$ is a pathwise Riemann-Stieltjes integral;
\item $X_{0}$ is a $d$ - dimensional random variable.
\item $b : [0,T] \times
\mathbb{R}^{d}\mathbb{\rightarrow}\mathbb{R}^{d}$, $\sigma : [0,T] \times\mathbb{R
^{d}\mathbb{\rightarrow R}^{d}$ are continuous functions.
\end{itemize}
Remark that the fractional Brownian motion has the following property:
For every $0<\varepsilon<H$ and $T>0$ there exists a positive random variable $\eta_{\varepsilon,T}$ such that
$\mathbb{E}(|\eta_{\varepsilon,T}|^{p})<\infty$, for all $p\in[1,\infty)$ and for all $s,t\in[0,T]$
\[
|B^{H}(t)-B^{H}(s)|\leq\eta_{\varepsilon,T}|t-s|^{H-\varepsilon} \quad a.s.
\]
And from~\cite{pm} proposition 1.7.1(see also in \cite{pn}), we have for every $t_{0}\in[0,+\infty)$,
\begin{equation}\label{equ2}
\mathbb{P}\left\{\limsup_{t\rightarrow t_{0},~t\ge
t_{0}}\Big|\frac{B^{H}(t)-B^{H}(t_{0})}{t-t_{0}}\Big|=+\infty\right\}=1
\end{equation}
Using the same method we can easily proof that
\begin{equation}\label{equ3}
\mathbb{P}\left\{\limsup_{t\rightarrow t_{0},~t\ge
t_{0}}\frac{B^{H}(t)-B^{H}(t_{0})}{t-t_{0}}=+\infty\right\}=\mathbb{P}\left\{\liminf_{t\rightarrow
t_{0},~t\ge
t_{0}}\frac{B^{H}(t)-B^{H}(t_{0})}{t-t_{0}}=-\infty\right\}=\frac{1}{2}
\quad .
\end{equation}
\subsection{Assumptions and Notations}
For the function and coefficients appearing in the equation (\ref{equ2.2}),
we make the following standard assumptions which we will use throughout the paper:
\begin{description}
\item[$\left( \mathbf{H}_{1}\right) $]$\sigma(t,x)$ is differentiable in $x\in\mathbb{R}^{d}$,
and there exist some constants $\beta, \delta, 0<\beta,\delta\leq1$, and for every $R>0$ there exists $M_{R}>0$
such that the following properties hold for all $t\in[0,T]$,
\[
(H_{\sigma}):\;\left\{
\begin{array}
[c]{rl
i)\quad & \left\vert \sigma(t,x)-\sigma(s,y)\right\vert \leq M_{0}\left(
|t-s|^{\beta}+|x-y|\right) ,\quad\forall x,y\in\mathbb{R}^{d},\medskip\\
ii)\quad & |\nabla_{x}\sigma(t,y)-\nabla_{x}\sigma(s,z)|\leq M_{R}\left(
|t-s|^{\beta}+|y-z|^{\delta}\right) ,\quad\forall\left\vert y\right\vert
,\left\vert z\right\vert \leq R,
\end{array}
\right.
\]
where $\nabla_{x}\sigma(t,x)=(\nabla_{x}\sigma^{i}(t,x))_{i=\overline{1,d}}$ and
\[
|\nabla_{x}\sigma(t,x)|^{2}=\sum_{l=1}^{d}\sum_{i=1}^{d}|\partial_{x_{l}}\sigma^{i}(t,x)|^2
\]
Remark that for all $x\in\mathbb{R}^{d}$
\[
|\sigma(t,x)|\leq|\sigma(0,0)|+M_{0}(|t|^{\beta}+|x|)\leq M_{0,T}(1+|x|)
\]
where $M_{0,T}=|\sigma(0,0)|+M_{0}+M_{0}T$.
Let
\[
\alpha_{0}=\min\left\{ \frac{1}{2},\beta,\frac{\delta}{1+\delta}\right\} .
\]
\item[$\left( \mathbf{H}_{2}\right) $] There exist $\mu\in(1-\alpha_{0},1]$
and for every $R\geq0$ there exists $L_{R}>0$ such that the following properties hold for all $t\in\left[
0,T\right] ,$
\[
(H_{b}):\;\left\{
\begin{array}[c]{rl}
i)\quad & \left\vert b(r,x)-b(s,y)\right\vert \leq L_{R}\left(|r-s|^{\mu}+|x-y|\right) ,\quad\quad\forall\left\vert x\right\vert
,\left\vert y\right\vert \leq R,\medskip\\
ii)\quad & \left\vert b(t,x)\right\vert \,\leq\,L_{0}(1+|x|),\quad\forall
x\in\mathbb{R}^{d}.
\end{array}
\right.
\]
\end{description}
Finally, we introduce some notations which will be used later.
Let $d,k\in\mathbb{N}^{*}$. Given a matrix $A=(a^{i,j})_{d\times k}$
and a vector $y=(y^{i})_{d\times 1}$, we denote
$|A|^{2}=\sum_{i,j}|a^{i,j}|^{2}$ and $|y|=\sum_{i}|y^{i}|^{2}$.
\medskip
Let $t\in\left[ 0,T\right] $ be fixed. Denote
\begin{itemize}
\item $W^{\alpha,\infty}(t,T;\mathbb{R}^{d}),$ $0<\alpha<1,$ the space of
continuous functions $f:[t,T]\rightarrow\mathbb{R}^{d}$ such tha
\[
\left\Vert f\right\Vert _{\alpha,\infty;\left[ t,T\right] }:=\sup
_{s\in\lbrack t,T]}\left( |f(s)|+\int_{t}^{s}\dfrac{\left\vert f\left(
s\right) -f\left( r\right) \right\vert }{\left( s-r\right) ^{\alpha+1
}dr\right) <\infty.
\]
An equivalent norm can be defined b
\[
\left\Vert f\right\Vert
_{\alpha,\lambda;\left[ t,T\right] }:=\sup_{s\in\lbrack
t,T]}e^{-\lambda s}\left( |f(s)|
{\displaystyle\int_{t}^{s}}
\dfrac{\left\vert f\left( s\right) -f\left( r\right) \right\vert }{\left(
s-r\right) ^{\alpha+1}}dr\right)\quad\forall\lambda\geq0.
\]
\item $\tilde{W}^{1-\alpha,\infty}(t,T;\mathbb{R}^{d}),$ $0<\alpha<\frac{1}{2}.$ the space of
continuous functions $g:[t,T]\rightarrow\mathbb{R}^{d}$ such that
\[
\left\Vert g\right\Vert _{\tilde{W}^{1-\alpha,\infty}(t,T;\mathbb{R}^{d
)}:=\left\vert g\left( t\right) \right\vert +\sup_{t<r<s<T}\left(
\frac{|g(s)-g(r)|}{(s-r)^{1-\alpha}}+\int_{r}^{s}\frac{|g(y)-g(r)|
{(y-r)^{2-\alpha}}dy\right) <\infty.
\]
\item $C^\mu([t,T];\mathbb{R}^d)$, $0<\mu<1$, the space of $\mu$-H\"{o}lder continuous functions $f:[t,T]\to\mathbb{R}^d$, equipped with the norm
\[\left\Vert f\right\Vert
_{\mu;\left[ t,T\right] }:=\left\Vert f\right\Vert
_{\infty;\left[ t,T\right] }+\sup_{t\leq r<s\leq T}\dfrac{\left\vert f\left( s\right) -f\left( r\right) \right\vert }{\left(
s-r\right) ^{\mu}}<\infty
\]
where $\left\Vert f\right\Vert_{\infty;\left[ t,T\right]}:=\sup_{s\in[t,T]}\lvert f(s)\rvert$. We have, for all $0<\varepsilon<\alpha$
\[C^{\alpha+\varepsilon}([t,T];\mathbb{R}^d)\subset W^{\alpha,\infty}(t,T;\mathbb{R}^{d})
\]
\end{itemize}
\begin{itemize}
\item $W^{\alpha,1}(t,T;\mathbb{R}^{d})$ the space of measurable functions $f$
on $[t,T]$ such that
\[
\left\Vert f\right\Vert _{\alpha,1;\left[ t,T\right] }:=
\int_{t}^{T}\left[ \frac{|f(s)|}{\left( s-t\right) ^{\alpha}}+\int_{t
^{s}\frac{|f(s)-f(y)|}{(s-y)^{\alpha+1}}dy\right] ds<\infty.
\]
Clearly
\[
W^{\alpha,\infty}(t,T;\mathbb{R}^{d})\subset W^{\alpha,1}(t,T;\mathbb{R
^{d}).
\]
\end{itemize}
\subsection{Generalized Stieltjes integral}
Denoting
\[
\Lambda_{\alpha}(g;\left[ t,T\right] ):=\frac{1
{\Gamma(1-\alpha)}\sup_{t<r<s<T}\left\vert \left( D_{s-}^{1-\alpha
g_{s-}\right) (r)\right\vert .
\]
where
\[
\Gamma(\alpha)=\int_{0}^{\infty}s^{\alpha-1}e^{-s}ds
\]
is the Gamma function and
\[(D_{s-}^{1-\alpha}g_{s-})(r)=\frac{e^{i\pi(1-\alpha)}}{\Gamma(\alpha)}\left(
\frac{g(s)-g(r)}{(s-r)^{1-\alpha}}+(1-\alpha)\int_{r}^{s}\frac{g(r)-g(y)
{(y-r)^{2-\alpha}}dy\right)1_{(t,s)}(r).
\]
we have
\[
\Lambda_{\alpha}(g;\left[ t,T\right] )\leq\frac{1}
{\Gamma(1-\alpha)\Gamma(\alpha)}\left\Vert g\right\Vert _{\tilde{W}^{1-\alpha,\infty}(t,T;\mathbb{R}^{d}
)}
\]
Note that
\[
\Lambda_{\alpha}(g;\left[ t,T\right] )\leq\Lambda_{\alpha}(g;\left[ 0,T\right] )\Big(:=\Lambda_{\alpha}(g)\Big).
\]
We also introduce the notation
\[
(D_{t+}^{\alpha}f)(r)=\frac{1}{\Gamma(1-\alpha)}\left(
\frac{f(r)}{(r-t)^{\alpha}}+\alpha\int_{t}^{r}\frac{f(r)-f(y)
{(r-y)^{\alpha+1}}dy\right)1_{(t,T)}(r).
\]
\begin{definition}\label{def2.1}
Let $0<\alpha<\frac{1}{2}$. If $f\in W^{\alpha,1}(t,T;\mathbb{R}^{d\times k})$ and
$g\in\tilde{W}^{1-\alpha,\infty}(t,T;\mathbb{R}^{k})$, then defining
\[
\int_{t}^{s}f\left( r\right) dg\left( r\right) :=\left(
-1\right) ^{\alpha}\int_{t}^{s}\left( D_{t+}^{\alpha}f\right) (r)\left(
D_{s-}^{1-\alpha}g_{s-}\right) \left( r\right) dr.
\]
the integral
{\displaystyle\int_{t}^{s}}
fdg
$ exists for all $s\in[t,T]$ and
\begin{eqnarray*}
\left\vert
{\displaystyle\int_{t}^{T}}
f\left( r\right) dg\left( r\right) \right\vert
& \leq &
\sup_{t\leq r<s\leq T}\lvert(D_{s-}^{1-\alpha}g_{s-})(r)\rvert
{\displaystyle\int_{t}^{T}}
\lvert(D_{t+}^{\alpha}f)(s)ds\rvert{}
\nonumber\bigskip \\
& \leq & {}
\Lambda_{\alpha
}(g;\left[ t,T\right] )\left\Vert f\right\Vert _{\alpha,1;\left[
t,T\right] .}
\end{eqnarray*}
\end{definition}
It is known that when $H\in(\frac{1}{2},1)$ and $1-H<\alpha<\frac{1}{2}$, then the random variable
\[
G=\Lambda_{\alpha}(
B^{H})=\frac{1}{\Gamma(1-\alpha)}\sup_{t<s<r<T}|(D_{r-}^{1-\alpha}B_{r-})(s)|
\]
has moments of all order. As a consequence, if $u=\{u_{t}, t\in[0,T]\}$ is a
stochastic process whose trajectories belong to the space $W^{\alpha,1}(t,T;\mathbb{R}^{d})$, with $1-H<\alpha<\frac{1}{2}$,
the pathwise integral $
{\displaystyle\int_{0}^{T}}
u_{s}dB_{s}^{H}$ exists in the sense of Definition \ref{def2.1} and we have the estimate
\[
\Big|
{\displaystyle\int_{0}^{T}}
u_{s}dB^{H}_{s}\Big|
\leq G\|u\|_{\alpha,1}.
\]
This is the reason why in the SDE (\ref{equ2.2}) the integral with respect to $B^{H}$ is a pathwise Riemann-Stieltjes integral.
D. Nualart and\ A. R\u{a}\c{s}canu have proved in~\cite{pl}
that under the assumpations $\left(
\mathbf{H}_{1}\right) $ and $\left( \mathbf{H}_{2}\right) ,$ with $\beta
>1-H$ and $\delta>\frac{1}{H}-1$ the SDE
\begin{equation*}
X_{s}^{t,\xi}=\xi+\int_{t}^{s}b(r,X_{r}^{t,\xi})dr+\int_{t}^{s}\sigma\left(
r,X_{r}^{t,\xi}\right) dB^{H}_{r},\,\;s\in\left[ t,T\right] ,
\end{equation*}
has a unique solution $X^{t,\xi}\in L^{0}\left( \Omega
,\mathcal{F},\mathbb{P\,};W^{\alpha,\infty}(t,T;\mathbb{R}^{d})\right)
,$ for all $\alpha\in\left( 1-H,\alpha_{0}\right) .$
Moreover, for $\mathbb{P}$-almost all $\omega\in\Omega$,
$X\left( \omega,\cdot\right) \in C^{1-\alpha}\left( 0,T;\mathbb{R}^{d}\right) .$
\subsection{Fractional Viability}
In this subsection we recall the notion of the viability property for SDE driven by fractional Brownian motion.
On the other hand we will present the main result of~\cite{pi} which is very useful for our results.
Consider the stochastic differential equation driven by fractional Brownian motion $B^{H}$ with Hurst parameter $\frac{1}{2}<H<1$,
\begin{equation}\label{equ2.3}
X_{s}^{t,x}=x+{\displaystyle\int_{t}^{s}}b(r,X_{r}^{t,x})dr+{\displaystyle\int_{t}^{s}}\sigma(r,X_{r}^{t,x})dB_{r}^{H}
,\quad s\in[t,T].
\end{equation}
\begin{definition}
Let $\mathcal{K}=\{ K(t) :t\in[0,T]\}$
be a family of subsets of $\mathbb{R}^{d}$. We say that $\mathcal{K}$ is viable (weak invariant) for the equation
(\ref{equ2.3}) if, starting at any time $t\in[0,T]$ and
from any point $x\in K(t)$, there exists at least one of its solutions $\{
X_{s}^{t,x}:s\in[t,T]\}$ which satisfies
\[X_{s}^{t,x}\in K( s) \quad \text{for all}\quad s\in[t,T].
\]
\end{definition}
\begin{definition}
The family $\mathcal{K}$ is invariant (strong invariant) for the equation
(\ref{equ2.3}) if, for any $t\in[0,T] $ and for any starting point $x\in K(t)$, all solutions
$\{ X_{s}^{t,x}:s\in[t,T] \} $ of the fractional stochastic differential equation
(\ref{equ2.3}) have the property
\[
X_{s}^{t,x}\in K(s) \quad \text{for all} \quad s\in[t,T] .
\]
\end{definition}
Remark that, in the case when the equation has a unique solution (which is the case for equation (\ref{equ2.3})
under the assumptions $(\mathbf{H}_{1})$ and $(\mathbf{H}_{2})$), viability is equivalent to invariance.
\smallskip
Assuming that the mappings $b$ and $\sigma$ from the equation (\ref{equ2.3}) satisfy $(\mathbf{H}_{1})$ and $(\mathbf{H}_{2})$.
\begin{definition}
Let $t\in\left[ 0,T\right] $ and
$x\in K\left( t\right) .$ Let $\frac{1}{2}<1-\alpha<H.$
\medskip
The $\left( 1-\alpha\right) $-fractional $B^{H}$-contingent set to
$K\left( t\right) $ in $x$ is the set of the pairs $(u,v)$, such
that there exist random variable $\bar{h}=\bar{h}^{t,x}>0$ and a
stochastic process $Q=Q^{t,x}:\Omega\times\left[ t,t+\bar{h}\right]
\rightarrow \mathbb{R}^{d}$, and for every $R>0$ with $\left\vert
x\right\vert \leq R$ there exist two random variables
$H_{R},\tilde{H}_{R}>0$ independent of $(t,\bar{h})$ and a constant
$\gamma=\gamma_{R}(\alpha,\beta)\in(0,1)$ such that for all
$s,\tau\in[t,t+\bar{h}]$, $\mathbb{P}$-a.s.
\[
\left\vert Q\left( s\right) -Q\left( \tau\right) \right\vert \leq
H_{R}\left\vert s-\tau\right\vert ^{1-\alpha},\quad\left\vert
Q\left( s\right) \right\vert \leq\tilde{H}_{R}\left\vert s-t\right\vert
^{1+\gamma
\]
and
\[
x+\left( s-t\right) u +v \left[
B_{s}^{H} -B_{t}^{H} \right] +Q\left( s\right) \in
K\left( s\right) ,
\]
where the constants $H_{R}$, $\tilde{H}_{R
$ depend only on $R$, $L_{R
$, $M_{0,T}$,$M_{0}$, $L_{0}$,
$T$, $\alpha$, $\beta$, $\Lambda_{\alpha}\left(
B^{H}\right) $.
\end{definition}
\begin{definition}\label{def2.5}
Let $t\in\left[ 0,T\right] $ and
$x\in K(t).$ Let $\frac{1}{2}<1-\alpha<H.$
The $\left( 1-\alpha\right) $-fractional $B^{H}$-tangent set
to $K(t) $ in $x$, denoted by $S_{K(t)}(t,x)$, is the set of the pairs $(u,v)$, such that there exist random variable $\bar{h}=\bar{h}^{t,x}>0$
and two stochastic process
\[
\begin{array}
[c]{ll}
U=U^{t,x}:& \left[ t,t+\bar{h}\right] \rightarrow\mathbb{R}^{d} ,~\ U(t)=0
\\
V=V^{t,x}:& \left[ t,t+\bar{h}\right] \rightarrow\mathbb{R}^{d} ,~\ V(t)=0
\end{array}
\]
and for every $R>0$ with $\left\vert x\right\vert \leq R$ there
exsit two random variables $D_{R},\tilde{D}_{R}>0$ independent of
$(t,\bar{h})$ such that for all $s,\tau\in\left[ t,t+\bar{h}\right]
$, $\mathbb{P}$-a.s.
\[
\left\vert U\left( s\right) -U\left( \tau\right) \right\vert \leq
D_{R}\left\vert s-\tau\right\vert ^{1-\alpha},\quad\left\vert
V\left( s\right) -V\left( \tau\right) \right\vert \leq\tilde{D}_{R}\left\vert s-\tau\right\vert
^{min\{\beta,1-\alpha\}
\]
and
\[
x+\int_{t}^{s}(u+U(r))dr+ \int_{t}^{s} (v+V(r))dB_{r}^{H} \in K(s) ,
\]
where the constants $D_{R}$, $\tilde{D}_{R
$ depend only on $R$, $L_{R
$,$M_{0,T}$, $M_{0}$, $L_{0}$,
$T$, $\alpha$, $\beta$, $\Lambda_{\alpha}\left(
B^{H}\right) $.
\end{definition}
\textbf{Remark.}
$\bullet$ From~\cite{pi}, we can always assume that
$0<\bar{h}\leq1$.
$\bullet$ The definition of $S_{\varphi(K(t))}(t,\varphi(x))$ is the same to $S_{K(t)}(t,x)$, only changes the condition
\[
x+\int_{t}^{s}(u+U(r))dr+ \int_{t}^{s} (v+V(r))dB_{r}^{H} \in K(s) ,
\]
\qquad to
\[
\varphi(x)+\int_{t}^{s}(u+U(r))dr+ \int_{t}^{s} (v+V(r))dB_{r}^{H} \in \varphi(K(s)).
\]
Now we recall the main result of~\cite{pi} concerning the stochastic
viability.
\begin{theorem}\label{th2.6}
Let $\mathcal{K}=\left\{ K\left( t\right) :t\in\left[ 0,T\right] \right\}
$\textit{, }$K\left( t\right) =\overline{K\left( t\right) }\subset
\mathbb{R}^{d}$. Assume that $(\mathbf{H}_{1})$ and $(\mathbf{H}_{2})$ are satisfied with $\frac{1}{2}<H<1,~ 1-H<\beta, ~\delta>\frac{1-H}{H}$.
Let $1-H<\alpha<\alpha_{0}$. Then the following assertions are equivalent:
\begin{itemize}
\item[(I)] $\mathcal{K}$ is viable for the fractional SDE, i.e. for
all $t\in\left[ 0,T\right] $ and for all $x\in K\left( t\right)
$ there exists a solution $X^{t,x}\left( \omega,\cdot\right) \in
C^{1-\alpha}\left( \left[ t,T\right] ;\mathbb{R}^{d}\right) $ of
the equation
\[
X_{s}^{t,x}=x+\int_{t}^{s}b(r,X_{r}^{t,x})dr+\int_{t}^{s
\sigma(r,X_{r}^{t,x})dB_{r}^{H},\;\;s\in\lbrack t,T],\ a.s.\;\omega\in\Omega,
\]
and
\[
X_{s}^{t,x}\in K\left( s\right) ,\quad\quad\forall~s\in\left[ t,T\right]
.
\]
\item[(II)] For all $t\in\left[ 0,T\right] $ and all $x\in
K\left( t\right) ,$ $\left( b\left( t,x\right) ,\sigma\left( t,x\right) \right) $\textit{ is
}$\left( 1-\alpha\right) $-fractional $B^{H}$-contingent to $K\left( t\right) $ in $x$ .
\item[(III)] For all $t\in\left[ 0,T\right] $ and all $x\in
K(t) ,$\textit{ }$\left( b\left( t,x\right) ,\sigma\left( t,x\right) \right) $ is
$\left( 1-\alpha\right) $-fractional $B^{H}$-tangent to $K(t) $ in $x$ .
\end{itemize}
\end{theorem}
\textbf{Remark.} The assertion (III) is given only for the
deterministic case in~\cite{pi}. In fact we can obtain the
stochastic case from the deterministic one in the same manner as
that (II) is obtained.
\smallskip
Under the same assumptions in Theorem \ref{th2.6}, it follows:
\begin{corollary}\label{cor2.7}
If $K$ is independent of $t$, the following assertions are equivalent:
\begin{itemize}
\item[(j)] $K$ is viable for the fractional SDE (\ref{equ2.3}).
\item[(jj)] For all $t\in\left[ 0,T\right] $ and all $x\in
\partial K ,$ $\left( b\left( t,x\right) ,\sigma\left( t,x\right) \right) $ is
$\left( 1-\alpha\right) $-fractional $B^{H}$-contingent to $K$ in $x$ .
\item[(jjj)] For all $t\in\left[ 0,T\right] $ and all $x\in
\partial K ,$\textit{ }$\left( b\left( t,x\right) ,\sigma\left( t,x\right) \right) $ is
$\left( 1-\alpha\right) $-fractional $B^{H}$-tangent to $K$ in $x$ .
\end{itemize}
\end{corollary}
\textbf{Proof.}
When $K$ is independent of $t$, just using Theorem \ref{th2.6}, it's obvious that $(j)\Rightarrow(jj)\Rightarrow(jjj)$.
Now we only need prove $(jjj)\Rightarrow(j)$, In fact we will prove $(jjj)\Rightarrow(III)$, and then we will get our result.
Let $t\in[0,T] $ and $\forall x\in K\setminus \partial K$, Since
$X^{t,x}$ is continuous, then there exists a random variable
$\bar{h}$, such that for all $s\in [t,t+\bar{h}]$,
\begin{equation*}
X_{s}^{t,x}=x+{\displaystyle\int_{t}^{s}}b(r,X_{r}^{t,x})dr+{\displaystyle\int_{t}^{s}}\sigma(r,X_{r}^{t,x})dB_{r}^{H}\in K.
\end{equation*}
we have for all $s\in [t,t+\bar{h}]$,
\begin{equation*}
X_{s}^{t,x}=x+{\displaystyle\int_{t}^{s}}[b(t,x)+U(r)]dr+{\displaystyle\int_{t}^{s}}[\sigma(t,x)+V(r)]dB_{r}^{H}\in K.
\end{equation*}
where
\[U(r)=b(r,X_{r}^{t,x})-b(t,x), \quad V(r)=\sigma(r,X_{r}^{t,x})-\sigma(t,x)
\]
clearly that $\left( b\left( t,x\right) ,\sigma\left( t,x\right)
\right) $ is $\left( 1-\alpha\right) $-fractional $B^{H}$-tangent
to $K $ in $x$. Together with $(jjj)$, we have that for all
$t\in\left[ 0,T\right] $ and all $x\in K ,$\textit{ }$\left(
b\left( t,x\right) ,\sigma\left( t,x\right) \right) $ is $\left(
1-\alpha\right) $-fractional $B^{H}$-tangent to $K$ in $x$. This is
just $(III)$ for the case that $K$ is independent of $t$.
\begin{flushright}
$\Box$
\end{flushright}
\section{Results and Applications}\label{sec3}
The next two theorems are our main theorems, firstly we extend \textit{Stochastic Tangent Sets to Direct Images}
which is introduced by J.P.Aubin, and G.Da Prato~\cite{pc} (1990) to the fBM framework.
\begin{theorem}\label{th3.1}
Assume that $(\mathbf{H}_{1})$ and $(\mathbf{H}_{2})$ are satisfied. Let
$K\left( t\right) =\overline{K\left( t\right) }\subset
\mathbb{R}^{d}, t\in\left[ 0,T\right]$ and $S_{K(t)}(t,x)$ the $\left( 1-\alpha\right) $-fractional $B^{H}$-tangent set to $K$ in $x$.
Let $\varphi$ be a $C^{2}$ map from $\mathbb{R}^d$ to $\mathbb{R}^m$ with a bounded second derivative.
If
\[(b(t,x),\sigma(t,x))\in S_{K(t)}(t,x)
\]
then
\[(\varphi^{\prime}(x)b(t,x),\varphi^{\prime}(x)\sigma(t,x))\in S_{\varphi(K(t))}(t,\varphi(x)).
\]
\end{theorem}
\medskip
Also we can prove the \textit{ Stochastic Tangent Sets to Inverse Images} in the fBM form.\\
We introduce a space $\mathcal{H}$ of the functions $\varphi:
\mathbb{R}^{d}\rightarrow\mathbb{R}^{m}$ of class $C^{2}$, with a
bounded and Lipschitz continuous second derivative and there exist
$a_{\varphi}<b_{\varphi}$ and some constants $M>0$, $L>0$ such that
for all $a_{\varphi}\leq|x|\leq b_{\varphi}$, the matrix
$\varphi^{\prime}(x)$ has a right inverse denoted by
$\varphi^{\prime}(x)^{+}$ satisfying
\[
\begin{array}
[c]{ll}
(1) \; \qquad \left|[\varphi^{\prime}(x)^{+}]^{\prime}\right|\leq M,\\
(2) \;
\qquad\left|[\varphi^{\prime}(x)^{+}]^{\prime}-[\varphi^{\prime}(y)^{+}]^{\prime}\right|\leq
L|x-y|.
\end{array}
\]
\begin{theorem}\label{th3.2}
Assume that $(\mathbf{H}_{1})$ and $(\mathbf{H}_{2})$ are satisfied.
Let $K\left( t\right) =\overline{K\left( t\right) }\subset
\mathbb{R}^{d}, t\in\left[ 0,T\right]$ and
$\varphi\in \mathcal{H}$,
\textit{then for} every $\varepsilon>0$ and $a_{\varphi}+\varepsilon\leq|x|\leq b_{\varphi}-\varepsilon$, then
\[(b(t,x),\sigma(t,x))\in S_{\varphi^{-1}(\varphi(K(t)))}(t,x)
\]
if and only if
\[(\varphi^{\prime}(x)b(t,x),\varphi^{\prime}(x)\sigma(t,x))\in S_{\varphi(K(t))}(t,\varphi(x)).
\]
\end{theorem}
\medskip
Using Theorem \ref{th3.2}, we can get the deterministic sufficient
and necessary conditions for viability when $K$ takes some particular forms. Firstly we give some Lemmas.
\begin{lemma}\label{lem3.3}
Let $K$ be the unit sphere, then for all $x\in K$, $(b(t,x),\sigma(t,x))\in S_{K}(t,x)$ if and only if
\[\langle x,b(t,x)\rangle=0, \qquad \langle x,\sigma(t,x)\rangle=0.
\]
\end{lemma}
\begin{lemma}\label{lem3.4}
Let $K=\{x\in \mathbb{R}^{d}; r\leq|x|\leq R\}$ then for all $x$, such that $|x|=R$, $(b(t,x),\sigma(t,x))\in S_{K}(t,x)$ if and only if
\[\langle x,b(t,x)\rangle\leq 0, \qquad \langle x,\sigma(t,x)\rangle=0
\]
and for all $x$, such that $|x|=r$, $(b(t,x),\sigma(t,x))\in S_{K}(t,x)$ if and only if
\[\langle x,b(t,x)\rangle\ge 0, \qquad \langle x,\sigma(t,x)\rangle=0.
\]
\end{lemma}
\begin{lemma}\label{lem3.5}
Let $K$ be the unit ball, then for all $x$, such that $|x|=1$, $(b(t,x),\sigma(t,x))\in S_{K}(t,x)$ if and only if
\[\langle x,b(t,x)\rangle\leq 0, \qquad \langle x,\sigma(t,x)\rangle=0.
\]
\end{lemma}
\medskip
Just as Corollary \ref{cor2.7} said, considering that if we want to get the conditions for the viability of $K$,
we only need to think about the starting point $x\in\partial K$. Then together with Lemma \ref{lem3.3} and \ref{lem3.5}, it is obviously that
\begin{proposition}
Let $(\mathbf{H}_{1})$, $(\mathbf{H}_{2})$ be satisfied, $1-H<\alpha<\alpha_{0}$ and $K$ is the unit sphere.
Then the following assertions are equivalent:
\begin{itemize}
\item[(I)] $K$ is viable for the fractional SDE (\ref{equ2.3}).
\item[(II)] For all $t\in\left[ 0,T\right] $ and all $x\in
K$ ,
\[
\langle x,b(t,x)\rangle=0,\quad \langle x,\sigma(t,x)\rangle=0.
\]
\end{itemize}
\end{proposition}
\begin{proposition}\label{pro}
Let $(\mathbf{H}_{1})$, $(\mathbf{H}_{2})$ be satisfied, $1-H<\alpha<\alpha_{0}$ and $K$ is the unit ball.
Then the following assertions are equivalent:
\begin{itemize}
\item[(I)] $K$ is viable for the fractional SDE (\ref{equ2.3}).
\item[(II)] For all $t\in\left[ 0,T\right] $ and all $|x|=1$,
\[
\langle x,b(t,x)\rangle\leq 0,\quad \langle x,\sigma(t,x)\rangle=0.
\]
\end{itemize}
\end{proposition}
\begin{corollary}\label{cor3.8}
Consider the SDE on $\mathbb{R}$,
\[
X_{s}=x+\int_{t}^{s}b(r,X_{r})dr+\int_{t}^{s}\sigma\left( r,X_{r}\right)
dB_{r}^{H},\,\;s\in\left[ t,T\right] .
\]
$B^{H}=\left\{ B^{H}_{t},t\geq0\right\} $ is a
fractional Brownian motion. $b, \sigma$ satisfy the assumptions $(\mathbf{H}_{1}), (\mathbf{H}_{2})$. Then for any $t\in[0,T]$ and
every $x\ge0$ the equation has a positive solution if and only if
\[b(t,0)\geq0, \quad\sigma(t,0)=0, \quad\forall t\in[0,T].
\]
\end{corollary}
\textbf{Proof.}
In fact we take $K=[0,+\infty)$, the problem is just that $K$ is viable for the fractional SDE.
We can use $x=\tan\frac{\pi}{4}(y+1)$ and we get $y=\frac{4}{\pi}\arctan x-1$,
it just maps $[0,+\infty)$ to $[-1,1]$, and using Proposition \ref{pro} and It\^{o} formula of fractional SDE (see~\cite{pk}), we have
\[b(t,0)\geq0, \quad\sigma(t,0)=0, \quad\forall t\in[0,T].
\]
\begin{flushright}
$\Box$
\end{flushright}
The most interesting application is the characterization of comparison theorem.\\
Let us firstly consider the linear case.
\begin{corollary}
Consider the linear two dimensional decoupled system
\[
\left\{
\begin{array}
[c]{l
X_{s}^{t,x}=x+{\displaystyle\int_{t}^{s}}(f(r)X_{r}^{t,x}+f_{1}(r))dr+{\displaystyle\int_{t}^{s}}(g(r)X_{r}^{t,x}+g_{1}(r))dB^{H}_{r},\smallskip\;
s\in[t,T] \medskip\\
Y_{s}^{t,y}=y+{\displaystyle\int_{t}^{s}}(f(r)Y_{r}^{t,y}+f_{2}(r))dr+{\displaystyle\int_{t}^{s}
(g(r)Y_{r}^{t,y}+g_{2}(r))dB^{H}_{r},\smallskip\;s\in[t,T]
\end{array}
\right.
\]
then
\[\textit{for any}~t\in[0,T] ~\textit{and every}~x\leq y, ~ X_{s}^{t,x}\leq Y_{s}^{t,y},~\forall s\in[t,T].\]
\[\Longleftrightarrow ~f_{1}(t)\leq f_{2}(t),~ g_{1}(t)=g_{2}(t),~\forall t\in[0,T] .
\]
\end{corollary}
\textbf{Proof.} In fact we set
$Z_{s}^{t,z}=Y_{s}^{t,y}-X_{s}^{t,x}$, where $z=y-x\ge0$, then we
can change the problem to $Z_{s}^{t,z}\ge0$, it means that for any
$t\in[0,T]$ and every $z\ge0$ the fractional SDE of $Z_{s}^{t,z}$
has a positive solution. Then using Corollary \ref{cor3.8}, we can
easily prove this corollary.
\begin{flushright}
$\Box$
\end{flushright}
In general, we have
\begin{theorem}\label{th3.10}(Comparison theorem)
Consider the two dimensional decoupled syste
\[
\left\{
\begin{array}
[c]{l
X_{s}^{t,x}=x+{\displaystyle\int_{t}^{s}}(b_{1}(r,X_{r}^{t,x}))dr+{\displaystyle\int_{t}^{s}}(\sigma_{1}(r,X_{r}^{t,x}))dB^{H}_{r},
\smallskip\;s\in[t,T] \medskip\\
Y_{s}^{t,y}=y+{\displaystyle\int_{t}^{s}}(b_{2}(r,Y_{r}^{t,y}))dr+{\displaystyle\int_{t}^{s}}(\sigma_{2}(r,Y_{r}^{t,y}))dB^{H}_{r},
\smallskip\;s\in[t,T]
\end{array}
\right.
\]
then
\[\textit{for any}~ t\in[0,T] ~\textit{and every}~x\leq y,~X_{s}^{t,x}\leq Y_{s}^{t,y},~\forall s\in[t,T]
\]
\[\Longleftrightarrow b_{1}(t,z)\leq b_{2}(t,z),~ \sigma_{1}(t,z)=\sigma_{2}(t,z),~\forall t\in[0,T], ~\forall z\in\mathbb{R}.
\]
\end{theorem}
we will give the proof of this result in Section \ref{sec5}.
\section{Proofs of main results}
This section is devoted to the proofs of the main results which have been given in Section \ref{sec3}.
Firstly we present some auxiliary Lemmas which will be used in the
sequel.
\subsection{Auxiliary Results}
\begin{lemma}\label{lem4.1}
Given two stochastic process
\[
\begin{array}
[c]{ll}
U=U^{t,x}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R}^{d} ,~\ U(t)=0
\\
V=V^{t,x}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R}^{d} ,~\ V(t)=0
\end{array}
\]
such that for all $s,\tau\in\left[ t,t+\bar{h}\right] $ and
for every $R>0$ with $\left\vert x\right\vert \leq R:$
\[
\begin{array}
[c]{ll}
\left\vert U\left( s\right) -U\left( \tau\right) \right\vert & \leq
D_{R}\left\vert s-\tau\right\vert ^{1-\alpha},\\
\left\vert
V\left( s\right) -V\left( \tau\right) \right\vert & \leq\tilde{D}_{R}\left\vert s-\tau\right\vert
^{min\{\beta,1-\alpha\}}.
\end{array}
\]
then for all $t\leq\tau\leq s\leq t+\bar{h}$,
\[
\begin{array}
[c]{ll}
\left( a\right) \; & \left\vert
{\displaystyle\int_{\tau}^{s}}
U(r)dr\right\vert \leq
D_{R}\left( s-t\right) ^{1-\alpha}(s-\tau)\\
\left( b\right) \; & \left\vert
{\displaystyle\int_{\tau}^{s}}
V(r) dB^{H}_{r}\right\vert \leq
C_{R}(\alpha,\beta)\tilde{D}_{R}\Lambda_{\alpha}(B^{H})(s-t)^{\min\{\beta,1-\alpha\}}\left(
s-\tau\right) ^{1-\alpha}.
\end{array}
\]
where $C_{R}(\alpha,\beta)$ depends only on $R$, $\alpha$, and
$\beta$.
\end{lemma}
\textbf{Proof.}
$(a)$ we have
\begin{eqnarray*}
\left\vert
{\displaystyle\int_{\tau}^{s}}
U(r) dr\right\vert
& = &
\left\vert
{\displaystyle\int_{\tau}^{s}}
[U(r)-U(t)] dr\right\vert
{}
\nonumber\\
& \leq & {} D_{R}\left\vert
{\displaystyle\int_{\tau}^{s}}
(r-t)^{1-\alpha} dr\right\vert {}
\nonumber\\
& \leq & {} D_{R}(s-t)^{1-\alpha}(s-\tau).
\end{eqnarray*}
$(d)$
\begin{eqnarray*}
\left\vert
{\displaystyle\int_{\tau}^{s}}
V(r) dB_{r}^{H}\right\vert
& = &
\left\vert
{\displaystyle\int_{\tau}^{s}}
[V(r)-V(t)] dB_{r}^{H}\right\vert
{}
\nonumber\\
& \leq & {}
\Lambda_{\alpha}(B^{H})\|V\|_{\alpha,1;[\tau,s]}
{}
\nonumber\\
& \leq & {}
\Lambda_{\alpha}(B^{H})\int_{\tau}^{s}\left[ \frac{|V(r)-V(t)|}{\left( r-\tau\right) ^{\alpha}}+\int_{\tau
^{r}\frac{|V(r)-V(y)|}{(r-y)^{\alpha+1}}dy\right] dr {}
\nonumber\\
& \leq & {}
\tilde{D}_{R}\Lambda_{\alpha}(B^{H})\int_{\tau}^{s}\left[ \frac{(r-t)^{\min\{\beta,1-\alpha\}}}{\left( r-\tau\right) ^{\alpha}}+\int_{\tau
^{r}\frac{(r-y)^{\min\{\beta,1-\alpha\}}}{(r-y)^{\alpha+1}}dy\right]
dr {}
\nonumber\\
& \leq & {}
\tilde{D}_{R}\Lambda_{\alpha}(B^{H})\Big[\frac{1}{1-\alpha}(s-t)^{\min\{\beta,1-\alpha\}}\left(
s-\tau\right) ^{1-\alpha}
{}
\nonumber\\
& & {}
\qquad\qquad\qquad\qquad+\int_{\tau}^{s}\int_{\tau
^{r}(r-y)^{\min\{\beta-\alpha,1-2\alpha\}-1}dy dr {}\Big]
\nonumber\\
& \leq & {}
C_{R}(\alpha,\beta)\tilde{D}_{R}\Lambda_{\alpha}(B^{H})(s-t)^{\min\{\beta,1-\alpha\}}\left(
s-\tau\right) ^{1-\alpha}.
\end{eqnarray*}
\begin{flushright}
$\Box$
\end{flushright}
\textbf{Remark.} From $(a)$ and $(b)$, just taking $\tau=t$, we have
\[
\begin{array}
[c]{ll} \left( a^{\prime}\right) \; & \left\vert
{\displaystyle\int_{t}^{s}}
U(r)dr\right\vert \leq
D_{R}\left( s-t\right) ^{2-\alpha}\\
\left( b^{\prime}\right) \; & \left\vert
{\displaystyle\int_{t}^{s}}
V(r) dB^{H}_{r}\right\vert \leq
C_{R}(\alpha,\beta)\tilde{D}_{R}\Lambda_{\alpha}(B^{H})(s-t)^{1+\min\{\beta-\alpha,1-2\alpha\}}.
\end{array}
\]
\begin{lemma}\label{lem4.2}
Given two stochastic process $U=U^{t,x}, V=V^{t,x}$ which satisfy the conditions in Lemma \ref{lem4.1}, and $\varphi\in\mathcal{H}$, let
\begin{eqnarray*}
f(r,y) & = & \varphi^{\prime}(y)^{+}\left[U(r)-(\varphi^{\prime}(y)-\varphi^{\prime}(x))b(t,x)\right]\\
g(r,y) & = & \varphi^{\prime}(y)^{+}\left[V(r)-(\varphi^{\prime}(y)-\varphi^{\prime}(x))\sigma(t,x)\right]
\end{eqnarray*}
then for $\alpha\in\left( 1-H,\alpha_{0}\right)$ and for every
$\delta_{0}>0$ there exists a random variable
$\bar{h}_{1}=\bar{h}_{1}^{t,x}$ such that for
$a_{\varphi}+2\delta_{0}\leq|x|\leq b_{\varphi}-2\delta_{0}$ and
$\mathbb{P}$-a.s. $\omega\in\Omega$, the following SDE
\[
\xi_{s}=x+\int_{t}^{s}(b(t,x)+f(r,\xi_{r}))dr+\int_{t}^{s}(\sigma(t,x)+g(r,\xi_{r}))dB^{H}(r),
~s\in[t,t+\bar{h}_{1}],
\]
has a unique solution $\xi_{\cdot}\left( \omega\right) \in
L^{0}\left( \Omega,\mathcal{F},\mathbb{P\,};
W^{\alpha,\infty}(t,T;\mathbb{R}^{d})\right).$ \\Moreover
$\mathbb{P}$-a.s. $\xi_{\cdot}\left( \omega\right) \in
C^{1-\alpha}\left( t,t+\bar{h}_{1};\mathbb{R}^{d}\right)$.
\end{lemma}
\textbf{Proof.} From \cite{po} Theorem(the partition of unity) p.61,
we have that for every $\delta_{0}>0$, there exists one function
$\alpha(x)\in C^{\infty}(\mathbb{R}^{d})$ such that $\alpha(x)=1$
for $a_{\varphi}+\delta_{0}\leq|x|\leq b_{\varphi}-\delta_{0}$ and
$\alpha(x)=0$ for $|x|\ge b_{\varphi}~ or ~|x|\leq a_{\varphi}$,
then we define
\[\tilde{f}(t,y)=\alpha(y)f(t,y)=\left\{
\begin{array}
[c]{ll
f(t,y), \quad & a_{\varphi}+\delta_{0}\leq|y|\leq b_{\varphi}-\delta_{0}\\
\alpha(y)f(t,y) \quad & a_{\varphi}\leq|y|\leq a_{\varphi}+\delta_{0}, ~or ~b_{\varphi}-\delta_{0}\leq|y|\leq b_{\varphi}\\
0, \quad & |y|\ge b_{\varphi}, ~or~|y|\leq a_{\varphi}.
\end{array}
\right.
\]
and we define $\tilde{g}(t,y)$ in the same method and then we consider the following SDE
\begin{equation}\label{equ1}
\tilde{\xi}_{s}=x+\int_{t}^{s}(b(t,x)+\tilde{f}(r,\tilde{\xi}_{r}))dr+\int_{t}^{s}(\sigma(t,x)+\tilde{g}(r,\tilde{\xi}_{r}))dB^{H}(r),
~s\in[t,t+\bar{h}].
\end{equation}
Since $\varphi\in\mathcal{H}$ and $U=U^{t,x}, V=V^{t,x}$ satisfy the
conditions in Lemma \ref{lem4.1}, we can verify that for
$\alpha\in\left( 1-H,\alpha_{0}\right),$ $\tilde{f}(t,y)$, and
$\tilde{g}(t,y)$ satisfy the conditions in
($\mathbf{H}_{1}$),($\mathbf{H}_{2}$) in ~\cite{pi} where the
constants $M_{0}, M_{R},L_{0}, L_{R}$ depend on $\omega$, then the
SDE (\ref{equ1}) has a unique solution $\tilde{\xi}_{\cdot}\left(
\omega\right) \in L^{0}\left(
\Omega,\mathcal{F},\mathbb{P\,};W^{\alpha,\infty}(t,T;\mathbb{R}^{d})\right)$
for all $\alpha\in\left( 1-H,\alpha_{0}\right) .$ And moreover
$\mathbb{P}$-a.s. $\tilde{\xi}_{\cdot}\left( \omega\right) \in
C^{1-\alpha}\left( t,t+\bar{h};\mathbb{R}^{d}\right)$. Since
$a_{\varphi}+2\delta_{0}\leq|x|\leq b_{\varphi}-2\delta_{0}$ then
there exists a random variable $\bar{h}_{1}=\bar{h}_{1}^{t,x}$, such
that $\mathbb{P}$-a.s. $a_{\varphi}+\delta_{0}\leq|\tilde{\xi}|\leq
b_{\varphi}-\delta_{0}$, then for $s\in[t,t+\bar{h}_{1}]$, the SDE
(\ref{equ1}) becomes $\mathbb{P}$-a.s.
\begin{equation*}
\tilde{\xi}_{s}=x+\int_{t}^{s}(b(t,x)+f(r,\tilde{\xi}_{r}))dr+\int_{t}^{s}(\sigma(t,x)+g(r,\tilde{\xi}_{r}))dB^{H}(r),
~s\in[t,t+\bar{h}_{1}].
\end{equation*}
just taking $\xi_{s}=\tilde{\xi}_{s}, s\in[t,t+\bar{h}_{1}]$, and together with the uniqueness of $\tilde{\xi}_{s}$, then we finish our proof.
\begin{flushright}
$\Box$
\end{flushright}
\subsection{Proof of Theorem 3.1 and Theorem 3.2}
\textbf{Proof of Theorem \ref{th3.1}}
Since $(b(t,x),\sigma(t,x))\in S_{K(t)}(t,x)$, then
there exist a random variable $\bar{h}=\bar{h}^{t,x}>0,$ and two stochastic process
\[
\begin{array}
[c]{ll}
U=U^{t,x}: & \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R}^{d} ,~\ U(t)=0
\\
V=V^{t,x}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R}^{d} ,~\ V(t)=0
\end{array}
\]
such that for all $s,\tau\in\left[ t,t+\bar{h}\right] $ and for
every $R>0$ and $\left\vert x\right\vert \leq R:$
\[
\left\vert U\left( s\right) -U\left( \tau\right) \right\vert \leq
D_{R}\left\vert s-\tau\right\vert ^{1-\alpha},\quad\left\vert
V\left( s\right) -V\left( \tau\right) \right\vert \leq\tilde{D}_{R}\left\vert s-\tau\right\vert
^{min\{\beta,1-\alpha\}
\]
and
\[
x+\int_{t}^{s}(b(t,x)+U(r))dr+ \int_{t}^{s} (\sigma(t,x)+V(r))dB^{H}(r) \in K(s) ,
\]
where $D_{R}$, $\tilde{D}_{R}$, depend only on $R$, $L_{R
$, $M_{0,T}$, $M_{0}$, $L_{0}$,
$T$, $\alpha$, $\beta$, $\Lambda_{\alpha}(B^{H})$.
Let
\[\eta_{s}=x+\int_{t}^{s}(b(t,x)+U(r))dr+\int_{t}^{s}(\sigma(t,x)+V(r))dB^{H}(r)
\]
and from Lemma \ref{lem4.1} and $H-\epsilon$ H\"{o}lder continuous property of fractional Brownian motion,
it follows that for all $s,\tau\in\left[ t,t+\bar{h}\right] $,
\[|\eta_{s}-\eta_{\tau}|\leq \zeta(s-\tau)^{1-\alpha}.
\]
According to the fractional It\^{o} formula (see Yuliya
S.Mishura~\cite{pk}), We have for all $s\in\left[ t,t+\bar{h}\right]
$
\[
\varphi\Big(x+\int_{t}^{s}(b(t,x)+U(r))dr+\int_{t}^{s}(\sigma(t,x)+V(r))dB^{H}(r)\Big)
\]
\[
=\varphi(x)+\int_{t}^{s}[\varphi^{\prime}(\eta_{r})(b(t,x)+U(r))]dr+\int_{t}^{s}[\varphi^{\prime}(\eta_{r})(\sigma(t,x)+V(r))]dB^{H}(r)
\]
\[
=\varphi(x)+\int_{t}^{s}[\varphi^{\prime}(x)b(t,x)+U_{1}(r)]dr+\int_{t}^{s}[\varphi^{\prime}(x)\sigma(t,x)+V_{1}(r)]dB^{H}(r)
\]
where
\begin{eqnarray*}
U_{1}(r) & = & \varphi^{\prime}(\eta_{r})U(r)+(\varphi^{\prime}(\eta_{r})-\varphi^{\prime}(x))b(t,x)\\
V_{1}(r) & = & \varphi^{\prime}(\eta_{r})V(r)+(\varphi^{\prime}(\eta_{r})-\varphi^{\prime}(x))\sigma(t,x)
\end{eqnarray*}
Then
\[
\varphi(x)+\int_{t}^{s}[\varphi^{\prime}(x)b(t,x)+U_{1}(r)]dr+\int_{t}^{s}[\varphi^{\prime}(x)\sigma(t,x)+V_{1}(r)]dB^{H}(r)
\]
\[=
\varphi\Big(x+\int_{t}^{s}(b(t,x)+U(r))dr+\int_{t}^{s}(\sigma(t,x)+V(r))dB^{H}(r)\Big)\in\varphi(K(s))
\]
and it's easy to verify that
\[U_{1}(t)=0, \quad V_{1}(t)=0
\]
For all $s,\tau\in\left[ t,t+\bar{h}\right] $ and for every $R>0$
and $\left\vert x\right\vert \leq R,$ Using the Lipschitz continuity
of $\varphi^{\prime}$ and ($\mathbf{H}_{2}$), we obtain that
\begin{eqnarray*}
|U_{1}(s)-U_{1}(\tau)|&\leq& |\varphi^{\prime}(\eta_{\tau})||U(s)-U(\tau)|+(|U(s)|+|b(t,x)|)|\varphi^{\prime}(\eta_{s})-\varphi^{\prime}(\eta_{\tau})|\\
&\leq& \theta_{1}|s-\tau|^{1-\alpha}+ \theta_{2}|\eta_{s}-\eta_{\tau}|\\
&\leq& \theta|s-\tau|^{1-\alpha}
\end{eqnarray*}
Similarly we can proof that
\begin{eqnarray*}
|V_{1}(s)-V_{1}(\tau)|\leq
\tilde{\theta}|s-\tau|^{min\{\beta,1-\alpha\}}
\end{eqnarray*}
The H\"{o}lder constants $\theta$, $\tilde{\theta}$ are random variables which
depend only on $R$, $L_{R}$, $M_{0}$, $M_{0}$, $L_{0}$,
$T$, $\alpha$, $\beta$, $\Lambda_{\alpha}\left(
B^{H}\right) $.
This means that
\[(\varphi^{\prime}(x)b(t,x),\varphi^{\prime}(x)\sigma(t,x))\in S_{\varphi(K(t))}(t,\varphi(x)).
\]
\begin{flushright}
$\Box$
\end{flushright}
\textbf{Proof of Theorem \ref{th3.2}}
We shall only have to prove that from
$(\varphi^{\prime}(x)b(t,x),\varphi^{\prime}(x)\sigma(t,x))\in S_{\varphi(K(t))}(t,\varphi(x)),$
we infer that $(b(t,x),\sigma(t,x))\in S_{\varphi^{-1}(\varphi(K(t)))}(t,x)$.
Since
\[(\varphi^{\prime}(x)b(t,x),\varphi^{\prime}(x)\sigma(t,x))\in S_{\varphi(K(t))}(t,\varphi(x)).
\]
then for $x\in K(t)$, there exist a random variable
$\bar{h}=\bar{h}^{t,x}>0$ and two stochastic process,
\[
\begin{array}
[c]{ll} U_{1}=U_{1}^{t,x}:& \Omega\times\left[ t,t+\bar{h}\right]
\rightarrow\mathbb{R}^{m} ,~\ U_{1}(t)=0,
\\
V_{1}=V_{1}^{t,x}:& \Omega\times\left[ t,t+\bar{h}\right]
\rightarrow\mathbb{R}^{m} ,~\ V_{1}(t)=0
\end{array}
\]
such that for all $s,\tau\in\left[ t,t+\bar{h}\right] $ and for every $R>0$ and $\left\vert x\right\vert \leq R,$
\[
\left\vert U_{1}\left( s\right) -U_{1}\left( \tau\right) \right\vert \leq
D_{R}\left\vert s-\tau\right\vert ^{1-\alpha},\quad\left\vert
V_{1}\left( s\right) -V_{1}\left( \tau\right) \right\vert \leq\tilde{D}_{R}\left\vert s-\tau\right\vert
^{min\{\beta,1-\alpha\}
\]
and
\[
\varphi(x)+\int_{t}^{s}(\varphi^{\prime}(x)b(t,x)+U_{1}(r))dr+ \int_{t}^{s} (\varphi^{\prime}(x)\sigma(t,x)+V_{1}(r))dB^{H}(r) \in \varphi(K(s)).
\]
Let
\begin{eqnarray*}
f(r,y) & = & \varphi^{\prime}(y)^{+}\left[U_{1}(r)-(\varphi^{\prime}(y)-\varphi^{\prime}(x))b(t,x)\right],\\
g(r,y) & = &
\varphi^{\prime}(y)^{+}\left[V_{1}(r)-(\varphi^{\prime}(y)-\varphi^{\prime}(x))\sigma(t,x)\right],
\end{eqnarray*}
where $\varphi^{\prime}(y)^{+}$ is the right inverse of
$\varphi^{\prime}(y)$. By Lemma \ref{lem4.2}, for every
$\delta_{0}>0$ and $a_{\varphi}+2\delta_{0}\leq|x|\leq
b_{\varphi}-2\delta_{0}$, there exists a random variable
$\bar{h}_{1}$ such that for $\mathbb{P}$-a.s. $\omega\in\Omega$ the
following SDE
\[
\xi_{s}=x+\int_{t}^{s}(b(t,x)+f(r,\xi_{r}))dr+\int_{t}^{s}(\sigma(t,x)+g(r,\xi_{r}))dB^{H}(r),
~s\in[t,t+\bar{h}_{1}],
\]
has a unique solution $\xi_{\cdot}\left( \omega\right)$. Then with
\begin{eqnarray*}
U(r) & = & \varphi^{\prime}(\xi_{r})^{+}\left[U_{1}(r)-(\varphi^{\prime}(\xi_{r})-\varphi^{\prime}(x))b(t,x)\right] \quad\text{and}\\
V(r) & = & \varphi^{\prime}(\xi_{r})^{+}\left[V_{1}(r)-(\varphi^{\prime}(\xi_{r})-\varphi^{\prime}(x))\sigma(t,x)\right]
\end{eqnarray*}
according to the fractional It\^{o} formula, we have for all
$s\in\left[ t,t+\bar{h}_{1}\right] $
\[
\varphi\Big(x+\int_{t}^{s}(b(t,x)+U(r))dr+\int_{t}^{s}(\sigma(t,x)+V(r))dB^{H}(r)\Big)
\]
\[=
\varphi(x)+\int_{t}^{s}(\varphi^{\prime}(x)b(t,x)+U_{1}(r))dr+\int_{t}^{s}(\varphi^{\prime}(x)\sigma(t,x)+V_{1}(r))dB^{H}(r)\in\varphi(K(s)).
\]
Clearly that
\[U(t)=0,\quad V(t)=0.
\]
Since $\varphi\in\mathcal{H}$ and $\xi_{\cdot}\left( \omega\right)
\in C^{1-\alpha}\left( t,t+\bar{h}_{1};\mathbb{R}^{d}\right)$ and
together with ($\mathbf{H}_{1}$) and ($\mathbf{H}_{2}$), it easily
follows
\[
\left\vert U\left( s\right) -U\left( \tau\right) \right\vert \leq
\theta\left\vert s-\tau\right\vert ^{1-\alpha},\quad\left\vert
V\left( s\right) -V\left( \tau\right) \right\vert \leq\tilde{\theta}\left\vert s-\tau\right\vert
^{min\{\beta,1-\alpha\}}
\]
Then it means that
\[(b(t,x),\sigma(t,x))\in S_{\varphi^{-1}(\varphi(K(t)))}(t,x).
\]
\begin{flushright}
$\Box$
\end{flushright}
\subsection{Proof of Lemmas 3.3, 3.4, 3.5}.
\textbf{Proof of Lemma \ref{lem3.3}}
Firstly, we take $\varphi(x)=|x|^2$, and it's easy to verify that for $\frac{1}{4}\leq|x|\leq4$,
\[\varphi^{\prime}(x)^{+}=\frac{x}{2|x|^2}.
\]
and we can verify that $\varphi\in\mathcal{H}$ taking
$a_{\varphi}=\frac{1}{4}$, $b_{\varphi}=4$,
$\varepsilon=\frac{1}{4}$, then for $x\in K$ we have
$\frac{1}{2}\leq|x|=1\leq4-\frac{1}{4}$, by Theorem \ref{th3.2} we
have
\[(b(t,x),\sigma(t,x))\in S_{K}(t,x)\Leftrightarrow (\langle 2x,b(t,x)\rangle,2x^{\ast}\sigma(t,x))\in S_{1}(t,x^2)
\]
So now it's equivalent to prove
\[(\langle 2x,b(t,x)\rangle,\langle 2x,\sigma(t,x)\rangle)\in S_{1}(t,|x|^2)\Leftrightarrow \langle x,b(t,x)\rangle=0,
\quad \langle x,\sigma(t,x)\rangle=0
\]
\textit{Sufficient.} If $\langle x,b(t,x)\rangle=0,~\langle
x,\sigma(t,x)\rangle=0$, we can take $U(r)\equiv 0, ~V(r)\equiv 0$,
and we have $\forall s\in[t,t+\bar{h}]$ and $|x|=1$
\[|x|^2+\int_{t}^{s}(\langle 2x,b(t,x)\rangle+U(r))dr+\int_{t}^{s}(\langle 2x,\sigma(t,x)\rangle)+V(r))dB^{H}(r)=1,
\]
This means that $(\langle 2x,b(t,x)\rangle, \langle 2x,\sigma(t,x)\rangle)\in S_{1}(t,|x|^2)$.
\textit{Necessary.} Since $(\langle 2x,b(t,x)\rangle, \langle
2x,\sigma(t,x)\rangle)\in S_{1}(t,|x|^2)$. then there exist a random
variable $\bar{h}=\bar{h}^{t,x}>0,$ and two stochastic process
\[
\begin{array}[c]{ll}
U=U^{t,x}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R} ,~\ U(t)=0
\\
V=V^{t,x}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R} ,~\ V(t)=0
\end{array}
\]
such that for all $s,\tau\in\left[ t,t+\bar{h}\right] $ and for
every $R>0$, $\left\vert x\right\vert \leq R:$
\[
\left\vert U\left( s\right) -U\left( \tau\right) \right\vert \leq
D_{R}\left\vert s-\tau\right\vert ^{1-\alpha},\quad\left\vert
V\left( s\right) -V\left( \tau\right) \right\vert \leq\tilde{D}_{R}\left\vert s-\tau\right\vert
^{min\{\beta,1-\alpha\}
\]
and
\begin{equation}\label{equ4.1}
|x|^2+\int_{t}^{s}(\langle 2x,b(t,x)\rangle+U(r))dr+\int_{t}^{s}(\langle 2x,\sigma(t,x)\rangle+V(r))dB^{H}(r)=1, ~\mathbb{P}-a.s.
\end{equation}
where $D_{R}$, $\tilde{D}_{R
$, depend only on $R$, $L_{R
$, $M_{0}$, $L_{0}$,
$T$, $\alpha$, $\beta$, $\Lambda_{\alpha}\left(
B^{H}\right) $.
Since $|x|^{2}=1$, then from the equation (\ref{equ4.1}) we clearly
have
\begin{equation}\label{equ4.2}
\left|\langle
2x,b(t,x)\rangle+\left[\int_{t}^{s}U(r)dr+\int_{t}^{s}V(r)dB^{H}(r)\right]\frac{1}{s-t}\right|=
\left|\langle
2x,\sigma(t,x)\rangle\frac{B^{H}(s)-B^{H}(t)}{s-t}\right|
\end{equation}
By (\ref{equ2}), there exists $\Omega_{0}\subset\Omega$ with
$\mathbb{P}(\Omega_{0})=1$ such that $\forall \omega\in\Omega_{0}$,
(\ref{equ4.1}) is satisfied and
\[\limsup_{t\rightarrow t_{0},~t\ge
t_{0}}\Big|\frac{B^{H}_{t}(\omega)-B^{H}_{t_{0}}(\omega)}{t-t_{0}}\Big|=+\infty.
\]
Let $\omega_{0}\in\Omega_{0}$. Then there is a subsequence
$r_{n}=r_{n}(\omega_{0})\downarrow t$ when $n\rightarrow\infty$,
such that
\begin{equation}\label{equ4.10}
\lim_{n\rightarrow\infty}\Big|
\frac{B^{H}_{r_{n}}(\omega_{0})-B^{H}_{t}(\omega_{0})}{r_{n}-t}\Big|=+\infty.
\end{equation}
Setting in \ref{equ4.2}
$s=r_{n}\wedge(t+\bar{h}(\omega_{0}))\in[t,t+\bar{h}(\omega_{0})]$
and passing to limit as $n\rightarrow\infty$, the left member, via
Lemma \ref{lem4.1}, has limit $2\langle x,b(t,x)\rangle$.
Consequently, noting (\ref{equ4.10}), we must have
\[\langle x,\sigma(t,x)\rangle=0.
\]
and therefore
\[\langle x,b(t,x)\rangle=0.
\]
The proof of Lemma \ref{lem3.3} is complete.
\begin{flushright}
$\Box$
\end{flushright}
\textbf{Proof of Lemma \ref{lem3.4}}
Like the analysis in the proof of Lemma \ref{lem3.3}, the proof of Lemma \ref{lem3.4} is reduced to the following equivalent:
$\forall x$ such that $|x|=R$
\[(\langle 2x,b(t,x)\rangle,2x^{\ast}\sigma(t,x))\in S_{\varphi(K)}(t,|x|^2)\Leftrightarrow \langle x,b(t,x)\rangle\leq 0,
\quad \langle x,\sigma(t,x)\rangle=0
\]
$\forall x$, such that $|x|=r$,
\[(\langle 2x,b(t,x)\rangle,2x^{\ast}\sigma(t,x))\in S_{\varphi(K)}(t,|x|^2)\Leftrightarrow \langle x,b(t,x)\rangle\ge 0,
\quad \langle x,\sigma(t,x)\rangle=0.
\]
We only prove in the case $|x|=R$, the other one is similar.
\textit{Sufficient.} If $\langle x,b(t,x)\rangle\leq0, ~\langle x,\sigma(t,x)\rangle=0$, taking $U(r)\equiv 0, ~V(r)\equiv 0$,
and we can choose $\bar{h}$ small enough such that
$\forall s\in[t,t+\bar{h}]$,
\[r^{2}\leq |x|^2+\int_{t}^{s}(\langle 2x,b(t,x)\rangle+U(r))dr+\int_{t}^{s}(2x^{\ast}\sigma(t,x)+V(r))dB^{H}(r)\leq R^{2},
\]
This means that $(\langle 2x,b(t,x)\rangle,\langle 2x,\sigma(t,x)\rangle)\in S_{\varphi(K)}(t,|x|^2)$.
\textit{Necessary.} Since $(\langle 2x,b(t,x)\rangle,\langle
2x,\sigma(t,x)\rangle)\in S_{\varphi(K)}(t,|x|^2)$. Then there exist
random variable $\bar{h}=\bar{h}^{t,x}>0,$ and two stochastic
process
\[
\begin{array}[c]{ll}
U=U^{t,x}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R} ,~\ U(t)=0
\\
V=V^{t,x}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R} ,~\ V(t)=0
\end{array}
\]
such that for all $s,\tau\in\left[ t,t+\bar{h}\right] $ and for
every $R>0$, $\left\vert x\right\vert \leq R:$
\[
\left\vert U\left( s\right) -U\left( \tau\right) \right\vert \leq
D_{R}\left\vert s-\tau\right\vert ^{1-\alpha},\quad\left\vert
V\left( s\right) -V\left( \tau\right) \right\vert \leq\tilde{D}_{R}\left\vert s-\tau\right\vert
^{min\{\beta,1-\alpha\}
\]
and
\[r^{2}\leq |x|^2+\int_{t}^{s}(\langle 2x,b(t,x)\rangle+U(r))dr+\int_{t}^{s}(\langle 2x,\sigma(t,x)\rangle+V(r))dB^{H}(r)\leq R^{2},
\]
where $D_{R}$, $\tilde{D}_{R
$, depend only on$R$, $L_{R
$, $M_{0}$, $L_{0}$,
$T$, $\alpha$, $\beta$, $\Lambda_{\alpha}\left(
B^{H}\right) $.
Since $|x|=R$, then we yield
\begin{equation}\label{equ4.3}
\langle 2x,b(t,x)\rangle(s-t)+\langle 2x,\sigma(t,x)\rangle(B^{H}(s)-B^{H}(t))+\int_{t}^{s}U(r)dr+\int_{t}^{s}V(r)dB^{H}(r)\leq0
\end{equation}
By (\ref{equ3}), there exists $\Omega_{0}\subset\Omega$ with
$\mathbb{P}(\Omega_{0})=\frac{1}{2}$ such that for each
$\omega_{0}\in\Omega_{0}$, (\ref{equ4.3}) is satisfied and there is
a sequence $t\leq s_{n}=s_{n}(\omega_{0})\leq
t+\bar{h}(\omega_{0})$, $s_{n}\downarrow t$, such that
\begin{equation}\label{equ4.11}
\lim_{s_{n}\downarrow
t}\frac{B^{H}_{s_{n}}(\omega_{0})-B^{H}_{t}(\omega_{0})}{s_{n}-t}=+\infty.
\end{equation}
Then we have
\begin{equation}\label{equ4.4}
\langle2x,\sigma(t,x)\rangle\frac{B^{H}_{\omega_{0}}(s_{n})-B^{H}_{\omega_{0}}(t)}{s_{n}-t}+
\left[\int_{t}^{s_{n}}U(r)dr+\int_{t}^{s_{n}}V(r)dB^{H}_{\omega_{0}}(r)\right]\frac{1}{s_{n}-t}
\leq -\langle 2x,b(t,x)\rangle.
\end{equation}
By Lemma \ref{lem4.1}
\[
\left[\int_{t}^{s_{n}}U(r)dr+\int_{t}^{s_{n}}V(r)dB^{H}_{\omega_{0}}(r)\right]\frac{1}{s_{n}-t}\rightarrow
0.
\]
and noting (\ref{equ4.11}), we derive that
\[\langle x,\sigma(t,x)\rangle\leq0.
\]
Similarly we can prove $\langle x,\sigma(t,x)\rangle\ge0$, choosing
$\omega_{0}^{\prime}$ and a sequence $t\leq
r_{n}=r_{n}(\omega_{0}^{\prime})\leq t+\bar{h}(\omega_{0}^{\prime})$
and $r_{n}\downarrow t$ such that $\lim_{r_{n}\downarrow t}
\frac{B^{H}_{r_{n}}(\omega_{0}^{\prime})-B^{H}_{t}(\omega_{0}^{\prime})}{r_{n}-t}=-\infty$.
So
\[\langle x,\sigma(t,x)\rangle=0.
\]
Then from (\ref{equ4.3}), we have
\[\langle
2x,b(t,x)\rangle+\left[\int_{t}^{s}U(r)dr+\int_{t}^{s}V(r)dB^{H}(r)\right]\frac{1}{s-t}\leq0.
\]
and passing to limit $s\rightarrow t$, it follows, via Lemma
\ref{lem4.1},
\[\langle x,b(t,x)\rangle\leq0
\]
The proof of Lemma \ref{lem3.4} is finished.
\begin{flushright}
$\Box$
\end{flushright}
\textbf{Proof of Lemma \ref{lem3.5}} It is very similar to the proof
of Lemma \ref{lem3.4}, therefore we omit it.
\begin{flushright}
$\Box$
\end{flushright}
\section{Proof of the Comparison Theorem}\label{sec5}
\textbf{Proof of Theorem \ref{th3.10}}
We write the two dimensional decoupled syste
\[
\left\{
\begin{array}
[c]{l
X_{s}^{t,x}=x+{\displaystyle\int_{t}^{s}}(b_{1}(r,X_{r}^{t,x}))dr+{\displaystyle\int_{t}^{s}}(\sigma_{1}(r,X_{r}^{t,x}))dB^{H}_{r},
\smallskip\;s\in[t,T] \medskip\\
Y_{s}^{t,y}=y+{\displaystyle\int_{t}^{s}}(b_{2}(r,Y_{r}^{t,y}))dr+{\displaystyle\int_{t}^{s}}(\sigma_{2}(r,Y_{r}^{t,y}))dB^{H}_{r},
\smallskip\;s\in[t,T]
\end{array}
\right.
\]
as
\[Z_{s}^{t,z}=z+{\displaystyle\int_{t}^{s}}(b(r,Z_{r}^{t,z}))dr+{\displaystyle\int_{t}^{s}}(\sigma(r,Z_{r}^{t,z}))dB^{H}_{r},\smallskip\;s\in[t,T]
\]
where
\[Z_{s}^{t,z}={X_{s}^{t,x}\choose Y_{s}^{t,y}}, ~z={x\choose y},~ b(r,Z_{r}^{t,z})={b_{1}(r,X_{r}^{t,x})\choose b_{2}(r,Y_{r}^{t,y})},
~\sigma(r,Z_{r}^{t,z})={\sigma_{1}(r,X_{r}^{t,x})\choose \sigma_{2}(r,Y_{r}^{t,y})}.
\]
we take $\varphi(z)=\varphi(x,y)=y-x$, then for every
$z\in\mathbb{R}^{2}$, $\varphi^{\prime}(z)=(-1,1)$, and
\[
\varphi^{\prime}(z)^{+}=\frac{1}{2}{-1 \choose1}
\]
so $\varphi\in\mathcal{H}$ and if we set $K=\{(x,y)\mid y-x\ge0\}$,
we have $\varphi(K)=\mathbb{R}^{+}$ and
$\varphi^{-1}(\varphi(K))=K$, then using the same method of the
proof of Theorem \ref{th3.2} we have
\[(b(t,z),\sigma(t,z))\in S_{K}(t,x)\Leftrightarrow (\varphi^{\prime}(z)b(t,z),\varphi^{\prime}(z)\sigma(t,z))\in S_{\mathbb{R}^{+}}(t,\varphi(z)).
\]
In fact, considering Theorem \ref{th3.1}, we only need prove
\[(\varphi^{\prime}(z)b(t,z),\varphi^{\prime}(z)\sigma(t,z))\in S_{\mathbb{R}^{+}}(t,\varphi(z))\Rightarrow (b(t,z),\sigma(t,z))\in S_{K}(t,x).
\]
Since
\[(\varphi^{\prime}(z)b(t,z),\varphi^{\prime}(z)\sigma(t,z))\in S_{\mathbb{R}^{+}}(t,\varphi(z))
\]
then for $z\in K$, there exist a random variable
$\bar{h}=\bar{h}^{t,z}>0,$ and two stochastic process,
\[
\begin{array}
[c]{ll}
U_{1}=U_{1}^{t,z}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R} ,~\ U_{1}(t)=0
\\
V_{1}=V_{1}^{t,z}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R} ,~\ V_{1}(t)=0
\end{array}
\]
such that for all $s,\tau\in\left[ t,t+\bar{h}\right] $ and for every $R>0$ and $\left\vert z\right\vert \leq R,$
\[
\left\vert U_{1}\left( s\right) -U_{1}\left( \tau\right) \right\vert \leq
D_{R}\left\vert s-\tau\right\vert ^{1-\alpha},\quad\left\vert
V_{1}\left( s\right) -V_{1}\left( \tau\right) \right\vert \leq\tilde{D}_{R}\left\vert s-\tau\right\vert
^{min\{\beta,1-\alpha\}
\]
and
\[
\varphi(z)+\int_{t}^{s}(\varphi^{\prime}(z)b(t,z)+U_{1}(r))dr+ \int_{t}^{s} (\varphi^{\prime}(z)\sigma(t,z)+V_{1}(r))dB^{H}(r) \in \varphi(K(s)).
\]
Let
\begin{eqnarray*}
f(r,y) & = & \varphi^{\prime}(y)^{+}\left[U_{1}(r)-(\varphi^{\prime}(y)-\varphi^{\prime}(z))b(t,z)\right]\\
g(r,y) & = &
\varphi^{\prime}(y)^{+}\left[V_{1}(r)-(\varphi^{\prime}(y)-\varphi^{\prime}(z))\sigma(t,z)\right].
\end{eqnarray*}
It's obviously that $f(r,y)$, $g(r,y)$ are independent of $y$.
Let
\[
\xi_{s}=z+\int_{t}^{s}(b(t,z)+f(r,\xi_{r}))dr+\int_{t}^{s}(\sigma(t,z)+g(r,\xi_{r}))dB^{H}(r), ~s\in[t,t+\bar{h}], ~|z|\leq R.
\]
Then we take
\begin{eqnarray*}
U(r) & = & \varphi^{\prime}(\xi_{r})^{+}\left[U_{1}(r)-(\varphi^{\prime}(\xi_{r})-\varphi^{\prime}(z))b(t,z)\right]\\
V(r) & = & \varphi^{\prime}(\xi_{r})^{+}\left[V_{1}(r)-(\varphi^{\prime}(\xi_{r})-\varphi^{\prime}(z))\sigma(t,z)\right]
\end{eqnarray*}
According to the fractional It\^{o} formula, we have for all
$s\in\left[ t,t+\bar{h}\right] $
\[
\varphi\Big(z+\int_{t}^{s}(b(t,z)+U(r))dr+\int_{t}^{s}(\sigma(t,z)+V(r))dB^{H}(r)\Big)
\]
\[=
\varphi(z)+\int_{t}^{s}(\varphi^{\prime}(z)b(t,z)+U_{1}(r))dr+\int_{t}^{s}(\varphi^{\prime}(z)\sigma(t,z)+V_{1}(r))dB^{H}(r)\in\mathbb{R}^{+}
\]
Clearly that
\[U(t)=0,\quad V(t)=0.
\]
Since for every $z\in\mathbb{R}^{2}$, $\varphi^{\prime}(z)=(-1,1)$,
$\varphi^{\prime}(z)^{+}=\frac{1}{2}{-1 \choose1}$ and together with
($\mathbf{H}_{1}$), ($\mathbf{H}_{2}$), it is clear that
\[
\left\vert U\left( s\right) -U\left( \tau\right) \right\vert \leq
\theta\left\vert s-\tau\right\vert ^{1-\alpha},\quad\left\vert
V\left( s\right) -V\left( \tau\right) \right\vert \leq\tilde{\theta}\left\vert s-\tau\right\vert
^{min\{\beta,1-\alpha\}}
\]
This means that
\[(b(t,z),\sigma(t,z))\in
S_{\varphi^{-1}(\varphi(K))}(t,x)=S_{K}(t,x).
\]
Just as Corollary \ref{cor2.7} said, if we want to get the
conditions for the viability of $K$, we only need to think about the
starting point $x\in\partial K$. Then the comparison theorem is
equivalent to prove that for all $t\in[0,T]$ and for any
$z={x\choose y}$ such that $x=y$, and $|z|\leq R$,
\[(\varphi^{\prime}(z)b(t,z),\varphi^{\prime}(z)\sigma(t,z))\in S_{\mathbb{R}^{+}}(t,\varphi(z))\Leftrightarrow b_{1}(t,x)\leq b_{2}(t,y),
\sigma_{1}(t,x)=\sigma_{2}(t,y).
\]
\textit{Sufficient}. If $b_{1}(t,x)\leq b_{2}(t,y), \sigma_{1}(t,x)=\sigma_{2}(t,y)$, for $x=y$, we can take $U(r)\equiv 0, V(r)\equiv 0$,
and we have $\forall s\in[t,t+\bar{h}]$, and $z={x\choose y}$, such that $x=y$,
\[y-x+\int_{t}^{s}(\varphi^{\prime}(z)b(t,z)+U(r))dr+\int_{t}^{s}(\varphi^{\prime}(z)\sigma(t,z)+V(r))dB^{H}(r)=(b_{2}(t,y)- b_{1}(t,y))(s-t)\ge0,
\]
This means that $(\varphi^{\prime}(z)b(t,z),\varphi^{\prime}(z)\sigma(t,z))\in S_{\mathbb{R}^{+}}(t,\varphi(z))$.
\textit{Necessary}. Since $(\varphi^{\prime}(z)b(t,z),\varphi^{\prime}(z)\sigma(t,z))\in S_{\mathbb{R}^{+}}(t,\varphi(z))$,
then there exist random variable $\bar{h}=\bar{h}^{t,z}>0,$ and two stochastic process
\[
\begin{array}[c]{ll}
U=U^{t,z}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R} ,~\ U(t)=0
\\
V=V^{t,z}:& \Omega\times\left[ t,t+\bar{h}\right] \rightarrow\mathbb{R} ,~\ V(t)=0
\end{array}
\]
such that for all $s,\tau\in\left[ t,t+\bar{h}\right] $ and for every $R>0$ and
$\left\vert z\right\vert \leq R:$
\[
\left\vert U\left( s\right) -U\left( \tau\right) \right\vert \leq
D_{R}\left\vert s-\tau\right\vert ^{1-\alpha},\quad\left\vert
V\left( s\right) -V\left( \tau\right) \right\vert \leq\tilde{D}_{R}\left\vert s-\tau\right\vert
^{min\{\beta,1-\alpha\}
\]
and
\[y-x+\int_{t}^{s}((b_{2}(t,y)-b_{1}(t,x))+U(r))dr+\int_{t}^{s}((\sigma_{2}(t,y)-\sigma_{1}(t,x))+V(r))dB^{H}(r)\ge0,
\]
Since $y=x$, then we get
\begin{equation*}
(b_{2}(t,x)-b_{1}(t,x))(s-t)+(\sigma_{2}(t,x)-\sigma_{1}(t,x))(B^{H}(s)-B^{H}(t))+\int_{t}^{s}U(r)dr+\int_{t}^{s}V(r)dB^{H}(r)\ge0.
\end{equation*}
With the same analysis in the proof of Lemma \ref{lem3.4}, we obtain
that for every $R>0$
\[b_{1}(t,x)\leq b_{2}(t,x),\quad \sigma_{1}(t,x)=\sigma_{2}(t,x), \quad\forall |x|\leq
R.
\]
This complete the proof of Comparison Theorem.
\begin{flushright}
$\Box$
\end{flushright}
\textbf{Acknowledgement}\quad The authors express special thanks
to Rainer Buckdahn and Lucian Maticiuc for their useful
suggestions and discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,564 |
Hamlet este un film britanic din 1948 regizat, produs și scris de Laurence Olivier. Rolurile principale au fost interpretate de actorii Laurence Olivier și Basil Sydney. A câștigat Premiul Oscar pentru cel mai bun film. Este bazat pe o piesă de teatru omonimă de William Shakespeare.
Prezentare
Distribuție
Laurence Olivier: Hamlet
Basil Sydney: Claudius
Eileen Herlie: Gertrude
Jean Simmons: Ophelia
Felix Aylmer: Polonius
Terence Morgan: Laertes
Norman Wooland: Horatio
John Laurie: Francisco
Esmond Knight: Bernardo
Anthony Quayle: Marcellus
Peter Cushing: Osric
Primire
Într-un sondaj din 1999 al Institutului Britanic de Film (British Film Institute - BFI) filmul a fost desemnat ca fiind al 69-lea cel mai bun dintr-o listă de 100 filme britanice.
Vezi și
Listă de filme străine până în 1989
Note
Legături externe
Filme din 1948
Filme britanice
Filme în limba engleză
Filme regizate de Laurence Olivier
Filme premiate cu Oscar | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,487 |
Financial Accounting and Advisory Services (FAAS)
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IFRS Insights into IFRS 3 - Recognition principle
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,411 |
package binding
import (
"context"
glbinformer "knative.dev/eventing-gitlab/pkg/client/injection/informers/bindings/v1alpha1/gitlabbinding"
"knative.dev/pkg/client/injection/ducks/duck/v1/podspecable"
"knative.dev/pkg/client/injection/kube/informers/core/v1/namespace"
"knative.dev/pkg/reconciler"
corev1 "k8s.io/api/core/v1"
"k8s.io/apimachinery/pkg/labels"
"k8s.io/apimachinery/pkg/types"
"k8s.io/client-go/kubernetes/scheme"
"k8s.io/client-go/tools/cache"
"k8s.io/client-go/tools/record"
"knative.dev/eventing-gitlab/pkg/apis/bindings/v1alpha1"
"knative.dev/pkg/apis/duck"
"knative.dev/pkg/configmap"
"knative.dev/pkg/controller"
"knative.dev/pkg/injection/clients/dynamicclient"
"knative.dev/pkg/logging"
"knative.dev/pkg/webhook/psbinding"
)
const (
controllerAgentName = "gitlabbinding-controller"
)
// NewController returns a new GitLabBinding reconciler.
func NewController(
ctx context.Context,
cmw configmap.Watcher,
) *controller.Impl {
logger := logging.FromContext(ctx)
glbInformer := glbinformer.Get(ctx)
dc := dynamicclient.Get(ctx)
psInformerFactory := podspecable.Get(ctx)
namespaceInformer := namespace.Get(ctx)
c := &psbinding.BaseReconciler{
LeaderAwareFuncs: reconciler.LeaderAwareFuncs{
PromoteFunc: func(bkt reconciler.Bucket, enq func(reconciler.Bucket, types.NamespacedName)) error {
all, err := glbInformer.Lister().List(labels.Everything())
if err != nil {
return err
}
for _, elt := range all {
enq(bkt, types.NamespacedName{
Namespace: elt.GetNamespace(),
Name: elt.GetName(),
})
}
return nil
},
},
GVR: v1alpha1.SchemeGroupVersion.WithResource("gitlabbindings"),
Get: func(namespace string, name string) (psbinding.Bindable, error) {
return glbInformer.Lister().GitLabBindings(namespace).Get(name)
},
DynamicClient: dc,
Recorder: record.NewBroadcaster().NewRecorder(
scheme.Scheme, corev1.EventSource{Component: controllerAgentName}),
NamespaceLister: namespaceInformer.Lister(),
}
impl := controller.NewContext(ctx, c, controller.ControllerOptions{
Logger: logger,
WorkQueueName: "GitLabBindings",
})
glbInformer.Informer().AddEventHandler(controller.HandleAll(impl.Enqueue))
namespaceInformer.Informer().AddEventHandler(controller.HandleAll(impl.Enqueue))
c.Tracker = impl.Tracker
c.Factory = &duck.CachedInformerFactory{
Delegate: &duck.EnqueueInformerFactory{
Delegate: psInformerFactory,
EventHandler: controller.HandleAll(c.Tracker.OnChanged),
},
}
return impl
}
func ListAll(ctx context.Context, handler cache.ResourceEventHandler) psbinding.ListAll {
fbInformer := glbinformer.Get(ctx)
// Whenever a GitLabBinding changes our webhook programming might change.
fbInformer.Informer().AddEventHandler(handler)
return func() ([]psbinding.Bindable, error) {
l, err := fbInformer.Lister().List(labels.Everything())
if err != nil {
return nil, err
}
bl := make([]psbinding.Bindable, 0, len(l))
for _, elt := range l {
bl = append(bl, elt)
}
return bl, nil
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,219 |
{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/cpaa.2011.10.397","text":"# American Institute of Mathematical Sciences\n\nMarch\u00a0 2011,\u00a010(2):\u00a0397-414. doi:\u00a010.3934\/cpaa.2011.10.397\n\n## Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $R^2$\n\n 1 Department of Mathematics University of Toronto, 100 St. George St, Room 4072 Toronto, Ontario M5S 3G3 2 Department Of Mathematics, University of California, Los Angeles, CA, USA Government\n\nReceived\u00a0 June 2010 Revised\u00a0 October 2010 Published\u00a0 December 2010\n\nWe prove global well-posedness for the $L^2$-critical cubic defocusing nonlinear Schr\u00f6dinger equation on $R^2$ with data $u_0 \\in H^s(R^2)$ for $s > \\frac{1}{3}$. The proof combines a priori Morawetz estimates obtained in [4] and the improved almost conservation law obtained in [6]. There are two technical difficulties. The first one is to estimate the variation of the improved almost conservation law on intervals given in terms of Strichartz spaces rather than in terms of $X^{s,b}$ spaces. The second one is to control the error of the a priori Morawetz estimates on an arbitrary large time interval, which is performed by a bootstrap via a double layer in time decomposition.\nCitation: J. Colliander, Tristan Roy. Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $R^2$. Communications on Pure & Applied Analysis, 2011, 10 (2) : 397-414. doi: 10.3934\/cpaa.2011.10.397\n##### References:\n [1] J. Bourgain, Refinement of Strichartz inequality and applications to $2D-NLS$ with critical nonlinearity,, Internat. Math. Res. Notices, 5 (1998), 253.\u00a0 doi:\u00a0doi:10.1155\/S1073792898000191. \u00a0Google Scholar [2] J. Bourgain, \"Global Solutions of Nonlinear Schr\u00f6dinger Equations,\", American Mathematical Society, (1999).\u00a0 \u00a0Google Scholar [3] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schr\u00f6dinger equation in $H^s$,, Non. Anal. TMA, 14 (1990), 807.\u00a0 \u00a0Google Scholar [4] J. Colliander, M. Grillakis and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schr\u00f6dinger equation on $R^2$ ,, Int. Math. Res. Not., 23 (2007).\u00a0 \u00a0Google Scholar [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schr\u00f6dinger equation,, Math. Res. Letters, 9 (2002), 659.\u00a0 \u00a0Google Scholar [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the $I$-method for the cubic nonlinear Schr\u00f6inger equation on $Bbb R^2$,, Discrete Contin. Dyn. 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Weissler, The Cauchy problem for the critical nonlinear Schr\u00f6dinger equation in $H^s$,, Non. Anal. TMA, 14 (1990), 807.\u00a0 \u00a0Google Scholar [4] J. Colliander, M. Grillakis and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schr\u00f6dinger equation on $R^2$ ,, Int. Math. Res. Not., 23 (2007).\u00a0 \u00a0Google Scholar [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schr\u00f6dinger equation,, Math. Res. Letters, 9 (2002), 659.\u00a0 \u00a0Google Scholar [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the $I$-method for the cubic nonlinear Schr\u00f6inger equation on $Bbb R^2$,, Discrete Contin. Dyn. Syst., 21 (2008), 665.\u00a0 doi:\u00a0doi:10.3934\/dcds.2008.21.665. \u00a0Google Scholar [7] Y. Fang and M. Grillakis, On the global existence of rough solutions of the cubic defocusing Schr\u00f6dinger equation in $R^{2+1}$,, J. Hyperbolic Differ. Equ., 4 (2007), 233.\u00a0 \u00a0Google Scholar [8] R. Killip, T. Tao and M. Visan, The cubic nonlinear Schr\u00f6dinger equations in two dimensions with radial data,, Journ. Eur. Math. Soc. (JEMS), 11 (2009), 1203.\u00a0 \u00a0Google Scholar\n [1] Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schr\u00f6dinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1905-1926. doi: 10.3934\/dcds.2013.33.1905 [2] Ademir Pastor. On three-wave interaction Schr\u00f6dinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934\/cpaa.2019100 [3] Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schr\u00f6dinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934\/cpaa.2016.15.831 [4] Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schr\u00f6dinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934\/cpaa.2014.13.1563 [5] Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schr\u00f6dinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934\/dcds.2007.17.181 [6] Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schr\u00f6dinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934\/cpaa.2010.9.261 [7] Daniela De Silva, Nata\u0161a Pavlovi\u0107, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schr\u00f6dinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 37-65. doi: 10.3934\/dcds.2007.19.37 [8] Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schr\u00f6dinger equation in $H^{\\frac 12} (\\mathbb{R} )$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 257-264. doi: 10.3934\/dcds.2017010 [9] Daniela De Silva, Nata\u0161a Pavlovi\u0107, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schr\u00f6dinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934\/cpaa.2007.6.1023 [10] Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schr\u00f6dinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934\/dcds.2013.33.1389 [11] Benjamin Dodson. Improved almost Morawetz estimates for the cubic nonlinear Schr\u00f6dinger equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 127-140. doi: 10.3934\/cpaa.2011.10.127 [12] Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schr\u00f6dinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934\/eect.2014.3.15 [13] Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schr\u00f6dinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934\/cpaa.2014.13.273 [14] Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schr\u00f6dinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934\/cpaa.2018027 [15] Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schr\u00f6dinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934\/cpaa.2007.6.997 [16] Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schr\u00f6dinger type equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1093-1105. doi: 10.3934\/dcds.2010.27.1093 [17] Takeshi Wada. A remark on local well-posedness for nonlinear Schr\u00f6dinger equations with power nonlinearity-an alternative approach. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934\/cpaa.2019066 [18] Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schr\u00f6dinger equations with non-algebraic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3479-3520. doi: 10.3934\/dcds.2019144 [19] Younghun Hong. Scattering for a nonlinear Schr\u00f6dinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934\/cpaa.2016003 [20] J. Colliander, Justin Holmer, Monica Visan, Xiaoyi Zhang. Global existence and scattering for rough solutions to generalized nonlinear Schr\u00f6dinger equations on $R$. Communications on Pure & Applied Analysis, 2008, 7 (3) : 467-489. doi: 10.3934\/cpaa.2008.7.467\n\n2018\u00a0Impact Factor:\u00a00.925","date":"2020-06-07 03:39:20","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\": 0.7012410759925842, \"perplexity\": 3642.789650819968}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590348523476.97\/warc\/CC-MAIN-20200607013327-20200607043327-00232.warc.gz\"}"} | null | null |
Q: Proof: $z$ is a limit point of $(z_n)_{n\geq 1} \quad \iff \quad z$ is a limit point of $(z_n+w_n)_{n\geq 1}$ Let $z\in \mathbb{C}$, $(z_n)_{n\geq 1} \subset \mathbb{C}$ be a sequence and $(w_n)_{n\geq 1} \subset \mathbb{C}$ a null sequence (sequence tending to zero).
Proof that: $z$ is a limit point of $(z_n)_{n\geq 1} \iff z$ is a limit point of $(z_n+w_n)_{n\geq 1}$
My attempt:
For "$\Rightarrow$":
Let $z$ be the limit point of $(z_n)_{n\geq 1}$. We know that $(w_n)_{n\geq 1}$ is a null sequence, thus $w_n\overset{n\to \infty}\longrightarrow 0$. $0$ is a neutral element regarding addition, hence $z$ is a limit point of $(z_n+w_n)_{n\geq 1}$
A: You surely have the right idea, but you might want to elaborate that if $z$ is a limit point of $(z_n)_n$ then $z_{n_k} \to z$ for some subsequence $(z_{n_k})_k$ of $(z_n)_n$. It follows that $(z_{n_k} + w_{n_k})_k$ is a subsequence of $(z_n + w_n)_n$ with
$$
z_{n_k} + w_{n_k} \to z + 0 = z
$$
so that $z$ is a limit point of $(z_n + w_n)_n$.
For the opposite direction "$\Leftarrow$" assume that a limit point of $(z_n + w_n)_n$. Applying the $\Rightarrow$ part to $\tilde z_n = z_n + w_n$ and $\tilde w_n = -w_n$ implies that $z$ is a limit point of
$$
\tilde z_n + \tilde w_n = (z_n + w_n) + (-w_n) = z_n \, .
$$
you can apply the same argument to $z_n = (z_n + w_n) - w_n$.
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Q: Change cookie value on click issue I'm currently using this code:
$(function () {
$(".mainmenu a").click(function () {
$.cookie('value', '0', {
expires: 365,
path: '/'
});
});
});
This works fine on one directory.
I try to run the same code on another page two directories up, and when clicking on a menu anchor link, this is not changing the value of the cookie.
A: Try setting domain for the cookie :
domain: '*.mydomain.com',
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\section{Introduction}
\label{sec:intro}
Since the 1990'ies, electroweak precision data
from LEP and SLD~\cite{ALEPH:2005ema} were used together with accurate
Standard Model (SM) calculations to predict parameters of the theory.
A first impressive confirmation of the predictive power of global fits
in high-energy physics (HEP) was the discovery of the top quark
at the Tevatron~\cite{Abachi:1995iq, Abe:1995hr} in 1995, with a mass
in agreement with the predictions from global fits.
Knowledge of the top quark mass ($m_t$) made it possible
to constrain the mass of the Higgs boson ($M_H$). Increasing experimental
and theoretical precision and the inclusion of constraints from direct
Higgs boson searches from LEP and Tevatron narrowed the allowed mass
range over time~\cite{lepppe,pdgweb,GfitterWeb,HEPFitterWeb,Flacher:2008zq}.
The discovery of the Higgs boson at the
Large Hadron Collider (LHC)~\cite{Aad:2012tfa,Chatrchyan:2012xdj} with a mass around
$125\:$GeV impressively confirmed the SM at the quantum level.
The historical development of the constraints is
illustrated in Figs.~\ref{fig:mt_history} and \ref{fig:mh_history},
where the predictions of, respectively, $m_t$ and $M_{H}$ as derived from various global
fits and direct measurements~\cite{Abachi:1995iq, Abe:1995hr, tevewwg,
ATLAS:2014wva, Aad:2015nba, Khachatryan:2015hba, TevatronElectroweakWorkingGroup:2016lid, Abazov:2017ktz, Aad:2012tfa,Chatrchyan:2012xdj,Aad:2015zhl, ATLAS-CONF-2017-046, Sirunyan:2017exp, ATLAS:2017lqh}
are shown versus time.
With the measurement of $M_{H}$ the electroweak sector of
the SM is overconstrained and the strength of global fits can be
exploited to predict key observables such as the $W$ boson mass and
the effective electroweak mixing angle, with a precision exceeding that of the
direct measurements~\cite{Baak:2012kk}. Since the last update of our
fit~\cite{Baak:2014ora} improved experimental results have become
available that allow for more accurate tests of the internal
consistency of the SM. Among these are the first
determination of the $W$ boson mass at the LHC
by the ATLAS collaboration~\cite{Aaboud:2017svj},
new combined results of the top quark mass by the
LHC experiments~\cite{Khachatryan:2015hba, ATLAS:2017lqh},
a new combination of measurements of the
effective leptonic electroweak mixing angles from the Tevatron
experiments~\cite{Aaltonen:2018dxj}, a
Higgs boson mass combination released by the ATLAS and
CMS collaborations~\cite{Aad:2015zhl},
and an updated value
of the hadronic contribution to the running of the electromagnetic
coupling strength at the $Z$ boson mass~\cite{Davier:2017zfy}.
In the first part of this paper we present an update of the electroweak
fit including these new experimental
results and up-to-date theoretical predictions.
While the Higgs boson measurements so far agree
with a minimal scalar sector as implemented in the SM, the question
remains whether a more complex scalar sector may be realised in
nature, possibly featuring a variety of Higgs boson states.
Two-Higgs-doublet models (2HDM)~\cite{Haber:1978jt} are a popular SM
extension in which an additional $SU(2)_L\times
U(1)_Y$ scalar doublet field with hypercharge $Y=1$ is added to the SM
leading to the existence of five physical Higgs boson states, $h$, $H$,
$A$, $H^+$, and $H^-$, where the neutral $h$ may be identified
with the discovered 125$\:$GeV Higgs boson as is assumed in
this paper. The scalar $H$ boson has CP-even quantum number, $A$ is
a CP-odd pseudo-scalar, and $H^+$ and $H^-$ carry opposite electric
charge but have identical mass.
No experimental hint for additional scalar states has been observed
so far in direct searches~\cite{Abbiendi:2013hk,
Abazov:2008rn, Abazov:2009aa, Aaltonen:2009ke,
Aad:2013hla,Aad:2014kga,Aad:2015typ,Aad:2015nfa, Aaboud:2016cre,
Aaboud:2017rel,Aaboud:2017cxo,Aaboud:2017sjh,Aaboud:2017hnm,
Chatrchyan:2012vca,Khachatryan:2015qxa, Khachatryan:2015uua,
Khachatryan:2016are}.
In this situation global 2HDM fits, exploiting observables sensitive
to these additional Higgs boson states via quantum corrections,
can be used to constrain the allowed mass ranges and 2HDM mixing
parameters. In the second part of this article such constraints are
derived from a global fit using a combination of electroweak precision
data, flavour physics observables, the anomalous magnetic moment of the muon, and
measurements of the Higgs boson coupling strength to SM particles.
\begin{figure}[p]
\begin{center}
\includegraphics[width=\defaultSingleFigureScale\textwidth]{SMFigs/Historymtop.pdf}
\end{center}
\vspace{-0.3cm}
\caption[]{Prediction of the top quark mass versus year as obtained by various analysis
groups using electroweak precision data (grey~\cite{pdgweb}, light blue~\cite{lepppe},
green~\cite{GfitterWeb}).
The bands indicate the 68\% confidence level.
The direct $m_t$ measurements after the top quark discovery are displayed by the data points
(orange~\cite{Abachi:1995iq, Abe:1995hr, tevewwg, TevatronElectroweakWorkingGroup:2016lid, Abazov:2017ktz}, red~\cite{Aad:2015nba, Khachatryan:2015hba, ATLAS:2017lqh}, black~\cite{ATLAS:2014wva}). }
\label{fig:mt_history}
\end{figure}
\begin{figure}[p]
\begin{center}
\includegraphics[width=\defaultSingleFigureScale\textwidth]{SMFigs/HistoryMH.pdf}
\end{center}
\vspace{-0.3cm}
\caption[]{Prediction of the Higgs boson mass versus year as obtained by various analysis
groups using electroweak precision data (grey~\cite{pdgweb}, light blue~\cite{lepppe}, dark
blue~\cite{GfitterWeb}) and including direct search results (green~\cite{GfitterWeb}).
The bands indicate the 68\% confidence level.
The direct $M_H$ measurements after the Higgs boson discovery are displayed by the red
data points~\cite{Aad:2012tfa,Chatrchyan:2012xdj,Aad:2015zhl, ATLAS-CONF-2017-046, Sirunyan:2017exp}. }
\label{fig:mh_history}
\end{figure}
\section{Update of the global electroweak fit}
\label{sec:ewfit}
The updated global electroweak fit presented in this section uses the Gfitter
framework. For a detailed discussion of the experimental
data, the implementation of the theoretical predictions, and the
statistical procedure employed by Gfitter we refer the reader to our previous
publications~\cite{Flacher:2008zq,Baak:2011ze,Baak:2012kk,Baak:2014ora}.
A detailed list of all the observables, their values and uncertainties used
in the fit, is given in the first two columns of Table~\ref{tab:results}.
The description below discusses recent changes in the input quantities
and calculations.
\subsection{Input measurements and theoretical predictions}
\label{sec:impr}
The electroweak precision data measured at the $Z$ pole and their
correlations~\cite{ALEPH:2005ema} as well as the width of the $W$
boson have not changed since
our last analysis~\cite{Baak:2014ora}. The update to the
most recent world average values for the running $c$ and $b$ quark
masses~\cite{Olive:2016xmw} has negligible impact on the fit
result. This is also the case for the Run-1 LHC average
of the Higgs boson mass, $M_H=125.09\pm0.21\pm0.11\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$~\cite{Aad:2015zhl},
which we use now instead of a simple weighted average.\footnote{The Run-1 result on $M_H$
was confirmed by ATLAS and CMS measurements at
$\sqrt{s} = 13\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$~\cite{ATLAS-CONF-2017-046,Sirunyan:2017exp}.}
New results are available for several observables with
high sensitivity and potentially significant impact on the fit.
We include new measurements of the $W$ boson and top quark
masses as described in the following sections. For the
first time we include as a separate fit input (assuming no correlation with other measurements) the latest combination
of measurements of the effective leptonic electroweak mixing angle
from the Tevatron experiments\footnote{The \ensuremath{\seffsf{\ell}}\xspace measurements of ATLAS
($\ensuremath{\seffsf{\ell}}\xspace=0.2308\pm0.0012$~\cite{Aad:2015uau}) and CMS ($\ensuremath{\seffsf{\ell}}\xspace=0.23101\pm0.00052$~\cite{CMS:2017zzj}) are not
included in the fit because of their presently insufficient precision and unknown correlations.},
$\ensuremath{\seffsf{\ell}}\xspace=0.23148\pm0.00033$~\cite{Aaltonen:2018dxj},
and we use an updated value for the five quark flavour hadronic contribution to the
running of the electromagnetic coupling strength at $M_Z$,
$\ensuremath{\Delta\alpha_{\rm had}^{(5)}(M_Z^2)}\xspace=(2760\pm9)\cdot10^{-5}$~\cite{Davier:2017zfy}.
\subsubsection*{{\em W} boson mass}
The ATLAS collaboration has recently released the first LHC measurement
of the mass of the $W$ boson~\cite{Aaboud:2017svj}. Analysing their
$7\:\ensuremath{\mathrm{Te\kern -0.1em V}}\xspace$ dataset ATLAS measures $M_W = 80\,370 \pm 7_{\rm stat} \pm 11_{\rm exp\;syst} \pm
14_{\rm model}\:\ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$. We include this result in the
fit by combining it with the Tevatron
($M_W = 80\,387\pm 16\:\ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$~\cite{TevatronElectroweakWorkingGroup:2012gb})
and LEP combinations
($M_W = 80\,376 \pm 25_{\rm stat} \pm 22_{\rm syst}\:\ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$~\cite{LEPEWWG}) as follows.
Using information from Ref.~\cite{TevatronElectroweakWorkingGroup:2012gb}
we estimate the composition of individual statistical, experimental systematic
and modelling uncertainties in the combined Tevatron
result by $\pm8_{\rm stat} \pm 8_{\rm exp\;syst} \pm 12_{\rm model}\:\ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$.
All statistical and experimental systematic uncertainties are assumed to be
uncorrelated among the three input results (ATLAS, Tevatron, LEP) as is the
modelling uncertainty from LEP.
The impact of the unknown correlation
among the modelling uncertainties affecting the ATLAS and Tevatron measurements
has been studied by varying its value between zero and one. For a large range of
correlations we observe a stable
average of $M_W = 80\,379 \pm 13\:\ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$, which we use in the fit.\footnote{A
central value of $80\,379\:\ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$ is obtained for all possible
values of the model correlation, except for
coefficients exceeding 0.9 for which a value of $80\,380\:\ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$ is
found. A combined uncertainty of $13\:\ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$ is obtained for
correlations between 0.4 and 0.9, while smaller and larger correlation
values yield $12\:\ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$ and $14\:\ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$, respectively. These values have
been consistently calculated using the Best Linear Unbiased
Estimate (BLUE)~\cite{Valassi:2003mu} and the least-squares
averaging implemented in Gfitter~\cite{Flacher:2008zq}.}
\subsubsection*{Top quark mass}
For lack of a recent \ensuremath{m_{t}}\xspace world average, we attempt here for the purpose
of the fit a conservative combination
of the most precise kinematic \ensuremath{m_{t}}\xspace measurements obtained at the LHC.
We combine the $\ensuremath{m_{t}}\xspace$ averages from ATLAS
($172.51 \pm 0.27_\mathrm{stat} \pm 0.42_\mathrm{syst}\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$)~\cite{ATLAS:2017lqh}
and CMS
($172.47 \pm 0.13_\mathrm{stat} \pm 0.47_\mathrm{syst}\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$)~\cite{Khachatryan:2015hba},
which are based on 7 and 8\:\ensuremath{\mathrm{Te\kern -0.1em V}}\xspace data. These averages include results from the
dilepton~\cite{Aaboud:2016igd,
Chatrchyan:2011nb, Chatrchyan:2012ea},
lepton+jets~\cite{Aad:2015nba, Chatrchyan:2012cz} and
fully hadronic~\cite{Chatrchyan:2013xza} channels.
Assuming the overlapping fraction of the systematic uncertainties
to be fully correlated (which corresponds to a correlation coefficient
of 72\% between the two measurements)
we obtain the combined value
$\ensuremath{m_{t}}\xspace = 172.47 \pm 0.46\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$ ($p$-value of $0.84$), which we use as input in the fit.
The latest average from the D0 collaboration $\ensuremath{m_{t}}\xspace = 174.95 \pm
0.40_\mathrm{stat} \pm 0.64_\mathrm{syst}\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$~\cite{Abazov:2017ktz}
is barely compatible with the aforementioned average of the LHC
measurements. A combination of the D0 average with the LHC average
would result in $p$-values between $5\cdot10^{-3}$ and $3\cdot10^{-5}$,
depending on the assumed correlation between the systematic uncertainties.
The result from the CDF collaboration, $\ensuremath{m_{t}}\xspace = 173.16
\pm 0.57_\mathrm{stat} \pm
0.74_\mathrm{syst}\,\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$~\cite{CDF:mtopcombo}, agrees with
the LHC average, with $p$-values between $0.40$ and $0.51$ depending
on the correlation.
As in our previous work~\cite{Baak:2014ora} we assign an additional theoretical
uncertainty of $0.5\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$ to the value of $\ensuremath{m_{t}}\xspace$ from hadron collider
measurements due to the ambiguity in the kinematic top quark mass
definition~\cite{Hoang:2008yj,Hoang:2008xm,Buckley:2011ms,Moch:2014tta,
top13mangano}, the colour structure of the fragmentation
process~\cite{Skands:2007zg,Wicke:2008iz}, and the perturbative
relation between pole and \ensuremath{\overline{\rm MS}}\xspace mass currently known to three-loop
order~\cite{Chetyrkin:1999qi, Melnikov:2000qh, Hoang:2000yr}.
\subsubsection*{Theoretical calculations}
The theoretical higher-order calculations used in Gfitter have not changed since
our last update~\cite{Baak:2014ora}, except for new bosonic two-loop corrections
to the $Zb\overline{b}$ vertex~\cite{Dubovyk:2016aqv}.
For the effective weak mixing angle \ensuremath{\seffsf{f}}\xspace we use the parametrisations provided
in~\cite{Awramik:2004ge, Awramik:2006uz, Dubovyk:2016aqv}, which include full
two-loop electroweak~\cite{Awramik:2004ge, Awramik:2006uz} and partial three-loop and
four-loop QCD corrections~\cite{Avdeev:1994db, Chetyrkin:1995ix, Chetyrkin:1995js,
vanderBij:2000cg, Faisst:2003px, Schroder:2005db, Chetyrkin:2006bj, Boughezal:2006xk}.
For bottom quarks, the calculations from Refs.~\cite{Dubovyk:2016aqv, Awramik:2008gi} are used.
The new bosonic two-loop corrections are numerically small. They shift the prediction
of the forward-backward asymmetry for $b$ quarks $A_{\rm FB}^{0,b}$ by $1.3\cdot10^{-5}$,
which is two orders of magnitude smaller than the experimental uncertainty and thus
does not alter the fit results.
We use the parametrisation of the full two-loop result~\cite{Awramik:2003rn}
for predicting the mass of the $W$ boson, where we also include four-loop QCD
corrections~\cite{Schroder:2005db, Chetyrkin:2006bj, Boughezal:2006xk}.
Full fermionic two-loop corrections for the partial widths and branching ratios
of the $Z$ boson and the hadronic peak cross section $\sigma^0_{\rm had}$
are used~\cite{Freitas:2014hra, Freitas:2013dpa, Freitas:2012sy}.
The dominant contributions from final-state QED and QCD radiation
are included in the calculations~\cite{Chetyrkin:1994js, Baikov:2008jh, Baikov:2012er,
Kataev:1992dg, Czarnecki:1996ei, Harlander:1997zb}.
The width of the $W$ boson is known up to one electroweak
loop order, where we use the parametrisation given in Ref.~\cite{Cho:2011rk}.
The size and treatment of theoretical uncertainties are unchanged with respect
to our last analysis~\cite{Baak:2014ora}.
\subsection{Results}
\label{sec:sm}
\input Result_Table
The fit uses as input observables the quantities and values given in the left rows
of Table~\ref{tab:results}. The fit parameters are $M_H$, $M_Z$, $m_c$, $m_b$,
\ensuremath{m_{t}}\xspace, $\ensuremath{\Delta\alpha_{\rm had}^{(5)}(M_Z^2)}\xspace$, \ensuremath{\alpha_{\scriptscriptstyle S}}\xspace, as well as ten theoretical uncertainty ({\em nuisance})
parameters constrained by Gaussian functions (see Ref.~\cite{Baak:2014ora}
for more details).
The fit results in a minimum $\chi^2$ value of $18.6$ for $15$ degrees
of freedom, corresponding to a $p$-value of $0.23$. The
results of the full fit for each observable are
given in the fourth column of Table~\ref{tab:results}, together with
the uncertainties estimated from their $\Delta \chi^2 = 1$
profiles. The fifth column in Table~\ref{tab:results} gives the
results obtained without using the experimental measurement
corresponding to that row in the fit ({\it indirect determination}
of the observable).
The last column in Table~\ref{tab:results} corresponds to
the fits of the previous column but ignoring all theoretical
uncertainties~\cite{Baak:2014ora}.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{SMFigs/PullPlot.pdf}\hspace{0.3cm}
\includegraphics[width=0.46\textwidth]{SMFigs/TablePlot.pdf}
\end{center}
\vspace{-0.1cm}
\caption[]{Left: comparison of the fit results with the input
measurements in units of the experimental uncertainties.
Right: comparison of the fit results and the input measurements
with the indirect determinations in
units of the total uncertainties. Analog results for the indirect determinations illustrate the impact of their uncertainties on the total uncertainties.
The indirect determination of an
observable corresponds to a fit without using the constraint from
the corresponding input measurement.}
\label{fig:pulls2}
\end{figure}
The left-hand panel of Fig.~\ref{fig:pulls2} displays the pulls each
given by the difference of the global fit result of an observable (fourth column of
Table~\ref{tab:results}) and the corresponding input measurement
(second column of Table~\ref{tab:results}) in units of the measurement
uncertainty. The right-hand panel of Fig.~\ref{fig:pulls2} shows
the difference between the global fit result (fourth column of
Table~\ref{tab:results}) as well as the input measurements
(first column of Table~\ref{tab:results}) with the
indirect determination (fifth column of Table~\ref{tab:results})
for each observable in units of the total uncertainty obtained by adding
in quadrature the uncertainties of the indirect determination and the
input measurement.
The analog result using the value of the indirect
determination, trivially centered around zero, are shown to illustrate the impact of its
uncertainty on the total uncertainty.
As in our previous fits, a tension is observed in the
leptonic and hadronic asymmetry observables, which is largest in
the forward-backward asymmetry of the $b$ quarks, $A^{0,b}_{\rm FB}$. The
impact of the new Tevatron $\ensuremath{\seffsf{\ell}}\xspace$ measurement on the fit
result is small due to yet insufficient precision.
\begin{figure}
\begin{center}
\includegraphics[width=0.60\textwidth]{SMFigs/MainObservables.pdf}
\end{center}
\vspace{-0.5cm}
\caption[]{Comparison of the constraints on $\ensuremath{M_{H}}\xspace$ obtained indirectly
from individual observables with the fit result and the direct
LHC measurement. For the indirect determinations among the four
observables providing the strongest $M_H$ constraints (namely
\ensuremath{\seffsf{\ell}}\xspace, $M_W$, $A^{0,b}_{\rm FB}$ and $A_\ell$) only the one indicated in
a given row of the plot is included in the fit. The results shown are not fully independent. }
\label{fig:mainobs}
\end{figure}
Figure~\ref{fig:mainobs} displays the indirect determination of the Higgs boson
mass from fits in which among the four observables providing
the strongest $M_H$ constraints (namely \ensuremath{\seffsf{\ell}}\xspace, $M_W$, $A^{0,b}_{\rm FB}$ and $A_\ell$) only the one indicated in a given row of the plot
is included. The results are compared to the direct $M_H$ measurement
as well as to the result of a fit including all data except the direct $M_H$
measurement. This latter fit gives the indirect determination
\begin{equation}
M_H = 90^{\,+21}_{\,-18}\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace\,,
\label{eq:mh}
\end{equation}
which is in agreement with the direct measurement within $1.7$
standard deviations. The value is lower by
$3\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$ than in our previous result ($93^{\,+25}_{\,-21}\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$)~\cite{Baak:2014ora}
due to the lower value of $\ensuremath{m_{t}}\xspace$ used here. The reduced uncertainty of
$^{\,+21}_{\,-18}\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$ compared to $^{\,+25}_{\,-21}\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$ previously,
is due to the smaller uncertainty in $m_t$.
When assuming perfect knowledge of $m_t$, $\ensuremath{\Delta\alpha_{\rm had}^{(5)}(M_Z^2)}\xspace$ and $\ensuremath{\alpha_{\scriptscriptstyle S}}\xspace(M_{Z}^{2})$,
the uncertainty is reduced by $\,^{\,+4.5}_{\,-3.5}$, $\,^{\,+5}_{\,-4}$ and
$\pm2\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$, respectively.
The predictions of $M_H$ using $A_\ell$, $A^{0,b}_{\rm FB}$ and $M_W$
(LEP and Tevatron) concur with earlier findings~\cite{Flacher:2008zq}.
The predictions derived from the ATLAS $M_W$ and
Tevatron \ensuremath{\seffsf{\ell}}\xspace measurements are in agreement with the direct
$M_H$ measurement.
An important consistency test of the SM is the simultaneous indirect
determination of \ensuremath{m_{t}}\xspace and $M_W$. A scan of the confidence level (CL)
profile of $M_W$ versus $m_t$ is shown in Fig.~\ref{fig:mwmt} for the
scenarios where the direct $M_H$ measurement is included in the fit
(blue) or not (grey). Both contours agree with the direct measurements
(green bands and ellipse for two degrees of freedom).
\begin{figure}
\begin{center}
\includegraphics[width=\defaultSingleFigureScale\textwidth]{SMFigs/Scan2D_MWvsmt.pdf}
\end{center}
\vspace{-0.3cm}
\caption[]{Contours at 68\% and 95\% CL obtained from scans of $M_W$
versus $m_t$ for the fit including (blue) and excluding the $M_H$
measurement (grey), as compared to the direct measurements (green vertical
and horizontal $1\sigma$ bands, and two-dimensional $1\sigma$ and $2\sigma$ ellipses).
The direct measurements of $M_W$ and $m_t$ are excluded from the fits. }
\label{fig:mwmt}
\end{figure}
Figure~\ref{fig:scans} displays \DeltaChi fit profiles for the indirect
determination of some of the electroweak observables.\footnote{The indirect determination
profiles are obtained by excluding the input measurement of the respective observable
from the fit (see figure legends).}
The results are shown for fits
including (blue) and excluding (grey) the direct $M_H$ measurement
highlighting the strong impact of the $M_H$ measurement on the fit
constraints. The direct measurement of each observable with its $1\sigma$ uncertainty
are indicated by the data points at $\Delta\chi^2=1$. The detailed predictions of the
fit are given in Table~\ref{tab:results}.
\begin{figure}
\begin{center}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{SMFigs/WMassScan.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{SMFigs/TopScan.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{SMFigs/Sin2ThetaScan2.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{SMFigs/HiggsScan.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{SMFigs/dalphahadScan.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{SMFigs/AlphasScan.pdf}
\end{center}
\vspace{-0.1cm}
\caption[]{Scans of \DeltaChi as a function of $M_W$ (top left),
$m_t$ (top right), \ensuremath{\seffsf{\ell}}\xspace (middle left), $M_H$ (middle right),
$\ensuremath{\Delta\alpha_{\rm had}^{(5)}(M_Z^2)}\xspace$ (bottom left) and $\ensuremath{\alpha_{\scriptscriptstyle S}}\xspace(M_{Z}^{2})$ (bottom right),
under varying conditions. The results of the fits without and with the
measurement of \ensuremath{M_{H}}\xspace as input are shown in grey and blue colours,
respectively. The solid and dotted lines represent the results when
including or excluding the theoretical uncertainties.
The data points with uncertainty bars indicate the
direct measurements of a given observable. }
\label{fig:scans}
\end{figure}
The fit indirectly determines the $W$ mass to be
\begin{eqnarray}
M_W &=& 80.3535
\pm 0.0027_{m_t} \pm 0.0030_{\delta_{\rm theo} m_t} \pm 0.0026_{M_Z} \pm 0.0026_{\ensuremath{\alpha_{\scriptscriptstyle S}}\xspace} \nonumber \\
& & \phantom{80.3535}
\pm 0.0024_{\Delta\alpha_{\rm had}} \pm 0.0001_{M_H} \pm 0.0040_{\delta_{\rm theo} M_W}\:{\rm GeV}\,, \nonumber \\[0.2cm]
&=& 80.354 \pm 0.007_{\rm tot} \:{\rm GeV} \,,
\label{eq:mw}
\end{eqnarray}
and the effective leptonic weak mixing angle as
\begin{eqnarray}
\ensuremath{\seffsf{\ell}}\xspace &=& 0.231532
\pm 0.000011_{m_t} \pm 0.000016_{\delta_{\rm theo} m_t} \pm 0.000012_{M_Z} \pm 0.000021_{\ensuremath{\alpha_{\scriptscriptstyle S}}\xspace} \nonumber \\
& & \phantom{0.231508}
\pm 0.000035_{\Delta\alpha_{\rm had}} \pm 0.000001_{M_H} \pm 0.000040_{\delta_{\rm theo} \ensuremath{\seffsf{\ell}}\xspace} \,, \nonumber \\[0.2cm]
&=& 0.23153 \pm 0.00006_{\rm tot} \:.
\label{eq:sin2t}
\end{eqnarray}
When evaluating $\ensuremath{\seffsf{\ell}}\xspace$ through the parametric formula from Ref.~\cite{Awramik:2006uz}, an
upward shift of $2\cdot 10^{-5}$ with respect to the fit result is observed, mostly due to the
inclusion of $M_W$ in the fit. Using the parametric formula the total
uncertainty is larger by $0.6\cdot 10^{-5}$, as the global fit exploits
the additional constraint from $M_W$. The fit also constrains
the nuisance parameter associated with the theoretical uncertainty in the calculation of
$\ensuremath{\seffsf{\ell}}\xspace$, resulting in a reduced theoretical uncertainty
of $4.0\cdot 10^{-5}$ compared to the $4.7\cdot 10^{-5}$ input uncertainty.
The mass of the top quark is indirectly determined to be
\begin{eqnarray}
m_t &=& 176.4\pm 2.1\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace\,,
\end{eqnarray}
with a theoretical uncertainty of 0.6\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace induced by the theoretical uncertainty on
the prediction of $M_W$. The largest potential to improve the precision of the
indirect determination of $m_t$ is through a more precise measurement of $M_W$.
Perfect knowledge of $M_W$ would result in an uncertainty on $m_t$ of 0.9\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace.
The strong coupling strength at the $Z$-boson mass scale is determined to be
\begin{eqnarray}
\ensuremath{\alpha_{\scriptscriptstyle S}}\xspace(M_{Z}^{2}) &=& 0.1194\pm 0.0029\,,
\end{eqnarray}
which corresponds to a determination at full next-to-next-to leading order (NNLO) for
electroweak and strong contributions, and partial strong next-to-NNLO (NNNLO) corrections.
The theory uncertainty of this result is $0.0009$, which is
shared in equal parts between missing higher orders in the calculations of the radiator
functions and the partial widths of the $Z$ boson.
The most important constraints on $\ensuremath{\alpha_{\scriptscriptstyle S}}\xspace(M_{Z}^{2})$ come from the measurements of
$R^{0}_{\l}$, $\Gamma_{Z}$ and $\sigma_{\rm had}^{0}$, also shown in Fig.~\ref{fig:scans}.
The values of $\ensuremath{\alpha_{\scriptscriptstyle S}}\xspace(M_{Z}^{2})$ obtained
from the individual measurements are $0.1237\pm0.0043$ ($R^{0}_{\l}$),
$0.1209\pm0.0049$ ($\Gamma_{Z}$) and $0.1078\pm0.0076$ ($\sigma_{\rm had}^{0}$).
A fit to all three measurements results in a value of $0.1203 \pm 0.0030$, which is
only slightly less precise than the result of the full fit.
The results obtained for $\ensuremath{\alpha_{\scriptscriptstyle S}}\xspace(M_{Z}^{2})$ are stable with respect to additional
invisible beyond-the-standard-model contributions to $\Gamma_{Z}$.
No significant deviation from the direct measurements is observed in
any of these predictions.
The indirect determinations of $M_W$ and $\ensuremath{\seffsf{\ell}}\xspace$ outperform the
direct measurements in precision while the indirect determinations of $m_t$ and
$\ensuremath{\alpha_{\scriptscriptstyle S}}\xspace(M_{Z}^{2})$ are competitive to other experimental results.
\subsubsection*{Oblique parameters}
Using the updated SM reference values $M_{H,{\rm ref}}=125$\:GeV and $m_{t,{\rm
ref}}=172.5$\:GeV we obtain for the {\it oblique parameters}
denoted $S$, $T$, $U$ ~\cite{Peskin:1990zt,Peskin:1991sw} the following values:
\begin{equation}
S= \SParam\,, \hspace{0.5cm}
T= \TParam\,, \hspace{0.5cm}
U=\UParam\,,
\label{eq:stu}
\end{equation}
with correlation coefficients of $\STParamCor$ between $S$ and $T$, $\SUParamCor$ ($\TUParamCor $) between $S$
and $U$ ($T$ and $U$).
Fixing $U=0$ one obtains $S|_{U=0}= \SParamNU$ and $T|_{U=0}= \TParamNU$, with a correlation coefficient of
$\STParamCorNo$.
The constraints on $S$ and $T$ for a fixed
value of $U=0$ are shown in Fig.~\ref{fig:STU}.
\begin{figure}
\begin{center}
\includegraphics[width=\defaultSingleFigureScale\textwidth]{SMFigs/SM_S_vs_T_U0.pdf}
\end{center}
\vspace{-0.3cm}
\caption[]{Constraints in the oblique parameters $S$ and $T$, with the $U$ parameter fixed to zero,
using all observables (blue). Individual constraints are shown from the asymmetry and
direct \ensuremath{\seffsf{\ell}}\xspace measurements (yellow), the $Z$ partial and total widths (green)
and $W$ mass and width (red), with confidence levels drawn for one degree of freedom.
\label{fig:STU}}
\end{figure}
\pagebreak
\section{Global fits in the two-Higgs-doublet model}
\label{sec:2hdm}
Combining information from the electroweak precision data, Higgs boson coupling
measurements, flavour observables and the anomalous magnetic moment of the muon
we derive in this section constraints on parameters of various 2HDM scenarios.
Besides the four mass parameters for the scalars, $M_{h}$,
$M_{H}$, $M_{A}$, and $M_{H^{\pm}}$, the 2HDM introduces the angle $\alpha$,
which describes the mixing of the two neutral Higgs fields $h$ and $H$,
and the angle $\beta$ that fixes the ratio of the vacuum expectation values
of the two Higgs doublets, $\ensuremath{\tan\!\beta}\xspace=v_2/v_1$. We only
consider 2HDM scenarios with a $\mathbb{Z}_2$
symmetric potential with a dimension-two softly broken term
proportional to the scale parameter $M_{12}^2$.
Depending on the Yukawa couplings of the two Higgs doublets, the 2HDM may
introduce dangerous flavour-changing neutral currents (FCNCs) and CP
violating interactions. CP conservation can be maintained by fixing the
Higgs boson couplings for up-type quarks, down-type quarks, and leptons to
specific values~\cite{Haber:1978jt,Branco:2011iw}.
In this work, four CP conserving 2HDM scenarios are studied. In the {\em Type-I} scenario,
only one of the two Higgs doublets is allowed to couple to fermions, while the
other couples to the gauge bosons. The {\em Type-II} scenario is defined by a
separation of the Yukawa interactions: one Higgs doublet couples only to up-type
quarks and the other only to down-type quarks and charged leptons. The Type-II
2HDM resembles the Higgs sector in the Minimal Supersymmetric Standard Model.
The third, {\em lepton specific} scenario is similar to the Type-I model with the
difference that leptons only couple to the other Higgs doublet that does not
interact with the quarks. Finally, the fourth, {\em flipped} scenario is the same
as the Type-II model with swapped lepton couplings to the Higgs doublets.
Throughout this section the lightest scalar Higgs boson, $M_{h}$, is identified
with the observed Higgs boson with mass fixed to $125.09\pm0.24$~GeV~\cite{Aad:2015zhl}.
If not stated otherwise, all other 2HDM model parameters are allowed to vary
within the intervals: $130<M_{H},M_{A}<1000\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$, $100<M_{H^{\pm}}<1000\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$,
$0 \le \beta-\alpha \le \pi$, $0.001 < \ensuremath{\tan\!\beta}\xspace < 50$,
and $-8\cdot10^5< M_{12}^2<8\cdot10^5\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace^2$. No contribution from new physics
other than the 2HDM is assumed.
Direct searches for additional Higgs bosons in collider experiments can be
interpreted in the context of the 2HDM (see, for example, Ref.~\cite{Arbey:2017gmh}).
However, due to the large freedom in the choice of the 2HDM parameters, these search results
provide only weak absolute exclusion limits on the masses of the scalars.
From searches for a charged Higgs boson by the LEP experiments~\cite{Abbiendi:2013hk}
a lower limit of $M_{H^{\pm}}>72.5\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$ was reported for the Type-I scenario, while
a limit of $M_{H^{\pm}}\gtrsim150\:\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$ can be derived from searches at the LHC for
the Type-II scenario~\cite{Arbey:2017gmh}. Stronger mass limits mainly on $M_{H^{\pm}}$
can be obtained for specific regions of $\ensuremath{\tan\!\beta}\xspace$.
\subsection{Constraints from Higgs boson coupling measurements}
A second Higgs doublet modifies the coupling strengths of the lightest
neutral Higgs boson $h$ to SM particles compared to those of the SM Higgs boson.
The modifications depend on the 2HDM scenario and parameters in particular
the angles $\alpha$ and $\beta$. Constraints on $h$ are derived from the joint
ATLAS and CMS Higgs boson coupling analysis~\cite{Khachatryan:2016vau}
in which measurements sensitive to
five Higgs boson production modes (ggF, VBF, $WH$, $ZH$, $t\overline{t}H$)
and five decay modes ($\gamma\gamma$, $WW$, $ZZ$, $\tau\tau$, $b\overline{b}$) were
combined.
We make use of the relative signal strengths $\mu_{ij}$ defined as the ratio
of measured over predicted cross section times branching ratio,
$\mu_{ij}=(\sigma_i\cdot\ensuremath{{\mathcal B}}\xspace_j)/(\sigma_i^{\rm SM}\cdot\ensuremath{{\mathcal B}}\xspace_j^{\rm SM})$.
We include the 20 (out of the 25 possible) $\mu_{ij}$ parameters determined by ATLAS and CMS together with their uncertainties
and correlations.
A validation of our results is discussed in the Appendix on
page~\pageref{appendix}.
The corresponding SM predictions and uncertainties are taken from
Ref.~\cite{Heinemeyer:2013tqa}. The signal strength measurements are compared
with the theory predictions for the 2HDM calculated with the program
\texttt{2HDMC}~\cite{Eriksson:2009ws}.\footnote{\texttt{2HDMC} computes the couplings
of all five Higgs boson states to SM particles for a given set of parameters in a
CP conserving 2HDM with general Yukawa structure. From these couplings, production
and decay rates of the Higgs boson states can be derived. Most decay widths are
calculated at leading QCD order in \texttt{2HDMC}. Higher order QCD corrections
are included for couplings to fermion and gluon pairs.}
In the calculation of the $\mu_{ij}$ for the 2HDM also the denominator $\sigma_i^{\rm SM}\cdot\ensuremath{{\mathcal B}}\xspace_j^{\rm SM}$
is determined using \texttt{2HDMC} for consistency.
Since more precise theory predictions for the SM cross sections and branching ratios exist and are used
for the normalisation of the results in~\cite{Khachatryan:2016vau},
theory uncertainties in the SM prediction from~\cite{Heinemeyer:2013tqa} are taken into account as additional
scaling (nuisance) parameters of the $\mu_{ij}$.
The constraints from the Higgs boson signal strength measurements on the
four 2HDM scenarios are shown as 68\% and 95\% CL allowed regions
in the $\ensuremath{\tan\!\beta}\xspace$ versus $\cos(\beta-\alpha)$
plane in Fig.~\ref{fig:bma_tanb}.\footnote{Theoretical
bounds from positivity of the Higgs potential, tree-level unitarity, and
perturbativity of the quartic Higgs boson couplings as implemented in
\texttt{2HDMC} were found to give no additional constraints in these
figures.}
The angles $\alpha$ and $\beta$ are highly constrained in all 2HDM scenarios
except for Type-I. The allowed parameter regions are concentrated in two bands
corresponding to solutions with $\beta\pm\alpha=\pi/2$. For $\beta-\alpha=\pi/2$,
the Yukawa structure of the SM is reproduced ({\it alignment limit}). The case $\beta+\alpha=\pi/2$
differs from the SM-like Yukawa couplings by a sign flip that is still allowed by the
combined coupling strengths measurements. These constraints are differently
pronounced in the four 2HDM scenarios as they depend on the Yukawa coupling strengths.
In the Type-I scenario (top left panel in Fig.~\ref{fig:bma_tanb}) the Yukawa couplings
of $h$ to all fermions are proportional to $\cos\alpha/\sin\beta$. The constraints
are stronger in the other three scenarios as the Yukawa coupling for at least
one fermion type is proportional to $-\sin\alpha/\cos\beta$. In the flipped scenario
(bottom right panel) only the Yukawa coupling to down-type quarks is given by
$-\sin\alpha/\cos\beta$, which is constrained by the measurements of
$H\to{b\overline{b}}$. Measurements of $H\to\tau^+\tau^-$ give stronger bounds
in the Type-II (top right panel) and lepton specific (bottom left panel) scenarios
where the Yukawa couplings to leptons is given by $-\sin\alpha/\cos\beta$.
In all scenarios, the measurements of Higgs boson decays to $W$ and $Z$ boson pairs
disfavour large values of $\cos(\beta-\alpha)$.
Similar constraints have been obtained by the ATLAS collaboration~\cite{Aad:2015pla}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/THDM_betaminusalpha_vs_tanBeta_TypeI.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/THDM_betaminusalpha_vs_tanBeta_TypeII.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/THDM_betaminusalpha_vs_tanBeta_TypeIV.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/THDM_betaminusalpha_vs_tanBeta_TypeIII.pdf}
\vspace{-0.8cm}
\end{center}
\caption[]{Results from 2HDM fits using the ATLAS and CMS combined Higgs coupling
strength measurements. Shown are allowed parameter regions (68\% and 95\% CL) for the four
2HDM scenarios from scans of $\ensuremath{\tan\!\beta}\xspace$ versus $\cos(\beta-\alpha)$:
Type-I (top left), Type-II (top right), lepton specific (bottom left) and
flipped (bottom right) 2HDMs. The figure insets show a zoom of the
region with $\ensuremath{\tan\!\beta}\xspace<1$.
}
\label{fig:bma_tanb}
\end{figure}
\subsection{Constraints from flavour observables}
Because tree-level FCNC transitions are forbidden by construction in the four 2HDM
scenarios considered, flavour violation only arises at loop level by the exchange of a
charged Higgs boson with observable
strength depending on the parameters $M_{H^{\pm}}$ and $\ensuremath{\tan\!\beta}\xspace$.
\subsubsection*{Experimental input data and theory calculation}
\begin{table}[t]
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcc}
\hline\noalign{\smallskip}
Observable & Value & Reference \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\ensuremath{{\mathcal B}}\xspace(B\rightarrow X_s\gamma)$ for $E_{\gamma}>1.6$~GeV & $(3.32\pm0.15_{\rm stat+syst})\cdot10^{-4}\pm7\%_{\rm theo}$ & \cite{HFLAV,Misiak:2006zs,Misiak:2015xwa}\\
\noalign{\smallskip}\hline\noalign{\smallskip}
$R(D)$ & $0.407\pm0.039_{\rm stat}\pm0.024_{\rm syst}\pm0.008_{\rm theo}$ & ~\cite{HFLAV,Aoki:2016frl} \\
\noalign{\smallskip}
$R(D^{*})$ & $0.304\pm0.013_{\rm stat}\pm0.007_{\rm syst}\pm0.003_{\rm theo}$ & \cite{HFLAV,Fajfer:2012vx}\\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\ensuremath{{\mathcal B}}\xspace(B\rightarrow\tau\nu)$ & $(1.06\pm0.19)\cdot10^{-4}$ & \cite{HFLAV} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\ensuremath{{\mathcal B}}\xspace(B_s\rightarrow\mu\mu)$ (CMS) & $(2.8^{\,+1.0}_{\,-0.9})\cdot10^{-9}$ & \cite{CMS:2014xfa} \\
$\ensuremath{{\mathcal B}}\xspace(B_s\rightarrow\mu\mu)$ (LHCb)& $(3.0\pm{0.6_{\rm stat}}^{\,+0.3}_{\,-0.2_{\rm syst}})\cdot10^{-9}$ & \cite{Aaij:2017vad} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\ensuremath{{\mathcal B}}\xspace(B_d\rightarrow\mu\mu)$ (CMS)& $(4.4^{\,+2.2}_{\,-1.9})\cdot10^{-10}$ & ~\cite{CMS:2014xfa} \\
$\ensuremath{{\mathcal B}}\xspace(B_d\rightarrow\mu\mu)$ (LHCb) & $(1.5^{\,+1.2}_{\,-1.0_{\rm stat}}{}^{\,+0.2}_{\,-0.1_{\rm syst}})\cdot10^{-10}$ & ~\cite{Aaij:2017vad} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\ensuremath{{\mathcal B}}\xspace(D_s\rightarrow\mu\nu)$ & $(5.54\pm0.20_{\rm stat}\pm0.13_{\rm syst})\cdot10^{-3}$ & \cite{HFLAV}\\
$\ensuremath{{\mathcal B}}\xspace(D_s\rightarrow\tau\nu)$ & $(5.51\pm0.18_{\rm stat}\pm0.16_{\rm syst})\cdot10^{-2}$ & \cite{HFLAV}\\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\Delta m_{d}$ & $(0.5065\pm0.0019)$~ps$^{-1}$ & \cite{HFLAV}\\
$\Delta m_{s}$ & $(17.757\pm0.021)$~ps$^{-1}$ & \cite{HFLAV}\\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\ensuremath{{\mathcal B}}\xspace(K\rightarrow \mu\nu)/\ensuremath{{\mathcal B}}\xspace(\pi\rightarrow \mu\nu)$ & $0.6357\pm0.0011$ & \cite{Olive:2016xmw}\\
\noalign{\smallskip}\hline
\end{tabular*}
\caption{Flavour physics observables and values used in the 2HDM fit.}
\label{tab:flav_obs}
\end{table}
The flavour physics observables taken into account in our analysis are listed in
Table~\ref{tab:flav_obs} and briefly described below.
For the branching fraction of the radiative decay
$\ensuremath{{\mathcal B}}\xspace(B\rightarrow X_s\gamma)$ with $E_{\gamma}>1.6$~GeV we use the value of
the Heavy Flavour Averaging Group (HFLAV)~\cite{HFLAV} which combines
measurements from the BABAR~\cite{Aubert:2007my,Lees:2012ym,Lees:2012wg},
Belle~\cite{Limosani:2009qg,Saito:2014das,Belle:2016ufb}, and CLEO~\cite{Chen:2001fja}
experiments. The prediction for $\ensuremath{{\mathcal B}}\xspace(B\rightarrow X_s\gamma)$ has been
adopted from Ref.~\cite{Misiak:2015xwa} and includes QCD corrections up
to NNLO~\cite{Hermann:2012fc}. We make use of the code implementation kindly provided by M.~Misiak.
HFLAV also combined measurements of the semileptonic decay ratios of neutral $B$ mesons
$R(D^{(*)})=\ensuremath{{\mathcal B}}\xspace(\Bzb\to D^{(*)+}\tau^-\ensuremath{\overline{\nu}}\xspace)/\ensuremath{{\mathcal B}}\xspace(\Bzb\to D^{(*)+}\ell^-\ensuremath{\overline{\nu}}\xspace)$
by BABAR~\cite{Lees:2012xj,Lees:2013uzd},
Belle~\cite{Huschle:2015rga,Sato:2016svk,Hirose:2016wfn}, and
LHCb~\cite{Aaij:2015yra} with a correlation of $-0.23$ between the two
observables that is taken into account in the fit.
The prediction of $R(D^{(*)})$~\cite{Aoki:2016frl,Fajfer:2012vx,Enomoto:2015wbn} includes
tree-level contributions of a charged Higgs boson and is based on form
factors evaluated in Heavy-Quark Effective Theory.
Variations of the parameters $\rho^2_{R(D)}$, $\rho^2_{R(D^*)}$,
$R_1(1)$, and $R_2(1)$ are included in the fit with
values and correlations taken from Ref.~\cite{HFLAV}.
For the branching ratio $\ensuremath{{\mathcal B}}\xspace(B\rightarrow\tau\nu)$ we use the HFLAV
average~\cite{HFLAV} of measurements from BABAR~\cite{Lees:2012ju}
and Belle~\cite{Kronenbitter:2015kls}.
For the prediction of $\ensuremath{{\mathcal B}}\xspace(B\rightarrow\tau\nu)$ in the 2HDM we use the calculation
from Ref.~\cite{Isidori:2006pk}, which contains tree-level contributions of a charged Higgs boson where the
leading $\ensuremath{\tan\!\beta}\xspace$ corrections are resummed to all orders~\cite{Isidori:2006pk}.
The theoretical uncertainties in $|V_{ub}|$ and $f_{B_d}$ (see below) are included.
The latest measurements of $\ensuremath{{\mathcal B}}\xspace(B_s\rightarrow\mu\mu)$ and $\ensuremath{{\mathcal B}}\xspace(B_d\rightarrow\mu\mu)$
from LHCb ~\cite{Aaij:2017vad} are combined in our fits with the
CMS result~\cite{CMS:2014xfa},
assuming them uncorrelated. Their theoretical predictions in the 2HDM include NLO corrections
given in Refs.~\cite{Li:2014fea,Cheng:2015yfu}. The SM contribution to these observables are known up
to three-loop level in QCD and include NLO electroweak
corrections~\cite{Hermann:2013kca,Bobeth:2013tba,Bobeth:2013uxa}.
The predictions depend on the CKM matrix elements $|V_{tb}|$ and $|V_{ts}|$ or $|V_{td}|$, respectively,
and on the respective hadronic parameters $f_{B_s}$ and $f_{B_d}$.
Uncertainties in these parameters are taken into account in the fit.
The charged Higgs boson of the 2HDM contributes to the leptonic decays
of $D_s$ mesons. For the observables $\ensuremath{{\mathcal B}}\xspace(D_s\rightarrow\mu\nu)$ and
$\ensuremath{{\mathcal B}}\xspace(D_s\rightarrow\tau\nu)$ we use the HFLAV averages~\cite{HFLAV} of
measurements from BABAR~\cite{delAmoSanchez:2010jg}, Belle~\cite{Zupanc:2013byn},
and CLEO~\cite{Onyisi:2009th,Alexander:2009ux,Naik:2009tk}.
For the 2HDM predictions we use the analytic expression for the 2HDM tree-level contribution
to $\ensuremath{{\mathcal B}}\xspace(D_s\rightarrow\ell\nu)$ from Ref.~\cite{Akeroyd:2009tn} that allows
us to vary the dependencies on $|V_{cs}|$ and $f_{D_s}$ in the fit.
The charged Higgs boson also contributes via box diagrams to the mixing of the neutral
$B_d$ and $B_s$ mesons altering the mixing frequencies
$\Delta m_{d}$ and/or $\Delta m_{s}$.
We use again the HFLAV~\cite{HFLAV} experimental averages for these quantities.
Their predictions in the 2HDM are obtained from analytic expressions of the full one-loop calculation of
Refs.~\cite{Chang:2015rva,Enomoto:2015wbn} neglecting small terms proportional
to $m_{b}^2/M_{W}^2$. The predictions depend on the CKM matrix elements
$|V_{td}|$ and $|V_{ts}|$, the bag parameters $\hat{B}_{d}$ and $\hat{B}_{s}$,
and the decay constants $f_{B_d}$ and $f_{B_s}$, respectively, and the correction
factor $\eta_B$.
Finally, the 2HDM contributes at leading order to the ratio
$\ensuremath{{\mathcal B}}\xspace(K\rightarrow \mu\nu)/\ensuremath{{\mathcal B}}\xspace(\pi\rightarrow \mu\nu)$ for which we use
a value adopted from Ref.~\cite{Olive:2016xmw}, based on the measurement of
the kaon decay rates~\cite{Ford:1967zz}, and the 2HDM prediction from Ref.~\cite{Enomoto:2015wbn}.
The ratio involves the CKM matrix elements $|V_{us}|$ and $|V_{ud}|$,
the decay constants $f_K$ and $f_\pi$, and an electromagnetic correction
$\delta_{\rm EM}^{K/\pi}$.
As input values for the unitarity CKM matrix we use the latest available
results for the all-orders Wolfenstein parameters $A$, $\lambda$, $\rhobar$, $\etabar$
from Ref.~\cite{Charles:2004jd,Charles:2015gya,CKMfit}, taking them uncorrelated.
A fully consistent analysis would
require a combined fit of the Wolfenstein and 2HDM parameters within the
2HDM~\cite{Deschamps:2009rh}, which is however beyond the scope of this paper.
Studies in Ref.~\cite{Enomoto:2015wbn} and by ourselves have shown that the
numerical impact of the 2HDM on the CKM parameters is modest.
For the CKM element $|V_{ub}|$, occurring mainly in the prediction of the leptonic
$B^\pm$ branching fraction, we take the average of inclusive and exclusive
measurements~\cite{Bevan:2014iga} instead of the CKM fit prediction to
allow for a more conservative uncertainty in view of the tension between
the inclusive and exclusive results.
The input parameters used in the fit are summarised in Table~\ref{tab:flav_input}.
\begin{table}[t]
\setlength{\tabcolsep}{0.0pc}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lc|lc}
\hline\noalign{\smallskip}
Parameter & Value & Parameter & Value \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$A$ & $0.8250^{\,+0.0071}_{\,-0.0111}$ & $f_{D_s}$ & $(248.2\pm 0.3_{\rm stat} \pm 1.9_{\rm syst})$~MeV\\
$\lambda$ & $0.22509^{\,+0.00029}_{\,-0.00028}$ & $f_{B_s}$ & $(225.6\pm 1.1_{\rm stat}\pm 5.4_{\rm syst})$~MeV\\
$\rhobar$ & $0.1598^{\,+0.0076}_{\,-0.0072}$ & $f_{B_s}/f_{B_d}$ & $1.205 \pm 0.004_{\rm stat} \pm 0.007_{\rm syst}$\\
$\etabar$ & $0.3499^{\,+0.0063}_{\,-0.0061}$ & $\hat{B}_{s}$& $1.320 \pm 0.017_{\rm stat} \pm 0.030_{\rm syst}$ \\
$|V_{ub}|$ & $0.00395\pm0.00038_{\rm exp}\pm0.00039_{\rm theo}$~~~ & $\hat{B}_{s}/\hat{B}_{d}$ & $1.023 \pm 0.013_{\rm stat} \pm 0.014_{\rm syst}$\\
$\rho^2_{R(D)}$ & $1.128\pm0.033$ & $\eta_B$ & $0.551 \pm 0.0022$ \\
$\rho^2_{R(D^*)}$ & $1.21\pm0.027$ & $f_{K}/f_{\pi}$ & $1.1952 \pm 0.0007_{\rm stat} \pm 0.0029_{\rm syst}$\\
$R_1(1)$ & $1.404\pm0.032$ & $\delta_{\rm EM}^{K/\pi}$ & $-0.0070 \pm 0.0018$\\
$R_2(1)$ & $0.854\pm0.020$ & & \\
\noalign{\smallskip}\hline
\end{tabular*}
\caption{Parameters used in the fit to the flavour observables. Most values are taken from
latest available version of the CKM fit~\cite{CKMfit}. For the CKM matrix element $|V_{ub}|$
we use the average of inclusive and exclusive measurements~\cite{Bevan:2014iga}, while
all other CKM matrix elements are calculated from the Wolfenstein parameters.
The parameters related to the $R(D^{(*)})$ measurements,
$\rho^2_{R(D)}$, $\rho^2_{R(D^*)}$, $R_1(1)$, $R_2(1)$ are taken from Ref.~\cite{HFLAV}.
Value and uncertainty for $\delta_{\rm EM}^{K/\pi}$ are taken from Ref.~\cite{Antonelli:2010yf}.}
\label{tab:flav_input}
\end{table}
\subsubsection*{Results}
Since most flavour observables are only sensitive to $M_{H^\pm}$ and $\ensuremath{\tan\!\beta}\xspace$, separate scans of
these parameters are performed for each observable. The other 2HDM parameters are ignored
in these scans, with the exception of $\ensuremath{{\mathcal B}}\xspace(B_{s/d}\rightarrow\mu\mu)$,
where in addition $M_{H}$, $M_{A}$, and $M_{12}^2$ are allowed to float freely within the bounds
defined in the introduction of Section~\ref{sec:2hdm} as these two observables
depend at NLO level on these parameters. In all fits the CKM matrix elements and the
other parameters given in Table~\ref{tab:flav_input} are allowed to vary within
their uncertainties.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/Scan2D_TypeI_tanBeta_vs_MHc_Combined.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/Scan2D_TypeII_tanBeta_vs_MHc_Combined.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/Scan2D_TypeLS_tanBeta_vs_MHc_Combined.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/Scan2D_TypeFlipped_tanBeta_vs_MHc_Combined.pdf}
\vspace{-0.8cm}
\end{center}
\caption[]{Excluded parameter regions (95\% CL) in the $\ensuremath{\tan\!\beta}\xspace$ versus $M_{H^\pm}$ plane
from individual observables for the four 2HDM scenarios considered:
Type-I (top left), Type-II (top right), lepton specific (bottom left), flipped (bottom right).}
\label{fig:TypeItoIV_flav}
\end{figure}
Figure~\ref{fig:TypeItoIV_flav} shows for the four 2HDM scenarios the one-sided 95\% CL excluded
regions in the $\ensuremath{\tan\!\beta}\xspace$ versus $M_{H^\pm}$ plane as obtained from fits using the most sensitive individual
flavour observables. The CLs are derived assuming a Gaussian behaviour of the test statistic with one
degree of freedom. The Type-I (top left) and lepton specific (bottom left) scenarios are only weakly
constrained allowing to exclude $\ensuremath{\tan\!\beta}\xspace<1$. Stronger constraints are obtained for the Type-II (top right)
and flipped (bottom right) scenarios in which in particular $\ensuremath{{\mathcal B}}\xspace(B\rightarrow X_s\gamma)$ allows to exclude
$M_{H^\pm}<590\;$GeV.\footnote{Our results are compatible with those of Ref.~\cite{Misiak:2017bgg},
where limits on $M_{H^\pm}$ between 570 and 800\;GeV are reported for the Type-II model,
depending on the statistical method used (the CL has a relatively weak gradient
versus $M_{H^\pm}$ and thus exhibits a strong numerical sensitivity to the details of the interpretation).
Similar exclusion limits on $M_{H^\pm}$ can be achieved in a
complex 2HDM (C2HDM), which features additional mixing between the neutral CP-even and CP-odd
Higgs bosons~\cite{Muhlleitner:2017dkd}.}
The measurements of $R(D)$ and $R(D^*)$ differ from their SM
predictions~\cite{Aoki:2016frl,Fajfer:2012vx, Na:2015kha}.
In the 2HDM only the Type-II scenario features a compatible parameter
region (at large $\ensuremath{\tan\!\beta}\xspace$ and relatively
small $M_{H^\pm}$, not shown in the upper right plot of Fig.~\ref{fig:TypeItoIV_flav}), which is,
however, excluded by several other observables. Similar results have been reported
in Ref.~\cite{Enomoto:2015wbn}. Because of this incompatibility $R(D)$ and $R(D^{(*)})$
are excluded from our analysis in the following.
\subsection{Constraints from the anomalous magnetic moment of the muon}
The measured value of the anomalous magnetic moment of the muon
$a_\mu=(g_\mu-2)/2$ shows a long-standing tension with the SM prediction
of $\Delta a_\mu=(268\pm63\pm43)\cdot10^{-11}$~\cite{Bennett:2006fi,Davier:2017zfy},
where the first uncertainty is due the the measurement and the second the prediction
(see also the recent reanalysis in Ref.~\cite{Keshavarzi:2018mgv}). Loops involving 2HDM
bosons can modify the coupling between photons and muons. We have adopted the two-loop
2HDM prediction of $\Delta a_\mu$ from Ref.~\cite{Cherchiglia:2016eui}, which
depends on all 2HDM parameters. We make use of the code implementation kindly provided
by H.~St\"ockinger-Kim.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/Scan2D_TypeI_tanBeta_vs_MHc_Delta_aMu_NEW_DHMZ.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/Scan2D_TypeII_tanBeta_vs_MHc_Delta_aMu_NEW_DHMZ.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/Scan2D_TypeIV_tanBeta_vs_MHc_Delta_aMu_NEW_DHMZ.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/Scan2D_TypeIII_tanBeta_vs_MHc_Delta_aMu_NEW_DHMZ.pdf}
\vspace{-0.8cm}
\end{center}
\caption[]{2HDM fits using the anomalous magnetic moment of the muon as input. Shown are
allowed 68\% and 95\% CL regions in the $\ensuremath{\tan\!\beta}\xspace$ versus $M_{H^\pm}$ plane for the four
2HDM scenarios considered: Type-I (top left), Type-II (top right), lepton specific (bottom left),
and flipped (bottom right).}
\label{fig:amu}
\end{figure}
Figure~\ref{fig:amu} shows the 68\% and 95\% CL allowed regions
in the $\ensuremath{\tan\!\beta}\xspace$ versus $M_{H^\pm}$ plane for the four 2HDM
scenarios using only $\Delta a_\mu$ as input.
All other parameters of the 2HDM are left free to vary within their respective bounds.
Compatibility is found in a narrow band with $\ensuremath{\tan\!\beta}\xspace\ll1$ and
$M_{H^\pm}$ below about 600\;GeV (depending on the scenario),
as well as for a region with larger $\ensuremath{\tan\!\beta}\xspace$ that broadens with
decreasing $M_{H^\pm}$. When combined with the constraints from
the other flavour observables (cf. Fig.~\ref{fig:TypeItoIV_flav}),
values of $\ensuremath{\tan\!\beta}\xspace$ above about 5$\sim$10 remain allowed.
\subsection{Constraints from electroweak precision data}
The electroweak precision data can be used to constrain the 2HDM via the
oblique parameters determined in Eq.~(\ref{eq:stu}). We use the predictions from
Refs.~\cite{Haber:1993wf,Haber:1999zh,Froggatt:1991qw} similar to our previous analysis~\cite{Baak:2011ze}.
The oblique corrections to electroweak observables in the 2HDM are independent of the Yukawa
interactions and their impact is identical in the four 2HDM scenarios considered.
Figure~\ref{fig:2HDM_STU} shows the 68\% and 95\% CL allowed parameter
regions in the neutral Higgs-boson mass plane $M_{A}$ versus $M_{H}$
for fixed charged Higgs-boson masses of 250, 500, and
750\;GeV as obtained from fits using only the oblique parameters as input. All
other parameters of the 2HDM (including $\beta-\alpha$) are free
to vary in these scans. While no information on the absolute mass scale of
the 2HDM bosons is obtained from the electroweak data, relative masses
are constrained. In our previous
analysis~\cite{Baak:2011ze} we showed that the oblique
parameters constrain the values of $M_{H}$ and $M_{A}$ to be close
to $M_{H^{\pm}}$ for fixed $\beta-\alpha=\pi/2$. Removing this
restriction (cf. Fig.~\ref{fig:2HDM_STU}) relaxes
the constraint to having either $M_{A}$ close to $M_{H^{\pm}}$,
or $M_{H}$ larger than $M_{H^{\pm}}$.
\begin{figure}
\begin{center}
\includegraphics[width=\defaultSingleFigureScale\textwidth]{Figures/THDM_MA_vs_MH0.pdf}
\end{center}
\vspace{-0.3cm}
\caption[]{2HDM fit results using the oblique $S$, $T$, $U$ parameters.
Shown are allowed 68\% and 95\% CL regions in the $M_{A}$ versus $M_{H}$
plane for fixed benchmark values of $M_{H^{\pm}}$.
The constraints are independent of the 2HDM scenario.}
\label{fig:2HDM_STU}
\end{figure}
\subsection{Combined fit}
\label{sec:combi}
We combine in this section the 2HDM constraints from the Higgs-boson coupling strength
measurements, flavour observables, muon anomalous magnetic moment, and electroweak precision data.
Figure~\ref{fig:MA_vs_MH0} shows for the four 2HDM scenarios considered
the resulting 68\% and 95 \% CL allowed regions in the $M_{A}$
versus $M_{H}$ plane for fixed (benchmark) charged Higgs-boson masses of 250, 500, and 750\;GeV.
All other 2HDM parameters are allowed to vary freely within their bounds.
Depending on the 2HDM scenario and $M_{H^{\pm}}$, the minimum $\chi^2$ values found
lie between 48 and 59 for $\Ndof=53$ (corresponding to $p$-values between 25\% and 68\%).
The combined fit leads in all four 2HDM scenarios to a strong alignment of either the $H$ or the
$A$ boson mass with that of the $H^{\pm}$ boson, owing to the constraint on $\beta-\alpha$ from
the Higgs coupling strength measurements (cf. Fig.~\ref{fig:bma_tanb}) in addition to those
from the electroweak precision data.
In this sense, the fit resembles the result from our previous analysis~\cite{Baak:2011ze},
but replacing the fixed restriction of $\beta-\alpha=\pi/2$ by the Higgs couplings strengths measurements.
The absolute mass limits on $M_{\rm H^{\pm}}$ obtained from the flavour observables in the Type-II
and flipped scenarios (cf. Fig.~\ref{fig:TypeItoIV_flav}) exclude the low-$M_{\rm H^{\pm}}$ benchmarks,
as indicated by the hatched regions in the two right-hand panels of Fig.~\ref{fig:MA_vs_MH0}
(where in addition different statistical assumptions are compared: one-sided versus two-sided test statistic and
one versus two degrees of freedom\footnote{The limits obtained for a two-sided test statistic and two
degrees of freedom have been verified with a pseudo Monte Carlo study based on randomly
drawn sets of the measurements used in the fit.}). For these two scenarios pairs of ($H$, $A$) masses below
$\sim$$400\;\ensuremath{\mathrm{Ge\kern -0.1em V}}\xspace$ are excluded for any set of values of the other 2HDM parameters.
For the Type-I and lepton specific scenarios no absolute limits on the Higgs boson masses
can be derived.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/THDM_MA_vs_MH0_withHBR_TypeI_withHB_2DPlots_logo.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/THDM_MA_vs_MH0_withHBR_TypeII_withHB_2DPlots_logo.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/THDM_MA_vs_MH0_withHBR_TypeIV_withHB_2DPlots_logo.pdf}
\includegraphics[width=\defaultDoubleFigureScale\textwidth]{Figures/THDM_MA_vs_MH0_withHBR_TypeIII_withHB_2DPlots_logo.pdf}
\vspace{-0.8cm}
\end{center}
\caption[]{2HDM fit results using a combination of constraints from the Higgs-boson coupling strength
measurements, flavour observables, muon anomalous magnetic moment, and electroweak precision data.
Shown are allowed 68\% and 95\% CL regions in the $M_{A}$ versus $M_{H}$ plane for fixed benchmark
values of $M_{H^{\pm}}$ and for the four 2HDM scenarios considered: Type-I (top left), Type-II (top right),
lepton specific (bottom left), and flipped (bottom right).}
\label{fig:MA_vs_MH0}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
We have presented results for an updated global fit of the electroweak
sector of the Standard Model using latest experimental and theoretical input.
We include new precise kinematic top quark and
$W$ boson mass measurements from the LHC, a \ensuremath{\seffsf{\ell}}\xspace measurement
from the Tevatron, and a new evaluation of the hadronic contribution
to $\alpha(M_Z^2)$.
The fit confirms the consistency of the Standard Model and slightly improves
the precision of the indirect determination of key observables.
Using constraints from Higgs-boson coupling strength measurements, flavour
observables, the muon anomalous magnetic moment, and electroweak precision data,
we studied allowed and excluded parameter regions of four CP conserving two-Higgs-doublet
models. Strong constraints on the extended Higgs boson masses
are found for the so-called Type-II and flipped scenarios.
\subsubsection*{Acknowledgements}
\label{sec:Acknowledgments}
\begin{details}
We are indebted to Mikolaj Misiak and Hyejung St\"ockinger-Kim for providing the implementation
of their calculations of $\ensuremath{{\mathcal B}}\xspace(B\rightarrow X_s\gamma)$ and the muon anomalous magnetic moment
in the two-Higgs-doublet models.
We thank Rui Santos for helpful discussions and feedback on early stages of the paper.
This work is supported by the German Research Foundation (DFG) in the Collaborative Research
Centre (SFB) 676 ``Particles, Strings and the Early Universe'' located in Hamburg.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,358 |
09/24/11 — We have an immediate search for a 79 year-old woman and her 31 year old son who is mentally and physically handicapped — he has a prostetic leg. The address to report is 1376 Magnolia Ridge Park, Woodville, TX. 75979. Your contact person is Scott Duckworth, and he can be reached at 409-48-9-6679. We need foot searchers and ATVs. The Search Coordinator will be Ron Overman.
To see news coverage of this case, follow link below.
Lettie Fisher, age 79, is 5'5″, 175 lbs., and was last seen wearing slacks and rubber boots eyes brown, grey hair brown shoulder length wig. John M. Fisher Jr, age 31, is 5'7″, and weighs 300 lbs. He has a prosthetic left leg (above the knee), brown eyes, short black hair, and was last seen wearing blue jeans and grey rubber boots, size 11. Both require medications. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,478 |
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The good news is that I have actually found the devices and the detox facilities in TX that can beat methadone withdrawal smoothly and conveniently.
Bluff Dale, Texas is on U.S. Highway 377 and the North Paluxy River in northeastern Erath County. It was originally called Bluff Springs by pioneers who settled nearby; Bluff Dale became the town name when the post office was established in 1877. In 1889 when the Fort Worth and Rio Grande Railway was built, Jack Glenn donated land for the development of a town. The town was incorporated in 1908. By 1915 a bank and newspaper had been developed. In 1936 Bluff Dale had 680 residents, 500 in 1940, 123 in 1980, and 300 in 1989. In 1989 the town had a Garden Club, three churches, a volunteer fire department, five historical markers, and a beautification committee. A gas station–convenience store was built around 2002. Around 2005, a bank was opened up. | {
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Cesare Emiliani (Bologna, 8 de diciembre de 1922 – Palm Beach Gardens, 20 de julio de 1995) fue un geólogo, micropaleontólogo e isotopista geoquímico italiano, fundador de la paleo-oceanografía.
Emiliani nació en Bolonia y sus padres fueron Luigi y Maria (Manfredidi) Emiliani.
Junto con Roger Revelle y muchos otros desarrolló activamente el programa JOIDES que se basó en el proyecto original LOCO (LOng COres).
Fue un científico pro-renacentista – políglota y hombre muy leído – y adversario militante de posturas dogmáticas al igual que de toda rigidez mental. Era por este motivo que en sus últimos años consagró mucho tiempo en despertar interés y propuso el Calendario holoceno, una reforma del calendario con origen en la época holocena. Su ambición fue eliminar el hiato entre la escala a. C. y la escala d. C.. Éste es igual a 1 año porque cada escala tiene origen 1 como consecuencia de la definición del Calendario Gregoriano.
Obras populares
Emiliani, C. 1992. Planet Earth : Cosmology, Geology, & the Evolution of Life & the Environment. Cambridge University Press. ISBN 0-521-40949-7
Emiliani, C. 1995. The Scientific Companion : Exploring the Physical World with Facts, Figures, and Formulas (Wiley Popular Science) (2ª ed.). Wiley. ISBN 0-471-13324-8
Emiliani, Cesare. 1993. Dictionary of Physical Sciences. Oxford University Press. ISBN 0-19-503652-2
Obra prominente
Emiliani C. 1954. Depth habitats of some species of pelagic foraminifera as indicated by oxygen isotope ratios. Am. J. of Science 252:149–158
Emiliani C. 1954. Temperature of Pacific bottom waters and polar superficial waters during the Tertiary. Science 119:853–855
Emiliani C. 1956. Oligocene and Miocene temperature of the equatorial and subtropical Atlantic Ocean. J. of Geology 64:281–288
Emiliani C. 1956. On paleotemperatures of Pacific bottom waters. Science 123:460–461
Emiliani C. 1957. Temperature and age analysis of deepsea cores. Science 125:383–385
Emiliani C. 1961. The temperature decrease of surface water in high latitudes and of abyssal-hadal water in open oceanic basins during the past 75 million years. Deep-Sea Research 8:144–147
Emiliani C. 1965. Precipitous continental clopes and considerations on the transitional crust. Science 147:145–148
Emiliani C. 1966. Isotopic paleotemperatures. Science 154: 851–857
Emiliani C. 1966. Paleotemperature analysis of Caribbean cores P6304-8 and P6304-9 and a generalized temperature curve for the past 425,000 years. Journal of Geology 74:109–124
Emiliani C. 1968. The Pleistocene epoch and the evolution of man. Current Anthropology 9:27–47
Emiliani C. 1969. Interglacials, high sea levels and the control of Greenland ice by the precession of the equinoxes. Science 166:1503–1504
Emiliani C. 1969. A new paleontology. Micropaleontology 15:265–300
Emiliani C. 1970. Pleistocene paleotemperatures. Science 168:822–825
Emiliani C. 1971. The amplitude of Pleistocene climatic cycles at low latitudes and the isotopic composition of glacial ice. In: Turekian KK (ed) Late Cenozoic Glacial Ages. New Haven, CO: Yale University Press, pp 183–197
Emiliani C. 1971. Depth habitats and growth stages of pelagic formanifera. Science 173:1122–1124
Emiliani C. 1971. Paleotemperature variations across the Plio-Pleistocene boundary at the type section. Science 171:600–602
Emiliani C. 1978. The cause of the ice ages. Earth and Planetary Science Letters 37:347–354
Emiliani C. 1981. A new global geology. In: Emiliani C (ed) The Oceanic Lithosphere. The Sea (8th edn). Vol. 7. New York: Wiley Interscience, pp 1687–738
Emiliani C. 1982. Extinctive evolution. Journal of Theoretical Biology 97:13–33
Emiliani C. 1987. Dictionary of Physical Sciences. Oxford: Oxford University Press
Emiliani C. 1988. The Scientific Companion. New York: Wiley
Emiliani C. 1989. The new geology or the old role of the geological sciences in science education. J. of Geological Education 37:327–331
Emiliani C. 1991. Avogadro number and mole: a royal confusion. Journal of Geological Education 39:31–33
Emiliani C. 1991. Planktic et al. Marine Micropaleontology 18:3
Emiliani C. 1991. Planktic/planktonic, nektic/nektonic, benthic/benthonic. Journal of Paleontology 65:329
Emiliani C, Ericson DB. 1991. The glacial/interglacial temperature range of the surface water of the ocean at low latitudes. In: Taylor HP, O'Neil JR, Kaplan IR (eds) Special Publication: Stable Isotope Geochemistry: A Tribute to Samuel Epstein. Pennsylvania: Geochemical Society, University Park, pp 223–228
Emiliani C. 1992. The Moon as a piece of Mercury. Geologische Rundschau 81:791–794
Emiliani C. 1992. Planet Earth: Cosmology, Geology, and the Evolution of Life and Environment. New York: Cambridge University Press
Emiliani C. 1992. Pleistocene paleotemperatures. Science 257:1188–1189
Emiliani C. 1993. Milankovitch theory verified; discussion. Nature 364:583
Emiliani C. 1993. Calendar reform. Nature 366:716
Emiliani C. 1993. Extinction and viruses. BioSystems 31:155–159
Emiliani C. 1993. Paleoecological implications of Alaskan terrestrial vertebrate fauna in latest Cretaceous time at high paleolatitudes: Comment. Geology 21:1151–1152
Emiliani C. 1993. Viral extinctions in deep-sea species. Nature 366:217–218
Emiliani C. 1995. Redefinition of atomic mass unit, Avogadro constant, and mole. Geochimica et Cosmochimica Acta 59:1205–1206
Emiliani C. 1995. Tropical paleotemperatures: discussion. Science 268:1264
Emiliani C, Edwards G. 1953. Tertiary ocean bottom temperatures. Nature 171:887–888
Emiliani C, Elliott I. 1995. Vatican confusion. Nature 375:530
Emiliani C, Epstein S. 1953. Temperature variations in the lower Pleistocene of Southern California. J. of Geology 61:171–181
Emiliani C, Gartner S, Lidz B. 1972. Neogene sedimentation on the Blake Plateau and the emergence of the Central American Isthmus. Palaeogeography, Palaeoclimatology, Palaeoecology 11:1–10
Emiliani C, Gartner S, Lidz B, Eldridge K, Elvey DK, Huang PC, Stipp JJ, Swanson M (1975) Paleoclimatological analysis of late Quaternary cores from the northwestern Gulf of Mexico. Science 189:1083–1088
Emiliani C, Geiss J. 1959. On glaciations and their causes. Geologische Rundschau 46:576–601
Emiliani C, Harrison CG, Swanson M. 1969. Underground nuclear explosions and the control of earthquakes. Science 165:1255–1256
Emiliani C, Kraus EB, Shoemaker EM. 1981. Sudden death at the end of the Mesozoic. Earth and Planetary Science Letters 55:327–334
Emiliani C, Mayeda T, Selli R. 1961. Paleotemperature analysis of the Plio-Pleistocene section at le Castella, Calabria, southern Italy. Geological Society of America Bulletin 72:679–688
Emiliani C, Milliman JD. 1966. Deep-sea sediments and their geological record. Earth Science Reviews 1:105–132
Emiliani C, Price DA, Seipp J. 1991. Is the Postglacial artificial? In: Taylor HP, O'Neil JR, Kaplan IR (eds) Special Publication: Stable Isotope Geochemistry: A Tribute to Samuel Epstein. Pennsylvania: Geochemical Society, University Park, pp 229–231
Emiliani C, Shackleton NJ. 1974. The Brunhes Epoch: paleotemperature and geochronology. Science 183:511–514
Véase también
Calendario holoceno
Geólogos de Italia
Paleontólogos de Italia
Geólogos de Estados Unidos
Catedráticos de la Universidad de Miami
Emigrantes italianos hacia Estados Unidos
Científicos de Italia del siglo XX
Nacidos en Bolonia
Fallecidos en Florida
Premiados con la Medalla Vega | {
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Ajijic (ahi-HEEK) is considered by many the jewel in the crown of small towns that ring the northern side of Lake Chapala, Mexico's largest natural fresh-water lake. It is located in the state of Jalisco in central Mexico, which also includes the much-lauded, yet tourist-laden and overbuilt resort town of Puerto Vallarta.
Jalisco is also the home the country's second-largest city, Guadalajara, home of tequila and mariachis, but also some of the finest universities and performing arts venues in North America. Lake Chapala is situated at an elevation of just over 5,000 feet, with a temperate climate that is considered among the best in the world.
Ajijic has been a draw for North Americans, predominantly, since the 1950's, attracting a compendium of talented writers, performers, artists, retirees and world-weary wanderers of all stripes. To read more about this lovely town, go to: Facts – Ajijic.
Expats have made a significant contribution to the area via The Lake Chapala Society (Facts – The Lake Chapala Society), a group of part-time and full-time residents who have established their presence with an enormous array of activities, services, resources and educational opportunities. Their campus is located directly across the street from my board and breakfast, The Hotel Casa Blanca. The society recently celebrated their 60th anniversary and I'm proud to announce that I've joined the ranks of their 3,000+ members.
Having only arrived here a little over two weeks ago, I've spent a good part of my days walking for miles on the famed cobble-stoned streets and along the lakeside Malecon. The village is filled with charm, color, art and smells of coffee, chocolate and fresh tortillas!
One large church and a smaller chapel are visible reminders of the Catholic Church's impact on Mexican life. Their bells ring out daily and serve as wake-up alarms to the town, animals, roosters – and all living things.
The center of town is the village square, or plaza. It was especially festive this past week during Holy Week, or Semana Santa, as the days leading up to Easter are known. This time of year is "bigger than Christmas", as it combines the religious holiday with spring break and family gatherings. The plaza is festooned with banners, food stalls, musical groups and an abundant array of both beautifully detailed, handmade artwork and more mundane mass produced items, often for children and young adults.
Mexicans know how to celebrate – and they're really good at it. During holidays, the wealthy gentry and the working class come together to drink, eat (the food!), listen to music and toast to health and happiness for all. Granted, the venues and level of grandeur may differ, but everyone shares the same key elements of the season.
As I've said, color is everywhere in Lakeside, the inclusive name for the cities situated along this side of Lake Chapala. The plaza and neighborhood leave no color underutilized – and often exhibited in bright and creative ways.
Along the lake is the Ajijic Malecon, a wide, elevated walkway dotted with play areas, food and art stalls and even a sculpture garden, newly opened just this past year. It is a unique delight to walk along the malecon any time of day, with the sunsets being a well-attended tradition for locals, expats and tourists alike.
A local photographer's work is currently being featured in display all around the central plaza in the village. I found each one exceedingly charming, keeping in mind that the artistic blending of fresh young faces and local produce isn't exactly a combo that should necessarily ensure such a stunning end result. I hope you'll find them as enchanting as I do.
Of course, the Lakeside community wouldn't exist without….ah….THE LAKE! Lake Chapala is 50 miles long and roughly 8 miles wide. It's a shallow lake with average depths of 10-15 feet and has had challenges retaining both optimal levels and water quality. A major initiative was put into effect this past decade to improve and maintain the lake's sustainability and ecological standards. It's not used widely for water activities or boating, although locals can often be seen wadding, swimming and fishing along the shore.
Birds are abundant, both on the water and in low-flying flocks that skim the shoreline.
As I mentioned before, seeing the sun set from the pier on the malecon is a memorable evening delight. I have done it twice now and especially love to see the parents who bring their young children down to view it for (maybe) the very first time. It's a memory they won't soon forget.
I admit it. I'm really smitten with Ajijic and the Lakeside area. I've just secured a six month rental of a lovely house, into which I will settle on April 8. And I'm buying a smart car….no, not the brand…it's smart because it's the first simple sedan I've owned in a long while; cost-effective, non-glitzy, inexpensive to operate and maintain and with a good resale value. A Nissan with only 1000 miles on it, it was bought by an elderly expat who dropped dead two months later. I feel certain that she'd be happy for me, bless her soul.
I've begun to meet people who strike me as smart, sophisticated, culturally aware, socially adept, involved and well-travelled. I've attended the opening of the latest English-language theatre production here (Facts – Lakeside Little Theatre), a choral concert (Facts – Los Cantantes del Lago) and have been in contact with others who are well-connected to the community arts and philanthropy, both of which are huge. Whether on stage as a performer, or off, as a volunteer board member or fundraiser, opportunities abound. There is a magic realism to to this area that has drawn expats for decades. Whether they stay here six months a year, or relocate here permanently, they make a contribution to the quality of life in immeasurable ways.
But let us never forget that this is Mexico, with it's own extraordinary culture and (at times) frustrating quirkiness. It's a foreign country after all, in the coolest meanings of the phrase, with both a heavy European and indigenous Indian presence and influence. Expats have added greatly to the mix in towns like Ajijic, San Miguel and others, but the magic of Mexico predates them all.
I'll leave you with one of my favorite shots taken as I roamed the streets of this quaint village of just over 10,000 residents (up to 15,000 by some current estimates). It's the spirit of the people and the smile (or in this case, quizzical look) of the children that capture the best of any country. Both are in ample supply here in Ajijic, my new home.
Until next time….hasta que nos encontremos de nuevo.
Ah Steve, you are living the life I dream of living some day – For a dozen or so years I've spend a week or two in Izaptan de la Sal in Mexico and loved every minute of it. Enjoy my friend.
Thanks, VJ. I'm not familiar with that town but of course will have to do my research post haste to see where it is. Hope the desert is treating you well. There are lots of similarities to Ajijic that I'm finding which is great fun! Best to you.
I enjoyed the photography exhibition and the stunning sunset photos. It sounds as though you've found yourself a milieu that will fulfill you in different ways than Chang Mai and Asia. I can't wait to see photos of your house. How fun to finally have a home to settle into after the last six or seven months. More on Ajijic, Mr. Browning. Please sir! | {
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{"url":"https:\/\/zbmath.org\/?q=an%3A1008.54030","text":"# zbMATH \u2014 the first resource for mathematics\n\nSome new common fixed point theorems under strict contractive conditions. (English) Zbl\u00a01008.54030\nLet $$(X,d)$$ be a metric space and $$S,T:X\\to X$$ two mappings. The authors define a new property for $$(S,T)$$ which generalizes the concept of noncompatible mappings and give some common fixed point theorems under strict contractive conditions.\n\n##### MSC:\n 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems\n##### Keywords:\nfixed point; weakly compatible mapping\nFull Text:\n##### References:\n [1] Jachymski, J., Common fixed point theorems for some families of maps, Indian J. pure appl. math., 25, 925-937, (1994) \u00b7 Zbl\u00a00811.54034 [2] Jungck, G.; Moon, K.B.; Park, S.; Rhoades, B.E., On generalization of the meir \u2013 keeler type contraction maps: corrections, J. math. anal. appl., 180, 221-222, (1993) \u00b7 Zbl\u00a00790.54055 [3] Pant, R.P., Common fixed points of sequences of mappings, Ganita, 47, 43-49, (1996) \u00b7 Zbl\u00a00892.54028 [4] Jungck, G., Compatible mappings and common fixed points, Internat. J. math. math. sci., 9, 771-779, (1986) \u00b7 Zbl\u00a00613.54029 [5] Pant, R.P., R-weak commutativity and common fixed points, Soochow J. math., 25, 37-42, (1999) \u00b7 Zbl\u00a00918.54038 [6] Pant, R.P., Common fixed points of contractive maps, J. math. anal. appl., 226, 251-258, (1998) \u00b7 Zbl\u00a00916.54027 [7] Sessa, S., On a weak commutativity condition of mappings in fixed point considerations, Publ. inst. math. (beograd), 32, 149-153, (1982) \u00b7 Zbl\u00a00523.54030 [8] Caristi, J., Fixed point theorems for mapping satisfying inwardness conditions, Trans. amer. math. soc., 215, 241-251, (1976) \u00b7 Zbl\u00a00305.47029\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-01-20 03:21:44","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\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6493843793869019, \"perplexity\": 3998.565506330975}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703519883.54\/warc\/CC-MAIN-20210120023125-20210120053125-00315.warc.gz\"}"} | null | null |
Q: Sum of Two Fields in Alv Report and Display Total in Third Field How to sum the fields amount and tax and display total in third field in alv report?
Table and fields are:
vbak-netwr
vbap-mwsbp
A: Firt of all, you need Fill the work_area of table fieldcat and after that you need APPEND work_area-fieldcat TO it_fieldcat, you need do it for all fields in your out internal table , When you want a sum in some field, you use wa_fieldcat-sum 'X'.
Examples:
wa_fieldcat-fieldname = 'NETWR'. "Name of field.
wa_fieldcat-tabname = 'IT_VBAK'. "Name of Internal Table.
wa_fieldcat-ref_fieldname = 'NETWR'. "Reference of field
wa_fieldcat-ref_tabname = 'VBAK'. "Reference of Table
wa_fieldcat-sum = 'X'. "Making a sum of Field
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Patronage Sainte-Anne – kongijski klub piłkarski grający w pierwszej lidze kongijskiej, mający siedzibę w mieście Brazzaville.
Sukcesy
I liga): mistrzostwo (2): 1969, 1986
wicemistrzostwo (2): 1988, 1995
Puchar Konga: zwycięstwo (1):''' 1988
finał (1): 2005
Występy w afrykańskich pucharach
Stadion
Swoje domowe mecze klub rozgrywa na stadionie o nazwie Stade Alphonse Massemba-Débat w Brazzaville, który może pomieścić 33 037 widzów.
Reprezentanci kraju grający w klubie od 1992 roku
Stan na styczeń 2023.
Przypisy
Kongijskie kluby piłkarskie
Brazzaville | {
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Q: Flex & Flash mobile application Developemt i want to learn how to develop mobile application in flash n flex
which book is good ?? n pls tell if any good online tutorial are available or not
i am just beginner ,i never desined any mobile application but have experi
A: You should go for Adobe's online tutorial. Try building your own application on mobile Its for free.
Some recommended books :
*
*Developing Android Applications with Flex 4.5
*Developing Android Applications with Adobe AIR (Adobe Developer Library)
*Professional Flash Mobile Development: Creating Android and iPhone Applications
*Developing Blackberry Tablet Applications with Flex 4.5
| {
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{"url":"http:\/\/www.komal.hu\/verseny\/feladat.cgi?a=feladat&f=I112&l=en","text":"Mathematical and Physical Journal\nfor High Schools\nIssued by the MATFUND Foundation\n Already signed up? New to K\u00f6MaL?\n\n# Problem I. 112. (October 2005)\n\nI.\u00a0112. Some unknown real numbers (at most 10) are denoted by the first few letters of the alphabet. The greater member of certain pairs of numbers is given. You should order the variables such that all given inequalities are satisfied. If there is no such ordering, print No\u00a0solution''.\n\nYour program should read the given relations from the standard input. Each line will contain a single inequality, without spaces between the variables and inequality sign.\n\nThe output should contain a possible ordering of the numbers in the same format as in the example.\n\n Input Output AC C\n\nThe source code of the program (i112.pas, i112.c, ...) should be submitted.\n\n(10\u00a0pont)\n\nDeadline expired on November 15, 2005.\n\n### Statistics:\n\n 19\u00a0students\u00a0sent\u00a0a\u00a0solution. 10\u00a0points: Balamb\u00e9r D\u00e1vid, Gy\u00f6r\u00f6k P\u00e9ter, Kisfaludi-Bak S\u00e1ndor, Ozsv\u00e1rt L\u00e1szl\u00f3, Szoldatics Andr\u00e1s, Vincze J\u00e1nos. 8\u00a0points: 2\u00a0students. 7\u00a0points: 2\u00a0students. 6\u00a0points: 2\u00a0students. 5\u00a0points: 3\u00a0students. 3\u00a0points: 4\u00a0students.\n\nProblems in Information Technology of K\u00f6MaL, October 2005","date":"2018-09-22 20:50:20","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\": 0.3808321952819824, \"perplexity\": 2741.0421710562528}, \"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-2018-39\/segments\/1537267158691.56\/warc\/CC-MAIN-20180922201637-20180922222037-00221.warc.gz\"}"} | null | null |
(August 29th, 2017) Earlier this month, the ERC announced its funding plans for 2018. Besides the familiar grant types, there's a new one.
They have become European scientists' favourite funding scheme – the European Research Council's research grants. In early August, the ERC published its Work Programme 2018, announcing the next grant competitions and how the Council intends to spend the earmarked €1.86 billion. Not much has changed to the previous years: the sole criterion for a successful proposal is still "excellence" - whatever that means. And researchers can again choose between Starting, Consolidator, Advanced or Proof-of-Concept Grants, depending on their career stage.
In addition, there's a new type of grant, or rather, a comeback kid: the Synergy Grant. Back in 2012/2013, the ERC already experimented with it in a pilot project. Now, they are perhaps here to stay. Not less than 250 million have been put aside for this grant, which is to support groups of two to four scientists, who tackle a scientific question together. The ERC awards this collaborative approach with up to ten million euros over a period of six years.
"The reintroduction of Synergy Grants in the 2018 Work Programme has been much anticipated. These grants can trigger unconventional collaborations, allow for the emergence of new fields of study and help put scientists working in Europe at the global forefront. By providing €250 million of funding for the Synergy Grant call, the ERC Scientific Council intends to make possible substantial advances at the frontiers of knowledge which would be impossible for researchers working alone," said ERC president, Jean-Pierre Bourguignon, in a press release.
With a similar budget to 2017, exactly these 250 million euros for the Synergy Grants have to be drawn off somewhere. Hence, compared to 2017, the Starting Grants were lowered by 24 million euros, from 605 to 581 million and the Consolidator Grants went down 25 million euros, from 575 to 550 million euros. It is, however, the Advanced Grants that had to endure the largest slash: 117 million euros less are available for senior researchers, down to 450 from 567 million euros last year. Calls for Starting and Synergy Grants are already open and close in October and November, respectively.
"This is the starting whistle for the next round of this champions' league of European research within the EU's Horizon 2020 research and innovation programme. I hope this new series of competitions for excellence in science will identify and reward potential breakthroughs, and will be an investment for the future of Europe," Commissioner for Research, Science and Innovation, Carlos Moedas, is optimistic. | {
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} | 6,599 |
{"url":"https:\/\/plainmath.net\/123\/matrix-multiplication-pretty-tough-meantime-compute-following-bmatrix","text":"# Matrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if A=begin{bmatrix}2&1&1 -1&-1&4 end{bma\n\nMatrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if\n$A=\\left[\\begin{array}{ccc}2& 1& 1\\\\ -1& -1& 4\\end{array}\\right],B=\\left[\\begin{array}{cc}0& 2\\\\ -4& 1\\\\ 2& -3\\end{array}\\right],C=\\left[\\begin{array}{cc}6& -1\\\\ 3& 0\\\\ -2& 5\\end{array}\\right],D=\\left[\\begin{array}{ccc}2& -3& 4\\\\ -3& 1& -2\\end{array}\\right]$\nIf the operation is not possible , write NOT POSSIBLE and be able to explain why\na)A+B\nb)B+C\nc)2A\nYou can still ask an expert for help\n\n\u2022 Live experts 24\/7\n\u2022 Questions are typically answered in as fast as 30 minutes\n\u2022 Personalized clear answers\n\nSolve your problem for the price of one coffee\n\n\u2022 Math expert for every subject\n\u2022 Pay only if we can solve it\n\nObiajulu\nStep 1\n\nSolution for question a:\nTo compute A+B:\nNote that two matrices may be added if and only if they have the same dimension, that is, they must have the same number of rows and columns.\nHere, Dimension of $A=2\u00d73$\nAnd, Dimension of $B=3\u00d72$\nMatrix A and B do not have the same dimension. Hence, matrix A and B cannot be added.\nTherefore it is not possible to perform A+B.\nStep 2\nSolution for question b:\nHere,\nDimension of matrix $B=3\u00d72$\nDimension of matrix $C=3\u00d72$\nBoth matrices B and C have the same dimension. Hence, matrix B and C can be added.\nFurther,\n$B+C=\\left[\\begin{array}{cc}0& 2\\\\ -4& 1\\\\ 2& -3\\end{array}\\right]+\\left[\\begin{array}{cc}6& -1\\\\ 3& 0\\\\ -2& 5\\end{array}\\right]$\n$=\\left[\\begin{array}{cc}0+6& 2+\\left(-1\\right)\\\\ -4+3& 1+0\\\\ 2+\\left(-2\\right)& \\left(-3\\right)+5\\end{array}\\right]$\n$=\\left[\\begin{array}{cc}6& 1\\\\ -1& 1\\\\ 0& 2\\end{array}\\right]$\nTherefore,\n$B+C=\\left[\\begin{array}{cc}6& 1\\\\ -1& 1\\\\ 0& 2\\end{array}\\right]$\nStep 3\nSolution for question c:\nTo compute 2A multiply each entry of the matrix A by 2.\n$2A=2\\left[\\begin{array}{ccc}2& 1& 1\\\\ -1& -1& 4\\end{array}\\right]$\n$=\\left[\\begin{array}{ccc}4& 2& 2\\\\ -2& -2& 8\\end{array}\\right]$\ntherefore,\n$2A=\\left[\\begin{array}{ccc}4& 2& 2\\\\ -2& -2& 8\\end{array}\\right]$\nJeffrey Jordon","date":"2022-06-26 13:49:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 18, \"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\": 0.8944109678268433, \"perplexity\": 1141.263909079545}, \"config\": {\"markdown_headings\": true, \"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-2022-27\/segments\/1656103269583.13\/warc\/CC-MAIN-20220626131545-20220626161545-00105.warc.gz\"}"} | null | null |
package namespace
import (
"k8s.io/kubernetes/pkg/fields"
"k8s.io/kubernetes/pkg/labels"
"k8s.io/kubernetes/pkg/watch"
)
// Watcher is a (k8s.io/kubernetes/pkg/client/unversioned).NamespaceInterface compatible
// interface which only has the Watch function. It's used in places that only need perform watches,
// to make those codebases easier to test and more easily swappable with other implementations
// (should the need arise).
//
// Example usage:
//
// var nsl NamespaceWatcher
// nsl = kubeClient.Namespaces()
type Watcher interface {
Watch(label labels.Selector, field fields.Selector, resourceVersion string) (watch.Interface, error)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,310 |
Q: How can I Block Attachments where file format is JPG, PNG using Trigger in lightning experience? I want to block the upload of attachments having file types png,jpg using trigger in Lightning Experience,
Thanks,
Karthi
A: This is a scenario I ran smack into yesterday afternoon. As Keith described in his answer, LEx treats uploads from the standard Notes & Attachments list as "Files". This means that any trigger code you have on the Attachment object won't fire. In our case, we welcome image Attachments, but use a third-party API to convert them to PDF format after they've been inserted.
Luckily, we know that Salesforce uses Content for storage of its Files, allowing users to update the same File with subsequent versions. As such, it is storing the File data in the ContentVersion object behind the scenes. As I said I ran into this problem yesterday so I don't have a complete solution in place, but we can add a trigger to this object and leverage its fields on newly-inserted records to determine whether or not a File is an image.
The fields are a bit different than what you're used to on the Attachment object, I'd imagine. A few notes on the differences:
*
*Content likes to clean up filenames (e.g. photo001.jpg becomes photo001 when displayed in the UI. To get the original filename, use the ContentVersion.PathOnClient field.
*If you'd rather not have to parse out the file extension on your own, you can leverage the ContentVersion.FileExtension field.
*Rather than Attachment.ContentType, Salesforce attempts to clean up the mimetype for you, which is stored in the ContentVersion.FileType field. Rather than seeing image/bmp here, you'll just see BMP.
So you should be able to add some quick trigger logic and throw an error if any image is uploaded; this does seem a bit strict, however, since this change would apply to all ContentVersion records. You may need to come up with a way to determine which File records should be prevented from being uploaded; e.g. you could check the parent record of the File being inserted (which would mean doing some digging in the ContentVersion.ContentDocumentId lookup). Otherwise, this would most likely prevent any images from being added to Chatter.
For my use case...rather than overwriting the Attachment.Body field with the results of a webservice callout, I'll most likely need to insert a new ContentVersion record.
Edit: I just stubbed out some code for this below. Disclaimer: this hasn't been tested or even compiled, this is just me in Notepad++. I'll attempt to test when I get some more free time, but it should accomplish what you're looking for. Obviously you'd want to keep your trigger simple and move this logic into a helper class.
trigger ContentVersionTrigger on ContentVersion (after insert) {
Set<Schema.SObjectType> objsPreventImageFiles = new Set<Schema.SObjectType>{
Account.SObjectType
};
Set<String> imageTypes = new Set<String>{'bmp', 'jpg', 'jpeg', 'png'}; // the list goes on...
// determine which files we're interested in (images), so we don't unnecessarily query ContentDocumentLink
Set<Id> contentDocIds = new Set<Id>();
List<ContentVersion> images = new List<ContentVersion>();
for ( ContentVersion cv : Trigger.new ) {
// have to use PathOnClient, since FileExtension field doesn't seem to be populated in this context
if ( imageTypes.contains(Utils.getFileExtension(cv.PathOnClient)) ) {
images.add(cv);
contentDocIds.add(cv.ContentDocumentId);
}
}
if ( !images.isEmpty() ) {
// query ContentDocumentLink and build a map of ContentDocumentId -> CDL, which will allow us to determine the parent object type
Map<Id, List<ContentDocumentLink>> contentDocToLinks = new Map<Id, List<ContentDocumentLink>>();
for ( ContentDocumentLink cdl : [select Id, ContentDocumentId, LinkedEntityId from ContentDocumentLink where ContentDocumentId in :contentDocIds] ) {
if ( !contentDocToLinks.containsKey(cdl.ContentDocumentId) ) {
contentDocToLinks.put(cdl.ContentDocumentId, new List<ContentDocumentLink>());
}
contentDocToLinks.get(cdl.ContentDocumentId).add(cdl);
}
// loop back through our images and determine which are linked to forbidden objects
for ( ContentVersion cv : images ) {
List<ContentDocumentLink> cdls = contentDocToLinks.get(cv.ContentDocumentId);
if ( cdls != null ) {
for ( ContentDocumentLink cdl : cdls ) {
// use the nifty Id.getSObjectType() field to determine if the image File is linked to an object we don't want users associating with
if ( cdl.LinkedEntityId != null && objsPreventImageFiles.contains(cdl.LinkedEntityId.getSObjectType()) ) {
cv.addError('You cannot upload an image File associated with this object (' + cdl.LinkedEntityId.getSObjectType() + ')');
break;
}
}
}
}
}
}
public class Utils {
public static String getFileExtension(String filename) {
String ext = '';
if ( filename != null ) {
List<String> splits = filename.split('\\.');
ext = splits.get(splits.size()-1);
}
return ext.toLowerCase();
}
}
A: When the Attachment object is inserted after uploading via a browser, its ContentType field will have a value of "image/png" or "image/jpeg" so you can call addError from your trigger to block the insert. I do not know, but presume that this error is presented appropriately in Lightning Experience.
(See e.g. The Complete List of MIME Types for more possible values.)
PS
From Files and Content: What's Not in Lightning Experience:
Files and attachments are two different types of objects, and always
have been. Unlike files, attachments are associated only with a
particular record and can't be shared further. In Lightning
Experience, uploads to the Notes & Attachments related list are files.
In Salesforce Classic prior to Spring '16, uploads to this related
list were always attachments instead of files. For existing orgs, this
means that the Notes & Attachments list can contain a mix of
attachments that were uploaded in Salesforce Classic and files added
in Lightning Experience.
Personally I'm unclear on what you can and can't do with a file - hopefully someone else can tell you.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,147 |
MOURINHO CONSOLES ESSIEN ON BIRTHDAY 'NO SHOW'
https://pbs.twimg.com/media/FAG765HUcAEdmal.jpg
Ghanaian midfielder Michael Essien has a very good relationship with Jose Mourinho dating back to their time together at Chelsea. Mourinho was the catalyst behind Chelsea paying a big fee to sign Michael Essien from Lyon in 2005.
Read more about MOURINHO CONSOLES ESSIEN ON BIRTHDAY 'NO SHOW'
ESSIEN: LUKAKU WILL BE VERY IMPORTANT THIS SEASON
The newest Chelsea striker, well, not entirely new, because he was once their player, Romelu Lukaku looks to be getting all the love, praise, and attention he needs as he continues to be hailed by the former players of the Blues. The likes of Didier Drogba and John Terry have already hailed the club's decision to bring back the player to Stamford Bridge and now, their former midfielder Michael Essien has also revealed how glad he is that the player has been brought back to the club by the manager Thomas Tuchel. Chelsea broke their club transfer record after paying £97 million to Italian Serie A giants Inter Milan for the signing of Lukaku and fans have been buzzing ever since.
Read more about ESSIEN: LUKAKU WILL BE VERY IMPORTANT THIS SEASON
LEGEND REVEALS BEST TEAM IN GHANA
Ghanaian football legend Michael Essien has celebrated Accra Hearts of Oak after they emerged the league champions in the top flight as far as Ghanaian club football is concerned. Michael Essien joined the Hearts of Oak celebrations via his official Instagram account.
Michael Essien described Accra Hearts of Oak as the best team in Ghana.
Hearts of Oak became the champions in the Ghana Premier League after securing an unassailable lead with one game left to play. The Phobianswere declared the Ghanaian champions with a game to spare in the 2020/21 season. Accra Hearts of Oak had not won the league title in Ghana for 12 years until the latest triumph.
Read more about LEGEND REVEALS BEST TEAM IN GHANA
Accra Hearts of Oak sealed the league title despite the draw with Liberty Professionals in theirmatchday 33 encounter as they ended their 12-year wait for a league title.
ESSIEN BEGINS COACHING CAREER AT NORDJAELLAND
Ex-Chelsea and Lyon midfielder Michael Essien has started his coaching career at FC Nordjaelland in the Danish league. Michael Essien has embarked on his first coaching role after being named the assistant manager at the Danish top flight club.
Michael Essien posted images of him on the touchline during preseason with FC Nordjaelland. The former Ghanaian national team player had been working with the Danish club since last season but his appointment as an assistant manager was not confirmed until this summer. Michael Essien has penned a two-year deal to become the FC Nordjaelland assistant manager until 2023.
Read more about ESSIEN BEGINS COACHING CAREER AT NORDJAELLAND
ESSIEN DELIGHTED WITH EKWAH OVER TRANSFER
Chelsea and Ghanaian football legend Michael Essien has expressed his delight after youngster Pierre Ekwah left Chelsea to join London rivals West Ham United earlier this month. Michael Essien sent his congratulatory message to 19-year-old Ekwah via Instagram as he wished him the best of luck with his move across London to West Ham United.
Ekwahthanked 'big brother' Michael Essien for his congratulatory message to him. He would be hoping to establish himself as a quality Premier League midfielder just like Michael Essien did almost two decades ago when he moved to the Premier League from Lyon in the French top flight.
Read more about ESSIEN DELIGHTED WITH EKWAH OVER TRANSFER
MOURINHO PROFESSES LOVE FOR GHANA
Newly appointed AS Roma manager Jose Mourinho has professed his love for Ghana and its people after his interaction with former player Michael Essien on social media. Michael Essien initiated the latest interaction between the two Chelsea icons after he posted a photo of himself and Mourinho while they were together.
Michael Essien posted a photo of himself and Mourinhoon his official Instagram page while they were laughing in Chelsea shirts and with the caption that says the picture was taken when the ex-Ghanaian national team star took the Portuguese tactician to a charity event in Ghana. Mourinho then replied that he enjoyed his stay in Ghana and that it was an amazing experience for him.
Read more about MOURINHO PROFESSES LOVE FOR GHANA
ESSIEN PREDICTS FA CUP FINAL WRONGLY
Chelsea legend Michael Essien has made a wrong prediction on the FA Cup final after he backed the Blues to beat Leicester City. In his latest interview granted shortly before the FA Cup final and monitored by popular English media portal Metro, Essien backed Chelsea to beat Leicester City to win the trophy.
In the end, Michael Essien was proved wrong as Leicester City stunned Chelsea in the FA Cup final played at the Wembley Stadium at the weekend to win the competition for the first time. Belgian international YouriTielemans was the hero for the Foxes as he produced a moment of brilliance to win the game. His unstoppable long range shot found the back in the net in the second half to give Leicester City a 1-0 win over Chelsea.Tielemans was rightly named the best player at the end of the game.
Read more about ESSIEN PREDICTS FA CUP FINAL WRONGLY
MOURINHO, ANCELOTTI SHARE MANAGERIAL SIMILARITIES - ESSIEN
Michael Essien has revealed that managerial greats Carlo Ancelotti and Jose Mourinho share some managerial similarities. The Ghanaian talked about the two managers in an exclusive interview with the official Everton podcast.
According to Essien, the two managers are similar in their management techniques. Michael Essien said Ancelotti and Mourinho are relentless when it comes to man-management and training techniques.
Michael Essien played for both Mourinho and Ancelotti at Chelsea during his playing career in the Premier League.Mourinho was responsible for his then club-record move from French giants Lyon to Stamford Bridge in 2005. Chelsea paid around £26m for his services.
Read more about MOURINHO, ANCELOTTI SHARE MANAGERIAL SIMILARITIES - ESSIEN
Ex-Ghanaian international player Michael Essienis currently a member of the coaching staff at Nordsjaelland in the Danish Superligahaving recently made the step up into football management. Nordsjaellandis the start for Michael Essien as far as his coaching career is concerned, while Bastia was the start of his professional football career back in 2000.
Michael Essien left Bastia for fellow French team Lyon after three years at the club. The Ghanaian midfielder rejected a more lucrative offer from another French club in PSG before moving to Lyon in the summer of 2003 for a fee believed to be around €7.8m. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,546 |
package betterwithmods.common.blocks.mechanical.tile;
import betterwithmods.common.BWRegistry;
public class TileEntityCauldron extends TileEntityCookingPot {
public TileEntityCauldron() {
super(BWRegistry.CAULDRON);
}
@Override
public String getName() {
return "inv.cauldron.name";
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,921 |
Induced reflection in Tamm plasmon systems
Hua Lu, Yangwu Li, Han Jiao, Zhiwen Li, Dong Mao, and Jianlin Zhao
Hua Lu,1,* Yangwu Li,1 Han Jiao,1 Zhiwen Li,1 Dong Mao,1 and Jianlin Zhao1,2
1MOE Key Laboratory of Material Physics and Chemistry under Extraordinary Conditions, and Shaanxi Key Laboratory of Optical Information Technology, School of Science, Northwestern Polytechnical University, Xi'an 710072, China
2jlzhao@nwpu.edu.cn
*Corresponding author: hualu@nwpu.edu.cn
Hua Lu https://orcid.org/0000-0003-2559-3597
H Lu
Y Li
H Jiao
Z Li
D Mao
J Zhao
Hua Lu, Yangwu Li, Han Jiao, Zhiwen Li, Dong Mao, and Jianlin Zhao, "Induced reflection in Tamm plasmon systems," Opt. Express 27, 5383-5392 (2019)
High-performance optical sensing based on electromagnetically induced transparency-like effect in Tamm plasmon multilayer structures (AO)
Graphene-tuned EIT-like effect in photonic multilayers for actively controlled light absorption of topological insulators (OE)
Strong longitudinal coupling of Tamm plasmon polaritons in graphene/DBR/Ag hybrid structure (OE)
Nonlinear effects
Organic solar cells
Spontaneous emission
Surface plasmon polaritons
2. Structure and model
3. Results and analysis
We present an induced reflection response analogue to electromagnetically induced transparency (EIT) in a novel Tamm plasmon system, consisting of a thin metal film and a Bragg grating with a defect layer. The results show that an induced narrow peak can be generated in the original broad reflection dip, which is attributed to the coupling and interference between the Tamm plasmon and defect modes in the grating structure. It is found that the EIT-like induced reflection is strongly dependent on the thickness of defect layer, grating period number between the metal and defect layers, thickness of Bragg grating layer, refractive index of defect layer, and thickness of metal film. Additionally, the induced reflection can be dynamically tuned by adjusting the angle of incident light. The numerical simulations agree extremely well with theoretical calculations. The coupling strength between the Tamm plasmon and defect modes is determined by the above parameters. These results will provide a new avenue for light field control and devices in multilayer photonic systems.
In recent years, surface plasmon polaritons (SPPs) have attracted broad attentions in the nanophotonics field because of their promising prospects for novel optical physics and functional devices [1–32]. However, the generation of SPPs requires the external assistance of prism, grating, or metal/fiber tips to compensate the wavevector mismatch between the SPPs and incident light [33]. In 2007, Kaliteevski et al. theoretically proposed a relatively new optical effect named Tamm plasmon polaritons (TPPs), which is an optical state formed between the metal film and dielectric Bragg grating [34]. Next year, they reported experimental observation of the TPP effect [35]. In contrast to SPPs, TPPs are polarization-independent and can be generated without the external compensation of wavevector [34]. This feasibly excited plasmon polaritons with strong localization and slow light properties have obtained rapid development and found crucial applications in optical manipulation and functionalities [35–55]. Particularly, the strong localization makes TPPs an exciting candidate for applications in strong coupling [36,40], nonlinear optical effect [37,44,46], spontaneous emission [38,50,53], enhanced absorption [39,43,54,55], and lasing [42,45]. For instance, Symonds et al. experimentally reported the observation of strong coupling regime generating between the TPP mode and exciton in Ga0.95In0.05As quantum wells (QWs) [36]. Liew et al. realized the integrated logical circuits based on bistable states in semiconductor microcavities with tunable Tamm-plasmon-exciton-polariton modes [37]. Gazzano et al. demonstrated the controlled spontaneous optical emission with the confined TPP mode [38]. Zhang et al. achieved enhanced broad-band light absorption in organic solar cells by the excitation of TPPs [43]. Symonds et al. proposed a novel kind of lasers based on the confined TPP modes in the metal-semiconductor structure [42,45]. Subsequently, the sensitivity of TPPs to refractive index of Bragg grating were proposed for the application in optical sensing [48,49,52]. Besides these, as a crucial issue, the coupling behaviors between TPP and cavity modes attracts special attentions [40,41,46,47,51]. Especially, the terahertz-range electromagnetically induced transparency (EIT)-like effect was realized by the coupling between Tamm and plasmonic defect states [47]. EIT is a counterintuitive phenomenon occurring in atomic systems owing to the destructive quantum interference between the possible pathways to the upper energy levels [56]. The coupling-based EIT-like effect operating in near-infrared region could find excellent applications in optical modulation [12], sensitive sensing [30], multi-channel filtering [31], and lasing [32]. Exploring the near-infrared EIT-like response in Tamm plasmon systems is particularly significant for the realization of novel functionalities based on TPPs.
Here, we report a near-infrared optical response analogue to the EIT-like effect in the Tamm plasmon system by introducing a defect layer into Bragg grating. The simulation results illustrate that an obvious peak can appear in the TPPs-induced broad reflection spectral dip. This EIT-like induced reflection is attributed to the coupling and interference between the TPP and defect modes in the multilayer photonic structure. We also find that the induced reflection is particularly dependent on the thickness of defect layer, grating period number between the metal and defect layers, thickness of Bragg grating, refractive index of defect layer, thickness of metal film, and incident angle of light. The numerical simulations are consistent with theoretical calculations. These results will open a new avenue for the generation of EIT-like effect and enrich the TPP control and applications.
As shown in Fig. 1, the proposed Tamm plasmon multilayer system is composed of a thin metal film and a Bragg grating with a defect layer. The metal is assumed as silver (Ag), whose relative permittivity can be described using Drude model: εa(ω) = ε∞-ωp2/[ω(iγ + ω)] [57], where ω = 2πc/λ represents the free-space angular frequency of light, ε∞ is the relative permittivity of metal at the infinite frequency, ωp is the bulk plasma frequency of metal, and γ is the electron collision frequency of metal. c is the light velocity in vacuum. The parameters for silver can be fitted as ε∞ = 3.7, ωp = 9.1 eV, and γ = 0.018 eV according to the experimental data [57]. The Bragg grating consists of periodically stacked SiO2 and Si3N4 layers, whose refractive indices can be set as nA = 1.45 and nB = 2.2, respectively. The refractive index of Al2O3 defect layer is set as ns = 1.76. The thicknesses of metal, SiO2, Si3N4, and Al2O3 layers are originally set as da = 30 nm, dA = 275 nm, dB = 160 nm, and ds = 0 nm, respectively. The grating period number N is set as 24. The TM-polarized light is incident with an angle θ = 0°. The characteristics of light propagation in multilayer photonic systems can be theoretically calculated by the transfer matrix method (TMM) [34]. In TMM, the matrixes Mj and Pj can be used to characterize the light propagation through the j-th boundary and layer, which are respectively described as
(1)Mj=12nj−1cosθj−1(nj−1cosθj+njcosθj−1nj−1cosθj−njcosθj−1nj−1cosθj−njcosθj−1nj−1cosθj+njcosθj−1),
(2)Pj=(exp(−i2πdjnjcosθj/λ)00exp(i2πdjnjcosθj/λ)),
where θj and nj stand for the angle of light propagation and the refractive index in the j-th layer, respectively. They satisfy the relation: nj-1sinθj-1 = njsinθj (θ0 = θ). dj represents the thickness of the j-th layer. The total matrix can be written as Q = M1P1M2P2…MN+2. The reflection of multilayer photonic system can be calculated by R = |Q21/Q11|2. There exists an electromagnetic state between the metal and dielectric multilayer, whose frequency is less than plasma frequency of the metal and close to the operating frequency of Bragg grating [34]. The condition of eigenmode in this multilayer structure can be written as rMrB = 1, where rM and rB are the amplitude coefficients of light reflection on the metal and Bragg grating from Si3N4, respectively. According to the Fresnel formula, rM can be written as rM = (nB-na)/(nB + na). When ω<<ωp and γ is small, na can be approximately expressed as na≈iωp/ω [34]. Thus, rM can be written as rM≈exp[i(π + 2nBω/ωp)]. With a large number of layers for Bragg grating, rB can be expressed as rB = exp[iη(ω-ω0)/ω0], where η = πnB/(nB-nA). na is the refractive index of the metal, and ω0 = πc/(nBdB + nAdA) is the Bragg frequency. By combing these equations, the frequency of this eigenmode (i.e., TPP frequency) can approximately be expressed as ωT = ω0/[1 + 2ω0(nB-nA)/(πωp)]. By substituting the above parameters, ωT is predicted to be 0.790 eV. This value agrees well with the frequency of the reflection dip (0.796 eV) obtained by the TMM method, as shown in Fig. 2(a). To verify the theoretical results, we use the finite-difference time-domain (FDTD) method (commercial package from FDTD Solutions) to simulate the optical response in the multilayer system. In FDTD method, the periodic boundary conditions are set on the top/bottom of unit cell, and the perfectly matched layer absorbing boundary conditions are set on the left/right sides [58]. The spatial mesh grids are set as Δx = Δy = 4 nm. The source is set as a pulse with a pulse length of 9.9 fs and center wavelength of 1550 nm. For the convergence of the results, the temporal step and simulation time are set as 0.00893 fs and 15000 fs, respectively. The FDTD simulations agree well with the theoretical results, as shown in Fig. 2(a).
Fig. 1 Schematic diagram of the Tamm plasmon multilayer system. The thicknesses of metal, SiO2, Si3N4, and Al2O3 layers are denoted by da, dA, dB, and ds, respectively. The grating period number is N. The incident angle of light is θ.
Fig. 2 (a) Reflection spectra of the multilayer photonic system without and with the defect layer (i. e., ds = 0 and 258 nm). The circles and curves stand for the FDTD simulation and TMM theoretical results, respectively. The inset shows the three-level system. (b)-(c): Field distributions of |E|2 at the wavelength of 1556 nm in the multilayer systems without and with the defect layer. Here, da = 30 nm, dA = 275 nm, dB = 160 nm, N = 24, and θ = 0°.
As shown in Fig. 2(b), the strongly confined eigenmode (TPP mode) is formed between the metal and Bragg grating at the dip of reflection spectrum in Fig. 1 [34]. TPPs are different from surface Bloch wave generated between the isotropic medium and Bragg grating [59]. The enhanced field of TPP mode could find crucial applications in perfect absorption [54,60–64]. As the thickness of defect layer is set as ds = 258 nm, an obvious narrow peak appears at the center of the broad TPPs-induced reflection dip. The defect mode in the middle of the grating can be excited at the reflection peak (λp = 1556 nm). This spectral response can be regarded as a typical EIT-like profile [23,31]. The physical mechanism can be understood through the analogy to EIT in atomic systems [23,65–67]. For a typical three-level system, the ground state is denoted by |0> and the upper states are denoted by |1> and |2> (i.e., dark state), as shown in the inset of Fig. 2(a) [23,67]. The light is incident on the metal and excites the TPP mode, which can be analogue to the transition from |0> to |1> under the condition of "probe laser" [65]. The narrow defect mode in Bragg grating will be excited (by the evanescent field of TPPs) and couple with the TPP mode, which can be in analog to the transition between |1> and |2> in the absence of "pump light" [65]. The coupling strength κ corresponds to the Rabi frequency [67]. Thus, the two possible transition processes, namely |0>→|1> and |0>→|1>→|2>→|1>, will generate the destructive interference owing to a π-phase difference [23,66]. The destructive interference contributes to the disappearance of TPP field and the generation of narrow induced peak in a broad reflection dip, as depicted in Fig. 2. the dissipation rates of TPP and defect modes are denoted by γ1 and γ2, respectively. The defect mode in Bragg grating relies on the thickness of defect layer ds [68]. Here, we investigate the dependence of induced reflection spectrum on ds. Figure 3(a) shows that there exist obvious EIT-like peaks when ds is around 258 nm. As depicted in Fig. 3(b), the wavelength of induced reflection peak exhibits a linear red-shift with the increase of ds around 258 nm. The resonance wavelength of defect cavity in the Bragg grating can be expressed as λ0 = 4πnsds/(2kπ-φ1-φ2), where φ1 and φ2 are reflective phase shifts on two sides of defect cavity [68]. The integer k stands for the order of resonant mode. From this formula, it is found that the wavelength of defect mode linearly increases with ds. According to the interference mechanism of EIT-like effect, the wavelength of defect mode will determine the position of reflection peak [29]. Thus, we can see that the reflection peak has a linear red-shift with increasing ds. The simulation results are in good agreement with theoretical calculations. The coupling strength is a crucial factor to influence the EIT-like response, which can be controlled by changing the coupling distance between the TPP and defect modes. Figure 4 depicts the reflection spectra with different grating period numbers between the metal and defect layers. It is found that the spectral width of induced reflection becomes sharper with increasing the period number, while the height of reflection peak gradually drops, thus providing a trade-off between the width and height of reflection peak. The numerical simulations are consistent with theoretical results. This behavior is similar to the EIT-like response in plasmonic systems [22,29]. The coupling strength κ increases with the decrease of coupling distance [29,31]. In the strong-driving region with κ>>κT (κT = (γ1-γ2)/4), the induced reflection in the transparency window will be attributed to Autler-Townes splitting (ATS) with a symmetric doublet [67]. When the grating period number between metal and defect layer is 8, ATS will appear with κ = 7.97 × 1012 rad/s and κT = 1.06 × 1012 rad/s.
Fig. 3 (a) Evolution of reflection spectrum with the defect layer thickness ds. (b) Wavelengths (λp) of induced reflection peak with different ds. Here, da = 30 nm, dA = 275 nm, dB = 160 nm, N = 24, and θ = 0°.
Fig. 4 FDTD simulation (a) and TMM theoretical (b) results of reflection spectra with different grating period numbers between the defect and metal layers. Here, da = 30 nm, dA = 275 nm, dB = 160 nm, ds = 258 nm, and θ = 0°.
Subsequently, we investigate the influence of grating layer thickness on the induced reflection. As shown in Fig. 5(a), the wavelength of reflection dip possesses a linear red-shift with increasing the Si3N4 thickness dB. According to the TPP theory, the TPP frequency can also be expressed as ωT = πc/[nBdB + nAdA + 2(nB-nA)c/ωp]. Thus, the TPP wavelength can be obtained as λT = 2[nBdB + nAdA + 2(nB-nA)c/ωp], which linearly increases with dB. Thus, we can observe the linear red-shift of reflection dip with increasing dB. When dB reaches around 160 nm, the narrow induced refection peaks emerge in the spectral dip. The results in Fig. 5(b) show that the wavelength of reflection peak has a red-shift with increasing dB, which is attributed to the increase of the wavelength of defect mode. The similar response will be generated when we change the SiO2 thickness dA (not shown here). The selection of geometrical parameters can offer the effective means for the tunability of induced reflection spectrum. Figure 6(a) shows the evolution of reflection spectrum with the refractive index of defect layer ns. It is found that the wavelength of induced reflection peak has a linear red-shift with increasing ns. According to λ0 = 4πnsds/(2kπ-φ1-φ2), we can see that the wavelength of defect mode linearly increases with ns. Thus, the induced reflection peak linearly red-shifts as ns increases. It is shown in Fig. 6(b) that the wavelength of induced reflection peak remains constant, and the steeper slope moves from the left to right side of reflection peak when da increases. From Ref [39], we can find that the TPP wavelength possesses a blue-shift with the increase of metal film thickness da. The wavelength of defect mode is mainly dependent on the parameters of Bragg grating, while not sensitive to da. Thus, we can see the unchanged position of induced reflection peak in Fig. 6(b). By combining with the blue-shift of TPP reflection dip, it is not difficult to understand the change of spectral lineshape for induced reflection peak with da. Finally, the dependence of induced reflection spectrum on the incident angle θ is theoretically studied. As depicted in Fig. 7(a), the wavelength of induced reflection peak has a blue-shift as θ increases. The TPP wavelength decreases when the angle of incident light increases [35]. Therefore, the reflection dip possesses a blue-shift with increasing θ. We can see from Ref [69] that the wavelength of defect mode also decreases with the increase of θ. Thus, it is also not difficult to understand the blue-shift of reflection peak with increasing θ. Moreover, it is found that the peak position gradually deviates from the center and approaches the short-wavelength dip. It illustrates that the wavelength of TPP mode exhibits slower shift than that of defect mode. When the incident light is changed into TE polarization, we can observe the similar response for the induced reflection spectrum, as shown in Fig. 7(b). In contrast to TM polarization, the blue-shift range is smaller for TE polarization. The theoretical positions of induced reflection peaks are consistent with the wavelengths of reflection peaks obtained by the simulations. The adjustment of incident angle provides an essential way for the dynamical control of the EIT-like reflection response. It is worth noting that the above parameters play an important role in determining the coupling strength κ due to the variation of electromagnetic field. Figure 8 depicts the dependence of κ on the thickness of defect layer ds, period number of Bragg grating between metal and defect layers, thickness of Si3N4 layer dB, refractive index of defect layer ns, thickness of metal film da, incident angle of light θ. These values are obtained by fitting the results in Figs. 3–7 using equations in Ref [29]. We can see in Figs. 8(a) and 8(d) that κ exhibits a slight decrease when ds and ns deviate from 258 nm and 1.76, respectively. κ drops distinctly with increasing the period number of Bragg grating and da, as shown in Figs. 8(b) and 8(e). This verifies the above analysis. κ raises slowly when dB increases around 160 nm. κ increases slightly as θ increases from 0° to 12° for TM-polarized light, while decreases for TE-polarized light. The variations of electric field intensities in the defect layer at the induced peaks are opposite with the change of θ for TM/TE-polarized light (not shown here).
Fig. 5 (a) Evolution of reflection spectrum with the thickness of Si3N4 layer dB. (b) Wavelengths (λp) of induced reflection peak with different dB. Here, da = 30 nm, dA = 275 nm, ds = 258 nm, N = 24, and θ = 0°.
Fig. 6 (a) Evolution of reflection spectrum with the refractive index of defect layer ns when da = 30 nm. (b) Evolution of reflection spectrum with the thickness of metal film da when ns = 1.76. The circles denote the positions of induced reflection peak obtained by FDTD simulations. Here, dA = 275 nm, dB = 160 nm, ds = 258 nm, and N = 24.
Fig. 7 Evolution of reflection spectrum with the incident angle θ for TM (a) and TE (b) polarized light. The circles denote the positions of induced reflection peak obtained by FDTD simulations. Here, da = 30 nm, dA = 275 nm, dB = 160 nm, ds = 258 nm, ns = 1.76, and N = 24.
Fig. 8 Dependence of coupling strength κ on the thicknesses of defect layer ds (a), period number of Bragg grating between metal and defect layers (b), thickness of Si3N4 layer dB (c), refractive index of defect layer ns (d), thickness of metal film da (e), incident angle of light θ (f). The structural parameters in (a)-(f) are the same as those in Figs. 3–5, 6(a), 6(b), and 7, respectively.
We have theoretically and numerically investigated an near-infrared EIT-like induced reflection effect in a new Tamm plasmon multilayer system consisting of the thin silver film and SiO2/Si3N4 layers stacked Bragg grating with an Al2O3 defect layer. The results illustrate that the induced reflection peaks can be generated when the thickness of grating defect layer approaches special values due to the coupling and destructive interference between the TPP and defect modes in the multilayer photonic structure. It is also found that the position of induced reflection peak is particularly dependent on the thickness of defect layer and Bragg grating layer as well as the refractive index of defect layer. The spectral width and height of induced reflection can be dynamically tailored by controlling the grating period number between the metal and defect layers. The spectral lineshape of induced reflection is sensitive to the thickness of metal film. Additionally, the induced reflection spectrum can be flexibly tuned by adjusting the incident angle for both the TM and TE polarized light. The above parameters together determine the coupling strength between the TPP and defect modes in this system. Our results will open a new pathway for the EIT-like spectral response as well as the TPP manipulation and devices (e.g. optical switches, filters, and sensors) in multilayer photonic systems.
National Key R&D Program of China (2017YFA0303800); National Natural Science Foundation of China (11774290, 11634010, and 61705186); Natural Science Basic Research Plan in Shaanxi Province of China (2017JQ1023); Technology Foundation for Selected Overseas Chinese Scholar of Shaanxi Province (2017007); Fundamental Research Funds for the Central Universities (3102018zy039 and 3102018zy050).
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J. Wu, J. Guo, X. Wang, L. Jiang, X. Dai, Y. Xiang, and S. Wen, "Dual-band infrared near-perfect absorption by Fabry-Perot resonances and surface phonons," Plasmonics 13(3), 803–809 (2018).
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Abdala, N. L.
Aberra Guebrou, S.
Abram, R.
Aizin, G.
Alaee, R.
Allen, S.
Angelomé, P. C.
Argyropoulos, C.
Auguié, B.
Bartal, G.
Baumann, V.
Baumberg, J.
Beere, H.
Bellessa, J.
Bethke, D.
Bloch, J.
Boller, K.-J.
Boyd, R.
Bozhevolnyi, S.
Brand, S.
Braun, T.
Brolo, A.
Brückner, R.
Brucoli, G.
Cai, B.
Chamberlain, J.
Chang, H.
Chen, C.
Chen, J.
Chen, K.
Chen, W.
Christmann, G.
Christy, R.
Coulson, C.
Dai, L.
Dai, X.
de Vasconcellos, S. M.
Deng, J.
Deng, Z. L.
Dong, J. W.
Du, L.
Duan, X.
Dyer, G.
Ebbesen, T. W.
Egorov, A. Yu.
Eigenthaler, U.
Fainstein, A.
Fan, D.
Fang, H.
Fang, Y.
Farrer, I.
Feng, J.
Frob, H.
Fuertes, M. C.
Fuller, K.
Gan, X.
Garrido Alzar, C. L.
Gauthron, K.
Gazzano, O.
Genet, C.
Giannini, V.
Gladden, C.
Gómez Rivas, J.
Gong, Q.
Gong, Y.
Gramotnev, D.
Greffet, J. J.
Grine, A.
Grossmann, C.
Gu, M.
Gubaydullin, A. R.
Gutowski, J.
Harris, S. E.
Hintschich, S.
Hirscher, M.
Hofling, S.
Homeyer, E.
Hommel, D.
Hong, M.
Hossain, M. M.
Hou, Y.
Hu, F.
Huang, K.
Huang, S.
Hugonin, J. P.
Iff, O.
Imamoglu, A.
Iorsh, I.
Ishihara, T.
Ivanov, K. A.
Jeng, S.
Jia, B.
Jiang, L.
Jiang, X.
Jiang, Y.
Jiao, J.
Johnson, P.
Jomaa, M. H.
Kaliteevski, M.
Kaliteevski, M. A.
Kalitteevski, M.
Kavokin, A.
Kim, K.
Klein, T.
Klembt, S.
Kolykhalova, E. D.
Laverdant, J.
Lederer, F.
Lee, K. J.
Lei, D. Y.
Lei, T.
Lemaitre, A.
Lemaître, A.
Leo, K.
Leong, E. S.
Le-Van, Q.
Lheureux, G.
Li, G. C.
Li, Y.
Liew, T.
Liew, T. Y.
Liu, J. Q.
Liu, L.
Liu, S. D.
Liu, X.
Long, Y.
Lu, H.
Lu, S.
Lukiyanchuk, B.
Lyssenko, V.
Ma, Q.
Ma, R. M.
Maier, S. A.
Mao, D.
Martinez, M. A. G.
Mesch, M.
Mikhrin, V.
Min, C.
Ming, H.
Ning, T.
Nori, F.
Nussenzveig, P.
Ong, H. C.
Ostatnický, T.
Oulton, R. F.
Özdemir, S. K.
Peng, B.
Peyronel, T.
Plenet, J.
Pozina, G.
Qin, F.
Qiu, C.
Rahman, S. S.
Rahmani, M.
Ramezani, M.
Ranjbar, M.
Rao, Y. J.
Reineck, P.
Ren, H.
Reno, J.
Ritchie, D.
Rockstuhl, C.
Rosenberger, A.
Sasin, M.
Sasin, M. E.
Schneider, C.
Scholz, R.
Sebald, K.
Seisyan, R.
Senellart, P.
Shaner, E.
Shelykh, I.
Shen, J.
Smith, D.
Smolyaninov, I.
Soler Illia, G. J. A. A.
Song, J.
Sönnichsen, C.
Sorger, V. J.
Sudzius, M.
Sun, B.
Sun, H.
Symonds, C.
Tang, C.
Tang, D.
Teng, J. H.
Tian, J.
Tiecke, T. G.
Van Hoof, N.
Vasil'ev, A.
Voisin, P.
Wang, F.
Wang, G.
Wang, H.
Wang, L. L.
Wang, P.
Wang, X.
Wen, S.
Wen, S. C.
Wu, C.
Wu, J.
Wu, J. W.
Wu, L.
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Xiang, Y.
Xiao, J.
Xiao, S.
Yang, D.
Yang, G.
Yang, H.
Yang, L.
Yang, X.
Yi, H.
You, Q.
Yu, P.
Yu, R.
Yuan, G.
Yuan, X.
Yue, S.
Yue, Z.
Zakhidov, A.
Zang, X.
Zayats, A.
Zeng, C.
Zentgraf, T.
Zhai, X.
Zhang, H.
Zhang, Q.
Zhang, W. L.
Zhang, Z.
Zhao, J.
Zheng, J.
Zhou, X.
Zhou, Y.
Zhou, Z.
Zhu, S.
ACS Nano (1)
AIP Adv. (1)
Am. J. Phys. (1)
Eur. Phys. J. Appl. Phys. (1)
J. Opt. A, Pure Appl. Opt. (1)
J. Phys. D Appl. Phys. (1)
Nano Lett. (4)
Nat. Commun. (2)
Opt. Mater. Express (1)
Phys. Rev. B (1)
Phys. Rev. B Condens. Matter Mater. Phys. (3)
Prog. Electromagnetics Res. (1)
Sci. Adv. (1)
(1) M j = 1 2 n j − 1 cos θ j − 1 ( n j − 1 cos θ j + n j cos θ j − 1 n j − 1 cos θ j − n j cos θ j − 1 n j − 1 cos θ j − n j cos θ j − 1 n j − 1 cos θ j + n j cos θ j − 1 ) ,
(2) P j = ( exp ( − i 2 π d j n j cos θ j / λ ) 0 0 exp ( i 2 π d j n j cos θ j / λ ) ) , | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,288 |
<?php
namespace Light\ObjectService\Resource\Selection;
use Light\ObjectAccess\Type\ComplexTypeHelper;
final class NestedComplexSelection extends RootSelection
{
/** @var Selection */
private $parent;
public function __construct(Selection $parent, ComplexTypeHelper $typeHelper)
{
parent::__construct($typeHelper);
$this->parent = $parent;
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,065 |
module Gitlab
module CycleAnalytics
module Summary
class Issue < Base
def initialize(project:, from:, current_user:)
@project = project
@from = from
@current_user = current_user
end
def title
'New Issue'
end
def value
@value ||= IssuesFinder.new(@current_user, project_id: @project.id).execute.created_after(@from).count
end
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,030 |
Ukrainian MP Nadiya Savchenko visited the anti-terrorist operation (ATO) zone, eastern Ukraine, to conduct checks on the state of corruption there.
She said this at a press conference following her trip to the ATO zone, an Ukrinform correspondent reports.
"What was this trip for? First of all, I as an MP wanted to personally check [if there is] corruption in the military unit in the ATO zone. I have special materials, I will submit deputy letters of inquiry, I will fight against corruption there as it should not exist during wartime," she said.
Savchenko also noted that among topical problems are also pensions and utility bills.
"This is another question which I will deal with in the near future," the MP said. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,554 |
Taxis are available at the airports and in Roseau, and can be arranged all over the island. They are easily identified by the letters, H, HA or HB preceding the registration numbers on the number plates. There are standard fees from the city to both airports. The fare from the Douglas Charles Airport to Canefield/ Roseau/ Newtown Portsmouth/ Picard is EC$65 or US $26. From Douglas Charles Airport to the Carib Territory/ Concorde/ Atkinson/ Calibishie is EC$40 or US$15; to Castle Comfort/ Loubiere/ Wallhouse is EC$70 or US$28 and to Salisbury/ Batalie/ Coulibistrie/ Colihaut is EC$100 or US$40. From Canefield to Roseau EC$25.
There are a number of car rental agencies on the island offering vehicles for rent. But before you get on the road, you will need to obtain a driver's license which cost $30 (US$12). You must be between 25 and 65 years old, with two years' driving experience to qualify for a driver's permit. Traffic use the left side of the road, most of which are well maintained.
Dominica has a reliable public transportation system consisting of primarily private minibus operators. Bus stops can be found at designated points throughout the city depending on your destination. The bus fares are standardized and ranges from EC$1.50 to EC$10.25 according the specific route. Bus rotation is fairly frequent throughout the day, but this method of transportation is not suitable for night travel. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,881 |
Q: How can I compare two lists with xunit test I am currently trying to compare two lists, with the same items in it, with xUnit but getting an error while running.
Assert.Equal(expectedList, actualList);
Error:
"Assert.Equal() Failure"
Expected: List<myObject> [myObject { modifier = '+', name = "name", type = "string" }, myObject { modifier = '+', name = "age", type = "int" }]
Actual: List<myObject> [myObject { modifier = '+', name = "name", type = "string" }, myObject { modifier = '+', name = "age", type = "int" }]
A: This has to do with object equality.
MyObject does not implement the Equals method. By default you get a reference equality. I assume you have two different objects for MyObject.
Meaning it does not matter that your List holds the similar object(meaning with same values) they are not of the same reference, so your test checks that, this is why it fails.
internal class MyObject
{
{
public char Modifier { get; set; }
public string Name { get; set; }
public string Type { get; set; }
}
}
[Fact]
public void ListMyObject()
{
var list1 = new List<MyObject>
{
new MyObject{ }
};
var list2 = new List<MyObject>
{
new MyObject{ }
};
Assert.Equal(list1, list2); // Fails
}
When we update our class to this.
internal class MyObject
{
public char Modifier { get; set; }
public string Name { get; set; }
public string Type { get; set; }
//When i add this to my class.
public override bool Equals(object obj)
{
return this.Name == ((MyObject)obj).Name;
}
}
Also as mentioned in the comments by Jonathon Chase.
It is a good idea to override the GetHashCode() method as well. It is preferred to inherit from IEquatable<T> so you can avoid casting.
Everything goes green.
[Fact]
public void ListMyObject()
{
var list1 = new List<MyObject>
{
new MyObject{ Name = "H" }
};
var list2 = new List<MyObject>
{
new MyObject{ Name = "H" }
};
Assert.Equal(list1, list2); //Passes
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,577 |
from BaseHTTPServer import BaseHTTPRequestHandler, HTTPServer
from SocketServer import ThreadingMixIn
from subprocess import Popen, PIPE
class ThreadedHTTPServer(ThreadingMixIn, HTTPServer):
pass
class WHandler(BaseHTTPRequestHandler):
def do_POST(self):
length = self.headers['content-length']
data = self.rfile.read(int(length))
p = Popen(['/docker/createtemp.sh'], stdout=PIPE)
p.wait()
tmp_file = p.stdout.read()
with open(tmp_file, 'w') as fh:
fh.write(data.decode())
p = Popen(['/docker/runcsharp.sh', tmp_file], stdout=PIPE)
p.wait()
self.send_response(200)
self.send_header('Content-type', 'text/json')
self.end_headers()
self.wfile.write(p.stdout.read().encode())
server = ThreadedHTTPServer(('', 49150), WHandler)
server.serve_forever() | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,149 |
The Dark Side of the Moon - Essays and Articles by Jim Freeman: WHEN THE TARGETING EVIDENCE FAILS--"CONTINUE TO BELIEVE"
WHEN THE TARGETING EVIDENCE FAILS--"CONTINUE TO BELIEVE"
ISLAMABAD, Pakistan, Aug. 26 -- United Nations officials in Afghanistan said Tuesday that there was "convincing evidence" at least 90 civilians -- two-thirds of them children -- were killed in a U.S.-led airstrike last week that caused the Afghan government to call for a review of U.S. and NATO military operations in the country.
Kai Eide, the top U.N. official in Afghanistan, said local officials and residents in the western province of Herat corroborated reports that 60 children and 30 adults had been killed in an Aug. 21 military operation led by U.S. Special Operations forces and the Afghan army.
. . . U.S. forces in Afghanistan have increased their reliance on air power since last year, causing a corresponding increase in civilian deaths.
The Herat assault appears to have caused the largest civilian loss of life attributed to U.S. forces since the war began in late 2001.
Pentagon spokesman Bryan Whitman said military commanders in Afghanistan continued to believe that the attack in Herat "was a legitimate strike on a Taliban target."
Sixty children dead. War is indeed hell, but war by remote control air strike is not even remotely acceptable. This story comes after a similar attack killed 25 wedding attendees, including the bride.
Yet American commanders 'continue to believe' in the viability of the targets. Who selects these targets in a country where no Americans speak the language and operations are carried out through interpreters? Who vouches for targeting and who vouches for those who vouch?
Is it possible we are being tipped by those who do not have our best interests at heart? Is it just possible that, in a country where we hardly know friend from foe, an occasional foe is giving us false targeting to besmirch an already shaky image?
We are liberating Iraq and Afghanistan and will continue to liberate until no one is left and the name American joins Attilla the Hun in infamy. | {
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Jaden Smith Says Tyler The Creator Is His "Boyfriend" And Has Been For A While
Jaden Smith Says Tyler The Creator Is His "Boyfriend" And Has Been For A While
Maybe Fans of Jaden Smith and Tyler, the Creator area unit in all probability scratching their heads when this weekend. On Sunday, whereas functioning on stage at the Camp Flog Gnaw music competition in la, Smith, 20, sky-high told concertgoers that Tyler, 27, is his beau. "I just wanna say Tyler, the Creator is the best friend in the world and I love him so much,"
Smith said in a video recorded by a fan. "And I wanna tell you guys something. I wanna tell you Tyler doesn't wanna say, but Tyler's my boyfriend!" While the group greeted the claim with cheers and sounds of shock, Smith quipped, "And he's been my boyfriend my whole life. Tyler, the Creator is my boyfriend! It's true!"
Tyler, World Health Organization was within the crowd, looked taken aback by the announcement, waving his finger at the audience to signal he wasn't on board with it. It is very unclear that if the buds simply have a deep bromance or an actual romance.
Also read: Selena Gomez on Katy Perry And Orlando Bloom Putting Off Marriage Plans
Tyler, World Health Organization created the Camp Flog Gnaw competition, has danced around his sexuality on multiple occasions in the past, leaving fans wondering if he was serious. Requests for comment from each Smith and Tyler's groups weren't straight off came.
On Monday, the musician hosted the pop-up I Love You Restaurant, a food truck on Skid Row that provided free vegan meals to the homeless for the day.
"The I Love You Restaurant Is A Movement That Is All About Giving People What They Deserve, Healthy, Vegan Food for Free," he captioned photos of the truck. "Today we tend to Launch Our 1st in the future Food Truck Pop-Up in Downtown LA." Smith also posted a video of the truck's first patrons grabbing their pre-packaged meals. His philanthropic deed came just days after he released his latest album, "ERYS," on the Fourth of July.
See also: Shane Gillis- The Famous Comedian Fired From 'Saturday Night Live' For Racist Remarks
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} | 8,014 |
{"url":"https:\/\/space.stackexchange.com\/questions\/2021\/nuclear-thermal-rocket-specific-impulse-calculation-uses-1-amu-is-that-wrong","text":"Nuclear thermal rocket specific impulse calculation uses 1 amu, is that wrong?\n\nFollowing the parameters in the Wikipedia nuclear thermal rocket article, it seems to paint somewhat of a shaky world view. Consider these quotes:\n\nCurrent (2010) 25,000 pound-thrust reference designs (NERVA-Derivative Rockets, or NDRs) are based on the Pewee, and have specific impulses of 925 seconds.[citation needed]\n\nand\n\nA solid-core nuclear thermal rocket's fuel elements are unlikely to spread over a wide area because the elements are designed to withstand very high temperatures (up to 3500K) and high pressures (up to 200 atm)\n\nI seriously doubt any of these actual parameters are realistic, but that's not important, I just want to talk about them in an academic sense. I think I understand the general idea of how a temperature can be turned into a specific impulse:\n\n$$\\frac{3}{2} k T = \\frac{1}{2} m v^2$$\n\n$$I_{sp} = \\frac{ \\sqrt{ \\frac{3 k T}{m} } }{g}$$\n\nIf I use the above temperature, I can reproduce their specific impulse. Like so, and Google gives 950 seconds. I'm sure there are some other factors that could easily reduce that by 25. But in order to get that, I had to plug in $m= 1 \\text{ amu}$.\n\nThat clearly can't be right! A nuclear thermal rocket heats cryogenic Hydrogen to produce Hydrogen gas, a diatomic gas with the formula $H_2$, not $H_1$. The molecular weight of the diatomic gas is obviously $2 \\text{ amu}$, and there is no way to get that specific impulse (or anywhere close) using that mass.\n\nSo what's going on here? Did NASA engineers of the 60s demonstrate that heating of the Hydrogen gas would disassociate the molecule, or did some high school kid blindly plug numbers into the equation without thinking?\n\nYes\n\nthey considered the disassosciation of hydrogen\n\nAccording to this source :\n\nPrevious testing used a maximum temperature of 2,750\u00b0 K, short of the 3000+\u00b0 K design temperature for the NCPS. The NTREES facility is designed to test fuel elements and materials in hot flowing hydrogen, reaching pressures up to 1,000 pounds per square inch and temperatures of nearly 5,000\u00b0 F (2,760\u00b0 C) \u2013 conditions that simulate space-based nuclear propulsion systems to provide baseline data critical to the research team.\n\nUnder that conditions the hydrogen would disassociate into atomic hydrogen\n\nexternal source\n\nat 5,000\u00b0K about 95% of the molecules in a sample of hydrogen are dissociated into atoms\n\n\u2022 Your first source gives a temperature of almost half your second source. Do you think this is because of the high pressure lowering the dissociation T? \u2013\u00a0AlanSE Sep 18 '13 at 15:03\n\u2022 @AlanSE yes the second source is conducted experiment in lab. The high temperature decrease the disassociation temperature of hydrogen \u2013\u00a0Hash Sep 19 '13 at 6:49\n\nSome of the hydrogen will be disassociated. For the reaction mass that is not dissociated, and passes through the engine in the form of diatomic hydrogen, in addition to the three translational degrees of freedom, heat energy is also put into the vibrational and rotational energy of the hydrogen molecule. So the energy stored is 6\/2kT, not 3\/2 kT as in the first formula you quote. (The vibrational and rotational energy is converted into exhaust kinetic energy as the hydrogen cools as it expands through the nozzle. Some of it, however, isn't converted into kinetic energy because there isn't time. This is known as \"frozen flow loss\". Frozen flow losses depend on details of the engine.)","date":"2019-11-22 02:37:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7712998390197754, \"perplexity\": 1082.7163563829633}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496671106.83\/warc\/CC-MAIN-20191122014756-20191122042756-00459.warc.gz\"}"} | null | null |
Emiliano's family has already raised 350,000$ to continue the adventure of the former striker Nantes. Many football stars answered the call.
Since Saturday morning very early, the search for the plane of Emiliano Sala disappeared Monday night over the Channel have resumed. Two boats went to sea in an area north of the island of Guernsey. David Mearns, who made his name by discovering several wrecks at sea, is participating in the research that was privately funded by the player's family through an online prize pool on gofundme.com.
Set up Friday, it has been a huge success. At 10:00 this Sunday, the bar of 350,000$ has been crossed. "The temporary ceiling of the collection has already been almost reached, we decided to raise it (this will be the one and only time): the costs related to research being subject to variables (duration, weather conditions, towing, etc. .) We hope to cover the maximum expenses related to our quest, "said the family, thanking everyone who took part in this momentum of solidarity.
Among the 3680 people who made a donation, we find the names of many footballers. The National Union of Footballers in France (UNFP) had sent SMS to all professional players in France to encourage them to inflate the pot, revealed RMC. A successful call since several big names in Ligue 1 has obviously made their contribution.
Sunday morning, a donation of 30,010 euros was paid by a certain Kylian Mbappe. The day before, Adrien Rabiot, whose identity was confirmed by the platform, had donated € 25,000. Donors under the names of the Marseilles Lucas Ocampos (5000 €) and Dimitri Payet (10.000 €), "the Kita family" (6000 €) or the Nantais coach Vahid Halilhodzic (2000 €) have also been generous alongside other stars of the ball and hundreds of anonymous. | {
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M7 är en motorväg i Ungern som går mellan Budapest och gränsen till Kroatien. Motorvägen går via Siófok, Balatonkeresztúr, Zalakomár, Nagykanizsa och Letenye. Detta är en för Ungern mycket betydelsefull motorväg eftersom den binder ihop Budapest med Balatonsjön (som är en populär badsjö och en mycket stor turistattraktion i Ungern) och Kroatien. Den sista biten av M7 blev klar i slutet på oktober 2008. Motorvägen utgör en förbindelse mellan Budapest och Zagreb och det är motorväg hela vägen mellan städerna.
Galleri
Se även
Motorväg
Motorvägar i Ungern
A4 (motorväg, Kroatien)
Externa länkar
Motorvägar i Ungern | {
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Video Shows this Cop Shoot at Minivan Full of Children, How is He Still a Cop?
Precedent set, shooting innocent until proven guilty people in the back, as they flee. is justifiable.
Taos County, NM -- In 2013, a mother and her children feared for their lives when a stop for a speeding ticket turned into much more.
Oriana Farrell, 39, of Memphis, was bringing her children across the country on a family road trip, when things took a turn for the worse.
After Farrell had been given a citation, she decided to drive away from police. Several short chases then followed, before one of the cops, Elias Montoya, unloaded his pistol into the minivan as it drove off.
Common sense prevailed, for a short time, and Montoya was fired from his job as a New Mexico State Trooper. However, he was just rehired with the Taos County Sheriff's Department as a deputy.
According to Taos Co, Sheriff Jerry Hogrefe, Montoya was chosen from 17 different applicants to fill just one of two available positions.
Did the other sixteen applicants actually kill children? Is that why a person who shot at children was hired?
Insidiously enough, after the incident got swept under the rug, the State Troopers actually tried to hire Montoya back, but he refused.
Apparently Sheriff Hogrefe thinks that shooting at a van full of children is just fine and dandy and has stated that Montoya has a clean record.
"Nothing from DPS [or] the training academy pertaining to his certification or qualifications…so he has a clean work history," Hogrefe said.
According KOB, the sheriff defended Montoya's actions in the dash cam video, saying Montoya didn't know there were kids in the van and that he believed the driver had a gun.
"I cannot fault anyone for making a split-second decision," Hogrefe said.
According to Montoya's new boss, shooting innocent until proven guilty people in the back as they flee is justifiable, as it was a "split-second decision." Seems like Montoya will fit right in.
Below is the raw video from the original incident. | {
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妖怪 ("bewitching spectre") may refer to:
Yōkai, supernatural monsters in Japanese folklore
Yaoguai, supernatural monsters in Chinese folklore | {
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Q: Salesforce1 scripts problem lightning So I have implemented a script that uses angular as well, however when I use {{1+1}} it does nothing, also it shows me a warning that is created inside the Javascript file(we created it too show that it works).
The code for the lightning static resource is:
<aura:component implements="flexipage:availableForAllPageTypes">
<ltng:require
styles="/resource/bootstrap"
scripts="/resource/BookingWidget"
/>
<html ng-app="">
<body >
{{1+1}}
</body>
</html>
ng-app="" even being empty it should run {{1+1}}
I left it for five minutes and it gave me a warning
"WARNING: Tried to load angular more than once"
| {
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import { Meteor } from 'meteor/meteor';
import { Roles } from 'meteor/alanning:roles';
import { Accounts } from 'meteor/accounts-base';
if (!Meteor.isProduction) {
const users = [{
email: 'admin@admin.com',
password: 'password',
profile: {
name: { first: 'Carl', last: 'Winslow' },
},
roles: ['admin'],
}];
users.forEach(({ email, password, profile, roles }) => {
const userExists = Meteor.users.findOne({ 'emails.address': email });
if (!userExists) {
const userId = Accounts.createUser({ email, password, profile });
Roles.addUsersToRoles(userId, roles);
}
});
}
| {
"redpajama_set_name": "RedPajamaGithub"
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Sri Jain Sthanak
Sri Jain Sthanak is located inside the premises of the Jain Temple in Mattancherry, which was built in 1960. The Jain Sthanak is a large prayer hall. It is locally known as Vilakkillatha Ambalam or a temple without lamp.
Sri Jalaram Dham
Jalaram Dham in Mattancherry was built by the devotees of Saint Jalaram. Saint Jalaram, popularly known as Jalaram Bapa was from Virpur near Rajkot. Born in the year 1799, the Gujaratis and the Marwadis of Mattancherry consider him as their godman.
Sri Nagakaleeswari Amman Temple
Traditionally, the Sri Nagakaleeshwari Amman Temple at Kappalandimukku in Mattancherry was owned by a Telugu-speaking community named '24 Manai Telugu Chetiyars' (24 MTC).
Sri Navneeta Krishna Mandir
This is one of the temples in Kochi that was established by the Gujaratis. The Gujarati Vaishnav Mahajan group manages this temple.
Sri Ram Mandir
Sri Ram Mandir, built in the year 1982, is on Gujarati Road in Mattancherry. The temple is owned by the Aggarwal community in Kochi and is famous for its Diwali and Holi celebrations. Aggarwals migrated to Kochi from different parts of North India, especially Gujarat, Rajasthan, and Haryana.
Sri Samudri Mandir
This is one of the temples at Kochi owned by Gujarati migrants. The Bhatias, a group among the Gujaratis, pray at this temple and the main deity is goddess Durga.
Sri Sivamariamman Kovil
Sri Sivamariamman Kovil is a temple on Dhobi Street, near Veli at Fort Kochi, owned by the Tamil-speaking migrant community of washermen, the Vannans. Legend has it that the Vannans were invited to Kochi by the King of Kochi, who granted the community land to settle down near Veli.
St. Francis Church Cemetery Chapel
This cemetery is owned by St. Francis Church at Fort Kochi, which was the first European Church in India, built by the Portuguese in 1503. This church is famous as the place where Vasco da Gama the Portuguese sailor was buried.
St. Francis Church
St. Francis Church, well-known for its beautiful architecture and ambience, is believed to be one of the oldest churches built by the Europeans in India.
St. Francis Church L P School
With a history of two centuries, St. Francis L.P. School located at Fort Kochi, is one of the oldest English-medium educational institutions in Kerala. This school was established in the year 1817.
St. John Deere Brito's A I I H School
St. John De Britto's Anglo Indian Higher Secondary School was formed on 15th August 1945 by the Benedictine Fathers. The school is named after John De Britto, a 17th-century Portuguese missionary. It started with seven teachers and ninety one students in 1945.
St. John Pattom Church
This chapel opposite the Holy Face Emmanuel Church at Fort Kochi, has Jesus Christ as the main deity. Believers claim that a miracle was witnessed at Holy Face Emmanuel Church a few years ago.
St. Peter And St. Paul's Orthodox Syrian Church
St. Peter's and St. Paul's Church was destroyed in 1868 during a festival at the church, when a massive fire broke out due to the fireworks. Established once more in 1867 by St. Thomas, the Syrian church is also referred to as the 'Church of Keralites'.
Tagore Library and Reading Room
Founded before Indian Independence, Tagore Library is one the active libraries of Kochi, with nearly 1100 members.
Tea Bungalow
This heritage building, constructed in 1921, was the office of a British Company trading in coir and spices. In 1950, the building was bought by the British Tea Company Brooke Bond, to be used as a staff guest house for its visiting managers.
Thakur House
Thakur House, situated by the Dutch Cemetery road is an 18th century Dutch building. It was built on the Gelderland Bastion of the erstwhile Dutch fort, which was built around the bastions of the demolished Portuguese fort.
Thakkyavu Mosque
Thakkyavu Mosque at Kochangadi in Mattancherry, is a building that was built in the 16th century and displays Arabic influence in its architecture."Thakkyavu in Arabic means a place where people gather in the presence of God", says Hashim Kochukoya Thangal, the head of the Thangal family
The Delta Study
The Delta Study located on Tower Road in Fort Kochi is a co-educational senior secondary institution affiliated to the Central Board of Secondary Education (CBSE). This institution's building was formerly a warehouse built in 1808 during British rule.
The Indo-Portuguese museum
The Indo-Portuguese museum, in the compound of Bishop House at Fort Kochi, is a heritage museum where one can be acquainted with the Portuguese connections in Kochi's past.
Thuruthy Island
Thuruthy, is geographically an island on the banks of the Calvetti Canal and is considered to be important to the history of Kochi. In Malayalam, the regional language of Kerala, the word
The Tower House Hotel
The Tower House Hotel is a centuries-old heritage building which was built during the colonial era. This European-style building was used by the British firm, Pierce Leslie Company during the British Period. It is a heritage hotel at present.
Udyaneshwara Siva Temple
Located near the Cochin Thirumala Devaswom (TD) temple at Cherlai in Mattancherry, Udyaneshwara temple is one of the few temples where Lord Siva is worshipped as the 'God of the Garden'.There is an interesting story behind this temple.
Vasco House
Vasco House is a privately owned homestay at Fort Kochi near St. Francis Church. It is believed that Vasco da Gama, the legendary Portuguese explorer stayed in this building.
Vasco da Gama Square
Vasco da Gama Square in Fort Kochi, near the sea shore, is an ideal place to spend the evenings. One can see stalls that sell nuts, tender coconut, spices, and other local delicacies.
Veli Ground
The Veli Ground at Fort Kochi is a football-field owned by the government. There is no one authentic account on the history of the Veli Ground. Veli Ground is associated with the Vannans, the Tamil-speaking washermen community, in local oral histories.
VOC Gate
Parade Ground, the four acre open land in Fort Kochi, is one of the landmarks of Kochi. This ground, at which military parades were carried out during the Portuguese, the Dutch and the British periods, is surrounded by many historical sites and monuments.
Women's Madrasa
Women's Madrasa is a remarkable contribution made by the Cutchi Memon migrant community in Kochi, who commonly speak Kutch within the community. Built in the year 1930, it was the first madrasa for women in Kerala.
Saraswath Association
The Saraswath communtiy is one of the Konkani-speaking migrant communities in Kochi, and the organization of their people is referred to as the 'Saraswath Association'. Originally, this community was called the Saraswath Abhramana.
Saraswath Association Library
Saraswath Association Library is an initiative by the association of the Saraswath people, a Konkani speaking migrant community. Saraswath Abrahmana is what the community was originally referred to as, and there are currently about 100 Saraswath families living in Mattancherry.
Santa Cruz School
Santa Cruz School is a heritage building. In the 16th and 17th centuries Fort Kochi was known as Santa Cruz. Built by the Portuguese, it was the first European city in India. The School received its name from the name of this city. | {
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{"url":"http:\/\/math.stackexchange.com\/questions\/288348\/applications-of-hypergeometric-continued-fractions","text":"# Applications of hypergeometric continued fractions\n\nUsing a technique due to Gauss a lot of special functions can be expressed as continued fractions.\n\nWhat applications of this are there within mathematics and number theory?\n\n-\nOne application mentioned in that page is: Analytic continuation of $_2 F_1$ type. \u2013\u00a0 user58512 Jan 27 '13 at 21:17","date":"2015-03-26 19:11:49","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\": 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\": 0.9344057440757751, \"perplexity\": 1065.1121153909214}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-14\/segments\/1427131292567.7\/warc\/CC-MAIN-20150323172132-00099-ip-10-168-14-71.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/educegh.com\/how-to-find-lowest-common-multiple-examples-steps\/","text":"# How to find lowest common multiple? Examples\n\nThe lowest common multiple (LCM) of given numbers is the lowest or least multiple that the given numbers have in common and it is exactly divisible by all the given numbers. The lowest common multiple is also referred to as the least common multiple (LCM).\n\n## How to find the lowest common multiple of numbers?\n\nIn finding the lowest common multiple (LCM) or least common multiple, you can use these two methods which will be discussed in this article.\n\n\u2022 Using multiples\n\n\u2022 Using product of primes\n\nSteps in finding the lowest common multiple of a number\n\nTo find the Lowest common multiple using the method of multiples;\n\n\u2022 list all the multiples of the given numbers\n\n\u2022 pick out all the common multiples among the multiples of the given numbers.\n\n\u2022 The lowest multiple among the common multiples is the lowest common multiple (LCM) of the numbers.\n\nBefore we look at some examples of this method, let\u2019s try to know how to find the multiples of numbers.\n\nMultiples of a number\n\nmultiple of a given number is found by multiplying the number with other whole numbers. For instance, 5 \u00d7 8 = 40. In this case, 40 is a multiple of 5 and also a multiple of 8.\n\nIn general, when a whole number is exactly divisible by another whole number, then the first number is a multiple of the second number. For example, 12 is divisible by 3. Hence 12 is a multiple of 3.\n\nExample 1: find the LCM of 3 and 4\n\nanswer: as we discussed in the steps above list all the multiples of 3 and 4\n\nREAD:\u00a0 How to find the area of a sector, formula | examples.\n\nMultiples of 3 = {3,6,9,12,15,18,21,24,27,30,33,36\u2026}\n\nMultiples of 4 = {4,8,12,16,20,24,28,32,36,40,44,48\u2026}\n\nPick out the common multiples of 3 and 4\n\ncommon multiples of 3 and 4 = {12,24,36}. In this case, 12 is the least or smallest multiple among the common multiples. Hence the LCM of 3 and 4 is 12.\n\nThat\u2019s pretty cool, right? Let\u2019s look at another example of this method.\n\nExample 2: find the LCM of 9, 15, and 18.\n\nAnswer: list all the multiples of 9, 15 and 18\n\nMultiples of 9 = {9,18,27,36,45,54,63,72,81,90,99\u2026}\n\nMultiples of 15 = {15,30,45,60,75,90,105\u2026}\n\nMultiples of 18 = {18,36,54,72,90,108\u2026}\n\nThe common multiples of 9, 15, and 18 = {90}. Therefore the LCM of 9, 15, and 18 is 90.\n\nThis is pretty good and simple. Now let\u2019s turn our attention to the second method (product of primes)\n\n## Steps in finding the Lowest common multiple using the product of primes.\n\nTo find the Lowest common multiple (LCM) of given numbers using the product of primes follow the steps\n\n\u2022 Write down each of the given numbers as a product of primes (index notation form)\n\n\u2022 pick out the highest power of each prime number that has occurred and multiply them all together.\n\nThis will be clearly explained in the examples here.\n\nExample 1. Find the LCM of 12 and 15\n\nAnswer: write the numbers given as a product of primes\n\n\\begin{aligned} &12 = 2\\times 2\\times3 =2^2\\times3\\\\\n&15 = 3\\times5 \\end{aligned}\\\\\n\\text{Therefore, the LCM of 12 and 15 } = 2^2\\times3\\times5=4\\times 3\\times5 = 60\n\nThis is so simple, right? Let\u2019s run over another example under this method.\n\nExample 2. Find the LCM of 12, 28, and 45\n\nAnswer: write down the given numbers as a product of primes\n\n\\begin{aligned}&12 = 2\\times2\\times3 = 2^2\\times3\\\\\n&28 = 2\\times2\\times7=2^2\\times7\\\\\n&45=3\\times3\\times5=3^2\\times5\\\\\\end{aligned}\\\\\n\\text{Therefore, the LCM of 12, 28 and 45} = 2^2\\times3^2\\times5\\times7= 4\\times9\\times5\\times7=1260\n\nSolving practical problems involving Lowest common multiple (LCM)\n\nREAD:\u00a0 How to multiply fractions with mixed numbers;\n\nThe lowest common multiple (LCM) can be used to solve practical problems. The problem will not ask you to \u201cfind the LCM of\u201d. There are clues to look out to. Let\u2019s discuss the clues\n\n\u2022 when the question says something like \u201crepeating pattern of the numbers\u201d. This clue means to find the Multiples.\n\n\u2022 Then again when the question says something \u201cto find the first time the match after the start\u201d this clue means find the LCM.\n\nLet\u2019s go ahead to look at some example\n\nExample 1. Esther belongs to three committees in a high school. One of the committees meets every 4 days, another committee meets every 5 days and the third Committee meets every 6 days. They all meet on the first day of the school term. How many days after this will they meet on the same day?\n\nAnswer: This question is talking about the LCM of 4,5 and 6. Using a product of primes, write down the given numbers as a product of primes\n\n\\begin{aligned}&4 =2\u00d72= 2^2\\\\\n&5= 5\\\\\n&6= 2\u00d73\\\\\\end{aligned}\\\\\n\\text{Therefore, the LCM of 4 5 and 6} = 2^2\u00d73\u00d75=4\u00d73\u00d75=60\\\\\n\\text{Hence,the three committees will together on the 60th day.}","date":"2023-01-29 16:03:31","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\": 0.7011089324951172, \"perplexity\": 481.16593840800897}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499744.74\/warc\/CC-MAIN-20230129144110-20230129174110-00328.warc.gz\"}"} | null | null |
HSN Community / Forums / Jewelry / Does jewelry free ship over $75 not apply when also using a 20% off?
Does jewelry free ship over $75 not apply when also using a 20% off?
Can someone help? I don't order that often so I'm not sure how this works.
I've been thinking about ordering two jewelry items since I found a 20% off a single item in my email inbox. I was hoping to also get free shipping since the jewelry items (after any discounts) total over $75. When I apply the code, the free shipping disappears from the cart.
There is something else I am curious about, and hoping someone with more shopping experience can shed some light. I noticed, with the FS over $75 jewelry deal, that some of the clearance jewelry does not get free shipping in my cart, when combined with other jewelry pieces to total over $75. Are the clearance items supposed to be excluded from the FS over $75 deal?
Yes, you have to choose either free S&H or 20% off. Can't get both deals. I had that happen when I used a coupon. For any coupon, if you go online there is always a "click her for more details", and in the fine print it will state that the use of the coupon basically will render any other special offer null and void. So I choose whichever one saves me the most. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,835 |
Product Tag - friendship
I'm a Mitzvah (2014)
I'm a Mitzvah (2014)
A short film about a young American man who spends one last night with his deceased friend while stranded in rural Mexico.
That Awkward Moment (2014)
Best pals Jason and Daniel indulge in casual flings and revel in their carefree, unattached lives. After learning that the marriage of their friend Mikey is over, they gladly welcome him back into their circle. The three young men make a pact to have fun and avoid commitment. However, when all three find themselves involved in serious relationships, they must keep their romances secret from one another.
Gunday (2014)
Based in Calcutta during its most unsettled times in the 1970s, the film deals with the inseparable life of Bikram and Bala. The story of 2 boys, boys who became refugees. Refugees who became gun couriers. Gun Couriers who became coal bandits, coal bandits who became Calcutta's Most Loved, Most Celebrated, Most Reckless, Most Fearless, Most Powerful! A story of two happy-go-lucky renegades who came to be known as…GUNDAY!
Nunca regreses (2014)
Two friends from the countryside decide to take a job, without measuring the consequences they will have to face. They can't go back to the way they were.
Hide Your Smiling Faces (2014)
Tommy and his older brother Eric live in the midst of vast remote forests. The death of their friend pushes them close to the edge. Eric doesn't know how to channel his energy. All at once, nature's vastness feels stifling.
A Brony Tale (2014)
Vancouver-based voice artist Ashleigh Ball has been the voice of numerous characters in classic cartoons such as Care Bears, Strawberry Shortcake, Cinderella and more. When Ashleigh was hired to voice Apple Jack and Rainbow Dash for Hasbro's fourth series to use the My Little Pony name – My Little Pony: Friendship Is Magic – she had no idea she would become an Internet phenomenon and major celebrity to a worldwide fan-base of grownups. Bronies are united by their belief in the show's philosophy. This documentary gives an inside view of the Pony fan-world, and an intimate look at the courage it takes to just be yourself…even when that means liking a little girls' cartoon.
The List Film (2014)
A film of the award winning live theatre production The List. A woman struggles to adjust to rural life with a young family in Quebec. Increasingly isolated, she keeps life in order through obsessive list making. As her marriage struggles she befriends Caroline. When Caroline requests a favour she adds it to her list. The difference between remembering to do it, and neglecting to take it seriously, becomes the difference between life and death. Produced in association with Screen Academy Scotland at Edinburgh Napier University.
Viejos amigos (2014)
Balo, Villarán and Domingo are lifelong friends in their 80s, attending the funeral of their fourth comrade, Quique. On a whim, and fueled by their animosity towards the departed's shrewish wife, they steal the urn containing the ashes and are thrust into a series of misadventures.
Fashion Statements (2014)
Wearing an old pair of sunglasses from the thrift shop, Sherry realises that she can read everybody's minds… up to a point. She can see why they chose the clothes they wear and what insecurities they are covering up. It seems that almost everyone has a deep-rooted hatred of how they look and the sunglasses may be part of the antidote. When she meets her friend Caitlyn, for coffee, she realises her glasses could help her and others.
Not Cool (2014)
NOT COOL follows former prom king and college freshman Scott (Shane Dawson) who has just returned home for Thanksgiving break only to be dumped by his eccentric, long-term girlfriend. With his world turned upside down, Scott strikes an unlikely friendship with former classmate Tori (Cherami Leigh), an ugly duckling who blossomed in her first semester of college. Together, the two embark on an outrageous adventure through their hometown. But when Scott and Tori find their friendship turning into something deeper, they realize that a few months away may have changed them more than they realized.
My Little Pony: Equestria Girls - Rainbow Rocks (2014)
My Little Pony: Equestria Girls – Rainbow Rocks (2014)
Music rules and rainbows rock as Twilight Sparkle and pals compete for the top spot in the Canterlot High "Mane Event" talent show. The girls must rock their way to the top, and outshine rival Adagio Dazzle and her band The Dazzlings, to restore harmony back to Canterlot High.
[EX-RENTAL] Home DVD 2014 (Original)
When Earth is taken over by the overly-confident Boov, an alien race in search of a new place to call home, all humans are promptly relocated, while all Boov get busy reorganizing the planet. But when one resourceful girl, Tip, manages to avoid capture, she finds herself the accidental accomplice of a banished Boov named Oh. The two fugitives realize there's a lot more at stake than intergalactic relations as they embark on the road trip of a lifetime. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,872 |
var nconf = require('nconf');
var defaults = {
V1_TENANTID: '[tenant-id]',
V1_APPPRINCIPALID: '[app-principal-id]',
V1_SYMMETRICKEY: '[symmetric-key]',
V1_UPN: '[user-upn]',
V2_WAAD_TENANTDOMAIN: '[tenant-domain]',
V2_WAAD_CLIENTID: '[client-id]',
V2_WAAD_CLIENTSECRET: '[client-secret]',
V2_UPN: '[user-upn]',
USER_OBJECT_ID: '[user-object-id]',
USER_DISPLAY_NAME : '[user-display-name]',
USER_GROUPS: '[user-groups]',
INVALID_EMAIL: '[invalid-email]'
};
nconf
.env()
.file('./test/env.json')
.required(Object.keys(defaults)); // Require this file exist in order to run tests - fail hard and fast
var config = nconf.get.bind(nconf);
module.exports = {
// for v1 access_tokens tests
v1: {
TENANTID: config('V1_TENANTID'),
APPPRINCIPALID: config('V1_APPPRINCIPALID'),
SYMMETRICKEY: config('V1_SYMMETRICKEY'),
UPN: config('V1_UPN')
},
// for waad v2 tests
v2: {
WAAD_TENANTDOMAIN: config('V2_WAAD_TENANTDOMAIN'),
WAAD_CLIENTID: config('V2_WAAD_CLIENTID'),
WAAD_CLIENTSECRET: config('V2_WAAD_CLIENTSECRET'),
UPN: config('V2_UPN')
},
user: {
objectId: config('USER_OBJECT_ID'),
displayName : config('USER_DISPLAY_NAME'),
groups: config('USER_GROUPS').split(','),
allGroups: config('USER_GROUPS').split(',')
},
invalid_email: config('INVALID_EMAIL')
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,089 |
\section{Introduction}
The problem of an adequate description and understanding of the behavior of spin
systems with strong disorder has been studied for around forty years by a large
number of scientists in statistical and condensed matter physics as well as,
increasingly, researchers in adjacent fields such as computer science and mathematics
\cite{kawashima:03a}. It is a hard problem in that many of the well-developed tools
of the theory of critical phenomena, such as the renormalization group, fail to
satisfactorily describe all important aspects of these models, and in that the
standard techniques of numerical simulations are faced with diminishing efficiency in
view of exploding relaxation times and the massive computational demand of the
average over quenched disorder. But it is also a good and fruitful problem in that
the questions it poses are deeply rooted in the foundations of statistical mechanics
\cite{binder:86a} and the simplicity of the models has led to applications ranging
from the physics of structural glasses to error correcting codes and neural networks
\cite{nishimori:book}.
While even fundamental questions such as the values of the lower and upper critical
dimensions of such models are still under active debate \cite{viet:09,fernandez:09a,
viet:10,sharma:11,beyer:12}, there is consensus that a spin-glass phase appears at
non-zero temperatures for short-ranged systems of Ising spins in at least three
dimensions, but no spin-glass order occurs beyond ground states in two-dimensional
(2D) systems \cite{hasenbusch:08,ohzeki:09,fernandez:16}. While such 2D geometries
might hence appear less useful for modeling experimentally realized spin-glass
phases, the physics of these systems is in fact rather interesting in its own
right. One intriguing aspect is that for sufficiently asymmetric coupling
distributions a long-range ferromagnetic phase can exist at non-zero temperatures,
and it is found that the phase boundary at low temperatures shows re-entrance or
inverse melting, that is, on further cooling a system in the ferromagnetic phase,
order is lost in favor of a paramagnetic state \cite{toldin:08,thomas:11a}. Another
facet is the question of universality regarding the distribution of exchange
couplings: at zero temperature, the bimodal model has extensive ground-state
degeneracies leading to behavior rather different from the case of continuous
coupling distributions \cite{hartmann:01a}. The resulting entropy of volatile spin
clusters was long believed to lead to power-law correlations at zero temperature, but
there is now evidence of true long-range spin-glass order \cite{jorg:06,roma:10}. The
behavior of this model at low temperatures is determined by a delicate interplay of
the distinct fixed points of the universality classes of discrete and continuous
coupling distributions, respectively
\cite{thomas:11,toldin:11a,joerg:12,jinuntuya:12}, and there is still no complete
consensus about universality at finite temperatures \cite{fernandez:16,lundow:16}. It
is the subtle role played by entropic fluctuations which makes this model relevant to
the finite-temperature transitions observed in three dimensions \cite{thomas:11}.
Apart from such theoretical considerations, interest in the 2D models has been fueled
by the relative ease in numerical tractability as compared to higher-dimensional
systems. This goes beyond the general advantage of systems in low dimensions of
providing larger linear system sizes at the same number of sites: 2D systems in zero
external field are an exception to the {\em NP\/} hardness of ground-state problems
found in systems of higher dimensions \cite{barahona:82}. Ground states on planar
graphs can be determined in polynomial time from the mapping to a minimum-weight
perfect matching problem \cite{bieche:80a}. This allows to treat significantly larger
lattice sizes than those accessible to simulation methods. The restriction to planar
graphs, and hence periodic boundary conditions in at most one direction, has been
rather inconvenient for certain types of studies \cite{hartmann:02} and, in general,
leads to relatively larger finite-size corrections. Polynomial-time algorithms also
exist for the more general problem of determining the partition function
\cite{blackman:91,saul:93,galluccio:00}. These methods, based on the evaluation of
Pfaffians, have the advantage of allowing for periodic boundary conditions, but they
are technically more demanding than the ground-state computations and thus restricted
to smaller system sizes. Only recent advances have allowed to extend these approaches
to system sizes $L\gtrsim 100$. \cite{thomas:09} In parallel, exact sampling
techniques for Ising spin glasses at non-zero temperatures based on the application
of ``coupling-from-the-past'' \cite{propp:96} or sampling of dimer coverings
\cite{wilson:97} have recently been suggested, that are either restricted to or only
efficient in 2D \cite{chanal:08,thomas:09}.
A wide range of aspects of 2D spin glasses has been found to be consistent with
droplet theory \cite{mcmillan:84,bray:87a,fisher:88}. Droplet and domain-wall
excitations can be directly inserted in zero-temperature configurations. Domain-wall
energies are found to scale as a power law
$E_\mathrm{def}\sim L^{\theta_\mathrm{DW}}$ for the Gaussian model with
$\theta_\mathrm{DW}\approx -0.28$. \cite{hartmann:01a} Roughly consistent values are
found for the scaling of droplet energies if scaling corrections are taken into
account \cite{hartmann:03a}. No power-law scaling of domain-wall energies is found
for bimodal couplings \cite{hartmann:01a}, but droplets in this model show
$\theta\approx-0.29$, possibly compatible with the Gaussian case
\cite{hartmann:08}. As the spin-glass phase is confined to zero temperature for 2D
models, ground-state calculations give direct access to the {\em critical\/} behavior
of the spin-glass transition. In this case, the correlation length exponent is
expected to follow from $\nu = -1/\theta$. \cite{bray:87a} As $\eta = 0$ at least for
the Gaussian model \cite{fernandez:16}, this is the only relevant critical exponent
(but see Ref.~\onlinecite{hartmann:08} for the bimodal case). Domain walls and
droplet interfaces are found to be fractal curves with dimension $d_\mathrm{f} < 2$,
i.e., not space filling \cite{bray:87}. At least for the Gaussian case, these fractal
curves appear to be compatible, under certain conditions, with a description in terms
of stochastic Loewner evolution \cite{amoruso:06a,bernard:07,khoshbakht:12}. Such
consistence together with further assumptions would suggest a relation between
stiffness exponent and fractal dimension, $d_\mathrm{f} = 1+3/[4(3+\theta)]$.
\cite{amoruso:06a} For the bimodal model, on the other hand, the fractal dimension is
possibly different \cite{melchert:07,weigel:06a,gusman:08}, but calculations are
complicated by sampling problems since the ground-state algorithms do not produce the
degenerate ground states with the correct weights. These subtle differences between
results for different coupling distributions and excitation types call for
high-precision studies to distinguish random from systematic coincidences. Some
previous results for $\theta$ and $d_\mathrm{f}$ in the Gaussian model are collected
in Table \ref{theta_articles}.
Here we combine a formulation of the ground-state problem on planar graphs in terms
of Kasteleyn cities \cite{gregor:07,thomas:07} with a recently suggested efficient
implementation of the Blossom algorithm for minimum-weight perfect matching
\cite{kolmogorov:09}. This allows us to determine ground states for systems of up to
$10\,000 \times 10\,000$ spins on commodity hardware. To extend these results to the
case of periodic boundaries with the smaller scaling corrections expected there, we
introduce a hierarchical optimization procedure using windows, alike to the patchwork
dynamics discussed in Ref.~\onlinecite{thomas:08}, which allows to determine ground
states of fully periodic samples with a constant relative increase in computational
effort as compared to the matching technique for planar samples. To treat the case of
bimodal couplings correctly, we use a new approach based on an exact decomposition of
the ground-state manifold into rigid clusters that are then sampled within a parallel
tempering framework that guarantees uniform sampling of ground states to high
precision.
The rest of this paper is organized as follows. In Sec.~\ref{sec:model} we outline
the matching algorithm based on Kasteleyn cities, introduce the windowing technique
that allows to generalize the method to systems with fully periodic boundaries, and
evaluate the performance of these algorithms. Section \ref{sec:gauss} is devoted to
the system with Gaussian coupling distribution, and we report our results for the
average ground-state and defect energies, the domain-wall fractal dimension as well
as the probability distributions of these quantities for different boundary
conditions. In Sec.~\ref{sec:bimodal} we analyze these quantities for the bimodal
model, introducing a new uniform-sampling technique for the degenerate ground states
in this case that allows us to provide an unbiased estimate of the domain-wall
fractal dimension. Finally, Sec.~\ref{sec:conclusions} discusses the compatibility of
our results with the conjecture $d_\mathrm{f} = 1+3/[4(3+\theta)]$ of
Ref.~\onlinecite{amoruso:06a} and contains our conclusions.
\begin{table}[tb!]
\caption{Previous estimates of the spin-stiffness exponent $\theta$ and the fractal
dimension $d_\mathrm{f}$ of the 2D Ising spin glass with Gaussian bound distribution.}
\begin{ruledtabular}
\begin{tabular}{lt{4}t{4}c}
Ref. &\multicolumn{1}{c}{$\theta$} & \multicolumn{1}{c}{$d_\mathrm{f}$} & \multicolumn{1}{c}{max. system size} \\ \hline
\onlinecite{mcmillan:84a} & -0.281(5) & \multicolumn{1}{c}{\textemdash} & $8\times8$ \\
\onlinecite{palassini:99} & -0.285(2) & \multicolumn{1}{c}{\textemdash} & $30\times30$ \\
\onlinecite{bray:84} & -0.294(9) & \multicolumn{1}{c}{\textemdash} & $12\times 12$ \\
\onlinecite{bray:87} & -0.29(1) & 1.26(3) & $120\times13$ \\
\onlinecite{rieger:96} & -0.281(2) & 1.34(10) & $30\times30$ \\
\onlinecite{hartmann:01a} & -0.282(2) & \multicolumn{1}{c}{\textemdash} & $480\times480$ \\
\onlinecite{middleton:01} & \multicolumn{1}{c}{\textemdash} & 1.25(1) & $256\times256$ \\
\onlinecite{weigel:06a} & -0.284(4) & 1.273(3) & $256\times256 $ \\
\onlinecite{bernard:07} & \multicolumn{1}{c}{\textemdash} & 1.28(1) & $720\times 360$ \\
\onlinecite{hartmann:02a} & -0.287(4) & \multicolumn{1}{c}{\textemdash} & $16\times1024$ \\
\onlinecite{carter:02a} & -0.282(3) & \multicolumn{1}{c}{\textemdash} & $12\times384$ \\
\onlinecite{hartmann:04a} & -0.281(7) & \multicolumn{1}{c}{\textemdash} & $64\times64$\\
\onlinecite{amoruso:06a} & -0.285(5) & 1.27(1) & $300\times300 $ \\
\onlinecite{melchert:07} & -0.287(4) & 1.274(2) & $320\times320$ \\
\hline
This work & -0.2793(3) & 1.27319(9) & $10\,000\times10\,000$ \\
\end{tabular}
\end{ruledtabular}
\label{theta_articles}
\end{table}
\section{Model and algorithms\label{sec:model}}
\begin{figure*}
\includegraphics[width=0.95\textwidth]{matching_v2_landscape}
\caption{
(Color online)
Mapping of the Ising spin-glass ground-state problem to a minimum-weight perfect
matching. An auxiliary graph is constructed by expanding each plaquette of the
dual lattice into a complete graph $K_4$ of four nodes (left). Additional rows and
columns of $K_4$ nodes are added instead of the outer plaquette to make the
auxiliary graph more regular. Edge weights on the auxiliary graph are $J_{ij}$
for each bond that crosses a bond $(i,j)$ of the original graph and zero
otherwise. Then, a minimum-weight perfect matching is determined on the auxiliary
graph (middle). By contracting the $K_4$ vertices again, the matching reduces to
a minimum cut on the spin lattice, i.e., a set of closed loops surrounding
islands of down spins in a sea of up spins or vice versa (right). Dashed bonds on
the spin lattice correspond to antiferromagnetic couplings $J_{ij} < 0$, solid
bonds to ferromagnetic ones, $J_{ij} > 0$.
\label{fig:kasteleyn}
}
\end{figure*}
\subsection{The model}
We consider the random-exchange, zero-field Ising model with Hamiltonian
\begin{equation}
{\cal H} = -\sum_{\langle i,j\rangle} J_{ij} s_i s_j.
\label{eq:hamiltonian}
\end{equation}
Here, $\langle i,j\rangle$ denotes summation over pairs of nearest neighbors. For the
purposes of this study, the underlying lattice is chosen to have square elementary
plaquettes, but the techniques described here are applicable {\em mutatis mutandis\/}
to any regular planar graph (see, for instance,
Refs.~\onlinecite{weigel:06a,melchert:11}). The case of non-planar graphs is discussed
in Sec.~\ref{sec:window} below.
The couplings $J_{ij}$ are quenched random variables. At zero temperature, two
distinct types of behavior are expected, one for discrete and commensurate allowed
coupling values and a second class for distributions with incommensurate or
continuous support \cite{amoruso:03a,jorg:06,thomas:11,toldin:11a,joerg:12}. We
consider one representative of each class, namely the symmetric bimodal ($\pm J$)
distribution,
\begin{equation}
P(J_{ij}) = \frac{1}{2}\delta(J_{ij}-J)+\frac{1}{2}\delta(J_{ij}+J),
\label{eq:bimodal}
\end{equation}
for the commensurate class and the symmetric Gaussian,
\begin{equation}
J_{ij} \sim {\cal N}(0,1),
\label{eq:gaussian}
\end{equation}
as example of the continuous class of distributions.
\subsection{Matching with Kasteleyn cities}
\label{sec:kasteleyn}
It was initially noted by Toulouse that the model \eqref{eq:hamiltonian} could be
dualized and the trivial up-down symmetry of the states removed by considering the
interactions around an elementary plaquette \cite{toulouse:77a}. Each plaquette with
an odd number of antiferromagnetic bonds is inherently {\em frustrated\/}, such that
in each spin configuration at least one of the elementary interactions around the
plaquette will be unsatisfied. The energy of the ground state of such a system will
hence be elevated above the ground-state energy of a ferromagnet by an amount
proportional to the total weight of such {\em broken\/} bonds. If edges of the {\em
dual\/} lattice are used to indicate the broken bonds, these link together to form
defect lines on the dual lattice, emanating and ending in frustrated plaquettes
\cite{bieche:80a}. The search for a ground state is thus (for a planar lattice)
equivalent to the determination of a {\em minimum-weight perfect matching\/} (MWPM)
on the complete graph of frustrated plaquettes, where the edge weights correspond to
the shortest paths (on the dual lattice) between each pair of frustrated
plaquettes. For details see, e.g., Refs.~\onlinecite{bieche:80a,weigel:06a}. As MWPM
is a polynomial problem which is solved efficiently using the so-called blossom
algorithm \cite{edmonds:65a}, it was first noted by Bieche {\em et al.\/}
\cite{bieche:80a} that this allows to calculate exact ground states for relatively
large systems.
\begin{figure}
\begin{center}
\includegraphics[width=2.8 in, height=3.0 in, angle=0]{windowing_technique_color}
\caption{%
(Color online) Schematic representation of the windowing technique to determine
ground states for toroidal systems. The dashed square shows
the window and the blue diamonds represent the sites whose spins will be updated next by
the windowing technique, as they are contained within the current
window. Red squares indicate sites whose spins are fixed in their current orientation with
strong bonds, indicated by the thick black lines. As a result, the MWPM problem
will be solved for the system of red and blue spins with using free
boundary conditions.
}
\label{fig:windowing_technique}
\end{center}
\end {figure}
In practice, however, the outlined mapping has certain disadvantages. The weighted
distance between each pair of frustrated plaquettes needs to be determined before the
matching can proceed. Since each plaquette could be matched up with any other, a
solution is sought for the {\em complete\/} graph of frustrated plaquettes. The
average number of such plaquettes is $F=\alpha N$, where $N$ is the number of spins
and $\alpha$ is a disorder-dependent constant that equals $\alpha = 1/2$ for the
symmetric distributions considered here. The number of edges, however, is $F(F-1)/2$,
increasing quadratically in the system volume. The original implementation of the
blossom algorithm has complexity $O(V^2E)$, where $V$ is the number of vertices in
the auxiliary graph and $E$ the number of edges \cite{edmonds:65a}. For the present
problem, this corresponds to $O(L^8)$ scaling. Memory requirements are
$O(L^4)$. While a number of algorithms with improved worst-case complexity have been
proposed, not all of them are fast and hence useful in practice. We use here the
currently fastest publicly available algorithm due to Kolmogorov
\cite{kolmogorov:09}. As it is unlikely that edges with a very large weight are part
of the minimum-weight matching, in practice only edges up to a certain weight are
retained \cite{bieche:80a}. One has to proceed carefully here, however, to ensure
high success rates also for larger system sizes. Strictly speaking, the resulting
algorithm is merely quasi-exact.
A polynomial-time solution to the Ising spin-glass ground state problem on planar
graphs based on a somewhat different mapping was proposed in
Refs.~\onlinecite{thomas:07,gregor:07}. This is a rather direct implementation of the
interpretation of the Ising ground-state search as a maximum/minimum-cut
problem. Splitting the Hamiltonian \eqref{eq:hamiltonian} into three terms as
follows,
\begin{equation}
-{\cal H} = W^++W^--W^\pm = K-2W^\pm,
\end{equation}
where $K = \sum_{\langle ij\rangle}J_{ij}$ and
\begin{equation}
\begin{split}
W^+ &= \sum_{\stackrel{\langle ij\rangle}{s_i=s_j=+1}}J_{ij},\\
W^- &= \sum_{\stackrel{\langle ij\rangle}{s_i=s_j=-1}}J_{ij},\\
W^\pm &= \sum_{\stackrel{\langle ij\rangle}{s_i\ne s_j}}J_{ij},
\end{split}
\end{equation}
it is clear that the energy is minimized for a configuration that minimizes $W^{\pm}$
which is the weight of the {\em cut\/} or, in more physical terms the interface,
separating up-spins from down-spins. Note that the interface can consist of more than
one connected component. As it turns out, such cuts can be related one-to-one to
perfect matchings in an auxiliary graph. To see this, consider the example shown in
Fig.~\ref{fig:kasteleyn}. The right panel shows a configuration of up and down spins
on a patch of the square lattice with free boundaries together with the corresponding
cut of anti-aligned neighboring spins. The cut forms a set of closed loops on the
dual lattice (red lines). To represent it as a matching, consider the auxiliary graph
shown on the left of Fig.~\ref{fig:kasteleyn} that replaces each plaquette of the
original lattice (i.e., each node of the dual lattice) by a complete graph of four
nodes, a ``Kasteleyn city''. To create a regular lattice graph, the single outer
plaquette of the dual graph is replaced by $4L$ individual plaquettes surrounding the
original lattice. The cut on the right can then be represented as a perfect matching
on the auxiliary graph as is shown in the middle panel of
Fig.~\ref{fig:kasteleyn}. Here, vertices that do not have cut lines adjacent to them
will have all four vertices of the associated Kasteleyn city matched by the internal
edges, such that after contracting back the Kasteleyn cities to regular vertices one
ends up with the graph shown on the right, that represents the cut in spin
language. To ensure that a MWPM corresponds to a minimum cut, we assign edge weights
in the auxiliary graph that are equal to the coupling $J_{ij}$ of the bond in the
original graph that is crossed by the bond in the auxiliary graph. For bonds in the
auxiliary graph that do not correspond to edges in the original graph, in particular
the internal bonds of Kasteleyn cities as well as bonds between the additional
external plaquettes, the weight is set to zero. Finally, a spin configuration
consistent with the loops on the dual graph found in this way is constructed by
flipping the spin orientation each time a loop line is crossed
\cite{thomas:07,gregor:07}.
\begin{figure*}
\includegraphics[width=0.95\textwidth]{WT_convergence_with_window_position}
\caption{%
Application of the windowing method to find a ground state of a sample with
toroidal boundaries. Spins on white lattice sites are consistent with the ground-state
orientation $s_i^0$, i.e., $s_i s_i^0 = +1$, black spins are oppositely oriented,
i.e., $s_i s_i^0 = -1$. In a random initial
configuration the spins have $s_i s_i^0 = \pm 1$ uniformly at random (top
left). Exact ground states are found in windows of size $(L-2)\times (L-2)$
placed at a random location (red dotted lines), with the remaining spins acting as fixed
boundaries. After a few iterations all spins have the ground-state orientation
(bottom right).
\label{fig:window-example}
}
\end{figure*}
As the auxiliary graph used here has only $4(L+1)^2$ vertices and
$6(L+1)^2+2L(L-1) = 8L^2+10L+6$ edges (for periodic-free boundaries) as compared to
the $O(L^2)$ vertices and $O(L^4)$ edges of Bieche's approach \cite{bieche:80a}, it
is significantly more efficient and, due to the smaller storage requirements, this
approach allows to treat much larger systems sizes. In practice, we use the Blossom V
implementation introduced in Ref.~\onlinecite{kolmogorov:09} to perform the MWPM
calculations. The method can be easily generalized to other planar graphs, for
instance $L\times L$ graphs with periodic boundaries in one direction. In this case,
the two additional lines of external plaquettes in either the horizontal or vertical
direction can be removed, otherwise the algorithm proceeds in the same way. A
generalization to non-planar graphs is not possible, however, as then the one-to-one
mapping between solutions of the MWPM problem and ground states of the spin system
breaks down \cite{thomas:07}: if the solution to the MWPM leads to loops that wrap
around the lattice it is possible to find an odd number of loop lines in a given row
or column of the lattice. In this case, it is not possible to find a spin
configuration that is consistent with the lines.
\subsection{Windowing technique for toroidal systems}
\label{sec:window}
Such a configuration with an odd number of line segments in a given row or column of
an $L\times L$ system with fully periodic boundaries can be repaired by changing the
boundary conditions in the corresponding direction from periodic to antiperiodic,
corresponding to an extra loop wrapping around the lattice, thus resulting in an even
number of lines again. In this sense, as discussed in Ref.~\onlinecite{thomas:07},
the approach outlined above finds an {\em extended ground state\/} for a system where
the boundary conditions are added to the dynamical degrees of freedom. While this can
be quite useful, it is not immediately applicable to the calculation of defect
energies and domain walls, where specific, fixed boundary conditions need to be
applied.
Nevertheless, a method for finding ground states for a fixed choice of periodic
boundary conditions can be constructed from the MWPM approach outlined above, as we
will now show. To achieve this, we successively determine exact ground states in
square windows of size $L'\times L'$, $L' \le L$, with free boundary conditions,
while the spin configuration outside of the window remains unchanged. By moving this
window randomly over the full $L\times L$ lattice, the exact ground state is
typically found after a moderate number of iterations. The sequence is started by
initializing the system in a random spin configuration $\{s_i\}$. The origin of the
window is then chosen randomly at one of the lattice sites, and the exact ground
state of the spins inside of the window is determined using MWPM, subject to the
additional constraint of a layer of fixed spins surrounding it. These spins are fixed
by placing very strong bonds with couplings $J_\mathrm{strong}$ between them that
cannot be broken in the solution of the MWPM, for instance by choosing
$|J_\mathrm{strong}| > \sum_{\langle ij\rangle}|J_{ij}|$. We choose
$J_{ij} = +|J_\mathrm{strong}|$ for parallel spins along the boundary of the window
and $J_{ij} = -|J_\mathrm{strong}|$ for antiparallel ones to ensure that these spins
do not change their relative orientation as a result of the MWPM run. This setup is
illustrated in Fig.~\ref{fig:windowing_technique}.
\begin{table}[tb!]
\caption{%
The average probability $\overline{P}_{n}$ of finding the ground
state (success probability) for $20\leq L \leq 1000$,
and for different numbers $n$ of iterations. Results are averaged over
100 disorder realizations.
\label{tab:parameters}
}
\begin{ruledtabular}
\begin{tabular}{lcccccc}
$L \backslash n $ & 5 & 10 & 15 & 20 & 25 & 30 \\
\hline
20 & 0.276 & 0.561 & 0.671 & 0.728 & 0.762 & 0.782 \\
50 & 0.317 & 0.603 & 0.705 & 0.756 & 0.790 & 0.805 \\
80 & 0.315 & 0.592 & 0.700 & 0.752 & 0.783 & 0.806 \\
100 & 0.315 & 0.594 & 0.700 & 0.745 & 0.779 & 0.789 \\
150 & 0.326 & 0.611 & 0.714 & 0.768 & 0.797 & 0.821 \\
200 & 0.323 & 0.610 & 0.712 & 0.765 & 0.792 & 0.814 \\
350 & 0.340 & 0.628 & 0.729 & 0.789 & 0.822 & 0.833 \\
500 & 0.317 & 0.589 & 0.683 & 0.740 & 0.771 & 0.801 \\
700 & 0.329 & 0.612 & 0.723 & 0.770 & 0.782 & 0.818 \\
1000 & 0.322 & 0.609 & 0.713 & 0.764 & 0.779 & 0.807 \\
\end{tabular}
\end{ruledtabular}
\end{table}
As the spins at window boundaries are fixed and the resulting constraint optimization
problem is solved exactly, each iteration of the windowing method decreases the
energy of the total system or leaves it invariant. We observe convergence of the
method after a moderate number $n$ of iterations. The process is illustrated in
Fig.~\ref{fig:window-example}, where we display the overlap $s_i s_i^0$ with the
exact ground state $s_i^0$ for an example disorder configuration of linear size
$L=200$ with Gaussian couplings starting from a random initial spin configuration. It
is seen how even the first optimization with a window of size $L'=L-2=198$ leaves
only a single (large) cluster excitation over the ground state. As is seen from the
following panels, such excitations can only be fully relaxed if the window does not
intersect them. Hence the time until convergence is a random variable. To determine a
good set of parameters we performed test runs for different sizes $L$ and $L'$ of the
system and the window, respectively, and with a varying number of iterations. The
results show that the necessary number of iterations depends both on $L'$ and the
initial spin configuration, such that larger $L'$ needs smaller $n$, and if the
initial spin configuration is changed, $n$ will also change. As is intuitively
plausible, we find best results for the largest windows, and so we fixed the window
size to its maximum $L'=L-2$ for all runs. To decide whether a given run arrives in
one of the ground states, we compared against exact results for system sizes
$L\le 100$ produced by the branch-and-cut method implemented in the spin-glass server
\cite{sgs}. For larger system sizes we used the lowest energy found in a sequence of
independent runs as an estimate of the ground-state energy and measured the success
probability $P_{n}(\{J_{ij}\})$ as the proportion of runs that ended in this
lowest-energy state found or in the exact ground-state for the system sizes treated
by the spin-glass server. The resulting success probability data, estimated from
between $250$ ($L\ge 700$) to $2000$ ($L\le 150$) runs for different initial spin
configurations for each disorder realization, is collected in Table
\ref{tab:parameters}. As is clearly seen, the success probabilities are rather high
such that for $n = 20$, for instance, they are consistently above $70\%$. There is
almost no size dependence of the average success probability $\overline{P}_{n}$, so
the hardness of finding ground states for the fully periodic torus lattices with the
proposed method does not increase with system size.
Still, from the data presented in Table \ref{tab:parameters}, it is clear that not
every run of the windowing method converges to the ground state. To further increase
the success probability of the method, we use repeated runs and pick the lowest
energy found there \cite{weigel:06b}. If the success probability for a given sample
in runs of ${n}$ iterations is $P_{n}(\{J_{ij}\})$, then the probability of finding
the ground state at least once in $m$ independent runs is
\begin{equation}
P_s(\{J_{ij}\})=1-[1-P_{n}(\{J_{ij}\})]^m,
\label{eq:success}
\end{equation}
and this can be tuned arbitrarily close to unity by increasing $m$. If we set a
desired success probability of, say, $P_s = 0.999$, we can use Eq.~\eqref{eq:success}
to determine the required number $m$ of repetitions. For each realization we hence
find
\[
m(\{J_{ij}\}) = \log[1-P_s]/\log[1-P_{n}(\{J_{ij}\})].
\]
In Table \ref{tab:parameters2} we show the values of $\overline{m}$ averaged over 100
disorder realizations as a function of $L$ and $n$. Clearly, the dependence on system
size is weak. The total computational effort of such repeated runs is proportional to
$m \times n$. From the values of $n$ tested in Table \ref{tab:parameters2}, this
effort is found to be minimal for $n = 10$, and we use $m=8$ repetitions independent
of system size to find the exact ground state in approximately $99.9\%$ of the
samples. As an additional protection against potential outliers we demand that the
lowest-energy state found in these $m=8$ runs must have occurred at least three out
of these $8$ times. If this is not the case, another $8$ runs are performed etc. This
adds only a tiny fraction of extra average runtime, but it will be able to catch a
few of the $0.01\%$ of samples where the ground state would otherwise not be
found. As a test, we applied this combined technique to the samples for $L\le 100$
where the exact ground-state energy is known and it arrived in a ground state in all
cases.
\begin{table}[tb!]
\caption{%
The average number $\overline{m}$ of repetitions required according to
Eq.~\eqref{eq:success} for runs of the windowing technique with $n$ random
placements of the window per run to ensure an overall success probably of $P_s =
0.999$.
\label{tab:parameters2}
%
}
\begin{ruledtabular}
\begin{tabular}{lcccccc}
$L \backslash n $ & 5 & 10 & 15 & 20 & 25 & 30 \\
\hline
20 & 23.5 & 9.3 & 6.9 & 5.8 & 5.3 & 4.9 \\
50 & 19.7 & 8.2 & 6.2 & 5.3 & 4.7 & 4.5 \\
80 & 20.2 & 8.6 & 6.4 & 5.4 & 4.9 & 4.4 \\
100 & 20.0 & 8.5 & 6.4 & 5.6 & 4.9 & 4.5 \\
150 & 19.2 & 8.1 & 6.0 & 5.0 & 4.5 & 4.0 \\
200 & 19.2 & 8.3 & 6.1 & 5.3 & 4.7 & 3.6 \\
350 & 18.4 & 7.4 & 5.8 & 4.8 & 3.9 & 3.8 \\
500 & 21.2 & 8.5 & 7.0 & 6.0 & 5.0 & 4.6 \\
700 & 20.3 & 7.9 & 6.3 & 5.8 & 4.6 & 5.0 \\
1000 & 20.0 & 8.5 & 6.4 & 5.1 & 5.0 & 4.5 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{Performance of the algorithm}
\label{sec:performance}
It is interesting to see how the matching based on Kasteleyn cities for planar
instances as well as the windowing method outlined above for toroidal graphs fare in
computational efficiency as compared to the more general approaches implemented in
the spin-glass server \cite{sgs}. The run times in seconds on standard hardware are
shown for periodic-free boundary conditions (PFBC) and for periodic-periodic
(toroidal) boundaries (PPBC) as compared to the corresponding results of the
spin-glass server for system sizes $L\le 100$ in Table \ref{windowing_time}. For PFBC
the matching approach is always much faster than the method used by the spin-glass
server, which is based on a modified exact numeration technique known as
branch-and-cut. For PPBC the windowing technique introduces a certain overhead,
such that a crossover is observed with branch-and-cut being faster for
$L \lesssim 20$ and the windowing method winning out for $L\gtrsim 20$.
\begin{table}[tb!]
\caption{%
Average run time (in seconds) for determining a ground state of samples with
periodic-free boundaries (PFBC) and periodic-periodic boundaries (PPBC),
respectively, using the minimum-weight perfect matching (MWPM) approach based on
Kasteleyn cities for PFBC and the windowing technique (WT) for PPBC as compared to
the times reported by the spin-glass server (SGS) on the same samples.
}
\begin{ruledtabular}
\begin{tabular}{lllll}
$L$ & \multicolumn{2}{c}{PFBC} & \multicolumn{2}{c}{PPBC} \\
\cline{2-3} \cline{4-5}
& SGS & MWPM & SGS & WT \\
\hline
8 & 0.00228 & 0.000203 & 0.00560 & 0.02468 \\
10 & 0.01330 & 0.000424 & 0.01950 & 0.04462 \\
20 & 0.18330 & 0.002361 & 0.22820 & 0.19119 \\
50 & 3.38740 & 0.024184 & 3.93040 & 2.18788 \\
80 & 31.0738 & 0.069104 & 35.7004 & 6.42005 \\
100 & 150.218 & 0.115761 & 189.501 & 9.81247 \\
\end{tabular}
\end{ruledtabular}
\label{windowing_time}
\end{table}
The scaling of run times with system size is illustrated in
Fig.~\ref{fig:run_time}. The algorithm of the spin-glass server utilized here is
based on branch-and-cut \cite{liers:04}, which corresponds to a combination of a
cutting plane technique with the iterative removal of branches of the search tree
that cannot contain a solution. While this approach is quite efficient, and
outperforms other exact methods for hard problems, its run-time still scales
exponentially with system size. The super-polynomial behavior is clearly seen in the
doubly logarithmic representation of Fig.~\ref{fig:run_time}. For the matching
approach for PFBC, the implementation used here has $O(L^6)$ worst-case scaling
\cite{kolmogorov:09}. As the straight line indicates, we indeed see clear power-law
behavior, but the average run times probed here increase much more gently with system
size. A power-law fit of the form
\begin{equation}
\label{eq:power-law}
t(L) = A_t L^{\kappa}
\end{equation}
to the data yields $\kappa = 2.22(2)$, so the scaling is only slightly worse than
linear in the volume in the considered range of system sizes.
Finally, for the windowing technique built on top of MWPM for the PPBC samples, we
find an overhead that is to a very good approximation independent of system size,
such that calculations for PPBC are by a factor of $80$ more expensive that those for
samples with PFBC for the chosen confidence level of $P_s = 0.999$, corresponding to
the $n = 10$ iterations and $m=8$ repetitions. A fit of the form
\eqref{eq:power-law} to the data for PPBC yields $\kappa = 2.20(2)$, perfectly
consistent with the results for PFBC. The ratio of amplitudes $A_t$ is estimated as
$A_t = 83\pm 12$, consistent with the expected value of slightly above $80$ resulting
from the additional requirement of a threefold occurrence of the ground state.
\section{Results for Gaussian couplings\label{sec:gauss}}
For the Gaussian distribution \eqref{eq:gaussian} the set of couplings for which
exact degeneracies occur is expected to be of zero measure. The present techniques
based on matching hence directly yield the correct distribution of states at zero
temperature.
\subsection{Ground-state energies}
\begin{figure}
\begin{center}
\includegraphics[width=0.95\columnwidth]{run_time}
\caption{(Color online) Average time $t$ per sample to determine ground states of
systems with PFBC and PPBC for $L\times L$ samples using the minimum-weight
perfect matching (MWPM) method for periodic-free samples (PFBC), the windowing
technique (WT) for periodic-periodic samples (PPBC), and the spin-glass server
(SGS), respectively. The straight lines are fits of the form
\eqref{eq:power-law} to the data, whereas the lines for the SGS data are just
interpolations to guide the eye.}
\label{fig:run_time}
\end{center}
\end {figure}
The average ground-state energy per spin, $\langle e(L)\rangle_J$, depends on the
coupling distribution. Additionally, we expect finite-size corrections which in turn
are sensitive to the boundary conditions employed
\cite{bouchaud:03,hartmann:04a,weigel:06c}. Following Ref.~\onlinecite{hartmann:04a}
one expects a Wegner correction exponent $\omega(d) = (6-d)+\cdots$ to leading order,
whereas numerically one finds \cite{hasenbusch:08} $\omega \approx 1.0$ for Ising
spin glasses in $d=3$ and $\omega\approx 0.75$ for $d=2$. \cite{fernandez:16} As then
$-(d-\theta)+\omega \approx -3.03$ in two dimensions, this implies that non-analytic
corrections are substantially suppressed against the leading analytic ones in this
quantity. We hence assume the following general form for the size dependence of the
average ground-state energy,
\begin{equation}
\begin{split}
\langle e(L)\rangle_J = & e_\infty + A_E L^{-(d-\theta)} + C_E L^{-1} + \\
& D_E L^{-2} + E_E L^{-3} + \ldots.
\end{split}
\label{eq:energy_fit_form}
\end{equation}
The presence of a term proportional to $L^{-(d-\theta)}$ follows from standard
arguments about the scaling of the correlation length and the free-energy density
\cite{privman:privman}, taking additionally into account that for a $T=0$ critical
point the $1/\beta^2$ prefactor in the relation $e = (-1/\beta^2)\mathrm{d}\hspace*{-0.1ex}(\beta f)/\mathrm{d}\hspace*{-0.1ex} T$ is
critical, as well as making use of the relation $\nu = -1/\theta$.
\cite{hartmann:04a} Although this derivation should apply for any $T=0$ critical
point, for the spin glass it is tempting to attribute the occurrence of the
$L^{d-\theta}$ term to the presence of domain-wall defects that are trapped in the
system due to periodic boundary conditions. In Ref.~\onlinecite{hartmann:04a} it is
suggested to reduce the number of parameters in Eq.~\eqref{eq:energy_fit_form} by
considering the energy $\hat{e}(L)$ per bond instead of the energy $e(L)$ per
site. If one assumes that depending on the boundary conditions this quantity has a
$1/L$ correction for any free edge and a $1/L^2$ correction for any corner, for the
square lattice with its two bonds per site we expect
\[
2\langle \hat{e}(L)\rangle_J= e_\infty + \hat{A}_E L^{-(d-\theta)} + \hat{C}_EL^{-1} +
\hat{D}_EL^{-2}
\]
up to higher-order corrections. For free-free boundaries, one has $E(L) = L^2 e(L) =
(2L^2-2L)\hat{e}(L)$ and hence
\begin{equation}
\label{eq:energy_ffbc}
\begin{split}
\langle e(L)\rangle_J = & e_\infty + \hat{A}_E L^{-(d-\theta)} + (\hat{C}_E-e_\infty) L^{-1}\\
& +(\hat{D}_E-\hat{C}_E) L^{-2}-\hat{D}_E L^{-3},
\end{split}
\end{equation}
where a term of order $L^{-(d-\theta)-1}$ which for $\theta < 0$ is asymptotically
smaller than $1/L^3$ has been neglected. This is of the form of
Eq.~\eqref{eq:energy_fit_form}, but with the $1/L^3$ term merely being produced by
the $1/L^2$ correction in $\hat{e}(L)$, such that there are only five fit parameters
in \eqref{eq:energy_ffbc} as compared to six parameters in
Eq.~\eqref{eq:energy_fit_form}. For periodic-free boundaries there is a free edge
but no corners, such that $\hat{D}_E=0$ and $E(L) = (2L^2-L)\hat{e}(L)=L^2e(L)$, and
we find
\begin{equation}
\label{eq:energy_pfbc}
\begin{split}
\langle e(L)\rangle_J = & e_\infty + \hat{A}_E L^{-(d-\theta)} + (\hat{C}_E-e_\infty/2) L^{-1}\\
& -(\hat{C}_E/2) L^{-2},
\end{split}
\end{equation}
where again a term proportional to $L^{-(d-\theta)-1}$ was omitted. For
periodic-periodic boundaries, on the other hand, one should have
$\hat{C}_E = 0 = \hat{D}_E$, and hence only a correction proportional to
$L^{-(d-\theta)}$. We will test the validity of these assumptions for our data below.
Beyond the mean ground-state energy, it is interesting to study the shape of the
energy distribution over different disorder samples. It has been shown in
Ref.~\onlinecite{bouchaud:03}, based on results of Wehr and Aizenman \cite{wehr:90},
that the width of this distribution scales as $L^{\Theta_f}$ with $\Theta_f = -d/2$.
Below, we investigate the distribution shape by direct inspection and by analyzing
the scaling of its kurtosis defined by
\begin{equation}
\label{eq:kurtosis}
\operatorname{Kurt}[e] = \frac{\langle (e-\langle e\rangle_J)^4\rangle_J}
{[\langle (e-\langle e\rangle_J)^2\rangle_J]^2}
\end{equation}
with system size, where $\operatorname{Kurt}[\cdot] = 3$ for a Gaussian distribution.
\subsection{Domain-wall calculations}
\begin{table}[tb!]
\caption{The number of disorder realizations for different boundary conditions,
coupling distributions and system sizes.}
\begin{ruledtabular}
\begin{tabular}{l c c c}
$L$ & PFBC Gaussian & PPBC Gaussian & PFBC bimodal \\
\hline
8 & $1\times10^6$ & $1\times10^5$ & $1\times10^5$ \\
10 & $1\times10^6$ & $1\times10^5$ & $1\times10^5$ \\
20 & $1\times10^6$ & $1\times10^5$ & $1\times10^5$ \\
30 & $1\times10^6$ & $1\times10^5$ & $1\times10^5$ \\
40 & $1\times10^6$ & $1\times10^5$ & $1\times10^5$ \\
50 & $1\times10^6$ & $1\times10^5$ & $1\times10^5$ \\
80 & $1\times10^6$ & $8\times10^4$ & $1\times10^5$ \\
100 & $1\times10^6$ & $8\times10^4$ & $1\times10^5$ \\
150 & $1\times10^6$ & $1\times10^5$ & $1\times10^5$ \\
200 & $1\times10^6$ & $5\times10^4$ & $8\times10^4$ \\
350 & $5\times10^5$ & $5\times10^4$ & $8\times10^4$ \\
500 & $5\times10^5$ & $3\times10^4$ & $5\times10^4$ \\
700 & $5\times10^5$ & $1\times10^4$ & $3\times10^4$ \\
1000 & $3\times10^5$ & $1\times10^4$ & $1\times10^4$ \\
1500 & $1\times10^5$ & $7\times10^3$ & $5\times10^3$ \\
2000 & $5\times10^4$ & $1\times10^3$ & $3\times10^3$ \\
3000 & $3\times10^4$ & $640 $ & $1505 $ \\
4000 & $2\times10^4$ & & \\
5000 & $3\times10^3$ & & \\
7000 & $400 $ & & \\
8000 & $455 $ & & \\
10000 & $265 $ & & \\
\end{tabular}
\end{ruledtabular}
\label{tab:samples}
\end{table}
The analysis of defect energies provides a convenient way of studying the stability
of the ordered phase. In the most common approach one inserts system-spanning domain
walls into the system by a suitable change of boundary conditions
\cite{banavar:82a}. The energy of such excitations scales as a power of their linear
size \cite{bray:84},
\begin{equation}
E_{\mathrm{def}} \propto L^{\theta}, \label{eq:defE}
\end{equation}
where the spin-stiffness exponent $\theta$ depends on the symmetries of the model as
well as the lattice dimension $d$. In a simple generalization of Peierls' argument
for the stability of the ferromagnetic phase, one concludes that a spin-glass phase
is stable against thermal fluctuations up to some $T_c>0$ if $\theta > 0$ and
unstable for $\theta < 0$, with $\theta = 0$ denoting the marginal case. The
conceptually most direct way of inserting a domain-wall excitation is to compute a
ground-state for free boundaries in, say, the $x$ direction as a reference and to
then fix the boundary spins along the $x$ boundary in opposite relative orientations
as compared to this state for a second ground-state calculation. The excess energy in
the second run corresponds to the energy contained in the domain wall. This setup is
sometimes referred to as domain-wall boundary condition
\cite{hartmann:01a,carter:02a}. An alternative proposed initially by Banavar
\cite{banavar:82a} uses the difference between the ground-state energies for periodic
and for antiperiodic boundaries in $x$ direction. The resulting value of
$\Delta E = E_\mathrm{P}-E_\mathrm{AP}$ is potentially the difference of energies of
two configurations with such domain walls as the periodicity of both P and AP
boundaries can force a domain wall into the system \cite{kosterlitz:99a,weigel:05f},
but this difference is found to nevertheless scale with the same stiffness exponent
as for domain-wall boundaries \cite{carter:02a}.
\begin{figure}
\begin{center}
\includegraphics[width=0.95\columnwidth]{eGS_vs_L_PFBCs_Gaussian}
\includegraphics[width=0.95\columnwidth]{deltaE_vs_L_PFBCs_Gaussian}
\includegraphics[width=0.95\columnwidth]{domainwall_length_vs_L_PFBCs_Gaussian}
\caption{%
(Color online) (a) Disorder-averaged ground-state energy per site
$\langle e\rangle_J = \langle\bar{E}/L^2\rangle_J$ for PFBC and Gaussian
couplings together with a fit of the form \eqref{eq:energy_pfbc} to the data in
the range $L=10,\ldots,10\,000$. (b) Average defect energies
$\langle|\Delta E|\rangle_J$ for the same system as calculated from the
difference in ground-state energies between periodic and antiperiodic boundary
conditions in the $x$ direction. The points show our data for
$8 \leq L \leq 10\,000$ and the solid line represents a fit of the form
$\langle|\Delta E|\rangle_J(L) = A_\theta L^\theta + C_\theta/L^2$ to the data.
The inset shows the correction
$\langle|\Delta E|\rangle_J(L) - A_\theta L^\theta$ plotted against $1/L^2$
illustrating that this single term describes the corrections very well. (c)
Average length $\ell$ of the domain-wall in the overlap of ground states for
periodic and antiperiodic boundaries in $x$ direction and free boundaries in
$y$ direction (PFBC boundaries). The line shows a fit of the functional form
$\langle \ell\rangle_J = A_\ell L^{d_\mathrm{f}}$ to the data for
$L\ge L_\mathrm{min} = 40$. The inset shows a blow-up of the deviations for
small $L$.}
\label{fig:pfbc_G}
\end{center}
\end {figure}
For calculations based on MWPM alone one needs to apply free boundaries in $y$
direction in order to ensure planarity of the lattice. With the help of the windowing
technique it is also possible to implement this procedure for samples with
periodic-periodic boundaries, however. In general we expect the leading scaling to be
accompanied by scaling corrections of the form \cite{weigel:06c}
\begin{equation}
\label{eq:defect-energy-corrected}
\langle|\Delta E(L)|\rangle_J(L) = A_\theta L^{\theta} (1+B_\theta L^{-\omega})
+\frac{C_\theta}{L}+\frac{D_\theta}{L^2}+\cdots,
\end{equation}
where $\omega$ denotes the leading corrections-to-scaling exponent, and $1/L$ and
$1/L^2$ are analytic corrections \cite{privman:privman}. For the setup with
domain-wall boundary conditions significantly stronger corrections have been observed
than for the P-AP situation \cite{carter:02a} and we hence concentrate on the latter
approach here.
Apart from the energy density of domain walls or droplet boundaries another
contentious question is that of the geometric nature of excitations in spin
glasses. While it is not ultimately clear whether droplets or domain walls are the
fundamental objects in this system or rather some more esoteric form of excitations
such as sponges exist \cite{palassini:00,krzakala:00,newman:01}, it is interesting to
see whether domain-walls are stochastically fractal objects and if the corresponding
fractal dimension $d_\mathrm{f} < d$ or rather domain walls can be space-filling
\cite{marinari:00}. We determined the domain wall as the set ${\cal D}$ of all dual
bonds for which
\begin{equation}
[J_{ij} s_i s_j]^{(\mathrm{P})} [J_{ij} s_i s_j]^{(\mathrm{AP})} < 0.
\label{eq:dw-def}
\end{equation}
The inclusion of the couplings $J_{ij}$ in the product takes care of the fact that
across the edge where the boundary condition is changed from P to AP the spins will
be in different relative orientation before and after the change, but this is merely
a consequence of the flip $J_{ij} \to -J_{ij}$ of the couplings there and should not
be counted as a part of the induced domain wall. We denote by $\ell$ the number of
(dual) edges in the set ${\cal D}$. Following the usual box-counting argument,
scaling according to $\langle \ell\rangle_J \sim L^{d_\mathrm{f}}$ defines the
domain-wall fractal dimension $d_\mathrm{f}$. As for the defect energies we
anticipate the presence of corrections, leading to the scaling form
\begin{equation}
\langle\ell\rangle_J(L) = A_\ell L^{d_\mathrm{f}} (1+B_\ell L^{-\omega})
+\frac{C_\ell}{L}+\frac{D_\ell}{L^2}+\cdots.
\end{equation}
\subsection{Periodic-free boundaries}
For the periodic-free setup (PFBC) we used the MWPM approach for periodic and
antiperiodic boundaries in $x$ direction and system sizes ranging from $L=8$ up to
$L=10\,000$. For $L\le 350$ we generated $10^6$ disorder configurations, while for
larger systems the number of replicas is gradually reduced down to about $300$ for
$L=10\,000$, see the details collected in Table \ref{tab:samples}. We used the MIXMAX
random number generator \cite{savvidy:91,savvidy:15} which has provably good
statistical properties and also passes all of the tests in the suite TestU01
\cite{lecuyer:07}. As an additional check in view of the high-precision nature of the
present study, part of our calculations were repeated with Mersenne twister
\cite{matsumoto:98}. All results were found to be perfectly consistent within error
bars.
\begin{figure}
\begin{center}
\includegraphics[width=0.95\columnwidth]{sampleDW}
\caption{%
(Color online) Overlap configuration of the ground states for P and AP
boundaries for a $L=10\,000$ disorder realization of the PFBC Gaussian
system. The red line demarcates the domain wall which traverses
$\ell = 233\,141$ dual links. }
\label{fig:sampleDW}
\end{center}
\end {figure}
We start by considering the ground-state energies. Here, we use the results for both
P and AP boundary conditions. They differ from each other, on average, by far less
than the statistical errors would suggest, but this is due to the fact that for each
sample both energies are highly correlated. For studying the average ground-state
energy, we hence calculated the average $\bar{E} =(E_\mathrm{P}+E_\mathrm{AP})/2$ and
estimated statistical errors for $\langle \bar{E}\rangle_J$ through the variation
over disorder samples. As the data in panel (a) of Fig.~\ref{fig:pfbc_G} show, the
finite-size corrections to scaling are relatively small, with the result for $L=10$
only being about 4\% above the asymptotic value. Due to the large range of system
sizes and high statistics in disorder samples we get a stable result for the full
non-linear five parameter fit of the form \eqref{eq:energy_pfbc} to the data with a
quality-of-fit\footnote{$Q$ is the probability that a $\chi^2$ as poor as the one
observed could have occurred by chance, i.e., through random fluctuations, although
the model is correct \cite{young:12}.} of $Q=0.81$. For the asymptotic ground-state
energy we find
\[
e_\infty = -1.314\,787\,6(7),
\]
while the spin-stiffness exponent $\theta = -0.273(65)$ from this fit\footnote{Note
that hence the form \eqref{eq:energy_pfbc} is found to describe the data perfectly
well, in contrast to the corresponding form used in Ref.~\onlinecite{hartmann:04a},
cf.\ Eq.~(22) there, which is not consistent with the equation derived here.}. If
we fix $\theta$ at the value $\theta = -0.2793$ found below from the defect energy
calculations for the PFBC boundaries, the asymptotic ground-state estimate $e_\infty$
is unaltered from the above value up to the given number of digits. On gradually
increasing $L_\mathrm{min}$ we find statistically consistent fits that, however,
become less and less stable as the number of degrees of freedom is reduced. The
resulting estimate of $e_\infty$ is unaltered within statistical errors.
Our data for the defect energies are shown in Fig.~\ref{fig:pfbc_G}(b). We find
scaling corrections to be small and a pure power-law fit without corrections yields a
quality-of-fit $Q = 0.37$ for $L \ge L_\mathrm{min} = 50$. The corresponding
estimate of the stiffness exponent is $\theta = -0.2798(4)$. Corrections can hence
only be clearly resolved for $L\lesssim 50$. There, we find that the data are very
well described by a single correction term proportional to $1/L^2$, cf.\ the inset of
Fig.~\ref{fig:pfbc_G}(b), where we show the residual contribution
$\langle |\Delta E|\rangle_J - A_\theta L^\theta$ plotted against $1/L^2$. Our
$\theta$ estimate from this fit is
\[
\theta = -0.2793(3)
\]
with $Q=0.16$ when including all lattice sizes. Gradually increasing $L_\mathrm{min}$
does not reveal any discernible drift in the estimate for $\theta$. Since we have one
free boundary one might have expected the presence of a $1/L$ correction, which is
clearly present in the ground-state energy itself according to the fit following
Eq.~\eqref{eq:energy_pfbc}. In the energy difference $\Delta E$, however, this
contribution cancels out since the couplings along the free edge are absent in both
samples. If we nevertheless include such a term in the fit, its amplitude is found to
be consistent with zero. We are not able to clearly resolve a Wegner correction
$\propto L^{-\omega}$, which is not surprising since as discussed above we expect it
to be clearly weaker than $1/L^2$.
We finally turn to the domain-wall length. Fig.~\ref{fig:sampleDW} shows a sample
configuration with $L=10\,000$ illustrating the meandering nature of the domain
wall. For the average domain-wall length we find very clean scaling for PFBC as is
seen from our data depicted in Fig.~\ref{fig:pfbc_G}(c). A fit of the pure power-law
form $\langle \ell\rangle_J = A_\ell L^{d_\mathrm{f}}$ yields a fit quality of
$Q=0.56$ for $L_\mathrm{min} = 40$. The corresponding estimate of the fractal
dimension is
\[
d_\mathrm{f} = 1.273\,19(9).
\]
The deviations from a pure power law visible for system sizes $L < 20$ are rather
small and not well described by a single correction term. We hence prefer to take
them into account by simply omitting data from the small-$L$ side instead of
performing corrected fits. On systematically varying $L_\mathrm{min}$ in these fits,
we find a drift only for $L_\mathrm{min} \le 30$ and mutually consistent results for
larger $L_\mathrm{min}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.95\columnwidth]{eGS_vs_L_PPBCs_Gaussian}
\includegraphics[width=0.95\columnwidth]{deltaE_vs_L_PPBCs_Gaussian}
\includegraphics[width=0.95\columnwidth]{domainwall_length_vs_L_PPBCs_Gaussian}
\caption{%
(Color online) (a) Average ground-state energies for PPBC and Gaussian
couplings together with a fit of the form \eqref{eq:energy_ppbc} to the data in the
range $L \ge L_\mathrm{min} = 16$. (b) Scaling of defect energies for the Gaussian
model with fully periodic boundary conditions. The solid line shows a fit of the form
$\langle|\Delta E|\rangle_J(L) = A_\theta L^\theta + C_\theta/L^2$ to the data. The
inset shows the correction $\langle|\Delta E|\rangle_J(L) - A_\theta L^\theta$
plotted against $1/L^2$ illustrating that this single term describes the corrections
very well. (c) Scaling of the length of the domain wall between P and AP ground
states for the Gaussian PPBC case. The solid lines shows a fit of the form $\langle
\ell\rangle_J = A_\ell L^{d_\mathrm{f}}$ to the data with $L_\mathrm{min} = 40$. The
inset shows a detail of the main plot for small $L$.
}
\label{fig:ppbc_G}
\end{center}
\end {figure}
\subsection{Periodic-periodic boundaries}
For fully periodic or toroidal boundaries (PPBC) we use the windowing technique
discussed above in Sec.~\ref{sec:window} to find exact ground states in more than
99.9\% of the cases. Due to the increase in effort by the constant factor of $80$
resulting from the windowing technique, we reduced the maximum system size a bit and
considered lattices in the range $8\le L\le 3000$. Additionally, the number of
disorder realizations considered was reduced correspondingly, the exact numbers are
shown in Table~\ref{tab:samples}.
Our data for the ground-state energies for PPBC are shown in
Fig.~\ref{fig:ppbc_G}(a), illustrating that finite-size corrections in this case are
tiny, even much weaker than for the PFBC case. According to the discussion above, for
the ground-state energies we do not expect the presence of analytic corrections for
PPBC, and so we assume a scaling form
\begin{equation}
\label{eq:energy_ppbc}
\langle e\rangle_J = e_\infty + A_EL^{-(2-\theta)}.
\end{equation}
Fits of this form work very well and yield fit qualities of $Q>0.4$ for all
$L_\mathrm{min} \ge 10$. For $L_\mathrm{min} = 16$ we find
\[
e_\infty = -1.314\,788(3)
\]
as well as $\theta = -0.35(14)$ and $A=1.51(65)$ with a good $Q=0.60$. This fit is
shown together with the data in panel (a) of Fig.~\ref{fig:ppbc_G}.
For the defect energies, the data again show clear power-law scaling with $L$, see
Fig.~\ref{fig:ppbc_G}(b). For $L \ge L_\mathrm{min} = 50$ we get an excellent fit
($Q=0.74$) for the pure power-law $\langle|\Delta E|\rangle_J = A_L L^\theta$ with
$\theta = -0.2778(14)$. Regarding scaling corrections, it turns out that the size
range where they are visible is rather small. As the inset of
Fig.~\ref{fig:ppbc_G}(b) shows, corrections are well described by a single
$1/L^2$ term, consistent with the findings for the PFBC case. A corresponding fit for
$L_\mathrm{min} = 10$ yields high quality with $Q=0.92$ and
\[
\theta = -0.2778(11).
\]
A systematic trend on successively increasing $L_\mathrm{min}$ is not visible.
Regarding the domain-wall length, we again find only tiny scaling corrections, which
cannot be resolved for any $L > 20$. To avoid any risk from spurious remnant
corrections, we take $L_\mathrm{min} = 40$ for the uncorrected fit
$\langle \ell\rangle_J = A_\ell L^{d_\mathrm{f}}$ and arrive at
\[
d_\mathrm{f} = 1.2732(5).
\]
which yields $Q=0.73$. This fit is shown together with the data in
Fig.~\ref{fig:ppbc_G}(c). Comparing the results for $\theta$ and $d_\mathrm{f}$
between the PFBC and PPBC cases we see that they are in perfect agreement with each
other, indicating that the results truly probe the asymptotic regime and acting as an
{\em ex post\/} verification of the correctness of the windowing technique for the
PPBC case.
\begin{figure}
\begin{center}
\includegraphics[width=0.95\columnwidth]{kurtosis_eGS_PFBC_Gaussian}
\includegraphics[width=0.95\columnwidth]{distribution_deltaE_PFBC_Gaussian}
\includegraphics[width=0.95\columnwidth]{distribution_domainwall_PFBC_Gaussian}
\caption{%
(Color online) (a) Scaling of the kurtosis $\operatorname{Kurt}[e]$ of the
distribution of ground-state energies per spin for the Gaussian model with PFBC
as a function of system size. For $L\ge 20$ it is consistent with the value
$\operatorname{Kurt}[e] = 3$ of a normal distribution. (b) Distribution of
defect energies $|\Delta E|$ for the same model, rescaled with the expected
asymptotic behavior $\propto L^{\theta}$ with $\theta = 0.2793$. The solid line
shows a Gaussian distribution of the same mean and variance. (c) Distribution
of domain-wall lengths $\ell$ for the PFBC Gaussian case, rescaled according to
the limiting form $\propto L^{d_\mathrm{f}}$ with $d_\mathrm{f} =
1.273\,19$.
The solid line represents a lognormal distribution fitted to the empirical
data.
}
\label{fig:distributions}
\end{center}
\end {figure}
\subsection{Probability distributions}
When investigating the ground-state and defect energies as well as the domain-wall
lengths, besides looking at the average values reported above it is also instructive
to study the full distributions of these quantities over disorder samples. The width
$\langle (e-\langle e\rangle_J)^2\rangle_J$ of the distribution of ground-state
energies per spin shows power-law scaling according to $L^{\Theta_f}$, where we find
$\Theta_f = -0.9995(3)$ for PFBC and $\Theta_f = -1.002(1)$ for PPBC, consistent with
the theoretical expectation \cite{bouchaud:03} $\Theta_f = -d/2$. The latter follows
from a standard argument of decomposition of the system into effectively uncorrelated
subsystems, such that the total energy is a sum of independent contributions. As a
result, in the thermodynamic limit the distribution narrows to a delta peak,
consistent with the fact that the ground-state energy is self-averaging
\cite{dotsenko:17}. To investigate the shape of the distribution, we studied its
kurtosis defined in Eq.~\eqref{eq:kurtosis}. $\operatorname{Kurt}[e]$ is shown in
Fig.~\ref{fig:distributions}(a) for the PFBC case, where it is found to be consistent
with $3$ to within statistical errors for all lattice sizes $L\ge 20$, indicating
that the distribution of ground-state energies is in fact Gaussian
\cite{aspelmeier:03a}. This is in contrast to systems with long-range interactions
such as the Sherrington-Kirkpatrick model, where non-Gaussian distributions are found
\cite{bouchaud:03}.
For symmetric coupling distributions the histogram of defect energies for P and AP
boundaries is also symmetric and so has zero mean. It is expected that the standard
deviation $\sigma(E)$ has the same asymptotic scaling behavior as the modulus
$|\Delta E|$, and this is consistent with our observations. Considering the data for
$\sigma(\Delta E)$ for PFBC, we use a pure power-law fit with
$L\ge L_\mathrm{min} = 30$ to find $\theta = -0.2793(3)$ ($Q=0.55$). For PPBC, on the
other hand, the same analysis yields $\theta = -0.279(2)$ and $Q=0.81$ for the same
range. In Fig.~\ref{fig:distributions}(b) we show the defect energy distribution for
PFBC systems for a number of different lattice sizes, rescaled by the factor
$L^\theta$ with $\theta = -0.2793$ describing the decay in width. As the Gaussian
distribution with the same mean and width shows, the defect energy distribution is
clearly not normal, but instead has much heavier tails\footnote{We note that there
might be a relation between the behavior of the defect-energy distribution at
vanishing energies and the question of a multiplicity of states in spin glasses
\cite{vaezi:17}.}. This is confirmed by an inspection of the distribution kurtosis,
$\operatorname{Kurt}[\Delta E]$, which is found to be consistent with
$\operatorname{Kurt}[\Delta E] = 4.70(2)$ for systems of size
$L\ge L_\mathrm{min} = 20$.
The standard deviation of the distribution of domain-wall lengths is found to have
the same scaling as the mean, i.e., it is asymptotically proportional to $L^{d_\mathrm{f}}$,
suggesting a complementary way of determining the fractal dimension. This approach
yields estimates of $d_\mathrm{f} = 1.2740(3)$ for PFBC ($L_\mathrm{min} = 40$, $Q=0.34$) and
$d_\mathrm{f} = 1.276(2)$ for PPBC ($L_\mathrm{min} = 50$, $Q=0.61$), respectively. The result
for PFBC is slightly high as compared to the result from the mean, but still
statistically consistent: the deviation is $2.6$ times the combined error bar, but
this does not take into account that the two error estimates are correlated and so
the combined fluctuation is likely higher than the naive estimate
\cite{weigel:09,weigel:10}. The two PPBC estimates are fully consistent. The
distribution of domain wall lengths is found to be clearly non-Gaussian, with a
kurtosis that is consistent with $\operatorname{Kurt}[\ell] = 3.656(4)$ for systems
of size $L\ge L_\mathrm{min} = 20$. It was suggested in Ref.~\onlinecite{melchert:07}
that the distribution might be in fact lognormal. Our data for the distribution of
$\ell$ for PFBC are shown in Fig.~\ref{fig:distributions}(c), together with a fit to
a lognormal distribution. As is apparent, it describes the data reasonably well close
to the mode, but there are significant deviations in the tails.
\section{Results for bimodal couplings\label{sec:bimodal}}
For bimodal couplings there is a huge ground-state degeneracy. As has been
demonstrated with numerical calculations \cite{blackman:91,saul:93} and also shown
rigorously \cite{avron:81}, this model even has a finite ground-state entropy,
indicating that the number of ground states grows exponentially with system size. It
turns out to be a challenge to fulfill the equilibrium requirement of ensuring that
all such states are sampled with equal probability.
\subsection{Uniform sampling of ground states}
\label{sec:gaussian-noise}
For the case of systems with ground-state degeneracies, the solution to the matching
problem described in Sec.~\ref{sec:window} is not unique. There are several, possibly
many solutions to the matching problem that have the same minimal weight. In
practice, the implementation of the matching algorithm used will return an arbitrary
solution out of this set, where the state chosen depends on the specific
implementation of the algorithm used (for instance on the order in which nodes and
edges are examined) and the state returned might or might not be reproducible between
runs\footnote{The energy of the state returned, on the other hand, is of course
always the same.}. Clearly, this setup is not suitable for sampling such states
with a prescribed probability weight.
One way of solving this problem and ensuring uniform sampling of states might be to
break the degeneracy in a way such that each ground state is preferred the same
number of times by a chosen procedure. If one examines a pair of ground states, one
will find that they differ by the overturning of a set of disjoint, but singly
connected clusters of spins. As, by definition, this procedure does not change the
overall energy, this corresponds to a set of ``free'' spins \cite{khoshbakht:17}. The
degeneracy can be lifted by adding some small perturbation to the bonds, i.e.,
\begin{equation}
\label{eq:gaussian_noise}
J_{ij}(\kappa) = J_{ij} + \kappa \epsilon_{ij},
\end{equation}
with a continuous, symmetric distribution of the random variables $\epsilon_{ij}$, a
natural choice being the standard normal distribution,
$\epsilon_{ij} \sim {\cal N}(0,1)$. As the spectrum of states for the bimodal model
is gapped \cite{saul:93}, if $\kappa$ is chosen sufficiently small the ground state
of the system with couplings $J_{ij}(\kappa)$ will also be a ground state of the
system with $\kappa = 0$. Considering a cluster of free spins for a symmetric
distribution of $\epsilon_{ij}$, the sum of the noise terms $\epsilon_{ij}$ along the
bonds on the cluster boundary will have either sign with the same probability of
$1/2$. Hence one half of the realizations of $\epsilon_{ij}$ should lead to this
cluster being in one orientation and the other half to it being in the reversed
orientation, implying uniform sampling of degenerate ground states. A similar
approach was used in Refs.~\onlinecite{cieplak:90,gusman:08}. As we discuss elsewhere
\cite{khoshbakht:17}, however, clusters that touch each other are not independent and
hence the procedure leads to a strongly non-uniform distribution of sampled states.
Uniform sampling is achieved via a new technique based on a combination of
combinatorial optimization in the form of the MWPM algorithm and Markov chain Monte
Carlo \cite{khoshbakht:17}. We use MWPM to exactly determine the set of {\em rigid\/}
clusters in the ground-state manifold, i.e., the set of connected regions such that
the spins inside of them have the same relative orientation in all ground states. In
a second step, we then perform a parallel tempering simulation \cite{hukushima:96a}
with updates that are a combination of flipping individual rigid clusters and a
non-local cluster-update move \cite{houdayer:01}. Details of the procedure as well as
benchmarks will be presented elsewhere \cite{khoshbakht:17}.
\subsection{Ground-state and defect energies}
For the ground-state energy the presence of degeneracies and sampling bias is not
relevant. We hence used the regular MWPM procedure to determine ground-state energies
for pairs of samples with periodic and antiperiodic boundaries and the resulting
defect energies. For these quantities we restricted our calculations to the case of
PFBC as this allows for treating larger system sizes, but studies of PPBC would also
be possible using the windowing technique. The range of system sizes and number of
realizations for each size are summarized in the fourth column of
Table~\ref{tab:samples}. The average ground-state energy per spin is shown in
Fig.~\ref{fig:pfbc_B}(a). Inspecting the general scaling ansatz
\eqref{eq:energy_pfbc} and taking into account that we expect $\theta = 0$ for this
model (see below), we should only have analytical corrections proportional to $1/L$
and $1/L^2$ up to $O(L^{-3})$, and indeed we find a good fit ($Q=0.18$) of this
functional form for the range $L \ge L_\mathrm{min} = 20$, yielding
\[
e_\infty = -1.401\,922(3).
\]
This fit is shown together with the data in Fig.~\ref{fig:pfbc_B}(a). No drift of
$e_\infty$ is visible on further increasing $L_\mathrm{min}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.95\columnwidth]{eGS_vs_L_PFBCs_bimodal}
\includegraphics[width=0.95\columnwidth]{deltaE_vs_L_PFBCs_bimodal}
\includegraphics[width=0.95\columnwidth]{distribution_deltaE_PFBC_bimodal}
\caption{(Color online) (a) Average ground-state energies for bimodal couplings
and PFBC boundaries, together with a fit of the functional form
\eqref{eq:energy_pfbc} with $\theta = 0$ to the data for the range
$L=20,\ldots,3000$. (b) Defect energies for systems with bimodal couplings and
PFBC boundaries. Clearly, $\langle|\Delta E|\rangle_J$ converges to a non-zero
value as $L\to\infty$, indicating that $\theta = 0$. The line shows a fit of
the form $\langle|\Delta E|\rangle_J =\Delta{E_\infty} + B_\theta L^{-\omega}$
to the data with $L\ge L_\mathrm{min} = 10$ yielding
$\Delta{E_\infty} = 0.960(5)$. (c) Probability distribution over disorder of
the defect energies for the PFBC $\pm J$ model and different system sizes. For
$L\to\infty$ the distribution approaches a limiting shape close to the $L=1000$
case shown here.}
\label{fig:pfbc_B}
\end{center}
\end {figure}
The defect energies resulting from this procedure are shown in
Fig.~\ref{fig:pfbc_B}(b), indicating that for this model $\langle|\Delta E|\rangle_J$
converges to a finite value instead of decaying away to zero. This is consistent with
previous findings \cite{hartmann:01a,amoruso:03a}. If we assume a power-law decay as
prescribed by Eq.~\eqref{eq:defect-energy-corrected} and ignore the correction terms,
i.e., we use a pure power-law form $\langle|\Delta E|\rangle_J = A_\theta L^\theta$,
a good fit is achieved for $L\ge L_\mathrm{min} = 150$, resulting in
$\theta = -0.012(4)$, marginally compatible with $\theta = 0$. Additionally, the
modulus of $\theta$ systematically drops as $L_\mathrm{min}$ is increased. The defect
energy in this case hence does not decay to zero, but attains a non-zero value in the
thermodynamic limit. We therefore make the scaling ansatz
\begin{equation}
\langle|\Delta E|\rangle_J = \Delta{E_\infty} + B_\theta L^{-\omega}.
\label{eq:defect-energy-bimodal}
\end{equation}
We find an excellent fit with $Q=0.99$ already for $L_\mathrm{min} = 10$, resulting
in
\[
\Delta{E_\infty} = 0.960(5)
\]
and $\omega = 0.67(4)$. An alternative fit form including analytic corrections
proportional to $1/L$ and $1/L^2$ but omitting the $ L^{-\omega}$ term is found to be
of significantly lower quality.
Studying the distributions of both ground-state and defect energies, we again find a
Gaussian shape for the ground-state energies, the kurtosis being compatible with that
of a normal distribution for all system sizes studied. The standard deviation of the
defect energy shows analogous behavior to $\langle|\Delta E|\rangle_J$, settling down
at a finite value as $L\to\infty$. A fit of the form \eqref{eq:defect-energy-bimodal}
yields an asymptotic $\sigma_\infty(\Delta E) = 1.1564(4)$ ($L_\mathrm{min} = 16$,
$Q=0.41$). The disorder distribution of defect energies is shown in
Fig.~\ref{fig:pfbc_B}(c), illustrating that it approaches a limiting shape as
$L\to\infty$ in which about 57\% of domain walls have zero energy, 38\% have
$\Delta E = 2$, 4\% have $\Delta E = 4$, and higher defect energies occur in less
than 1\% of the cases.
\subsection{Domain walls}
\begin{figure}
\begin{center}
\includegraphics[width=0.95\columnwidth]{domain_wall}
\caption{%
Left: Schematic representation of the set of dual bonds satisfying the
condition \eqref{eq:dw-def} for the case of bimodal couplings. Besides the
domain wall it contains isolated loops enclosing free clusters of spins as well
as bubbles of free spins attached to the domain wall. Removing the isolated
free loops one arrives at the set ${\cal D}_\mathrm{long}$, which we denote as
the ``long'' domain wall. Right: After the additional removal of bubbles one
arrives at the set ${\cal D}_\mathrm{short}$ of dual bonds comprising the
``short'' domain wall of the configuration.%
}
\label{fig:domain_wall}
\end{center}
\end {figure}
The presence of free clusters of spins in the manifold of degenerate ground states
complicates the identification of domain walls for the bimodal model
\cite{gusman:08}. A possible difference in configuration between the ground state for
a disorder configuration with P boundaries and a ground state for the same
realization with AP boundary conditions is schematically depicted in the left panel
of Fig.~\ref{fig:domain_wall}. We see that in this case the set of domain-wall bonds
satisfying condition \eqref{eq:dw-def}, i.e., different relative orientations of
spins at both ends for the P and AP configurations, does not only contain the actual
domain wall but also a set of closed loops detached from the wall. These correspond
to free clusters that can be overturned at zero energy cost and so happen to be in
one orientation in the P ground state, but in the opposite orientation in the AP
configuration. Conceptually, these bonds do not belong to the domain wall. We remove
them by only counting the system spanning part of the set ${\cal D}$. We refer to the
corresponding set, denoted as ${\cal D}_\mathrm{long}$, as the ``long'' domain wall
and its length as $\ell_\mathrm{long} = |{\cal D}_\mathrm{long}|$. Additionally,
however, it is possible for such free clusters to be attached to the domain wall as
is also depicted in the example of Fig.~\ref{fig:domain_wall}. Such ``bubbles''
attached to corners of the wall are somewhat arbitrary additions and removing them by
only considering the shortest path in the set ${\cal D}$ connecting opposite ends of
the system defines the reduced set ${\cal D}_\mathrm{short}$ with
$\ell_\mathrm{short} = |{\cal D}_\mathrm{short}|$. Clearly we have that
${\cal D}_\mathrm{short} \subseteq {\cal D}_\mathrm{long } \subseteq {\cal D}$. Note
that even after these removals the set ${\cal D}_\mathrm{short}$ is not unique for a
given bond configuration, and the additional degeneracy is connected to zero-energy
loops that share (at least) one bond with the domain wall (instead of only sharing a
corner) and hence can be interpreted as diversions of the wall. In order to probe the
equilibrium properties, we must sample from such walls with equal probability.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=0.95\columnwidth]{domainwall_length_vs_L_PFBCs_bimodal_short}
\includegraphics[width=0.95\columnwidth]{domainwall_length_vs_L_PFBCs_bimodal_long}
\includegraphics[width=0.95\columnwidth]{domainwall_length_vs_L_PFBCs_bimodal_compare}
\caption{(Color online)%
(a) Average length $\langle\ell_\mathrm{short}\rangle_J$ of the short domain
wall for the bimodal model as a function of linear system size $L$ for the
three different algorithms employed. The inset shows the deviation of each data
set from the fit of the power law
$\langle\ell\rangle_J = A_\ell L^{d_\mathrm{f}}$ to the uniform sampling data
for $L\ge L_\mathrm{min} = 16$, which results in $d_\mathrm{f} = 1.279(2)$
($Q=0.33$). (b) Average length $\langle\ell_\mathrm{long}\rangle_J$ of the long
domain walls for the different algorithms. The inset shows the deviation of
each data set from the fit of a pure power law to the uniform data, yielding
$d_\mathrm{f} = 1.281(3)$ for $L_\mathrm{min} = 16$ ($Q=0.97$). (c) Ratio of
the average lengths of long and short domain walls as estimated from the
different algorithms. In all cases, the ratio approaches a constant, in line
with the identical estimates of fractal dimension for $\ell_\mathrm{short}$ and
$\ell_\mathrm{long}$.
}
\label{fig:uniform-sampling}
\end{center}
\end {figure}
Regarding the sampling of domain-wall lengths for the bimodal model we have produced
data from three different algorithms:
\begin{enumerate}
\item Our implementation of the MWPM algorithm calculates a ground-state for each
sample with both P and AP boundary conditions, and comparing these we can determine
the lengths $\ell_\mathrm{short}$ and $\ell_\mathrm{long}$ of the related domain
walls. It is clear that this does not correspond to a fair sampling of ground
states, but the nature of the bias depends on internal details of the MWPM
implementation \cite{kolmogorov:09} and is not clear on a physical level. This
technique allows to treat large system sizes and we applied it to the data set of
sizes $8\le L\le 3000$ described in the third column of Table \ref{tab:samples}. In
the following, we denote this as the ``matching'' algorithm.
\item The Gaussian noise technique described in Sec.~\ref{sec:gaussian-noise} is
designed to break the degeneracy in a systematic way. For each realization it only
requires an additional run of the MWPM algorithm per boundary condition, and we
hence applied it to the same set of samples with $8\le L\le 3000$. As discussed in
Sec.~\ref{sec:gaussian-noise} it also does not provide uniform samples,
however. This technique is referred to as ``Gaussian noise'' in the following.
\item The new algorithm based on a cluster decomposition and parallel tempering
outlined in Ref.~\onlinecite{khoshbakht:17} provides uniform samples, but it is
much more demanding computationally, such that only smaller system sizes can be
treated reliably. We have studied systems of edge lengths $L=10$, $16$, $20$, $24$,
$28$, $32$, $48$, $64$, $80$, $100$, and $128$ for this method, using $1000$
samples per size and producing ten independent ground-state configurations per
sample.
Data from this algorithm are labeled ``uniform sampling''.
\end{enumerate}
Figure \ref{fig:uniform-sampling}(a) shows the three data sets for the scaling of the
lengths of short domain walls. On the scale of the domain-wall lengths
themselves, all data appear to fall on top of each other, but a closer inspection
reveals that this is in fact not the case. The data from uniform sampling show very
clean scaling behavior and a pure power law $\langle\ell\rangle_J = A_\ell
L^{d_\mathrm{f}}$ describes the data for $L\ge L_\mathrm{min} = 16$ well. No drift of
the exponent value is observed on omitting further values on the small-$L$ side. The
fractal dimension is estimated from this fit as
\[
d_\mathrm{f} = 1.279(2)
\]
with $Q=0.33$. As the inset of Fig.~\ref{fig:uniform-sampling}(a) shows, there are
statistically significant deviations of the data from the other two sampling
techniques from this result. The samples generated by the Gaussian noise technique
show clean scaling as well, but with a significantly larger exponent
$d_\mathrm{f} = 1.323(3)$ ($L_\mathrm{min} = 16$, $Q=0.86$). The data from the
matching approach alone, on the other hand, show somewhat inconsistent behavior for
successive system sizes, and they are compatible with a pure power law only for
$L \ge L_\mathrm{min} = 80$, yielding $d_\mathrm{f} = 1.2802(5)$ ($Q=0.18$). This
slightly unsteady statistical behavior is probably connected to the fact that the
matching technique does not use a stochastic sampling technique, and due to internal
design decisions the behavior of the algorithm might change discontinuously at
certain system sizes. Somewhat surprisingly, however, the results for the pure
matching technique are closer to the correct result represented by uniform sampling
than the samples produced by Gaussian noise, see also the inset of
Fig.~\ref{fig:uniform-sampling}(a).
We move on to considering the results for the long domain walls. The data are
summarized in Fig.~\ref{fig:uniform-sampling}(b). While for each data set, the values
of $\langle \ell_\mathrm{long}\rangle_J$ are somewhat larger than those of
$\langle \ell_\mathrm{short}\rangle_J$ the relative behavior of the three data sets
for the long domain walls is very similar to that found for the short walls. From the
uniform sampling data, a pure power-law fit for $L_\mathrm{min}=16$ yields
$d_\mathrm{f} = 1.281(3)$ ($Q=0.97$) which is statistically consistent with the
result from the short domain walls. For comparison, matching and Gaussian noise yield
$d_\mathrm{f} = 1.2797(5)$ and $d_\mathrm{f} = 1.325(3)$, respectively, for the same
ranges that were used for the short walls. It hence appears that for the scaling of
domain-wall length with system size, there is no difference between the short and
long definitions of domain walls. This impression is corroborated by the data shown
in Fig.~\ref{fig:uniform-sampling}(c) of the ratios of long and short lengths of
domain walls, averaged over disorder, for the three different techniques. It is clear
that this ratio settles down to a finite value as $L\to\infty$, and a fit of the
function form
$\langle\ell_\mathrm{long}/\ell_\mathrm{short}\rangle_J = \kappa + A_\kappa
L^{-\omega}$
to the uniform sampling data yields $\kappa = 1.021(6)$ and $\omega = 0.85(16)$ with
$Q=0.18$ ($L_\mathrm{min}=10$).
\begin{figure}
\begin{center}
\includegraphics[width=0.95\columnwidth]{distribution_domainwall_PFBC_bimodal}
\caption{(Color online)%
Distribution of the lengths $\ell_\mathrm{long}$ of the long domain walls for
$\pm J$ couplings and PFBC boundaries as resulting from the uniform sampling
approach. The re-scaling of the axes is with respect to the fractal dimension
$d_\mathrm{f} = 1.281(3)$ estimated from the data in
Fig.~\ref{fig:uniform-sampling}.
}
\label{fig:bimodal_DW_distribution}
\end{center}
\end {figure}
It is worthwhile to compare these estimates of the fractal dimension to those found
previously: Melchert and Hartmann \cite{melchert:07} used combinatorial optimization
methods to find minimal and maximal domain walls in the manifold of degenerate
ground-state pairs, yielding lower and upper bounds for $d_\mathrm{f}$, namely
$1.095(2) \le d_\mathrm{f} \le 1.395(3)$. Our estimates are clearly compatible with
these, and it is interesting to note that the actual value is much closer to the
upper than to the lower limit which corresponds to almost flat walls. Risau-Gusman
and Rom\'a \cite{gusman:08} estimate $d_\mathrm{f} = 1.323(3)$ using non-uniform
sampling resulting from employing the bare MWPM algorithm; this is compatible with
our ``matching'' results, but too large compared to the unbiased estimate from
uniform sampling. Studying domain walls in a hexagonal lattice, Weigel and Johnston
\cite{weigel:06a} find $d_\mathrm{f} = 1.283(11)$, but again not using unbiased
sampling. Analyzing the behavior of the ground-state entropy, Fisch \cite{fisch:08}
estimates $d_\mathrm{f} = 1.22(1)$ which is strongly incompatible with our results,
which could be a sign of the relation $d_\mathrm{f} = 2\theta_S$ on which Fisch's
estimate is based, where $\theta_S$ is the scaling exponent of the ground-state
entropy, not being valid in two dimensions.
We finally tend to the distribution of domain-wall lengths for this case. As is
illustrated in Fig.~\ref{fig:bimodal_DW_distribution} for the long domain walls,
these follow the scaling form
$P(\ell) = L^{d_\mathrm{f}} \hat{P}(\ell L^{-d_\mathrm{f}})$ already observed for the
case with Gaussian couplings, cf.\ Fig.~\ref{fig:distributions}(c), where now
$d_\mathrm{f} = 1.281(3)$. The fit to a log-normal distribution also shown in
Fig.~\ref{fig:bimodal_DW_distribution} works quite well over the full range of the
distribution, in contrast to the case of Gaussian couplings, where deviations could
be seen in the right tail, cf.~Fig.~\ref{fig:distributions}(c). Very similar results
are obtained for the distribution of short domain walls also (not shown).
\section{Conclusions\label{sec:conclusions}}
We used an exact algorithm based on minimum-weight perfect matching to calculate
ground states for the square-lattice Ising spin glass with Gaussian and bimodal
couplings and lattice sizes of up to $10\,000\times 10\,000$ ($10^8$) spins,
employing periodic boundary conditions in one direction and free boundaries in the
other. For systems with full periodic boundaries, we developed a quasi-exact
algorithm that can find true ground-states with arbitrarily high probability and a
computational effort that is a constant time larger than for the planar graphs, and
we used it to study systems of up to $3\,000\times 3\,000$ spins. Our estimates of
the ground-state energies $e_\infty = -1.314\,787\,6(7)$ (Gaussian model) and
$e_\infty=-1.401\,922(3)$ (bimodal model) are compatible with, but up to 100 times
more precise than the estimates in the careful study of
Ref.~\onlinecite{hartmann:04a} using exact ground-state methods and the recent work
Ref.~\onlinecite{perez-morelo:13} using Monte Carlo. For Gaussian couplings, we also
determined the spin-stiffness exponent and the fractal dimension of domain walls with
unprecedented precision, yielding $\theta = -0.2793(3)$ and
$d_\mathrm{f} = 1.273\,19(9)$. These estimates are one to two orders of magnitude
more precise than previous results, see the data collected in Table
\ref{theta_articles}. We note that this value is also consistent with the most recent
estimate of $1/\nu = -\theta = 0.283(6)$ in Ref.~\onlinecite{fernandez:16}, but the
zero-temperature result has 10-fold increased precision. For bimodal couplings, we
find $\theta = 0$, in agreement with previous studies. Due to the large degeneracy of
the ground state for bimodal couplings, methods based on matching do not allow to
sample states with the proper statistical weight, and as a result unbiased estimates
of the domain-wall fractal dimension have not been possible previously. Using a newly
developed algorithm \cite{khoshbakht:17} allowed us to sample exact ground states for
this case uniformly, here up to system size $L=128$. The resulting estimates of the
fractal dimension, $d_\mathrm{f} = 1.279(2)$ and $d_\mathrm{f} = 1.281(3)$ for
``short'' and ``long'' domain walls, respectively, are marginally consistent with
$d_\mathrm{f}$ for the Gaussian couplings, the deviation being 3 and 4 standard
deviations, respectively.
In 2006, Amoruso {\em et al.\/} \cite{amoruso:06a} used results from stochastic
Loewner evolution (SLE) to conjecture that the 2D spin glass with Gaussian couplings
is described by a non-unitary conformal field theory with central charge $c < -1$,
related to the SLE parameter $\kappa$ as \cite{henkel:12}
$c = (6-\kappa)(3\kappa-8)/\kappa$, and they numerically determined a value
$\kappa \approx 2.1$. Further, it was assumed that the scaling dimension
$x_t = d-y_t = d-1/\nu = d+\theta = 2+\theta$ of the energy operator should be
represented in the corresponding Kac table \cite{henkel:book}, and a numerically
close value was found in tentatively identifying
$x_t = 2\Delta_{1,2} = (6-\kappa)/\kappa$. Together with the relation
$d_\mathrm{f} = 1+\kappa/8$ for the fractal dimension, this yields the equation
\cite{amoruso:06a}
\begin{equation}
\label{eq:sle-relation}
d_\mathrm{f} = 1+\frac{3}{4(3+\theta)}.
\end{equation}
We note that additional to the assumption of a CFT representation, the identification
of conformal weights with items in the Kac table is only supposed to work for
rational values of $\kappa$, which does not appear to be the case
here. Eq.~\eqref{eq:sle-relation} was found to be consistent with previous estimates
of $\theta$ and $d_\mathrm{f}$. \cite{amoruso:06a} Our most accurate results are for
PFBC boundaries. The corresponding estimate $d_\mathrm{f} = 1.273\,19(9)$ would imply
via Eq.~\eqref{eq:sle-relation} that $\theta = -0.2546(9)$ which does not seem
consistent with the estimate $\theta = -0.2793(3)$ from the defect energies. More
systematically, if \eqref{eq:sle-relation} is to hold, the difference
\[
d_\mathrm{f} -1 -\frac{3}{4(3+\theta)} = -0.00247(9)
\]
must be consistent with zero. Here, we used the estimates for $d_\mathrm{f}$ and
$\theta$ from PFBC and standard error propagation \cite{brandt:book}. The difference
from zero corresponds to about 27 standard deviations, so based on the usual
confidence limits one would need to reject the hypothesis that our data are
consistent with \eqref{eq:sle-relation}. This neglects the fact, however, that our
estimates for $d_\mathrm{f}$ and $\theta$ are correlated as they are derived from the
same set of disorder realizations \cite{weigel:10}. To correct for this effect, we
divided the disorder samples for PFBC such that one half is used to estimate
$\theta=-0.2795(3)$ ($Q=0.47$) and the other half is used to estimate
$d_\mathrm{f} = 1.273\,22(12)$ ($Q=0.32$) using the same fit functions and ranges as
for the full data set. With these estimates, we find
\[
d_\mathrm{f} -1 -\frac{3}{4(3+\theta)} = -0.00246(12),
\]
where the deviation from zero is still about 20 standard deviations, corresponding to
the expected reduction by halving the statistics, so the correlation effect appears
to be weak. As an alternative analysis, we also attempted to perform a simultaneous
fit of power laws to the scaling of $|\Delta E|$ and $\ell$ while enforcing the
relation \eqref{eq:sle-relation} between the scaling exponents. Independent of
whether we use the full or the split data set, a fit quality $Q> 0.01$ is only
achieved for $L_\mathrm{min} \ge 1000$, which is way above the range of lattice sizes
where scaling corrections are visible above the statistical errors (recall that both
the defect energies and domain-wall lengths are fully consistent statistically with
pure power-laws for $L > L_\mathrm{min} = 40$). The conclusions from considering the
independent data set for PPBC are similar, with the deviation from
Eq.~\eqref{eq:sle-relation} being $-0.00231(47)$, corresponding to 5 standard
deviations. The values for the deviations for PFBC and PPBC are statistically
consistent, the appearance of better consistency for PPBC is due to the smaller
statistics there. While it is always difficult to reject or confirm an exact (but
non-rigorous) relation based on numerics, it appears safe to say that our data do not
appear to be consistent with Eq.~\eqref{eq:sle-relation}\footnote{While we tried to
take careful account of scaling corrections by including additional terms in the
fit functions and/or monitoring the dependence of the results on the choice of
$L_\mathrm{min}$, it is not possible to completely exclude the possibility of
spurious systematic corrections leading to the observed deviations from
Eq.~\eqref{eq:sle-relation}.}. It is worthwhile to note that, on the other hand,
our values for $\theta$ and $d_\mathrm{f}$ are fully consistent with previous
estimates, cf.\ the data compiled in Table \ref{theta_articles}, and it is only due
to the increased accuracy resulting from the bigger systems and larger numbers of
disorder samples considered here that the inconsistency with
Eq.~\eqref{eq:sle-relation} arises.
Our results for the fractal dimension of the bimodal model are marginally consistent
with those for the Gaussian model, and it remains an interesting question for further
studies whether universality between the two models holds in this respect.
\begin{acknowledgments}
We are grateful to Frank Beyer, Giorgio Parisi, Michael Moore, and Jacob Stevenson
for useful discussions. MW acknowledges extensive discussions with Zohar Nussinov,
Gerardo Ortiz and Mohammad-Sadegh Vaezi on the physics of the spin-glass phase. We
acknowledge funding from the DFG in the Emmy Noether Programme (WE4425/1-1) and
funding from the European Commission through the IRSES network DIONICOS
(PIRSES-GA-2013-612707).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,100 |
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"redpajama_set_name": "RedPajamaC4"
} | 5,816 |
AKTIBE. KAZINFORM The bodies of the Kazakhstanis killed in the road accident on Orenburg-Sol-Iletsk highway, will be brought to Kazakhstan today, Kazinform correspondent reports.
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"redpajama_set_name": "RedPajamaC4"
} | 4,534 |
Mugilogobius littoralis är en fiskart som beskrevs av Helen K. Larson 2001. Mugilogobius littoralis ingår i släktet Mugilogobius och familjen smörbultsfiskar. Inga underarter finns listade i Catalogue of Life.
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} | 474 |
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How Is The Bowstring Different?
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Other Reasons Korean Archery Training Is Different
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Korea seems to be blessed by the perks of archery.
It was a key element to have been used during the Three Kingdoms Period.
It has helped them secure their land from attacks from the North, the Chinese from the west, and the Japanese from the south and the East.
It was used primarily as a weapon until now.
To protect its mountain fortresses and strengthen its military forces, Korea has used the bow on multiple occasions.
During the 14th century, i.e., now known as the Chosun era, most of the soldiers were selected based on their excellence in archery.
It was assumed that a good archer is likely to rise higher in society and become a military officer.
Traditionally, archers would use the thumb draw while pulling the string right past the ear.
The target range earlier was around 145 meters, but now it's mostly between 30 to 90 meters.
From then to now, Korean archers are the most trained and skilled archers around the world.
Their bow is often referred to as the Korean Bow.
Koreans use two different types of bows.
One is the modern bow, and the other is the traditional bow or the Gakgung.
Gakgung refers to the nation bow and is built with various materials.
It is a water buffalo horn-based complex bow with its core made of bamboo.
This bamboo is also called the arrow bamboo.
The makers have to be very careful in selecting the type and age of the wood used to make the arrow.
This bow was intended to provide maximum efficiency and grip to eliminate the attackers even before they reach close.
All the parts of the bow are glued together.
It requires a thumb draw, which means that using a thumb ring while playing the sport is common.
This thumb ring is different from the ones used by other countries in terms of its availability for both males and females.
With such a glorious history, archery in Korea became such an important part of Korea that it led to various arrows' invention.
One of the most famous arrows to be recognized is the Pyeonjeon.
Pyeonjeon is considered the most famous type of arrow in Korean Archery.
This arrow is termed as powerful and has secured the bow to save the country.
It is also considered the Korean Army's secret weapon.
It's small in size and is fired through a Gakgung.
Due to its size, it is almost impossible for the enemies to reuse the same arrow and allow the Korean Army to buy time to deliver the next shot with no haste.
The Pyeonjeon was highly useful in defending against the Japanese invasion of Korea in 1952 during the Imjin war.
The one aspect that separates the Korean bowstring from the bows used by other countries, namely the Mongolian or the Turkish bows, is geometry.
One could be derailed by how small it looks, but deep down, this bow is very raw and highly efficient.
The extreme tightness could only be felt once the bow is unstrung.
Korea wished to exclude itself during the 19th century from the other parts of the world but was soon under attack by the west.
Their initial effort to avoid trading had failed over time, and they were eventually forced to upgrade themselves into being modernized while trading with the west.
After this era, the Korean military was equipped with modern firearms on the battlefield, and the use of archery was forever lost.
With the end of archery in the military field, it has still maintained its importance in culture and athletics.
There are reasons why Korean archers are simply the best.
This includes their intense training, systematic coaching, and even structured support to help the archers reach the top.
In most other countries, archers initially engage in archery in the form of recreation and later take it up as a serious sport, but it is of utmost importance in Korea.
There is a lot of pride and elegance within Korean Archery.
They require no differentiation between the rich and the poor.
They invest a lot of money in setting up schools for beginners, who can train in the afternoons from good professional coaches.
Most of the students are highly prepared for the sport since they are trained within the school itself.
In many schools, students are trained so well that they could learn how to shoot a bow and its basics in just three months before firing a live round.
They do not believe in private coaching for only the specifics.
It is a sport suitable for all.
They depend on how well they are in procuring the required skills and are also expected to have a sense of representation.
The idea that 'I can represent my country if I shoot a bow well' is embedded within them.
Kim Sam-Soo, a coach and the former director of the Korean Archery Association, said in 2017, "Archery is a fierce competitor for middle and high school athletes."
He further adds that "There are some players who are student geniuses who have been on the national team pretty much since high school."
Korean athletes are provided with the best, full time, and first-class training, enabling them to develop more confidence over the other players.
They may not aim to win but will foreshadow their expertise on it.
Korean archery has been considered one of the most difficult sports.
Along with its need to attain perfection, the training is longer and harder than any other participants in the world.
Their focus is extremely high.
So much that to increase their mental abilities, they watch videos on archery and enhance their learning.
The two-time Olympian Ko Bo Bae told the Korean Times, "Looking at the simulation on screen; I could improve my concentration by checking which motions I should take in each phase of the game."
For them, the Olympic games are way easier than their national trials.
It is believed that the players throw a minimum of 100,000 arrows a year to represent themselves nationally.
Very few countable athletes train at this size outside of Korea.
Korean athletes are so efficient in their training that it is almost difficult to make a National team for their tournaments.
Archers use many visualization techniques with unloaded bows.
This helps them critically analyze their moves and how well they have been or could have done.
The selection process is termed as the hardest part in Korea compared to any other country.
The best candidate to be a part of the national team may have to survive with consistency for as long as seven months.
No tournament can go badly; if it does, the archers don't have a way to redeem themselves.
The toughest trial takes place in March, which adds the best 24 athletes to a group of 16 already existing from last year in a week.
This group then undergoes rigorous training full time in Jincheon with the National team coaches.
The experience afforded by these trials is very advantageous not only in terms of the sport and its efficiency but also the prior knowledge of the various skills required to be the best.
It reaches out as a learning objective while the athlete is actively in the field.
After the eight best archers are selected, they are further divided into two groups, A and B.
The top four contestants officially compete in World Cups and other important worldly events, while the bottom four are awarded a consolation prize of participating in the Annual Championship.
Every participant is expected to follow the rules and regulations of the game.
Any manipulation of any sort by any athlete is thoroughly discouraged.
There are 28 recurve men's and 29 recurve women's teams in Korea.
The dominance of Korean archery at International levels is mostly because the primary reason is funding.
It is believed that archery equipment is very expensive.
A single arrow could cost at least $40.
Thus, when the government funds archery, the archers are fully invested in molding their craft.
The Korean government provides systematic funding that helps more than 130 full-time athletes to continue to pursue their training as Archers.
Local or private schools or Universities mostly sponsor funding.
These teams are selected through the schools or Universities they belong to.
The total number of professional archers are less in Korea because they filter out only the best.
Thus, the athletes start participating from school and keep playing unless they are eliminated.
They do not have a specific academy to enroll themselves; rather, they are trained solely to participate in competitions.
The real secret of Korean archery is the Korean Tax System.
The government implies a specific tax system to regulate the smooth running of archery as a sport.
Being an Olympic Archer in Korea can considerably upgrade an athlete's life.
If won, there could be a lot of concessions involved.
In 2014 Asian games in Incheon, the Korean Archers and their coaches have collectively received a sum of 880 million won which is around $500,000 in an event sponsored by Hyundai.
In this game they won a total of five Gold, three Silver and a Bronze medals.
The maximum rate during that time was around 70 million won for the Gold medal, for Silver it was 60 million won and for Bronze it was 50 million.
Korean archery not only dates back from the early ages but is of primary importance.
Archery is a part of Korean education as well.
It is taken seriously enough and is believed to have achieved a dedicated two hours each day in schools.
The general public can't have access to this game in the professional clubs.
It is solely made for tournament purposes and is considered as an Elite game.
Korea believes in the best coaching.
Coaches start training from the beginning of high school and college.
They believe that the archer needs to develop a bond and team between the bow and the arrow.
This shall only be achieved with a higher range of daily training.
The only aim they have while training is to make sure that they make it to the professional team, and if they achieve the highest excellency, they will make it to South Korea's national team.
There is a complete competition that goes on, and the archers often take part in World Archery and Olympic stages.
Making a National team is difficult because the performance is judged by a range of full-time students, professional athletes, sponsored and even governed by various schools or companies.
Korean Archers not only play the game rather they also study it.
In the 1980s the Americans and the Russians were the top Archers in the World.
Their approach towards the sport was primarily based on their own techniques of timings, line, and even repetition.
It is believed that a good technique can be developed if the fundamental techniques are clear and understood.
There is a continuous practice that takes place where the trainer is made to bow without shooting.
Some archers also use mirrors in order to track their performance.
Korea organizes the Idol Stra athletics Championship every two years on their Thanksgiving and their lunar new year.
They have a Korea Archery Association which is a member of the World archery association in the Republic of Korea.
It was founded in 1922 and was later affiliated to the World association in 1963.
It is the national representative body for archery in Korea.
The KAA has been organizing multiple trials after it got separated from the Traditional Archery Association in 1983.
Earlier the focus of the selection was based only on efficiency in Archery but now over the period of time and with multiple trials, selection is expected through various manners, like physical and mental fitness, leadership, etc.
Currently, the Korea National Sports University and Chungju City are the only two clubs that run all the four disciplines in their teams- Men, Women, Recurve and Compound.
All the other clubs focus on either any one or two or three.
30% of the KAA budget actually comes from the Olympic Committee.
Korea is not the only country to have been spending so much on Archery.
Americans Brady Ellison's recently joined hands with the United States Olympic Committee in order to build a $14m, 40,00 square foot, state of the art Archery field house in Chula Vista at the United States Olympic Training Centre.
There is a prize money of $1m to the athletes in Archery in India for Olympic gold.
Many nations are thoroughly supported by their government for the Olympic games.
The highest bidding used to be by Singapore with an offer of around a million Singaporean dollars if an Olympic gold medal is won.
Then came Malaysia with a million dollars to provide for the Gold medal.
In Korea, an Olympic Gold medal can help you earn a lifetime stipend.
These pensions were earlier cumulative until lately when Kim Soo Nyung won three gold medals.
In 2018, after South Korea was awarded the Winter Olympics, a large reorganization took place of the Olympic sports.
The training center for the National Archery team athletes shifted from Taerung on the outskirts of Seoul to a larger multi-sports complex in Jincheon.
Being the highest Olympic medal receivers and compensations, the Archery team got the best treatment and facilities available along with the indoor range.
The equipment and the facilities are such that it almost feels like science fiction made to acquire the members of the National team.
They have a separate space all to themselves while the compound team is provided with a different space in the same sports center.
These facilities are provided in order to maintain efficient performance by the Archers in the Olympic games.
After an inspection that took place after the Rio games in 2016, it came to notice that the sports center lacked in providing a space to relax for the athletes.
There were no housing arrangements done for them and the players were struggling with food and humidity.
This further developed a different plan where the KAA rented a separate garage and installed a proper facility with a high-end air-conditioned camper van equipped with proper food, beds, sofas, TV, and a Kitchen.
This project was so intense and it required a huge amount of funds to be utilized in order to make it a success.
Archery is still one of the most entertaining sports in Korea.
The training of the participants is so fierce and tough.
It takes more than ten hours to train for being an elite performer.
Korean Archery is the most renowned and important sport in Korea.
It is not just about being a skilled player.
It is a matter of pride and elegance.
Archery is now a part of their life and their culture.
The Archers train themselves seriously and for long hours.
There are a lot of ways that archery can be introduced or practiced.
Many countries have their own ways of attaining perfection.
We hope you enjoyed this article on Korean archery and how it's different.
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